Handbook Of Coastal And Ocean Engineering

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HANDBOOK OF COASTAL AND OCEAN ENGINEERING

edited by

Young C Kim California State University, Los Angeles, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF COASTAL AND OCEAN ENGINEERING Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-929-1 ISBN-10 981-281-929-0

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

YHwa - Hdbk of Coastal & Ocean.pmd

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Preface Although coastal and ocean engineering is a very ancient ﬁeld with the construction of Port A-ur near the mouth of the Nile in 3,000 BC, signiﬁcant advances in this ﬁeld have been made in the last several decades. The rise of interest in this ﬁeld can be seen from the number of attendees by academics and practitioners in international conferences. The ﬁrst International Conference on Coastal Engineering was held in Long Beach, California in 1950 with less than 100 people. When the same conference was held in San Diego, California, in 2006, over 1000 delegates attended. In the last several decades, the world has seen signiﬁcant coastal and ocean engineering projects, one of which is the Delta Project in the Netherlands. This project was designed to shorten and strengthen the total length of coast and dykes washed by the sea by closing oﬀ the sea arms in the Delta region. Other noteworthy coastal engineering projects include the Kansai Airport Project in Japan and, in recent years, the construction of mobile barriers at inlets to regulate tides in the Venice Lagoon known as the Venice Project. Interest in coastal and ocean engineering has arisen in recent years as humankind experiences coastal disasters that derive from coastal storm, hurricane and coastal ﬂooding and seismic activities such as tsunamis, and the impacts of climate change which result in sea-level rise. The tsunami activity in Sumatra in December 2004 aﬀected countries throughout the Indian Ocean and resulted in the loss of thousands of lives. Hurricane Katrina in New Orleans also claimed many lives with property damage exceeding $63 billion. Global warming and sea-level rise will aﬀect shoreline retreats, inundate low coastal areas, damage coastal structures, and accelerate beach erosion. The need for better understanding of our coastal and ocean environment has risen considerably in recent years. This handbook contains a comprehensive compilation of topics that are the forefronts of many technical advances in ocean waves, coastal and ocean engineering. It represents the most comprehensive reference available on coastal and ocean engineering to date, and it also provides the most up-to-date technical advances and latest research ﬁndings on coastal and ocean engineering. More than 70 internationally recognized authorities in the ﬁeld of coastal and ocean engineering contributed papers on their areas of expertise to this handbook. These international luminaries are from highly respected universities and renowned research and consulting organizations from all over the world.

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Preface

This handbook provides a comprehensive overview of shallow-water waves, water-level ﬂuctuations, coastal and oﬀshore structures, ports and harbors, coastal sediment processes, environmental problems, sustainable coastal development, coastal hazards, physical modeling, and coastal engineering practice and education. This book is an essential source of reference for professionals and researchers in the areas of coastal engineering, ocean engineering, oceanography, meteorology, and civil engineering, and as a text for graduate students in these ﬁelds. This handbook will be of immediate, practical use to coastal, ocean, civil, geotechnical, and structural engineers, and coastal planners and managers as well as marine biologists and oceanographers. It will also be an excellent source book for educational and teaching purposes, and would be a good reference book for any technical library. I would like to express my indebtedness to those who guided me and supported me as a mentor and a colleague throughout my professional life. They are: Professor Robert L. Wiegel, University of California, Berkeley Professor Joe W. Johnson, University of California Berkeley Professor Robert G. Dean, University of Florida Professor Fredric Raichlen, California Institute of Technology Professor Raymond C. Binder, University of Southern California Professor Shoshichiro Nagai, Osaka City University Dr Basil Wilson, Science Engineering Associates Dr Lars Skjelbreia, Science Engineering Associates Dr Bernard LeMehaute, University of Miami Professor Richard Silvester, University of Western Australia Mr Orville T. Magoon, Coastal Zone Foundation Professor Billy L. Edge, Texas A&M University Professor Michael E. McCormick, US Naval Academy Professor Yoshimi Goda, Yokohama National University and ECOH Corporation Professor Philip L.F. Liu, Cornell University Professor Forrest M. Holly, The University of Iowa Dr Etienne Mansard, National Research Council, Canada Professor J. Richard Weggel, Drexel University Mr Ronald M. Noble, Noble Consultants, Inc. I also wish to express my indebtedness to those who nurtured me from my early teen years and changed my course of life. They are: Dr Helen Miller Bailey, East Los Angeles College Mr H. Karl Bouvier, Jet Propulsion Laboratory I extend my gratitude to my wife, Janet, for her constant support, encouragement, patience, and understanding while I was undertaking this task and to my daughter, Susan Calix, for proofreading some of the materials.

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Finally, I wish to express my deep appreciation to Ms Kimberley Chua of World Scientiﬁc Publishing Company who gave me invaluable support and encouragement from the inception of this handbook to its realization. Young C. Kim Los Angeles, California January 2008

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Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Section 1: Shallow-Water Waves 1. Wave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and T. L. Walton

1

2. Wavemaker Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. T. Hudspeth and R. B. Guenther

25

3. Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. B. Guenther and R. T. Hudspeth

57

4. Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Goda

87

5. Aeration and Bubbles in the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori, S. Kakuno and D. T. Cox

115

6. Freak Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori

131

7. Short-Term Wave Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Kimura

151

Section 2: Water-Level Fluctuations 8. Generation and Prediction of Seiches in Rotterdam Harbor Basins . . . . . M. P. C. de Jong and J. A. Battjes

179

9. Seiches and Harbor Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. B. Rabinovich

193

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10. Finite Diﬀerence Model for Practical Simulation of Distant Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. B. Yoon

237

Section 3: Coastal Structures 11. Tsunami-Induced Forces on Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Nistor, D. Palermo, Y. Nouri, T. Murty and M. Saatcioglu

261

12. Nonconventional Wave Damping Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci

287

13. Wave Interaction with Breakwaters Including Perforated Walls . . . . . . . . K.-D. Suh

317

14. Prediction of Overtopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. van der Meer, T. Pullen, W. Allsop, T. Bruce, H. Sch¨ uttrumpf and A. Kortenhaus

341

15. Wave Run-Up and Wave Overtopping at Armored Rubble Slopes and Mounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Sch¨ uttrumpf, J. van der Meer, A. Kortenhaus, T. Bruce and L. Franco

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16. Wave Overtopping at Vertical and Steep Structures . . . . . . . . . . . . . . . . . . . T. Bruce, J. van der Meer, T. Pullen and W. Allsop

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17. Surf Parameters for the Design of Coastal Structures . . . . . . . . . . . . . . . . . . D. H. Yoo

441

18. Development of Caisson Breakwater Design Based on Failure Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Takahashi

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19. Design of Alternative Revetments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Pilarczyk

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20. Remarks on Coastal Stabilization and Alternative Solutions . . . . . . . . . . . K. Pilarczyk

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21. Geotextile Sand Containers for Shore Protection. . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci and J. Recio

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22. Low Crested Breakwaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Lamberti and B. Zanuttigh

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23. Hydrodynamic Behavior of Net Cages in the Open Sea . . . . . . . . . . . . . . . . Y.-C Li

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Section 4: Oﬀshore Structures 24. State of Oﬀshore Structure Development and Design Challenges . . . . . . . S. Chakrabarti

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Section 5: Ports and Harbors 25. Computer Modeling for Harbor Planning and Design . . . . . . . . . . . . . . . . . . J.-J. Lee and X. Xing

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26. Prediction of Squat for Underkeel Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, M. Vantorre, K. Uliczka and P. Debaillon

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Section 6: Coastal Sediment Processes 27. Wave-Induced Resuspension of Fine Sediment . . . . . . . . . . . . . . . . . . . . . . . . . M. Jain and A. J. Metha

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28. Suspended Sand and Bedload Transport on Beaches . . . . . . . . . . . . . . . . . . . N. Kobayashi, A. Payo and B. D. Johnson

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29. Headland-Bay Beaches for Recreation and Shore Protection . . . . . . . . . . . J. R.-C. Hsu, M. M.-J. Yu, F.-C. Lee and R. Silvester

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30. Beach Nourishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and J. D. Rosati

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31. Engineering of Tidal Inlets and Morphologic Consequences . . . . . . . . . . . . N. C. Kraus

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Section 7: Environmental Problems 32. Water and Nutrients Flow in the Enclosed Bays . . . . . . . . . . . . . . . . . . . . . . . Y. Koibuchi and M. Isobe

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Section 8: Sustainable Coastal Development 33. Socioeconomic and Environmental Risk in Coastal and Ocean Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Losada, A. Baquerizo, M. Ortega-S´ anchez, J. M. Santiago and E. S´ anchez-Badorrey 34. Utilization of the Coastal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-H. Hwung

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Section 9: Coastal Hazards 35. Ocean Wave Climates: Trends and Variations Due to Earth’s Changing Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. D. Komar, J. C. Allan and P. Ruggiero

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36. Sea Level Rise: Major Implications to Coastal Engineering and Coastal Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Ewing

997

37. Sea Level Rise and Coastal Erosion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. F. Stive, R. Ranasinghe and P. J. Cowell

1023

38. Coastal Flooding: Analysis and Assessment of Risk . . . . . . . . . . . . . . . . . . . . P. Prinos and P. Galiatsatou

1039

Section 10: Physical Modeling 39. Physical Modeling of Tsunami Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, H. Yeh and D. T. Cox

1073

40. Laboratory Simulation of Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. P. D. Mansard and M. D. Miles

1107

Section 11: Coastal Engineering Practice and Education 41. Perspective on Coastal Engineering Practice and Education . . . . . . . . . . . J. W. Kamphuis

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Contributors Jonathan C. Allan Coastal Field Oﬃce Oregon Department of Geology and Mineral Industries Newport, Oregon [email protected] William Allsop Technical Director HR Wallingford Wallingford, UK [email protected] Elena Sanchez Badorrey Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Asuncion Baquerizo Associate Professor CEAMA — Universidad de Granada Granada, Spain Jurjen A. Battjes Emeritus Professor Environmental Fluid Mechanics Section Delft University of Technology Delft, The Netherlands [email protected] Michael J. Briggs Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected]

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Tom Bruce School of Engineering and Electronics University of Edinburgh Edinburgh, UK [email protected] Subrata Chakrabarti Joint Professor, Civil and Mechanical Engineering University of Illinois at Chicago Chicago, Illinois [email protected] Peter J. Cowell Associate Professor School of Geosciences Institute of Marine Science University of Sydney Sydney, Australia Daniel T. Cox Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Robert G. Dean Graduate Research Professor of Coastal Engineering, Emeritus Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]ﬂ.edu Pierre Debaillon Research Hydraulic Engineer Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF) Compiegne, France [email protected] Martijn P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology Presently at Delft Hydraulics Delft, The Netherlands

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Lesley Ewing Senior Coastal Engineer California Coastal Commission San Francisco, California [email protected] Leopoldo Franco Professor of Coastal Engineering Department of Civil Engineering University of Rome 3 Rome, Italy [email protected] Panagiota Galiatsatou Research Associate Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Yoshimi Goda Professor Emeritus Yokohama National University Adviser to ECHO Corporation Tokyo, Japan [email protected] Ronald B. Guenther Professor Emeritus Department of Mathematics Oregon State University Corvallis, Oregon [email protected] John Rong-Chung Hsu Professor Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Honorary Research Fellow School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Robert T. Hudspeth Professor and Director, Emeritus Coastal and Ocean Engineering Program Oregon State University Corvallis, Oregon [email protected] Hwung-Hweng Hwung Professor of Hydraulic and Ocean Engineering Director of Tainan Hydraulics Laboratory Department of Hydraulic and Ocean Engineering National Cheng Kung University Tainan, Taiwan [email protected] Masahiko Isobe Professor and Special Adviser to the President Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Mamta Jain Coastal Engineer Halcrow Inc. Tampa, Florida [email protected] Bradley D. Johnson Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi Shohachi Kakuno Professor and Vice President Department of Civil Engineering Osaka City University Osaka, Japan [email protected] J. William Kamphuis Professor of Civil Engineering, Emeritus Department of Civil Engineering Queen’s University Kingston, Ontario, Canada [email protected]

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Akira Kimura Professor Department of Social Systems Engineering Tottori University Tottori, Japan [email protected] Nobuhisa Kobayashi Professor and Director Center for Applied Coastal Research University of Delaware Newark, Delaware [email protected] Yukio Koibuchi Assistant Professor Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Paul D. Komar Professor of Oceanography College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, Oregon [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics Technical University of Braunschweig Braunschweig, Germany [email protected] Nicholas C. Kraus Senior Scientist Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected] Alberto Lamberti Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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Contributors

Fang-Chun Lee Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Jiin-Jen Lee Professor of Civil and Environmental Engineering Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California [email protected] Yu-Cheng Li Professor School of Civil Engineering Dalian University of Technology Dalian, China [email protected] Miguel A. Losada Professor Research Group on Environmental Flux Dynamics CEAMA — Universidad de Granada Granada, Spain [email protected] Etienne P. D. Mansard Executive Director Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada [email protected] Ashish J. Mehta Professor of Coastal Engineering Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]ﬂ.edu Michael D. Miles Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada

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Nobuhito Mori Associate Professor Disaster Prevention Research Institute Kyoto University Kyoto, Japan [email protected] Tad S. Murty Adjunct Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Ioan Nistor Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Younes Nouri Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada Miquel Ortega Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Hocine Oumeraci University Professor Leichtweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany [email protected] Dan Palermo Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected]

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Andres Payo Graduate School of Science and Technology University of Kumamoto Kumamoto, Japan Krystian W. Pilarczyk (Former) Manager, Research and Development Hydraulic Engineering Institute Rykswaterstaat Delft, The Netherlands HYDROpil Consultancy Zoetermeer, The Netherlands [email protected] Panayotis Prinos Professor of Hydraulic Engineering Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Tim Pullen Senior Engineer HR Wallingford Wallingford, UK [email protected] Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology Russian Academy of Sciences Moscow, Russia Department of Fisheries and Oceans Institute of Ocean Sciences Sidney, B.C., Canada [email protected]. Roshanka Ranasinghe Associate Professor UNESCO-IHE/Delft University of Technology Delft, The Netherlands [email protected] Juan Recio Leichweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany

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Julie D. Rosati Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Corps of Engineers Mobile, Alabama [email protected] Peter Ruggiero Assistant Professor Department of Geosciences Oregon State University Corvallis, Oregon [email protected] Murat Saatcioglu Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Juan M. Santiago Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Holger Schuttrumpf Professor and Director Institute of Hydraulic Engineering and Water Resources Management RWTH — Aachen University Aaachen, Germany [email protected] Richard Silvester Professor Emeritus School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Marcel J. F. Stive Professor and Director Delft Water Research Centre Department of Hydraulic Engineering Delft University of Technology Delft, The Netherlands [email protected] Kyung-Duck Suh Professor Department of Civil and Environmental Engineering Seoul National University Seoul, Korea [email protected] Shigeo Takahashi Executive Researcher and Director Tsunami Research Center Port and Airport Research Institute Yokosuka, Japan takahashi [email protected] Klemens Uliczka Research Hydraulic Engineer Federal Waterways Engineering and Research Institute (BAW) Hamburg, Germany [email protected] Jentsje van der Meer Principal Van der Meer Consulting Heerenveen, The Netherlands [email protected] Marc Vantorre Professor Division of Maritime Technology Ghent University Ghent, Belgium [email protected]

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Contributors

Todd L. Walton Director Beaches and Shore Resource Center Florida State University Tallahassee, Florida [email protected] Xiuying Xing Graduate Research Assistant Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California Harry Yeh Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Dong Hoon Yoo Professor Department of Civil Engineering Ajou University Suwon, Korea [email protected] Sung Bum Yoon Professor Department of Civil and Environmental Engineering Hanyang University Ansan, Korea [email protected] Melissa Meng-Jiuan Yu Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Barbara Zanuttigh Assistant Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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The Editor Young C. Kim, PhD, is currently a Professor of Civil Engineering, Emeritus at California State University, Los Angeles. Other academic positions held by him include a Visiting Scholar of Coastal Engineering at the University of California, Berkeley (1971); a NATO Senior Fellow in Science at the Delft University of Technology in the Netherlands (1975); and a Visiting Scientist at the Osaka City University for the National Science Foundations’ US–Japan Cooperative Science Program (1976). For more than a decade, he served as Chair of the Department of Civil Engineering and recently he was Associate Dean of the College of Engineering. For his dedicated teaching and outstanding professional activities, he was awarded the university-wide Outstanding Professor Award in 1994. Dr Kim was a consultant to the US Naval Civil Engineering Laboratory in Port Hueneme and became a resident consultant to the Science Engineering Associates where he investigated wave forces on the Howard-Doris platform structure, now being placed in Ninian Field, North Sea. Dr Kim is the past Chair of the Executive Committee of the Waterway, Port, Coastal and Ocean Division of the American Society of Civil Engineering (ASCE). Recently, he served as Chair of the Nominating Committee of the International Association of Hydraulic Engineering and Research (IAHR). Since 1998, he served on the International Board of Directors of the Paciﬁc Congress on Marine Science and Technology (PACON). He currently serves as the President of PACON. Dr Kim has been involved in organizing 10 national and international conferences, has authored three books, and has published 52 technical papers in various engineering journals.

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Chapter 1

Wave Setup Robert G. Dean Department of Civil and Coastal Engineering University of Florida, Gainesville, FL, USA [email protected] Todd L. Walton Beaches and Shores Resource Center Florida State University, Tallahassee, FL, USA [email protected] Wave setup is the increase of water level within the surf zone due to the transfer of wave-related momentum to the water column during wave-breaking. Wave setup has been investigated theoretically and under laboratory and ﬁeld conditions, and it includes both static and dynamic components. Engineering applications include a signiﬁcant ﬂooding component due to severe storms and oscillating water levels that can increase hazards to recreational beach goers and can contribute to undesirable oscillations of both constructed and natural systems including harbors and moored ships. This chapter provides a review of the knowledge regarding wave setup and presents preliminary recommendations for design. It will be shown that wave setup is not adequately understood quantitatively for engineering design purposes.

1.1. Introduction Wave setup was brought to the attention of coastal engineers and scientists in the 1960s (i.e., see Ref. 1, p. 245) after the initial theoretic developments of LonguetHiggins2 and Longuet-Higgins and Stewart3,4 along with limited ﬁeld observations and laboratory studies supported the existence of wave setup, the magnitude of which was observed to be in the order of 10–20% of the incident wave height. It was noted in early ﬁeld observations that water levels on the beach were higher than those recorded by a tide gauge at the end of a pier suggesting a wave setup physically forced by wind waves and swell. 1

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Fig. 1.1. Deﬁnition sketch. Energy and momentum are transferred from winds to waves in the generating area. The waves convey energy and momentum to the surf zone where the waves break. Upon breaking, the energy is dissipated and the momentum is transferred to the water column resulting in longshore and onshore forces exerted on the water column.

Wave setup is the additional water level that is due to the transfer of wave-related momentum to the water column during the wave-breaking process. As waves approach the shoreline, they convey both energy and momentum in the wave direction. Upon breaking, the wave energy is dissipated, as is evident from the turbulence generated; however, momentum is never dissipated but rather is transferred to the water column resulting in a slope of the water surface to balance the onshore component of the ﬂux of momentum (see Fig. 1.1). If waves are irregular, in addition to a steady wave setup, the setup includes a dynamic component that oscillates with the wave group period and there may be a weak resonance within the nearshore amplifying this oscillating component. These have been termed infragravity waves and are more dominant for narrow banded spectra both in frequency and in directional spreading. The oscillatory component is denoted “dynamic wave setup” in this chapter. This chapter discusses the signiﬁcance of wave setup to coastal engineering design, provides a review of the classical linear wave theory of wave setup, reviews results from laboratory and ﬁeld studies, summarizes results and recommends preliminary design approaches for the static component. To provide a “look ahead,” we will see that the phenomenon of wave setup is not yet adequately understood for satisfactory engineering calculations and that the eﬀects of proﬁle slope are very

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signiﬁcant. The interested reader is also referred to an earlier review article on wave setup by Holman.5 1.2. Engineering Significance of Wave Setup Wave setup (both static and dynamic components) is relevant to a number of engineering applications. The contributions of wave setup under extreme storm events can be substantial, adding several feet to the elevated water levels. The interaction of wave setup with vegetation diﬀers from wind surge and thus it is important to diﬀerentiate the two components, for example in ascertaining the beneﬁts of wetlands in reducing wave setup. Finally, the oscillating component of wave setup is relevant to beach safety in some locations and to many natural and constructed coastal systems that have the capability to resonate including harbors and moored ships. 1.3. Terminology and Related Considerations Standard terminology deﬁnes the water level in the absence of wave eﬀects as “still water level,” whereas wave setup will cause a departure from the still water level and this water level including the eﬀects of the waves is the “mean water level.” As implied, the mean water level is determined as the average of the ﬂuctuating water level over a suitable time frame usually taken as a number of multiples of the short wave period, say the spectral peak. In considering wave setup, often the location of interest is that of the maximum wave setup at the shoreline. This raises the question of whether wave setup is deﬁned at elevations above the maximum rundown, say on the beach face where the water is present over only a portion of the wave period. Since wave setup is deﬁned as the mean water level, over what period should the water surface be averaged on the beach face which is “wetted” over only a portion of the wave period? If the time average is over only the portion of the period that water is present, in the upper limit, the maximum setup will be the maximum runup. For purposes here, wave setup will usually be deﬁned only for conditions where water is present over a full wave period. When calculating wave runup on a structure such as a levee or revetment, the question arises whether it is appropriate to ﬁrst calculate wave setup and then add the wave runup which is usually empirically based on model results. In the more recent empirical results (e.g., the TAW method, see Ref. 6), the runup is expressed as a proportion of the signiﬁcant wave height at the base of the steeper slope (e.g., at a revetment or levee). The wave runup determined in the model on which the method was based generally included some wave setup (or setdown) seaward of the toe of the slope and included wave setup landward of the toe of the slope. Thus, in the application of interest, the most appropriate approach is to calculate and include wave setup at the toe of the slope; however, recognizing that the measured landward runup includes setup, no additional setup should be added explicitly landward of the toe of the slope.

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1.4. A Brief Review of Wave Setup Mechanics 1.4.1. Static wave setup for monochromatic waves Longuet-Higgins2 and Longuet-Higgins and Stewart3,4 were the ﬁrst to formalize the notion of wave momentum ﬂux and its relationship to wave setup. The momentum ﬂux, Sij , is a second-order tensor given by 1 , Sxx = E n(cos2 θ + 1) − 2 1 2 (1.1) Syy = E n(sin θ + 1) − , 2 E sin 2θ, 2 where E is the wave energy density, n is the ratio of wave group velocity to wave celerity and θ is the angle between the wave direction and the x-axis. The term Sxy reads “the ﬂux per unit width, in the x-direction, of the y-component of momentum,” etc. The steady-state equations of motion obtained by time averaging over the short wave period are, including the eﬀects of wind stress and bottom friction: 1 ∂Sxx ∂Sxy ∂ (η wind + ηwave ) =− + − τsx + τbx ∂x ρg(h + η) ∂x ∂y Sxy = Syx =

and ∂ (η wind + η wave ) 1 =− ∂y ρg(h + η)

(1.2)

∂Syy ∂Syx + − τsy + τby . ∂y ∂x

In the above, ηwind is the surge component due to the wind stress, ηwave is the wave setup, ρ is the mass density of water, g is the gravitational constant, h is the local water depth, τsx and τbx are the surface and bottom shear stresses, respectively, and similarly for the y-direction. The coordinate direction, x is oriented shoreward and a right-handed coordinate system is considered. The most simple solution is for waves propagating directly shoreward (Sxy = 0) in which the surface and bottom stresses are considered negligible, and all variables are considered uniform in the y-direction. The resulting equation is 1 ∂Sxx ∂η wave =− . (1.3) ∂x ρg(h + η) ∂x To proceed, we need to determine a boundary condition for η wave ,a at the seaward end of the surf zone. Longuet-Higgins7 has shown that in the absence of energy dissipation, the following general relationship for η applies η=C−

1 2 (u − w2 )η=0 , 2g

(1.4)

a For purposes of convenience, hereafter the subscript on η wave will be omitted such that the wave setup is simply denoted as η.

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where u2 and w2 represent the time averages of the square of the ﬁrst-order horizontal and vertical wave velocities evaluated at the mean water surface, respectively. Equation (1.4) is a type of a Bernoulli equation for unsteady ﬂows which, when evaluated at the break point and considering no wave setup in deep water to evaluate the constant, C = 0, the setup is negative (setdown) and given by ηb = −

Hb2 kb , 8 sinh 2kb hb

(1.5)

where Hb is the breaking wave height and kb is the wave number at breaking. For shallow water conditions and depth limited breaking (Hb = κ(hb + η b )), Eq. (1.5) yields ηb = −

κHb . 16

(1.6)

As an example, for a κ value of 0.78, the wave setdown is approximately 5% of the breaking wave height. With the seaward boundary condition now established, for the case of shallow water wave-breaking and the consideration of depth limited breaking across the surf zone, the wave setup is η=−

3κ2 /8 κHb + (hb − h) . 16 (1 + (3κ2 /8))

(1.7)

It is noted that in the above equation, the bottom shear stress has been taken as zero and that a shoreward directed bottom shear stress on the water column as would occur due to undertow would increase the wave setup. As examples, the ratio of wave setup to breaking height at the still water line (h = 0) and at the location of maximum wave setup (η = − h) for a κ value of 0.78 are 5κ η(h = 0, κ = 0.78) = = 0.198 (1.8) F0 |κ=0.78 ≡ Hb 16(1 + (3κ2 /8)) κ=0.78 and Fmax |κ=0.78

η(h = − η, κ = 0.78) ≡ = F0 Hb

3κ2 5κ 1+ = = 0.244 . 8 κ=0.78 16 κ=0.78 (1.9)

It is seen that the wave setup is strongly dependent on the value of the breaking ratio κ which will be shown to decrease with decreasing beach slope. Figure 1.2 presents the ratios, F0 and Fmax versus κ. It is useful to relate κ in an approximate manner to beach slope. Although there is not a one-to-one correspondence, Fig. 1.3 is based on the Dally et al.8 wave-breaking model and provides an approximate correspondence between uniform proﬁle slope and the associated κ value. It is evident that the Dally et al. model provides reasonable κ values for smaller beach slopes (say less than about 0.06), but the κ values are too large for steeper slopes.

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0.4

Fo Fmax

Fo and Fmax

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Kappa Fig. 1.2.

Values of F0 and Fmax versus wave-breaking index, κ (kappa).

2.0

Kappa

1.5

1.0

0.5

0.0 0.00

0.02

0.04

0.06

0.08

0.10

Profile Slope

Fig. 1.3. Relationship between proﬁle slope and κ (kappa) value. Based on Dally et al.8 wavebreaking model.

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1.4.2. Eﬀects of wave nonlinearity Wave nonlinearity depends on the following parameters: H/L0 and h/L0 . The nonlinearity is exhibited in the wave proﬁle by peaked crests and ﬂatter troughs and increases with wave height and shallow water. Somewhat surprisingly, the momentum ﬂuxes in shallow water are less for nonlinear waves than for linear waves of the same height. This is primarily because the momentum ﬂuxes are proportional to wave energy (Eq. (1.1)) and the wave energy is proportional to the root-mean square of the water surface displacement that is less for nonlinear waves with long troughs and peaked wave crests. Figure 1.4 presents the ratio of nonlinear to linear momentum ﬂuxes as determined by Stream Function wave theory.9–11 The reason that the quantities for nonlinear waves are greater in deep water than for linear waves is that the nonlinear calculations extend up to the actual free surface whereas the linear quantities only extend up to the mean free surface. 1.4.3. Role of wave directionality

1.2

b =1

H/ H

b

/H H

b=

b =0

0.6

H/ H

.2 5

0. 50

=0 .7 5

0.8

.0

1.0

H/ H

Ratio of Nonlinear to Linear Momentum Flux

Equation (1.1) demonstrates that for a given wave height, the maximum shoreward ﬂux of onshore momentum occurs for normally incident waves (θ = 0◦ ). Thus as expected for directional waves, the Sxx term is reduced. However, this reduction is relatively small as can be demonstrated by considering a breaking wave direction of 30◦ relative to a beach normal (this represents a reasonably large wave obliquity

0.4

0.2

0.0 10-3.000

2

3

4 5 6

10-2.000

2

3

4 5 6

10-1.000

2

3

4 5 5

100.000

2

3

4 5 6

101.000

h/Lo Fig. 1.4. Ratio of nonlinear to linear wave momentum ﬂux, Sxx , for forty stream function wave combinations.12

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at breaking). The reduction in Sxx for a wave of given height and for shallow water conditions is 16.7%. 1.4.4. Eﬀects of vegetation The eﬀects of vegetation have been shown to result in a reduced setup and, in some cases, may cause a setdown.12 For linear waves, vegetation protruding through the water surface experiences a net drag force (quadratically related to velocity) on the vegetation in the direction of wave propagation and, of course, there must be an equal and opposing force exerted on the water column. This opposing force acting on the water column partially counteracts the force due to momentum transfer and thus reduces the wave setup (similar to an oﬀshore directed wind stress). For linear waves and vegetation which is submerged during the entire wave passage, no net vegetation-related force exists on the water column and thus there is no eﬀect on the wave setup. However, due to the character of nonlinear waves with higher and shorter shoreward velocities under the wave crests, even if the vegetation is fully submerged during the passage of the wave, a net drag force is induced on the vegetation in the wave propagation direction again resulting in a reduction in the wave setup and, for some cases, a wave setdown. 1.4.5. Dynamic wave setup It is noted that theoretical formulations of the dynamic wave setup must include the time dependent terms in the counterparts of Eq. (1.2). The dynamic wave setup or “surf beat” was ﬁrst identiﬁed through ﬁeld observations and measurements by Munk13 and Tucker.14 A number of theoretical treatments of dynamic wave setup based on various hypotheses have been developed with each focusing on a diﬀerent mechanism. These include Symonds et al.15 (time-varying breakpoint), Symonds and Bowen16 (trapping of long waves by longshore bars), Schaﬀer and Svendsen17 (reinforcement of incoming and reﬂected long waves), etc. Kostense18 conducted laboratory experiments to investigate the dynamic setup component and found that the results were in qualitative agreement with the theory of Symonds et al.15 We can apply the results for monochromatic waves to investigate the approximate dynamic wave setup for a simple irregular wave case. Consider a bichromatic wave system with wave heights H1 and H2 (H1 > H2 ) and a small frequency difference between the two components. If the resulting wave group varies so slowly that static conditions occur within the surf zone, Eq. (1.7) applies and is written as η = F Hb ,

(1.10)

where Hb is the breaking wave height and F is a proportionality factor depending on whether the referenced setup is at the still water shoreline or the maximum wave setup (see Eqs. (1.8) and (1.9)). The maximum and minimum wave setup values are: η max =

F (H1 + H2 ) ,

ηmin =

F (H1 − H2 ) .

(1.11)

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Table 1.1. Static and dynamic wave setup characteristics for a biharmonic wave system. H2 /H1 0.2 0.4 0.6 0.8 1.0

η max /F H1

η avg /F H1

(η max − η avg )/η avg

1.2 1.4 1.6 1.8 2.0

1.01 1.04 1.09 1.16 1.27

0.19 0.35 0.47 0.55 0.57

It can be shown that the average wave setup depends on the ratio H2 /H1 as shown in Table 1.1. The fourth column presents the ratios of the maximum dynamic wave set amplitude to the average wave setup component. In the case with H2 = H1 , the dynamic wave setup displacement from the mean setup equals 57% of the average wave setup (Table 1.1, Column 4). In the above, we have examined the dynamic wave setup for the case of a simple bichromatic wave system in which the diﬀerence in frequencies of the two components was fairly small. For the case of a wave spectrum, the situation is much more complex with, for the case of a narrow spectrum, the group envelope varying according to the Rayleigh distribution. For the case of a wide spectrum, the dynamic component is reduced considerably.

1.5. Laboratory and Field Measurements of Wave Setup Having reviewed the theory of wave setup and its relationship to various factors, the two sources available for evaluation are laboratory and ﬁeld data. 1.5.1. Laboratory experiments on wave setup Many laboratory investigations of static and dynamic wave setup have been conducted. The results of an early laboratory investigation with monochromatic waves by Bowen et al.19 are shown in Fig. 1.5. For this study, the ratio of maximum wave setdown and wave setup on the beach face to breaking wave height are − 0.035 and + 0.316, respectively, compared with − 0.049 and + 0.244 on the beach face for a κ value of 0.78. The eﬀect of beach slope has been noted earlier and the relatively large beach slope of 0.082 in these experiments is undoubtedly a contributor to the large setup value. Later, laboratory investigations have included examination of irregular waves including measurements of water particle velocities and pressures which form the basis of the Sxx momentum ﬂux component. Battjes20 conducted one of the earliest laboratory studies of wave setup due to irregular waves. Setup was measured through bottom mounted manometers and it was found that the wave setup was less than predicted. It was hypothesized that this diﬀerence was possibly due to air in the water column of the manometers. The entire setup was shifted landward relative to the theoretical and this delay was later attributed to a “roller” that is transported along with the wave crest region and

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Fig. 1.5.

Measured wave setup and setdown in the laboratory.19

conveys wave energy and momentum landward prior to transfer of the momentum to the water column and the associated wave setup.21 Later, Stive and Wind22 conducted a very detailed laboratory investigation in which they demonstrated the role of wave nonlinearity. In this study, the momentum ﬂux components (velocities and pressures) were measured to the degree possible and it was found that the calculated wave setup based on nonlinear wave theories was in much better agreement with measured wave setup than calculations based on linear wave theories. In these comparisons, it was not necessary to introduce the roller concept. The two laboratory studies reviewed above have focused on static setup and it has been noted that irregular waves also produce dynamic wave setup. Hedges and Mase23 have presented an interesting reanalysis of earlier runup laboratory measurements by Mase24 in which irregular waves provided the forcing.b The planar slopes represented in the data were: 1:5, 1:10, 1:20, and 1:30. Figure 1.6 presents an example of the form in which the data were plotted where the horizontal axis is b Walton25

was the ﬁrst to analyze the Mase data to extract the static wave setup.

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Fig. 1.6.

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Variation of nondimensional runup with Iribarren number.23

the Iribarren number, ξ0 deﬁned as tan α ξ0 = H/L

(1.12)

in which tan α is the proﬁle slope. The interpretation of Fig. 1.6 is that for a zero slope (zero Iribarren number), there would be no short wave runup; therefore, the intercept represents the sum of the static and dynamic components of wave runup. Equations of the following form were ﬁt to plots of the type of Fig. 1.6: Rchar Schar = + cchar ξ0 , H1/3 H1/3

(1.13)

where the subscript “char” refers to the percent associated with the variable; for example, the 2% runup is deﬁned as R2% . It was found that both Schar and cchar were Rayleigh distributed with S1/3 and c1/3 equal to 0.27 and 1.04, respectively, where only the ﬁrst term represents wave setup and is of interest here. The results for Schar can be interpreted in terms of the static and dynamic wave setup components. As an example, Smean = 0.17 and S2% = 0.37. Thus, the 2% value of the nondimensional dynamic setup deﬁned here as ∆S2% is ηdyn,2% = (S2% − Smean ) = (0.37 − 0.17) = 0.20 . (1.14) ∆S2% = H1/3 Thus, the mean setup at the still waterline is 17% of the signiﬁcant wave height measured at the toe of the slope and the 2% dynamic component at the still waterline is 20% of the signiﬁcant wave height at the toe of the slope or slightly larger than the mean wave setup. These results are interesting and of reasonable magnitudes; however, there are two problems with recommending them for universal application. First, we know that the mean setup depends on the slope (through the κ dependency

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as discussed earlier), and the second is that the oscillating wave setup component should depend on the width of the input spectrum. Referring to Fig. 1.6, which is one of several similar plots presented in the Hedges and Mase paper, since each plot may include a mix of beach slopes, the slope dependency is not resolved in the Smean results which of course are derived from the y-intercept of these graphs. Secondly, these experiments were not designed to evaluate the eﬀect of spectral width and the spectral characteristics included in the experiments are not known. However, it is of interest to identify the “representative” κ value and beach slope associated with a Smean = η avg /H1/3 of 0.17. Referring to Fig. 1.2, we see that the associated κ value is approximately 0.63 for F0 . Based on the Dally et al.8 breaking wave model, the associated beach slope from Fig. 1.3 is 1:29 compared to the beach slopes in the Mase experiments ranging from 1:30 to 1:5. 1.5.2. Field experiments on wave setup The paragraphs below describe several ﬁeld experiments and observations of wave setup. An early study of wave setup comprised a pair of observations at an exposed coastal site (Narragansett Pier, RI) and a calmer water site (Newport, RI) where, at the latter, wave action was assumed not a factor and was found to show an approximate 3 foot water level diﬀerence during the peak of the 1938 hurricane storm surge.26 In a second early ﬁeld experiment on wave setup at Fernandina Beach, Florida, Dorrestein27 placed transparent plastic tubes with lightweight ﬂoats to track the water surface on the beach in the zone of wave setup. To obtain the setup records, 16 mm movie ﬁlm recorded the tracked surface of the ﬂoats. A ﬂoat type tide gage on the end of a ﬁshing pier provided oﬀshore water level records. Through analysis of the tide gage records and the beach placed setup gages, Dorrestein27 evaluated the setup (with respect to the end of the pier) and compared observational results to existing setup theory. He found the measured setup in four of ﬁve experiments to be larger than the computed setup. One shortcoming of Dorrestein’s work is that the water level records were only 72 s in length and thus subject to considerable scatter and large standard deviation as later noted by Holman and Sallenger.28 Although rationale was provided by Dorrestein27 for possible diﬀerences between measured and computed setup in this early experiment, large discrepancies between measured and analytically or numerically computed setup still exist today. A North Sea ﬁeld wave setup experiment was conducted on the Island of Sylt by Hansen.29 Utilizing a combination of ultrasonic wave gages and pressure sensor wave gages out to a distance of 1280 m from shore (10 m depth), Hansen29 found good correspondence of data to an empirical expression provided by: η = 0.3Hos = 0.42Horms .

(1.15)

Hansen also noted the maximum wave setup to be approximately 50% of the significant breaking wave height. It is not clear as to the methodology utilized to obtain η max in this ﬁeld experiment.

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A wave setup ﬁeld experiment was conducted as part of the Nearshore Sediment Transport Study at Torrey Pines Beach, San Diego, California by Guza and Thornton.30 The Torrey Pines Beach face was gently sloping (beach slope ≈ 0.02) and the beach material was a moderately sorted ﬁne grain sediment (≈ 0.1 mm). A dual wire resistance runup meter was used for the recording and estimation of the wave setup. It should be noted that the measurements of the wave setup were considered to be the average runup determined from wires placed approximately 3 cm above the beach level rather than an actual water level at one location in these experiments. Oﬀshore pressure sensors outside the surf zone at mean depths of 7 to 10.5 m were used for estimating wave height with recording lengths of 4096 s. Guza and Thornton30 note speciﬁc problems in the data set, which are typical of ﬁeld measurements, i.e., the diﬃculty in obtaining a common datum for the oﬀshore wave measurements and the beach wave setup measurements. Results of their measurement program suggest an empirical relationship as follows: η = 0.17Hos = 0.24Horms

(1.16)

with scatter that suggests η/Hos ranging approximately from 0.05 to 0.50 for individual experiments. Holman and Sallenger28 conducted a ﬁeld experiment for measuring wave setup as well as other surf zone parameters at the U.S. Army Corps of Engineers ﬁeld research pier in Duck, NC, USA. Data on water level at the shoreline were collected using longshore looking time lapse photography from Super-8 movie cameras mounted on the research pier scaﬀolding. The beach at the experiment site had a very steep foreshore (∼ 1 on 10) while the oﬀshore proﬁle slope is much milder (∼ 1 on 100). Beach material was bimodal in size with a median sand size of 0.25 mm and a coarse fraction of 0.75 mm. Results of the experiments showed considerable scatter and dependence on tide level. Regression lines were ﬁt to the data (segmented by tide levels) with results as follows for high tide and mid-tide data: η = 0.35ξ0 + 0.14 (high tide) , (1.17) Hs η = 0.46ξ0 + 0.06 (mid-tide) . (1.18) Hs As most of the data fell in a range of ξ0 = 1 to 2, the maximum setup was noted to be of the same order as the signiﬁcant wave height in many of the experiments, much higher than theoretically suggested values. Note that in terms of Horms (based on consideration of monochromatic theory results) the setup would be much higher than most other studies show or suggest. Although Holman and Sallenger28 conclude from their experiments that the setup is dependent on the Iribarren number, it is not entirely clear from their data, especially for higher waves (i.e., see Fig. 1.4, Ref. 28). An additional problem that must be considered when computing the Iribarren number for real beaches and irregular waves is how to deﬁne beach slope. It should be noted that video camera (visual) approaches estimate setup via the measurement of the water surface elevation on the beach (similar to the Guza and Thornton measurements) rather than an actual vertically ﬂuctuating water level. The anomaly between dependence of

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setup on Iribarren number as noted by Holman and Sallenger28 is likely due to the aforementioned relationship between the wave-breaking coeﬃcient, κ, and beach slope. Nielsen31 and Davis and Nielsen32 conducted a novel setup experiment on Dee Why Beach, in New South Wales, Australia using a set of manometer tubes as shown in Figs. 1.7 and 1.8 from Davis and Nielsen.32 The tubes were deployed throughout the beach face and surf zone. A total of 120 setup proﬁles were measured in 11 days. Wave heights Horms ranged from 0.6 to 2.6 m in height and signiﬁcant wave periods (Ts ) ranged from 5.8 to 12.1 seconds. A shoreline setup of about 40% of Horms was found although Davis and Nielsen32 point out that there is reason to believe that the surf zone characteristics inﬂuence the relationship between wave height and setup magnitude, and also note a problem of deﬁning beach slope via the Iribarren number. Nielsen31 and Davis and Nielsen32 also observe that a major portion of the setup occurred on the beach face as shown in Fig. 1.9. Nielsen31 points out that previous ﬁeld investigations have typically measured the mean water level elevation on the beach as opposed to the average ﬂuctuating mean water level in the vertical plane (i.e., the wave setup as usually deﬁned), and that the two measurements are often diﬀerent in part due to the beach permeability, which in turn is related to beach material size. The issue of extracting wave setup from runup and rundown on the beach is illustrated in Fig. 1.10. King et al.33 collected wave setup data at Woolacombe Beach in North Devon, U.K. which faces the North Atlantic Ocean. The beach face slope varied between 1 on 40 at high tide and 1 on 70 at mid-tide level with a tidal range of 3 m at neap and as much as 9 m at springs. Beach face material consisted of ﬁne sand with

Fig. 1.7.

Manometer setup of Davis and Nielsen32 for measuring setup.

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Fig. 1.8.

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Schematic diagram of apparatus (from Ref. 32).

Fig. 1.9. Dimensionless setup versus total depth where much of the setup occurs on the beach face (from Ref. 32). In this ﬁgure, B and D are equal to η and h as used in this chapter, respectively.

90% in the 0.125 mm to 0.25 mm size range. Pressure transducers were utilized to collect wave and setup information at various stations across the beach and also in a longshore direction to assess the spatial variability of the mean setup. Both tripod mounted and buried pressure transducers were utilized. The buried pressure transducers were 50 to 80 cm below the beach surface and were protected by a porous cover. Instruments collected pressure data which were then transformed to water level data over 4096 second intervals. Data did not include sampling in very shallow water and the maximum wave setup was estimated by extrapolating the

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Fig. 1.10. Illustration of diﬀerences between mean water level (MWL) shoreline and mean water line on beach.

water surface from the most shoreward water stations. Wave setup estimated from the data showed the wave setup to be roughly: η = 0.10H os = 0.14H orms

(1.19)

with most of the values of η/H orms between 0.11 and 0.15. The authors do not speculate as to why such low values of setup (compared to analytical results) were found in this measurement program. Yanagishima and Katoh34 discuss ﬁeld measurements of mean water level near the shoreline on the Paciﬁc Coast of Japan as measured by an ultrasonic wave gage mounted on a pier where the mean depth of water was ∼ 0.4 m. The setup was determined via a multiple regression approach on 1305 sets of (20 minute records) data taking into account astronomical tide, wind setup, and atmospheric pressure head components of mean water level. Their data included 91 records in which the oﬀshore wave height was above 3 m. Yanagishima and Katoh’s34 regression analysis suggested the following relationship: 0.2 Los η = 0.0520H os , (1.20) Hos which can be formulated in terms of Iribarren number for their beach slope (1 on 60) to the following: η = 0.27H os (ξ0 )0.4 = 0.38H orms (ξ0 )0.4 .

(1.21)

Yanagishima and Katoh34 noted reasonable agreement with the theory of Goda35 (to be discussed later). Even higher values of setup would be expected on the beach face in accord with theory and ﬁndings of other researchers. Greenwood and Osborne36 conducted ﬁeld measurements on a Georgian Bay Beach, in Lake Huron, Ontario, Canada. Lake Huron has no measurable tide and the beach proﬁle at the site had a slope of 0.015 with a steeper sloped (0.031 to 0.047) inshore bar. Setup was measured using surface piercing resistance wire wave staﬀs with the shoreward most gage being in approximately 0.4 m of water depth. Measured setup values were found as follows: η = 0.19H os = 0.27H orms .

(1.22)

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It is again noted that even higher values of setup would be expected on the beach face in accord with theory and experience of other researchers. Further work by Hanslow and Nielsen37,38 utilized the manometer tube deployment shown in Fig. 1.7 on three additional beaches (Seven Mile, Palm, and Brunswick) in New South Wales, Australia. With beach face slopes ranging from 0.03 to 0.16 and mean grain sizes of swash zone beach material ranging from 0.18 to 0.5 mm, shoreline beach setup was measured using 20 minute record averages. Using the data from these three beaches as well as earlier measurements at Dee Why Beach (see Refs. 31 and 32), linear least square relationships were ﬁt to the data as follows: η = 0.27H os = 0.38H orms

with R = 0.65

(1.23)

η = 0.040 Hos L0 = 0.048 Horms L0

with R = 0.77 ,

(1.24)

or

where a somewhat higher value of explained regression was noted using wave height and wave period. Data and regression lines for these two relationships are shown in Figs. 1.11 and 1.12. The improvement in ﬁt due to inclusion of the deep water wavelength is not evident visually. A signiﬁcant ﬁnding of these studies was that a major portion of the setup occurred on the beach face (see Fig. 1.9). Further measurements on wave setup at two river entrances is also discussed in Hanslow and Nielsen37 and Dunn et al.39 with the result that the wave setup at river entrances was found to be (somewhat surprisingly) negligible. Lentz and Raubenheimer40 report on a ﬁeld experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 11 pressure sensor gages and 10 sonar altimeters extended across the surf zone from 2 to 8 m of water depth. Close agreement with Longuet-Higgins radiation stress theory for wave setup was noted

Fig. 1.11.

Empirical relationship between setup and wave height (from Ref. 38).

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Fig. 1.12.

Empirical relationship between setup and wave parameters (from Ref. 38).

although the lack of setup measurements in shallow water (< 2 m) did not allow conclusions regarding the maximum setup that might be expected on the beach. Raubenheimer et al.41 report on a second ﬁeld experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 12 buried pressure sensor gages were employed across the surf zone from the shoreline to 5 m of water depth. Again good agreement with Longuet-Higgins and Stewart42 radiation stress theory was noted by integration of the cross-shore momentum equation to estimate the wave setup for water depths greater than 1 m but the theory was found to under-predict wave setup in shallow water (h < 1 m). The lack of setup measurements on the beach face did not allow conclusions regarding the maximum setup that might be expected on a beach although an empirical equation was provided to estimate wave setup at the SWL line as follows: η SWL −1/3 = 0.019 + 0.003βf , Hos

(1.25)

where βf is the average slope across the surf zone. Raubenheimer et al.41 suggest that theory under-predicts the setup by a factor of 2 for water depths less than 1 m. Stockdon et al.43 using video shoreline water level time series determined wave setup and wave runup results during 10 diverse ﬁeld experiments (four from Duck, NC, USA; four from West Coast beaches in California/Oregon, USA; and two from Terschelling, The Netherlands). These wave setup results were analyzed to provide empirical parameterizations for wave setup under many natural beach conditions as follows: η = 0.385βf H0 L0 , (1.26) which, assuming that tan βf ≈ βf can be expressed in terms of the Iribarren number as η = 0.385ξ0 H0

(1.27)

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and for extremely dissipative beaches η 0.043 = , H0 H0 /L0

(1.28)

where H0 is the eﬀective deep water wave height, L0 is the deep water wave length associated with the peak spectral period and βf is the average slope over a depth range deﬁned in terms of the standard deviation of the water surface displacement. It should again be noted that the video camera (visual) approach estimates setup via the mean of measurements of the water surface elevation on the beach rather than the mean of ﬂuctuating water levels at one location. Results from nine of the ﬁeld experiments presented here have been analyzed to determine the average ratio of wave setup at the still water line to signiﬁcant wave height and its associated standard deviation. The ratios at the still water line were determined to be 0.191±0.100. Several caveats apply to these results. In cases where the beach slope and/or the deep water wave steepness was incorporated into the expression presented, these were taken as 0.01 and 0.04, respectively. Some of the published expressions were in terms of the breaking wave height and some in terms of the deep water wave height and no attempt was made to diﬀerentiate between breaking and deep water wave heights. The Holman and Sallenger results were not included in these results as they appeared to be anomalously high. Finally, the wave setup ratio at the intersection of the mean water line intersection with the beach proﬁle would be greater than the average ratio (0.191) above. Also, although not examined in detail here, the dynamic wave setup which increases with energetic narrow spectra, would also contribute to the total wave setup. It is relevant to note that results from ﬁeld measurements are often not consistent, possibly due to: (1) Limited measurement distances across the nearshore. (2) Use of many diﬀerent approaches to measure/evaluate setup (i.e., videos, pressure sensors, runup gages, manometers, etc.). (3) Inherent diﬃculties in obtaining a consistent datum for nearshore measurements and oﬀshore measurements. (4) A clear deﬁnition of setup on the beach face is lacking due to the nature of the permeable beach and the diﬃculty of sub-aerial setup measurements. 1.6. Published Guidance on Wave Setup for Engineering Applications Several sources of wave setup recommendations are available; two are reviewed here. The U.S. Army Corps of Engineers 1984 Shore Protection Manual (SPM) presents a graphical method to calculate wave setup at mid-depth of the surf zone. This method, developed for irregular waves, is presented in Fig. 1.13 in which the normalized setup has been multiplied by a factor of 2 to transfer approximately the results to the still water shoreline. The eﬀect of beach slope and deep water wave steepness in Fig. 1.13 are evident.

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Fig. 1.13. Nondimensional wave setup versus deep water wave steepness and proﬁle slope by the 1984 Shore Protection Manual recommendations as incorporated in Appendix D of FEMA44 Guidelines. Note that the normalized setup has been multiplied by a factor of 2 to transfer the setup from the mid-depth of the surf zone as it appears in SPM to the approximate still water level contour. Note: S in this ﬁgure is equal to η in this chapter.

Fig. 1.14. Nondimensional wave setup by Goda versus deep water wave steepness and relative water depth within the surf zone. Proﬁle slope = 1:100.

Goda35 has presented guidance for static and dynamic wave setups due to irregular waves. The guidance for static setup and a proﬁle slope of 1:100 is shown in Fig. 1.14. The eﬀects of various deep water wave steepness values are illustrated in Fig. 1.14.

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Table 1.2. Comparison of nondimensional wave setup by SPM and Goda methods for irregular waves. η/H0 H0 /L0

SPM

Goda

0.005 0.01 0.02 0.04 0.08

0.154 0.135 0.120 0.103 0.097

0.122 0.102 0.083 0.065 0.049

Note: Values in SPM method have been multiplied by 2.0 to transfer from surf zone mid-depth to still water line.

Table 1.2 presents a comparison of ratios of nondimensional wave setup values at the still water shoreline as recommended by SPM and Goda. In examining the results in Table 1.2, recall that an additional wave setup occurs from the still water line to the location where the maximum setup intersects the beach proﬁle.

1.7. Summary and Recommendations The reviews of theory, laboratory and ﬁeld data, and published guidance for engineering applications presented here have identiﬁed static and dynamic wave setup components as contributing to the deviation from still water level in the surf zone and their relevance to engineering design. Examination of the static wave setup has reinforced the eﬀect of beach slope on wave setup. The theory presented here does not account for the onshore bottom stress acting on the water column due to undertow. The available ﬁeld measurement results exhibit a wide range of wave setup to wave height ratios. Some of this variability is undoubtedly due to the eﬀect of proﬁle slope, which is not accounted for explicitly in some of the analyses and part is due to the eﬀect of wave-breaking in depths greater than the shallow water limit. Design methodology should account for the static and dynamic wave setup components. In determining the wave setup to include in design, the characteristics of the particular application of interest should be compared with those of the various ﬁeld and laboratory experiments available including those referenced here. The dominant role of beach slope should be recognized. The preliminary results presented here of η/Hs = 0.191 ± 0.100 may serve as a useful guide for the static wave setup component. It is hoped that further research with improved instrumentation, modern surveying techniques, and more diverse ﬁeld site studies will help to clarify both the static and dynamic wave setup components for future design applications.

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References 1. A. T. Ippen, Estuary and Coastline Hydrodynamics (McGraw-Hill Book Company, 1966), 744 pp. 2. M. S. Longuet-Higgins, Radiation stress and mass transport in gravity waves with application to “surf beats”, J. Fluid Mech. 13(4), 481–504 (1962). 3. M. S. Longuet-Higgins and R. W. Stewart, A note on wave set-up, J. Marine Res. 21, 4–10 (1963). 4. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves; a physical discussion, with applications, Deep-Sea Res. 11, 529–562 (1964). 5. R. A. Holman, Wave set-up, Handbook of Coastal and Ocean Engineering, Vol. 1, ed. J. Herbich (Gulf Publishing, 1990), Chapter 11, pp. 635–646. 6. van der Meer, TAW Technical Report: Wave Run-up and Wave Overtopping at Dikes, Technical Advisory Committee for Flood Defence, Delft (2003). 7. M. S. Longuet-Higgins, On the wave-induced diﬀerence in mean sea level between the two sides of a submerged breakwater, J. Marine Res. 25(2), 148–153 (1967). 8. W. R. Dally, R. G. Dean and R. A. Dalrymple, Wave height variation across beaches of arbitrary proﬁle, J. Geophys. Res. 90(C6), 11917–11927 (1985). 9. R. G. Dean, Stream function representation of nonlinear ocean waves, J. Geophys Res. 79(30), 4489–4504 (1965). 10. R. G. Dean, Evaluation and Development of Water Wave Theories for Engineering Application; Volume I: Presentation of Research Results; Volume II: Tabulation of Dimensionless Stream Function Variables, Special Report No. 1, Published by U.S. Army Corps of Engineers, Coastal Engineering Research Center (1974). 11. R. A. Dalrymple, A ﬁnite amplitude wave on a linear shear current, J. Geophys. Res. 87(C1), 483–491 (1974). 12. R. G. Dean and C. J. Bender, Static wave setup with emphasis on damping eﬀects by vegetation and bottom friction, Coast. Eng. 13, 149–156 (2006). 13. W. R. Munk, Surf beats, Trans. Am. Geophys. Union 30, 849–854 (1949). 14. M. J. Tucker, Surf beats: Sea waves of 1 to 5 minutes period, Proc. Roy. Soc. A 202, 565–573 (1950). 15. G. Symonds, D. A. Huntley and A. J. Bowen, Two-dimensional surf beat: Long wave generation by a time-varying breakpoint, J. Geophys. Res. 87(C1), 492–498 (1982). 16. G. Symonds and A. J. Bowen, Interaction of nearshore bars with wave groups, J. Geophys. Res. 89(C2), 1953–1959 (1984). 17. H. A. Schaﬀer and I. A. Svendsen, Surf beat generation on a mild-slope beach, Proc. ASCE Int. Conf. Coastal Engineering (1988), pp. 1058–1072. 18. J. K. Kostense, Measurements of surf beat and set-down beneath wave groups, Proc. ASCE Int. Conf. Coastal Engineering (1984), pp. 724–740. 19. A. J. Bowen, D. L. Inman and V. P. Simmons, Wave set-down and set-up, J. Geophys. Res. 73, 2569–2577 (1968). 20. J. A. Battjes, Set-Up Due to Irregular Waves, Report No. 72–2, Communications on Hydraulics, Delft University of Technology, Department of Civil Engineering (1972), 13 pp. (Paper also presented at the 13th Int. Conf. Coastal Engineering, Vancouver, B.C.). 21. I. A. Svendsen, Wave heights and set-up in a surf zone, Coast. Eng. 8, 302–329 (1984). 22. M. J. F. Stive and H. G. Wind, A Study of Radiation Stress and Set-Up in the Nearshore Zone, Publication No. 267, Waterlopkundig Laboratorium, Delft Hydraulics Laboratory (1982), 25 pp.

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23. T. S. Hedges and H. Mase, Modiﬁed Hunt’s equation incorporating wave setup, J. Waterway Port Coast. Ocean Eng. 130(3), 109–113 (2004). 24. H. Mase, Random wave runup height on gentle slopes, J. Waterway Port Coast. Ocean Div. 115(WW2), 649–661 (1989). 25. T. L. Walton, Interim guidance for prediction of wave run-up on beaches, Ocean Eng. 19(2), 199–207 (1999). 26. T. Saville, Experimental determination of wave set-up, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce Washington, D.C. (1961), pp. 242–252. 27. R. Dorrestein, Wave set-up on a beach, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce, Washington, D.C. (1961), pp. 230–241. 28. R. A. Holman and A. H. Sallenger, Setup and swash on a natural beach, J. Geophys. Res. 90(C1), 945–953 (1985). 29. U. A. Hansen, Wave setup and design water level, J. Waterway Port Coast. Ocean Div. 104(WW2), 227–240 (1978). 30. R. T. Guza and E. B. Thornton, Wave set-up on a natural beach, J. Geophys. Res. 96(C2), 4133–4137 (1981). 31. P. Nielsen, Wave setup: A ﬁeld study, J. Geophys. Res. 93(C12), 15643–15652 (1988). 32. G. A. Davis and P. Nielsen, Field measurement of wave set-up, ASCE Int. Conf. Coastal Engineering, Malaga, Spain (1988), Chapter 38, pp. 539–552. 33. B. A. King, M. W. L. Blackley, A. P. Carr and P. J. Hardcastle, Observations of wave induced setup on a natural beach, J. Geophys. Res. 95(C12), 22289–22297 (1990). 34. S. Yanagishima and K. Katoh, Field observation on wave setup near the shoreline, Proc. 22nd Int. Conf. Coastal Engineering, Vol. 1, ASCE, New York, N.Y. (1990), Chapter 7, pp. 95–108. 35. Y. Goda, Random Seas and Design of Maritime Structures (World Scientiﬁc Publishing Co., 2000), 443 pp. 36. B. Greenwood and P. D. Osborne, Vertical and horizontal structure in cross-shore ﬂows: An example of undertow and wave setup on a barred beach, Coast. Eng. 14, 543–580 (1990). 37. D. J. Hanslow and P. Nielsen, Wave setup on beaches and in river entrances, 23rd Int. Conf. Coastal Engineering, Venice, Italy (1992), pp. 240–252. 38. D. J. Hanslow and P. Nielsen, Shoreline set-up on natural beaches, J. Coast. Res. SI15, 1–10 (1993). 39. S. L. Dunn, P. Nielsen, P. A. Madsen and P. Evans, Wave setup in river entrances, Proc. 27th Int. Conf. Coastal Engineering, ASCE, New York, Sydney, Australia (2000), pp. 3432–3445. 40. S. Lentz and B. Raubenheimer, Field observations of wave setup, J. Geophys. Res. 104(C11), 25867–25875 (1999). 41. B. Raubenheimer, R. T. Guza and S. Elgar, Field observations of wave-driven setdown and setup, J. Geophys. Res. 106(C3), 4629–4638 (2001). 42. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves: A physical discussion with applications, Deep Sea Res. 11(4), 529–562 (1964). 43. H. F. Stockdon, R. A. Holman, P. A. Howd and A. H. Sallenger, Jr., Empirical parameterization of setup, swash, and runup, Coast. Eng. 53, 573–588 (2006). 44. FEMA, Guidelines and Specifications for Flood Hazard Mapping Partners, Appendix D: Guidance for Coastal Flooding Analysis and Mapping, Map Modernization Program, Washington, D.C. (2003). Also available at: www.fema.gov/fhm/ dl cgs.shtm.

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Chapter 2

Wavemaker Theories Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] The fundamental solutions to the wavemaker boundary value problem (WMBVP) are given for 2D channels, 3D basins, and circular basins. The solutions are given in algebraic equations that replace integral and diﬀerential calculus. The solutions are generic and apply to both full- and partial-draft piston and hinged wavemakers; to double-articulated wavemakers, and to directional wave basins. The WMBVP is solved by conformal mapping and by domain mapping. The loads on a wavemaker are connected to the radiation boundary value problem for semiimmersed bodies and demonstrate the connection of these loads to the added mass and radiation damping coeﬃcients required to compute the dynamic response of large Lagrangian solid bodies.

2.1. Introduction Wavemaker theories play several important roles in coastal and ocean engineering. The most important role is the application to laboratory wavemakers for both wavemaker designs and wave experiments. A second role for wavemaker theories is to compute a scalar radiated wave potential to compute the wave-induced loads on large solid bodies applying potential wave theory. The displacements and rotations of a semi-immersed six degrees-of-freedom large Lagrangian solid body are related to the displacements and rotations of wavemakers. The boundary between a planar wavemaker and an ideal ﬂuid requires special care because the ﬂuid motion

25

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is an Eulerian field with time and space as the independent variables, and the planar wavemaker is a Lagrangian solid body with time and the wavemaker as the independent variables. Consequently, the kinematic boundary condition will be different from the free surface boundary that separates two Eulerian ﬂuid ﬁelds of air and water. The boundary between the ﬂuid and wavemaker separates an Eulerian ﬁeld (the ﬂuid) from a Lagrangian body (the wavemaker), and the wavemaker kinematic boundary condition (WMKBC) must convert the Lagrangian wavemaker motion to a Eulerian ﬁeld motion in order that the independent variables for both dependent motion variables are equivalent. This may be accomplished by multiplying the Lagrangian motion of the wavemaker by the unit normal to the boundary. Because the unit normal is a function of space and the Lagrangian wavemaker motion is a function of time, the product will produce a motion that is a function of both space and time that are the independent variables of the Eulerian ﬂuid ﬁeld. Although this fact is not central to the WMBVP, it is an important connection between the WMBVP and the radiation potential boundary value problem for semi-immersed large Lagrangian solid bodies.1 The formulae for computing the two fundamental ﬂuid unknowns for an incompressible ﬂuid of the velocity q(x, z, t) and the pressure p(x, z, t) from a scalar velocity potential Φ(x, z, t) are given ﬁrst. The classical linear WMBVP for dimensionless 2D planar wavemaker is reviewed for two types of double-articulated planar wavemakers. The sway X1 (t) displacement of a full-draft piston wavemaker and the roll Θ5 (t) rotation of a hinged wavemaker are connected directly to the sway displacement and the roll rotation of a semi-immersed large Lagrangian solid body. In this review, integral calculus formulae for computing the integrals that are required to compute the coeﬃcients of the eigenseries for the ﬂuid motion, to compute the loads on the wavemaker and the average power required to generate the propagating waves are replaced by generic algebraic formulae. For example, an integral equation that is required to compute the nth eigenseries coeﬃcient Cn for the nth eigenfunction Ψn (Kn , z/h) from a wavemaker shape function χ(z/h) may be computed symbolically and expressed by a dimensionless algebraic formula In (α, β, b, d, Kn ), that is given by Cn = h

0

−1

χ(z/h)d(z/h)Ψn (z/h)d(z/h) = In (α, β, b, d, Kn ).

(2.1)

The coeﬃcient in (2.1) may then be computed very eﬃciently by substitution into algebraic formulae in all subsequent applications. Next a dimensionless theory for both amplitude-modulated (AM) and phase-modulated (PM) circular wavemakers is reviewed. Then, a dimensionless theory for double-actuated wavemakers is reviewed. Following that, a dimensionless directional wavemaker theory for large wave basins based on a WKBJ approximation1 is reviewed. Next, a theory for sloshing waves due to transverse motions of a segmented wavemaker in a narrow wave channel is reviewed. Then, 2D planar wavemakers are mapped to a unit circle by conformal mapping and to a ﬁxed rectangular domain by domain mapping; and both the linear and nonlinear wavemaker solutions are computed numerically.

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2.2. Planar Wavemaker in a 2D Channel Two generic planar wavemaker conﬁgurations are shown in Figs. 2.1(a) and 2.1(b). The ﬂuid motion may be obtained from the negative gradient of a dimensional scalar velocity potential Φ(x, z, t) according to q(x, z, t) = u(x, z, t)ex + w(x, z, t)ez = −∇2 Φ(x, z, t), where the 2D gradient operator in (2.2a) is given by ∇2 (•) =

∂(•) ∂(•) ex + ez . ∂x ∂z

Fig. 2.1(a).

Definition sketch for a Type I planar wavemaker.

Fig. 2.1(b).

Definition sketch for a Type II planar wavemaker.

(2.2a)

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The total pressure ﬁeld P (x, z, t) may be computed from the unsteady Bernoulli equation according to

∂Φ(x, z, t) 1 2 P (x, z, t) = p(x, z, t) + pS (z) = ρ − |∇Φ(x, z, t)| + Q(t) − ρgz, ∂t 2 (2.2b) where Q(t) = the Bernoulli constant; and the free surface elevation η(x, t) for zero atmospheric pressure according to 1 η(x, t) = g

∂Φ(x, η, t) 1 2 − |∇Φ(x, η, t)| + Q(t) ; ∂t 2

x ≥ ξ(η, t);

z = η(x, t). (2.2c)

The scalar velocity potential must be a solution to the Laplace equation ∇22 Φ = 0;

x ≥ ξ(z, t);

−h ≤ z ≤ η(x, t),

(2.3a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

x ≥ ξ(−h, t);

z = −h.

(2.3b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC): ∂ 2Φ 1 ∂ ∂Φ dQ − − ∇Φ · ∇ |∇Φ|2 + = 0; + g ∂t2 ∂z ∂t 2 dt

x ≥ ξ(η, t);

z = η(x, t). (2.3c)

Kinematic WaveMaker Boundary Condition (KWMBC): A Stokes material surface for planar wavemaker is W (x, z, t) = x − ξ(z, t), and the Stokes material derivative gives the KWMBC from ∂Φ ∂ξ ∂Φ ∂ξ DW = + − = 0; Dt ∂x ∂t ∂z ∂z

x = ξ(z, t);

−h ≤ z ≤ η(t).

(2.3d)

Kinematic Radiation Boundary Condition (KRBC): A KRBC is required as x → +∞ for uniqueness to insure that propagating waves are only right progressing or that evanescent eigenmodes are bounded. For a temporal dependence proportional to exp ±iωt, the KRBC may be expressed by lim

x→+∞

∂ ± iKn Φ(x, z, t) = 0. ∂x

(2.3e)

A velocity potential ϕ(x, z) may be deﬁned by the real part of Φ(x, z, t) = Re{ϕ(x, z) exp −i(ωt + ν)},

(2.4)

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where Re{•} means the real part of {•}; and ν = arbitrary phase angle. The linearized WMBVP for kh = O(1) is1 ∇22 ϕ(x, z) = 0;

0 ≤ x < +∞;

∂ϕ(x, z) = 0; ∂z

0 ≤ x + ∞;

∂ϕ(x, z) − k0 ϕ(x, z) = 0; ∂z lim

x→+∞

η(x, t) = Re

z = −h,

(2.5b)

z = 0,

∂ − iKn ϕ(x, z) = 0, ∂x x = 0;

−iω ϕ(x, 0) exp −i(ωt + ν) ; g

p(x, z, t) = ρ

(2.5a)

0 ≤ x < +∞;

∂ϕ(x, z) ∂ξ(z, t) exp −i(ωt + ν) = − ; ∂x ∂t

−h ≤ z ≤ 0,

∂Φ(x, z, t) ; ∂t

(2.5d)

−h ≤ z ≤ 0, x ≥ 0;

0 ≤ x < +∞;

(2.5c)

z = 0,

z = 0,

(2.5e)

(2.5f)

(2.5g)

where k0 = ω 2 /g. Because the boundary conditions deﬁned by (2.5b)–(2.5e) are prescribed on boundaries with constant values of the independent variables x and z, a solution by the method of separation of (independent) variables may be computed.1 The instantaneous wavemaker displacement ξ(z, t) from its mean position x = 0 is assumed to be strictly periodic in time with period T = 2π/ω, and may be expressed by S χ(z/h) exp −i(ωt + ν) ξ(z/h, t) = Re i (∆/h) S = χ(z/h) sin(ωt + ν). (2.6) (∆/h) The speciﬁed shape function χ(z/h) for the Type I wavemaker shown in Fig. 2.1(a) is valid for either a double-articulated piston or hinged wavemaker of variable draft and is given by the following dimensionless equation for a straight line2 : χ(z/h) = [α(z/h) + β][U (z/h + 1 − d/h) − U (z/h + b/h)],

(2.7a)

where α, β = dimensionless constants; U (•) = the Heaviside step function with two boundary conditions given by [S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h + ∆/h) = S,

(2.7b)

[S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h) = Sb ,

(2.7c)

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that may be solved simultaneously for the dimensionless coeﬃcients α, β to obtain α = (1 − Sb /S);

β = ∆/h + α(1 − d/h − ∆b /h − ∆/h).

(2.7d,e)

The coeﬃcients α and β for the speciﬁed shape function χ(z/h) in (2.7a) may be obtained for the Type II wavemaker shown in Fig. 2.1(b) by substituting ˆ S = S¯ + S;

¯ Sb = S;

∆=h−b−d

into the following boundary conditions2 in (2.7b) and (2.7c): ˆ (S¯ + S) ˆ χ(z/h = −b/h) = S¯ + S, 1 − b/h − d/h

ˆ (S¯ + S) ¯ χ(z/h = −1 + d/h) = S, 1 − b/h − d/h

(2.8a)

(2.8b)

that may be solved simultaneously for the constant coeﬃcients α, β to obtain ¯ d b S Sˆ ; β =1− − . (2.8c,d) α= ¯ ˆ ¯ ˆ h S+S S+S h 2.2.1. Eigenfunction solution to the WMBVP Because all of the boundary conditions deﬁned by (2.5b)–(2.5e) are now prescribed for constant values of the independent variables (x, z) and the dimensionless parameter kh = O(1), a solution by separation of independent variables1 is suggested according to ϕ(x, z) = X(x) • Z(z).

(2.9)

The eigenseries solution may be written compactly as1,3,4

Φ(x, z, t; Kn ) = Cn cosh Kn (z + h) exp +i(Knx − ωt + ν),

(2.10a)

n=1

where Kn = k for n = 1 and Kn = +iκn for n ≥ 2 provided that ko h − kh tanh kh = ko h + κn h tan κn h = 0;

n > 2.

(2.10b)

The eigenseries (2.10a) may be separated into a propagating Φp (x, z, t; k) and evanescent eigenmodes Φe (x, z, t; κn) or “local ” wave components3 according to Φ(x, z, t; Kn ) = Φp (x, z, t; k) + Φe (x, z, t; κn )

Cn cos κn (z + h) exp +i(Kn x − ωt + ν). = C1 cosh k(z + h) + n=2

(2.10c)

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The wave number k = 2π/λ where λ = wavelength. Because the numerical value of kh must be computed from an eigenvalue problem in the vertical z coordinate, equivalence of the eigenvalue k to the wave number 2π/λ requires a pseudo-horizontal boundary condition of periodicity given by k = 2π/λ and ϕ(x + λ, z) = ϕ(x, z). It is computationally eﬃcient to normalize the eigenseries in (2.10a) according to cosh Kn h(1 + z/h) ; n = 1, 2, . . . , Nn where the nondimensional normalizing constant Nn is  2kh + sinh 2kh   0 ; n = 1,  4kh Nn2 = cosh2 Kn h(1 + z/h)d(z/h) =  2κn h + sin 2κn h −1   ; n ≥ 2. 4κn h Ψn (Kn , z/h) =

(2.11a)

(2.11b) (2.11c)

The eigenseries in (2.10a) may be written as an orthonormal eigenseries by Φ(x, z, t; Kn ) =

∞

Cn Ψn(Kn , z/h) exp i(Knx − ωt − ν),

(2.12)

n=1

where the orthonormal eigenfunction Ψn (•,•) is dimensionless. 2.2.2. Evaluation of Cn by WM vertical displacement χ(z/h) The following dimensionless coeﬃcient computed from (2.5e) will replace integral calculus with algebraic substitution for the coeﬃcients Cn in the eigenseries (2.12): −b/h [α(z/h) + β]Ψn (Kn , z/h)d(z/h) In (α, β, b, d, Kn ) = −1+d/h

  d b     sinh K K h 1 − d − K b sinh K h 1 −   n n n  n h h  α =  (Kn h)2 Nn  b     − cosh Kn h 1 − + cosh Kn d   h b β (2.13) + sinh Kn h) 1 − − sinh Kn d (Kn h)Nn h that is dimensionless when α and β are given by (2.7d) and (2.7e) or (2.8c) and (2.8d). The coeﬃcients Cn may be computed algebraically by (2.13) from the KWMBC (2.5e) to obtain Sωh In (α, β, b, d, Kn ), Kn ∆ and the orthonormal eigenseries (2.12) is given by Cn = i

(2.14)

Φ(x, z, t; Kn ) =

∞

iSωh In (α, β, b, d, Kn )Ψn (Kn , z/h) exp i(Kn x − ωt − ν). Kn ∆ n=1

(2.15)

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2.2.3. Decay distance of evanescent eigenmodes n ≥ 2 Numerical solutions and experimental measurements of ocean and coastal designs require that the KRBC (2.5d) be applied far enough away so that only the propagating eigenmode for n = 1 in (2.12) is measurable. The evanescent eigenseries in (2.12) for n ≥ 2 will decay spatially at least as fast as the smallest evanescent eigenvalue κ2 . This eigenvalue must be κ2 h > (n − 3/2)π = π/2. If the smallest value for κ2 h > π/2, then κ2 > π/2h and ϕ(x, z) ∝ exp −(πx/2h). For the values of the evanescent eigenseries to be less than 1% of their values at the wavemaker, ϕ(x, z) ∝ exp −(πxd /2h) = 0.01 and πxd /(2h) = 4.6 ≈ 3π/2, and the minimum decay distance is xd ≥ 3h. 2.2.4. Transfer function for wave amplitude from wavemaker stroke The average rate of work or power done by a wavemaker of width B is1 ˙ τ = Pτ = B W

τ +1

0

h

p(x, z, τ )u(x, z, τ )d(z/h)dτ,

τ

(2.16a)

−1

where the temporal averaging operator is deﬁned by •τ =

τ +1

(•)dτ,

(2.16b)

τ

and ˙ τ = Pτ = W

ρω 3 S 2 Bh4 ∆2 2kh

I12 (α, β, b, d, k),

(2.16c)

so that all of the average power from a wavemaker is transferred to only the propagating eigenmode. The average energy ﬂux in a linear wave is given by1 ˙ τ = E

ρgBA2 2

CG ,

(2.16d)

where the group velocity CG is given by1 CG =

2kh C 1+ . 2 sinh 2kh

(2.16e)

Equating (2.16c) to (2.16d) gives the following transfer function for a planar wavemaker: A ko h = Ψ1 (k, 0)I1 (α, β, b, d, k). (2.16f) S kh

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2.2.5. Hydrodynamic pressure loads (added mass and radiation damping) The wave loads on a planar wavemaker may be estimated by integrating the total pressure over the wetted surface of the wavemaker, i.e., F n = dS, (2.17a,b) P M r×n 0

S

where the outward pointing unit normal n points from the wavemaker into the ﬂuid, and the pseudo-unit normal n for the rotational modes is given by n = r × n = (z + h − d)nx ey = ny ey .

(2.17c)

Force. For the Type I piston wavemaker of total width B, the horizontal component of the pressure force on the fluid side only may be computed from the real part of −b/h

Cn Ψn (Kn , z/h)d(z/h) exp −i(ωt + ν) F1 (t) = Re iρωBh n=1

−1+d/h

= −F1 cos(ωt + ν − α1 ),

(2.18a)

where the static component of the pressure force on the fluid side only is Fs = −

ρgBh2 [1 − 2(d/h) + (d/h)2 − (b/h)2 ]. 2

(2.18b)

The hydrodynamic component of F1 (t) may be separated linearly into a propagating and an evanescent component that are related to the piston wavemaker translational velocity and acceleration, respectively, from the real part of F1 (t) = −Re{[λ11 (Sω) + µ11 (−iSω 2 )] exp −i(ωt + ν)} = −µ11 (−Sω 2 sin(ωt + ν)) − λ11 (Sω cos(ωt + ν))

(2.18c)

¨ 1 (t) − λ11 X˙ 1 (t)}, = Re{−µ11X

(2.18d)

where the added mass coeﬃcient µ11 may be computed from the evanescent eigenmodes only, and the radiation damping coeﬃcient λ11 may be computed from the propagating eigenmode only. The average power may be computed from λ11 (Sω)2 . −F1 X˙ 1 t = 2

(2.19a)

Equating (2.19a) to (2.16d) yields λ11 =

A1 S1

2

ρBh CG , ko h

(2.19b)

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that relates the radiation damping coeﬃcient to the square of the ratio of the radiated wave amplitude to the amplitude of the wavemaker displacement. Moment. For the Type I wavemaker of width B, the dynamic pressure moment on one side only of the wavemaker may be computed from the real part of −b/h

d z 2 Cn M5 (t) = Re iρωBh 1+ − h h −1+d/h n=1 z × Ψn (Kn , z/h)d exp −(ωt + ν) h = −M5 cos(ωt + ν − α5 ),

(2.20a)

and the static component of the pressure moment on the fluid side only is 2 3 2 3 2 d d b ρgBh3 b d b −3 . 1− 1− +2 +3 − Ms = 6 h h h h h h (2.20b) The pressure moment M5 (t) in (2.20a) may be separated linearly into a propagating and an evanescent component that are related to the rotational velocity and acceleration from the real part of1 Sω 2 M5 (t) = −Re µ55 −i ∆(1 + ∆b /∆) Sω + λ55 −i exp −i(ωt + ν) , ∆(1 + ∆b /∆) ¨ 5 (t) − λ55 Θ ˙ 5 (t), M5 (t) = −µ55 Θ

(2.21a)

where µ55 = ρBh4

I 2 (α, β, b, d, κn ) n , κn h n=2

λ55 = ρωBh4

I12 (α, β, b, d, k) . kh

(2.21b)

(2.21c)

2.3. Circular Wavemaker Havelock5 applied Fourier integrals to develop a theory for surface gravity waves forced by circular wavemakers in water of both inﬁnite and ﬁnite depth. The ﬂuid motion may be obtained from the negative gradient of a scalar velocity potential Φ(r, θ, z, t) according to q(r, θ, z, t) = −∇Φ(r, θ, z, t),

(2.22a)

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where the 3D gradient operator ∇(•) in polar coordinates is ∇(•) =

∂ er + ∂r

1 ∂ ∂ eθ + e3 (•). r ∂θ ∂z

(2.22b)

The total pressure ﬁeld P (r, θ, z, t) may be computed from the unsteady Bernoulli equation in polar coordinates according to P (r, θ, z, t) = p(r, θ, z, t) + pS (z) ∂Φ(r, θ, z, t) 1 − |∇Φ(r, θ, z, t)|2 + Q(t) − ρgz, =ρ ∂t 2

(2.22c)

where Q(t) = the Bernoulli constant, and the free surface elevation η(r, θ, t) for zero atmospheric pressure according to 1 η(r, θ, t) = g

Q(t) ∂Φ(r, θ, η, t) 1 − |∇Φ(r, θ, η, t)|2 + ∂t 2 ρ

r ≥ b + ξ(θ, η, t);

;

z = η(r, θ, t).

(2.22d)

The scalar velocity potential Φ(r, θ, z, t) must be a solution to the continuity equation ∇2 Φ =

1 ∂ r ∂r

∂Φ 1 ∂2Φ ∂2Φ r + 2 + = 0, ∂r r ∂θ2 ∂z 2

r ≥ b + ξ(θ, z, t);

0 ≤ θ ≤ 2π;

−h ≤ z ≤ η(r, θ, t),

(2.23a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

r ≥ b + ξ(θ, −h, t);

0 ≤ θ ≤ 2π;

z = −h.

(2.23b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC): ∂Φ dQ ∂ 1 ∂ 2Φ + g − − ∇Φ · ∇ |∇Φ|2 + = 0; ∂t2 ∂z ∂t 2 dt r ≥ b + ξ(θ, η, t);

0 ≤ θ ≤ 2π;

z = η(r, θ, t).

(2.23c)

Kinematic WaveMaker Boundary Condition (KWMBC): ∂Φ ∂ξ 1 ∂Φ ∂ξ ∂Φ ∂ξ + − − = 0; ∂r ∂t r2 ∂θ ∂θ ∂z ∂z

r = ξ(θ, z, t);

−h ≤ z ≤ η(b, θ, t).

(2.23d)

Two types of circular cylindrical wavemaker displacements ξ(θ, z, t) may be analyzed, viz., amplitude-modulated (AM) and phase-modulated (PM) wavemakers.

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The distinction between these two types is in the azimuthal θ dependency of the wavemaker displacement ξ(θ, z, t) from its mean position r = b, given by cos mθ sin(ωt + ν) mS (2.23e) χ(z/h) . ξ(θ, z, t) = (∆/h) sin(ωt + ν + mθ) (2.23f) Kinematic Radiation Boundary Condition (KRBC): ∂ ± iKn Φ(r, θ, z, t) = 0; r → ∞. √ lim |Kn r|→+∞ ∂r

(2.23g)

Finally, physically realizable solutions to (2.23a) must be periodic in θ; i.e., Φ(r, θ, z, t) = Φ(r, θ + 2π, z, t).

(2.23h)

1

The dimensional WMBVP may be scaled and linearized by expanding the variables in perturbation series with a dimensionless perturbation parameter ε = kA. A scalar radiated velocity potential ϕ(r, θ, z) may be deﬁned by the real part of Φ(r, θ, z, t) = Re{ϕ(r, θ, z) exp −i(ωt + ν)}.

(2.24)

A linearized WMBVP may be obtained by setting the dimensionless parameter kA = ε = 0 and by requiring that kh = O(1). This linearized WMBVP is ∇2 ϕ(r, θ, z) = 0;

b ≤ r < +∞; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

∂ϕ(r, θ, z) = 0; ∂z

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = −h,

∂ϕ(r, θ, z) − ko ϕ(r, θ, z) = 0; ∂z √

lim

|Kn r|→+∞

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = 0, ∂ − iKn ϕ(r, θ, z) = 0, ∂r

∂ϕ(r, θ, z) ∂ξ(θ, z, t) exp −i(ωt + ν) = − ; ∂r ∂t η(r, θ, t) = Re

(2.25b)

(2.25c)

(2.25d)

r = b; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

−iωϕ(r, θ, z) exp −i(ωt + ν) ; g

(2.25e) b ≤ r < ∞; 0 ≤ θ ≤ 2π; z = 0,

P (r, θ, z, t) = {p(r, θ, z, t)} + ps (z) = Re{−iωρϕ(r, θ, z) exp −i(ωt + ν)} − ρgz, ϕ(r, θ, z) = ϕ(r + λ, θ, z);

(2.25a)

ϕ(r, θ, z) = ϕ(r, θ + 2π, z).

(2.25f) (2.25g) (2.25h,i)

The speciﬁed wavemaker shape function χ(z/h) is valid for either a doublearticulated piston or hinged circular AM or PM wavemaker of variable draft that

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Definition sketch for circular wavemaker.

is shown in Fig. 2.2 is identical to (2.7) for a 2D planar wavemaker with the dimension b replaced with a and the stroke S replaced with the azimuthal stroke m S. The solution to the WMBVP (2.25) may be compactly expressed by the following orthonormal eigenseries: m ϕ(r, θ, z)

=

∞

(1) Cmn Ψn (Kn , z/h)Hm (Kn r)MA(P ) (mθ),

(2.26a,b)

n=1

where the azimuthal mode function is cos mθ ; MA(P ) (mθ) = exp −imθ

m ≥ 0 and integer,

(2.26c,d)

and where (2.26a) represents an AM wavemaker; (2.26b) represents a PM wavemaker; Ψn (Kn , z/h) = the orthonormal eigenseries deﬁned in (2.11); (1) Hm (Kn r) = the Hankle function of the ﬁrst kind. When K1 = k and Kn = iκn for n ≥ 2 and integer, (1) Hm (iκn r) = Jm (iκn r) + iYm (iκn r) =

2 −(m+1) i Km (κn r), π

(2.26e)

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where Km (•) = the Modiﬁed Bessel (or Kelvin) function of the second kind of order m. The coeﬃcients Cmn may be computed by expanding the KWMBC in an eigenseries following the procedure in (2.14) and obtaining Cmn = −

m Sj hω

In (α, β, a, d, Kn )

(1) Kn ∆ Ln (Hm (Kn b))

Ln (Zm (ζn )) =

;

n ≥ 1 and integer,

(2.26f)

dZm (ζn ) 1 = {Zm−1 (ζn ) − Zm+1 (ζn )}; dζn 2

(1) (ζn ). Zm (ζn ) = Jm (ζn ), Ym (ζn ), Km (ζn ), Hm

(2.26g)

The solution to (2.25) is given by the real part of the following eigenseries expansion: m Φj (r, θ, z, t)

[m Sj hω]

=

m Φpj (r, θ, z, t)

+ m Φej (r, θ, z, t) [m Sj hω]

  (1)   I1 (α, β, a, d, k) Ψ1 (k, z/h)Hm (kr)    (1)  k∆   L1 (Hm (kb))   = −Re  ∞ 

In (α, β, a, d, κn ) Ψn (κn , z/h)Km (κn r)       +  κn ∆ Ln (Km (κn b)) n=2       −i(ωt+ν) . (2.27a,b) × MA(P ) (mθ)e      Because the asymptotic behavior of the evanescent eigenseries Km (κn r) depends on the mode m(1) , it is not possible to specify a minimum distance from the wavemaker equilibrium boundary at r = b where the evanescent eigenvalues are less than 1% of their value at the circular wavemaker boundary. The wave ﬁeld must be computed far away from the wavemaker, and it is understood that far away must be computed uniquely for each radial mode m for either an AM or PM circular wavemaker. The evaluation of the power, forces, and moments, and added mass and radiation damping coeﬃcients for both AM and PM circular wavemakers are given by Hudspeth.1 2.4. Directional Wavemakers Directional wavemakers are vertically segmented wavemakers that undulate sinuously and, consequently, are also called snake wavemakers. Segmented directional wavemakers may be driven either in the middle of each vertical segment or at the joint between vertical segments. Because of these two methods of wave generation, parasitic waves are formed along the wavemaker due to either the discontinuity of

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the wavemaker surface (middle segment driven) or of the derivative of the wavemaker surface (joint driven). A dimensional scalar spatial velocity potential ϕ(x, y, z) may be deﬁned by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.28a)

The dimensional linear ﬂuid dynamic pressure ﬁeld p(x, y, z, t) and 3D ﬂuid velocity vector ﬁeld may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) , ∂t

q(x, y, z, t) = −∇3 Φ(x, y, z, t), ∇3 (•) =

∂(•) ∂(•) ∂(•) ex + ey + ez . ∂x ∂y ∂z

(2.28b) (2.28c) (2.28d)

The dimensional WMBVP for directional waves is given by ∇23 ϕ(x, y, z) = 0;

x ≥ 0;

B ≤ y ≤ +B;

−h(x, y) ≤ z ≤ 0,

(2.29a)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.29b)

∂ϕ = 0; x ≥ 0; y = ±B; −h(x, y) ≤ z ≤ 0, ∂y  x = 0, ∂Φ(x, y, z, t) ∂ξ(y, z, t)  =− ; −B ≤ y ≤ +B,  ∂x ∂t −h(0, y) ≤ z ≤ 0, lim

x→+∞

∂ − iKn ϕ(x, y, z) = 0, ∂x

∂ϕ(x, y, z) = −∇2 ϕ(x, y, z) · ∇2 h(x, y); z = −h(x, y), ∂z 2 ∂ ∂2 (•), ∇22 (•) = , ∂x2 ∂y 2 ˆ z) U(y, ξ(y, z, t) = Re i [∆U (y, a)][∆U (z, b, d)] exp −i(ωt + ν) , ω ˆ (y, z) = U

Sω ∆/ho

(2.29c)

(2.29d)

(2.29e)

(2.29f) (2.29g)

(2.29h)

Γ(y)χ(z/ho ),

∆U (y, a) = U (y + a− ) − U (y − a+ ), ∆U (z, b, d) = U (z + h − d) − U (z + b),

(2.29i)

(2.29j)

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Fig. 2.3.

Definition sketch for rectangular directional wave basin.

where ko = ω 2 /g and where a± denotes the (possibly nonsymmetric) transverse ends of the directional wavemaker in the transverse y-direction in Fig. 2.3. The solution to the WMBVP in (2.29) is given by the following set of orthonormal eigenfunctions: ϕ(x, y, z) = i

g

ζn (x, y)Υn (Kn , z/h), ω n=1

  Ψ1 (k, z/h) ; Ψ1 (k, 0) Υn (Kn , z/h) =  Ψ1 (κn , z/h); ko h = Kn h tanh Kn h = 0;

(2.30a)

n = 1,

(2.30b)

n ≥ 2,

(2.30c)

n = 1, 2, 3, . . . ,

(2.30d)

where K1 = k and Kn = +iκn for n ≥ 2 and Ψn (•) is deﬁned in (2.11). The orthonormal eigenfunctions (2.30b) and (2.30c) are applicable strictly only for constant depth wave basins; however, they may be applied to slowly varying depth wave basins if (2.30b) and (2.30c) are considered to be evaluated only locally over relatively small horizontal length scales (e.g., several wavelengths λ), where the depth may be considered to be locally equal to a constant by a Taylor series expansion of the depth.1 Substituting (2.30a) into (2.29a) yields the following 2D Helmholtz equation: ∇22 ζ(x, y) + Kn2 ζ(x, y) = 0;

x ≥ 0;

−B ≤ y ≤ +B.

(2.31)

Alternatively, for wave basins with mildly sloping bottoms, the mild slope equation may be applied according to1 ∇2 • (CCG ∇2 ζ(x, y)) + Kn2 CCG ζ(x, y) = 0,

(2.32)

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where the wave group velocity CG is given by (2.16e). If the product CCG is a constant, (2.32) reduces to the 2D Helmholtz equation (2.31). Applying the WKBJ approximation1 for the x-dependent solution in the method of separation of variables to (2.32) yields the following solution1 : g Φ(x, y, z, t) = Re i ζ(x, y)Υn (Kn , z/h) exp −i(ωt + ν) ω   M   g

[C(x)CG (x)]x=0     Ξm (µm , y/B) Amn i  ω m=0 n=1 C(x)CG (x) = Re x        × Υn(Kn , z/h) exp i Qmn dξ exp − i(ωt + ν)  (2.33a)

Ξm (µm , y/B) =

cos µm B(y/B − 1) ; Mm

Qmn =

Mm = 1 + δm0 ;

Kn2 − µ2m ;

m > 0;

µm =

mπ 2B

K1 = k > µ2m .

(2.33b–d)

(2.33e)

The coeﬃcients Am may be computed from (2.29d) by expanding the wavemaker shape function in orthonormal eigenfunctions1 and are given by k

Amn

S Qmn (∆/hx0 )

= −Ψn (Kn , 0)In (α, β, b, d, k) ×

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B).

(2.34)

2.4.1. Full-draft piston wavemaker The prescribed transverse y-component of the snake displacement of a full-draft (b = d = 0) piston (α = 0 and β = 1) wavemaker may be expressed as Γj (qj , y/B) =

+∞

c˜j exp i[qj B(y/B + νy )],

(2.35a)

j=−∞

where the coeﬃcients c˜j may be computed from the integral in (2.34) by cmj = =

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B)

Ra+ ,a− + iIa+ ,a− . mB(qj2 − µ2m )(1 + δm0 )

(2.35b)

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If the full-draft piston snake wavemaker spans the entire width of the wave basin so that a± = ±B, then (2.35b) reduces to the integral in (2.34) and 4qj B cmj =

sin[qj B(νj − 1)] + (−1)m sin[qj B(νj + 1)] −i{cos[qj B(νj − 1)] + (−1)m cos[qj B(νj + 1)]} ((qj B − (mπ)2 )2 )(1 + δm0 )

.

(2.35c)

Values for cmj for (possibly nonsymmetric) values for a± are given by Hudspeth.1

2.5. Sloshing Waves in a 2D Wave Channel A long rectangular wave channel with a horizontal ﬂat bottom, two rigid vertical sidewalls, and a wavemaker may generate either 2D, long-crested progressive waves or two types of transverse waves, viz., (1) sloshing waves that are excited directly by transverse motion of the wavemaker or (2) cross waves that are excited parametrically by the progressive waves at a sub-harmonic of the wavemaker frequency. The WMBVP for 3D sloshing waves is identical to (2.5) for planar 2D wavemakers except for the KWMBC at x = 0 and an additional kinematic boundary condition on the sidewalls of the 2D wave channel at y = ±B in Fig. 2.4. The kinematic and dynamic wave ﬁelds may be computed from a dimensional 3D scalar

Fig. 2.4.

Definition sketch for a sloshing wave channel.

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43

velocity potential Φ(x, y, z, t). The ﬂuid velocity q(x, y, z, t) may be computed from the negative directional derivative of a scalar velocity potential by q(x, y, z, t) = −∇Φ(x, y, z, t). (2.36a) The dimensional ﬂuid dynamic pressure ﬁeld p(x, y, z, t) may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) . ∂t

(2.36b)

A spatial velocity potential ϕ(x, y, z) may be deﬁned by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.36c)

The WMBVP for sloshing waves is given by the following: ∇2 ϕ(x, y, z) = 0;

x ≥ 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0,

(2.37a)

∂ϕ(x, y, −h) = 0; x ≥ 0; −B ≤ y ≤ +B; z = −h, ∂z

(2.37b)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.37c)

∂Φ(x, y, z, t) ∂ξ(y, z, t) =− ; x = 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0, ∂x ∂t ∂ϕ = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y ∂ lim − iKn ϕ(x, y, z) = 0, x→+∞ ∂x   x ≥ 0, −iω −B ≤ y ≤ +B, Φ(x, y, 0, t) ; η(x, y, t) = Re  g z = 0.

(2.37d)

(2.37e)

A solution to (2.37) is given by the following eigenfunction expansions1 :

˜ Φ(x, y, z, t) = Re ψn (x, y)Ψn (Kn , z/h) exp −i(ωt + ν) ,

(2.37f)

(2.37g)

(2.38a)

n=1

ϕ(x, y, z) = ψ˜n (x, y)Ψn (Kn , z/h),

η(x, y, t) = Re ζn (x, y) exp −i(ωt + ν) ,

(2.38b) (2.38c)

n=1

ω ζn (x, y) = −i ψ˜n (x, y)Ψn (Kn , 0), g

(2.38d)

g ζn (x, y) , ψ˜n (x, y) = i ω Ψn (Kn , 0)

(2.38e)

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where ζn (x, y) is sometimes referred to as a displacement potential. The scalar potential (2.38a) may be expressed from (2.38d) and (2.38e) as

g Ψn (Kn , z/h) Φ(x, y, z, t) = Re exp −i(ωt + ν) . (2.39) i ζn (x, y) ω Ψn (Kn , 0) n=1 The WMBVP may be expressed in terms of a displacement potential ζn (x, y) by   x ≥ 0, 2 2 ∂ ζn (x, y) ∂ ζn (x, y) 2 −B ≤ y ≤ +B, (2.40a) + + K ζ (x, y) = 0; n n  ∂x2 ∂y 2 −h ≤ z ≤ 0.   x = 0,

∂ζn (x, y) Ψn (Kn , z/h) ω (2.40b) = i U (y, z) B ≤ y ≤ +B,  ∂x Ψn (Kn , 0) g n=1 −h ≤ z ≤ 0. ∂ (2.40c) lim − iKn ζn (x, y) = 0. x→+∞ ∂x ∂ζn (x, y) = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y

(2.40d)

where (2.40a) is the 2D Helmholtz equation.9,10 Because the boundary conditions are prescribed on boundaries that are constant values of (y,z), a solution to the WMBVP (2.40) may be computed by the method of separation of variables and is given by1 Φ(x, y, z, t)                   g = Re i  ω                

 M

 Ψ1 (k, z/h)   Cm1 Υm (y/B)  exp iPm1 x   Ψ1 (k, 0)  m=0    

 Ψ1 (k, z/h)   exp + Cm1 Υm (y/B)    Ψ (k, 0)                 

                 

             1 m=M+1 exp −i(ωt + ν) ,   − Ξm1 x          

  Ψn(κn , z/h)    exp + Cmn Υm (y/B)       Ψ (κ , 0) n n  m=0 n=2     − Qmn x (2.41a)

where Υm (µm , y/B) =

cos µm B(y/B − 1) ; Mm µm =

mπ , 2B

2 Mm = 1 + δm0 ;

(2.41b)

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Pm1 =

K12 − µ2m = Ξm1 =

k 2 − µ2m ;

µ2m − k 2 ;

Q2mn = µ2m − Kn2 ; n ≥ 1: K1 = k > µm : Qm1 = i

k < µm

45

k > µm ; m ≤ M,

k < µm ; m ≥ 0;

(2.41d)

n ≥ 1,

(2.41e) m ≤ M,

n = 1: k = µm : Qm1 = 0 : Qm1 = µ2m − k 2 = Ξm1 > 0; m > M,

(2.41c)

m > M,

k 2 − µ2m = iPm ;

n ≥ 2 : Kn = iκn : Qmn =

FA

µ2m + κ2n > 0,

(2.41f)

(2.41g) (2.41h)

where M is the maximum integer value for m in order for µm < k. Substituting (2.41a) into (2.40b) yields the following solutions for Cmn 1 : ω Ψ1 (Kn , 0) +1 y 0 z z z y d d U y, Ψ1 K n , Υ m µm ; Cm1 = g Pm1 B −1 h h h B −1

Cm1 = −i

Cmn = −i

ω Ψ1 (Kn , 0) g Ξm1 ω Ψn (Kn , 0) g Qmn

+1

d

y B

−1

+1

d

y

−1

B

m≤M (2.42a) 0 z z z y U y, Ψ1 K n , Υm µm , ; d h h h B −1

m>M +1 (2.42b) z z 0 z y U y, Ψn K n , Υ m µm , ; d h h h B −1 m ≥ 0, n ≥ 2.

(2.42c)

The ﬁrst three transverse eigenmodes are illustrated in Fig. 2.5. 2.6. Conformal Mappings Conformal and domain mappings are applications of complex variables to solve 2D boundary value problems. Conformal mapping is an angle preserving transformation that will compute exact nonlinear solutions for surface gravity waves of constant

Fig. 2.5.

First three transverse eigenmodes in a 2D wave channel.

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form that may be treated as a steady ﬂow following a Galilean transformation from a ﬁxed inertial coordinate system to a noninertial moving coordinate system. Domain mapping is a transformation of the wavemaker geometry into a ﬁxed computational domain where a solution may be computed eﬃciently. 2.6.1. Conformal mapping1 Conformal mapping of the WMBVP provides a global solution that accurately accounts for the singular behavior at all irregular points. The irregular points in the physical wavemaker domain are transformed into both weak and strong singular kernels in a Fredholm integral equation. The two irregular points on the WMBVP boundary are located at (1) the intersection between the free-surface and the wavemaker boundary and (2) the intersection between the horizontal bottom and the wavemaker boundary. These two irregular points exhibit integrable weakly singular kernels. The far-ﬁeld radiation boundary exhibits a strongly singular kernel and signiﬁcantly aﬀects the solution. The irregular frequencies3,4 are included in the global solution by the Fredholm alternative. A theory for the planar WMBVP computes a global solution by a conformal mapping of the physical wavemaker boundary to a unit disk that includes the motion of an inviscid ﬂuid near irregular points that are illustrated in Fig. 2.6. A numerical solution to Laplace’s equation in a transformed unit disk may be computed from a Fredholm integral equation. The WMBVP deﬁned by (2.5) is transformed into complex-valued analytical functions where the complex-valued coordinates are deﬁned as z = x + iy. The coordinates for the semi-inﬁnite wave channel in Fig. 2.1(a) must be transformed to complexvalued coordinates z. The ﬂuid velocity q(x, y, t) and dynamic pressure p(x, y, t) may be computed from a scalar velocity potential Φ(x, y, t) according to q(x, y, t) = −∇Φ(x, y, t);

p(x, y, t) = ρ

∂Φ(x, y, t) . ∂t

(2.43a,b)

The WMBVP and Type I wavemaker shape function are given by (2.5)–(2.7). There are both Irregular (I) and Regular (R) points at the intersections between the Smooth (S) and Critical (C) boundaries B1 and B2 in the WMBVP as illustrated in Fig. 2.6 where these two boundary intersection points are identiﬁed as P1 and P2 . The classiﬁcation of the boundary points P1 and P2 in Fig. 2.6 depends on (1) the boundary conditions ϕi (Pj ) and (2) the continuity of the boundaries Bm and their derivatives where i, j, and m = 1 or 2. A conformal mapping of the semi-inﬁnite wave channel strip in the physical plane will yield a Fredholm integral equation,6,7 where these critical points may be transformed to singular points that are integrable over a smooth continuous mapped boundary. 2.6.1.1. Conformal mapping to the unit disk 1 Two conformal mappings are: (i) the physical z-plane to a semi-circle in the Z-plane shown in Fig. 2.7; and (ii) a semi-circle in the upper Z-plane to a unit disk in the

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Fig. 2.6. Combinations of Irregular (I) and Regular (R) boundary points P1 and P2 between Smooth (S) and Critical (C) boundaries B1 and B2 intersections in the WMBVP.8

Fig. 2.7. Mapping of the semi-infinite strip in the lower half x–y-plane in the physical z-plane to the upper half X–Y -plane in the Z-plane.8

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Fig. 2.8.

Mapping of the upper half-plane in the Z-plane to the unit disk in the Q-plane.

Q-plane shown in Fig. 2.8. The Schwarz–Christoﬀel transformation dz C1 √ =√ dZ Z +1 Z −1

(2.44a)

may be integrated to obtain z = x + iy h = − Log[−Z − Z 2 − 1], π

(2.44b)

where the Log[•] denotes the principal value of Log[•] and the argument of the Log[•] is −π ≤ arg < π. Inverting (2.44b) yields Z = X + iY = − cosh(πz/h),

(2.44c)

where ! πy " cos , h h ! πx " ! πy " Y = − sinh sin . h h

X = − cosh

! πx "

(2.44d) (2.44e)

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The radiation boundary in the z-plane transforms to a semi-circle in the Z-plane by 1 2πy 2πx cos + cosh , (2.44f) R2 = X 2 + Y 2 = 2 h h πx πy Y tan θ = = tanh tan . (2.44g) X h h Details of the transformation of the WMBVP are given by Hudspeth.1 2.6.1.2. Mapping Z-plane to a unit disk The upper-half-plane of the Z-plane may be mapped into a unit disk shown in Fig. 2.8 by the following bilinear transformation: i−Z Q = ξ + iζ = exp(iθ0 ) , (2.45a) i+Z that may be inverted to obtain 1−Q , Z = X + iY = i 1+Q

(2.45b)

and the mapping function coordinates are ξ=

1 − X2 − Y 2 , X 2 + (Y + 1)2

ζ=

2X , X 2 + (Y = 1)2

(2.45c,d)

that may be transformed into the cylindrical coordinates for the unit disk in Fig. 2.8 by 2X (X 2 + Y 2 − 1)2 − 4X 2 ζ . = arctan r2 = ξ 2 +ζ 2 = ; = arctan (X 2 + (Y + 1)2 )2 ξ 1 − X2 − Y 2 (2.45e,f) Details of the transformation are given by Hudspeth.1 A numerical solution to the transformed WMBVP may be computed from the following Fredholm integral equation1 : +π ˆ r , ) ∂G(r, r ˆ , , ) ˆ ∂ Φ(ˆ ˆ 2π ˆ ˆ r , ) ˆ + G(r, rˆ, , ) ˆ rˆd, ˆ π Φ(r, ) = − −π Φ(ˆ ∂ˆ r ∂ˆ r (2.46a) where G(r, rˆ, , ) ˆ = − ln[ρ(r, rˆ, , )]; ˆ

ρ2 (r, rˆ, , ) ˆ = r2 − 2rˆ r cos( ˆ − ) + rˆ2 . (2.46b,c)

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Fig. 2.9. Nodal points on the unit disk in the Q-plane and the corresponding nodal points on the wavemaker in the physical z-plane.8

Numerical solutions to (2.46a) may be computed by discretizing the unit disk boundary as shown in Fig. 2.9. The numerical details regarding the evaluation (2.46a) at the two weakly singular irregular points at B and D in the physical z-plane in Fig. 2.7 and the strongly singular point at ±π that is the vertical radiation boundary A–E at +∞ in the physical z-plane in Fig. 2.7 are tedious.8 Global numerical solutions may be computed for both the linear and the nonlinear WMBVPs. 2.6.1.3. Conformal mapping to the unit disk 2 The wavemaker geometry shown in Fig. 2.10 is mapped to the unit disk by two transformations. The WMBVP is given by (2.5) and the WM shape function is χ(y/h) = [α(y/h) + β][U (y/h + 1 − b0 /h) − U (y/h + a0 /h)].

(2.47)

In order to transform the wavemaker geometry to a Jacobian elliptic function, it must be rotated and translated as shown in Fig. 2.11. The 90◦ rotation to the z -plane is given by z = x + iy = iz = −y + ix.

(2.48a)

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Fig. 2.10.

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51

WMBVP11 with the six critical boundary points at a–a0 –b–b0 –c–d.

Fig. 2.11. Rotation and translation of the physical wavemaker rectangular strip in the z-plane to the w-plane.11

The horizontal shift to the left in the z -plane is given by z = x + iy = w − h/2 = −y − h/2 + ix.

(2.48b)

In order to map the WM geometry in the z-plane to the semi-circle in the Z-plane shown in Fig. 2.12 as a Jacobian elliptic function, the rotated and translated strip in the z -plane must have the dimensions of −K ≤ ξ ≤ +K and 0 ≤ ζ ≤ K , where K = h/2 and K = 3h = 6K. This requires a coordinate ampliﬁcation given by 2K (x + iy ) h h 2K −y − + ix . = h 2

w=

(2.48c)

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Fig. 2.12.

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Mapping of the wavemaker semi-circle in the Z-plane to the unit disk in the Q-plane.11

The Schwarz–Christoﬀel transformation from the w-plane to the Z-plane is dw =

ˆ CkdZ (a − Z)(b − Z)(c − Z)(d − Z)

.

(2.48d)

The following change of variables: ˆ dZ = adZˆ : κ = a/c; Cˆ = c, Z = aZ; modiﬁes (2.48d) to the following Jacobian elliptic function: Z dZˆ = sn−1 [Z, κ], w= 1 [(1 − Zˆ 2 )(1 − κ2 Zˆ 2 )] 2 0

(2.48e)

where sn[Z, κ] = the Jacobian elliptic function of modulus κ or sine amplitude function.9 Deﬁne kˆ = sn−1 [1, κ],

(2.48f)

and the mapping of the rectangle {x1 , x2 ; y1 , y2 } = {0, 3h; 0, h} is given by Z = X + iY "   # $ % & ! ˆ  sn − 2K  y + h2 , κ dn 2Kx h h ,k " ! =  1 − dn2 #− 2K $y + h % , κ& sn2 2Kx , kˆ  h 2 h ! " ! "  # $ % & # $ % & h 2K h 2Kx ˆ 2Kx ˆ   cn − 2K y + , κ dn − y + , κ , sn , k cn , k h 2 h 2 h h " ! , +i # 2K $ % &   h 2Kx ˆ 2 2 1 − dn − h y + 2 , κ sn h ,k (2.48g) where sn[•, •] in the copolar half-period trio in (2.48g) is deﬁned in (2.48e), and cn[•, •] and dn[•, •] are deﬁned by cn 2 [•, •] = 1−sn 2 [•, •], dn 2 [•, •] = 1−κ2 sn2 [•, •]. The mapping from the semi-circle in the Z-plane to the unit disk in the Q-plane

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Fig. 2.13.

53

Transformed boundary conditions mapped to arcs on the perimeter of the unit disk.11

is shown in Fig. 2.12; and the mapping to the unit disk in the Q-plane is shown in Fig. 2.13. The mapping of the Z-plane to the Q-plane is given by Q=

ˆ )2 + 2iZ(1 − α ˆ )2 i−Z −α ˆ (i + Z) (1 − α ˆ )2 − Z 2 (1 + α , = i+Z −α ˆ (i − Z) (1 − α ˆ 2 ) + Z 2 (1 + α) ˆ 2

(2.49a)

where −1 < α ˆ < +1. Changing variables to circular cylindrical coordinates by (1 − α ˆ 2 ) − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ )2 , (1 − α ˆ )2 − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ2 ) 2X(1 − α ˆ2) , θ(X, Y ) = arctan (1 − α) ˆ 2 − (X 2 + Y 2 )(1 + α) ˆ 2

R2 (X, Y ) =

(2.49b) (2.49c)

the unit disk may be transformed into functions of the copolar trio of Jacobian elliptic functions. The transformed WMBVP in circular cylindrical coordinates is given by Hudspeth.1 A general solution to the transformed WMBVP may be written as10 ϕ(R, θ) =

N

n=0

R

n

a ˆn ˆ cos nθ + bn sin nθ , 1 + δn0

(2.50)

where δij is the Kronecker delta function. Substituting (2.50) into the generic boundary conditions on each of the six arcs on the perimeter of the unit disk illustrated in Fig. 2.13, multiplying each of these six boundary conditions by a member of the set of the orthogonal series in (2.50), integrating over the interval of orthogonality −π ≤ θ ≤ +π yields the following matrix equation for each of the coeﬃcients a ˆn and ˆbn AB = H.

(2.51)

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Fig. 2.14.

Physical fluid domain.12

2.7. Domain Mapping Domain mapping of the WMBVP12 follows the theory by Joseph.13 The physical ﬂuid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a ﬁxed computational ﬂuid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank–Nicholson (C–N) method.14,15 At each iteration of the C–N method, the potential ﬁeld is computed by the conjugate gradient method.15 The wavemaker motion Ξ(y/h, t) is assumed to be periodic with period T = 2π/ω, and the WMBVP with the surface tension Tˆ is given by 0 ≤ y ≤ Γ(x, t), 2 ∇ Φ(x, y, t) = ∆Φ(x, y, t) = 0; (2.52a) Ξ(y/h, t) ≤ x ≤ L. Tˆ ∂ 2 Γ(x, t) ∂Φ(x, y, t) 1 ρ ∂x2 + |∇Φ(x, y, t)|2 − + gΓ(x, t) = 0. 2 3/2 ∂t 2 ∂Γ(x, t) 1+ ∂x

(2.52b)

∂Φ(x, y, t) ∂Γ(x, t) ∂Φ(x, y, t) ∂Γ(x, t) − + = 0; ∂y ∂x ∂x ∂t Ξ(Γ(x, t), t) ≤ x ≤ L; ∂Φ(x, y, t) = 0; ∂y ∂Φ(x, y, t) = 0; ∂x

y = Γ(x, t).

Ξ(Γ(x, t), t) ≤ x ≤ L;

x = L;

(2.52c) y = 0.

0 ≤ y ≤ Γ(L, t).

∂Φ(x, y, t) ∂Ξ(y/h, t) ∂Ξ(y/h, t) ∂Φ(x, y, t) =− + ; ∂x ∂t ∂y ∂y

x = Ξ(y/h, t), 0 ≤ y ≤ Γ(0, t).

(2.52d)

(2.52e)

(2.52f)

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The initial conditions for t = 0 are Γ(x, 0) = H(x); ∂Γ(x, t) = 0;

(2.52g)

Ξ(x, 0) ≤ x ≤ L.

Φ(x, Γ(x, 0), 0) = 0;

(2.52h) (2.52i)

The physical ﬂuid domain shown in Fig. 2.14 may be mapped into a dimensionless ﬁxed rectangle of dimensions 0 ≤ ξ ≤ 1 by 0 ≤ ζ ≤ 1 by the transforms ξ=

x ; L

ζ=

y ; Γ(x, t)

τ = ωt;

Γ(x, t) , h

(2.53a–d)

P (x, y, t) , ρA2 ω 2

(2.53e,f)

γ(ξ, τ ) =

and dimensionless variables by q(ξ, ζ, τ ) = −

ϕ(ξ, ζ, τ ) =

∇Φ(x, y, t) ; Aω

Φ(x, y, t) ; Ahω

w(ζ, τ ) =

p(ξ, ζ, τ ) =

Ξ(y/h, t) ; S

ˆ= T

˜ T . ρALhω 2

(2.53g–i)

Because ζ is a function of both x and y in (2.53b), transforming partial derivatives with respect to x must be done with some care.12 Details of these lengthy transformations and the transformed WMBVP in the ﬁxed mapped domain are given by Hudspeth.1 References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientiﬁc, Singapore, 2006). 2. R. T. Hudspeth, J. M. Grassa, J. R. Medina and J. Lozano, J. Hydraulic Res. 32, 387 (1994). 3. F. John, Commun. Pure Appl. Math. 2, 13 (1949). 4. F. John, Commun. Pure Appl. Math. 3, 45 (1950). 5. T. H. Havelock, Phil. Mag. 8, 569 (1929). 6. P. R. Garabedian, Partial Diﬀerential Equations (Wiley, Inc., New York, 1964). 7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). 8. Y. Tanaka, Irregular points in wavemaker boundary value problem, PhD thesis, Oregon State University (1988). 9. G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Theory and Technique (McGraw-Hill Book Co. Inc., New York, 1966). 10. R. B. Guenther and J. Lee, Partial Diﬀerential Equations of Mathematical Physics and Integral Equations (Dover Publications, Inc., New York, 1996). 11. P. J. Averbeck, The boundary value problem for the rectangular wavemaker, MS thesis, Oregon State University (1993). 12. S. J. DeSilva, R. B. Guenther and R. T. Hudspeth, Appl. Ocean Res. 18, 293 (1996).

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13. D. D. Joseph, Arch. Rational Mech. Anal. 51, 295 (1973). 14. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied Numerical Methods (John Wiley and Sons, New York, 1965). 15. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (SpringerVerlag, 1984).

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Chapter 3

Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] The Wiggins–Holmes extension of the generalized Melnikov method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to nonautonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in order to demonstrate that cross waves are chaotic. The GMM is a global perturbation analysis about a manifold of fixed points that are connected by separatrices for higher dimensional nonlinear dynamical systems. The Luke Lagrangian, density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates that are the time-dependent components of three velocity potentials that represent three standing waves. The Hamiltonian for these cross waves is homomorphic to the Hamiltonian for a parametrically excited pendulum that is an example of a Floquet oscillator that may be approximated by the Mathieu equation. Neutral stability curves measured from wave tank data motivated the inclusion of dissipation in the Luke Lagrangian density function for cross waves. An integral containing a generalized dissipation function that is proportional to the Stokes material derivative of the free surface is added to the Luke Lagrangian integral so that dissipation is correctly incorporated into the dynamic free surface boundary condition. The generalized momenta are computed from the Lagrangian; and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains nonautonomous components and is transformed by three canonical transformations in order to obtain a suspended system for the application of the Kolmogorov–Arnold–Moser (KAM) averaging

57

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operation and the GMM. The system of nonlinear nonautonomous evolution equations determined from Hamilton’s equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic orbits are computed from the unperturbed autonomous system. The nondissipative perturbed Hamiltonian system with surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic orbit; and, consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents. A chaos diagram of the largest Liapunov exponent demonstrates regions in the Floquet forcing parameter space of possible chaotic motions; and the range of values of the Floquet parametric forcing parameter ε and of the wavemaker dissipation parameter α in the α−ε space where chaotic motions may exist.1,3

3.1. Introduction The cross waves shown in Fig. 3.1 are excited parametrically by progressive wavemaker waves at a sub-harmonic of the wavemaker frequency.1 Parametrically excited standing cross waves that oscillate in a direction transverse to the wavemaker forcing with crests perpendicular to the wavemaker are analyzed by the generalized Melnikov method (GMM) and by the Generalized Herglotz Algorithm (GHA) extended to nonautonomous systems. The cross wave wavelengths and wave modes

Fig. 3.1.

Mode 2 cross wave in a 2D wave channel.1–3

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possible are determined by the channel width Lc = 2(channel width)/n, where n = the mode number and is equal to the number of half-wavelengths across the channel. Floquet instabilities occur when the cross wave wavelengths also satisfy the frequency dispersion equation for surface gravity waves. Energy is transferred from the progressive wavemaker waves to the parametrically forced cross waves through the spatial mean motion of the free surface and their growth is due to the rate of working of the transverse stresses of the progressive wavemaker waves. The cross wave instability is shown to be homomorphic to the Floquet oscillator instability2 by constructing the neutral Floquet stability diagram shown in Fig. 3.2. The simultaneous generation of a primary resonance (ωp : ωc = 2 : 1) and a secondary resonance (ωp : ωc = 1 : 1) may be observed in the cross wave data. The criteria for horseshoe maps and for homoclinic/heteroclinic orbits are selected to test whether or not cross waves are a chaotic dynamical system. Specifically, the GMM provides local criterion for the transverse intersection of stable and unstable manifolds of the perturbed system and for the resulting chaotic motion near the unperturbed (undamped and unforced) homoclinic/heteroclinic orbits. In order to apply the GMM to a suspended dynamical system that will survive the KAM averaging operation, the nonautonomous Hamiltonian is transformed by three separate canonical transformations by applying the GHA for nonautonomous transformations. Three canonical transformations are applied in order to (1) eliminate cross product terms by a rotation of axes; (2) to eliminate two degrees of freedom, and (3) to suspend the nonautonomous terms in the wavemaker forcing component of the Hamiltonian by applying the Hamilton–Jacobian transformation. The Liapunov characteristic exponents represent an alternative criterion for diagnosing the chaotic behavior of a dynamical system by measuring the mean rate of exponential divergence of nearby trajectories, and is computed numerically for the perturbed dissipative Hamiltonian when the GMM fails to predict chaos.

Fig. 3.2. Neutral Floquet stability diagram for mode 2 cross waves (o = mode 2 cross waves and = no cross waves).1,2

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Fig. 3.3.

Definition sketch of fluid domain.

3.2. Hamilton’s Principle for Cross Waves The ﬂuid is assumed to be incompressible and inviscid and the ﬂow to be irrotational. The dimensional ﬂuid particle velocities u and the dimensional total pressure in the ﬂuid P are computed from ¯ ϕ ; u = − ∇

P 1 ¯ 2 = −g z + ϕ t − |∇ ϕ| ; ρ 2

(3.1a,b)

where ϕ = a velocity potential; ρ = the ﬂuid mass density; and g = the gravitational acceleration. The ﬂuid domain is the 3D rectangular wave channel shown in Fig. 3.3. 3.2.1. Variational principle The Lagrangian L with kinematic surface tension T is given by L =

V (t )

(ζ − 1) 1 ¯ 2 dSη ; |∇ ϕ | − ϕt + g z dV + T 2 ζ S

(3.2a)

η

where η = the free surface proﬁle and where dSη = ζ dx dy ;

¯ ϕ |2 ]1/2 . ζ = [1 + |∇

(3.2b,c)

Generalized Hamilton’s principle with dissipation.4 The ﬁrst variational of (3.2) is δ

t2 t1

L dt +

t2

t1

F

Dη Dt

dt = 0;

(3.3a)

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where the generalized dissipation function per unit mass density F (Dη /Dt ) is F

Dη Dt

= −α

g κ

Sη

Dη δη dSη ; Dt ζ

(3.3b)

where α = a dimensionless damping parameter and κ = the cross wave wavenumber. The dimensional boundary-value-problem may be obtained by requiring that the variation of φ and η vanish at the arbitrary values t1 and t2 , and is given by δL +

d dt

1 2 |∇ ϕ | −ϕt +g η −T ∇ (∇ η /ζ ) δη dx dy −b χη 2 z =η 2 ˆ s ) − (∇ ϕ )δϕ dV + (ϕn + Us · n V (t ) S ηn (3.4) × δϕ dSs + T δη /ζ dSη ,

δϕ dV =

(t )

b

Sη

where n denotes a normal derivative. The Lagrangian in (3.2) reduces to1 1 ˆ s ϕ dS ϕ ∇ ϕ dV + ϕ + Us · n 2 n V S b 1 b + g η 2 dx dy + T (ζ − 1)dx dy . 2 −b χη −b χ

1 L =− 2

2

(3.5)

η

The velocity potential ϕ is the ﬁeld variable, and the free surface η may be expressed in terms of ϕ by the free surface boundary conditions and the contact line conditions at the vertical sidewalls.1,3 The velocity potential ϕ and the free surface displacement η are assumed to be linear combinations of the progressive wavemaker wave and the parametrically excited cross wave given by the following: ϕ (x , y , z , t ) = ϕp (x , z , t ) + ϕc (y , z , t ),

η (x , y , t ) =

ηp (x , t )

+

ηc (y , t ).

(3.6a) (3.6b)

The dimensional variables may be scaled by the following scales1,5: x =

x ; k

y =

h=κh; ωp2 = g k τλ ; τ12 = 1 + τ ;

y ; κ

z =

z ; κ

t ; g κ

(3.7a–g)

ξ=k;

b=κb; τ = T κ 2 /g ;

τλ = 1 + (τ /λ4r ); λ2r = κ /k ;

t = √

ω c = g κ τ12 ; 2

(3.7h–m)

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ϕp

=

ϕp ap

ηc

=

g ; k

ac ηc ;

ϕc

=

χ =

ϕc ac

aw χ;

g ; κ

ηp = ap ηp ;

La2 g L = c . kκ

(3.7n–s)

Scaling by (3.7) yields the following dimensionless scaling parameters1,3: ε = κ ac ;

β = ωc /ωp ;

γ = k aw ;

Γ = ap /ac ;

(3.8a–d)

and the following dimensionless ﬁeld variables: ϕ(x, y, z, t) = ϕc (y, z, t) + Γλr ϕp (x, z, t); η(x, y, t) = ηc (y, t) + Γηp (x, t).

(3.9a,b)

The independent variation of ϕ and η vanish at arbitrary temporal values t1 and t2 in Hamilton’s principle (3.3) yielding the following scaled boundary value problem: 1 ϕxx + ϕyy + ϕzz = 0; γχ ≤ x ≤ ξ, y ≤ |b|, −h ≤ z ≤ εη, (3.10a) λ4r 1 z = εη, γχ ≤ x ≤ ξ, y ≤ |b|, ϕ η + ϕ η ϕz = −τ1 ηt + ε x x y y ; λ4r (3.10b) ε 1 2 1 ϕ + ϕ2y + ϕ2z − τ1 ϕt + η = τ ηxx + ηyy − αϕz ; 2 λ4r x λ4r γχ ≤ x ≤ ξ, (3.10c) y ≤ |b|, z = εη ˆ = 0; ∇ϕ · n

y = |b|,

−h ≤ z ≤ εη,

γχ ≤ x ≤ ξ

z = −h,

y ≤ |b|,

γχ ≤ x ≤ ξ,

γ ϕx = − λ4r τ1 χt + γλ4r ϕz χz ; ε

x = γχ,

y ≤ |b|,

−h ≤ z ≤ εη,

(3.10d,e)

(3.10f)

plus an appropriate radiation condition at x = ξ. The free surface curvature requires a dynamical constraint that is given either by the contact line condition ηn = 0 or by the edge constraint boundary condition η = 0.1,3 The wavemaker perturbation forcing parameter γ is smaller than the Floquet parametric forcing parameter ε because experiments demonstrate that the standing cross wave amplitude becomes larger than the wavemaker forcing amplitude as t → ∞. The parameter ordering is 0 < γ 2 < εγ < ε2 < γ < ε < 1 or

0<

γ2 γ < γ < < 1. ε ε

(3.11a,b)

The higher order terms that will be neglected are O(ε2 ), O(εγ), and O(γ 2 ); but terms O(γ/ε) will be retained. Neglecting these higher order terms, the nondimensional Lagrangian integrals may be approximated by the following linear

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combination1,3 : Lχ = −

γτ1 2ε b

+ −b

b −b

0

−h

[(ϕc + Γλr ϕp )χt ]x=0 dzdy

[εχt (ηc + Γηp )(ϕc + Γλr ϕp )]x,z=0 dy,

b 0 Γ [ϕpx (ϕc + Γλr ϕp )]x=ξ dzdy Lξ = − 3 2λr −b −h b + [εϕpx (ηc + Γηp )(ϕc + Γλr ϕp )]x=ξ,z=0 dy, Lη =

1 2

−b b

−b

ξ

(3.12a)

(3.12b)

{τ1 (ηc + Γηp )t [(ϕc + Γλr ϕp )

0

+ ε(ηc + Γηp )(ϕc + Γλr ϕp )r ] + (ηc + Γηp )2 }z=0 dxdy γ b − {χ[τ1 (ϕc + Γλr ϕp )(ηc + Γηp )t + (ηc + Γηp )2 ]}x,z=0 dy, (3.12c) 2 −b b 2 τ b ξ Γ2 2 Γ 2 2 2 Lτ = η + ηcy dxdy − η + ηcy dy. γχ 2 −b 0 λ4r px λ4r px −b x,z=0 (3.12d) The wavemaker forcing χ is speciﬁed to be t χ = f (z) sin β

f (z) = 1 + f (z) = 1,

z , for a full draft hinge, h for a full draft piston;

(3.13a,b)

and the deepwater cross wave and wavemaker wave are approximated by ϕc (y, z; t) = q(t) cos(y − b) exp[z], ϕp (x, z; t) = [Q1 (t) cos x +

Q2 (t) sin x] exp[z/λ2r ],

(3.14a) (3.14b)

where the variables q(t), Q1 (t), and Q2 (t) are the generalized coordinates. The freesurface displacement η is a solution to the linearized inhomogeneous boundary value problem given by L η η = [Lpη ηp − fp ] + [Lcη ηc − fc ] ≡ 0; Lpη ηp − fp = 0; ηpx = 0;

x = 0,

(3.15a)

Lcη ηc − fc = 0;

(3.15b,c)

ηcy = 0;

(3.15d,e)

y = |b|,

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where (3.15d and 3.15e) are the contact line conditions and where Lpη ηp =

τ ηp − ηp , λ4r xx

fp = αλr ϕpz − τ1 λr ϕpt ;

Lcη ηc = τ ηcyy − ηc ,

fc = αϕcz − τ1 ϕct ;

z = 0,

z = 0.

(3.15f,g) (3.15h,i)

Substituting (3.14a) and (3.15h,i) into (3.15c) yields ηc =

1 (−qα + τ1 q) ˙ cos(y − b). τ12

Substituting (3.14b) and (3.15f,g) into (3.15b) yields −Q2 α + λ2r τ1 Q˙ 2 −Q1 α + λ2r τ1 Q˙ 1 cos x + ηp = λr τλ λr τλ

√ λ2p τ . × sin x + 2 exp − √ x λp τ

(3.16)

(3.17)

The Lagrangian may be decomposed into the following components: L = L0 + Lε + Lγ + O(ε2 , εγ, γ 2 );

(3.18a)

where   bΓ2  2    (t) + Q22 (t)) + Q1 (t)Q2 (t)] [ξ(Q −   1   2         2 2 bτ Γ 1 4 2 2 ˙ ˙ ˙ ˙ L0 = ; + [ξλ τ ( Q (t) + Q (t)) + 4τ Q (t) Q (t)] 1 2 r λ 1 2   2λ2r τλ2             bξ 2 2   − (q (t) − q˙ (t)) 2 εbΓτ1 τ εbΓ Lε = − 3 q 2 Q˙ 2 − q qQ ˙ 2; 2λr τλ 2τ1 λ3r

(3.18b)

(3.18c)

√ t q q˙ Γλr τ1 Q1 Γ2 τ12 2 ˙ ˙ Q1 (λr Q1 + τ Q2 ) + f1 + Lγ = −γb cos β εβ βτλ 2β √ t Γ2 τ 2 −q˙2 + q2 Γ2 Q21 − 2 12 (λ4r Q˙ 21 + λ2r τ Q˙ 1 Q˙ 2 + τ Q˙ 22 ) + , + γb sin β λr τλ 2 (3.18d) where f1 is an integral of (3.13a or 3.13b), L0 represents the free oscillations, Lε represents the Floquet parametric forcing of the cross wave by the progressive wave, and Lγ represents the completely nonautonomous perturbation. The analogy to the nonautonomous Hamiltonian for a Floquet oscillator follows.

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3.2.2. Legendre transform The Legendre transform of the Lagrangian yields the following Hamiltonian: ˙ t), H(q, p, t) = pq˙ + P1 Q˙ 1 + P2 Q˙ 2 − L(q, q,

(3.19)

where p = (p, P1 , P2 ) are the conjugate momenta corresponding to the generalized coordinates q = (q, Q1 , Q2 ). The conjugate momenta are computed from pi =

∂(L0 + Lε ) ; ∂ q˙i

p1 = p = bξ q˙ −

i = 1, 2, 3, εbΓ qQ2 ; 2τ1 λ3r

2τ ξ Q˙ 1 + 4 Q˙ 2 ; λr τλ 2 2 2 εbτ1 Γτ 2 bτ1 Γ λr 2τ ˙ ˙ ξ Q2 + 4 Q1 . p3 = P2 = − 3 q + 2λr τλ τλ λr τλ p2 = P1 =

bτ12 Γ2 λ2r τλ

(3.20a) (3.20b) (3.20c) (3.20d)

Inverting (3.20b–3.20d) yields q˙1 = Q˙ 1 =

p εΓqQ2 ; + bξ 2τ1 ξλ3r

λ4r τλ P1 −

Q˙ 2 =

2τ ξ P2 2 bτ1 Γ2 λ6r ξ

λ4r τλ P2 −

−

2τ ξ P1 2 2 bξτ1 Γ λ6r

(3.21a)

ετ 2 q 2 ; Γλ9r ξ 2 τ1 τλ

(3.21b)

ετ q 2 . 2Γξλ5r τ1

(3.21c)

+

Substituting (3.18), (3.20), and (3.21) into (3.19) yields the following Hamiltonian: H(q, p, t) = H0 (q, p) + Hε (q, p) + Hγ (q, p, t) + O(ε2 , εγ, γ 2 ),

(3.22a)

where the free oscillation component is H0 =

bΓ2 1 p2 (ξ(Q21 + Q22 ) + Q1 Q2 ) + bξq 2 + 2 2 2bξ +

2τ P1 P2 (P12 + P22 ) − 2 2 6 2; 2bβ 2 Γ2 ξ bτ1 Γ λr ξ

(3.22b)

the Floquet parametric forcing component is Hε =

εΓ ετ ετ 2 2 qpQ + q P − q 2 P1 ; 2 2 2τ1 ξλ3r 2τ1 Γξλ5r τ1 Γξ 2 λ9r τλ

(3.22c)

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and the nonautonomous perturbed component from the wavemaker forcing is   √   Q1 (λ2r P1 + τ P2 ) bΓλr τ1 Q1     +  f1      εβ βξλ2r   t       γ cos   √     2 β   2τ Q1 (λr P2 + τ P1 ) pq − + . Hγ = 2 6 βξ λr τλ 2βξ          √      (λ2r P1 + τ P2 )2 1 2 p2 t   2 2  −γ sin  + bq − bΓ Q1 − 2 2 2 6 β bτ1 Γ ξ λr 2 2bξ 2

(3.22d)

Although the nonautonomous Hamiltonian in (3.22d) is homomorphic to a nonautonomous Floquet Hamiltonian, a subtle but important distinction exists. The nonautonomous component in a Floquet Hamiltonian is due to parametric forcing; while, in contrast, the nonautonomous component in (3.22d) is due to external wavemaker boundary forcing that generates the progressive wave that then parametrically forces the cross waves. This sequence of external forcing to parametric forcing distinguishes cross wave parametric forcing from the simple pendulum Floquet parametric forcing. Damping forces. The scaled Hamilton’s principle (3.3) is given by1,3 δ

t2

t1

L dt =

t2

t1

3

Di δqi dt = −

i=1

t2

t1

b −b

ξ

(αϕz δη)dxdydt,

(3.23a)

0

where D = (D1 , D2 , D3 ) is a set of generalized components of the damping force corresponding to the set of generalized coordinates q = (q, Q1 , Q2 ) and where

δη =

3 ∂η i=1

∂qi

δqi +

∂η δ q˙i . ∂ q˙i

(3.23b)

Integrating by parts, the last term1,3 reduces D to the canonical variables q, p ∂ ∂η ∂η + Di = α −ϕz dxdy; i = 1, 2, 3, ϕz ∂qi ∂t ∂ q˙i −b 0 2 αεbΓ α α bξ p+ qQ2 + q, D1 = τ1 2τ12 λ3r τ12 2 α 2ατ αεbΓτ 2 α bξΓ2 2 D2 = P1 − P2 − q + Q1 , λ2r τ1 ξλ6r τ1 τλ λ9r ξτλ λ2r τλ 2 2 α αεbΓτ 2α bτ Γ2 α bξΓ2 2 P2 + q + Q1 + Q2 . D3 = τ1 λ2r 2λ5r τλ λ6r τλ2 λ2r τλ

b

ξ

(3.24a) (3.24b) (3.24c) (3.24d)

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3.3. Hamilton’s Dynamic Equations and Canonical Transformations Hamilton’s equations of motion are given by q˙i =

∂H ; ∂pi

p˙ i = −

∂H − Di ; ∂qi

i = 1, 2, . . . , N.

(3.25a,b)

If the canonical transformation Q = Q(q, p, t), P = P(q, p, t) is computed from a generating function F (u, U, t) computed from the GHA-Type I (Appendix A), then the Hamiltonian K for the new set of variables is given by K(Q, P, t) = H[q(Q, P, t), p(Q, P, t), t] +

∂F (u, U, t) ; ∂t

(3.26a)

and the transformed Hamilton’s equations of motion are N

∂K ˜ ∂qj Q˙ i = + ; Dj ∂Pi j=1 ∂Pi

i = 1, 2, . . . , N,

N ∂K ˜ j ∂qj ; P˙i = − − D ∂Qi j=1 ∂Qi

i = 1, 2, . . . , N,

˜ i (Q, P, t) = Di [q(Q, P, t), p(Q, P, t)]; D

i = 1, 2, . . . , N.

(3.26b)

(3.26c) (3.26d)

3.3.1. Three canonical transformations and analyses by the GMM In order to apply the GMM to the dynamical system in (3.26), canonical transformations are required ﬁrst to simplify terms in H0 ; second to simplify terms in Hε ; and third to suspend nonautonomous terms in Hγ . The GHA will be applied to two of the following three canonical transformations, because the rotation of axes transformation is well known so that the GHA will not be applied for that transformation even though it is still applicable. (1) Rotation of axes. The cross product terms in (3.22b) may be eliminated by q = q˜,

p = p˜,

(3.27a,b)

˜ 1 cos θ + Q ˜ 2 sin θ, Q1 = Q

P1 = P˜1 cos θ + P˜2 sin θ,

(3.27c,d)

˜ 2 cos θ − Q ˜ 1 sin θ, Q2 = Q

P2 = P˜2 cos θ − P˜1 sin θ,

(3.27e,f)

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that satisfy the Poisson bracket requirements in Appendix A. The rotation angle θ = π/4 eliminates the cross product terms, and the transformed Hamiltonian is given in component form by ˜ε + H ˜ γ (t) + O(ε2 , εγ, γ 2 ), ˜ =H ˜0 + H H→H where

  2   bξ 2 p˜2 P˜12 bΓ ξ ˜ 2     q˜ + + Q1 +    2 2   2 2bξ 2 2bβ Γ ξ           

    2 2 ˜2 P ξ bΓ 2 ˜ ˜ , + Q2 + H0 =   2 2bβ 2 Γ2 ξ          

    2 2 2 ˜ ˜   τ ( P − P ) bΓ   1 2 2 2  ˜  ˜ + (Q2 − Q1 ) +    2 4 bτ1 Γ2 λ6r ξ 2 

(3.28a)

(3.28b)

4 ˜2 − Q ˜ 1 ) − ετ√(2τ + ξλr τλ ) q˜2 P˜1 ˜ ε = √ εΓ q˜p˜(Q H 2 2τ1 ξλ3r 2 2τ1 Γξ 2 λ9r τλ

+

ετ (−2τ + ξλ4r τλ ) 2 ˜ √ q˜ P2 , 2 2τ1 Γξ 2 λ9r τλ

(3.28c)

   ˜1 + Q ˜ 2) bΓλr τ1 (Q     √ f   1      2εβ             √ √     2 2  ˜ 2 )((λr − τ )P˜1 + (λr + τ )P˜2 )  ˜1 + Q t ( Q       γ cos     +   2     β 2βξλ   r           √ √   2 2 ˜ ˜ ˜ ˜ ˜ τ (Q1 + Q2 )((λr + τ )P2 + (−λr + τ )P1 ) p˜q˜ . Hγ = − +   2 λ6 τ   βξ 2βξ   r λ         √ √     2 2 2 2 ˜ ˜   ((λ − τ ) P + (λ + τ ) P ) bΓ   1 2 r r 2   ˜ ˜  + Q ) −  ( Q 1 2   2 2 2 6   2 t   2bτ1 Γ ξ λr       −γ sin     2   β 1 p ˜   2   + b˜ q − 2 2bξ 2 (3.28d) The ﬁrst three energy square brackets [•] in (3.28b) are the action variables in the next canonical transformation. (2) Action/Angle Transformation. The three new canonical variables are given by pˆ =

bξ 2 p˜2 q˜ + , 2 2bξ

bΓ2 ξ ˜ 2 P˜12 Pˆ1 = , Q1 + 2 2bβ 2 Γ2 ξ

bΓ2 ξ ˜ 2 P˜22 , Q2 + Pˆ2 = 2 2bβ 2 Γ2 ξ

(3.29a–c)

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that satisfy the Poisson brackets. The Herglotz auxiliary functions Xi are the ratios p; X1 = q˜/˜

˜ 1 /P˜1 ; X2 = Q

˜ 2 /P˜2 . X3 = Q

(3.30a–c)

The angle variables are qˆi =

3

q˜j

j=1

∂F ∂ p˜j + ; ∂ pˆi ∂ pˆi

i = 1, 2, 3,

(3.31a)

where qˆ = tan−1 (bξX1 );

ˆ 1 = β tan−1 (bβξΓ2 X2 ); Q

ˆ 2 = β tan−1 (bβξΓ2 X3 ). Q (3.31b–d)

The canonical transformation to action/angle variables is given by the following: q˜ =

˜1 = Q

˜2 = Q

2Pˆ1 sin bξΓ2

2Pˆ2 sin bξΓ2

ˆ1 Q β

ˆ2 Q β

2ˆ p sin qˆ, bξ

p˜ =

2bξ pˆ cos qˆ,

,

P˜1 = Γβ

,

P˜2 = Γβ

2bξ Pˆ1 cos

2bξ Pˆ2 cos

(3.32a,b)

ˆ1 Q β

ˆ2 Q β

,

(3.32c,d)

,

(3.32e,f)

that satisfy the Poisson brackets.1,3 The transformed Hamiltonian is ˜ ⇒H ˆ =H ˆ0 + H ˆε + H ˆ γ (t) + O(ε2 , εγ, γ 2 ), H ˆ1 2β 2 τ Q 2β 2 τ 2 ˆ P1 cos − λ6r ξτ12 β λ6r ξτ12 ˆ2 ˆ1 ˆ2 ˆ1 ˆ2 Q Q Q P P 2 2 2 × Pˆ2 cos − sin + sin , β 2ξ β 2ξ β

(3.33a)

ˆ 0 = [ˆ p + Pˆ1 + Pˆ2 ] + H

(3.33b)

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 

ˆ2 ˆ2   Q Q   ˆ   P2 cos 2ˆ − cos 2ˆ q+ q−     β β   εˆ p ˆ Hε =

 4 bξ 3 λ3r τ1    ˆ1 ˆ1   Q Q   ˆ   − cos 2ˆ q+ − P1 cos 2ˆ q−   β β εβτ pˆ − 2 bξ 5 λ9r τ1 τλ  

ˆ1 ˆ1   Q 1 Q   4   (2τ + ξλr τλ ) Pˆ1 cos − cos 2ˆ q+       β 2 β          

   ˆ2  ˆ1 Q 1 Q 4 ˆ , × q− + (2τ − ξλr τλ ) P2 cos − cos 2ˆ  2 β β                ˆ ˆ   Q 1 Q 1   2 2   cos 2ˆ q + − cos 2ˆ q − −     2 β 2 β (3.33c) √    ˆ1 ˆ2 bλr f1 τ1  Q Q     √ + Pˆ2 sin Pˆ1 sin       β β βε ξ                   √     2 4   ˆ   + τ )(−2τ + ξλ τ ) (λ Q   λ 1 r r   ˆ     P + cos 1   6 2     λ ξ τ β λ   t r     γ cos          β   √     2 4   ˆ + τ )(−2τ + ξλ τ )  (λ  Q λ 2   r r   ˆ   P + cos   2   6 ξ2τ  λ β    λ  r                  pˆ sin(2ˆ q)   + ˆ . Hγ = 2βξ    

      ˆ   √ −β 2   2   ˆ1 cos Q1 P (λ − τ )     r   6τ 2   λ β     r 1             2       ˆ √   Q   2 2   ˆ  γ  + τ ) cos + (λ P   2 t   r     β   − sin       ξ β      

2       ˆ ˆ     Q Q 1 2   ˆ1 sin ˆ2 sin     + + P P        β β            2 2 + pˆ(sin qˆ − cos qˆ) (3.33d)

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The nonautonomous perturbed component in (3.33d) will not survive the KAM averaging theorem, and some of the nonautonomous terms may be suspended by applying the following Hamilton–Jacobian nonautonomous canonical transformation. (3) Hamilton–Jacobian transformation. The following nonautonomous transformations: qˆ =

t + q + Q1 , 2β

ˆ 1 = t + 2βQ1 , Q

ˆ 2 = 3t + 2β(q + Q1 + Q2 ) Q

(3.34a–c)

satisfy the Poisson brackets. The Herglotz auxiliary functions Xi = Xi (p, q) satisfy the nonzero Jacobian condition given by ∂(ˆ q, X) 4β 2 = 0. = ∂(p, q) qQ1 Q2

(3.35)

Solving (3.35) for qi = qi (p, X), substituting qi (p, X) into (3.34), and inverting yields p=

X1 ˆ 1 ); P1 = −X2 (t − Q ˆ 1 ); (2β qˆ − Q 2β 2β −X3 ˆ 2 ). P2 = (2t + 2β qˆ − Q 2β

(3.36a–c)

The generating function F (•, •, •) for the nonautonomous canonical transformation may be computed from the following indeﬁnite integral: F (ˆ q, p(ˆ q, X, t), t) 3 Xi ∂P1 ∂P2 ∂p =− q + Q1 ∂X + Q2 ∂X dXi ∂X i i i i=1 =

−1 ˆ 1 )2 + X2 (t − Q ˆ 1 )2 + X3 (2t + 2β qˆ − Q ˆ 2 )2 ]. [X1 (2β qˆ − Q 4β 2

(3.37)

Computing the three remaining conjugate momenta from pˆi = −

3 j=1

qj

∂F ∂pj − ; ∂ qˆi ∂ qˆi

i = 1, 2, 3

(3.38a)

1 Pˆ2 = P2 . 2β

(3.38b–d)

gives pˆ = p − P2 ,

1 Pˆ1 = (P1 − p), 2β

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The transformations in (3.34) and (3.38) satisfy the Poisson brackets, and the ﬁrst square bracket term [•] in (3.33b) may be transformed to [ˆ p + Pˆ1 + Pˆ2 ] =

1 P1 1− (p − P2 ) + . 2β 2β

(3.39)

The primary Floquet resonance condition is, approximately, ωc − ωp ≈ O(ε) so that a detuning parameter Ω may be deﬁned by ωp 1 1− = 1 − = εΩ; 2ωc 2β

(3.40)

and the primary parametric Floquet resonance is ωp : ωc = 2:1. A generating function is now given by 1 ˆ 1 ˆ 1 ) + 1 P2 (2t + 2β qˆ − Q ˆ 2 ); F (ˆ q, p, t) = p −ˆ q+ P1 (t − Q Q1 + 2β 2β 2β

(3.41)

and the transformed Hamiltonian H(p, q, t) with autonomous and nonautonomous terms by ˆ − ∂F ⇒ H = H0 + H0 (t) + Hε + Hε (t) H ∂t + Hγ + Hγ (t) + O(ε2 , εγ, γ 2 ),

(3.42a)

where βτ 1 βτ 1 1 (P P2 , + + − − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12 √ ε(p − P2 ) P1 − p Hε = εΩ(p − P2 ) + [(2βτ 2 − ξλ4r τλ (λ2r − βτ )) cos(2q)], 4 2bβξ 5 λ9r τ1 τλ √ γ bf1 λr τ1 2β − 1 P1 − p sin(2Q1 ) + γ Hγ = 4βξ 2ε 2ξβ 3 H0 =

× (P2 − p) sin 2(q + Q1 ),

(3.42b) (3.42c)

(3.42d)

and where the nonautonomous terms are given by Fadel.5 (4) Transformation to original variable (qorig , porig ) ⇒ (q, p). In order to compute the generalized damping forces in (3.24), the original canonical variables (qorig , porig ) must be expressed as functions of the ﬁnal transformed canonical

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variables (q, p) by

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2 t , p − P2 sin q + Q1 + bξ 2β t , porig = 2bξ p − P2 cos q + Q1 + 2β 1 t Q1orig = √ P1 − p sin 2Q1 + β Γ 2bβξ 3t + P2 sin 2(q + Q1 + Q2 ) + , β bβξ t P1orig = Γ P1 − p cos 2Q1 + 2 β 3t , + P2 cos 2(q + Q1 + Q2 ) + β 1 t √ Q2orig = − P1 − p sin 2Q1 + β Γ 2bβξ 3t + P2 sin 2(q + Q1 + Q2 ) + , β bβξ t − P1 − p cos 2Q1 + P2orig = Γ 2 β 3t + P2 cos 2(q + Q1 + Q2 ) + . β qorig =

(3.43a) (3.43b)

(3.43c)

(3.43d)

(3.43e)

(3.43f)

3.3.2. Transformed damping forces The transformed damping forces are1,3  √ t 2bβξα     p − P2 cos q + Q1 +     τ1 2β         √     2   t 2bξα     + p − P sin q + Q +   2 1 2   τ 2β   1         εα t , D1 = √ + p − P sin q + Q + 2 1  2β  2λ3r ξ βτ12                 3t     P sin 2(q + Q + Q ) +   2 1 2     β         ×           t     − P1 − p sin 2Q1 + β

(3.44a)

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 √ t bβαΓ(2τ + λ4r ξτλ )     √ P − p cos 2Q + 1 1   6 2ξ   β τ τ λ   1 λ r       √     4   3t bβαΓ(−2τ + λ ξτ ) λ   r   √ + P cos 2(q + Q + Q ) +   2 1 2   6 2ξ β   τ τ λ 1 λ r           2 2Γεατ t − 9 2 2 (p − P2 ) sin 2(q + Q1 ) + , D2 =   λr ξ τλ 2β         √   2   Γ bξ α t     √   + P − p sin 2Q + 1 1   2τ   β λ 2β   r λ       √   2     Γ bξ α 3t     + √ P sin 2(q + Q + Q ) + 2 1 2 2 β λr τλ 2β (3.44b)     t     − P − p cos 2Q + 1 1  √      β   αΓ bβξ      √        2τ     2λ   3t r 1     + P cos 2(q + Q + Q ) +   2 1 2   β             εαΓτ t   + ) sin 2(q + Q ) + (p − P 2 1 5 D3 = , λ ξτ 2β r λ         √     t α2 Γ b(2τ − λ4r ξτλ )      √  P − p sin 2Q + + 1 1   2 6τ   β λ 2βξ   r λ         √   2 4   3t Γ b(2τ + λ ξτ ) α   λ r     √ P sin 2(q + Q + Q ) + + 2 1 2  2 6 β  λr τλ 2βξ (3.44c) that are completely autonomous and will survive the KAM averaging operation. Averaging H(q, p, t) over the dimensionless cross wave period 2π yields

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 ) βτ 1 βτ 1 1 (P P2 + + − = − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12

√ (p − P2 ) P1 − p 2 4 2 + ε Ω(p − P2 ) + [(2βτ − ξλr τλ (λr − βτ )) cos(2q)] 4 2bβξ 5 λ9r τ1 τλ

√ 2β − 1 bf1 λr τ1 (P2 − p) sin 2(q + Q1 ) . P1 − p sin(2Q1 ) + +γ 4βξ 2ε 2ξβ 3 (3.44d)

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The averaged system of autonomous equations is1,3   2P1 + P2 − 3p     √ −a cos[2q] + εΩ + εa 1 3     P − p   1         sin[2Q1 ] 2 − γc2 sin[2(q + Q1 )] + α d1 , − γc1 √ q˙ = ε P1 − p             d (P − p) + d (P − p)   2 1 3 2     √ sin[2q] − αε P1 − p     4εα3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )]   p˙ = , α     + αd4 P2 − p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1   4γ     − P − p cos[2Q ] − 2γc (P − p) cos[2(q + Q )] c 1 1 2 2 1   ε 1 ˙ , P1 = 2 (4λr − 1)(P2 − p) − P1       +α − αεd P − p(p − P ) cos[2q] 5 1 2 2λ2r τ1 τ − λ4r ξτλ P˙ 2 = αP2 , λ6r ξτ1 τλ   (p − P2 ) sin[2Q1 ]     √ √ cos[2q] + γc + εa a 3 1   1 P1 − p ε P1 − p ˙ , Q1 =   P2 − p     sin[2q] + α2 d6 + αεd3 √ P1 − p    a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )]  , Q˙ 2 =   + α2 d7 + αεd2 P1 − p sin[2q]

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(3.45a)

(3.45b)

(3.45c)

(3.45d)

(3.45e)

(3.45f)

where ai , ci , and di are tabulated in Appendix B. 3.4. Application of the GMM and Largest Liapunov Characteristic Exponent The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped α = 0 and unperturbed γ = 0) system. The unperturbed vector ﬁeld may be computed by setting the perturbation (wavemaker forcing) parameter γ = 0 and the dissipation parameter α = 0 in (3.45), and are q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos(2q), P1 − p

(3.46a)

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p˙ = 4εa3

P1 − p(p − P2 ) sin(2q),

P˙1 = P˙2 = 0,

(3.46b) (3.46c,d)

(p − P2 ) Q˙ 1 = a1 + εa3 √ cos(2q), P1 − p Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos(2q),

(3.46e) (3.46f)

where the coeﬃcients a1 , a2 , and a3 are deﬁned in Appendix B. The unperturbed vector ﬁeld α = γ = 0 in (3.46) is a three degrees of freedom Hamiltonian system with (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . An important consequence of the Hamilton–Jacobian canonical transformation is that the unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) is independent of Q1 and Q2 . Hyperbolic saddle points. For every (P1 , P2 ) ∈ R2 , the (q − p) components of the unperturbed vector ﬁeld α = γ = 0 in (3.46) possess a hyperbolic saddle point that varies smoothly with P1 and P2 and is given by p0 (P2 ) = P2 ;

2q0 = nπ;

n = 0, 1, . . . ,

a1 − εΩ 1 √ , cos−1 2 2εa3 P1 − P2 2 a1 − εΩ P1 ≥ P2 + . 2εa3

q0 (P1 , P2 ) =

(3.47a,b)

(3.47c)

(3.47d)

The ﬁxed point (q0 , p0 ) is a hyperbolic saddle point if the determinant satisﬁes # # # ∂ q˙ ∂ q˙ # # # # ∂q ∂p # # # = (a1 − εΩ)2 − 4(P1 − P2 )ε2 a23 < 0. (3.48) # ∂ p˙ ∂ p˙ # # # # ∂q ∂p # (q=q0 ,p=p0 ) The symmetry properties of Hamiltonian systems imply that the stable and the unstable manifolds of the hyperbolic saddle point (q0 , p0 ) have equal dimensions in the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The unperturbed system α = γ = 0 has a 4D (R2 × T2 ) normally hyperbolic invariant manifold given by the union of the hyperbolic saddle points (q0 , p0 ) according to M = {(q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P1 , P2 ); p = p0 (P2 )}. (3.49) The normally hyperbolic invariant manifold M has 5D (R1 ×R2 ×T2 ) stable manifold W s (M) and unstable manifold W u (M) that coincide along the 5D heteroclinic manifold H = W s (M) ∩ W u (M) − M, where W s (M) and W u (M) are the set of

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initial conditions that approach the hyperbolic saddle points as t → ±∞ under the action of the unperturbed ﬂow. Dynamics on M. The unperturbed vector ﬁeld α = γ = 0 restricted to the normally hyperbolic invariant manifold M may be determined from1,3 P˙1 = 0;

P˙2 = 0;

Q˙ 1 = a1 ;

Q˙ 2 = a2 − a1 .

(3.50a–d)

The unperturbed vector ﬁeld α = γ = 0 restricted to M has the form of two degrees of freedom, 4D phase space (T1 × R1 × T2 ), completely integrable Hamiltonian system with the Hamiltonian given by the level energy surfaces H(P1 , P2 ) = H(q = q0 , p = p0 ; γ = 0) = a1 P1 + (a2 − a1 )P2 = E,

(3.51)

where E is a constant energy that allows the phase space motion to be reduced from four dimensions (R2 × T2 ) to three dimensions. Because P1 = P2 = 0, the constant energy surfaces may be reduced to a 1D surface in the 3D constant energy space. On this 1D surface, the angular motion is parameterized by the two frequencies σ1 = a1 , σ2 = a2 − a1 .

(3.52a,b)

The angular components of the motion on the normally hyperbolic invariant manifold M are given by Q1 (t) = a1 t + Q1 (0);

Q2 (t) = (a2 − a1 )t + Q2 (0).

(3.53a,b)

The normally hyperbolic invariant manifold M is a two-parameter (P1 , P2 ) family of 2D tori shown in Fig. 3.4. A 2D torus on M may be deﬁned by $ (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P¯1 , P¯2 ); ¯ ¯ Υ(P1 , P2 ) = , p = p0 (P¯2 ); P1 = P¯1 ; P2 = P¯2 (3.54) where each 2D torus is invariant. The level energy surface is a family of concentric tori as shown in Fig. 3.5. Because the frequencies of motion σ1 and σ2 in (3.52) are independent of P1 or P2 , they do not change from one concentric torus to another concentric torus. The motion on the surface of the invariant torus Υ ⊂ M is quasiperiodic. When the frequency ratio σ1 /σ2 = a1 /(a2 − a1 ) is an irrational number, the motion on the surface of the 2D invariant nonresonant torus may no longer be periodic; i.e., trajectories wind densely on the surface of the torus and never close on themselves. This 2D torus has a 3D (R1 × T1 ) stable manifold W S and unstable manifold W U that coincide along the 3D (R1 × T1 ) heteroclinic manifold H deﬁned for ﬁxed values of P1 and P2 , as shown in Fig. 3.4. On a constant level energy surface, neither the nonresonant invariant tori nor the stable and unstable manifolds may be isolated. In addition, the 2D nonresonant invariant torus has a 2D center manifold corresponding to nonexponentially expanding or contracting directions tangent to the normally hyperbolic invariant manifold M.

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Fig. 3.4.

3D unperturbed heteroclinic manifold.1

Fig. 3.5. Motion of a phase space point for an integrable Hamiltonian system with two degrees of freedom. (a) Invariant tori in a 3D constant energy space E; and (b) the flow on a 2D torus.1,3

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Heteroclinic orbits. For ﬁxed values of (P1∗ , P2∗ ) ∈ R2 , the (q, p) components of the unperturbed vector ﬁeld α = γ = 0 in (3.46) possess a 1D heteroclinic orbit connecting the hyperbolic saddle points (q0 , p0 ). The heteroclinic orbit lies on each of the level energy surfaces deﬁned by (3.51) and are solutions to

H(γ = 0) − (a1 P1∗ + (a2 − a1 )P2∗ ) = 0.

(3.55)

Values for q on the heteroclinic orbit are given by & % 1 q = cos−1 (a1 − εΩ)/2εa3 P1∗ − p . 2

(3.56)

In the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 ×R1 ×R2 ×T2 , the heteroclinic manifold H may be determined by substituting 2 a1 − εΩ a1 − εΩ √ √ , sin[2q] = 1 − , (3.57a,b) cos[2q] = 2εa3 P1 − p 2εa3 P1 − p into the unperturbed vector ﬁeld α = γ = 0 in (3.46), and integrating to obtain1,3 √ A ph (t) = P2 + sech2 [ At] + p(0), (3.58a) B

√ √ 1 A −1 (3.58b) − qh (t) = tan tanh( At) + q(0), 2 a1 − εΩ P1h (t) = P1 (0), P2h (t) = P2 (0),

(3.58c,d)

Q1h (t) = a1 t − qh (t) + Q1 (0) + q(0)

√ √ 1 A −1 tanh( At) + Q1 (0), = a1 t − tan − 2 a1 − εΩ Q2h (t) = (a2 − a1 )t + Q2 (0), A = [−(a1 − εΩ)2 + (P1 − P2 )B] > 0,

(3.58e) (3.58f)

B = 4ε2 a23 ,

(3.58g,h)

π q = qn (0) = (2n + 1) ; n = 0, 1, 2, . . . . (3.58i,j) 2 The trajectories of the unperturbed system α = γ = 0 along the 5D phase space (R1 × R2 × T2 ) heteroclinic manifold H may be expressed as p(0) = 0,

Ψ(P1 ,P2 ) = {qh (t), ph (t), P1 (0), P2 (0), Q1h (t), Q2h (t)}.

(3.59)

The 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T × R × R × T is a direct product between a 4D space with coordinates (q, p, P1 , P2 ) and a 2D torus with angular 1

1

2

2

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Fig. 3.6.

Unperturbed 4D phase space.

coordinates (Q1 , Q2 ). Because P1 and P2 are constants, the motion in the 6D phase space is reduced to the four dimensions (q, p, Q1 , Q2 ) shown in Fig. 3.6. Small perturbations are anticipated to break up the geometric structure of the unperturbed system α = γ = 0 and to separate the manifolds. The behavior of the perturbed systems γ > 0 and α > 0 near the unperturbed heteroclinic manifold H is required in order to apply the GMM. The distance between the stable and unstable manifolds of any surviving invariant set in the perturbed system must be computed at a point P on the unperturbed heteroclinic manifold H. Two perturbed systems are evaluated: (1) γ > 0, α = 0 and (2) γ > 0 and α > 0. 3.4.1. Geometric structure for γ > 0 and α = 0 The perturbed vector ﬁeld for nonzero wavemaker forcing γ > 0 and no dissipation α = 0 is given by q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos[2q] P1 − p

sin[2Q1 ] − γc1 √ − γc2 sin[2(q + Q1 )], ε P1 − p

   4εa3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )] , p˙ =  +αd4 P2 − α p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1 4γ P˙1 = − c1 P1 − p cos[2Q1 ] − 2γc2 (P2 − p) cos[2(q + Q1 )], ε

(3.60a)

(3.60b)

(3.60c)

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P˙2 = 0, (p − P2 ) sin[2Q1 ] Q˙ 1 = a1 + εa3 √ cos[2q] + γc1 √ , P1 − p ε P1 − p Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )],

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(3.60d) (3.60e) (3.60f)

where (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The perturbed vector ﬁeld is computed from 5D level energy surfaces deﬁned by

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 ).

(3.61)

Because the perturbation is Hamiltonian, the 3D level energy surfaces are preserved. In the 4D normally hyperbolic invariant manifold of the unperturbed space, the locally stable and unstable manifolds and the ﬂow describe the geometric structure of the perturbed phase space given by the perturbed normally hyperbolic locally invariant manifold, the locally stable and unstable manifolds, and the persistence of the 2D nonresonant invariant tori Υγ (P1 , P2 ). Persistence of M. The perturbed system γ > 0 and α = 0 possesses a 4D normally hyperbolic locally invariant manifold Mγ given by   (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :     Mγ = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ); .     p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.62)

On Mγ there are locally stable and unstable manifolds that are of equal dimensions and are close to the unperturbed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold Mγ intersects each of the 5D level energy surfaces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem.1,3 The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. KAM Theorem.1,3 The KAM theorem determines whether the recurrent motions occur on the perturbed normally hyperbolic locally invariant manifold Mγ and whether any of the two parameter families of 2D nonresonant invariant tori survive the Hamiltonian perturbation. The unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) satisﬁes the following nondegeneracy (or nonresonance) condition: # 2 # ∂ H # # ∂P 2 # 1 # 2 # ∂ H # # ∂P ∂P 2

1

# # # # (a1 − εΩ)2 # =− < 0. # 4(P1 − P2 )2 ∂ 2 H ## ∂P22 #(q=q0 ,p=p0 ;γ=0)

∂ 2 H ∂P1 ∂P2

(3.63)

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Most of the 2D nonresonant invariant tori Υ(P1 , P2 )) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold Mγ and are KAM tori. In the phase space of the perturbed system γ > 0 and α = 0, there are invariant tori that are densely ﬁlled with winding trajectories that are conditionally periodic with two independent frequencies σ1 and σ2 . The resulting conditionally-periodic motions of the perturbed system are smooth functions of the perturbation γ. A generalization of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds.1,3 In order to determine if chaos exists, two measurements are required in order to determine whether or not W s (Υγ ) and W u (Υγ ) intersect transversely. Melnikov integral. The distance between W s (Υγ ) and W u (Υγ ) at any point P ∈ H may be computed from

βn

∂H γ (P1 ,P2 ) (Ψ )dt, ∂Q1

(3.64a)

2c1 P1 − p sin(2Q1 ) + c2 (P2 − p) sin[2(q + Q1 )], ε

(3.64b)

M(Q1 (0)) = −

lim

αn ,βn →∞

−αn

where Hγ =

where Hγ = γH γ in (3.61), and where1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))]     4c1 βn   P1 − ph (cos(2qh − 2a1 t)) −  dt  ε lim   αn ,βn →∞ −αn − 2c2 (P2 − ph ) cos(2a1 t) + sin[2(Q1 (0) + q(0))]     4c1 βn   P1 − ph (sin(2qh − 2a1 t)) −  dt ,  ε lim   αn ,βn →∞ −αn + 2c2 (P2 − ph ) sin(2a1 t) (3.64c) that after retaining only even integrands reduces to1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))][I1 + I2 + I3 ], $ −2c1 (a1 − εΩ) βn I1 = lim cos(2a1 t)dt , n→∞ ε 2 a3 −αn ' ( √ 2c2 A ∞ cos(2a1 t)sech2 ( At)dt , I2 = B −∞ $ √ ∞ √ −2c1 A I3 = sin(2a1 t)tanh( At)dt . ε 2 a3 −∞

(3.65a) (3.65b)

(3.65c)

(3.65d)

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The Melnikov integral M(Q1 (0)) = 0 when ¯ 1n (0) = (2n + 1) Q1 (0) = Q

π − q(0); 4

n = 0, 1, 2, . . . ,

¯ 1n (0)) ∂M(Q ¯ 1n (0) + q(0))][I1 + I2 + I3 ] = 0, = −2 sin[2(Q ∂Q1 (0)

(3.66a)

n = 0, 1, 2, . . . . (3.66b)

Consequently, the stable W s (Υγ (P1 , P2 )) and unstable W u (Υγ (P1 , P2 )) manifolds of the KAM tori Υγ (P1 , P2 ) intersect transversely yielding Smale horseshoes1,3 on the appropriate 5D level energy surfaces. This implies multiple transverse intersections and the corresponding existence of chaotic dynamics in the perturbed system γ > 0 and α = 0. 3.4.2. Geometric structure for γ > 0 and α > 0 The perturbed dissipative system α > 0 and γ > 0 possesses a 4D normally hyperbolic locally invariant manifold Mγα that is given by

Mαγ

  (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :       = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ), .       p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.67)

The manifold Mγα has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds; and if these manifolds intersect transversely, then the Smale–Birkhoﬀ theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Υγα (P1 , P2 ) may be located on Mγα by averaging the perturbed dissipative vector ﬁeld γ > 0 and α > 0 restricted to Mγα over the angular variables Q1 and Q2 . The averaged equations have a unique stable hyperbolic ﬁxed point (P1 , P2 ) = (0, 0) with two negative eigenvalues provided that the determinant1,3 # # # ∂ P˙1 ∂ P˙ 1 # # # # ∂P ∂P2 ## ν 2 γ 2 (λ4r ξτλ − τ ) 1 # > 0. (3.68) # #= # ∂ P˙ ∂ P˙ # 2λ8r ξτ12 τλ 2 2 # # # # ∂P1 ∂P2 The Melnikov method fails to predict chaos for this dissipative system when α > 0.1,3 Liapunov characteristic exponents (LCE). Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain toward an invariant surface or an attractor of lower dimensionality than the original space.

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For the 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 , there are six real exponents that may be ordered as µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 ≥ µ6 , where µ1 is the largest Liapunov characteristic exponent (LCE). The LCE is calculated from the ﬁrst variation of the perturbed dissipative vector ﬁeld γ > 0 and α > 0 according to  ∂ q˙  ∂q   ∂ p˙     δ q˙  ∂q     δ p˙   ˙    ∂ P1  ˙     δ P1   ∂q    δ P˙  =  ∂ P˙2  2      δ Q˙   ∂q  1    ∂ Q˙ 1  δ Q˙ 2  ∂q    ∂ Q˙ 2 ∂q

∂ q˙ ∂p

∂ q˙ ∂P1

∂ q˙ ∂P2

∂ q˙ ∂Q1

∂ p˙ ∂p

∂ p˙ ∂P1

∂ p˙ ∂P2

∂ p˙ ∂Q1

∂ P˙ 1 ∂p

∂ P˙1 ∂P1

∂ P˙ 1 ∂P2

∂ P˙1 ∂Q1

∂ P˙ 2 ∂p

∂ P˙2 ∂P1

∂ P˙ 2 ∂P2

∂ P˙2 ∂Q1

∂ Q˙ 1 ∂p

∂ Q˙ 1 ∂P1

∂ Q˙ 1 ∂P2

∂ Q˙ 1 ∂Q1

∂ Q˙ 2 ∂p

∂ Q˙ 2 ∂P1

∂ Q˙ 2 ∂P2

∂ Q˙ 2 ∂Q1

∂ q˙  ∂Q2   ∂ p˙    ∂Q2    ∂ P˙ 1   ∂Q2    ∂ P˙ 2   ∂Q2    ˙ ∂ Q1   ∂Q2    ˙ ∂Q 



δq



   δp       δP1     δP  .  2    δQ   1 δQ2

(3.69)

2

∂Q2

For 20 diﬀerent values of the dimensionless damping parameter and of the dimensionless Floquet parametric forcing parameter, the largest LCEs were calculated and a chaos diagram for the positive values of the largest Liapunov exponents identiﬁed the parameter space in which chaotic motions exist.1,3

Appendix A The extension of the Herglotz algorithm to nonautonomous dynamical systems (GHA) signiﬁcantly reduces the eﬀort required to suspend the nonautonomous Hamiltonian component in (3.33d).

Generalized Herglotz algorithm (GHA) The Herglotz algorithm for autonomous dynamical systems4 may be generalized by: (1) including time t; and (2) deﬁning a generating function with a nonzero determinant of second derivatives. The GHA transforms a set of 2N variables (u, v) to a set of 2N new variables (U, V) by choosing N new variables Ui and then computing the remaining N new variables Vi uniquely from the chosen Ui so that the transformation (u, v) → (U, V) is canonical as shown by satisfying the Poisson bracket conditions.

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GHA:Type I Equate N new variables to N old variables Ui = Ui (u, v, t) such that the Poisson brackets satisfy [[Ui , Uj ]]uv = 0 and the determinant # ∂U # 1 # # ∂v1 # # # · # # # # ∂UN # ∂v1

·

·

·

∂U1 ∂vN

·

∂Ui ∂vj

·

·

·

·

·

∂UN ∂vN

# # # # # # # = 0. # # # # #

(3.A.1)

Equate N Herglotz auxiliary functions Xi (u, v) to the absolute value of the ratios of the old variables in either of the following forms: |ui /vi | Xi = Xi (u, v) = ; i = 1, 2, . . . , N, (3.A.2a,b) |vi /ui | such that the Jacobian of the new variables U and the Herglotz auxiliary functions X is nonzero ∂(U, X)/∂(u, v) = 0. Solve (3.A.2) for vi = vi (u, X); substitute vi (u, X) into Ui (u, v, t) and invert to obtain ui = ui (U, X, t). Compute the generating function F (u, U, t) from dF (u(U, X, t), U, t) =

N

(vi dui − Vi dUi ),

(3.A.3)

i=1

N ∂F ∂F dXi + dUi ∂Xi ∂Ui i=1   N N N ∂uj ∂uj  vj dXi + vj dUi − Vi dUi  . = ∂X ∂U i i i=1 j=1 j=1

(3.A.4)

Equate the coeﬃcients of like diﬀerentials and compute: (1) the generating function F (u, U, t) =

N i=1

N Xi j=1

vj

∂uj dXi + C(U, t), ∂Xi

(3.A.5)

and (2) the new N variables from Vi (u, v, t) =

N j=1

vj

∂F ∂uj − ; ∂Ui ∂Ui

i = 1, 2, . . . , N.

(3.A.6)

In order to compute the transformed Hamiltonian in terms of the new variables (U, V), the inverse canonical transformation (u(U, V, t), v(U, V, t)) must be computed,

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and the new Hamiltonian K(Q, P, t) is given by K(Q, P, t) = H(q(Q, P, t), p(Q, P, t), t) +

∂F (u, U, t) . ∂t

(3.A.7)

The GHA may also be applied to equate N old variables ui = ui (U, V, t) to obtain N new variables Ui = Ui (u, v, t).1,3 Appendix B

τ1 =

√ 1+τ

κτ = κ(1 + τ )

τλ = 1 + (τ /λ4r )

 h  2  λ 1 − exp − ; for a full draft piston,  0  r λ2r z f1 = f (z) exp 2 ds =  λr h −h  λ4r h  ; for a full draft hinge − 1 + exp − 2  h λ2r λr a1 = −

1 βτ + −8βξ 2ξλ6r τ12

c1 =

√ bf λ τ 1 r 1 4 2ξβ 3

d1 =

a2 = −

1 βτ 1 + − β 8βξ 2ξλ6r τ12

c2 =

2β − 1 4βξ

1 d2 = 8 2bβξ 3 λ3r τ12

2βτ 2 − ξλ4r τλ (λ2r − βτ ) 8 2bβξ 5 λ9r τ1 τλ

d3 =

τ (2τ + λ4r ξτλr ) 8 2bβξ 5 λ9r τλ2

d4 =

a3 =

τ τ 2 − λ2r τλ d5 = 1 2 2bβξ 3 λ5r τλ τ12

−τ + λ4r ξτλ d6 = 4βλ6r ξτλ2

1 4

τ − λ4r ξτλr 2 2 + τ1 βλ6r ξτλ2

τ + λ4r ξτλ (λ2r − 1) λ6r ξτ1 τλ

1 d7 = 4

−2 τ + λ4r ξτλ + τ12 βλ6r ξτλ2

References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. C. M. Bowline, R. T. Hudspeth and R. B. Guenther, Applicable Anal. 72, 287 (1999). 3. R. T. Hudspeth, R. B. Guenther and S. Fadel, Acta Mechanica 175, 139 (2005). 4. R. B. Guenther, H. Schwerdtfeger and G. Herglotz, Vorlesungen u ¨ber die Mechanik der Kontinua (1985). 5. S. Fadel, Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension, PhD dissertation, Oregon State University, USA (1998).

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Chapter 4

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone Yoshimi Goda ECOH CORPORATION, 2-6-4 Kita-Ueno Taito-ku, Tokyo 110-0014, Japan [email protected] A review is made on the statistical features of breaking waves in the nearshore waters. Inherent variability of the breaker index for regular waves is examined with the revised Goda’s formula. The incipient breaking height of signiﬁcant wave is about 30% lower than that of regular waves. Nonlinearity of random waves is strongest at the outer edge of surf zone, but it is destroyed by wave-breaking process inside the surf zone. The wave height distribution is the narrowest in the middle of the surf zone, but it returns to the Rayleigh near the shoreline. Large diﬀerences among various wave models are noted for prediction of wave heights in the surf zone.

4.1. Introduction Breaking waves exert strong actions on maritime and coastal structures, while wave dissipation through breaking plays a major role in the generation of nearshore currents. Without good understanding of wave-breaking process, we cannot pursue any study for coastal engineering works. Nevertheless, wave breaking is an elusive phenomenon. Not many people spend enough time to observe wonderful pictures of wave deformation by breaking and regeneration after breaking. One needs some kind of a pier at a beach to have a good look of waves that break and rush toward the shore. Otherwise, one should work for some hours in a laboratory to measure waves at various locations along a wave ﬂume. Our knowledge on wave breaking mostly comes from various literature based on previous research works. Quite a number of people use the formulas, diagrams, and other information listed there without examining the credibility of the information. For example, many people regard the breaker index, or the ratio of wave height to water depth at breaking, as a deterministic value without paying consideration to the fact that the breaker height exhibits large ﬂuctuations even for a given wave 87

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condition and that the breaker index has been obtained by drawing a mean curve among scattered data. Nowadays, many researchers are developing various numerical models of random wave transformations. They have to adopt some kind of wave-breaking criteria and energy dissipation mechanism so that the model can reproduce wave deformation in the nearshore waters. However, modelers seem to pick up whatever is readily available without deliberation on the physical features of the wave-breaking process and the appropriateness of the breaking model. The present chapter is a revised version of the author’s paper1 presented at the 4th International Conference on Asian and Pacific Conference 2007, in Nanjin, China. It aims at providing coastal engineers and researchers with the most advanced knowledge on the statistics of wave breaking in the nearshore waters so that they can make a correct approach to the problems related to wave-breaking processes.

4.2. Physical Definitions of Wave Breaking Waves are deﬁned as breaking when the crest starts to contain foams or when water particles jump out from the wave crest. It is a disruption of smooth water surface, and breaking waves are often classiﬁed into three types: spilling, plunging, and collapsing breakers. In the sea, there is no pure spilling breaker, because it is always accompanied with a small portion of plunging water. For example, we can observe from a window of an airplane such a scene of wave breaking in deep water that a crest of large wave makes plunging, leaves a patch of white foam behind, and moves forward with blue color. The following wave grows in height and breaks. It is a manifestation of wave energy being transported by the group velocity, which is one-half the phase velocity in deep water. Theoretically, three criteria can be cited. The ﬁrst is the condition that the horizontal velocity of the water particle at the wave crest becomes equal to or greater than the phase speed of wave proﬁle. The second is the upward vertical acceleration of the water particle at the wave crest to be equal to or to exceed the gravitational acceleration. The third is the vertical gradient of the total pressure at the wave crest to be zero or negative. The ﬁrst criterion has been employed by mathematicians to ﬁnd out the limiting waves of permanent form on a horizontal bed. Conventional perturbation techniques are ineﬀective to derive the limiting waves, and the approach speciﬁc to the limiting form needs to be employed. These limiting waves are characterized with the angular crest having the angle of 120◦ . Yamada and Shiotani2 have produced the most reliable computation results so far, which are summarized by Goda3 as reproduced in Table 4.1. The symbol H denotes the wave height, h is the water depth, L is the wavelength, L0 is the deepwater wavelength, C is the wave celerity, and ηc is the crest elevation. The subscripts “b” and “A” refer to the quantities at breaking and those of the small amplitude waves (Airy’s wave), respectively.

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Table 4.1. Characteristics of breaking waves of permanent type (after Goda3 based on Yamada and Shiotani2 ). Hb /L0

hb /LA a

hb /Lb

Cb /CA a

Hb /Lb

Hb /hb

ηc /Hb

inﬁnity 0.935 0.471 0.286 0.1856 0.1117 0.0763 0.0474 0.0284 0.01669 0.01095 0.00575 0.00239 0.001144 0.000437 0

inﬁnity 0.935 0.474 0.300 0.216 0.1510 0.1198 0.0915 0.0694 0.0525 0.0422 0.0306 0.01953 0.01351 0.00833 0

inﬁnity 0.7686 0.4011 0.2597 0.1885 0.1331 0.1050 0.07915 0.05909 0.04398 0.03499 0.2483 0.01570 0.01075 0.00660 0

1.193 1.189 1.181 1.154 1.143 1.134 1.141 1.156 1.174 1.193 1.207 1.231 1.244 1.257 1.263 1.285b

0.1412 0.1409 0.1386 0.1277 0.1115 0.08997 0.07410 0.05771 0.04430 0.03371 0.02720 0.01962 0.01260 0.00871 0.00538 0

0 0.1791 0.3456 0.4919 0.5912 0.6683 0.7059 0.7293 0.7496 0.7666 0.7774 0.7904 0.8028 0.8099 0.8160 0.8261b

0.6706 0.6765 0.6908 0.7165 0.7619 0.7939 0.8392 0.8766 0.9061 0.9242 0.9453 0.9649 0.9757 0.9849 1.0000b

Notes: a LA and CA denote the wavelength and celerity of small amplitude waves. b These values are those of solitary wave computed by Yamada et al.4

The limiting height of solitary wave is (H/h)b = 0.8261 by Yamada et al.,4 instead of the often-quoted value of (H/h)b = 0.78 by McCowan.5 The limiting wave steepness of (H/L)b = 0.142 by Miche6 is calculated for waves of nonlinear waves, and it becomes Hb /(L0 )A = 0.1684 when the linear deepwater wavelength (L0 )A is used. The second criterion is for the breaking of standing waves, but no theoretical computation of limiting standing waves has been made. The third criterion has been proposed by Nadaoka et al.,7 for deﬁning wave breaking in numerical time-domain computation. Zero vertical gradient of total pressure implies no presence of water below the wave crest, i.e., wave breaking.

4.3. Parameters Governing Breaker Index Because wave breaking attracts attention of many researchers, there have been proposed a number of formulas to describe the ratio of wave height to water depth. Kaminsky and Kraus8 called this ratio as the breaker height-to-depth index, but the present chapter employs the term of breaker index for the ratio of wave height to water depth at breaking for the sake of simplicity. Kamphuis9 has compared 11 formulas for testing of goodness-of-ﬁt with his 225 sets of hydraulic model tests. Rattanapitikon and Shibayama10 have collected 574 data points from 24 papers/reports and calculated the root-mean-square errors of 24 breaking index formulas. Both authors have proposed their own formulas by modifying some of the previous ones.

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Rattanapitikon et al.11 have further added 121 data points in large-scale ﬂume tests and proposed revised formulas. Similar to the approach by Rattanapitikon et al.,11 the breaker index formulas can be categorized into the following four functional forms: Hb /hb = f1 (0) = constant ,

(4.1)

Hb /hb = f2 (hb /L0 or hb /Lb ) ,

(4.2)

Hb /hb = f3 (s) ,

(4.3)

Hb /hb = f4 (s, hb /L0 or hb /Lb ) ,

(4.4)

where s denotes the bed slope. The wavelength L0 is the deepwater wavelength given by L0 = g/(2π)T 2 , where T is the wave period. Because the relative water depth hb /L0 is easily converted to hb /Lb through the dispersion relationship, the two relative depths hb /L0 and hb /Lb are interchangeable. The formula of Hb /hb = 0.78 is a typical example of Eq. (4.1). There are some other formulas using the parameter of deepwater wave steepness H0 /L0 . They can be useful for predicting the breaker height of regular waves. Nevertheless, they cannot be applied for breaking of random waves, because individual zero-crossing waves in a train of random waves are unrelated to individual waves in deepwaters. Thus, there is little room for the parameter H0 /L0 to function in the breaker index for random waves. Performance of a breaker index formula can be judged by the magnitude of the bias of the predicted breaker height from the observed heights. It should also be examined with either the root-mean-square error of predicted breaker heights or the correlation coeﬃcient between prediction and observation. The root-mean-square error analysis by Rattanapitikon and Shibayama10 is not conclusive in diﬀerentiating the merits of four functional forms, but they recommend a certain modiﬁcation of the slope eﬀect in the function f4 (s, hb /L0 ), apparently indicating their preference of this functional form. Kamphuis9 calculated the correlation coeﬃcients between 11 formulas and his laboratory data. By assigning the best-ﬁtting value to the proportionality coeﬃcient of each formula, he obtained the determination coeﬃcient R2 = 0.69 to f1 (0), R2 = 0.67 to f2 (hb /L0 ), R2 = 0.84 to f3 (s), and R2 = 0.88 to f4 (s, hb /L0 ). His result clearly suggests the necessity to include both the parameters of bed slope and relative water depth in the breaker index formula.

4.4. Breaker Index for Regular Waves and Its Scatter 4.4.1. Scatter of regular breaking waves In 1970, Goda3 presented a diagram of breaker index curves of regular waves for four bed slopes, based on the laboratory data from eight sources, which included his own large-scale tests with Hb = 0.43 to 0.93 m; Rattanapitikon et al.11 did not

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analyze this data set. Later, Goda12,13 approximated the breaker index curves with the following empirical formula: Hb hb A 4/3 1 − exp −1.5π 1 + 15s : A = 0.17 . (4.5) = hb hb /L0 L0 Equation (4.5) was derived by a graphical curve-ﬁtting technique without direct comparison with the original laboratory data. Rattanapitikon and Shibayama10 have recommended a modiﬁcation of the slope eﬀect term of (1 + 15s4/3 ) into (1.033 + 4.71s − 10.46s2 ). Upon reexamination of the original laboratory data, the author has also recognized a necessity of modifying the slope eﬀect term. The revised formula is as follows: hb Hb A 1 − exp −1.5π 1 + 11 s4/3 : A = 0.17 . (4.6) = hb hb /L0 L0 Muttray and Oumeraci14 found the best-ﬁtting coeﬃcient of Eq. (4.5) being 0.167 instead of 0.170 for the slope of 1/30. When Eq. (4.6) is applied to their data, the coeﬃcient would have the value of 0.173. Comparison of the laboratory data of breaker index with Eq. (4.6) is shown in Fig. 4.1 for ﬁve groups of bed slopes, i.e., 1/7 to 1/12, 1/20, 1/30, 1/50, and 1/200 to 1/200. (See Goda3 for description of the laboratory data employed here.) Recent data sets by Li et al.15 for s = 1/30 and s = 1/50, Li et al.16 for s = 1/200, and Lara et al.17 for s = 1/20 are also added. It is clear in Fig. 4.1 that the value of the breaker index increases as the bed slope becomes steep. Thus, it is absolutely necessary to incorporate the slope eﬀect into the breaker index formula. Because the experimental data are scattered around the index curves, the upper and lower bound curves with the range of 87%–115% of the value by Eq. (4.6) are drawn in Fig. 4.1. A quantitative evaluation of the degree of the scatter is made by means of the relative error of the breaker index, i.e., E = (1 − γmeas/γest ), where γmeas is the measured value of Hb /hb and γest is the predicted value by the breaker index formula of Eq. (4.6). The mean and the standard deviation of the relative error are calculated for each group of the bed slope. The mean Emean indicates a bias of the breaker index and the standard deviation of E represents the degree of scatter of the breaker index. Because E is deﬁned as the relative error, the standard deviation σ(E) is equivalent to the coeﬃcient of variation (CoV). A positive bias indicates a tendency of overestimate, while a negative bias shows an underestimate. Table 4.2 lists the bias and CoV of the breaker index of Eq. (4.6) for the data of various bed slopes. The slope data of 1/9 and 1/12 are excluded from the analysis because of their small sample sizes. The bias varies from −2.9% to +6.2% depending on the bed slope, but the formula of Eq. (4.6) can be regarded as yielding reasonable estimates of the breaker heights. The scatter of the data as expressed by CoV is about 5% to 7% for the bed slope of 1/200 to 1/50; it increases as the slope becomes steep, and it takes the value of 14% for the bed slope of 1/10. Such scatter of data represents an inherent stochastic nature of wave-breaking phenomenon. It resides in the data set itself, being independent of the breaker index formula being applied.

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2.0

Breaker Index, γ = Hb /hb

Slope = 1/9 to 1/12

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Kishi-Iohara 1/9 Iversen 1/10 Goda et al. 1/10 Galvin 1/10 Bowen et al. 1/12 Index curve 1/10 Upper 115% Lower 87%

0.01

0.1

Relative water depth, hb /L0

(a) Breaker index data for the slope of 1/9−1/12. 1.5 Breaker Index, γ = Hb /hb

Slope = 1/20 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Iversen 1/20 Galvin 1/20 Toyoshima 1/20 Lara 1/20 Index curve 1/20 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(b) Breaker index data for the slope of 1/20.

Breaker Index, γ = Hb /hb

1.2

Slope = 1/30

1.0 0.9 0.8 0.7 0.6 0.5 0.4

0.3 0.001

Iversen 1/30 Mitsuyasu 1/30 Toyoshima 1/30 Li Y.C. et al. 1/30 Index curve 1/30 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(c) Breaker index data for the slope of 1/30. Fig. 4.1.

Comparison of breaker index formula Eq. (4.6) with laboratory data of regular waves.

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Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7

Slope = 1/50

0.6

Iversen 1/50 Mitsuyasu 1/50 Li Y.C. et al. 1/50 Index curve 1/50 Upper 115% Lower 87%

0.5 0.4

0.3 0.001

0.01 Relative water depth, hb /L0

0.1

(d) Breaker index data for the slope of 1/50.

Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7 0.6

Slope = 1/100 to 1/200

0.5 0.4

0.3 0.001

Goda 1/100 Li Y.C. et al. 1/200 Index curve 1/100 Index curve 1/200 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(e) Breaker index data for the slope of 1/100–1/200. Fig. 4.1.

Table 4.2.

(Continued )

Bias and CoV of the breaker index formula of Eq. (4.6).

Bed slope

No. of data

Bias = Emean (%)

CoV = σ(E) (%)

1/10 1/20 1/30 1/50 1/100 1/200

29 47 73 28 32 19

−0.5 +3.9 +1.4 −2.9 +6.2 +3.5

14.0 11.3 8.6 4.8 5.5 7.4

Sources Iversen, Goda et al. Iversen, Galvin, Toyoshima et al., Lara et al. Iversen, Mitsuyasu, Toyoshima et al., Li et al. Iversen, Mitsuyasu, Li et al. Goda Li et al.

Even under a well-controlled laboratory test, the breaking point ﬂuctuates over some distance and the breaker height varies from wave to wave. Smith and Kraus18 reported on their regular wave tests that “despite care in conducting the tests and use of the average value of the given quantity (i.e., over 10 waves), wide scatter appeared in some quantities and must be considered inherent to the breaking process of realistic waves.” One cause of the data scatter is the presence of small-amplitude, long-period oscillations of water level in a laboratory ﬂume, but the breaking process itself is triggered by many small factors beyond the control of experimenters. We should regard the wave-breaking phenomenon as stochastic one and accept a certain range of natural ﬂuctuation. As listed in Table 4.2, the coeﬃcient of

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variation is large for steep slope and it becomes small for gentle slope. In any research work involving wave-breaking phenomena, due consideration should be given to such stochastic nature of breaker heights. 4.4.2. Scatter of random breaking wave heights The ﬁrst eﬀort to identify the breaker heights of individual waves among trains of irregular waves was undertaken by Kimura and Seyama19 and Seyama and Kimura,20 who analyzed about 1000 individual breaking waves for each slope of 1/10, 1/20, 1/30, and 1/50 with the aid of a video camera and wave gauges. The breaker index of individual waves varied over a wide range, which was equivalent to CoV of 18% to 23% (the present author’s visual inspection of scatter diagrams). In order to reduce the scatter of data, they proposed to employ an artiﬁcial water depth below the mid-level between the wave crest and trough of individual breaking waves and succeeded in reducing CoV to the values between 8% and 11%. Because Eq. (4.5) yielded the breaker index larger than most of the observed value, they proposed its modiﬁed version. When the revised breaker index formula of Eq. (4.6) is employed, however, the center line of the scatted data appears to be at the level of 85% (s = 1/10) to 95% (for other slopes) of the predicted value. Black and Rosenberg21 made observation of individual breaking waves on a natural beach with a depth of 1.0–1.5 m at Apollo Bay in southern Australia. The median value of the breaker index was about 84% and 87% of those by Eqs. (4.5) and (4.6), respectively. Another observation was made by Kriebel22 in a large wave ﬂume with a bed slope of 1/50 for waves with a signiﬁcant height of 0.46 m and peak period of 2.9 s. When he applied Eq. (4.5) to individual breaker heights, he found that the value of the proportionality coeﬃcient A ﬁtted to the data varied from about 0.09 to 0.18 (against the original value of A = 0.17) with the mean of 0.142 (84% of Eq. (4.5)) and the standard deviation of 0.017 (equivalent to CoV of 12%). Li et al.23 also reported the result of their measurements of random breaking waves on the slope of 1/50 and 1/200, recommending the coeﬃcient value of A = 0.150 with the standard deviation of 0.031. It should be recalled that the random wave breaking model by Goda24 in 1975 had already incorporated the variability of breaker heights by assigning a variable probability of individual wave breaking. The probability was assumed to increase linearly from 0 to 1 in the range of the wave height from 71% to 106% of the height predicted by Eq. (4.5), corresponding to A = 0.12 to 0.18. Therefore, it is expected that the median value of individual breaker heights would be smaller than those predicted by Eq. (4.5) or (4.6). 4.5. Breaker Index for Random Waves 4.5.1. Incipient breaking index of significant wave Equations (4.5) and (4.6) are examples of the breaker index for regular waves. There are some people who try to apply such breaker index formulas to coastal

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waves or random waves, but such application does not yield correct answers. The breaker index for regular waves may be utilized for the highest wave in an irregular wave train, but it cannot be applied for the signiﬁcant wave, the root-mean-square wave, or any other characteristic wave. When random waves approach the shore, breaking of individual waves occurs gradually with large waves ﬁrst at the far distance, medium waves next at some distance from the shore, and small waves near the shoreline. The variation of signiﬁcant wave height from the oﬀshore toward the shore is so gradual that we cannot employ the concept of wave-breaking line, which is so obvious in the case of regular waves. Against such diﬃculty of deﬁning the breaking point of signiﬁcant wave, Kamphuis9 introduced the deﬁnition of “incipient wave breaking.” He measured cross-shore variations of signiﬁcant wave height beyond and across the surf zone, drew a curve of wave shoaling trend in the outside of the surf zone and a curve of wave height decay within the surf zone, and called the condition at the cross-point of the two curves as the incipient wave breaking. By using the data of the significant wave height at incipient breaking, he calibrated 11 breaker index formulas and determined the best-ﬁtting proportionality coeﬃcient. For the formula of Eq. (4.5), he obtained the proportionality coeﬃcient of A = 0.12 for signiﬁcant wave height. Li et al.23 have also presented a data of the breaker index of (H1/3 /h)b on the slope of 1/200, which is ﬁtted to Eq. (4.5) with a modiﬁed constant value of 0.12 for the initial stage of breaking. Their breaking condition was some observation of large individual breaking waves in an irregular wave train. Goda24 has prepared a set of diagrams depicting variations of signiﬁcant wave height across the surf zone (reproduced as Figs. 3.29 to 3.32 in Ref. 25). The boundary lines of 2% decay in these diagrams approximately correspond to the breaker index with A = 0.11, and the water depth (h1/3 )peak at which the signiﬁcant wave takes a peak value within the surf zone (Fig. 3.34 in Ref. 25) corresponds to A = 0.11–0.13. Therefore, the incipient breaker index of the signiﬁcant wave can be expressed with the following formula: H1/3, b 0.12 (hb )incipient 4/3 = 1 − exp −1.5π 1 + 11 s . (4.7) hb hb /L0 L0 incipient Thus, the incipient breaker index of signiﬁcant wave is about 30% lower than that of regular waves. The incipient breaking of signiﬁcant wave corresponds to the condition that the high waves of upper several percent among individual waves have begun to break. 4.5.2. Laboratory data of breaker index of random waves After incipient breaking, the percentage of wave breaking increases as waves proceed across the surf zone. The ratio of the signiﬁcant wave height to the water depth gradually increases toward the shoreline. Ting26,27 made detailed laboratory investigations of random wave deformations on a uniform slope of 1/35, using frequency spectra of broad- and narrow-band with the peak enhancement factor of 3.3 and 100, respectively. Waves had the signiﬁcant height of Hs = 0.15 m and the spectral

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Breaker indices, H1/3 /h & Hrms /h

1.0

0.6 0.4 0.3 0.2

0.10

Fig. 4.2.

Slope = 1/35

0.8

Broad spec. H1/3 /h Narrow spec. H1/3 /h Breaker envlp. H1/3 /h Broad spec. Hrms /h Narrow spec. Hrms /h Breaker envlp. Hrms /h

0.01

0.1 0.03 Relative depth, h/L0

0.3

Breaker indices for H1/3 and Hrms on s = 1/35 with the data by Ting.26,27

peak period of Tp = 2.0 s. He recorded wave proﬁles at an oﬀshore station with the depth of 0.457 m and at six stations on the slope with the depth of 0.27–0.0625 m. Waves at the six stations on the slope had the percentage of breaking ranging from 5% to 94% (the case of broad-band spectrum). Wave records were analyzed by the zero-downcrossing method, and calculated results of characteristic wave heights and periods are presented in tabular forms. From these results, the ratios of H1/3 and Hrms to the local depth (inclusive of mean water level change) are calculated and plotted against the relative water depth h/L0 , as shown in Fig. 4.2. The curves denoted as breaker envelopes are calculated by Eq. (4.6) for s = 1/35 with the proportionality coeﬃcient of A = 0.145 (85% of regular waves) for the signiﬁcant height H1/3 and to A = 0.111 (65%) for the root-mean-square height Hrms . Because the percentage of breaking waves is high in these data, an A value higher than that for incipient breaking ﬁts to the data. The breaker index data for Hrms by Tick is higher than the value proposed by Sallenger and Holman,28 who gave an expression of Hrms /h = 3.2s + 0.32 without inclusion of the relative depth (h/L0 ) term. They converted the orbital velocity spectra to the surface wave spectra with the transfer function based on the linear theory, and estimated the energy-based Hrms , which must have been smaller than the statistical Hrms value based on direct measurement data of surface proﬁles. 4.5.3. Description of field data employed for analysis In the ﬁeld wave observation at a ﬁxed station, it is not feasible to judge whether individual waves are at the stage of breaking or not, unless simultaneous measurements with video cameras are taken. However, we may ﬁnd out an upper limit of signiﬁcant wave height for a given water depth by taking an envelope of many data at diﬀerent relative water depths. For this type of analysis, stationary wave

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Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone Table 4.3.

Summary of stationary coastal wave data employed in the present analysis.

Observation station Rumoi Port Yamase-Domari Port Tomakomai Port

Kanazawa Port Caldera Port, Costa Rica Sakata Port

97

Type of wave gauge

Water depth (m)

Sampling interval ∆t (s)

Signiﬁcant height H1/3 (m)

Signiﬁcant period T1/3 (s)

No. of data

Step-resistance ”

−11.5 −12.7

0.5 0.5

2.2–7.1 1.9–6.2

5.9–11.7 7.7–15.6

44 9

” ” Ultrasonic ” ”

−10.8 −13.8 −20 −20 −18

1.0 0.5 0.5 1.0 0.5

2.9–5.8 2.6–2.8 2.4–2.5 1.0–6.8 1.5–3.6

7.7–10.9 6.7 – 7.5 6.9–7.4 1.0–6.8 14.2–18.4

9 2 2 13 50

Pressure

−14.5 −10.5

0.5 0.5

1.7–9.7 1.7–6.1

6.3–13.4 6.5–15.0

123 123

(Source: Goda and Nagai,29 Goda24,30 ).

observation data analyzed by Goda and Nagai29 and the data of long-traveled swell recorded with an ultrasonic wave sensor reported by Goda30 were utilized. Table 4.3 lists the characteristics of these ﬁeld data. Waves were recorded by means of either step-resistance gauges or ultrasonic wave sensors so that they were reliable registration of surface wave proﬁles. The data at Tomakomai and Kanazawa as well as Caldera were measured with ultrasonic wave sensors. They are not analyzed for breaker limits but for wave nonlinearity eﬀects to be described in Sec. 4.6. Table 4.3 also lists the wave records at Sakata Port measured by means of pressure gauges, which were utilized by Goda24 for calibration of his random wavebreaking model. Although there remains a problem of pressure conversion to surface proﬁles, the conversion error would have been small because of the relatively shallow water depth at Sakata stations (10.5 and 14.5 m). They were included in the present analysis to increase the size of database. Other sources of nearshore waves are the photogrametric measurement data by Hotta and Mizuguchi31,32 as well as by Ebersole and Hughes.33 Hotta and Mizuguchi mobilized 11 motion-picture cameras set on top of a coastal observation pier at Ajigaura Beach, Ibaragi, Japan. They took ﬁlm pictures of instantaneous water surfaces simultaneously at some 60 surveyor’s poles erected in the nearshore waters on a line perpendicular to the shoreline stretched over a distance of about 120 m. Films of surface wave records were taken every 0.2 s for an eﬀective duration of 760 s. The beach proﬁle in September 1978 had a trough at the distance of 25 m from the shoreline and the slope of about 1/60 oﬀshore of the trough. The beach proﬁle in December 1978 was somewhat uniform without any bar or trough, and the slope was about 1/70. The water depth inclusive of tides at the poles varied from 0.1 to 2.7 m. Photogrametric measurements of nearshore waves were also executed by Ebersole and Hughes33 during the DUCK85 campaign in Duck, North Carolina, USA with the cooperation of Dr. Hotta who brought twelve cameras with him and took charge of ﬁlming. They referred to this technique as “the photopole method”; this terminology is employed in the present chapter. Over the distance of 64 m,

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Table 4.4.

Ajigaura DUCK85 SUPERDUCK

FA

Y. Goda

98

Name

b684-ch04

Summary of photopole measurements.

Date

No. of data

h (m)

H0 (m)

T1/3 (s)

H0 /L0

1978/09/05 1978/12/13–14 1985/09/04–05 1986/09/11–19

54 175 99 140

0.6–2.7 0.1–1.8 0.5–2.4 0.4–3.7

0.7 0.5–0.7 0.3–0.5 0.5–1.1

8.4 7.2–7.9 10.3–11.1 5.4–11.5

0.0064 0.0059–0.0081 0.0017–0.0028 0.0027–0.0254

12 poles were erected in the initial depth ranging from 0.4 to 1.9 m. Measurements were taken for nine runs during the fourth and ﬁfth days of September 1985. With the variation of the tide level, the actual water depth varied from 0.5 to 2.4 m. The eﬀective duration of wave recording was about 650 s, judging from the number of waves and average periods listed by Ebersole and Hughes.33 The beach proﬁle was nearly ﬂat for about 25 m from the shoreline with the depth of about 0.5 m below the mean sea level, and it had the slope of about 1/30 beyond that. Another series of photopole measurements were carried out during the SUPERDUCK campaign in 1986. Dr Hughes kindly supplied the author with the data ﬁles of measured wave statistics. The number of poles was increased to 20 and the water depth inclusive of tides varied from 0.4 to 3.7 m. The beach proﬁle during SUPERDUCK is not known, but it would have been nearly the same as DUCK85 because of the same season. All the photopole measurement data were analyzed by the zero-downcrossing method, and various statistical wave characteristics were calculated. Table 4.4 lists the summary of the photopole wave measurement conditions. The signiﬁcant wave period T1/3 has been converted from the pole-averaged values of either the mean period Tmean or the spectral period Tp by assumption of T1/3 = 1.05Tmean or T1/3 = 0.95Tp, which would be appropriate for swell of very low steepness. The oﬀshore wave height H0 was converted from the signiﬁcant height H1/3 measured at the most oﬀshore pole using the shoaling coeﬃcient; no refraction eﬀect was taken into consideration as no information of wave direction was available. All the waves were swell of very low steepness ranging from 0.0017 to 0.0081, except one case of SUPERDUCK with H0 /L0 = 0.0254. 4.5.4. Field data of breaker index for energy-based significant waves Coastal surface wave data, pressure-converted wave data, and three sets of photopole data are plotted together in Fig. 4.3 in the form of Hm0 /h versus h/L0 . The energybased signiﬁcant wave height Hm0 deﬁned by Eq. (4.8) is employed here instead of the zero-crossing height H1/3 , because the latter is greatly enhanced over Hm0 by strong eﬀects of wave nonlinearity and it is not representative of breaking-dissipated wave energy level; this aspect is discussed in Sec. 4.6. √ (4.8) Hm0 = 4.0ηrms = 4.0 m0 , where m0 denotes the zeroth moment of frequency spectrum.

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Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone

1.0

FA

99

Step-resistance ( h=11-14m) Presssure gage (h=10-14m) Photopole Ajigaura (h=0.1-2.7 m) Photopole DUCK85 (h=0.5-2.4m) Photopole SUPERDUCK (h=0.5-3.2 m) Breaker envelope (80% limit)

Wave height ratio, Hm0 /h

0.8 0.6 0.4

0.2

0.1 0.001

Fig. 4.3.

0.01 0.1 Relative depth, h/L0

1

Breaker index for Hm0 based on the ﬁeld wave data.

Coastal wave data, on the other hand, were not much aﬀected by wave nonlinearity eﬀects, and the zero-crossing signiﬁcant height H1/3 was almost the same as Hm0 . Many data points in Fig. 4.3 belong to nonbreaking condition, but what interests us is the upper envelope which provides an estimate of the upper limit of breaking wave height. The curve of dash-dot line in Fig. 4.3 has been calculated by Eq. (4.6) for the slope of s = 0.0143 (1/70) with the coeﬃcient being reduced to A = 0.136 (80%). Similar with the laboratory data shown in Fig. 4.2, the wave height ratio Hm0 /h is higher than the incipient breaker index of signiﬁcant wave expressed by Eq. (4.7). It is because the breaker index increases inside the surf zone as the percentage of breaking waves increase. It is seen that the energy-based signiﬁcant wave height Hm0 on gentle slopes does not exceed 0.7 times the local water depth except for the low-steepness swell in very shallow water. For the range of h/L0 > 0.03, the upper limit of signiﬁcant wave height is about 0.6 times the local water depth. Some data points above the dash-dot curve are those of DUCK85 and SUPERDUCK, which were conducted on the beach steeper than the beach in Ajigaura.

4.6. Evolution of Wave Nonlinearity Across Surf Zone 4.6.1. Variations of skewness and kurtosis across the surf zone Ocean waves are characterized with almost linear property, as evidenced by the Gaussian distribution of surface elevation. Wave linearity is the basis of spectral

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FA

Y. Goda

100

representation and analysis. Deviation from the linearity is measured with the values of the skewness and kurtosis of the surface elevation with reference to the mean water level during a wave record. The skewness is zero when a distribution is symmetric with respect to the mean, and takes a positive value when a distribution is asymmetric with a long tail toward the right side (large value). The kurtosis takes a value of 3.0 for the Gaussian distribution. When the mode of distribution has a sharp peak and the distribution has long tails in both the left and right sides, the value of kurtosis becomes much larger than 3.0. The degree of positive skewness and the deviation of kurtosis from 3.0 are the measure of the strength of wave nonlinearity. The skewness of ocean waves is less than 0.5, and the kurtosis is below 4.0 for most cases, and thus the nonlinearity of waves in deepwater is weak. The variations of the skewness and kurtosis of ﬁeld waves are examined with coastal surface waves listed in Table 4.3 (excluding the pressure sensor data of Sakata Port), and the photopole data in Ajigaura, DUCK85, and SUPERDUCK listed in Table 4.4. The data of skewness and kurtosis of September 5th in Ajigaura were kindly provided by Dr Hotta. The data of the photopole measurements of SUPERDUCK provided by Dr Hughes had the data of skewness only. As waves approach the shore, wave nonlinearity is enhanced and both the skewness and kurtosis increase signiﬁcantly. Figure 4.4(a) exhibits the increase of the skewness with the nonlinearity parameter Π1/3 , which was introduced by Goda34 with the following deﬁnition: Π1/3 =

H1/3 2πh coth3 , LA LA

(4.9)

where LA denotes the wavelength calculated by small amplitude wave theory or Airy’s theory. The data are grouped by the range of the oﬀshore wave steepness H0 /L0 : the ﬁrst group for 0.001 < H0 /L0 < 0.0029, the second group for 0.003 < H0 /L0 < 0.0049, the third group for 0.0050 < H0 /L0 < 0.0099, the fourth group for 0.010 < H0 /L0 < 0.029, and the ﬁfth group for 0.030 < H0 /L0 < 0.049 (Legends are shown with abbreviated ﬁgures). The data shown in the left diagram of Fig. 4.4 are those outside the surf zone. Because the boundary of surf zone is diﬃcult to be set for random waves, an arbitrary boundary of h/H0 = 2.5 is employed here to separate the wave data outside and inside the surf zone. As shown in the left diagram, the skewness outside the surf zone shows a clear correlation with the wave nonlinearity parameter. The skewness begins from the value of zero at Π1/3 = 0, increases almost linearly with Π1/3 , and attains the value of 2.0 around Π1/3 = 4. The dashed line represents a semitheoretical relationship, which is based on the analysis of ﬁnite amplitude regular wave proﬁles by Goda34 with the consideration of the probability of individual wave heights according to the Rayleigh distribution. Variation of the skewness inside the surf zone (h/H0 < 2.5) is shown in Fig. 4.4(b). The ordinate is the ratio of oﬀshore wave height to water depth, H0 /h, which increases rapidly as waves approach the shore. There is a clear trend of skewness decreasing toward β1 = 0 with the increase of the height-to-depth ratio

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FA

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone

2.5

2.5 0.001
h/H0 < 2.5

h/H0 > 2.5 2.0

1.5

Skewness, β 1

Skewness, β 1

2.0

1.0

1.0

0.5

0.0

0.0

0.1 1 Nonlinearity parameter, Π1/3

0.6

0.8

1

1.2

1.4

Height-to-depht ratio, H0 /h

(a) O utside the surf zone (h/H0 > 2.5) Fig. 4.4.

-0.5 0.4

10

0.001
1.5

0.5

-0.5 0.01

(b) Inside the surf zone (h/H0 < 2.5)

Variation of the skewness of surface elevation outside and inside the surf zone.

10.5

10.5

7.5

h/H0 > 2.5

h/H0 < 2.5 9.0

Kurtosis, β 2

0.001
9.0 Kurtosis, β 2

101

6.0 4.5 3.0

0.001
7.5 6.0 4.5 3.0

1.5 0.01

0.1 1 Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 2.5) Fig. 4.5.

10

1.5 0.4

0.6

0.8 1.2 1.0 Height-to-depht ratio, H0 /h

1.4

(b) Inside the surf zone (h/H0 < 2.5)

Variation of kurtosis of surface elevation outside and inside the surf zone.

H0 /h. Use of the height-to-depth ratio in Fig. 4.4(b) is to provide a kind of contrast of the increase and decrease of skewness in the outside and inside of the surf zone, respectively. While the increase of skewness outside the surf zone seems indiﬀerent to the wave steepness, the value of skewness inside the surf zone is much aﬀected by the wave steepness; waves of low steepness maintain large values of skewness, while waves of high steepness have small values only. Variation of the kurtosis of surface elevation is shown in Fig. 4.5; Fig. 4.5(a) shows kurtosis outside the surf zone and Fig. 4.5(b) shows kurtosis inside the surf zone. The pattern of variation is the same as that of skewness, though the available number of kurtosis data is smaller than the skewness data. The kurtosis starts from the value of 3.0 at Π1/3 = 0, increases as the nonlinearity parameter increases, and attains the value of 9 around Π1/3 = 4. In Fig. 4.5(b), the kurtosis inside the surf

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FA

Y. Goda

102

zone decreases toward β2 = 3.0 as waves approach the shoreline (i.e., increase of H0 /h). Waves with the steepness larger than 0.01 exhibit the kurtosis value much smaller than swell with very low steepness. It is because waves with large steepness cannot experience the full process of nonlinear shoaling owing to the early start of wave breaking. 4.6.2. Variations of zero-crossing wave heights across the surf zone Wave nonlinearity is also reﬂected in individual wave heights deﬁned by the zerocrossing method. In the oﬀshore, individual wave heights almost follow the Rayleigh distribution. Close examination of wave height distribution has revealed a distribution slightly narrower than the Rayleigh with the result of the mean ratio of H1/3 /Hm0 being 0.95. The diﬀerence is due to the energy spread over frequency as represented by wave spectrum, the degree of which can be measured with the spectral shape parameter as discussed by Goda and Kudaka.35 As waves propagate toward the shore, however, waves undergo nonlinear shoaling. In this process, wave proﬁles become skewed with high and sharp crests and low and ﬂat troughs. Because of skewed wave proﬁle, the potential energy contained in such a proﬁle is smaller than the energy of sinusoidal wave with the same height. In other words, nonlinear waves can have the height much larger than the linear (sinusoidal) waves for the same potential energy. Owing to the nonlinear shoaling, individual wave heights increase rapidly with the rate greater than that of linear shoaling. Signiﬁcant wave heights and other characteristic wave heights also grow rapidly. The nonlinearity eﬀect becomes most conspicuous around the outer edge of the surf zone. After waves enter the surf zone and begin to be attenuated through breaking process, the wave nonlinearity is gradually lessened. The degree of nonlinear shoaling eﬀect may be judged by a departure of the statistical wave height from the theoretical prediction by the Rayleigh distribution, which is expressed as below: H1/10 = 1.27Hm0 ,

H1/3 = Hm0 ,

Hrms = 0.707 Hm0 .

(4.10)

Figures 4.6–4.8 show the evolutions of the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 , respectively. The diﬀerence between H1/3 and Hm0 in the nearshore waters has been pointed by several researchers such as Thompson and Vincent36 and Ebersole and Hughes.33 In Figs. 4.6–4.8, the (a) parts are for those outside the surf zone and the (b) parts are for those inside the surf zone, similar to Figs. 4.4 and 4.5. The boundary of the surf zone is subjectively set at h/H0 = 2.5 in Figs. 4.6 and 4.7 and h/H0 = 1.5 in Fig. 4.8. In case of the root-mean-square wave height Hrms , the surf zone should be deﬁned in a shallower area than for the signiﬁcant wave, because Hrms is calculated with all individual waves. Thus, the boundary of Hrms was set at h/H0 = 1.5. The dashed lines in the left diagrams are semi-theoretical predictions based on the potential energy calculation of ﬁnite amplitude waves and the Rayleigh distribution of wave heights, by referring to the methodology employed by Longuet-Higgins.37

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FA

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone

2.2

2.2

Wave height ratio, H1/10 /Hm0

1.8

h/H0 > 2.5

h/H0 < 2.5 2.0 Wave height ratio, H1/10 /Hm0

0.001
2.0

1.6

1.4

1.2

0.001
1.8

1.6

1.4

1.2

1.0 0.01

1.0 0.1 1 Nonlinearity parameter, Π1/3

10

0.4

(a) Outside the surf zone (h/H0 > 2.5)

4 1 Height-to-depht ratio, H0 /h

10

(b) Inside the surf zone (h/H0 < 2.5)

Variation of wave height ratio H1/10 /Hm0 outside and inside the surf zone.

Fig. 4.6.

1.75

1.75 0.001
1.50

h/H0 > 2.5

h/H0 < 2.5 Wave height ratio, H1/3/Hm0

Wave height ratio, H1/3 /Hm0

103

1.25

1.00

0.75 0.01

0.1

1

Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 2.5) Fig. 4.7.

10

1.50

0.001
1.25

1.00

0.75

0.4

1

4

10

Height-to-depht ratio, H0 /h

(b) Inside the surf zone (h/H0 < 2.5)

Variation of wave height ratio H1/3 /Hm0 outside and inside the surf zone.

Outside the surf zone shown in the left diagrams of Figs. 4.6–4.8, the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 increase with the increase of the wave nonlinearity parameter. Although there is much scatter of data, they follow the trend of semi-theoretical curves. The one-tenth highest wave height H1/10 in Fig. 4.6 is given the initial value of 1.20Hm0 based on the trend of the data in consideration of the spectral width eﬀect. The maximum values of the ratios of the statistical wave heights to the energybased signiﬁcant wave height for swell of very low steepness are around 2.1 for H1/10 /Hm0 , 1.6 for H1/3 /Hm0 , and 1.15 for Hrms /Hm0 . With reference to the wave height ratios at the weak nonlinearity state, the statistical wave heights are

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Y. Goda

104

1.2

1.2 h/H0 > 1.5

h/H0 < 1.5

1.1 Wave height ratio, Hrms /Hm0

0.001
1.1 Wave height ratio, Hrms /Hm0

FA

1.0

0.9 0.8

0.7

0.001
1.0

0.9

0.8

0.7

0.6 0.01

0.6 0.1 1 Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 1.5) Fig. 4.8.

10

0.4

1 4 Height-to-depht ratio, H0 /h

10

(b) I nside the surf zone (h/H0 < 1.5)

Variation of wave height ratio Hrms /Hm0 outside and inside the surf zone.

enhanced by 1.75 times for H1/10 , 1.68 times for H1/3 , and 1.64 times for Hrms . Such enhancement of statistical wave heights are apparent ones without real increase of wave energy as discussed earlier. Inside the surf zone shown in the right diagrams of Figs. 4.6–4.8, the data exhibit large scatter but they indicate a clear tendency of decrease with the height-to-depth ratio H0 /h; the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 converge to 1.27, 1.0, and 0.70, respectively, as H0 /h becomes larger than 2. As for the eﬀect of wave steepness, waves with low steepness in the range of 0.001 to 0.0049 have the mean value of H1/3 /Hm0 = 1.4 around H0 /h = 0.4, while waves with the medium steepness of 0.005 to 0.009 starts from the mean value of H1/3 /Hm0 = 1.2. Waves with a high steepness of 0.010 to 0.049 have the H1/3 /Hm0 value less than 1.1. Decrease of the wave height ratio with decreasing water depth has been pointed out by Thompson and Vincent,36 who suggested earlier start of decrease for waves of high steepness compared with waves of low steepness. Such eﬀects of wave steepness on the nonlinear features of waves inside the surf zone are originated from the fact that waves of low steepness experience a high degree of nonlinear wave shoaling before they are attenuated by breaking, while waves of high steepness are attenuated much earlier before they experience strong nonlinear shoaling. It is seen in the left diagrams of Figs. 4.6–4.8 that waves with the steepness of 0.010–0.049 have the nonlinearity parameter up to Π1/3 = 0.3 only and waves are transferred into the right diagrams which represent waves inside the surf zone. Waves with the steepness of 0.005 to 0.0099 have the nonlinearity parameter up to Π1/3 = 1.2 and moves into the group of those inside the surf zone. Waves with the steepness below 0.0029 can have the nonlinearity parameter up to Π1/3 = 4 before they enter into the surf zone. Thus, the evolution of wave nonlinearity expressed in terms of the skewness β1 , kurtosis β2 , and wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 is summarized as follows. It is weak in the oﬀshore, increases gradually as waves propagate shoreward, becomes strongest in the outer half of the surf zone, begins

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Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone

FA

105

to decrease as breaking process of random waves become intensive, and is nearly lost at the shoreline.

4.7. Changes of Wave Height Distribution and Probability Density Function Across Surf Zone 4.7.1. Histogram of wave height distribution The ﬁrst formulation of random wave-breaking process in a numerical model was made by Collins38 by modifying the probability density function (pdf ) of the Rayleigh distribution. He assumed that all waves beyond the breaking limit will reduce their heights to the breaker height Hb , i.e., the pdf is truncated at H = Hb , and it has an inﬁnite intensity of delta function there. An early work of Battjes39 adopted the same approach, while Kuo and Kuo40 redistributed the portion of broken waves over the range of 0 < H ≤ Hb in proportion to the remaining probability density. On the other hand, Goda24 proposed to make a gradual cut over an upper one-third range of wave height below the breaker height as explained in Sec. 4.4.2. Goda’s proposal was based on the ﬁeld data of wave height histograms, which was analyzed by himself. Figure 4.9 is a replotting of the data of surface wave records of Rumoi, Yamase-Domari, Tomakomai, and Kanazawa Ports listed in Table 4.3. The data is presented in four groups according to the value of relative water height H1/3 /h. The group with H1/3 /h < 0.4 demonstrates the wave heights being in good agreement with the Rayleigh distribution. Though not discernible in the ﬁgure, there is a small number of waves in the class of H/Hmean = 3.25–3.50, corresponding to the probability density of the Rayleigh. The next group with 0.4 ≤ H1/3 /h < 0.5 does not have any wave in the class of H/Hmean > 3.25, and the histogram indicates a slight leftward shift. As the relative wave height increases, disappearance of large individual waves becomes clearer and so is the deviation from the Rayleigh distribution. However, there is no concentration of wave heights at the upper end as assumed by Collins38 and Battjes.39 Changes of the pdf of wave heights within the surf zone have been observed in laboratory data such as Goda,24 Baldock et al.,41 and Ting.42,43 Figure 4.10 is an example of the cumulative distribution of wave heights in the surf zone, which has been prepared for the data presented in Fig. 4.5 of Ting.42 The abscissa of Fig. 4.10 is the relative wave height H/H1/3 and the ordinate is the nonexceedance probability. When wave heights follow the Rayleigh distribution, their cumulative distribution is aligned on the heavy straight line of Fig. 4.10. The data at the relative depth of h/H0 = 1.90 deviates only slightly from the straight line of the Rayleigh. As the relative depth decreases, the data of cumulative distribution shift leftward with the largest relative height being Hmax /H1/3 = 1.3 or less. At the relative depth of h/H0 = 0.69 and 0.48, however, the cumulative distribution exhibits a tendency of returning to the Rayleigh with Hmax /H1/3 = 1.5. By referring to the original diagram of Fig. 4.5 of Tick,42 it is observed that deviation from the Rayleigh distribution is largest around h/H0 = 1.0. Closer to the

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Y. Goda

106

1.0

1.0

H1/3/h = 0.400 - 0.499 17 records with 1398 waves

0.8

Probability density, p (x)

Probability density, p (x)

H1/3/h = 0 - 0.399 52 records with 5111 waves

Observation Rayleigh

0.6

0.4

0.2

0.8 Observation Rayleigh

0.6

0.4

0.2

0

0 0

1.0 1.5 2.0 2.5 3.0 0.5 Normalized wave height, x = H /Hmean

0

3.5

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Normalized wave height, x = H /Hmean

1.0

1.0

H1/3/h = 0.500 - 0.599 7 records with 534 waves Observation Rayleigh

0.6

H1/3/h = 0.600 - 0.630 3 records with 188 waves

0.8 Probability density, p (x)

0.8 Probability density, p (x)

FA

0.4

0.2

Observation Rayleigh

0.6

0.4

0.2

0

0 0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized wave height, x = H /Hmean

Fig. 4.9.

3.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Normalized wave height, x = H /Hmean

Histograms of ﬁeld wave data in four groups of relative water depth (after Goda24 ).

shore, the range of wave height begins to spread owing to regeneration of waves after breaking, and the wave heights show a tendency to approach the Rayleigh again. Some researchers such as Battjes and Groenendijk44 have tried to apply certain statistical distributions to the wave heights in the surf zone and to empirically formulate the parameter values. As demonstrated in Figs. 4.9 and 4.10, however, the wave height distribution continues to vary throughout the surf zone, and any eﬀort to describe it with a combination of multiple distribution functions will be futile. 4.7.2. Variation of wave height ratios across the surf zone Change in the pdf of wave heights also yields variation of the ratios among characteristic wave heights. General tendency in the surf zone is that the values of Hmax /H1/3 , H1/20 /H1/3 , and H1/10 /H1/3 decrease as waves enter the surf zone and take the minimum values around at its middle, while the values of Hrms /H1/3 and Hmean /H1/3 increase. However, the wave height ratios come back to the values predicted by the Rayleigh distribution as waves approach the shoreline. Goda45 has made a prediction of such variation of wave height ratios by means of a numerical model based on the Parabolic Equation with a Gradational Breaker

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107

2.5 0.99

Rayleigh h/H0'=1.90 h/H0'=1.60 h/H0'=1.28 h/H0'=0.97 h/H0'=0.69 h/H0'=0.48

2.0

1.5

0.95 0.90 0.75

1.0 0.50 0.5

0.25

0

0.10 0.05 0.01 0

0.5

1.0

1.5

Nonexceedance Probability, P

Reduced variate, SQRT [ -ln (1 - P)]

Ting's Lab. Data

2.0

Relative wave height, H/H1/3 Fig. 4.10. Evolution of cumulative distribution of wave heights in the surf zone based on the data by Ting.42

Index for Spectral waves (PEGBIS). It has been demonstrated that the diﬀerence among characteristic wave heights gradually decreases as the relative water depth becomes shallow down to around h/H0 = 1, but the range of wave heights expands again as the water becomes much shallower toward the shoreline. Laboratory tests by Ting42,43 on the wave breaking on a uniform slope of 1/35 show a variation of the wave height ratio Hrms /H1/3, following the general trend described above. Similar variations of the wave height ratios H1/10 /H1/3 and Hrms /H1/3 are found in the photopole data at Ajigaura, DUCK85, and SUPERDUCK. Figure 4.11 presents a compilation of the laboratory and ﬁeld data concerning the wave height ratio changes. Also included in Fig. 4.11 is the prediction of the wave height ratios across the surf zone by means of the PEGBIS model applied for the Ajigaura beach condition. Although there are large scatters of data, the ratio H1/10 /H1/3 takes the minimum value of about 1.15 around h/H0 = 1.4 for the ﬁeld data, which indicate slightly larger values than the PEGBIS prediction, probably owing to the wave nonlinearity eﬀect. The ratio Hrms /H1/3 takes the maximum value of about 0.81 around h/H0 = 1.0 for both the laboratory and ﬁeld data. The DUCK85 data exhibit the Hrms /H1/3 being closer to 1 than the Ajigaura data. It might be due to lower wave steepness of the swell at DUCK85. Such changes of the wave height ratios across the surf zone are the result of the evolution of the pdf of wave heights discussed in Sec. 4.7.1. Narrowing of the wave height distribution at the middle of the surf zone provides a larger safety margin in the design of breakwaters and other structures than the case of applying the Rayleigh distribution, because the design wave height is speciﬁed with either Hmax (composite breakwaters) or H1/20 (mound breakwaters) for a prescribed oﬀshore

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H1/10 /H1/3: Ajigaura H1/10 /H1/3: DUCK85 H1/10 /H1/3: SUPERDUCK H1/10 /H1/3: PEGBIS Hrms /H1/3: Ajigaura

Hrms /H1/3: DUCK85 Hrms /H1/3: SUPERDUCK Hrms /H1/3: Ting's Lab. Hrms /H1/3: PEGBIS

Wave height ratios, H1/10 /H1/3 & Hrms /H1/3

1.50

1.25

1.00

0.75

0.50 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Relative depth, h/H0 Fig. 4.11. Variations of wave height ratios H1/10 /H1/3 and Hrms /H1/3 by photopole ﬁeld data and Hrms /H1/3 in laboratory tests by Ting together with prediction by PEGBIS model.

signiﬁcant wave height. Therefore, reliable information on the evolution of characteristic wave heights, not just Hm0 or Hrms , needs to be provided by numerical models of wave transformation with random breaking.

4.8. Incorporation of Wave Breaking Process in Numerical Wave Transformation Models For engineering applications, numerical wave transformation models are indispensable tools, and there have been many models, proposed by various researchers and institutions. The models must be capable of dealing with random waves and the process of individual wave breaking, because the majority of engineering problems are concerned with the situations within the surf zone. For the simplest case of random waves on a uniform slope, Goda46 compared performance of seven models listed in Table 4.5, which include those by Battjes,39 Kuo and Kuo,40 Goda,24,45 Battjes and Janssen,47 Thornton and Guza,48 and Larson and Kraus,49 though there are many more models for random wave breaking. All the models introduced the breaker index Hb /hb , but the degree of inclusion of the parameters of the relative depth hb /L0 and the slope s diﬀers among them. The inherent variability of breaker heights is taken into account by the models by

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Main characteristics of random wave-breaking models for comparative study. Factors aﬀecting breaking wave height

Models Battjes39

Energy dissipation mechanism

PDF of broken wave

pdf Delta function deformation at breaker height Kuo and ” Remove and Kuo40 adjust the remainder Goda24 ” ” Bore model Delta function Battjes at breaker and height Janssen47 Thornton ” Adjust with and weight Guza48 function Larson and Dally Delta function Kraus49 model at breaker height ” Remove and Goda45 (PEGBIS) adjust the remainder

Breaker variability

Waves spectrum

Hb /h

h/L

Bed slope

Yes

Yes

No

No

Yes

No

Yes

No

No

No

No

No

Yes Yes

Yes Yes

Yes No

Yes No

Yes Yes

No No

Yes

No

No

No

No

No

Yes

No

No

No

Yes

No

Yes

Yes

Yes

Yes

Partially

Yes

Wave setup

Goda24,45 only. The energy dissipation is dealt directly with either the bore analogy or the model by Dally et al.50 or indirectly through modiﬁcation of the pdf of wave heights. Figure 4.12 is an example of diﬀerences in the variations of the root-mean-square wave height in the nearshore waters predicted by the seven models on the slope of s = 0.02 (Goda46 ). Diﬀerences are very large beyond expectation and Goda46 shows other examples on the slope of s = 1/10, 1/30, and 1/100. Because the generation of longshore currents is governed by the spatial gradient of the radiation stresses, these models produce much diﬀerent cross-shore proﬁles of longshore current velocities as demonstrated by Goda.46 Smith51 has also demonstrated diﬀerent performances of ﬁve wave-breaking models with diﬀerent breaker indices against ﬁeld measurement of spectral waves. Because of the large diﬀerence in the prediction of nearshore wave heights, one should be careful in selecting a numerical model for engineering applications. Development of numerical models for time-domain wave transformation has been tried by many researchers, but random wave-breaking process has not been well reproduced in these models. One of the exceptions is the Boussinesq-type model developed by Hirayama et al.52 and Hirayama and Hiraishi,53 who employed the breaking criterion of the vertical pressure gradient by Nadaoka et al.7 They raised the threshold gradient from 0 to 0.5 to compensate the insuﬃciency in numerical accuracy due to the features of weak nonlinearity inherent to the Boussinesq equation. They have succeeded in reproducing the pdf variation across the surf zone.

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2.0

RMS Wave Height, Hrms (m)

Bottom Slope : s = 0.02 1.5

1.0 Battjes '72 Kuo & Kuo '74 Goda '75 Battjes & Janssen '78 Thornton & Guza: M1 Larson & Kraus '91 Goda '04 -spectr-

0.5

0 0

1.0

2.0

3.0

4.0

5.0

6.0

Depth, h (m) Fig. 4.12. Variations of wave heights Hrms in the surf zone on planar beaches computed by seven random wave-breaking models for (H1/3 )0 = 2.0 m, T = 8.004 s, and θ0 = 30◦ (Goda46 ).

Good reproduction of random wave-breaking process having the features discussed in the present chapter is a key factor for future development of reliable and practical numerical models for engineering applications.

4.9. Summary Wave-breaking phenomenon in the nearshore waters is characterized by the following features: (1) The breaker index formula by Goda13 for regular waves is revised by reducing the bed slope eﬀect for better agreement with laboratory data. (2) Depth-limited breaker heights have inherent variability with the coeﬃcient of variation of about 6% for the slope of 1/100 and 14% for the slope of 1/10, which increases as the bed slope becomes steep. (3) Breaker index or the ratio of breaker height to water depth is governed by both the relative water depth hb /L0 and the bed slope. (4) Incipient breaker index for the signiﬁcant height of random waves is smaller than that for regular waves by about 30%, and the signiﬁcant wave height has an upper limit of about 0.6 times the water depth on gentle slopes. (5) Wave nonlinearity expressed in terms of skewness, kurtosis, and the wave ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 increases with wave propagation toward the shore and is most enhanced just outside the surf zone. For swell of very low steepness, the skewness, kurtosis, and wave height ratio H1/3 /Hm0 may go up to 2.0, 9.0, and 1.6, respectively, at the wave nonlinearity parameter around 4.0.

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(6) For wind waves and swell of relatively large steepness, random wave breaking begins at large depths compared with swell of small steepness. Thus, the former waves do not exhibit conspicuous wave nonlinearity. (7) Wave nonlinearity is weakened by wave breaking inside the surf zone and is eventually lost near the shoreline with return to linear wave features. (8) Probability density function (pdf ) of wave heights gradually deviates from the Rayleigh as waves propagate toward the shore, with the largest deviation taking place at the middle of the surf zone. Near the shoreline, the wave height distribution approaches the Rayleigh again. (9) Diﬀerences between characteristic wave heights such as H1/10 , H1/3 , and Hrms decrease in the middle of the surf zone, but they return to those by the Rayleigh near the shoreline. The PEGBIS model can simulate such changes of characteristic wave heights. (10) Various existing models of random wave breaking produce quite diﬀerent prediction of wave heights in the surf zone. Careful selection of a random wave-breaking model should be made so that the model will yield reliable information for engineering applications.

Acknowledgment The author much appreciates kind cooperation by Dr Steven Hughes and Dr Shintaro Hotta for providing him with their valuable wave data.

References 1. Y. Goda, How much do we know about wave breaking in the nearshore waters, Proc. 4th Int. Conf. Asian and Paciﬁc Coast (APAC 2007), Nanjin, China, CD-RM (2007). 2. H. Yamada and T. Shiotani, On the highest water waves of permanent type, Bull. Disaster Prevention Res. Inst., Kyoto Univ. 18–2(135), 1–22 (1968). 3. Y. Goda, A synthesis of breaker indices, Trans. Japan Soc. Civil Eng. 2(2), 39–49 (1970). 4. H. Yamada, G. Kimura and J. Okabe, Precise determination of the solitary waves of extreme height on water of a uniform depth, Rep. Res. Inst. Appl. Mech., Kyushu Univ. XVI(52), 15–32 (1968). 5. J. McCowan, On the highest waves in water, Phil. Mag. Ser. 5, 36, 351–358 (1894). 6. R. Miche, Mouvements ondulatoires de lamer en profondeur ou d´ecroissante, Annales de Ponts et Chauss´ees 19, 370–406 (1944). 7. K. Nadaoka, O. Ono and H. Kurihara, Near-crest pressure gradient of irregular water waves approaching to break, Proc. Coastal Dynamics ’97 (1997), pp. 255–264. 8. G. M. Kaminsky and N. C. Kraus, Evaluation of depth-limited wave breaking criteria, Second Int. Symp. Ocean Waves Measurement and Analysis, WAVES 1995 (1995), pp. 180–193. 9. J. W. Kamphuis, Incipient wave breaking, Coastal Eng. 15, 185–203 (1991). 10. W. Rattanapitikon and T. Shibayama, Veriﬁcation and modiﬁcation of breaker height formulas, Coastal Eng. J. 42(4), 389–406 (2000).

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11. W. Rattanapitikon, T. Vivattanasirisak and T. Shibayama, A proposal of new breaker height formula, Coastal Eng. J. 45(1), 29–48 (2003). 12. Y. Goda, A new method of wave pressure calculation for the design of composite breakwaters, Rept. Port Harbour Res. Inst. 14(3), 59–106 (1973) (in Japanese). 13. Y. Goda, New wave pressure formulae for composite breakwaters, Proc. 14th Int. Conf. Coastal Eng., Copenhagen, ASCE (1974) pp. 1702–1720. 14. M. Muttray and H. Oumeraci, Wave transformation on the foreshore of coastal structure, Coastal Eng. 2000 (Proc. ICCE) (2000) pp. 2178–1191. 15. Y. C. Li, G. H. Dong and B. Teng, Wave breaker indices in ﬁnite water depth, China Ocean Eng. 5(1), 51–64 (1991). 16. Y. C. Li, Y. Yu, L. F. Cui and G. H. Dong, Experimental study of wave breaking on gentle slope, China Ocean Eng. 14(1), 59–67 (2000). 17. J. L. Lara, I. J. Losada and P. L.-F. Liu, Breaking waves over a mild gravel slope: Experimental and numerical analysis, J. Geophys. Res. 111(C11019), 1–26 (2006). 18. J. W. Smith and N. C. Kraus, Laboratory study of wave-breaking over bars and artiﬁcial reefs, J. Waterway, Port, Coastal, Ocean Eng. ASCE, 117(4), 307–325 (1991). 19. A. Kimura and A. Seyama, Breaking limit of irregular waves on slopes, Proc. 33rd Japanese Coastal Eng. Conf. (1986), pp. 174–178 (in Japanese). 20. A. Seyama and A. Kimura, The measured properties of irregular wave breaking and wave height change after breaking on the slope, Proc. 21st Int. Conf. Coastal Eng. (1988), pp. 419–497. 21. K. P. Black and M. A. Rosenberg, Semi-empirical treatment of wave transformation outside and inside the breaker line, Coastal Eng. 16, 313–345 (1992). 22. D. Kriebel, Breaking waves in intermediate-depths with and without current, Coastal Eng. 2000 (Proc. ICCE), (2000) pp. 203–215. 23. Y. C. Li, Y. Yu, L. F. Cui and G. H. Dong, Transformation and breaking of irregular waves on very gentle slope, China Ocean Eng. 14(3), 261–278 (2000). 24. Y. Goda, Irregular wave deformation in the surf zone, Coastal Eng. Jpn, JSCE 18, 13–26 (1975). 25. Y. Goda, Random Seas and Design of Maritime Structures, 2nd Edn. (World Scientiﬁc, Singapore, 2000). 26. F. C. K. Ting, Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone, Coastal Eng. 43, 183–208 (2001). 27. F. C. K. Ting, Laboratory study of wave and turbulence characteristics in a narrowbanded irregular breaking waves, Coastal Eng. 46, 291–313 (2002). 28. A. H. Sallenger and R. A. Holman, Wave energy saturation on a natural beach of variable slope, J. Geophys. Res. 90(C6), 11939–11944 (1985). 29. Y. Goda and K. Nagai, Investigation of the statistical properties of sea waves with ﬁeld and simulation data, Rept. Port Harbour Res. Inst. 13(1), 3–37 (1974) (in Japanese). 30. Y. Goda, Analysis of wave grouping and spectra of long-travelled swell, Rept. Port Harbour Res. Inst. 22(1), 3–41 (1983). 31. S. Hotta and M. Mizuguchi, A ﬁeld study of waves in the surf zone, Coastal Eng. Jpn, JSCE 23, 59–79 (1980). 32. S. Hotta and M. Mizuguchi, Statistical properties of ﬁeld waves in the surf zone, Proc. 33rd Jpn Coastal Eng. Conf. (1986) pp. 154–157 (in Japanese). 33. B. A. Ebersole and S. A. Hughes, DUCK85 photopole experiment, US Army Corps of Engrs., WES, Misc. Paper, CERC-87-18, (1987) pp. 1–165. 34. Y. Goda, A uniﬁed nonlinearity parameter of water waves, Rept. Port Harbour Res. Inst. 22(3), 3–30 (1983).

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35. Y. Goda and M. Kudaka, On the role of spectral width and shape parameters in control of individual wave height distribution, Coastal Eng. J. 49(3), 311–335 (2007). 36. E. F. Thompson and C. L. Vincent, Signiﬁcant wave height for shallow water design, J. Waterway, Port, Coastal, Ocean Eng. ASCE 111(5), 828–841 (1985). 37. M. S. Longuet-Higgins, On the distribution of the heights of sea waves: Some eﬀects of nonlinearity and ﬁnite band eﬀects, J. Geophys. Res. 85(C3), 1519–1523 (1980). 38. J. I. Collins, Probabilities of breaking wave characteristics, Proc. 12th Int. Coastal Eng. Conf., Washington, DC, ASCE (1970) pp. 399–414. 39. J. A. Battjes, Setup due to irregular wave, Proc. 13th Int. Conf. Coastal Eng., Vancouver, ASCE (1972) pp. 1993–2004. 40. C. T. Kuo and S. T. Kuo, Eﬀect of wave breaking on statistical distribution of wave heights, Proc. Civil Eng. Ocean, ASCE (1974) pp. 1211–1231. 41. T. E. Baldock, P. Holmes, S. Bunker and P. van Weert, Cross-shore hydrodynamics with an unsaturated surf zone, Coastal Eng. 34(3–4), 173–196 (1998). 42. F. C. K. Ting, Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone, Coastal Eng. 43, 183–208 (2001). 43. F. C. K. Ting, Laboratory study of wave and turbulence characteristics in a narrowbanded irregular breaking waves, Coastal Eng. 46, 291–313 (2002). 44. J. A. Battjes and H. W. Groenendijk, Wave height distributions on shallow foreshores, Coastal Eng. 40, 161–182 (2000). 45. Y. Goda, A 2-D random wave transformation model with gradational breaker index, Coastal Eng. J. 46(1), 1–38 (2004). 46. Y. Goda, Examination of the inﬂuence of several factors on longshore current computation with random waves, Coastal Eng. 53, 156–170 (2006). 47. J. A. Battjes and J. P. F. M. Janssen, Energy loss and set-up due to breaking of random waves, Proc. 16th Int. Conf. Coastal Eng., Hamburg, ASCE (1978), pp. 1–19. 48. E. B. Thornton and R. T. Guza, Transformation of wave height distribution, J. Geophys. Res. 88(C10), 5925–5938 (1983). 49. M. Larson and N. C. Kraus, Numerical model of longshore current for bar and trough beaches, J. Waterway, Port, Coastal, Ocean Eng., ASCE 117(4), 326–347 (1991). 50. W. R. Dally, R. G. Dean and R. A. Darlymple, Wave height variation across beaches of arbitrary proﬁle, J. Geophys. Res. 90(C6), 11917–11927 (1985). 51. J. M. Smith, Breaking in a spectral wave model, Ocean Wave Measurement and Analysis (Proc. WAVES 2001), ASCE (2001) pp. 1022–1031. 52. K. Hirayama, N. Hara and T. Hiraishi, Application of bore model to non-linear wave transformation, Proc. 13th Int. Oﬀshore Polar Eng. Conf. (2003), pp. 796–801. 53. K. Hirayama and T. Hiraishi, A Boussinesq model for wave breaking and run-up in a coastal zone, 1D, Fifth Int. Symp. Ocean Waves Measurement Analysis, WAVES 2005, CD-ROM, No. 151 (2005), pp. 1–10.

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Chapter 5

Aeration and Bubbles in the Surf Zone Nobuhito Mori Disaster Prevention Research Institute Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan [email protected] Shohachi Kakuno Osaka City University 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan [email protected] Daniel T. Cox Oregon State University, Corvallis, OR 97331-4501, USA [email protected] This chapter presents a brief summary of aeration in the surf zone, beginning with a review of air–water characteristics in surf zone waves. Second, measurements techniques of the bulk of air and bubbles induced by breaking waves in the surf zone are described, and third, the bulk of air and bubble characteristics are summarized based on the in situ and visualization laboratory measurements. Finally, the gas transfer in the surf zone is described and related to the wave characteristics.

5.1. Introduction Ocean surface gravity wave propagation from oﬀshore to shoreline is often regarded as a single phase ﬂow using potential ﬂow theories or solving the Navier–Stokes equation. Generally, the single phase ﬂow approach to ocean waves is successful for simulating wave transformation in the coastal area. However, this assumption breaks down because waves become steep and start to break owing to the bottom bathymetric eﬀects in the near shore. The wave-breaking process creates dense bubble plumes, dissipates energy and generates turbulence in the surf zone. Figure 5.1 shows an example of the crest of wave breaking on a gentle slope. Most of the waves 115

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Fig. 5.1.

Image of breaking wave crest at an initiation of breaking in the surf zone.

can be regarded as a single (liquid) phase ﬂow, and only the near-crest region is aerated by the jet attachment to the water surface. An accurate estimation of the bulk of the void fraction, bubble size, and population distributions in the surf zone are important for understanding the two-phase ﬂow characteristics, solving engineering problems, and elucidating environmental mechanisms of the coastal area.1 For example, Peregrine2 summarized that the entrapped air of breaking wave reduces the wave impact pressure owing to its greater compressibility compared with the single-phase approach. There is also a signiﬁcant diﬀerence between salt water and fresh water experiments on the wave impact pressure (bubbles generated by turbulence in saltwater are smaller than fresh water), and the entrapped air reduced the pressure by approximately 10%.3 The compressibility owing to air– water mixture decreases the velocity of sound and is being used to estimate largescale prototype impacts, since the usual Froude scaling is unlikely to be correct for engineering problems. Therefore, the connection between air-mixture, bubble distribution, and wave-breaking-induced turbulence is essential to understand gas– liquid interactions in the surf zone. The air bubble plume is always present in the surf zone breaking waves. Experimental data show that the entrained air is proportional to the energy dissipation.4 For these breaking waves, entrained air enhances the gas-exchange at the water surface. For example, the dissolved oxygen is associated with the breaking waves in the coastal waters,5,6 although the carbon dioxide gas transfer can be regarded as a function of wind speed. Understanding air entrainment and bubble distribution for the depth-limited breaking of surf zone waves is limited in comparison with the wind-wave breaking. The air bubble distributions under the wind-wave breaking in the upper layer of the ocean surface has been summarized in detail by Thorpe.7 The wind-wave bubble formation and its dependence on the turbulent dissipation rate of ﬂuids have been discussed,8 and it is proposed a −2 power-law scaling with bubble diameter. The −10/3 power-law scaling is alternatively proposed based on the discussion of bubble fragmentation due to strong turbulent shear ﬂow. These bubble size scaling laws are still under development.9 However, the knowledge of wind-wave breaking induced

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bubbles and associated gas transfer has become well-known issue over the last decade.10 Despite the fruitful knowledge of air entrainment of wind-wave breaking, few studies of air entrainment for surf zone wave breaking exist. It has been found that there is little diﬀerence in the bubble populations beneath the mechanically generated surface waves in saltwater and fresh water.11 On the other hand, it has been reported that the void fraction depends on the turbulent intensity in the bore region of surf zone waves.12 The large-scale bubbles entrained by breaking waves in the surf zone are split by strong local turbulent shear at the bubble scale.13 However, neither qualitative nor quantitative bubble characteristics in the surf zone and connections between bubble characteristics and wave breaking are well known owing to the lack of detailed observations. This detailed information of two-phase ﬂow characteristics is required for mathematical modeling to solve both basic scientiﬁc interests and applied engineering problems.

5.2. Measurements Techniques There are several measurement methods for bubble population and size distributions, i.e., in situ measurements,14,15 video or photographic measurements,16 laser measurements,17 and acoustic measurements.18 The bubble size measurements using lasers probe such as a phase Doppler anemometer show high accuracy in small-scale bubble measurements smaller than 500 µm but is of limited use in the presence of high void fractions and large-scale bubble measurements due to fundamental instrument limitations. Acoustic measurements of bubbles are useful in deepwater but have limitation for very shallow water region owing to the multi-reﬂection of sound beams. Therefore, conventional optical or resistivity type void probes are useful for the surf zone breaking waves. The electrical conductivity probe (so-called void probe) has been generally used for the high void fractions.19 This type of void probe is robust and easy to handle for both wave ﬂume experiments and ﬁeld observations. Therefore, the void probe measurements have been widely used for the twophase ﬂow measurement in the surf zone.12,14,20 However, the in situ measurement has clear limitation for inhomogeneous surf zone breaking phenomena. The photographic or imaging technique has been advanced to in situ measurements which can cover the spatial and temporal variations of the two-phase ﬂow characteristics in the surf zone. Recent photographic studies have illustrated the disintegration of entrapped air cavities divided into bubbles.13,21 Several imaging techniques have been proposed to measure the bubble characteristics in the surf zone. One is an application of Particle Image Velocimetry (PIV) method with forward and backward lighting.22,23 This method basically correlates the scattered light intensity and bubble plume intensity, and it can be used for high void fraction region over 20%, although it cannot measure the bubble shape itself. The other imaging technique is a bubble shape tracking method with backlighting.24 This method is a so-called shadow graph method because the bubble shape is enhanced by a backlight source toward a camera. The shadow graph method can measure both bubble shape and velocity simultaneously. However, the density of the bubbles has to be within a certain limit to separate individual bubbles from the image by the shadow graph method.

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In summary, there is no ideal method that can be used for universal purpose of bubble measurements and one can choose appropriate method(s) for a particular purpose.

5.3. Bulk Aeration Characteristics in the Surf Zone Figure 5.2 shows successive images of breaking wave taken by high-speed video camera in the surf zone. Figure 5.2(a) is just after attachment of jet on the water surface which corresponds to the initial stage of bubble generation. Figures 5.2(b) and 5.2(c) are 42 ms and 108 ms after image (a), respectively. The lifetime of the large volume of entrained air can be categorized into two phases. The ﬁrst phase is the initial stage of breaking waves [Figs. 5.2(a) and 5.2(b)]. The newly injected air column produces pluses of sound and, therefore, this phase is regarded as the acoustically active phase.21 In the active phase, the trapped volume of air between the overturning jet and water surface has fragmented and injected into the deeper region, and the air pocket is fragmented into small size bubbles. This phase occurs within 1 s of the initial stage of wave breaking. The second phase is the acoustically silent phase. Most of the bubbles are advected by the surface roller of the carrier waves with buoyancy, and some of them are trapped by oblique descending eddy in this phase [Figs. 5.2(c)]. In the active phase, the void fraction ratio is relatively high and there is no clear formation of bubbles in that region. Therefore, the only possible approach to this phase is to account for the total amount of air, which is the summation of the air pocket and bubbles. The silent phase has a lower void fraction and is more dispersive, and it is possible to discriminate individual bubbles. These injected air penetrate into a deeper region during the active phase. After that, the volume of air and/or bubbles are advected by the wave motion due to their buoyancy. Therefore, the gas phase characteristics in the surf zone, such as the void fraction, are time–space variables and are inhomogeneous in both time and space. The next part of this chapter discusses the total amount of air, bubble size distributions, and gas-transfer in the surf zone, respectively. 5.3.1. Temporal characteristics The aeration process by breaking wave in the surf zone is obviously time-dependent. Therefore, the temporal transition is important in understanding the phenomena in the beginning. Figure 5.3 shows an example of phase-averaged time series of surface elevation and turbulence components of the mechanically generated periodic wave breaking on a slope.20 The notations z and Hb denote the vertical axis positive upward from the mean water level and wave height at the breaking point, respectively. The initial wave height and period are 16.3 cm and 1.6 s, respectively. Using the phase-averaged method, the velocity data were divided into the mean, periodic, 2 and turbulence components. The notation of σu2 , σv2 , and σw are the turbulent stresses for the velocity components (u, v, w). The peak of the turbulent stress is located at the front of the crest and decreases linearly as time passes. This is a typical time series of turbulence characteristics of depth-limited breaking waves.

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(a)

(b)

(c)

Fig. 5.2. Formation of bubble plume under the breaking wave in the surf zone. (a) Attachment of jet on the surface (initiation of bubble plume generation); (b) Active phase of bubble plume generation [42 ms after (a)]; (c) Bubble plume in silent phase [108 ms after (a)].

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<η> [cm]

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1

t/T Fig. 5.3. Example of phase-averaged surface elevation η and turbulence components 1.0 m from B.P. in Case 1. (a) Free surface η (solid) and standard deviation η ± ση ; (b)–(d) temporal variation of σu (thick solid), σv (thin solid), and σw (dotted) and (e) the phase-averaged void fraction α.20

The highest void fraction occurs at the wave front, and the void fraction is decreased rapidly in time. The temporal variation of the void fraction depends on the air supply from the surface, advection, and buoyancy, and these eﬀects are diﬃcult to discriminate from each other. Although the process is complex, the normalized temporal characteristics show a universal behavior. Similar laboratory experiments were observed by several researchers.12,25,26 Furthermore, Cox and Shin12 found the self-similar exponential decay of the temporal series of void fraction normalized by the averaged value in diﬀerent types of breaking waves. In addition, they reported that the maximum void fraction in the time series was three to four times higher than its average value. 5.3.2. Vertical distribution of void fraction The vertical distribution of void fraction of the surf zone breaking waves was ﬁrst systematically measured by Hoque and Aoki.27 They showed a simple description of vertical void fraction distribution as follows. If a void fraction α simply follows a

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C0 0

0.1

C0

0.2

0

0.1

C0

C0 0

0.2

0.1

121

0

0.2

0.1

C0 0

0.2

0

0

0

0

0

-2

-2

-2

-2

-2

-4

-4

-4

-4

-4

-6

-6

-6

-6

-6

-8

-8

-8

-8

-8 x-xb = 0.4 m

-10 -12

z (cm)

z (cm)

Fig. 5.4.

x-xb = 0.6 m

-10

2

x-xb = 0.8 m

-10 -12

-12

z (cm)

z (cm)

x-xb = 1.1 m

-10

0.1

0.2

x-xb 1.35 m

-12

z (cm)

Vertical distribution of void fraction for plunging breaker.27

diﬀusion equation in steady state, we have ∂ ∂(αu) = ∂z ∂z

∂α D , ∂z

(5.1)

where z is the vertical axis, u is the water velocity, and D is the turbulent diﬀusivity of air bubble plume. Assuming a uniform velocity distribution and D independent in the vertical direction, we have ∂ 2α u0 ∂α , = 2 ∂z D ∂z

(5.2)

where u0 denotes the uniform velocity. The solution of the above equation is α(z) = α0 exp(kz),

(5.3)

where k = u0 /D is the newly introduced decay parameter of the vertical void fraction distribution and α0 denotes the void fraction at the mean water level (z = 0). Figure 5.4 shows the comparison between experimental data and Eq. (5.3) for plunging breakers on a gentle slope. The experimental data shows an exponential decay of the void fraction in the vertical direction. Furthermore, introducing k0 = kh, where h is the local depth, k0 can be regarded as a constant value ranging approximately from 3.75 to 4.0, independent of breaker type. Similar results were measured by a visualization technique23 after that. In addition, the linear decay of the maximum water depth of penetrated bubble shoreward is also observed as shown in Fig. 5.4.27 5.3.3. Horizontal distribution of void fraction The characteristics of the volume of air changes both horizontally, vertically and temporally. The void fraction and total kinetic energy show similar distributions, and Cox and Shin12 also reported the dependence of void fraction on turbulent intensity in the bore region of surf zone waves. The entrapped air induced by wave breaking generates strong turbulence near the free surface. Therefore, the relationship between void fraction and turbulence should be connected to each other.

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×10

2

15 z/H =0.033 b

10 5 0

2

⋅ 10

15 z/H =0.013 b

α×104, k [cm2/s2]

10 5 0

2

⋅ 10

15 z/H =0.0 b

10 5 0

⋅ 102

15 z/Hb=-0.013

10 5 0 0

2

4

6

8

10

12

14

xb/Hb Fig. 5.5. Horizontal relationship between time-averaged void fraction α (solid line) and timeaveraged TKE k (dotted line).20

Figure 5.5 shows experimental results of void fraction and total kinetic energy (TKE) variations from the breaking point xb to the shore line.20 The spatial variations of the time-averaged void fraction α and time-averaged TKE, k show similar tendency, although the peak of α is located approximately 0.2 × Hb ∼ 2 × Hb seaward of the peak of k, as shown in Fig. 5.5. The short spatial lag between α and k might be associated with the spatial diﬀerence of entrapped air and associated turbulence. The spatial variations of the turbulence and void fraction are highly correlated near the mean water level. The entrained bulk of air and bubbles generate turbulence, and turbulence splits the coarse bubbles into ﬁne bubbles in this area. Moreover, the TKE and void fraction increased by an order of magnitude from the trough level (z/Hb = −0.013) to above the mean water level (z/Hb = 0.033); thus, the two-phase ﬂow characteristics are remarkably strong near the wave crest. Neglecting the short spatial lag between the void fraction and TKE, the direct comparison between the time-averaged void fraction α and normalized timeaveraged TKE k of all the measuring locations is shown in Fig. 5.6. For small values of α or k, the relationship between the two shows a high correlation and becomes scattered at high void fraction or TKE region. Overall, the relationship between time-averaged void fraction and time-averaged TKE shows high correlation12 as does the incident wave-dependence on the relationship between α and k.

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0.14

0.12

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0.08

0.06

0.04

Case1 Case2 Case3

0.02

0

0

0.1

0.2 ( k/gh )

0.3

0.4

1/2

Fig. 5.6. Relationship between time-averaged TKE k and time-averaged void fraction α. (•: Case 1, +: Case 2, : Case 3).

The relation between the void fraction and TKE, the scale-dependence of void fraction, and further theoretical considerations will be required to ﬁnd the universal relationship between the turbulence characteristics of depth-limited breaking waves and air entrainment.

5.4. Bubble Characteristics in the Surf Zone Bubble generation is a small-scale process of aeration in the surf zone.28 There is a large body of literature dealing with the distribution of bubbles in wind-wave breaking.7,8,29,30 From the knowledge of wind-wave breaking, there are two modes of bubble generation.31 One is entrapment of air under the falling water jet by the wave overturning and another is air entrainment at the point where the falling water jet meets the vortex formed at the rear of the splash-up. Figure 5.7 shows the time series of bubble diameter distribution just beneath the mean water level near the breaking point under the same wave condition as in Fig. 5.5.24 The bubble diameter here means the averaged value of the major and minor diameter. The horizontal axis is the characteristic bubble diameter, the vertical axis is the temporal change, and the contour indicates number of bubbles per unit area. The plunging wave breaking not only entrains the bulk of air but also generates a large number of bubbles in comparison with the spilling breaker. As discussed earlier, the void fraction decreases exponentially in time. The time series

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Time series of bubble size distribution.24 (a) Spilling breaker; (b) Plunging breaker.

of the bubble number density distribution (Fig. 5.7) shows a similar tendency, but the small-scale bubbles survive longer than large-scale bubbles. Generally, entrapped air by breaking waves is split into small bubbles owing to turbulence shear of surrounding ﬂows. The early work of this problem for a general condition is discussed by Kolmogorov32 and Hinze33 in the middle of the last century, and a large bibliography has been generated.34 Longuet-Higgins35 proposed a crushing air cavity model for the bubble distributions after the formation of a bubble plume and shows good agreement of bubble aeration in general condition, qualitatively. Figure 5.8 shows the instantaneous and time-averaged bubble size spectra with the estimated power law measured by an imaging technique.24 The instantaneous bubble size spectra corresponds to the time of the maximum void fraction observed. There is no signiﬁcant diﬀerence of bubble size distribution

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2

Number of bubbles /cm

2

10

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10

10

0

−2

time-averaged instantaneous −1.0 d −3.4 d −0.9 d

−4

10

−1

Fig. 5.8.

0

10 diameter [mm]

10

1

Power law of bubble size distribution.

between instantaneous and time-averaged spectra at small scale (diameter < 1 mm). On the other hand, two scaling laws can be seen in the bubble size distribution. The small and large size bubble power scaling laws are d−1 and d−3–−4 , respectively. Garrett et al.9 proposed semi-empirical −10/3 power-law scaling for the bubble which is larger than the Hinze scale, based on the discussion of bubble fragmentation in the strong turbulence ﬂow below the trough level. On the other hand, Deane and Stokes21 empirically proposed a −3/2 power-law scaling, smaller than the Hinze scale in the acoustically active phase near the crest. The node point between the two power scaling corresponds to the Hinze scale of the bubble splitting theory.33 The measured power-law scalings of the small size bubble shown in Fig. 5.8 are close to −3/2 and −10/3 power laws, respectively. These power-law scalings can be seen in diﬀerent conditions in mechanically generated waves in the ﬂume.13,20,21 The Hinze scale dH 21 of bubble splitting can be given by ρw ε−2/5 , dH = 2−8/5 c (5.4) γ where ρw is the water density, γ is the surface tension coeﬃcient, ε is turbulence energy dissipation at the bubble diameter, and c = (W ec )3/5 , where W ec is the critical Weber number. The Weber number W e is deﬁned as ρw 2 δu d, We = (5.5) γ where δu is turbulence velocity at the bubble scale d. The turbulence velocity δu at the bubble scale in the inertial range can be given by δu = 2ε2/3 d2/3 . If the energy dissipation rate ε can be estimated by wave energy dissipation, the Hinze scale can be obtained. For example, if ε = 12 W/kg, the Hinze scale is 2 mm in diameter. These theoretical predictions agree fairly well with the experimental data,20,21 although the Hinze scale is formulated under the assumption of isotropic and homogeneous turbulence ﬂow.

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5.5. Gas Exchange in the Surf Zone For moderately solvable gases such as carbon dioxide and oxygen, the gas transfer across the air–sea interface is controlled by the aqueous condition, and is parameterized as a function of a gas transfer velocity kL kL = αSc−1/2 u∗ ,

(5.6)

where Sc = νκ is the Schmidt number, α is the empirical coeﬃcient, and u∗ is the friction velocity, respectively.36 The role of bubbles in the exchange of gas in the surf zone is poorly understood in comparison to the wind-wave breaking. An analogy of wind-wave breaking to surf zone breaking waves is similar, but surface renewal process and the bubble-mediated gas transfer are diﬀerent from each other. For example, the surface renewal by the wind stress is important for the windwave scenario, but the turbulence generated by the wave overturning is dominant in the surf zone. Moreover, the bubble-mediated gas transfer process depends on the bubble population and size distribution in the surf zone. For the wind-wave case, the large-scale bubbles with a diameter greater than approximately 1 mm play an important role in air–sea gas exchange.37,38 However, there are a few conﬁdent prediction models that exist for macro or microscopic bubble-mediated gas transfer for surf zone breaking waves. The oxygen gas transfer is important for ﬁsh and marine mammals near the coastal region. Therefore, oxygen gas transfer, rather than carbon dioxide, plays a vital role in aquatic environment in the surf zone.39 There are signiﬁcant diﬀerence between the wind-waves and surf zone waves generally. For example, the spatial homogeneity can be assumed in wind-waves, but there is a strong spatial dependence for surf zone waves. Therefore, a diﬀerent approach to predict the gas transfer is required. One may assume that the gas transfer is related to the total water mass volume of surf zone (Fig. 5.9). The general expression of the gas transfer in an aeration tank of constant volume can be formulated: ∂C A (5.7) = kL ∆C = k2 ∆C, ∂t V where C is the gas concentration, ∆C is the concentration diﬀerence between liquid and gas phase, A is the aeration (surface) area, kL is the gas transfer velocity,

Fig. 5.9.

Illustration of gas transfer in the surf zone.

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and k2 is the reaeration coeﬃcient. Eckenfelder40 proposed the bubble-mediated gas transfer velocity kL for aeration tank as kL =

αh2/3 Qa , A dm Sc1/2

(5.8)

where α is the empirical coeﬃcient, h is the water depth of aeration tank, Qa is the total amount of air ﬂow rate, dm is mean bubble diameter. Multiplying the surf zone area A by Eq. (5.8), it can be formulated for aeration by surf zone breaking waves as41 2/3

kL A =

αHb Vg 1/2

dm Sc T

,

(5.9)

where Hb is the wave height at breaking point, Vg is the total entrained air volume per wave period, and T is the wave period (Fig. 5.10). In this formulation, the gas transfer in the surf zone linearly depends on the total entrained air volume. Now, we introduce the gas transfer ﬂow rate kL A instead of the gas transfer velocity kL in Eq. (5.9). This is because the aeration area A is diﬃcult to be estimated in the surf zone, but the total water volume V is easily obtained. Assuming a uniform bottom slope, Kakuno41 empirically obtained a relationship between Vg and surf zone parameter as Vg ∝ Hb hb / tan θ, where hb is the water depth at the breaking point, and θ is the bottom slope. Substituting this expression into Eq. (5.9), we have the gas transfer ﬂow rate coeﬃcient kL A for slope beach41 : 5/3

kL A =

α2 Hb hb 1/2

T dm Sc

tan θ

.

(5.10)

Kakuno compared Eq. (5.10) with the fresh water experimental data and found that the empirical coeﬃcient α2 is approximately 2.3 × 10−3 ∼ 2.8 × 10−3 as shown in Fig. 5.11.41 Equation (5.10) expresses the gas transfer ﬂow rate as a function of breaking wave height Hb , water depth at breaking point hb , wave period T , mean bubble diameter dm , bottom slope θ, and Schmidt number, although salinity and scaling eﬀects are neglected. Furthermore, Kakuno41 obtained an empirical relation between the gas transfer velocity and wave energy dissipation as kL ∝ ε1/3 . wave1

wave2

wave3

Vg XTi Xb Fig. 5.10. Total bubble volume distribution in the surf zone (Xb : length of surf zone, Vg : bubble volume per wave period).

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Fig. 5.11. Gas transfer flow rate and bottom bathymetry in the surf zone (•: 1/10 slope, ◦: 1/20 slope, : 1/30 slope).

Research to estimate the gas transfer velocity kL from the wave characteristics through the wave energy dissipation is ongoing.

5.6. Conclusions and Future Study In this chapter, we summarize our general review of gas–water two-phase ﬂow characteristics in the surf zone. There are several measurement methods for the aeration in the surf zone, such as in situ measurements, video or photographic measurements, laser measurements, and acoustic measurements. There is no one appropriate method to measure the volume of air and bubbles in the surf zone. The spatial and temporal characteristics of void fraction are summarized, and the bubble size distribution is described with simple theories. Furthermore, the gas transfer prediction method is presented as a function of the bottom slope and wave characteristics for the surf zone waves. There are many other problems such as salinity eﬀect, scale eﬀects, and more mathematical modeling should be investigated in the future.

References 1. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof, Gas Transfer at Water Surfaces (AGU, Washington, DC, 2002). 2. D. Peregrine, Water wave impact on walls, Ann. Rev. Fluid Mechanics 35, 23–43 (2003). 3. B. G. N. A. Crawford, P. Hewson and P. Bird, Experiments on full-scale wave impact pressures, Coastal Eng. 8, 331–346 (1999). 4. S. Longo, M. Petti and I. J. Losada, Turbulence in the swash and surf zones: A review, Coastal Eng. 45, 129–147 (2002).

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5. D. Farmer, C. McNeil and B. Johnson, Evidence for the importance of bubbles in increasing air-sea gas ﬂux, Nature 361, 620–623 (1993). 6. D. Wallace and C. Wirick, Large air-sea ﬂuxes associated with breaking waves, Nature 356, 145–147 (1992). 7. S. Thorpe, On the clouds of bubbles formed by breaking wind waves in deep water, and their role in air-sea gas transfer, Phil. Trans. R. Soc. London A 304, 155–210 (1982). 8. S. Baldy, A generation-dispersion model of ambient and transient bubbles in the close vicinity of breaking waves, J. Geophys. Res. C10, 18277–18293 (1993). 9. C. Garrett, M. Li and D. Farmer, The connection between bubble size spectra and energy dissipation rates in the upper ocean, J. Phys. Oceanogr. 30, 2163–2171 (2000). 10. B. J¨ ahne and H. Haußecker, Air-water gas exchange, Ann. Rev. Fluid Mechanics 30, 443–468 (1998). 11. M. Loewen, M. O’Dor and M. Skafel, Bubbles entrained by mechanically generated breaking waves, J. Geophys. Res. 101, 20759–20769 (1996). 12. D. Cox and S. Shin, Laboratory measurements of void fraction and turbulence in the bore region of surf zone waves, J. Eng. Mechanics, ASCE 129(10), 1197–1205 (2003). 13. G. B. Deane and M. D. Stokes, Air entrainment processes and bubble size distributions in the surf zone, J. Phys. Oceanogr. 29, 1393–1403 (1999). 14. H. Chanson, S. Aoki and M. Maruyama, Unsteady air bubble entrainment and detrainment at plunging breaker: Dominant time scales and similarity of water level variation, Coastal Eng. 46, 139–157 (2002). 15. H. Chanson, Comments on ﬁber optic refectometer for velocity and fraction ratio measurements in multiphase ﬂows, Rev. Sci. Instrum. 75(1), 284–285 (2003). 16. I. Leifer and G. de Leeuw, Bubble measurements in breaking-wave generated bubble plumes during the LUMINY wind-wave experiment, in Gas Transfer at Water Surfaces, eds. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof (AGU, Washington, DC, 2002), pp. 303–310. 17. N. Mori, Experimental study of air bubble distributions induced by wind wave breaking, in Proc. Asia Pacific Coast 2003, JSCE, Makuhari (2003), CD–ROM. 18. G. B. Deane, Sound generation and air entrainment by breaking wave in the surf zone, J. Acoustic Soc. Am. 102, 2671–2689 (1997). 19. S. Vagle and D. Farmer, A comparison of four methods of bubble measurement, IEEE Oceanic Eng. 23(3), 211–222 (1998). 20. N. Mori, T. Suzuki and S. Kakuno, Experimental study of air bubbles and turbulence characteristics in the surf zone, J. Geophys. Res. 112(C05014) (2007), doi: 10.1029/2006JC003647. 21. G. B. Deane and M. D. Stokes, Scale dependence of bubble creation mechanisms in breaking waves, Nature 418, 839–844 (2002). 22. Y. Ryu, K.-A. Chang and H. J. Lim, User of bubble image velocimetry for measurement of plunging wave impinging on structure and associated greenwater, Meas. Sci. Technol. 16, 1945–1953 (2002). 23. O. Kimmoun and H. Branger, A particle image velocimetry investigation on laboratory surf-zone breaking waves over a sloping beach, J. Fluid Mech. 588, 353–397 (2007), doi: 10.1017/S0022112007007641. 24. N. Mori and S. Kakuno, Aeration and bubble measurements of the coastal breaking waves, Fluid Dynamics Res. 40(7–8), 616–626 (2008). 25. F. Ting and J. Kirby, Observation of undertow and turbulence in a laboratory surf zone, Coastal Eng. 24, 51–80 (1994).

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26. D. Cox, N. Kobayashi and A. Okayasu, Vertical variations of ﬂuid velocities and shear stress in surf zones, Proc. 24th Int. Conf. on Coastal Eng., ASCE (1994), pp. 98–112. 27. A. Hoque and S. Aoki, Distributions of void fraction under breaking waves in the surf zone, Ocean Eng. 32, 1829–1840 (2005). 28. M. Loewen, Inside whitecaps, Nature 418, 830 (2002). 29. R. Cipriano and D. Blanchard, Bubble and aerosol spectra produced by a laboratory breaking waves, J. Geophys. Res. C86, 8085–8092 (1981). 30. C. Garrett, M. Li and D. Farmer, The connection between bubble size and energy dissipation rates in the upper ocean, J. Phys. Oceanogr. 30, 2163–2171 (2000). 31. P. Bonmarin, Geometric properties of deep-water breaking waves, J. Fluid Mech. 209, 405–433 (1989). 32. A. Kolmogorov, On the breakage of drops in a turbulent ﬂow, Dokl. Akad. Navk. SSSR 66, 825–828 (1949). 33. J. Hinze, Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes, Am. Inst. Chem. Eng. J. 1, 289–295 (1955). 34. C. Mart´ınez-Baz´ an, J. Montan´es and J. Lasheras, On the breakup of air bubble injected into a fully developed turbulent ﬂow. Part I: Breakup frequency, J. Fluid Mech. 401, 157–182 (1999). 35. M. Longuet-Higgins, The crushing of air cavities in a liquid, Proc. Roy. Soc. London A 439, 611–626 (1992). 36. W. Melville, The role of surface-wave breaking in air-sea interaction, Ann. Rev. Fluid Mechanics 28, 279–321 (1996). 37. L. Memery and L. Marlivat, Modeling of gas ﬂux through bubbles at the air-water interface, Tellus 37B, 272–285 (1985). 38. R. Keeling, On the role of large bubbles in air-sea gas exchange and supersaturation in the ocean, J. Marine Res. 51, 237–271 (1993). 39. C. Moutzouris and E. Daniil, Water oxygenation in the vicinity of coastal structures due to wave breaking, Proc. 24th Int. Conf. Coastal Eng., ASCE (1994), pp. 3167–3177. 40. W. Eckenfelder Jr., Absorption of oxygen from air bubbles in water, J. Sanitary Eng. Div. Proc. ASCE, 89–99 (1959). 41. S. Kakuno, D. B. Moog, T. Tatekawa, K. Takemura and T. Yamagishi, The eﬀect of bubble on air-water oxygen transfer in the breaker zone, in Gas Transfer at Water Surfaces, eds. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof (AGU, Washington, DC, 2002), pp. 265–277.

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Chapter 6

Freak Wave Nobuhito Mori Disaster Prevention Research Institute Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan [email protected] In this chapter, we describe mechanisms that lead to the occurrence of freak waves in the ocean. It is now generally recognized that freak waves can be generated by any one of four possible mechanisms. The traditional mechanism is a simple linear superposition of waves, the theory of which is reviewed here. The newest mechanism attempts to include the third-order nonlinearities that depart from linear wave theory. Therefore, we present here a state-of-the-art review that is based on nonlinear wave dynamics.

6.1. Introduction The history contains many reports of encounters with monstrous waves.1 For example, the captain of the cargo vessel Junior reported a wave estimated to be 100 ft high and the famous reliable report was that of the wave encountered by the U.S.S. Ramapo in the North Paciﬁc in 1993; that wave was estimated to be 112 ft high. There are many more reports of encounter of similar waves in the history of the seas. Over the last decade, more evidence has accumulated for such extreme waves. This may be because of an increase in the number of measurement locations together with the development of more accurate ﬁeld measurements, for example, several extreme wave events, the so-called freak waves, recorded in the North Sea.2 A sample wave proﬁle is shown in Fig. 6.1, which was extracted from the data provided by wave radar shooting downward from the platform to the sea surface in the North Sea. The data were collected over 12–15 hours at a depth of 40 m. Since the early 1990s such an event has come to be called a freak wave or a rogue wave. More evidence has been obtained by analyzing ﬁeld data from the North Sea,3,4 the Sea of Japan,5,6 and the Gulf of Mexico.7 The ﬁrst clear deﬁnition of a freak wave was proposed by Klinting and Sand2 ; in its original form, the deﬁnition required a wave to satisfy three conditions to be

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Fig. 6.1. Freak wave recorded on the 24th November, 1981 at the Gorm Field in the Danish Sector of the North Sea.8

considered “freak wave”: (1) It has a wave height higher than twice the signiﬁcant wave height. (2) Its wave height is larger than two times of the foregoing and the following wave heights. (3) Its wave crest height is larger than 65% of of its wave height. Subsequently, these conditions were found to be overly stringent, and today the ﬁrst condition is generally considered to be suﬃcient to identify a freak wave.8 Freak wave research has two basic objectives. The ﬁrst is an understanding of the mechanisms that create freak waves; several mechanisms have been proposed to explain why extreme-wave events occur in the ocean. The second is to establish a reliable statistical model for the occurrence of freak waves. However, the problem is diﬃcult because a freak wave is only one wave, or perhaps just a few waves, in a wave train. Therefore, its occurrence exhibits statistical sensitivity. Moreover, an extreme wave event is transient: it forms and disappears quickly in both time and space. To address these problems, we should study wave populations that contain freak waves, rather than concentrate on the features of an individual wave proﬁle. (This is not to say, however, that observations of single events are unimportant.) Once such wave populations can be characterized, then it should be possible to estimate how often waves of any given size will occur. Over the last decades, researchers in coastal and ocean engineering, and the oceanography have dedicated much eﬀorts to the study of extreme wave events. Detail on the state of the art are available in the conference proceedings of Rogue Waves 2004.9 In this chapter, we provide an overview of freak waves. First, we review the four mechanisms that are now thought to generate freak waves in Sec. 6.2. Next, we describe the probability distribution for the populations of linear waves in Sec. 6.3. Third, we show how that probability distribution changes when nonlinear eﬀects are taken into account in Sec. 6.4. Fourth, we discuss the accuracy with which extreme wave events can be predicted in Sec. 6.5. Finally, we make some brief suggestions about future work. 6.2. Mechanisms for Generating Freak Waves It is now accepted that at least four mechanisms can explain the formation of freak waves. A problem common to these four is whether a freak wave obeys an ordinary

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statistical law or whether other physical mechanisms come into play. That is, if we are to be able to predict freak waves, then any underlying mechanism must lead to reliable statistical model for extreme wave events. First Mechanism. The ﬁrst mechanism is simply that a freak wave can be caused by a linear superposition of waves. In this case, the probability distribution for wave height in the limit of the narrow-band approximation obeys a Rayleigh distribution10 ; corrections due to ﬁnite spectral band width have been obtained.11–13 Wave crest statistics can be obtained by using a second-order theory.14 The probability distribution for wave crests has been found in a narrow-band approximation15 and for ﬁnite band width.16 This mechanism is theoretically well established and relatively easy to follow; the basic theory will be reviewed in Sec. 6.3. Second mechanism. The second mechanism is that freak wave can result when waves interact with currents. When this situation occurs, ray theory can be used to explain the formation of extreme waves. However, it is not yet known how the statistical properties of surface are related to the properties of currents. From the studies of freak waves generated over a random current, it has been found that wave focusing with 1 m/s velocity ﬂuctuations occurs in the uniform current.17 Third Mechanism. A third mechanism has its basis in crossing-sea states (i.e., two-sea systems). An example is a swell and a wind sea in which the two have diﬀerent directions coexisting over some region of the ocean, or diﬀraction of waves behind an island. Usually, both the wave-current mechanism and the two-sea state mechanism occur in speciﬁc locations, such as along the Angus current; therefore, freak waves caused by either of these mechanisms are relatively predictable. Fourth Mechanism. In the fourth mechanism, extreme wave events result from the high-order nonlinear interactions; speciﬁcally, the underlying mechanism is fourwave interactions.18,19 Numerical and experimental studies have demonstrated that freak-like waves can be generated frequently in a two-dimensional wave ﬂume, even in the absence of a current, or wave refraction, or wave diﬀraction18,20–22 ; see Fig. 6.2. Further, numerical studies clearly indicate that a freak wave having a single, steep crest can be generated in deep-water by the third-order nonlinear interactions.18 Freak wave generation is sometimes discussed in the context of the Benjamin– Feir instability in deep-water waves because of the similarity in the steep wave

Fig. 6.2.

Freak wave simulated by the high-order spectrum method.18

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proﬁle.18,22 Over the last two decades, the Benjamin–Feir instability of deepwater gravity waves has been studied using the nonlinear Schr¨ odinger-type equations,23–25 mode-coupling equations,26 pseudo-spectral methods,27 and experiments.28 However, the Benjamin–Feir instability vanishes when the wave spectrum is suﬃciently broad. This means, there is a discrepancy between the nonlinear behaviors of periodic waves and random waves. Janssen19 investigated freak waves that occur as a consequence of four-wave interactions, including the eﬀects of nonresonant four-wave interactions. Using Monte Carlo simulations of the Zakharov equation, he found that homogeneous nonlinear interactions give rise to deviations from the Gaussian distribution for the surface elevation. The fourth mechanism for freak wave generation requires additional knowledge before predictions can be made. Relationship between kurtosis and freak wave events has been studied using laboratory data,29 numerical results,30 and ﬁeld data.7 North Sea data shows that the maximum crest, normalized by its wave heights, correlates with skewness, while the wave heights, normalized by signiﬁcant wave height, correlate with the kurtosis.7 This is because the kurtosis is the fourth moment of the probability density function; hence, it measures the relevance of tails in a distribution. These results explain the general characteristics of a freak wave. (a) A secondorder nonlinearity causes wave asymmetry over the short-time evolution, O(1/ε2 ), of a wave train (Here ε is a small parameter that corresponds to the steepness of deep-water waves). (b) But the kurtosis is associated with the third-order nonlinear interactions.14 Therefore, it may be that third-order nonlinear interactions are the fundamental factors contributing to the occurrence of freak waves. However, the timescales for such interactions are longer than those for second-order interactions, O(1/ε4 ), and they are two orders of magnitude longer than those for second-order nonlinearities. The formal relation between the kurtosis and the exceedance probability for wave height have been discussed (Mori and Janssen31 ; see also Tayfun and Lo32 ; Mori and Yasuda33 ). The kurtosis enters into the distribution function as a correction to the Rayleigh distribution: when the kurtosis tends to three, the expected Gaussian values, and then the distribution function tends to the Rayleigh distribution. In this context, changes in the kurtosis can be evaluated once the evolution of the wave spectrum is known.19 The wave height exceedance probability has been derived as a function of the number of waves.31 This relation will be explained in a later part in this chapter. In all these studies, an ultimate goal is the determination of the shape of the exceedance probability function P (H). Once an analytical form for P (H) is found for all sea conditions (for example, as a function of the wave spectrum), then the probability of a wave exceeding a certain height can be estimated directly. One diﬃculty in studying extreme wave events is the veriﬁcation of theory. Statistical theories are evaluated under the assumptions of stationarity; however, the real seas change states both spatially and temporally. Therefore, statistical theories are diﬃcult to verify.34 In addition, freak wave prediction requires us to predict the maximum wave height distribution. However, although study of the maximum wave height distribution is necessary to verify a theory for freak wave

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Yura Y88121401 Surface Waves

Surface Elevation (m)

8 6 4 2 0 −2 −4 45000

45100

45200

45300 Time (s)

45400

45500

45600

Wavelet Spectrum of Yura Y88121401 Surface Waves

Frequency (Hz)

0.6 0.5 0.4 0.3 0.2 0.1

10

20

30

40

50

60

70

80

90

100

Yura Y88010901 Surface Waves

Surface Elevation (m)

5

0

−5 79400

79500

79600

79700 Time (s)

79800

79900

80000

Wavelet Spectrum of Yura Y88010901 Surface Waves

Frequency (Hz)

0.6 0.5 0.4 0.3 0.2 0.1

10

Fig. 6.3.

20

30

40

50

60

70

Freak wave recorded in 1988 in the Sea of Japan.34

80

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prediction, veriﬁcation of wave height distribution is not suﬃcient to completely verify a theory.3,31 Samples of freak wave proﬁles and their wavelet spectrum, recorded in the Sea of Japan, are shown in Fig. 6.3.34 The wave proﬁle shown in Fig. 6.3(a) is a 10 min time-series segment that contains a freak wave along with the corresponding contour plot for the wavelet spectrum. It appears that for the well-deﬁned freak wave shown in Fig. 6.3(a), the occurrence of the freak wave can be readily identiﬁed from the wavelet spectrum: the strong energy density in the spectrum instantly increases at the onset of the freak wave, at the same time, that increase in energy appears to carry over into high frequency components. However, for another freak wave time series shown in Fig. 6.3(b), no corresponding focused surge in energy appears in the wavelet spectrum. So in this case, it is uncertain whether a freak wave is begun, generated by linear or nonlinear wave focusing. Nevertheless, at the onset of the event in Fig. 6.3(a), the freak wave proﬁle appears rather asymmetric with respect to the mean level, whereas the freak wave proﬁle in Fig. 6.3(b) is generally symmetric. So, these diﬀerences in wavelet spectra might result from diﬀerences in the processes that generated these freak waves. In any case, time series alone may or may not be reliable guides for identifying freak wave events. These observations suggest that linear and nonlinear focusing eﬀects are coupled (that is, linear and nonlinear processes do not belong to totally diﬀerent populations), and this coupling must be understood if we are to be able to predict freak waves. 6.3. Linear Wave The surface gravity waves’ celerity depends on the wave length owing to dispersive eﬀects in deep-water. The nonlinear eﬀects slightly modify the dispersion relation in the order of wave steepness.35 The order of nonlinear eﬀects, both dispersion and wave proﬁle, are square or cubic of the wave steepness if there is no resonant interactions in the system. Therefore, in general, the ocean waves can be regarded as a linear, ergodic, homogeneous, stationary, Gaussian processes. Because of these assumptions, the probability density function for water surface elevation of unidirectional waves can be expressed as the Gaussian distribution 1 η2 p(η) = √ exp − , (6.1) m0 2πm0 ∞ 2 2 S(ω)dω , (6.2) σ = ηrms = m0 = √

0

and η is the surface elevation while σ = m0 is its variance. Let η(t) be the sea surface elevation at time t and let ζ(t) be an auxiliary variable such that η(t) and ζ(t) are not correlated. Assuming both η(t) and ζ(t) are real zero-mean functions with variance σ, we have Z(t) = η(t) + iζ(t) = A(t)eiφ(t) , A(t) = η 2 (t) + ζ 2 (t) ,

(6.3) (6.4)

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,

(6.5)

where A is the envelope of the wave train and φ is the phase. As there is no correlation between η and ζ, the joint probability density function of η and ζ becomes 1 1 p(η, ζ) = exp − (η2 + ζ 2 ) . (6.6) 2π 2 √ All variables will be normalized by the variance of the surface elevation σ = m0 . The probability density function (PDF) for the envelope A can now be obtained by integrating the joint probability distribution p(A, φ) = A p(η, ζ) ,

(6.7)

over φ; hence,

2π

p(A) =

dφ p(A, φ).

(6.8)

0

The results are the usual Rayleigh distributions for wave amplitude p(A) = Ae−(1/2)A

2

(6.9)

and wave height p(H) =

2 1 He−(1/8)H . 4

(6.10)

In the narrow-band approximation, the wave height H = 2A. The exceedance probability PH (H) for wave height now follows by integrating (6.10) from H to ∞: 2

PH (H) = e−(1/8)H .

(6.11)

√ In an exact, linear narrow-band random wave theory, H1/3 = 4.004 m0 , but for √ simplicity we will assume H1/3 = 4 m0 . In this notation, the freak wave condition √ H/H1/3 > 2 corresponds to H/ m0 > 8. According to Eq. (6.11), the probability of occurrence for a freak wave in a wave train is P (8) = e−8 = 3.35 × 10−4 . This is equivalent to a freak wave occurring once in every 2981 waves. Assuming a characteristic wave period of 10 s, this corresponds to a storm having a duration of 8.3 h. When this linear wave theory is extended to 3D space, the appearance probability for fully directional conditions (Smax = 10) increases by 45% over that for the unidirectional condition.36 The maximum wave height distribution becomes important when we examine statistical properties of freak waves over a series of recorded data. This is because a freak wave prediction is equivalent to estimating the maximum wave height and the exceedance probability (that is, Eq. (6.11) is insuﬃcient for prediction). The PDF for the maximum wave height pm in wave trains can be obtained once the

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138

PDF of wave height p(H) and exceedance probability of the wave height P (H) are known.37 Thus, pm (Hmax )dHmax = N [1 − P (Hmax )]N −1 p(Hmax )dHmax ,

(6.12)

where N is the number of waves in the wave train. For suﬃciently large N , we may use the approximation N

N

N →∞

N →∞

lim [1 − P (Hmax )]N lim exp [−N P (Hmax )] .

(6.13)

Substituting Eqs. (6.11) and (6.13) into Eq. (6.12), gives the PDF for the maximum wave height pm , 2 2 N (6.14) pm (Hmax )dHmax = Hmax e−Hmax /8 exp −N e−Hmax /8 dHmax , 4 and the exceedance probability of maximum wave height Pm , 2 (6.15) Pm (Hmax ) = 1 − exp −N e−Hmax /8 . Note that Eqs. (6.14) and (6.15) are evaluated as functions of N number of waves √ in the train. The freak wave condition in this study becomes Hmax / m0 > 8, and we obtain from Eq. (6.15) the following simple formula for probability of a freak wave occurring as a function of N ,

(6.16) Pfreak = 1 − exp −e−8 N . This is a general expression for the expected probability of freak waves in unidirectional linear random waves. Therefore, in the linear wave case, the occurrence probability for a freak wave depends only on the number of waves N . For example, if the number of waves equals 2981, then Eq. (6.16) gives Pfreak = 0.632, that is, the expected probability of a freak wave is 63.2% when we randomly sample from a population of 2981 waves. For N = 120 waves, the expected probability of a freak wave is 3.9% and for N = 10,000 waves, it is 96.5%. These results show that the number of waves in a wave train (or, equivalently, the duration of a storm) is important for predicting freak waves.

6.4. Nonlinear Interactions 6.4.1. Role of nonlinear interactions Since the beginning of freak wave research, high-order nonlinear eﬀects have been considered.18,30 As a result, the nonlinear occurrence of freak waves, including their mechanisms and detailed dynamic properties, are becoming better understood.21,22,34,38–40 The state of the art has been summarized at the Rogue Wave Conference, held in 2000 and 2004.9,41 The consensus from those conferences is that third-order nonlinear interactions not only enhance the appearance of freak wave, but they are also the primary cause of freak waves. The only exceptions occur during strong wave–current interactions or wave diﬀraction behind an island.

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6.4.2. Quasi-resonant four-wave interactions and high-order moments Investigations have been carried out on the creation of freak waves in response to four-wave interactions, including the eﬀects of nonresonant four-wave interactions.19,31 Those studies culminated in the formulation of a probability for the occurrence of freak waves in nonlinear wave ﬁelds. However, the modulational instability is a quasi-resonant interaction process, i.e., wave numbers and frequencies satisfy the following conditions: k1 + k2 − k3 − k4 = 0 , ω(k1 ) + ω(k2 ) − ω(k3 ) − ω(k4 ) ≤ 2 ,

(6.17) (6.18)

where is a small parameter that corresponds to the steepness of deep-water waves. The modulational instability takes place when two wave numbers are the same, k1 = k2 , while k3 and k4 are side bands separated from k1 by ∆k, which should be small in order to satisfy the conditions Eqs. (6.17) and (6.18). The standard kinetic equation that describes the evolution of the wave spectrum in time42 is formally valid only for exact resonances and large times, O(−4 ω0−1). It has been extended to include quasi-resonant interactions,19 where a kinetic equation, which should also be valid on the timescale of the modulational instability, has been derived. If we then consider the evolution of higher order moments, such as the kurtosis µ4 , we ﬁnd that the quasi-resonant interactions are responsible for deviations from the Gaussian distribution. The explicit relation between nonlinear interactions of free waves and the fourth-order moment of the surface elevation η(x, t) in deep-water has been investigated.19 The result can be expressed in terms of κ40 , which is the fourth-order cumulant of the surface elevation (κ40 = µ4 − 3), η 4 −3 m20 √ 12 dk1,2,3,4 T1,2,3,4 ω1 ω2 ω3 ω4 = 2 2 g m0 × δ1+2−3−4 Rr (∆ω, t)N1 N2 N3 .

κ40 =

√

(6.19)

Here, m0 is the variance of the surface elevation η, g is the acceleration of gravity, k is the wave number, ω is the angular frequency, N is the wave action spectral density, and T1,2,3,4 is the coupling coeﬃcient in the Zakharov equation (see Krasitskii43 for its analytical form), δ1+2−3−4 = δ(k1 + k2 − k3 − k4 ), dk1,2,3,4 = dk1 dk2 dk3 dk4 . The resonant function is Rr =

1 − cos(∆ωt) , ∆ω

(6.20)

with ∆ω = ω1 + ω2 − ω3 − ω4 .

(6.21)

In the limit of large times, Rr → P/∆ω, where P denotes the principle value of the integral to avoid singularity in the integral.

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140

In the narrow-band approximation, we assume that the spectrum E(ω) has a Gaussian shape, E(ω) =

2 m0 √ e−(1/2)ν , σω 2π

(6.22)

where ν = (ω − ω0 )/σω and σω is the spectral band width; the integral in Eq. (6.19) for large times becomes 2 2 2 242 dν1,2,3 e−(1/2)[ν1 +ν2 +ν3 ] P . (6.23) κ40 = ∆2 (2π)3/2 (ν1 + ν2 − ν3 )2 − ν12 − ν22 + ν32 √ Here, = k0 m0 is the steepness parameter and ∆ = σω /ω0 is the relative spectral band width. The integral can be evaluated analytically to obtain π κ40 = √ BF I 2 , 3

(6.24)

where BFI is deﬁned as by Janssen19: BFI =

√ 2. ∆

(6.25)

Equation (6.24) is the simpliﬁed predictive equation for the kurtosis of the surface elevation assuming a narrow-band, unidirectional wave train, but a full description requires evaluation of six-dimensional integral in wave number space Eq. (6.19). 6.4.3. Wave height distributions in a nonlinear wave field To include nonlinear eﬀects in the wave height distribution function, thereby possibly obtaining deviations from Rayleigh statistics, the standard approach is to use the Edgeworth series.44 The theory for wave height has been described,31–33 and the resulting distribution has been named the Modiﬁed Edgeworth Rayleigh (MER) distribution. The MER wave height and exceedance wave height distributions are given by Mori and Janssen31 : p(H)dH =

1 −(1/8)H 2 He [1 + κ40 AH (H)] dH , 4

(6.26)

2

(6.27) PH (H) = e−(1/8)H [1 + κ40 BH (H)] . √ Here, H is the wave height normalized by ηrms = m0 , κ40 is deﬁned in Eq. (6.23), while AH (H) and BH (H) are polynomials deﬁned as 1 (H 4 − 32H 2 + 128) , 384 1 BH (H) = H 2 (H 2 − 16) . 384 AH (H) =

(6.28) (6.29)

Note that the distributions describe the deviations from linear statistics under the hypothesis of a narrow-band, weakly nonlinear wave train. Further, in the case of linear waves (κ40 = 0), these distributions are equivalent to Eqs. (6.10) and (6.11).

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Second-order contributions, which are important for the distribution of wave crests, can be included by using an approach like that used by Tayfun.15,45 The probability distribution function, pm , and the exceedance probability, Pm , of the maximum wave height can also be expressed as a function of the fourth cumulant of the surface elevation κ40 and the number of waves N in the wave train, pm (Hmax ) =

2 N Hmax e−Hmax /8 [1 + κ40 AH (Hmax )] 4 2 × exp −N e−Hmax /8 [1 + κ40 BH (Hmax )] ,

2 Pm (Hmax ) = 1 − exp −N e−Hmax /8 [1 + κ40 BH (Hmax )] ,

(6.30) (6.31)

where Hmax is the maximum wave height normalized by ηrms . Qualitative agreement has been found between the theoretical wave height distribution, given by Eq. (6.30), and ﬁeld data.31 Equations (6.30) and (6.31) are evaluated as functions of N and κ40 (or µ4 ). When κ40 = 0, Eqs. (6.30) and (6.31) reduce to Eqs. (6.14) and (6.15), which are provided by the Rayleigh distribution. The occurrence probability of a freak wave as a function of N and κ40 , taking into account nonlinear eﬀects, now becomes (6.32) Pfreak = 1 − exp −e−8 N (1 + 8κ40 ) . Using Eq. (6.31) we see that the eﬀect of kurtosis becomes of the same order as the linear theory for κ40 = 1/8. This corresponds to µ4 = 3.125, which is not a strong nonlinear condition. Hence, both the eﬀects of ﬁnite kurtosis and the number of waves N are important for determining the probability of maximum wave height in the nonlinear wave train. It should be emphasized here that the above discussion does not require any empirical, ad hoc parameters. A diﬀerent approach, also based on the dynamics of a cubic nonlinearity, has been used for ﬁnding the exceedance probability for wave height.46 That approach involves an optimization problem, requiring that at a speciﬁc space and time, the Fourier phases are all equal, under the constraints that the conservation laws of the Zakharov equation are satisﬁed. To summarize, quasi-resonant four-wave interactions introduce deviations from linear statistics for surface elevation; in particular, for weakly nonlinear, narrowbanded and long-crested wave trains, the kurtosis evolves according to Eq. (6.19). In the narrow-band approximation, the kurtosis is related to the BFI [deﬁned in Eq. (6.25)]; the tail of the wave height distribution depends on the kurtosis/BFI and it increases as the kurtosis increases. Finally, the maximum wave height distribution depends on the number of waves in the wave train (record length) and on the kurtosis of the nonlinear wave ﬁeld, as shown in Eq. (6.30). 6.5. Accuracy of Freak Wave Prediction Figure 6.4 compares the linear (Rayleigh) theory with those from the present theory for the occurrence probability of a freak wave, Pfreak ; the plot shows how Pfreak

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142

0

Pfreak

10

−1

10

Rayleigh µ4 = 3.1 µ4 = 3.2 µ4 = 3.3 µ4 = 3.4 µ4 = 3.5 −2

10

2

3

10

10 N

4

10

Fig. 6.4. Occurrence probability for freak waves as a function of number of waves N and kurtosis µ4 .31

responds to changes in the number of waves N and in the strength of nonlinearities, as measured by the kurtosis µ4 . For N = 100 waves, the occurrence probability of a freak wave is predicted to be 3.3% by the linear theory, while it is 15.4% according to Eq. (6.30) with µ4 = 3.5. For N = 1000 waves, the occurrence probability of a freak wave is 28.5% according to linear theory, while it is 81.3% according to Eq. (6.31) with µ4 = 3.5. The number of waves N = 1000 corresponds to a duration of about 3 hours, assuming T1/3 = 10 s, which is not unusual in stormy conditions. Alternatively, deﬁning the threshold value for the occurrence probability of a freak wave as 50%, the expected number of waves that would include at least one freak wave is predicted to be 2000 waves from linear theory, but only 500 waves when Eq. (6.31) is used with µ4 = 3.5. Thus, in a strong nonlinear wave ﬁeld, freak waves can occur several times more frequently than in a linear ﬁeld. Figure 6.5 shows freak wave occurrence probability as a ratio Rfreak of the value predicted by the present approach to that predicted by the Rayleigh theory (κ40 = 0); the ratio depends on the kurtosis µ4 , Rnonlinear =

Pfreak −1. Pfreak |κ40 =0

(6.33)

For a small number of waves N ≤ 250, the ratio Rfreak depends linearly on µ4 . If µ4 = 3.1 and N ≤ 500, the occurrence probability of freak waves is 50% more from the nonlinear theory than from the linear theory. However, the rate of increase in Rfreak decreases as the number of waves increases. This is because, for very large number of waves, the maximum wave height almost always exceeds 2H1/3 , even in linear wave trains.

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350 300

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N=100 N=250 N=500 N=1000 N=5000

Rfreak [%]

250 200 150 100 50 0 3

Fig. 6.5.

3.1

3.2

µ4

3.3

3.4

3.5

Ratio of freak wave occurrence predicted by Eq. (6.31) to the Rayleigh theory.31

The consequences of the maximum wave height distribution [Eq. (6.30)] are generally hard to verify, not only because Eq. (6.30) depends on the number of waves but also because large amount of data are required to characterize nonlinear eﬀects. In the following discussion, we compare the above theories with a set of long time-series experimental data47 ; these data appear to be suﬃciently long to make useful comparisons. Figure 6.6 shows the probability of occurrence of freak waves Hmax /ηrms ≥ 8 for short time records, as a function of kurtosis. We assumed 100 waves per record and counted the number of freak waves. In the experimental data, the occurrence probability for freak waves shows a clear dependence on the kurtosis (this eﬀect of course is not described by the Rayleigh distribution). But the experimental data appear to depend quadratically on the kurtosis, while the nonlinear theory Eq. (6.31) predicts a linear dependence. This is probably because the nonlinear theory includes only the lowest-order correction for nonlinearity: it excludes high-order cumulants. Figure 6.7 compares the maximum wave height distribution from the experimental data with linear, Eq. (6.14), and nonlinear theories, Eq. (6.30), for N = 150. In Fig. 6.7(a), the comparison between theory and experiment is shown for a nearby linear wave condition (µ4 = 3.06). The peak in the observed maximum wave height distribution is larger than the Rayleigh Hmax distribution and the MER Hmax distribution with µ4 = 3.06, but the observed distribution is not as wide (we ascribe this diﬀerence to the eﬀects of the ﬁnite width of the spectrum). The maximum wave height distribution from the laboratory data departs from the Rayleigh distribution [Figs. 6.7(a)–6.7(c)] at some distance from the wavemaker, and the peak of the observed distribution shifts to larger wave heights. While the Rayleigh Hmax distribution, Eq. (6.14), is independent of the kurtosis, the MER Hmax distribution,

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N=11900 0.45 Exp. MER Rayleigh

0.4

Probability of Freak Wave [%]

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 3

3.5

4

4.5

µ4 Fig. 6.6. Occurrence probability for freak waves in long recorded wave data: N = 11, 900 (•: Experimental data, solid line: MER distribution from Eq. (6.31), dashed line: Rayleigh distribution).

Eq. (6.30), follows the behavior of the experimental data. The MER Hmax distribution seems to reproduce the experimental data reasonably well; for very small kurtosis, it overestimates the experimental data, and for large kurtosis it slightly underestimates the data. This is consistent with what we have seen in Fig. 6.6 where the occurrence probability of freak waves is plotted as a function of the kurtosis. Similar results have been obtained in other studies.31 We also consider the general behavior of the probability density function of maximum wave height in a nonlinear wave ﬁeld. To do so, we plot in Fig. 6.8 the expected value of the maximum wave height, indicated by angle brackets, as a function of µ4 and N . In the ﬁgure, we show the numerically integrated value of Eq. (6.30) together with the ensemble average of the experimental data. The quantity Hmax /H1/3 in Fig. 6.8 from the Rayleigh theory corresponds to √ H/ m0 > 8. The dependence of Hmax /H1/3 on µ4 and N is clear in both the experimental data and Eq. (6.30). Overall, Hmax /H1/3 from the experimental data is smaller than that from the MER Hmax distribution; nevertheless, the theoretical and experimental data show similar trends. The above evidence shows that freak waves can be predicted for given strengths of nonlinearities, as measured by the kurtosis. The remaining problems, therefore,

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0.8 Exp. MER Rayleigh

(a) µ4 =3.06 0.7

0.5

P(H

max

/η

rms

)

0.6

0.4 0.3 0.2 0.1 0 4

5

6

7

8

H

9

10

11

12

/η

max

rms

0.8 Exp. MER Rayleigh

(b) µ4 =3.43 0.7

rms

0.4

P(H

/η

0.5

max

)

0.6

0.3 0.2 0.1 0 4

5

6

7

8

H

9

10

11

12

/η

max

rms

0.8

Exp. MER Rayleigh

(c) µ4 =4.10 0.7

P(Hmax/ηrms)

0.6 0.5 0.4 0.3 0.2 0.1 0 4

5

6

7

8

9

10

11

12

H max/ηrms

Fig. 6.7. Comparison of maximum wave height distribution Hmax /ηrms with N = 150 (•: experimental data, solid line: MER Hmax distribution, dashed line: Rayleigh Hmax distribution).

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2.2

2.1 2

1.9 1.8 1.7 1.6 1.5

1.4 1000 800 600 400 200 N

3

3.2

3.4

3.6

3.8

4

µ4

2.2

2.1 2

1.9 1.8 1.7 1.6 1.5

1.4 1000 800 600 400 N

Fig. 6.8.

200

3

3.2

3.4

3.6

3.8

4

µ4

Comparison of expected value of Hmax /H1/3 .48

are developing accurate methods for estimating the kurtosis and learning how to take directional eﬀects into account for the application in the ocean. 6.6. Conclusion and Future Study This chapter summarized generally accepted mechanisms that cause freak waves and we have described linear and nonlinear mechanisms in more detail. Four-wave

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quasi-resonant interactions have been shown to play an important role in determining the statistical properties of surface elevation. This discovery, which is fairly new, is of some relevance to extreme wave forecasting. Previously, deviations from Gaussian behavior have been attributed only to bound modes. However, even in the most severe sea states, the kurtosis obtained from second-order theory rarely reaches values above 3.15. Moreover, contributions to wave height distribution from second-order theory are practically negligible (in the narrow-band approximation that are exactly zero). If waves are long-crested and suﬃciently steep, the dynamics of the free waves can cause very strong departures from Gaussian behavior. Large values of the kurtosis can substantially change the tails of the probability density function for wave height and can increase the occurrence of freak waves. Although nonlinear eﬀects have been found to be important, some issues remain unresolved. First, we need an evaluation of the accuracy with which the kurtosis is estimated from spectra. BFI originally describes the asymptotic behavior, given an initial value, but in reality, there is no initial condition in the ocean. This should be veriﬁed in the near future. Second, we need to account for wave directionality. In real sea states, directional eﬀects are important. Directional contributions to the evolution of the kurtosis, using a modiﬁed nonlinear Schr¨ odinger equation, have been described,49 and the eﬀects of four-wave interactions only appear close to longcrested waves conditions.50 A quantitative discussion of directional eﬀects should be presented in the near future.

References 1. L. Draper, “Freak” ocean waves, Marine Observer 35, 193–195 (1965). 2. P. Klinting and S. Sand, Analysis of prototype freak waves, in Coastal Hydrodynamics, ed. R. Darlymple (ASCE, 1987), pp. 618–632. 3. P. Stansell, J. Wolfram and S. Zachary, Horizontal asymmetry and steepness distributions for wind-driven ocean waves, Appl. Ocean Res. 25, 137–155 (2003). 4. G. Guedes Soares, Z. Cherneva and E. Ant¨ ao, Characteristics of abnormal waves in North Sea storm sea states, Appl. Ocean Res. 25, 337–344 (2003). 5. T. Yasuda and N. Mori, Occurrence properties of giant freak waves the sea area around Japan, J. Waterway, Port, Coastal Ocean Eng. 123(4), 209–213 (1997). 6. T. Yasuda, N. Mori and S. Nakayama, Characteristics of giant freak waves observed in the Sea of Japan, in Waves97 (ASCE, 1997), pp. 482–495. 7. G. Guedes Soares, Z. Cherneva and E. Antao, Abnormal waves during hurricane Camille, J. Geophys. Res. Ocean 109, C08008 (2004). 8. S. Sand, N. O. Hansen, P. Klinting, O. Gudmestad and M. Sterndorﬀ, Freak Wave Kinematics (Kluwer Academic Pub., 1990), pp. 535–549. 9. M. Olagnon (ed.), Rogue Waves 2004 (IFREMER, 2004). 10. M. Longuet-Higgins, On the statistical distribution of the heights of sea waves, J. Marine Res. 11, 245–266 (1952). 11. A. Næss, On the distribution of crest-to-trough wave heights, Ocean Eng. 12(3), 221–234 (1985). 12. P. Boccotti, On Mechanics of Irregular Gravity Waves, Chap. VIII (Atti Accademia Nazionale dei Lincei, Memorie, 1989), pp. 111–170.

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13. M. A. Tayfun, Distribution of crest-to-trough wave heights, J. Waterway, Port, Coastal Ocean Eng. 107(3), 149–158 (1981). 14. M. Longuet-Higgins, The eﬀect of non-linearities on statistical distirbutions in the theory of sea waves, J. Fluid Mechanics 17, 459–480 (1963). 15. M. A. Tayfun, Narrow band nonlinear sea waves, J. Geophys. Res. Ocean 85(C3), 1548–1552 (1980). 16. F. Fedele and F. Arena, Weakly nonlinear statistics of high non-linear random waves, Phys. Fluids 17, 026601 (2005). 17. B. White and B. Fornberg, On the chance of freak wave at sea, J. Fluid Mechanics 355, 113–138 (1998). 18. T. Yasuda, N. Mori and K. Ito, Freak waves in a unidirectional wave train and their kinematics, Proc. 23rd Int. Conf. Coastal Eng., Vo. 1, ASCE, Venice (1992), pp. 751–764. 19. P. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr. 33(4), 863–884 (2003). 20. C. Stansberg, Extreme waves in laboratory generated irregular wave trains, in Water Wave Kinematics, eds. A. Tørum and O. Gudmestad (Kluwer Academic Pub., 1990), pp. 573–590. 21. K. Trulsen and K. Dysthe, Freak waves — A three-dimensional wave simulation, Proc. 21st Symp. Naval Hydrodynamics (National Academy Press, 1997), pp. 550–558. 22. M. Onorato, A. Osborne, M. Serio and S. Bertone, Freak waves in random oceanic sea states, Phys. Rev. Lett. 86(25), 5831–5834 (2001). 23. H. C. Yuen and B. M. Lake, Nonlinear dynamics of deep-water gravity waves, Adv. Appl. Mech. 22, 67–229 (1982). 24. E. Caponi, P. Saﬀman and H. Yuen, Instability and conﬁned chaos in a nonlinear dispersive wave system, Phys. Fluids 25(12), 2159–2166 (1982). 25. K. Dysthe, Note on modiﬁcation to the nonlinear Schr¨ odinger equation for application to deep water waves, Proc. Roy. Soc. Lond. A 369, 105–114 (1979). 26. M. Stiassnie and L. Shemer, Energy computations for evolution of class I and II instabilities of Stokes waves, J. Fluid Mechanics 174, 299–312 (1987). 27. T. Yasuda and N. Mori, Roles of sideband instability and mode coupling in forming a water wave chaos, Wave Motion 26(2), 163–185 (1997). 28. M. Su, Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric patterns, J. Fluid Mechanics 124, 73–108 (1982). 29. N. Mori, T. Yasuda and K. Kawaguchi, Breaking eﬀects on waves statistics for deepwater random wave train, in Waves97 (ASCE, 1997), pp. 316–328. 30. T. Yasuda and N. Mori, High order nonlinear eﬀects on deep-water random wave trains, Int. Symp. Waves-Physical and Numerical Modelling, Vol. 2, Vancouver (1993), pp. 823–832. 31. N. Mori and P. Janssen, On kurtosis and occurrence probability of freak waves, J. Phys. Oceanogr. 36(7), 1471–1483 (2006). 32. M. A. Tayfun and J.-M. Lo, Nonlinear eﬀects on wave envelope and phase, J. Waterway, Port, Coastal Ocean Eng. 116(1), 79–100 (1990). 33. N. Mori and T. Yasuda, A weakly non-Gaussian model of wave height distribution for random wave train, Ocean Eng. 29(10), 1219–1231 (2002). 34. N. Mori, P. Liu and T. Yasuda, Analysis of freak wave measurements in the Sea of Japan, Ocean Eng. 29(11), 1399–1414 (2002). 35. M. Longuet-Higgins and O. Phillips, Phase velocity eﬀects internary wave interactions, J. Fluid Mechanics 12, 133–135 (1962).

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36. A. Kimura, Probability of the freak wave appearance in a 3-dimensional sea condition, Proc. 24th Int. Conf. Coastal Eng., Vol. 1, ASCE (1994), pp. 356–369. 37. Y. Goda, Random Seas and Design of Maritime Structures, 2nd Edn. (World Scientiﬁc, 2000). 38. I. Lavrenov, The wave energy concentration at the Agulhas current oﬀ South Africa, Natural Hazard 17, 117–127 (1998). 39. A. Osborne, M. Onorato, M. Serio and S. Bertone, The nonlinear dynamics of rogue waves and holes in deep water gravity wave trains, Phys. Lett. A 275(5–6), 386–393 (2000). 40. S. Haver, Evidences of the existence of freak waves, in Rogue Waves, eds. M. Olagnon and G. Atanassoulis (IFREMER, 2001), pp. 129–140. 41. M. Olagnon and G. Athanassoulis (eds.), Rogue Waves 2000 (IFREMER, 2000). 42. K. Hasselmann, On the nonlinear energy transfer in gravity-wave spectrum. I. General theory, J. Fluid Mechanics 12, 481–500 (1962). 43. V. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves, J. Fluid Mechanics 272, 1–20 (1994). 44. F. Edgeworth, On the representation of statistical ferquency by a series, J. Roy. Statist. Soc. 70, 102–106 (1907). 45. M. A. Tayfun, Statistics of nonlinear wave crests and groups, Ocean Eng. 33(11–12), 1589–1622 (2006). 46. F. Fedele and M. A. Tayfun, Explaining freak waves by a stochastic theory of wave groups, 25th Int. Conf. Oﬀshore Mechanics and Arctic Engineering 1 (2006). 47. M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini and C. T. Stansberg, Extreme waves, modulational instability and second order theory: Wave ﬂume experiments on irregular waves, Eur. J. Mech. B/Fluids 25(5), 586–601 (2006). 48. N. Mori, M. Onorato, P. A. E. M. Janssen, A. R. Osborne and M. Serio, On the extreme statistics of long crested deep water waves: Theory and experiments, J. Geophys. Res., Ocean 112, C09011 (2007), doi: 10.1029/2006JC004024. 49. H. Socquet-Juglard, K. Dysthe, K. Trulsen, H. Krogstad and J. Liu, Probability distribution of surface gravity waves during spectral changes, J. Fluid Mechanics 542, 195–216 (2005). 50. O. Gramstad and K. Krulsen, Inﬂuence of crest and group length on the occurrence of freak waves, J. Fluid Mechanics 582, 463–472 (2007).

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Chapter 7

Short-Term Wave Statistics Akira Kimura Department of Social Systems Engineering Faculty of Engineering, Tottori University, 680-8552 Koyama Minami 4-101, Tottori, Japan [email protected] This chapter deals with the short-term statistical properties of irregular sea waves. It begins with a description of an early study by Rice [Bell Syst. Tech. J. 23, 282–332; 24, 46–156 (1944, 1945)] about the relation between wave spectra and statistics of wave amplitudes. Next, an explanation is given of studies which deal with the eﬀect of spectrum width and wave nonlinearity on wave statistics as well as with joint statistical properties between wave heights and periods. Finally, the chapter closes with a look at the statistics of wave direction, length of wave crest, and spatial maximum amplitude (which are the 3D wave’s properties) as well as the time series of wave height.

7.1. Introduction Waves in sea conditions have certain irregular properties. These comprise properties such as wave heights and wave periods. For many years, engineers have known that it is not suﬃcient to know only statistically representative values such as the mean value. Larger wave heights can induce damage on coastal structures while longer wave periods may induce resonance phenomena. To understand their overall statistical properties, engineers and researchers have tried to clarify the statistical characteristics of populations for these properties with the intention of ﬁnding out their probability density function (pdf). In particular, their pdf have been studied in relation to the wave spectrum. The study of this development was initiated primarily by Rice.20 The application of this idea in coastal engineering was started notably by Longuet-Higgins.15 He developed pdf for surface elevation and gradient, maxima and minima of water surface, number of zero-crossings along a line, etc. Of course, sailors have carried out visual observations of waves since ancient times. From the theoretical and empirical knowledge of wave statistics, two types of statistical study have appeared: shortterm and long-term wave statistics. Since waves must be stationary stochastic to 151

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study short-term statistics, the duration of concern is 1 or 2 days at most. On the other hand, long-term statistics cover waves for time periods of at least dozens of years. In this type, the time series of water surface elevation is not used. Instead, representative wave properties such as signiﬁcant wave height and period are utilized. The wave height/period is estimated for coastal structures within their lifetime from long-term statistics, for example. The eﬀectiveness of results from long-term statistics is heavily dependent on short-term statistics. Both are inseparable for the eﬃcient use of wave statistics in coastal engineering.

7.2. Rice Model Rice20 proposed a model of the Fourier sum for irregular signals. The irregular signal I(t) is expressed by the summation of trigonometric functions as I(t) =

N

(an cos σn t + bn sin σn t) =

n=1

N

cn cos(σn t + εn ),

(7.1)

n=1

in which σn is the angular frequency and εn (= tan−1 (−bn /an )) is the phase of the nth component wave, respectively. an , bn , and cn are amplitudes of nth component waves which are distributed normally with zero mean. Their variances are 2 cn 2 2 an = bn = = S(σ)dσ , (7.2) 2 where means the ensemble mean, S(σ) is the energy spectrum of I(t). Rice deﬁned the envelope function for I(t) as I(t) =

N

cn cos(σn t + εn ) = Ic (t) cos σt − Is (t) sin σt ,

(7.3)

n=1

where σ is the mean frequency of S(σ), and Ic (t) and Is (t) are Ic (t) =

N

cn cos(σn t − σt + εn ),

n=1

Is (t) =

N

cn sin(σn t − σt + εn ) .

(7.4)

n=1

The envelope function R(t) is R(t) = {Ic2 (t) + Is2 (t)}1/2 .

(7.5)

The nth component wave in Eq. (7.4), cn cos(σn t − σt + εn ) or cn sin (σn t − σt + εn ), has a U-shape pdf of ﬁnite width.14 From the central limit theorem, the inﬁnite sum (N → ∞) of the trigonometric functions, Ic (t) and Is (t), are distributed normally. Their joint pdf is 2 1 Ic + Is2 p(Ic , Is )dIc dIs = dIc dIs , exp − (7.6) 2πϕ0 2ϕ0

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since Ic (t) and Is (t) are independent (Ic (t) · Is (t) = 0). ϕ0 is the variance of Ic (t), Is (t), and I(t), respectively: Ic2 = Is2 = I 2 = ϕ0 .

(7.7)

The variables R(t), I(t), Ic (t), and Is (t) are represented as R, I, Ic , and Is , respectively. It is convenient to introduce the variable θ, where Ic = R cos θ,

Is = R sin θ .

By substituting Eq. (7.8) into Eq. (7.6), the joint pdf of R and θ becomes R R2 dRdθ . p(R, θ)dRdθ = exp − 2πϕ0 2ϕ0

(7.8)

(7.9)

Since R and θ are independent and θ is distributed uniformly over the range 0 to 2π, the pdf of R is obtained as R R2 p(R)dR = exp − dR . (7.10) ϕ0 2ϕ0 Figure 7.1(a) shows the relation between I and R. The spectrum S(σ) for I is the following normalized Wallops type (r = 10)8 : −4 −r r σ σ exp . (7.11) · 1− S(σ) = σp 4 σp The maxima or minima of I almost coincide with R within a “zero-up/downcrossing” interval of I. If I takes extremes (∂I/∂t = 0) at t = t∗1 , t∗2 , . . . , wave amplitude can be approximated by R|t∗i , (i = 1, 2, . . .) [Fig. 7.1(b)]. The intervals between t∗1 , t∗2 , . . . are almost constant when the spectrum is very narrow. The pdf for R|t∗i is also the Rayleigh distribution approximately, since the local maxima/ minima is equal to the wave amplitude. Equation (7.10) becomes the pdf for wave amplitude14 approximately.

7.3. Pdf of Wave Heights 7.3.1. Nonlinear eﬀect on the pdf Since the study by Longuet-Higgins,14 the pdf for wave amplitudes has been used extensively as pdf for wave heights. Many measurements and numerical simulations have supported the Rayleigh distribution for the wave height distribution. The standard form of the widely used Rayleigh distribution is π H π H2 p(H)dH = exp − dH , (7.12) 2 H2 4 H2 in which H is the mean wave height. When the wave spectrum is very narrow, wave height distribution can be the Rayleigh distribution, since wave height can be approximated as H ≈ 2R .

(7.13)

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Fig. 7.1. (a) Wave proﬁle I(t) (solid line) and its envelope R(t) (dotted line). (b) Wave proﬁle I(t) (solid line) and envelope R(t) (dotted line) (small discrepancies between I(t) and R(t) at t = t∗1 , t∗2 , · · · ).

However, the zero-crossing wave height is deﬁned by H = Il max − Il min , where Il max and Il min are the consecutive local maximum and minimum of the wave proﬁle, respectively. The pdf for H may not be the Rayleigh distribution if the pdf for Il max and Il min are the Rayleigh distribution. Field measurements have shown that there are small discrepancies between the Rayleigh distribution p(H) and frequency distributions of measured data F (H): F (H) < p(H) for H < Hp and F (H) > p(H) for H > Hp , where p(Hp ) is a peak of p(H). In particular, as nonlinearity on the wave proﬁle becomes prominent, this characteristic may bring a considerable eﬀect on the pdf. Forristall3 used the following Weibull distribution to improve the goodness of ﬁt:

α −ς , (7.14) P (ς) = exp β where P (ς) is the Weibull distribution function and α = 2.126, β = 8.42, and ς=

H∗ 1/2

m0

,

(7.15)

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in which H ∗ is the diﬀerence between the heights of consecutive wave crest and trough within a zero-crossing interval of the wave proﬁle. Longuet-Higgins17 pointed out that the discrepancy should be improved using rms (root mean square) wave amplitude. The ith moment of a spectrum is determined as ∞ mi = σ i S(σ)dσ (i = 1, 2, . . .) . (7.16) 0

From the Khintchine theorem, the 0th moment is also deﬁned in another way as 2 , m0 = ηrms

(7.17)

where η is the wave proﬁle and ηrms is its rms. When the spectrum is narrow and the nonlinearity on the wave proﬁle is negligible, (arms )2 , (7.18) 2 in which arms is the rms of wave amplitude a. The Rayleigh distribution function given by Longuet-Higgins is

a2 P (a) = exp − 2 . (7.19) arms m0 =

a2rms can be replaced by 2m0 if the nonlinearity is negligible. He discussed the nonlinearity eﬀect on the wave proﬁle as follows. Applying the third-order Stokes wave theory, the 2nd moment of the periodic wave proﬁle is derived from Eq. (7.17) as

1 1 2 19 3077 2 4 6 (ak) − · · · , (7.20) m0 = a 1 − (ak) − (ak) − 2 2 12 720 in which k is the wave number. m0 for irregular waves is calculated by ∞ m0 = 2V (a)dp(a) ,

(7.21)

0

in which V (a) denotes the potential energy (2V = η 2 = m0 ) of the zero-crossing wave with an amplitude a given as Eq. (7.20), p(a) is the pdf of a. Applying Eq. (7.19), m0 becomes

amax a2 2V (a)d exp − 2 m0 = +δ, (7.22) arms 0 in which amax is the maximum wave amplitude of nonbreaking waves. He used the same V for the waves (ak)2 ≥ 0.196. δ is the remainder and is exponentially small for the large value of a2max /a2rms . The new m0 is

1 19 3077 (arms k)6 − · · · . (7.23) m0 = a2rms 1 − (arms k)2 − (arms k)4 − 2 2 30 Since arms k for irregular waves distributes roughly 0.158 < arms /k < 0.203,13 0.935 <

2m0 < 0.968 . a2rms

(7.24)

If wave proﬁles have a nonlinearity, rms wave amplitude between 2.066m0 < a2rms < 2.140m0 may give a correction for the Rayleigh distribution of wave amplitudes.

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7.3.2. Eﬀect of spectrum width Since the zero-crossing method for irregular waves does not have a physical basis, a mathematical treatment is diﬃcult except for the very narrow band spectrum. Cartwright and Longuet-Higgins2 derived a spectrum width parameter as

m22 ε= 1− : 0 < ε < 1, (7.25) m0 m4 in which m0 , m2 , and m4 are determined by Eq. (7.16). Pdf of amplitudes (local maxima) is the Rayleigh distribution when ε → 0. If ε = 0, however, waves must be periodic. Longuet-Higgins17 investigated the eﬀect of the spectrum width on pdf of wave amplitudes. He used a periodic wave proﬁle with small disturbance given as y = b cos(σt + ε) + an cos(σn t + εn ) , (7.26) n

in which an , σn , and εn are the amplitude, angular frequency, and phase of the disturbance, respectively. Their amplitudes are b >> an . The spectrum of the disturbances S (σ) is deﬁned as σn +dσ σn

The rms of y is

2 yrms = b2rms + 0

∞

1 2 a = S (σn )dσ . 2 n

(7.27)

πσ σ 2 sin2 S (σ)dσ , + 2σ σ

(7.28)

and the 0th moment of the spectrum of y is ∞ ∞ b2rms {S(σ) + S (σ)}dσ = S (σ)dσ , + m0 = 2 0 0

(7.29)

where S(σ) is the spectrum of the periodic wave and brms is the rms of the periodic wave proﬁle. From Eqs. (7.28) and (7.29), yrms is ∞ 2

σ 2 2 πσ = 2m0 + − 1 − cos (7.30) yrms S (σ)dσ. 2σ σ2 0 Using the relations  2 σ 2

σ σ    −1 + −1 −1 =2   σ σ σ2  

πσ 1 − cos π(σ/σ − 1) 2 − 1) π(σ/σ 2 cos = = + ···  2 4  2σ  ∞   i  (σ − σ) S (σ)dσ , µi = 

(7.31)

0

the next relation is obtained. 2 yrms (π 2 /8 − 1/2)µ2 µ1 − =1+ . 2m0 m0 σ m0 σ 2

(7.32)

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If the spectrum is symmetry for σ ¯, 2 yrms =1− 2m0

1 π2 − 8 2

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ν2 ,

(7.33)

µ2 m0 m2 − m21 = , m21 m0 σ 2

(7.34)

where ν2 =

in which ν is also the spectrum width parameter.18 Since b an , yrms ≈ brms , Eq. (7.33) may give a correction for the Rayleigh distribution of wave amplitudes when the spectrum is not a narrow band. Tayfun25 also investigated the eﬀect of spectrum width. A better deﬁnition for the zero-crossing wave height than Eq. (7.13) may be given as Hi = Rmi + Rmi+1 ,

(7.35)

where Rmi (i = 1, 2, . . .) are the amplitudes of the wave envelope when the wave proﬁle I takes the local extreme values (t = t∗1 , t∗2 , . . .) [Fig. 1(b)]. The joint pdf for the consecutive amplitudes Rmi , Rmi+1 is πκξ1 ξ2 π(ξ12 + ξ22 ) π2 ξ1 ξ2 p(ξ1 , ξ2 ) = exp − I , (7.36) 0 4(1 − κ2 ) 4(1 − κ2 ) 2(1 − κ2 ) in which I0 [ ] is the 0th order modiﬁed Bessel function, ξi = Rmi /(πm0 /2)1/2 , ξi+1 = Rmi +1 /(πm0 /2)1/2 . They are replaced as ξi = ξ1 and ξi+1 = ξ2 , respectively for simplicity. κ is the correlation parameter given by 1/2 , (7.37) k = ρ2 + λ2 in which ρ=

1 m0

∞ 0

τ S(σ) cos(σ − σ) dσ , 2

−1 ∞ τ λ= S(σ) sin(σ − σ) dσ , m0 0 2 m1 τ = , m0 where τ is the mean wave period. Normalized wave height is given as H ξ1 + ξ2 . ς= = Hm 2

(7.38)

Substituting the relation ξ2 = 2ς − ξ1 into Eq. (7.36) and the integration in terms of ξ1 from 0 to 2ς gives the pdf for ς: 2ς p(ξ1 , 2ς − ξ1 )dξ1 . (7.39) p(ς) = 0

Figure 7.2 shows p(ς) for two diﬀerent wave spectra [Wallops-type, Eq. (7.11): r = 5 (dotted line), r = 10 (broken line)] and the Rayleigh distribution (solid line). Kimura et al.10 showed that the discrepancy in Fig. 7.2 comes from the diﬀerence between I and R at (t = t∗1 , t∗2 , . . .) [Fig. 7.1(b)].

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1.2 1.0

r=5 r = 10

p (ζ)

0.8

Rayleigh 0.6 0.4 0.2 0.0 0.0

0.5

1.0

Fig. 7.2.

1.5 ζ

2.0

2.5

3.0

Pdf of wave height.

7.4. Maximum Wave Height Zero-crossing wave amplitude in Eq. (7.19) extends the range from 0 to inﬁnity if the number of waves is inﬁnity. When the population of waves is ﬁnite, such as data from 20 min measurements every 2 h, maximum wave amplitude may be ﬁnite. If a time series of wave amplitudes a1 , a2 , . . . , aN , aN +1 , . . . , a2N , a2N +1 , . . . , a3N , a3N +1 . . . , a4N , Group No. 1

2

3

4

is divided into groups with N waves, the maximum values (amax ) from individual groups show irregular properties. The pdf of the maximum wave amplitude is derived by Longuet-Higgins.14 The probability that one wave amplitude a is greater than a0 is P (a0 ) [Eq. (7.19)]. When the size of the group is N , the probability that all the wave amplitudes are a < a0 is {1 − P (a0 )}N . The probability that one amplitude, at least, exceeds a0 is 1 − {1 − P (a0 )}N . Therefore, the probability that maximum wave amplitude lies in the range amax ∼ amax + da is 1 − {1 − P (amax )}N − 1 + {1 − P (amax + da)}N = −d[1 − {1 − P (amax )}N ] dP (amax ) = −N {1 − P (amax )}N −1 da . da The pdf for the maximum wave amplitude is 2 2 N −1 amax −amax amax exp p(amax ) = 2N 1 − exp − . arms arms a2rms

(7.40)

(7.41)

Its mean value is derived as E(amax ) = (ln N )1/2 + γ(ln N )−1/2 + · · · arms

(7.42)

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with an approximation 2 N −θ ≈ e−e , 1 − e−(amax /arms )

(7.43)

in which θ =

amax arms

2 − ln N ,

(7.44)

and γ is the Euler constant (γ = 0.5772 . . .). Maximum wave height E(Hmax )/Hrms is usually compared also with Eq. (7.42).

7.5. Zero-Crossing Wave Period The problem of deriving the pdf of zero-crossing wave periods is very diﬃcult. Rice derived a pdf for the zero-crossing interval. He treated the interval where I(t) [Eq. (7.1)] crosses the 0 line upward at t = 0 and crosses the 0 line downward between τ and τ + dτ . 1 τ 1 τ τ p(τ ) = p0 (τ ) − p1 (r, τ )dr + p2 (r, s, τ )drds 1! 0 2! 0 0 1 τ τ τ p3 (r, s, u, τ )drdsdu + · · · , (7.45) − 3! 0 0 0 where p0 (τ ) is the probability that I(t) = 0 at t = 0 and I(t) = 0 lies between τ < t < τ + dτ , p1 (r, τ ) is the probability that I(t) = 0 at t = 0 and I(t) = 0 lies between r < t < r + dr and τ < t < τ + dτ (r < τ ), p2 (r, s, τ ) is the probability that I(t) = 0 at t = 0, and I(t) = 0 lies between s < t < s + ds, r < t < r + dr, and τ < t < τ + dτ (s < r < τ ), and so on. Since this series of integrations is inexcusable, Rice21 showed p(τ ) only applying the ﬁrst term of Eq. (7.45) as p(τ ) =

dτ 2π

1/2

−3/2 M23 2 ψ0 − ψ0 − ψτ2 {1 + H cot−1 (−H)} , ψ0 H

(7.46)

with 2 2 −1/2 − M23 , H = M23 M22 where M22 and M23 are the cofactors of elements µ22 and µ23 of the matrix, 

µ11  µ21 M =  µ31 µ41

µ12 µ22 µ32 µ42

µ13 µ23 µ33 µ43

  µ14 ψ0  0 µ24  = µ34   ψτ ψτ µ44

0 −ψ0 −ψτ −ψτ

ψτ −ψτ −ψ0 0

 ψτ −ψτ   0  ψ0

(7.47)

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and 1 ψτ = 2π ψτ = − ψτ = −

∞

S(σ) cos στ dσ ,

0

1 2π 1 2π

∞

σS(σ) sin στ dσ ,

0

(7.48) ∞

σ 2 S (σ) cos στ dσ ,

0

ψ0 = ψτ |τ =0 ,

ψ0 = ψτ |τ =0 .

Equation (7.46) has a peak around the mean value of τ . The second and third peaks appear for larger τ . This is because the higher terms in Eq. (7.45) are neglected. No successful analytical solution for the zero-crossing wave period has been derived at the present time. Instead of wave periods, Longuet-Higgins16 derived the pdf for wave lengths. The wave proﬁle of unidirectional irregular waves is expressed as ς = R cos {χ} ,

(7.49)

where χ = ux + φ,

χx = u + φx ,

(7.50)

in which R is the amplitude of the wave envelope, φ is the phase, u is the mean wave number in x-direction, and subscript x means a diﬀerentiation in terms of x. Joint pdf for χ and χx was derived by Longuet-Higgins15 as p (χ, χx ) = with m∗n

∞

=

µ∗2 = u=

(m∗0 /µ∗2 )

1/2

&3/2 , % 2 4π 1 + (χx − u) · m∗0 /µ∗2

(7.51)

E (u) un du ,

(7.52)

(n = 0, 1, . . .)

0

∞

2

E (u) (u − u) du ,

0 m∗1 /m∗0

,

(7.53) (7.54)

in which u is the wave number in the x-direction and E (u) is its spectrum. The event ς = 0 is equivalent to χ = (2r − 1/2) π or χ = (2r + 1/2) π (r: integer). The probability that χ = (2r − 1/2) π between x and x + dx is 1/2 ∞ 1 m∗2 H(χ)dx = p(χ, χx )χ=(2r−1/2)π |χx | dxdχx = dx . (7.55) 2π m∗0 −∞ Since the probability that χ = (2r + 1/2)π between x and x + dx is given by the same equation, Eq. (7.55), the probability that ς = 0 between x and x + dx is 2H(χ)dx. The average zero-crossing interval is ∗ 1/2 1/2 m0 π 1 + ν ∗2 =π = , (7.56) ∗ m2 u

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with ν ∗2 =

m∗0 m∗2 − m∗2 1 = m∗2 1

µ∗2 u2 µ∗0

.

(7.57)

If ν ∗ is small, ≈

π . u

(7.58)

The zero-up-crossing event of ς takes place when χ = (2r − 1/2)π with ∂χ/∂x > 0 or χ = (2r + 1/2)π with ∂χ/∂x < 0. The probability of the latter condition is

∞

−∞

p (χ, χx )χ=(2r+1/2)π |χx | dxdχx 1 = 4

m∗2 m∗0

1/2 −1/2 1 − 1 + ν ∗2 dx .

(7.59)

This probability is negligible for small values of ν ∗ . Between any two successive zero-crossing points x1 and x2 , χ must increase by π. Hence, 1 π = χ(x2 ) − χ(x1 ) = χx + 2 χxx + · · · . 2 The terms on the left of this equation, Taylor series expansion of χ(x2 ) around x = x1 is used. Since χxx = O(ν 2 ), χxxx = O(ν 3 ), . . . and they are very small, neglecting these terms, ≈

π . χx

(7.60)

Applying the same method by Rice,20 a mean zero-crossing number of ς per unit length, the conditional pdf of χx is p(χx )χ =

1 p(χ, χx )|χx | = H(χ) 2

m∗0 m∗2

1/2

|χx |(m∗0 /µ∗2 )1/2 , {1 + (χx − u)2 (m∗0 /µ∗2 )}3/2

(7.61)

in which |χx |/H(χ) is the probability that χ falls between x and x + dx. Since Eq. (7.61) does not involve χ, p(χx )χ directly becomes the pdf for χx . Substituting Eqs. (7.57), (7.58), and (7.60) into Eq. (7.61), p(ξ) =

1 , 2ν{1 + (ξ/ν ∗ )2 }3/2

(7.62)

in which ξ = ( − )/. If the frequency spectrum S(σ) is used instead of the wave number spectrum E(u), the pdf for the zero-crossing interval (half-wave period) is derived following the same method.

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7.6. Joint Pdf for Wave Height and Period Longuet-Higgins18 derived the joint pdf for wave amplitude and period. He used the following expression for the wave proﬁle: ς = Re {A exp (iσt)} ,

(7.63)

where Re means a real part and A = a exp{iφ} ,

(7.64)

in which a is the wave amplitude, φ is the phase of A, and σ is the mean angular frequency given by m1 σ= , (7.65) m0 where m0 and m1 are calculated by Eq. (7.16). The phase of ς is χ = σt + φ. Diﬀerentiation of the phase with t becomes χt = φt + σ .

(7.66)

When the spectrum is very narrow, the envelope of ς changes very slowly and φt σ. The wave period can be approximated as 2π 2π τ= = . (7.67) χt σt + φt Longuet-Higgins derived a joint pdf for a and φt :

2 a2 a (1/µ0 + φ2t /µ2 ) , p(a, φt ) = exp − 2 (2πµ20 µ2 )1/2

(7.68)

where µ0 and µ2 are calculated with Eq. (7.31), and ν is calculated by Eq. (7.34). With new variables τ a , T = , R= τ (2m0 )1/2 Eq. (7.68) is expressed as ' '

2 2 2 ' ∂(a, φt ) ' ' = 2R L exp − R [1 + (1 − 1/T ) ] , p(R, T ) = p(a, φt ) '' ∂(R, T ) ' π 1/2 νT 2 ν2

(7.69)

in which L is the correction factor to neglect the p(R, T ) in the region T < 0, L=

2 . 1 + (1 + ν 2 )−1/2

(7.70)

Figure 7.3 shows the contour map of p(R, T ). Solid curves are contours for p(R, T )/pmax = 0.99, 0.90, 0.70, 0.50, 0.30, 0.10 from inside to outside, respectively. The marginal distribution for normalized wave amplitudes and periods are R/ν 2L p(R) = 1/2 R exp(−R2 ) exp(−β 2 )dβ, (7.71) π −∞

−3/2 (1 − 1/T )2 L . (7.72) 1+ p(T ) = 2νT 2 ν2

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Fig. 7.3.

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Joint pdf for wave amplitude and period Eq. (7.69) (after Longuet-Higgins18 ).

Fig. 7.4.

Pdf for wave amplitude p(R) Eq. (7.71) (after Longuet-Higgins18 ).

Figures 7.4 and 7.5 show p(R) and p(T ), respectively. If ν = 0, p(R) is not the Rayleigh distribution. Peak of the p(R) slightly shifts to the larger side of R with increasing spectrum width ν. When ν = 0, p(T ) becomes a straight line (broken line, T : constant). Figure 7.6 shows the comparison of data and Eq. (7.69) by Goda.8 The number of waves within a grid of ∆H = 0.2H and ∆T = 0.2T are shown. H and T are the mean values of wave height and period, respectively. Many data sets, their correlation coeﬃcients between H and T distributed from 0.19 to 0.25, are used. The spectrum width parameter in Eq. (7.69) is v = 0.26.

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Fig. 7.5.

Fig. 7.6.

Pdf for wave period p(T ) Eq. (7.72) (after Longuet-Higgins18 ).

Joint distribution of wave height and period (after Goda5 ).

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7.7. Statistical Properties of 3D Irregular Waves Statistical properties of waves, such as length and direction of wave crest, are important in the design of coastal structures. Wave height and period are determined from a record of point measurement in the ordinary method. In contrast, a 3D wave proﬁle is necessary for the deﬁnition of wave crest length, direction, etc. As a result, very few studies have examined their properties. 7.7.1. Pdf for wave directions Kimura12 introduced a theoretical pdf for the wave direction. Figure 7.7 shows axes and wave crest lines (dotted line). Two wave gauges are installed along the y-axis. One is on the origin y1 and the other is on y2 (0, y0 ). Figure 7.8 shows the wave proﬁles measured at these points. ∆t1 , ∆t2 , ∆t3 , . . . are the time lags between corresponding wave crests. The phase lag of individual waves is determined as ∆ti εi = 2π , (i = 1, 2, . . .) , (7.73) Ti in which Ti is the zero-crossing wave period. The angle between the crest line and y-axis (Fig. 7.7) is

εi Li /2π −1 θi = sin , (i = 1, 2, . . .) , (7.74) y0 where Li is the wave length corresponding to the zero-crossing wave period Ti and θi is the direction of wave propagation measured from the x-axis.

Fig. 7.7.

Wave crest lines (dotted line) and axes.

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Fig. 7.8.

Wave proﬁles (solid line) at y1 and y2 .

Assuming y0 u < 1 in which u is the mean wave number in the x direction, wave proﬁles at these points are η1 (t) =

∞

ci cos(σi t + δi ) = ηc1 cos σt − ηs1 sin σt ,

(7.75)

ci cos(σi t + δi + vi y0 ) =ηc2 cos σt − ηs2 sin σt ,

(7.76)

i=1

η2 (t) =

∞ i=1

in which ci , σi , δi , and vi are the amplitude, the angular frequency, the initial phase, and the wave number in the y-direction of the ith component wave, respectively. σ is the mean angular frequency. ηc1 , ηs1 , ηc2 , and ηs2 are derived in the same way as Eq. (7.4). Their pdfs are the normal distribution with 0 mean, respectively. Their joint pdf is 1 p(ηc1 , ηs1 , ηc2 , ηs2 ) = (2π)2 (m20 − µ23 − µ24 )

1 2 − 2µ3 ηc1 ηc2 − 2µ4 ηc1 ηs2 × exp − (m0 ηc1 2(m20 − µ23 − µ24 ) 2 2 2 + m0 ηs1 + 2µ4 ηs1 ηc2 − 2µ3 ηs1 ηs2 + m0 ηc2 + m0 ηs2 ) , (7.77)

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with 2 2 2 2 = ηs1 = ηc2 = ηs2 , m0 = ηc1

µ3 = ηc1 ηc2 = ηs1 ηs2 , µ4 = ηc1 ηs2 = −ηs1 ηc2 ,

(7.78)

ηc1 ηs1 = ηs2 ηc2 = 0 . From the translations ηc1 = R1 cos(σt + φ1 ) = R1 cos(χ1 ) , ηs1 = R1 sin(σt + φ1 ) = R1 sin(χ1 ) , ηc2 = R2 cos(σt + φ2 ) = R2 cos(χ2 ) ,

(7.79)

ηs2 = R2 sin(σt + φ2 ) = R2 sin(χ2 ) , the joint pdf for R1 , R2 , χ1 , and χ2 is 1 p(R1 , R2 , χ1 , χ2 ) = (2π)2 (m20 − µ23 − µ24 )

1 (m0 (R12 + R22 ) × exp − 2(m20 − µ23 − µ24 )

− 2R1 R2 [µ3 cos(χ1 − χ2 ) − µ4 sin(χ1 − χ2 )]) ,

(7.80)

in which Ri and χi are the amplitude and phase of ηi (i = 1, 2), respectively.15 The crest of the wave proﬁle passes the origin of axes when χ1 = 2nπ (n = 0, 1, 2, . . .). The probability of the event that χ1 = 2nπ between t and t + dt is H(χ1 )dt = m1 /m0 dt. The conditional pdf for R1 , R2 , and χ2 when χ1 = 2nπ is ' σp(R1 , R2 , χ1 , χ2 ) '' p(R1 , R2 , χ2 | χ1 = 2nπ) = ' H(χ1 ) χ1 =2nπ

=

R1 R2 exp 2πm20 (1 − κ2 )

−

1 2 m20 (1

− κ2 )

[m0 (R12 + R22 )

− 2R1 R2 (µ3 cos(2nπ − χ2 ) − µ4 sin(2nπ − χ2 ))] , (7.81) in which κ2 =

µ23 + µ24 ≈ 1 − y02 u2 γ∗2 m20

(7.82)

and γ∗ is the long-crestedness parameter.15 Putting χ2 = 2nπ + ε, pdf for ε is p(R1 , R2 , ε|χ1 = 2nπ) p(R1 , R2 ) (

R1 R2 κ R1 R2 κ cos(ε − β) 2πI0 , = exp m0 (1 − κ2 ) m0 (1 − κ2 )

p(ε|R1 , R2 ) =

(7.83)

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where I0 [ ] is the 0th order modiﬁed Bessel function and β = tan−1 (µ4 /µ3 ). β is an angle of the dominant wave direction from x-axis. From Eq. (7.74), the pdf for the wave direction from x-axis becomes 2π y0 cos θ L (

2πy0 R1 R2 ) R 1 R2 ) 2 cos 2 , 1 − y 1 − y sin θ − β 2πI × exp 0 ∗ ∗ m0 y∗2 m0 y∗2 L (7.84)

p(θ|R1 , R2 ) =

in which L is the wave length corresponding to σ, and * ) µ2 + µ2 y∗ ≈ 1 − κ2 = 1 − 3 2 4 . m0

(7.85)

If the interval between two wave gauges is inﬁnitesimally small, i.e., y∗ → 0, R1 = R2 → R. Putting β = 0 and applying the relation z → ∞ ⇒ I0 (z) →

exp(z) , 2πz

(7.86)

Eq. (7.84) reduces to p(θ|R)dθ =

1 R2 R cos θ √ exp − 2 sin2 θ dθ . 2γ∗ m0 γ∗ 2πm0

(7.87)

Figure 7.9 shows p(θ|R) for several R. R is the mean of R. The 3D spectrum is the Bretschneider–Mitsuyasu type with Mitsuyasu-directional function modiﬁed by Goda.8 The peakedness parameter for the directional function is Smax = 10. The pdf for larger R shows a narrower width. 7.7.2. Mean length of 3D wave crest The length of 3D irregular wave crests may aﬀect the total wave force on coastal structures with ﬁnite width such as caisson-type breakwaters, for example. Since a deﬁnition of the wave crest length is diﬃcult, there are very few studies about its statistical properties. Goda6 studied its statistics numerically. He calculated a spatial wave proﬁle η(x, y) of 3D irregular waves. The black and white pattern in Fig. 7.10 shows η(x, y) > ηth , Black area , (7.88) η(x, y) < ηth , Other , in which ηth is the threshold value (ηth = 0.1H1/3 in Fig. 7.10). The 3D wave spectrum is the Bretschneider–Mitsuyasu type (H1/3 = 2.0 m, T1/3 = 8.01 s) with Mitsuyasu-directional function modiﬁed by Goda, and Smax = 25. The pattern recognition method is applied to determine the wave crest lengths. Four threshold values ηth = 0.2 m, 0.5 m, 1.0 m, and 2.0 m were used. The calculated

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Fig. 7.9.

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Pdf for wave directions: Eq. (7.87).

crest lengths were arranged in descending order. The top 0.4%, 1%, 5%, and 10% of total data for each ηth were used in the analysis, respectively. Figure 7.11 shows the distribution of the mean length of wave crests. The four lines in the ﬁgure show the results from the above four diﬀerent data. 7.7.3. Pdf of the local maximum of 3D wave amplitudes Figure 7.12 shows a short crested spatial wave proﬁle and a wave gauge. The ﬁxedtype wave gauges usually fail to record the peak wave proﬁle in the neighborhood. Wave height changes along a large object such as a ship, caisson-type breakwater, etc. If a large wave hits any part of these objects, it is recognized that the large wave attacked it regardless of its position. Since local damage on the caisson-type breakwater, for example, may lead to the whole failure, the size eﬀect on the pdf becomes important. Kimura et al.11 showed the pdf of the maximum wave amplitudes within a ﬁnite interval. x , y , and vertical z -axes are taken in the 3D wave ﬁeld. The vertical plane H is installed parallel to x -axis. Figure 7.13 schematically shows the envelope R(x ) (solid line) for the cross section of the 3D wave proﬁle on H. A wave gauge is installed at x = 0. The width of the object is 2∆L (−∆L < x < ∆L). R∗ is the local maximum amplitude of R(x ) (−∆L < x < ∆L), and R(0) is the amplitude of the envelope at x = 0. ∆R is the diﬀerence between R∗ and R(0). ∆R is approximately

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Fig. 7.10.

Fig. 7.11.

Wave crests (after Goda6,7 ).

Mean length of wave crest (after Goda6,7 ).

given as ' ' ' R (0) ' ' ' ' R (0) ' ≤ ∆L : ' ' ' R (0) ' ' ' ' R (0) ' > ∆L

∆R = −

{R (0)}2 , 2R (0)

(7.89)

R (0)(∆L)2 , 2

R (0) ≥ 0 :

∆R = R (0)∆L +

R (0) < 0 :

∆R = −R (0)∆L +

R (0)(∆L)2 , 2

(7.90) (7.91)

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Fig. 7.12.

Fig. 7.13.

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3D wave proﬁle and the vertical plane H.

Wave envelope (solid line) for 3D wave proﬁle.

in which R (0) and R (0) are the ﬁrst and second derivatives of R(x ) at x = 0, respectively. The probability that R∗ falls between c < R∗ < c + dR on condition R(0) is p[R (0), R (0) : R(0)]dS , (7.92) Prob(c < R∗ < c + dR)|R(0) = S

in which p[R (0), R (0) : R(0)] is a conditional joint pdf of R (0) and R (0) on condition R(0), and S is the region in which c < R∗ < c + dR in the R ∼ R plane. The probability that R∗ falls between c < R∗ < c + dR regardless of R(0) is Prob(c < R∗ < c + dR) = 0

c

{Prob(c < R∗ < c + dR)|R(0) }p[R(0)]dR(0) ,

(7.93)

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Fig. 7.14. Pdf of local maximum wave amplitudes (dotted lines) within three ﬁnite width ∆L = 0 (Rayleigh dist.), 0.05L1/3 , and 0.10L1/3 .

where p[R(0)] is the pdf of R at x = 0 (Rayleigh distribution). Connecting Eqs. (7.92) and (7.93), the pdf of the local maximum amplitudes within the ﬁnite width is given as R∗ p(R∗ )dR∗ = p[R(0), R (0), R (0)]dGdR(0) , (7.94) 0

G

in which p[R(0), R (0), R (0)] is the joint pdf of R(0), R (0), and R (0), and G is the region in which the local maximum amplitude falls within R∗ and R∗ + dR∗ on condition R(0). Rice20 formulated p[R(0), R (0), R (0)]. Figure 7.14 shows p(R∗ ) (dotted lines). The spectrum is the Bretschneider– Mitsuyasu-type with H1/3 = 5.5 m and T1/3 = 10 s and Goda’s directional function with Smax = 10, the water depth is h/L1/3 = 1.0, and the half-object width is ∆L = 0.050L1/3 and ∆L = 0.10L1/3, respectively. The Rayleigh distribution (solid line ∆L = 0) is also shown in the ﬁgure. Broken lines are the Weibull distribution with m ≈ 2.5 (shape factor) shows good agreement when ∆L/L1/3 > 0.05 (h/L1/3 = 1.0, Smax = 10). 7.8. Statistics of the Time Series of Wave Heights Sawhney23 and Goda4 started to investigate the statistical properties of the time series of irregular wave heights, . . . , Hi−2 , Hi−1 , Hi , Hi+1 , Hi+2 , . . . ,

(7.95)

since this property may aﬀect wave overtopping, slow drift oscillation of ﬂoating bodies, etc. This property is analyzed by applying the concept of “run.” Figure 7.15 shows a time series of wave heights. The run of high waves is deﬁned by waves, which are Hi > H∗ consecutively, in which H∗ is the threshold wave height. In the case shown in Fig. 7.15, the length of the high wave run is 3.

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Fig. 7.15.

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Run of wave heights (high waves, low waves, and total run).

Goda4 gave the pdf for the run of high waves as p(j1 ) = pj1 −1 q ,

(7.96)

in which j1 is the length of the high wave run, p is the probability that the event H > H∗ takes place and q = 1 − p. Since the pdf for wave heights is the Rayleigh distribution, p and q are given as

H2 , p = exp − ∗ 8m0 (7.97) q = 1−p. The mean and standard deviation of j1 are j1 =

1 , q

√

σ(j1 ) =

p , q

(7.98)

respectively. The recurrence interval (total run) of high waves is deﬁned as the sum of the high wave run and the consecutive low wave run. The run of low waves is deﬁned by waves, which are Hi < H∗ , consecutively. The total run in Fig. 7.15 is jt = 3 (j1 : high wave run) + 4 (j2 : low wave run) = 7. The pdf for the total run is pq p(jt ) = (pjt −1 − q jt −1 ) . (7.99) p−q The mean and standard deviation of jt are jt =

1 1 + , p q

σ(jt ) =

+

q p + 2, 2 q p

(7.100)

respectively. Equations from (7.96) to (7.100) are derived assuming that the wave height is independent: γ(Hm Hm+n ) = 0,

m, n = 1, 2, 3, . . . ,

(7.101)

where γ( ) is the correlation coeﬃcient of wave heights. From observation, Rye22 showed that there is a correlation between consecutive wave

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heights γ(Hm Hm+1 ) = 0. From numerical calculations, Kimura9 showed that γ(Hm Hm+1 ) = 0 but γ(Hm Hm+n ) ≈ 0 (n ≥ 2). He assumed the time series of wave heights from a Markov chain9 and introduced the pdf for the high wave run: p(j1 ) = pj221 −1 (1 − p22 ) .

(7.102)

p22 is the transition probability from the event Hi > H∗ to Hi+1 > H∗ . Its mean and standard deviation are √ p22 1 j1 = , σ(j1 ) = . (7.103) 1 − p22 1 − p22 The pdf for the total run is p(jt ) =

(1 − p22 )(1 − p11 ) jt −1 (p22 − pj11t −1 ) . p22 − p11

(7.104)

p11 is the transition probability from the event Hi < H∗ to Hi+1 < H∗ . The mean and standard deviation of jt are 1 1 jt = + , 1 − p22 1 − p11 (7.105) 1/2

p11 p22 σ(jt ) = + , 2 (1 − p11 ) (1 − p22 )2 where p11 and p22 are given by H∗ H∗ p(H1 , H2 )dH1 dH2 / p11 = 0

p22 =

0 ∞ H∗

p(H1 )dH1 ,

0

∞ H∗

p(H1 , H2 )dH1 dH2 /

with p(H1 , H2 ) =

H∗

H12 + H22 2 (κ2 − 1)Hrms

2H1 H2 . p(H1 ) = 2 exp − 21 Hrms Hrms 4H1 H2 exp 4 (1 − κ2 )Hrms

∞

H∗

I0

(7.106) p(H1 )dH1 ,

2κH1 H2 2 (1 − κ2 )Hrms

,

(7.107) (7.108)

Equations (7.107) and (7.108) are the 2D Rayleigh and Rayleigh distributions, respectively. k is the correlation parameter. The relation between κ and the correlation coeﬃcient γ(Hm Hm+1 ) is

1 π 1 2 γ(Hm Hm+1 ) = E(κ) − (1 − κ )K(κ) − , (7.109) 1 − π/4 2 4 in which E( ) and K( ) are complete elliptic integrals of the ﬁrst and second kind, respectively. Battjes et al.1 and Longuet-Higgins19 showed κ and the relation between κ and γ(Hm Hm+1 ) as 2 ∞ 2 ∞ 1 2 S (σ) cos στ0 dσ + S (σ) sin στ0 dσ (7.110) κ = m0 0 0

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Fig. 7.16. (a) Mean length of high wave runs (solid line: Eq. (7.103); circles: after Goda8 ). (b) Mean length of total runs (solid line: Eq. (7.105); circles: Goda used j2 for the total run, after Goda8 ).

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and κ2 ≈ γ(Hm Hm+1 ) κ2 < 0.6 , γ(Hm Hm+1 ) ≈ 1 −

1 − κ2 4−π

κ: very close to 1 .

Figures 7.16(a) and 7.16(b) show the comparison between j1 and Eq. (7.103), and jt and Eq. (7.105), respectively. The threshold wave height is H1/3 .

7.9. Remarks Statistical studies have shown us the entire structures of wave irregularities in sea conditions. A knowledge about their pdf has brought us an advance not only for designing coastal structures but also for understanding nearshore hydraulics of the wave and current system. However, we have to keep several things in mind when we use these wave statistics. The Rayleigh distribution for wave height is derived on the condition that the wave spectrum is narrow. The problem still remains to clarify a good correspondence between measured distributions and the Rayleigh distribution regardless of the spectrum. The assumption of the narrow band spectrum gives a considerable eﬀect on the pdf of wave periods. Although the theoretical models have the spectrum parameter in the equation, we have to be careful about the stretch use for the spectrum with ﬁnite band width. The range of the pdf may be changed by wave dynamics such as wave breaking. Extensive application of the theoretical pdf is sometimes risky. The applications of the periodic wave theory for zero-crossing wave properties raise problems. There have been very few studies about the physical properties of zero-crossing waves, of wave period–wavelength/celerity, of wave height/period — water particle velocities, etc. Finally, we have to take into account the statistical properties for 3D irregular waves. Common deﬁnitions for wave properties, ﬁrst of all, become necessary: for wave height, wave period, wavelength, celerity, wave direction, wave crest length, etc.

References 1. J. A. Battjes and G. Ph. van Vledder, Veriﬁcation of Kimura’s theory for wave group statistics, Proc. 19th ICCE (1984), pp. 642–648. 2. D. E. Cartwright and M. S. Longuet-Higgins, The statistical distribution of the maxima of a random function, Proc. R. Soc. Lond. A 237, 212–232 (1956). 3. G. Z. Forristall, On the statistical distribution of wave heights in a storm, J. Geophys. Res. 83(C5), 2353–2358 (1978). 4. Y. Goda, Numerical experiments on wave statistics with spectral simulation, Report PHRI 9(3), 3–57 (1970).

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5. Y. Goda, On the joint distribution of measured wave heights and periods, Tec. Note of PHRI, No. 272, (1977) 19pp. (in Japanese). 6. Y. Goda, Statistics of wave crest lengths based on directional wave simulations, J. Oﬀshore Mech. and Arctic Eng., Trans. ASME 116, 239–245 (1994). 7. Y. Goda, Numerical simulation of ocean waves for statistical analysis, J. Marine Res. 33, 5–14 (1999). 8. Y. Goda, Random Sea and Design of Maritime Structures, 2nd Edn. (World Scientiﬁc, 2000), p. 443. 9. A. Kimura, Statistical properties of random wave groups, Proc. 17th ICCE (1980), pp. 2955–2973. 10. A. Kimura and T. Ohta, Statistical property of the irregular wave heights deﬁned by the zero-crossing method, Proc. Coastal Eng. 40, 146–150 (1993) (in Japanese). 11. A. Kimura and T. Ohta, Probability distribution of the maximum wave height along a sea wall with ﬁnite width, Proc. 25th ICCE (1996), pp. 2272–2283. 12. A. Kimura, Probability distribution of 3-D irregular wave directions, Proc. Waves 97, 295–306 (1997). 13. B. M. Lake and H. C. Yuen, A new model for nonlinear wind waves, 1, Physical model and experimental evidence, J. Fluid Mech. 88, 33–62 (1978). 14. M. S. Longuet-Higgins, On the statistical distribution of the heights of sea waves, J. Marine Res. IX(3), 245–266 (1952). 15. M. S. Longuet-Higgins, The statistical analysis of a random, moving surface, Phil. Trans. Roy. Soc. Ser. A 249(966), 321–387 (1957). 16. M. S. Longuet-Higgins, On the intervals between successive zeros of a random function, Proc. Roy. Soc. Ser. A 246, 99–118 (1958). 17. M. S. Longuet-Higgins, On the distribution of the heights of sea waves: Some eﬀects of nonlinearity and ﬁnite band width, J. Geophys. Res. 85(C3), 1519–1523 (1980). 18. M. S. Longuet-Higgins, On the joint distribution of wave periods and amplitudes in a random wave ﬁeld, Proc. Roy. Soc. Ser. A 389, 241–258 (1983). 19. M. S. Longuet-Higgins, Statistical properties of wave groups in a random sea state, Phil. Trans. Roy. Soc. Ser. A 312, 219–250 (1984). 20. S. O. Rice, The mathematical analysis of random noise, Bell System Tech. J. 23, 282–332, 24, 46–156 (1944). 21. S. O. Rice, Selected papers on Noise and Stochastic Processes, ed. N. Wax (Dover, 1945), pp. 133–294. 22. H. Rye, Wave group formation among storm waves, Proc. 14th ICCE (1974), pp. 164–183. 23. M. D. Sawhney, A study of ocean wave amplitudes in terms of the theory of runs and a Markov chain process, Tech. Rep. of New York University (1962), p. 29. 24. R. J. Sobey, Correlation between individual waves in a real sea states, Coastal Eng. 27, 223–242 (1996). 25. M. A. Tayfun, Eﬀects of spectrum band width on the distribution of wave heights and periods, Ocean Eng. 10(2), 107–118 (1983).

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Chapter 8

Generation and Prediction of Seiches in Rotterdam Harbor Basins M. P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology; Presently at Deltares |Delft Hydraulics P. O. Box 177, 2600 MH, Delft, The Netherlands J. A. Battjes Environmental Fluid Mechanics Section Delft University of Technology, P. O. Box 5048, 2600 GA Delft, The Netherlands A resume is presented of recent work that has identiﬁed the mechanism through which the seiches are generated that occasionally occur in Rotterdam Harbor basins, and that has led to the development and implementation of an operational warning system for the occurrence of signiﬁcant seiche events. The generation is due to moving systems of atmospheric convection cells that can arise over the relatively warm water of the North Sea, behind a cold front. These cause ﬂuctuations in wind speed and atmospheric pressure that can generate long waves at sea, as these cells move toward the coast. The average variance spectrum of these long-wave surface elevations is found to have an approximately f −1.5 tail. As these long waves at sea approach the harbor mouth, they can be resonantly ampliﬁed inside certain semi-closed basins. The ratio of seiche amplitudes in two diﬀerent basins, derived through numerical simulations based on a 2DH mildslope model and an incident f −1.5 long-wave elevation spectrum, is in excellent agreement with the mean ratio derived from observations for the same locations. Finally, the principle of a possible system of prediction of the occurrence of seiche events is described. Its predictive potential is proven through a series of hindcasts. Based on these promising results, an operational forecasting system has been developed and implemented by the authorities involved.

8.1. Introduction Resonant harbor oscillations (seiches) occur in a number of basins of the Port of Rotterdam, The Netherlands, during rough weather episodes. Within the Rotterdam 179

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Fig. 8.1.

The Port of Rotterdam.

Harbor system, the highest seiches occur in the Caland Canal, with measured crest heights over 1 m at the closed end near Rozenburgse Sluis, ROZ (see Fig. 8.1). This semi-closed basin has a length of approximately 20 km and a depth of approximately 20 m. For almost all of the observed seiche events at ROZ, the dominant frequency (determined through wavelet analysis) equals the lowest eigenfrequency (quarter-wavelength mode), equivalent to a period of approximately 90 min. Smaller-amplitude seiches, though still quite signiﬁcant, also occur in other harbor basins. The seiche oscillations in Rotterdam Harbor give rise to some practical concerns. First, water levels used for the design of dikes (in particular their crest height) are required by law to contain an allowance for seiches, superposed on the astronomical high water level (spring tide) and wind setup. The appropriate measure of this allowance is not well known. Second, seiches aﬀect the design and management of a movable storm surge barrier (MSSB) in Rotterdam Waterway (Fig. 8.1), which protects the low areas behind it against ﬂooding. The MSSB must be closed during extreme storm conditions. In that case a semi-closed basin is created on its seaside. Seiches are expected to occur in this temporary basin, with a dominant eigenperiod of approximately 30 min. These contribute to the failure probability of the barrier, but their expected heights were not known. Third, the Rotterdam Port Management needs to guarantee a minimum depth (“nautical depth”) in the approach channel and throughout the harbor for deepdraught vessels. To this end, water levels are predicted operationally including tides and wind-induced setup. Until recently, no method was available for inclusion of seiche events in this short-range prediction, despite their practical importance: the trough of a seiche could cause ships to run aground in case of a narrow predicted tidal window. The aforementioned practical applications in which seiches play a role have led to a recent study on the origin of the seiche events in Rotterdam. The present chapter summarizes the ﬁndings. Results are also presented of an analysis of the response of Rotterdam Harbor basins to incoming wave energy in the seiche frequency band using measured and theoretical ratios of seiche amplitudes at the closed ends of two basins. Finally, a new system of routine prediction of seiche events is described,

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based on the newly identiﬁed generation mechanism, as well as an indication of its forecasting skill. The above-mentioned research and results have been more fully described by de Jong and Battjes.1,2

8.2. Measurements 8.2.1. Data acquisition The sea surface elevation measurements used in this study were obtained at three oﬀshore platforms located approximately 20–40 km from the harbor mouth: Europlatform (EUR), Lichteiland Goeree (GOE), and Meetpost Noordwijk (MPN). Inside the harbor, surface elevation measurements are available from multiple locations (more than 10 key-locations are monitored). These have been taken continuously since 1995 with ﬂoaters, sampled at a 60 s interval. In addition, sea surface temperature measurements have been obtained from 1995 to 2001 at platforms located several hundreds of kilometers to the north and north-west of the harbor mouth in the North Sea (along the path of the relevant storm systems approaching the Port of Rotterdam). These were available as 24-h averaged daily values. Vertical proﬁles of air temperature and other meteorological parameters, obtained from balloon measurements, were available at the North Sea at Ekofisk platform every 12 or 24 h. 8.2.2. Data analysis To identify seiche events, records of the surface elevations at ROZ were Fourier band-passed ﬁltered, passing the seiche frequency band (0.1–2.0 mHz). A wavelet analysis based on the Morlet wavelet3−6 has been applied to the ﬁltered surface elevation data and meteorological data. This technique is suited for the detection of ﬂuctuations that come in bursts. It has been used to identify the seiche events inside the harbor and the corresponding periods of increased levels of low-frequency energy at sea. Following ﬁrst identiﬁcation from the wavelet spectra, the time intervals with seiche events were reviewed in more detail.

8.3. Origin of the Seiches in Rotterdam Harbor 8.3.1. Generation of seiche events by atmospheric convection cells In this section, a short description is given of the mechanism generating the long waves at sea that cause seiche episodes inside the harbor (as mentioned above, a more detailed description can be found in the work of de Jong and Battjes1,2 ). Meteorological measurements and visual inspection of weather charts show that all seiche episodes in Rotterdam Harbor (arbitrarily deﬁned as a seiche-event in which the crest height exceeded 0.25 m at ROZ, a criterion that was applied in several previous studies of seiches at this location) coincided with low-pressure

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systems with cold fronts crossing the southern North Sea toward the Dutch coast. However, the converse does not hold: not all such cold front passages give rise to seiche events. Analysis of measurements and numerical simulations showed that the seiches in Rotterdam Harbor result from resonant ampliﬁcation of oscillations at sea, which in turn are generated by ﬂuctuations in atmospheric pressure and wind speed following a moving cold front. For the large majority of the seiche events, these ﬂuctuations are caused by atmospheric convection cells that can arise in the area behind a cold front. These cells are a mesoscale atmospheric phenomenon, with typical horizontal spatial scales of 30–100 km.7 Conditions conducive to the formation of convection cells occur when cold (polar) air moves over relatively warm sea water, which is likely to occur in fall and winter (also the seasons of signiﬁcant seiche episodes). Especially in the area behind the cold front, the relatively warm sea water causes heating of the air in the lower layers, which results in an unstable lower atmosphere that causes air to rise to higher altitudes in narrow plumes, (self-)organized into a more or less regular pattern of convection cells. If suﬃcient moisture is present in the air, condensation of water vapor can locally cause a release of latent heat, further driving the circulation of air. Narrow bands of clouds can then be formed surrounding clear skies, which can be seen on satellite images (an example is shown in Fig. 8.2) and, in case of precipitation,

Fig. 8.2. Satellite image of infrared frequency band (10:12 GMT, 11 January 1995), taken during a seiche event in the Port of Rotterdam. The image shows a pattern of convection cells that passes the southern North Sea from the northwest. The location of the Port of Rotterdam is indicated by the black and white dot. Source image: Dundee Satellite Receiving Station, Dundee University, Scotland.

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cold front advection speed

2−4 km

warm air

cold air 30 −100 km

Fig. 8.3. Sketch of convection cells in the area behind a cold front together with a theoretical surface elevation response.

also on radar precipitation images. These satellite and precipitation images showed that the episodes of convection cells passing the southern North Sea more or less coincided with the seiche episodes. The convection cells cause more or less spatiallyperiodic ﬂuctuations in atmospheric pressure and wind speed, which in a ﬁxed point will manifest themselves as temporal ﬂuctuations as the cells are advected with the weather system. Because the system of convection cells moves, these ﬂuctuations can generate low-frequency waves at sea, which in turn can cause seiches in the harbor basins. A sketch of the sea surface elevation response to the wind speed changes caused by the convection cells is depicted in Fig. 8.3. A 1D model based on the linearized shallow water equations was applied for a transect across the North Sea along the main path of the convection cells, forced by sinusoidal variations in atmospheric pressure and wind speed translating at a constant speed, with values that are typical for those observed. Negligibly low responses were found in the more remote, deeper parts of the North Sea. However, the model showed that the ﬂuctuations in wind speed and atmospheric pressure caused by the passing convection cells are particularly eﬃcient at generating long waves with signiﬁcant amplitudes in the relatively shallow part of the North Sea, the southernmost 400 km near the Dutch coast. This is because in this region, with typical depths of approximately 30 m, the phase speed of shallow-water waves is approximately 17 m/s, i.e., near the advection velocity of the system of cold front and convection cells (typically 15–20 m/s). The result is a near-resonant response of the long waves at sea to the moving pattern of atmospheric convection cells, with amplitudes up to 1–2 dm near the Dutch coast. These waves can in turn be ampliﬁed in semi-closed harbor basins through (a quarter-wave) resonance, occasionally forming seiches with observed amplitudes (crest heights) up to a meter. 8.3.2. Energy density spectrum of surface elevation at sea Most previous studies of the seiches in the Port of Rotterdam focused on the relative responses in diﬀerent basins. To this end, and for lack of better information, those studies assumed a uniform incident energy density spectrum (a “white noise” spectrum). Veraart8 ﬁrst studied the spectra of measured surface elevations at the

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23/24 February 2001 (m0 = 1.5 cm ) 2 30 October 2000 (m0 = 3.4 cm ) 2 21 March 2001 (m0 = 0.40 cm ) f −1.5 trend

10

3

10

2

2

variance density (cm /Hz) normalized by m0 (cm )

184

2

10

1

10

−1

10

10

0

Frequency (mHz) Fig. 8.4. Three normalized variance density spectra of surface elevation measurements at the EUR platform at sea. Nominal cut-oﬀ frequencies are indicated by the thick dashed vertical lines at 0.1 mHz and 2.0 mHz. The thick line represents the best-ﬁt power law of f −1.5 .

southern North Sea in the seiche frequency band, obtained during seiche events at ROZ. Based on a relatively small number of seiche episodes he found that, on average, the variance density spectrum in the seiche frequency band approximately followed a f −1.2 trend. From June 2000, numerous processed and validated surface elevation data have become available for a number of storm events. These have now been used to determine the spectrum in the seiche frequency band at sea in more detail (results described here are based on measurements from the platform EUR). The variance density spectra have been estimated from ﬁltered records (0.1– 2.0 mHz) of 4096 min (212 values) approximately centered around the seiche-event, covering the complete storm duration of each event. Spectra of a number of events normalized by the energy content of the seiche frequency band are shown in Fig. 8.4. A least-squares ﬁt of a power law to the averaged normalized spectral densities results in a f −1.5 trend.

8.4. Seiche Amplitudes at Diﬀerent Locations 8.4.1. Seiche events in diﬀerent basins of the Port of Rotterdam A total of 49 seiche events were registered at ROZ in the studied time interval (1995–2001). For this location, each such event is deﬁned as a period of enhanced

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seiche activity with amplitudes (crest heights) exceeding 0.25 m. During most of these events, the response at the lowest eigenperiod of the Caland Canal (90 min) was dominant. Surface elevation measurements at Europe Harbor (EH, see Fig. 8.1) have also been studied for the same time interval as the observations at ROZ (1995–2001). These time series were ﬁltered for the frequency band 0.3–2.0 mHz in order to isolate the main eigenfrequencies of this (shorter) harbor basin. Based on a wavelet analysis, and using 0.1 m as an amplitude threshold, 46 seiche events at EH were identiﬁed, all at the dominant eigenperiod of 50 min. Measurement data at ROZ were available during 43 of the 46 events that were identiﬁed at EH. Of these 43 events at EH, 86% coincided with a seiche-event at ROZ. This percentage depends to some extent on the thresholds for the identiﬁcation of a seiche-event. These have been chosen subjectively: in case a threshold of 0.15 m is chosen for the identiﬁcation of seiche events at EH, 100% of the events at EH coincide with a seiche-event at ROZ (with amplitude exceeding 0.25 m). Conversely, of the 49 events at ROZ, 82% coincided with a seiche-event at EH. In this case too, this result depends on the threshold of the deﬁnition of a seicheevent. In case a threshold of 0.40 m is applied at ROZ, 100% of the events at ROZ coincided with an event at EH (with a threshold of 0.10 m). The fact that large-amplitude seiche events simultaneously occur in basins of diﬀerent dimensions indicates that the same mechanism generates seiches in basins of diﬀerent natural frequencies. 8.4.2. Empirical amplitude ratio Surface elevation signals from ROZ and EH have been ﬁltered for the complete seiche frequency band (0.1–2.0 mHz) in order to compare the crest heights during coinciding seiche events in the Caland Canal and the Europe Harbor. For this purpose, a typical local seiche amplitude could be based on, for example, the standard deviation of the ﬁltered surface elevation signals. However, the time span for which this value needs to be determined (for each event separately) is not obvious. This is problematic in view of the transient nature of the seiche events. Therefore, the maximum crest height of an event at each location has been chosen. Moreover, in order to get a more robust estimate, the average of the ﬁve largest crest heights and the ﬁve largest trough depths of an event has also been used. The latter parameter is taken as a typical average seiche amplitude of an event. Figure 8.5 shows these average seiche amplitudes for the 30 coinciding events in 1995–2001 for ROZ and EH for which the average amplitude could be determined from at least ﬁve crests and ﬁve troughs. These data have a correlation coeﬃcient of 0.92. A linear trend line was least-squares ﬁtted to the data, indicating that on average the seiche amplitudes at EH are a factor of approximately 0.56 smaller than those at ROZ. A scatter plot based on the single maximum crest height for all coinciding events (not shown) gave a similar result, though with some more scatter (correlation coeﬃcient 0.7).

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average seiche amplitude EH (cm)

40

30

20

10

y=0.5596x 0 0

10

20 30 40 50 average seiche amplitude ROZ (cm)

60

70

Fig. 8.5. Scatter plot of average seiche amplitudes (the average of the ﬁve highest crests and ﬁve deepest troughs of an event) of coinciding seiche events at ROZ and Europe Harbor. Trend line based on least-squares estimate.

8.4.3. Theoretical amplitude ratio The calculation of theoretical seiche amplitude ratios (r) at two locations a and b is based on a combination of the energy supply at the harbor mouth and its ampliﬁcation according to S(f )Ra2 (f )df , (8.1) rab = S(f )Rb2 (f )df where S(f ) is the (normalized) energy spectrum incident from the sea at the harbor mouth and Ra (f ) and Rb (f ) are the amplitude ampliﬁcation spectra for the considered locations a and b. These had been determined previously using a 2DH mildslope numerical wave model (PHAROS).9 Eﬀects of bottom stress, inﬂow losses, and seaward radiation were included. Applying these response functions and the empirical seiche energy spectrum with a f −1.5 tail, the theoretical amplitude ratio between EH and ROZ is found to be 0.57, nearly equal to the value of 0.56 that was estimated from the measurements. This result, based on 30 seiche events, supports the validity of the theoretical approach, and it conﬁrms that the energy density at sea in the seiche frequency band typically follows a f −1.5 trend, not just in the three events from which it was estimated.

8.5. Prediction of Occurrence of Seiche Events 8.5.1. Principle of prediction and ﬁrst veriﬁcation As noted in the Introduction, practical needs related to shipping or to the movable storm surge barrier require real-time knowledge of expected occurrences of seiches

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in Rotterdam Harbor. However, operational prediction of these cannot be based on measurements at available platforms at the North Sea, because the response at the remote stations is too low, whereas the nearby platforms are too close to allow a useful prediction window. Since moving convection cells are at the basis of the generating mechanism of seiche events in Rotterdam Harbor, circumstances needed for these cells to arise can, in principle, be used to predict the occurrence, and also the absence, of seiche events. The feasibility of this approach was shown by de Jong and Battjes2 using historical data. Based on this promising academic result, a pilot study has been set up using real-time data and weather predictions, as a precursor of a fully operational system. This conﬁrmed that the method was indeed feasible and that it had suﬃcient forecasting skill to warrant implementation of an operational version. Such a system, in which predictions with windows of 12 h and 24 h are provided to relevant parties throughout the harbor, has now been in operation since 2005 and so far has functioned to satisfaction. The basis for the prediction system is as follows. Convection cells of suﬃcient strength arise only if the lower atmosphere is (suﬃciently) unstable. The criterion for this is that the adiabatic lapse rate (rate of decrease in temperature from the sea surface upward) exceeds some threshold, the value of which depends on the moisture content because of its consequence for possible cloud formation. In case the lapse rate is smaller than 5.5 to 7.5◦ C/km, the lower atmosphere is stable; here 6.5 C/km is used as a representative critical value. Between approximately 10◦ C/km and (say) 6.5◦ C/km, it is conditionally unstable (depending on the moisture content), and a temperature lapse rate larger than 10◦ C/km results in an unstable atmosphere.10 Prediction of the lapse rate and moisture content using multi-layered atmospheric models is part of routine weather forecast systems. This, therefore, also allows operational prediction of the occurrence of signiﬁcant seiche events on a routine basis, just like common weather prediction. To test the feasibility of this approach, several months of atmospheric temperature and pressure data as well as sea surface water temperature data have been used for “predictions” in a hindcast mode, comparing the outcomes to observations of occurrences (or not) of seiche events. Results of a three-month period are shown here as a typical example of the outcome. The lapse rate can in practice be estimated using the air temperature at sea level (provisionally taken equal to the sea surface water temperature) and at some higher elevation. In meteorology, standard reference levels in the boundary layer are used corresponding to an atmospheric pressure of 925 hPa (average altitude of about 700 m) and 850 hPa (average altitude of about 1500 m). At the latter altitude, cloud formation can occur, which would result in the release of latent heat, implying inﬂuence of moisture content on the possible generation of convection cells. Indeed, preliminary tests showed that correct predictions as well as false predictions are found if the average altitude of the reference pressure of 850 hPa is used and moisture is not taken into account. However, generally speaking, the air will not become saturated below the level of the 925 hPa pressure. Therefore, cloud formation will not occur below this level. In this case, no information about moisture content of the air is required because

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the uncertain range of conditional stability is excluded. A lower limit of the lapse rate for unconditional instability of 10◦ C/km then applies. The following “predictions” use this criterion and the estimated temperature gradient between sea level and the observed level of 925 hPa pressure. For comparison, results based on the average elevation of this pressure level (approximately 700 m) are shown as well. The latter would be simpler operationally, albeit at the cost of enhanced errors in the veriﬁcation of the temperature diﬀerence criterion. The results of the aforementioned approach are illustrated by presenting the data of January till March 1999. This time interval was selected because the quality of the data was relatively high during these months (only a small percentage of missing data due to, e.g., atmospheric soundings that were not available on certain days). Weather maps have been used to identify the days during which cold fronts have passed the North Sea that resulted in a ﬂow toward the Dutch coast (including possible successive days during which this ﬂow direction continued). This is obviously needed for seiche events to be generated in Rotterdam Harbor. Figure 8.6 shows the results. The bars in the top panel show the observed difference between the air temperature at the (observed) altitude of the reference pressure 925 hPa (at platform EKO) and the sea surface water temperature (at platform AUK) for each day (not restricted to days of a front passage). Gaps are present for those days for which no data were available. The lower panel shows the

temp. difference ( °C)

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10 15 20 25 Feb

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10 15 20 25 30 Mar

temp. difference ( °C)

15 unstable

10

5 stable

0

5

10 15 20 25 30 Jan

5

10 15 20 25 Feb

5

10 15 20 25 30 Mar

Fig. 8.6. The bars indicate the observed temperature diﬀerence between the surface sea water at AUK and the air at 925 hPa level at Ekoﬁsk for January till March 1999. Top panel: all days; bottom panel: days of occurrence of a cold front that moved from the North Sea toward the Dutch coast. Straight dashed lines: critical temperature diﬀerence for an unstable atmosphere based on average elevation of 925 hPa. Wiggly curves: values of the critical temperature diﬀerence for an unstable atmosphere from day to day, based on an observed elevation of 925 hPa.

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same data, now restricted to days during which cold fronts have passed the North Sea toward the Dutch coast. The straight dashed lines in both panels indicate the critical temperature difference for an unstable atmosphere based on the average altitude of the pressure level of 925 hPa (∆T = 7◦ C). The wiggly curves in both panels indicate the value of the critical temperature diﬀerence for an unstable atmosphere from day to day, based on the temperature at the measured altitude of the reference pressure of 925 hPa for each day. A seiche-event is predicted for those days in the lower panel for which the observed temperature diﬀerence (the bar) exceeds the critical one (indicated by the wiggly curve). The asterisks indicate the days for which seiche events have been identiﬁed at ROZ. (Consecutive days with seiche events indicated represent a single event that started during the ﬁrst day and ended on the last day indicated.) The bottom panel shows that an unstable atmosphere is found during a number of potentially seiche-prone cold-front situations, correctly predicting a seicheepisode. All the seiche events that occurred in this time interval are correctly predicted (no misses). A stable atmosphere is found during most other days indicated in the bottom panel and no convection cells are expected during these days (regardless of the availability of moisture in the air), which agrees with the fact that no seiche events occurred. Nevertheless, a limited number of false predictions is also found (17 January and 28 February). On these days, the temperature diﬀerence just slightly exceeded the critical value. The resulting false predictions could therefore easily be due to the criterion not being strict enough, or to the spatial variability of the temperature of the sea surface water and the air at the reference levels: although the nominal criterion is met locally, this does not necessarily have to be representative for the complete southern North Sea region since only observations from sparse locations are used. This will not be the case during the correctly predicted seiche events in view of the fact that the prediction on these days is based on relatively large temperature diﬀerences, which far exceed the critical values. 8.5.2. Operational seiche prediction The evidence presented above indicates that conditions for the existence of convection cells, together with the information regarding the passage and direction of cold fronts, can be used to identify and predict seiche-prone situations. In an operational setting, the predicted weather chart for the next 24 h is available with relatively high reliability, and the predicted air temperature can be obtained from numerical weather prediction models. The moisture content of the air could also be taken into account. Moreover, a more thorough analysis of the stability of the lower atmosphere is possible, since information of the complete vertical and for the complete southern North Sea area is available (possibly avoiding false alarms due to criteria that are only met locally). This makes the use of speciﬁc reference levels unnecessary. Therefore, all relevant parameters are available for the prediction of seiche-prone situations with a 24-h prediction window, suﬃcient for the port

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authorities and for the closure management of the movable storm surge barrier of Rotterdam (which typically require prediction windows of, say, 6 h). Based on the aforementioned ﬁndings, a pilot project was implemented, set up by the Ministry of Public Works and Waterways in cooperation with the Royal Dutch Meteorological Institute (KNMI) for operational prediction of the occurrence of seiche events in the Port of Rotterdam, using parameter values derived from operational numerical weather prediction models. This pilot project has shown that the system has suﬃcient predictive skill for operational use. Such a system has since been implemented on an operational basis. Experience shows that all major seiche events (above the seiche-amplitude threshold) have been successfully predicted. However, in some cases a false prediction was made in the sense that a warning was issued while no signiﬁcant seiches occurred. This is a drawback that needs repair, but even as it is, the system has merit because it does not miss signiﬁcant events. The false alarms mentioned above are related to an important limitation of the above-described method, viz, the fact that so far it only predicts seiche-prone situations but not the expected energy level of the seiche-event. In case additional parameters are taken into account, such as the (range in) sizes of the atmospheric convection cells, the advection velocity, the excess of the temperature diﬀerence above the critical value (not just the question whether it exceeds a critical value or not), and the direction of movement of the weather systems, prediction of the seiche energy level might become possible, for example, through the use of a neural network that can be trained with a set of observed values of these quantiﬁable parameters. This may open the possibility of operational forecasting of seiche-prone conditions together with expected seiche crest heights. In that case, a seiche prediction (warning) would only be issued when seiche-conducive atmospheric conditions are expected that can lead to a seiche-event with amplitudes which justify a warning.

8.6. Conclusions Fluctuations in atmospheric pressure and wind speed, induced by mesoscale atmospheric convection cells that can arise following a cold front passing southward over the North Sea, generate low-frequency gravity waves in the North Sea. These low-frequency waves can subsequently induce a resonant response in coastal harbor basins, thus causing the so-called seiche events. During episodes of enhanced seiche activity in Rotterdam Harbor, the surface elevation energy spectrum at sea in the frequency band corresponding to the typical eigenfrequencies in Rotterdam Harbor basins, say, 0.1–2.0 mHz, varies with frequency (f ) approximately as f −1.5 . An episode of enhanced wave energy at sea in the seiche frequency band causes simultaneous resonant seiche responses in basins with diﬀerent natural frequencies. The ratio between seiche amplitudes in two different basins can be accurately predicted using an f −1.5 excitation spectrum and numerically calculated spectral transfer functions. Operational prediction of the occurrence of a seiche-episode cannot be based on measurements at available platforms at sea, because at the more remote stations

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the signal is too weak, and for the nearby stations the available prediction window is too short to be of practical use. A hindcast study has shown that the occurrence of signiﬁcant seiche episodes in Rotterdam Harbor can be accurately predicted on the basis of the prediction of the occurrence of meteorological conditions conducive to the formation of mesoscale atmospheric convection cells behind a cold front. This is presently realized using output of air temperatures and moisture content from numerical atmospheric models that are used in operational weather forecasting. The outcomes of this prediction method are used by the port authorities for ship traﬃc control and, after further development and testing, can eventually be incorporated into the closuremanagement system of the storm surge barrier in Rotterdam Waterway.

Acknowledgments The ﬁrst author acknowledges the ﬁnancial support from the Dr Ir Cornelis Lely Foundation. A part of this study was ﬁnanced by the Directorate-General for Public Works and Water Management, South-Holland Directorate (Contract number: ZHA 23735). Furthermore, the authors thank the Directorate-General for Public Works and Water Management and the Municipal Harbor Organization, Rotterdam for providing the surface elevation measurements at sea and in the harbor, respectively. The authors thank Dr A. J. van Delden of Utrecht University, The Netherlands, for his extensive information on convection cells. Atmospheric sounding data have been obtained from the Department of Atmospheric Science, University of Wyoming, USA. Wavelet software was provided by Torrence and Compo,6 through URL: http://paos.colorado.edu/research/wavelets/. The ﬁgures in this chapter are reprinted from de Jong and Battjes,2 with permission from Elsevier.

References 1. M. P. C. de Jong and J. A. Battjes, Low-frequency sea waves generated by atmospheric convection cells, J. Geophys. Res. 109(C1), C01011 (2004), doi: 10.1029/2003JC001931. 2. M. P. C. de Jong and J. A. Battjes, Seiche characteristics of Rotterdam Harbour, Coastal Eng. 51(5–6), 373–386 (2004). 3. J. Morlet, G. Arens, E. Furgeau and D. Giard, Wave propagation and sampling theory — Part 2: Sampling theory and complex waves, Geophysics 47(2), 222–236 (1982). 4. M. Farge, Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24, 395–457 (1992). 5. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, San Diego, 1998). 6. C. Torrence and G. P. Compo, A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc. 79, 61–78 (1998). 7. B. W. Atkinson and J. W. Zhang, Mesoscale shallow convection in the atmosphere, Rev. Geophys. 34(4), 403–431 (1996).

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8. C. Veraart, Events of Long Waves in the North Sea, Technical Report RIKZ-94.034, Rijkswaterstaat/RIKZ and Delft University of Technology, Delft, The Netherlands (1994). 9. P. Van den Bosch and J. J. Veldman, Europoortkering-Beerdam, Technical Report H2438, WL | Delft Hydraulics, Delft, The Netherlands (1995). 10. J. R. Holton, An Introduction to Dynamic Meteorology, 3rd edn. (Academic Press, London, 1992).

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Chapter 9

Seiches and Harbor Oscillations Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences 36 Nakhimovsky Prosp., Moscow, 117997 Russia Department of Fisheries and Oceans, Institute of Ocean Sciences 9860 West Saanich Road, Sidney, BC, Canada V8L 4B2 [email protected] [email protected] This chapter presents an overview of seiches and harbor oscillations. Seiches are long-period standing oscillations in an enclosed basin or in a locally isolated part of a basin. They have physical characteristics similar to the vibrations of a guitar string or an elastic membrane. The resonant (eigen) periods of seiches are determined by basin geometry and depth and in natural basins may range from tens of seconds to several hours. The set of seiche eigen frequencies (periods) and associated modal structures are a fundamental property of a particular basin and are independent of the external forcing mechanism. Harbor oscillations (coastal seiches) are a speciﬁc type of seiche motion that occur in partially enclosed basins (bays, fjords, inlets, and harbors) that are connected through one or more openings to the sea. In contrast to seiches, which are generated by direct external forcing (e.g., atmospheric pressure, wind, and seismic activity), harbor oscillations are mainly generated by long waves entering through the open boundary (harbor entrance) from the open sea. Energy losses of seiches in enclosed basins are mostly due to dissipative processes, while the decay of harbor oscillations is primarily due to radiation through the mouth of the harbor. An important property of harbor oscillations is the Helmholtz mode (pumping mode), similar to the fundamental tone of an acoustic resonator. This mode is absent in a closed basin. Harbor oscillations can produce damaging surging (or range action) in some ports and harbors yawing and swaying of ships at berth in a harbor. A property of oscillations in harbors is that even relatively small vertical motions (sea level oscillations) can be accompanied by large horizontal motions (harbor currents), resulting in increased risk of damage of moored ships, breaking mooring lines as well as aﬀecting various harbor procedures. Tsunamis constitute another important problem: catastrophic destruction may occur when the frequencies of arriving tsunami waves match the resonant frequencies of the harbor or bay. Seiches, as natural resonant oscillations, are generated by a wide variety of mechanisms, including tsunamis, seismic ground waves, internal ocean waves, and 193

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jet-like currents. However, the two most common factors initiating seiches are atmospheric processes and nonlinear interaction of wind waves or swell. At certain places in the World Ocean, waves due to atmospheric forcing (atmospheric gravity waves, pressure jumps, frontal passages, squalls) can be responsible for signiﬁcant, even devastating harbor oscillations, known as meteorological tsunamis. They have the same temporal and spatial scales as typical tsunami waves and can aﬀect coasts in a similar damaging way.

9.1. Introduction Seiches are long-period standing oscillations in an enclosed basin or in a locally isolated part of a basin (in the Japanese literature they are commonly known as “secondary oscillations (undulations) of tides”).30,56,59 The term “seiches” apparently originated from the Latin word siccus which means dry or exposed (from the exposure of the littoral zone at the down-swing).31,95 Free-surface oscillations, known as seiches or seiching in lakes and harbors or as sloshing in coﬀee cups, bathtubs, and storage tanks, have been observed since very early times; a vivid description of seiching in Lake Constance, Switzerland, was given in 1549, and the ﬁrst instrumental record of seiches obtained in 1730 in Lake Geneva.46,95 Korgen34 describes seiches as “the rhythmic, rocking motions that water bodies undergo after they have been disturbed and then sway back-and-forth as gravity and friction gradually restore them to their original, undisturbed conditions.” These oscillations occur at the natural resonant periods of the basin (so-called “eigen periods”) and physically are similar to vibrations of a guitar string and an elastic membrane. The resonant (eigen) periods of seiches are determined by the basin geometry and depth94,95 and in natural basins may be from a few tens of seconds to several hours. The oscillations are known as natural (or eigen) modes. The mode with the lowest frequency (and thus, the longest period) is referred to as the fundamental mode.41 The set of seiche eigen frequencies (periods) and associated modal structures are a fundamental property of a particular basin and are independent of the external mechanism forcing the oscillations. In contrast, the amplitudes of the generated seiches strongly depend on the energy source that generates them, and can therefore have pronounced variability.31 Resonance occurs when the dominant frequencies of the external forcing match the eigen frequencies of the basin. Harbor oscillations (coastal seiches according to Ref. 19 are a speciﬁc type of seiche motion that occur in partially enclosed basins (gulfs, bays, fjords, inlets, ports, and harbors) that are connected through one or more openings to the sea.41,94 Harbor oscillations diﬀer from seiches in closed water bodies (for example, in lakes) in three principal ways68 : (1) In contrast to seiches generated by direct external forcing (e.g., atmospheric pressure, wind, and seismic activity), harbor oscillations are mainly generated by long waves entering through the open boundary (harbor entrance) from the open sea.

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(2) Energy losses of seiches in closed basins are mostly associated with dissipation, while the decay of harbor oscillations is mainly due to radiation through the mouth of the harbor. (3) Harbor oscillations have a speciﬁc fundamental mode, the Helmholtz mode, similar to the fundamental tone of an acoustic resonator.54 This modes is absent in closed basins. Because harbor oscillations can produce damaging surging (or range action) — yaw and swaying of ships at berth in a harbor — this problem has been extensively examined in the scientiﬁc and engineering literature.4,11,12,36,41,46,47,62,67,68,70,78,79, 94,95 One of the essential properties of oscillations in harbors is that even relatively small vertical motions (sea level oscillations) can be accompanied by large horizontal water motions (harbor currents); when the period of these motions coincides with the natural period of sway, or yaw of a moored ship, further resonance occurs, which can result in considerable motion and possible damage of a moored ship.82,94 Harbor oscillations can also break mooring lines, cause costly delays in loading and unloading operations at port facilities, and seriously aﬀect various harbor procedures.79,80 Tsunamis constitute another important problem that have greatly stimulated investigations of harbor oscillations. Professor Omori (Japan) was likely the ﬁrst to notice in 1902 that the dominant periods of observed tsunami waves are normally identical to those caused by ordinary long waves in the same coastal basin (see Ref. 30). His explanation was that the bay or portion of the sea oscillates like a ﬂuid pendulum with its own period, i.e., the arriving tsunami waves generate similar seiches as those generated by atmospheric processes and other types of external forcing (see also Ref. 30). Numerous papers on the spectral analysis of tsunami records for various regions of the world ocean have conﬁrmed this conclusion.13,48,69,74,75,84,90 Catastrophic destruction may occur when the frequencies of arriving tsunami waves match the resonant frequencies of the harbor or bay. One of the best examples of strong tsunami ampliﬁcation due is the resonant response of Port Alberni (located at the head of long Alberni Inlet on the Paciﬁc coast of Vancouver Island, Canada) to the 1964 Alaska tsunami.28,54 9.2. Hydrodynamic Theory The basic theory of seiche oscillations is similar to the theory of free and forced oscillations of mechanical, electrical, and acoustical systems. The systems respond to an external forcing by developing a restoring force that re-establishes equilibrium in the system. A pendulum is a typical example of such a system. Free oscillations occur at the natural frequency of the system if the system disturbed beyond its equilibrium. Without additional forcing, these free oscillations retain the same frequencies but their amplitudes decay exponentially due to friction, until the system eventually comes to rest. In the case of a periodic continuous forcing, forced oscillations are produced with amplitudes depending on friction and the proximity of the forcing frequency to the natural frequency of the system.84

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9.2.1. Long and narrow channel Standing wave heights in a closed, long and narrow nonrotating rectangular basin of length, L, and uniform depth, H, have a simple trigonometric form35,95 : ζ(x; t) = A cos kx cos ωt,

(9.1)

where ζ is the sea level elevation, A is the wave amplitude, x is the along-basin coordinate, t is time, k = 2π/λ is the wave number, λ is the wavelength, ω = 2π/T is the angular wave frequency, and T is the wave period. The angular frequency and wave number (or the period and wavelength) are linked through the following well-known relationships: ω = kc, λ T = , c

(9.2a) (9.2b)

√ where c = gH is the long-wave phase speed and g is the gravitational acceleration. The condition of no-ﬂow through the basin boundaries (x = 0; x = L) yields the wave numbers: k=

π , L

2π , L

3π nπ ,..., , L L

(9.3)

which are related to the speciﬁc oscillation modes [Fig. 9.1(a)], i.e., to the various eigen modes of the water basin. The fundamental (n = 1) mode has a wavelength equal to twice the length of the basin; a basin oscillating in this manner is known as a half-wave oscillator.34 Other modes (overtones of the main or fundamental

Closed basin

Open-ended basin

n=1

n=0

n=2

n=1

n=3

n=2

n=4

n=3

0.00

1.00 0.00

L

1.00

L

Fig. 9.1. Surface proﬁles for the ﬁrst four seiche modes in closed and open-ended rectangular basins of uniform depth.

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√ Table 9.1. Normalized periods, Tn∗ = Tn gH/(2L), for a closed and open-mouth rectangular basin of uniform depth. Mode Basin

n=0

n=1

n=2

n=3

n=4

— 2

1 2/3

1/2 2/5

1/3 2/7

1/4 2/9

Closed Open-mouthed

“tone”) have wavelengths equal to one-half, one-third, one-fourth, and so on, of the wavelength of the fundamental mode (Fig. 9.1(a), Table 9.1). The fundamental mode is antisymmetric: when one side of the water surface is going up, the opposite side is going down. Maximum sea level oscillations are observed near the basin borders (x = 0; x = L), while maximum currents occur at the nodal lines, i.e., the lines where ζ = 0 for all time. Positions of the nodal lines are determined by xm n =

(2m − 1)L , 2n

m, n = 1, 2, . . . ; m ≤ n.

(9.4)

Thus, for n = 1, there is one nodal line: x11 = L/2 located in the middle of the basin; for n = 2, there are two lines: x12 = L/4 and x22 = 3L/4; for n = 3: x13 = L/6, x23 = 3L/6 = L/2 and x33 = 5L/6 and so on. The number of nodal lines equals the mode number n [Fig. 9.1(a)], which is why the ﬁrst mode is called the uninodal mode, the second mode is called binodal mode, the third mode the trinodal mode, etc.31,95 The antinode positions are those for which ζ attains maximum values, and are speciﬁed as xjn =

jL , n

j = 0, 1, 2, . . . , n.

(9.5)

For example, for n = 2 there are three antinodal lines: x02 = 0, x12 = L/2, and = L. Maximum currents occur at the nodal lines, while minimum currents occur at the antinodes. Water motions at the seiche nodes are entirely horizontal, while at the antinodes they are entirely vertical. The relationships (9.2) and (9.3) yield the well-known Merian’s formula for the periods of (natural) in a rectangular basin of uniform depth68,78 : x22

2L , Tn = √ n gH

(9.6)

where n = 1, 2, 3, . . .. Merian’s formula (9.6) shows that the longer the basin length (L) or the shallower the basin depth (H), the longer the seiche period. The fundamental (n = 1) mode has the maximum period; other modes — the overtones of the main fundamental — “tone” — have periods equal to one-half, one-third, one-fourth, and so on, of the fundamental period (Fig. 9.1(a), Table 9.1). The fundamental mode and all other odd modes are antisymmetric, while even modes are symmetric; an antinode line is located in the middle of the basin.

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The structures and parameters of open-mouth basins are quite diﬀerent from those of closed basins. Standing oscillations in a rectangular bay (harbor) with uniform depth and open entrance also have the form (9.1) but with a nodal line located near the entrance (bay mouth). In general, the approximate positions of nodal lines are determined by the following expressions (Fig. 9.1(b), Table 9.1): xm n =

(2m + 1)L , 2n + 1

m, n = 0, 1, 2, . . . ; m ≤ n,

(9.7)

j, n = 0, 1, 2, . . . ; j ≤ n.

(9.8)

while antinodes are located at xjn =

2jL , 2n + 1

In particular, for n = 1 there are two nodal lines: x01 = L/3 and x11 = L and two antinodal lines: x01 = 0 and x11 = 2L/3; for n = 2 there are three nodal lines: x02 = L/5, x12 = 3L/5 and x22 = L, and three antinodal: x02 = 0, x12 = 2L/5, and x22 = 4L/5. The most interesting and important mode is the lowest mode, for which n = 0. This mode, known as the Helmholtz mode, has a single nodal line at the mouth of the bay (x = L) and a single antinode on the opposite shore (x = L). The wavelength of this mode is equal to four times the length of the bay; a basin oscillating in this manner is known as a quarter-wave oscillator.34 The Helmholtz mode, which is also called the zeroth mode,a the gravest mode and the pumping mode (because it is related to periodic mass transport — pumping — through the open mouth,36,41 is of particular importance for any given harbor. For narrow-mouthed bays and harbors, as well as for narrow elongated inlets and fjords, this mode normally dominates. The periods of the Helmholtz and other harbor modes can be approximately estimated as84,95 Tn =

4L √ , (2n + 1) gH

for mode n = 0, 1, 2, 3, . . . .

(9.9)

Using (9.9) and (9.6), the fundamental (Helmholtz) mode in a rectangular open√ mouth basin of uniform depth H is found to have a period, T 0 = 4L/ gH, which √ is double the period of the gravest mode in a similar but closed basin, T 1 = 2L/ gH. Normalized periods of various modes (for n ≤ 4) are shown in Table 9.1. Expressions (9.4)–(9.9), Table 9.1, and Fig. 9.1 are all related to the idealized case of a simple rectangular basin of uniform depth. This model is useful for some preliminary estimates of seiche parameters in closed and semi-closed natural and artiﬁcial basins. Analytical solutions can be found for several other basins of simple geometric form and nonuniform depth. Wilson95 summarizes results that involve common basin shapes (Tables 9.2 and 9.3), which in many cases are quite a In many papers and text books,8,94,95 this mode is considered the “ﬁrst mode”. However, it is more common to count nodal lines only inside the basin (not at the entrance) and to consider the fundamental harbor mode as the “zeroth mode”.41,68,78,80,84 This approach is physically more sound because this mode is quite speciﬁc and markedly diﬀerent from the ﬁrst mode in a closed basin.

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good approximations to rather irregular shapes of natural lakes, bays, inlets, and harbors. The main concern for port operations and ships and boats in harbors is not from the sea level seiche variations but from the strong currents associated with the seiche. As noted above, maximum horizontal current velocities occur at the nodal lines. Therefore, its locations in the vicinity of the nodes that are potentially most risky and unsafe. Maximum velocities, Vmax , can be roughly estimated as84 : g , (9.10) Vmax = An H where An is the amplitude of the sea level oscillation for the mode. For example, if An = 0.5 m and H = 6 m, Vmax ≈ 0.64 m/s. 9.2.2. Rectangular and circular basins If a basin is not long and narrow, the 1D approach used above is not appropriate. For such basins, 2D eﬀects may begin to play an important role, producing compound or coupled seiches.95 Two elementary examples, which can be used to illustrate the 2D structure of seiche motions, are provided by rectangular and circular basins of uniform depth (H). Consider a rectangular basin with length L (x = 0, L) and width l (y = 0, l). Standing oscillations in the basin have the form35,41 ζ(x, y; t) = Amn cos

nπy mπx cos cos ωt, L l

where m, n = 0, 1, 2, 3, . . .. The eigen wave numbers (kmn ) are 1/2 mπ 2 nπ 2 kmn = + , L l and the corresponding eigen periods are78 −1/2 m 2 n 2 2 Tmn = √ + . L l gh

(9.11)

(9.12)

(9.13)

For n = 0 expression (9.13) becomes equivalent to the Merian’s formula (9.6); the longest period corresponds to the fundamental mode (m = 1, n = 0) which has one nodal line in the middle of the basin. In general, the numbers m and n denote the number of nodal lines across and along the basin, respectively. The normalized eigen ∗ periods Tmn = Tmn /T10 and spatial structure for the diﬀerent modes are shown in Table 9.4. For oscillations in a circular basin of radius r = a, it is convenient to use a polar coordinate system (r, θ) with the origin in the center: x = r cos θ,

y = r sin θ,

where θ is the polar angle. Standing oscillations in such basins have the form ζ(x, y; t) = Js (kr)(As cos sθ + Bs sin sθ) cos ωt,

(9.14)

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Table 9.2.

Modes of free oscillations in closed basins of simple geometric shape and constant width (after Ref. 95). Periods of free oscillation Mode ratios Tn /T1 values for n

Basin type Description

Dimensions

2

3

4

2L/(gh0 )1/2

1.000

0.500

0.333

0.250

h(x) = h0 (1 − 2x/L)

1.305[2L/(gh0 )1/2 ]

1.000

0.628

0.436

0.343

h(x) = h0 (1 − 4x2 /L2 )

1.110[2L/(gh0 )1/2 ]

1.000

0.577

0.408

0.316

h(x) = h0 (1 − 4x2 /L2 )2

1.242[2L/(gh0 )1/2 ]

1.000

0.686

0.500

0.388

L 0

Triangular (isosceles)

h0

h0

L0 h(x)

h0

Quartic

h0

x

∠ 2

L Parabolic

x

h(x)

0 h(x)

x

∠ 2

L

0 h(x)

∠ 2

x

h(x) = h0

(Continued)

A. B. Rabinovich

Rectangular

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Fundamental T1

Proﬁle equation

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Table 9.2.

(Continued ) Periods of free oscillation Mode ratios Tn /T1 values for n

Basin type Dimensions

Proﬁle equation

2

3

4

1.640[2L/(gh1 )1/2 ]

1.000

0.546

0.377

0.288

1.000

0.546

0.377

0.288

L1 /L2 = 1/2 4L2 /(gh2 )1/2

1.000

0.500

0.250

0.125

L1 /L2 = 1/3 3.13L2 /(gh2 )1/2 L1 /L2 = 1/4 2.73L2 /(gh2 )1/2 L1 /L2 = 1/8 2.31L2 /(gh2 )1/2

1.000 1.000 1.000

0.559 0.579 0.525

0.344 0.367 0.371

0.217 0.252 0.279

L 0

Triangular (right-angled)

h1

Trapezoidal

h0

h(x)

h1

h(x) = h1 x/L

L

0

h(x) = h0 + mx m = (h1 − h0 )/L

x h(x) h1

L1 Coupled, rectangular

x

L2 0

x h(x) h 2

h(x) = h1 (x < 0) h(x) = h2 (x < 0) h1 /h2 = 1/4

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Description

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Modes of free oscillations in semi-closed basins of simple geometric shape (modiﬁed after Ref. 95).

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Table 9.3.

Periods of free oscillation Mode ratios Ts /T1 [n = (s + 1)/2]

Basin type Description

Dimensions 0

0

b(x)

b1

x

h(x)

n=0

1

2

3

h(x) = h1

2.000[2L/(gh1 )1/2 ]

1.000

0.333

0.200

0.143

h(x) = h1 x/L

2.618[2L/(gh1 )1/2 ]

1.000

0.435

0.278

0.203

h(x) = h1 (1 − x2 /L2 )

2.220[2L/(gh1 )1/2 ]

1.000

0.409

0.259

0.189

b(x) = b1 x/L h(x) = h1

1.308[2L/(gh1 )1/2 ]

1.000

0.435

0.278

0.230

x

h1

0

b(x)

b1

x

0

h1

h(x)

L Rectangular

Triangular L

x

b(x)

b1

0

x

h1

h(x)

L Rectangular

Semi-parabolic

b1

L Triangular

x

0

h(x)

h1

x

Rectangular

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b(x)

0

0

A. B. Rabinovich

L x

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L Rectangular

L Rectangular

Fundamental T0

Proﬁle equation

(Continued)

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Table 9.3.

(Continued ) Periods of free oscillation Mode ratios Ts /T1 [n = (s + 1)/2]

Basin type Description

Dimensions

1

2

3

b(x) = b1 x/L h(x) = h1 x/L

1.653[2L/(gh1 )1/2 ]

1.000

0.541

0.374

0.283

b1 /L = 2 = 4/3 =1 = 2/3

2.220[2L/(gh1 )1/2 ]

1.000

0.707 0.554 0.447 0.317

0.578 0.493 0.468 0.455

0.378 0.323 0.264 0.185

h(x) = h1 (1 − r 2 /L2 )

2.220[2L/(gh1 )1/2 ]

1.000

0.707

0.578

0.500

L

b(x)

0

b1

x

0

h1

h(x)

L Triangular

x

Triangular L

0

b1

x

0

L Semi-elliptic

h1

x

Semi-paraboloidal

L

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n=0

Seiches and Harbor Oscillations

Fundamental T0

Proﬁle equation

L 0 b 1

0

r h(x)

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Semi-circular

Semi-paraboloidal

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FA

Mode parameters for free oscillations in uniform depth basins of rectangular and circular geometric shape.

Rectangular basin (l = 0.5L)

1

2

1

2

1

2

0.500

2

0

0.500

0

1

0.447

1

1

0.354

2

1

+

+

+ + + +

+

+

0

2

0.243

1

2

ωa/c

Tsm /T10

—

—

1.841

1.000

—

—

3.054

0.603

+

0.628

—

3.832

0.480

0.719

—

5.331

0.345

0.766

—

6.706

0.275

0.343

0.787

7.016

0.262

0.449

0.820

8.536

0.216

+

+

+

+

0.250

r2

+ +

+

+ +

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0

0

Relative period

+

1

1

Normalized frequency

r1

2a

+

1

1.000

l

+

Circular nodal lines

+

0

m

+

0

s

+

2

Tmn /T10

+

0

L

Mode forms

+

1

Relative period

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n

Mode numbers

Mode forms

A. B. Rabinovich

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Circular basin

+

Mode numbers

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Table 9.4.

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where Js is the Bessel function of an order s, As and Bs are arbitrary constants, and s = 0, 1, 2, 3, . . .35,41 These oscillations satisfy the boundary condition: Js (kr)|r=a = Js (ka) = 0.

(9.15)

The roots of this equation determine the eigen values ksm (m, s = 0, 1, 2, 3, . . .), with corresponding eigen modes described by Eq. (9.14) for various k = ksm . Table 9.4 presents the modal parameters and the free surface displacements of particular modes. As illustrated by Table 9.4, there are two classes of nodal lines, “rings” and “spokes” (diameters). The corresponding mode numbers m and s give the respective exact number of these lines. Due to mass conservation, the mode (0, 0) does not exist in a completely closed basin.41 For the case s = 0, the modes are symmetrical with respect to the origin and have annular crest and troughs.35 In particular, the ﬁrst symmetrical mode (s = 0, m = 1) has one nodal ring r = 0.628a (Table 9.4). When the central part of the circular basin (located inside of this ring) is going up, the marginal part (located between this ring and the basin border) is going down, and vice versa. The second symmetrical mode (s = 0, m = 1) has two nodal rings: r = 0.343a and r = 0.787a. For s > 0, there are s equidistant nodal diameters located at an angle ∆θ = π/s from each other; i.e., 180◦ for s = 1, 90◦ for s = 2, 60◦ for s = 3, etc. Positions of these diameters are indeterminate, since the origin of θ is arbitrary. The indetermatability disappears if the boundary deviates even slightly from a circle. Speciﬁcally, the ﬁrst nonsymmetrical mode (s = 1, m = 0) has one nodal diameter (θ = π/2), whose position is undeﬁned; but if the basin is not circular but elliptical, the nodal line would coincide with either the major or minor axis, and the corresponding eigen periods would be unequal.35 The ﬁrst unsymmetrical mode has the lowest frequency and the largest eigen period (Table 9.4); in this case, the water sways from one side to another relative to the nodal diameter. This mode is often referred to as the “sloshing” mode.78 Most natural lakes or water reservoirs can support rather complex 2D seiches. However, the two elementary examples of rectangular and circular basins help to understand some general properties of the corresponding standing oscillations and to provide rough estimates of the fundamental periods of the basins. 9.2.3. Harbor resonance Let us return to harbor oscillations and consider some important resonant properties of semi-closed basins. First, it is worthy to note that expressions (9.7)–(9.9) and Table 9.3 for open-mouth basins give only approximate values of the eigen periods and other parameters of harbor modes. Solutions of the wave equation for basins of simple geometric forms are based on the boundary condition that a nodal line (zero sea level) is always exactly at the entrance of a semi-closed basin that opens onto a much larger water body. In this case, the free harbor modes are equivalent to odd (antisymmetric) modes in a closed basin, formed by the open-mouth basin and its

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206

mirror image relative to the mouth.b However, this condition is not strictly correct because it does not take into account wave energy radiation through the mouth into the open sea. The exact solutions may be obtained based on the Sommerfeld radiation condition of free wave radiation through the open boundary.35,41 Following application of the appropriate mouth correction (α), the nodal line is located close to but outside the entrance. In other words, the eﬀect of this correction is to increase the eﬀective length of the basin.95 The mouth correction depends on two parameters: the basin aspect ratio q = l/L, which relates the width of the basin (l) to its length (L); and the aperture ratio ϑ = b/l, in which b is the actual width of the mouth. Mathematical determination of α is rather complicated but, as a rule, it increases with increases of q and ϑ. For example, the fractional correction to Eq. (9.9) for the fundamental mode in a rectangular basin of uniform depth and open mouth (ϑ = 1.0) is determined as30,95 q 3 πq α= − γ − ln , (9.16) π 2 4 where γ = 0.5772 is Euler’s constant. Roughly speaking, radiation into the external basin and the mouth correction are important when the semi-closed basin is broad and has a large open entrance, and negligible when the basin is long and narrow (i.e., when q is small); in the latter case, expressions (9.7)–(9.9), as well as those presented in Table 9.3, are quite accurate. The character of natural oscillations in a bay or harbor is strongly controlled by the aperture ratio ϑ = b/l, which can vary from ϑ = 1.0 to ϑ = 0.0. These two asymptotic cases represent a fully open harbor and a closed basin, respectively. It is evident that the smaller is ϑ (i.e., the smaller the width of the entrance), the slower water from the external basin (open sea) penetrates into the harbor. Thus, as ϑ decreases, the periods of all harbor modes for n ≥ 1 in Table 9.1 increase, tending to the periods of the corresponding eigen modes for a closed basin, while the period of the fundamental (Helmholtz) harbor mode tends to inﬁnity.c This is one of the important properties of harbor oscillations. Another important property is harbor resonance. The ampliﬁcation factor for long waves impinging on a harbor from the open sea is H 2 (f ) =

(1 − f /f0

)2

1 , + Q−2 (f /f0 )2

(9.17)

where f is the frequency of the long incoming waves, f0 is the resonant frequency of the harbor, and Q is the quality factor (“Q-factor”), which is a measure of energy damping in the system.47,95 Speciﬁcally, Q−1 =

b This

c This

dE/dt = 2β, ωE

approach is used for numerical computation of eigen modes in natural 2D basins.70 is the reason for calling this the “zeroth mode.”

(9.18)

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where E = E0 e−2βωt is the energy of the system as it decays from an initial value E0 , β is a dimensionless damping coeﬃcient, and ω = 2πf is the angular frequency. The power ampliﬁcation factor attains the value Q2 at resonance (f = f0 ), decreases to unity at f = 0 and goes to zero as f goes to inﬁnity. Therefore, Q for harbor oscillations plays a double role: as a measure of the resonant increase of wave heights for waves arriving from the open ocean and as an index of the time decay rate of wave heights inside the harbor. The higher the Q, the stronger will be the ampliﬁcation of the incoming waves and the slower the energy decay, i.e., the longer the “ringing” of seiche oscillations inside the harbor. In closed basins, like lakes, bottom friction is the main factor controlling energy decay. Normally, it is quite small, so in lakes with fairly regular topographic features (low damping), a high Q-factor may be expected. Consequently, even a small amount of forcing energy at the resonant frequency can produce signiﬁcant seiche oscillations that persist for several days.31,95 In contrast, the main factor of energy decay in semi-closed water basins, such as gulfs, bays, fjords, inlets and harbors, is wave radiation through the entrance. In their pioneering work, Miles and Munk47 concluded that narrowing the harbor entrance would increase the quality factor Q and, consequently, the ampliﬁcation of the arriving wave. This means that the construction of dams, dikes, and walls to protect the harbor from wind waves and swell could so constrict the entrance width that it leads to strong ampliﬁcation of the resonant seiche oscillations inside the harbor. Miles and Munk47 named this harbor paradox. As pointed out by Miles and Munk,47 there are two limitations to the previous conclusions: (1) A time of order Q/π cycles is necessary for the harbor oscillations to adjust to the external forcing. This means that harbors with high Q would not respond to a strong but short-lived incoming disturbance. In most cases, this limitation is not of major concern because atmospheric disturbances (the major source of open-sea long waves inducing harbor oscillations) are likely to last at least for several hours. Even tsunami waves from distant locations “ring” for many hours, resonantly “feeding” harbor seiches and producing maximum oscillations that have long (12–30 h) durations that persist well after the arrival of the ﬁrst waves.74,75 This contrasts with the case for near-ﬁeld sites, where tsunamis normally arrive as short-duration impulsive waves. Such tsunamis are much more dangerous at open coastal regions than in bays or harbors, as was observed for the coast of Thailand after the 2004 Sumatra tsunami.86 (2) As the harbor mouth becomes increasingly narrower, the internal harbor dissipation eventually exceeds energy radiation through the mouth. At this stage, further narrowing does not lead to a further increase in the Q-factor. However, normally internal dissipation is small compared to the typical radiative energy losses through the entrance. Originally, Miles and Munk47 believed that their “harbor paradox” concept was valid for every harbor mode provided the corresponding spectral peak was sharp and well deﬁned. Further thorough examination of this eﬀect37,46,78 indicated that the harbor paradox is only of major importance for the Helmholtz mode, while for higher modes frictional and nonlinear factors, not accounted for in the theory,

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dampen this eﬀect.95 However, the Helmholtz mode is the most important mode in natural basins and is normally observed in bays, inlets, and harbors with narrow entrance, i.e., in semi-closed basins with high Q-factor. Signiﬁcant problems with the mooring and docking of ships (and the loading and unloading of their cargo) in ports and harbors are often associated with this fundamental mode and most typically occur in ports with high Q.4,6,41,62,66–68,78–80 Rabinovich67 suggested reducing these negative eﬀects in ports by artiﬁcially increasing the internal dissipation. The idea is the same as that widely used in rocket technology to damp eigen oscillations in fuel tanks.44,45 Radial piers in ports and harbors play the same role as internal rings and ribs in rocket tanks, eﬃciently transforming wave energy into vortical motions which reduce the wave energy and therefore the intensity of the seiches and their associated horizontal currents. As shown by Rabinovich,67 the logarithmic attenuation factor, δ0 = π/Q, for the Helmholtz mode associated with the jth pier, is given by ˆ ∆E0j 1 j bj ω0 ζ0 j = Cx , (9.19) δ0 = 2E0 6 r0 h0 σ0 where E0 is the energy of the mode inside the harbor, ∆E0j is the energy dissipated at the pier over the mode period (T0 = 2π/ω0 ), bj is the length of the pier, r0 and h0 are the mean radius and depth of the harbor, ζˆ0 is the mean amplitude of the Helmholtz mode in the harbor, Cxj is a dimensionless resistance coeﬃcient, and σ0 = (gh0 )1/2 /(πr0 ). Thus, the rate of damping of oscillations in a harbor depends on the number of piers (N ) and a number of dimensionless parameters: speciﬁcally, the relative amplitudes of the oscillations, ξ0 = ζˆ0 /h0 ; the normalized harbor frequency, Ω0 = ω0 /σ0 ; the relative lengths of the piers, Bj = bj /r0 ; and the coeﬃcient Cxj . The parameter ξ0 depends on the intensity of the external forcing while the two other parameters Ω0 and Bj do not depend on forcing but only on the characteristics of the harbor. The coeﬃcient Cxj strongly depends on the Keulegan– Carpenter (KC) number which relates hydraulic resistance in oscillating ﬂows to those for stationary currents.32 For typical values Bj = 0.3, ξ0 = 0.1, Ω0 = 1.0, N = 8, and Cxj = 10, we ﬁnd δ0 ≈ 0.4 and Q ≈ 8. Another important aspect of the harbor oscillation problem is that changes in port geometry, and the construction of additional piers and dams can signiﬁcantly change the natural (eigen) periods of the port, thereby modifying considerably the resonant characteristics of the basin.6 Helmholtz resonators in acoustics are used to attenuate sound disturbances of long wavelengths, which are diﬃcult to reduce using ordinary methods of acoustical energy dissipation. Similarly, side channel resonators are suggested as a method for attenuating incident wave energy in harbors.6,66,78 In general, estimation of the Q-factor is a crucial consideration for ports, harbors, bay, and inlets. For a rectangular basin of uniform depth and entirely open mouth (ϑ = b/l = 1.0), this factor is easily estimated as: Q=

L , l

(9.20)

which is inversely proportional to the aspect ratio q = l/L. This means that high Q-factors can be expected for long and narrow inlets, fjords, and waterways.

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Honda et al.30 and Nakano and Unoki59 examined coastal seiches at more than 110 sites on the coast of Japan and found that strong and highly regular seiche oscillations are most often observed in such elongated basins and that the periods of these oscillations are in good agreement with the approximate period (9.9) for the Helmholtz mode (n = 0): 4L T0 = √ . gH

(9.21)

If the aperture ratio ϑ < 1.0, corresponding to a partly closed entrance, it is more diﬃcult to estimate the Q value and the resonant mode periods analytically. In practice, special diagrams for a rectangular basin with various q and ϑ are used for these purposes.80,84 For natural basins, these parameters can be estimated numerically or from direct observations. If the respective spectral peak in observational data is isolated, sharp and pronounced enough, then we can assume that Q 1. In this case, it follows from (9.17) that the half-power frequency points (f1/2 ) are given by the following expression47: 1 ± f1/2 = f0 1 ± , (9.22) 2Q and the relative frequency bandwidth is simply ∆f = Q−1 , f0

(9.23)

+ − where ∆f = f1/2 − f1/2 and f0 = 1/T0 is the resonant frequency. This is a useful practical method for estimating the Q-factor and ampliﬁcation for coastal basins based on results of spectral analysis of observational data. However, the spatial structure of diﬀerent modes, the distribution of currents, and sea levels inside a natural basin, inﬂuence harbor reconstruction based on changes in these characteristics, and many other aspects of harbor hydrodynamics, are diﬃcult to estimate without numerical computations. Numerical modeling has become a common approach that is now widely used to examine harbor oscillations.4,14,70,92

9.2.4. Harbor oscillations in a natural basin Some typical features of harbor oscillations are made more understandable using a concrete example. Figures 9.2 and 9.3 illustrate properties of typical harbor oscillations and results of their analysis and numerical modeling. Several temporary cable bottom pressure stations (BPS) were deployed in bays on the northern coast of Shikotan Island, Kuril Islands in 1986–1992.13,14,68,70 All BPSs were digital instruments that recorded long waves with 1-min sampling. One of these stations (BPS-1) was situated at the entrance of False Bay, a small bay with a broad open mouth [Fig. 9.2(a)]. The oscillations recorded at this site were weak and irregular; the respective spectrum [Fig. 9.2(b)] was “smooth” and did have any noticeable peaks, probably because of the closeness of the instrument position to the position of the entrance nodal line. Two more gauges (BPS-2 and BPS-3) were located inside Malokurilsk Bay, a “bottle-like” bay with a maximum width of about

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210

Sea of Okhotsk

45°

K

Hokkaido I.

sla il I ur

s nd

Iturup I.

4

Kunashir I. Shikotan I. 5

1 2

False Bay

3

Malokurilsk Bay

Pacific Ocean

Shikotan I. 0.5 km

(a) 40° 140°

145°

150° 10 5

Spectra (cm 2 cpm -1)

10 4

18.6

10 4 1

10 4 2 10 3

10 2

95%

10 3

18.6 3

10 2

4

10 0

10 -2 10 -3

95%

95%

10 2

(b)

10 1 10 -2

10 -1

10 -3

10 1

(c)

10 0 10 -2

10 -1

(d) 10 -2

10 -1

Frequency (cpm)

Fig. 9.2. (a) Location of cable bottom pressure stations near the northern coast of Shikotan Island (Kuril Islands) and sea level spectra at (b) BPS-1, (c) BPS-2 (both in autumn 1986) and (d) BPS-3 and BPS-4 (October–November 1990).

1300 m and a narrow neck of 350 m [Fig. 9.2(a)]. The oscillations recorded by these instruments were signiﬁcant, highly regular and almost monochromatic; the corresponding spectra [Figs. 9.2(c) and 9.2(d)] have a prominent peak at a period of 18.6 min. An analogue tide gauge [#5 in Fig. 2(a)] situated on the coast of this bay permanently measure oscillations with exactly the same period.70 It is clear that this period is related to the fundamental mode of the bay. The Q-factor of the bay, as estimated by expression (9.23) based on spectral analysis of the tide gauge data for sites BPS-2 and BPS-3, was 12–14 and 9–10, respectively. The high Qfactors are likely the main reason for the resonant ampliﬁcation of tsunami waves that arrive from the open ocean. Such tsunami oscillations are regularly observed in this bay.13,69 In particular, the two recent Kuril Islands tsunamis of 15 November 2006 and 13 January 2007 generated signiﬁcant resonant oscillations in Malokurilsk Bay of 155 cm and 72 cm, respectively, at the same strongly dominant period of 18.6 min.76

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n=0 18.9 min

n=1 6.5 min

n=2 3.8 min

n=3 3.4 min

n=4 2.6 min

n=5 2.3 min

Fig. 9.3. Computed eigen modes and periods of the ﬁrst six modes in Malokurilsk Bay (Shikotan Island). Black triangles indicate positions of the BPS-2 and BPS-3 gauges (from Ref. 70).

Figure 9.3 shows the ﬁrst six eigen modes for Malokurilsk Bay.70 The computations were based on numerical conformal mapping of the initial mirror reﬂected domain on a circular annulus (for details, see Ref. 77) and the following application of Ritz’s variational method to solve the eigenvalue problem. The computed period of the fundamental (Helmholtz) mode (18.9 min) was close to the observed period of 18.6 min. The spectra at BPS-2 and BPS-2 indicate weak spectral peaks (three orders of magnitude less than the main peak) with periods 4.1, 3.3, and 2.9 min (the latter only at BPS-3), thought to be related to modes n = 2, 3, and 4. The ﬁrst mode (n = 1), with period of 6.5 min, was not observed at these sites apparently because the nodal line for this mode passes through the positions of BPS-2 and BPS-3. Thus, the computed periods of the bay eigen modes are in good agreement with observation; plots in Fig. 9.3 give the spatial structure of the corresponding modes. However, this approach does not permit direct estimation of the bay response to the external forcing and the corresponding ampliﬁcation of waves arriving from the open ocean. In actuality, the main purpose of the simultaneous deployments at sites BPS-3 and BPS-4 [Fig. 9.2(a)] in the fall of 1990 was to obtain observed response parameters that could be compared with numerically evaluated values.14 The spectrum at BPS-4, the station located on the outer shelf of Shikotan Island near the entrance to Malokurilsk Bay [Fig. 9.2(d)], contains a noticeable peak with period of 18.6 min associated with energy radiation from the bay. This peak is about 1.5 orders of magnitude lower than a similar peak at BPS-3 inside the bay. The ampliﬁcation factor for the 18.6 min period oscillation at BPS-4 relative to that at BPS-3 was found to be about 4.0. Numerical computations of the response characteristics for Malokurilsk Bay using the HN-method14 gave resonant periods which

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were in close agreement with the empirical results of Rabinovich and Levyant70 (indicated in Fig. 9.3). Resonant ampliﬁcation of tsunami waves impinging on the bay was found to be 8–10. 9.2.5. Seiches in coupled bays A well-known physical phenomenon are the oscillations of two simple coupled pendulums connected by a spring with a small spring constant (weak coupling). For such systems, the oscillation energy of the combined system systematically moves from one part of the system to the other. Every time the ﬁrst pendulum swings, it pulls on the connecting string and gives the second pendulum a small tug, so the second pendulum begins to swing. As soon as the second pendulum starts to swing, it begins pulling back on the ﬁrst pendulum. Eventually, the ﬁrst pendulum is brought to rest after it has transferred all of its energy to the second pendulum. But now the original situation is exactly reversed, and the ﬁrst pendulum is in a position to begin “stealing” energy back from the second. Over time, the energy repeatedly switches back and forth until friction and air resistance eventually remove all of the energy out of the pendulum system. A similar eﬀect is observed in two adjacent bays that constitute a coupled system. Nakano56 was probably the ﬁrst to investigate this phenomenon based on observations for Koaziro and Moroiso bays located in the Miura Peninsula in the vicinity of Tokyo. The two bays have similar shapes and nearly equal eigen periods. As was pointed out by Nakano, seiches in both bays are very regular, but the variations of their amplitudes are such that, while the oscillations in one bay become high, the oscillations in the other become low, and vice versa. Nakano56 explained the eﬀect theoretically as a coupling between the two bays through water ﬂowing across the mouths of each bay. More than half a century later Nakano returned to this problem60 and, based on additional theoretical studies and hydraulic model experiments, demonstrated that two possible regimes can exist in the bays: (1) co-phase oscillations when seiches in the two bays have the same initial phase; and (2) contraphase when they have the opposite phase. The superposition of these two types of oscillations create beat phenomenon of time-modulated seiches, with the opposite phase modulation, such that “while one bay oscillates vigorously, the other rests”. Nakano and Fujimoto suggested the term “liquid pendulums” for the coupled interaction of two adjacent bays. A more complicated situation occurs when the two adjacent bays have signiﬁcantly diﬀerent eigen periods. For example, Ciutadella and Platja Gran are two elongated inlets located on the west coast of Menorca Island, one of the Balearic Islands in the Western Mediterranean [the inlets are shown in the inset of Fig. 9.5(a)]. Their fundamental periods (n = 0) are 10.5 min and 5.5 min, respectively.50,71,73 As a result of the interaction between these two inlets, their spectra and admittance functions have, in addition to their “own” strong resonant peaks, secondary “alien” peaks originating from the other inlet.38 This means that the mode from Ciutadella “spills over” into Platja Gran and vice versa. The two inlets are regularly observed to experience destructive seiches, locally known as “rissaga”.18,22,49–51,85 Speciﬁc aspects of rissaga waves will be discussed later (in

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Sec. 9.4); however, it is worth noting here that the coupling between the two inlets can apparently amplify the destructive eﬀects associated with each of the inlets individually.38

9.3. Generation Because they are natural resonant oscillations, seiches are generated by a wide variety of mechanisms (Fig. 9.4), including tsunamis,14,28,54,69 seismic ground waves,2,15,34,40 internal ocean waves,8,19,21 and jet-like currents.30,54,57 However, the two most common factors initiating these oscillations in bays and harbors are atmospheric processes and nonlinear interaction of wind waves or swell (Fig. 9.4).62,68,95 Seiches in lakes and other enclosed water bodies are normally generated by direct external forcing on the sea surface, primarily by atmospheric pressure variations and wind.31,95 In contrast, the generation of harbor oscillations is a two-step process involving the generation of long waves in the open ocean followed by forcing of the

Period (sec) 10 5

10 4

10 3

10 2

101 sec

AT M O S P H E R E

Storm surges

Scattering nonstationarity

Nonlinearity

Long waves

Meteorological waves

Swell Wind waves

Infragravity waves

Tsunami waves

OCEAN Seismicity

BOTTOM 10 6

10 5

10 4

10 3

10 2

10 1

10 0

Wavelength (m) Fig. 9.4.

Sketch of the main forcing mechanisms generating long ocean waves.

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harbor oscillations as the long waves arrive at the harbor entrance where they lead to resonant ampliﬁcation in the basin. Seiche oscillations produced by external periodic forcing can be both free and forced. The free oscillations are true seiches (i.e., eigen oscillations of the corresponding basin). However, if the external frequency (σ) diﬀers from the eigen frequencies of the basin (σ = ω), the oscillations can be considered forced seiches.95 Open-ocean waves arriving at the entrance of a speciﬁc open-mouth water body (such as a bay, gulf, inlet, fjord, or harbor) normally consist of a broad frequency spectrum that spans the response characteristics of the water body from resonantly generated eigenfree modes to nonresonantly forced oscillations at other frequencies. Following cessation of the external forcing, forced seiches normally decay rapidly, while free modes can persist for a considerable time. Munk53 jokingly remarked that “the most conspicuous thing about long waves in the open ocean is their absence.” This is partly true: the long-wave frequency band, which is situated between the highly energetic tidal frequencies and swell/wind wave frequencies, is relatively empty (Fig. 9.5). For both swell/wind waves and tides, the energy is of order 104 cm2 , while the energy contained throughout the entire intermediary range of frequencies is of order 1–10 cm2 . However, this particular frequency range is of primary scientiﬁc interest and applied importance (Walter Munk himself spent approximately 30 years of his life working on these “absent” waves!). Long waves are responsible for formation and modiﬁcation of the coastal zone and shore morphology;5,68 they also can strongly aﬀect docking and loading/unloading of ships and construction in harbors, causing considerable damage.41,78,79,96 Finally,

100

24 12 5

hr

Period

2 1 30 10 5

Tides

min

2 1 30 10 5

sec

2 1

Swell, wind waves

10 3

10 1 -2

10 0 Storm surges

Meteorological waves IG- waves

10 -1

Variance (cm 2 )

10 2 -5

10 -2 Tsunami 10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 -3

Frequency (cpm) Fig. 9.5. Spectrum of surface gravity waves in the ocean (modiﬁed from Ref. 68). Periods (upper scale) are in hours (h), minutes (min), and seconds (s).

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and probably the most important, are tsunamis and other marine hazardous long waves, which are related to this speciﬁc frequency band. The recent 2004 Sumatra tsunami in the Indian Ocean killed more than 226,000 people, triggering the largest international relief eﬀort in history and inducing unprecedented scientiﬁc and public interest in this phenomenon and in long waves in general.86 Because of their resonant properties, signiﬁcant harbor seiches can be produced by even relatively weak open ocean waves. In harbors and bays with high Qfactors, seiches are observed almost continuously. However, the most destructive events occur when the incoming waves have considerable energy at the resonant frequencies, especially at the frequency of the fundamental mode. Such a situation took place in Port Alberni located in the head of long Alberni Inlet on Vancouver Island (Canada) during the 1964 Alaska tsunami, when resonantly generated seiche oscillations in the inlet had trough-to-crest wave heights of up to 8 m, creating total economic losses of about $10 million (1964 dollars).28,54 9.3.1. Meteorological waves Long waves in the ocean are the primary factor determining the intensity of harbor oscillations. If we ignore tsunamis and internal waves, the main source of background long waves in the ocean are atmospheric processes (Fig. 9.4).10,53 There are three major mechanisms to transfer the energy of atmospheric processes into long waves in the ocean68 : (1) Direct generation of long waves by atmospheric forcing (pressure and wind) on the sea surface. (2) Generation of low-frequency motions (for example, storm surges) and subsequent transfer of energy into higher frequencies due to nonlinearity, topographic scattering and nonstationarity of the resulting motions. (3) Generation of high-frequency gravity waves (wind waves and swell) and subsequent transfer of energy into larger scale, lower frequency motions due to nonlinearity. Long waves generated by the ﬁrst two mechanisms are known as atmospherically induced or meteorological waves.d Typical periods of these waves are from a few minutes to several hours, typical scales are from one to a few hundreds of kilometers. The ﬁrst mechanism is the most important because it is this mechanism that is responsible for the generation of destructive seiche oscillations (meteorological tsunamis) in particular bays and inlets of the World Ocean (Sec. 9.4). “Meteorological waves” can be produced by the passages of typhoons, hurricanes or strong cyclones. They also have been linked to frontal zones, atmospheric pressure jumps, squalls, gales, wind gusts and trains of atmospheric buoyancy waves.10,59,68,71,87,95 The most frequent sources of seiches in lakes are barometric ﬂuctuations. However they can also be produced by heavy rain, snow, d The Russian name for these waves is “anemobaric”68 because they are induced by atmospheric pressure (“baric”) and wind (“anemos”) stress forcing on the ocean surface.

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or hail over a portion of the lake, or ﬂood discharge from rivers at one end of the lake.27,31,95 9.3.2. Infragravity waves Long waves generated through the nonlinear interaction of wind waves or swell are called infragravity waves.5,63 These waves have typical periods of 30 s to 300–600 s and length scales from 100 m to 10 km. The occurrence of relatively high-frequency long waves, highly correlated with the modulation of groups of wind or swell waves, was originally reported by Munk52 and Tucker.89 Because the waves were observed as sea level changes in the nearshore surf zone, they became known as surf beats. Later, it was found that these waves occur anywhere there are strong nonlinear interacting wind waves. As a result, the more general term infragravity waves (proposed by Kinsman33 ) became accepted for these waves. Recent ﬁeld measurements have established that infragravity waves (IG waves) dominate the velocity ﬁeld close to the shore and consist of superposition of free edge waves propagating along the shore, free leaky waves propagating in the oﬀshore direction, and forced bound waves locked to the groups of wind waves or swell propagating mainly onshore.3,5,68 Bound IG waves form the set-down that accompanies groups of incident waves, having troughs that are beneath the high short waves of the group and crests inbetween the wave groups.39 They have the same periodicity and the same lengths as the wave groups and travel with the group velocity of wind waves, which is significantly smaller than the phase speed of free long waves with the same frequencies. Free edge IG waves arise from the trapping of swell/wind wave generated oscillations over sloping coastal topography, while free leaky waves are mainly caused by the reﬂection of bound waves into deeper water.5,63 The general mechanisms of the formation of IG waves are shown in Fig. 9.6.e IG waves are found to be responsible for many phenomena in the coastal zone, including formation of rip currents, wave setup, crescentic bars, beach cusps and other regular forms of coastal topographies, as well as transport of sediment materials. Being of high-frequency relative to meteorological waves, IG waves can induce seiches in comparatively small-scale semi-closed basins, such as ports and harbors, which have natural periods of a few minutes and which may pose a serious threat for large amplitude wave responses. Certain harbors and ports are known to have frequent strong periodic horizontal water motions. These include Cape Town (South Africa), Los Angeles (USA), Dakar (Senegal), Toulon and Marseilles (France), Alger (Algeria), Tuapse and Sochi (Russia), Batumi (Georgia) and Esperance (Australia). Seiche motions in these basins create unacceptable vessel movement which can, in turn, lead to the breaking of mooring lines, fenders and piles, and to the onset of large amplitude ship oscillations and damage.62,67,68,82,94–96 Known as surging or range action,78,79 this phenomenon has well-established correlations with (a) harbor oscillations, (b) natural oscillations of the ship itself, and (c) intensive swell or wind waves outside the e Figure 9.6 does not include all possible types of IG waves and mechanisms of their generation; a more detailed description is presented by Bowen and Huntley5 and Battjes.3

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Fig. 9.6.

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Generation mechanisms for infragravity waves in the coastal zone.

harbor. Typical eigen periods of a harbor or a moored ship are the order of minutes. Therefore, they cannot be excited directly by wind waves or swell, having typical periods on the order of seconds.96 However, these periods exactly coincide with the periods of wave groups and IG waves. So, it is conventional wisdom that surging in harbors is the result of a triple resonance of external oscillations outside the harbor, natural oscillations within the harbor, and natural oscillations of a ship. The probability of such triple resonance is not very high, thus surging occurs only in a limited number of ports. Ports and harbors having large dimensions and long eigen periods (>10 min) are not aﬀected by surging because these periods are much higher than the predominant periods of the IG waves and the surging periods of the vessels. On the other hand, relatively small vessels are not aﬀected because their natural (eigen) periods are too short.82 The reconstruction of harbors and the creation of new harbor elements, can signiﬁcantly change the harbor resonant periods, either enhancing or, conversely, reducing the surging.f Another important aspect of the problem is that ship and mooring lines create an entirely separate oscillation system.79 Changing the material and the length of the lines and their position, f A famous example of this kind is the French port Le Havre. Before World War II it was known for very common and strong surging motions that created severe problems for ships. During the war a German submarine torpedoed by mistake a rip-rap breakwater, creating a second harbor opening of 20–25 m width. After this, the surging in the port disappeared.68

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changes the resonant properties of the system (analogous to changing the material and the length of a pendulum). It is important to keep in mind that each oscillation mode has a speciﬁc spatial distribution of sea level variability and associated current (as emphasized in Sec. 9.2.1, maximum currents are observed near the nodal lines). The intensity of the currents varies signiﬁcantly from place to place. Moreover, topographic irregularities within the harbor and the presence of structure elements (dams, dykes, piers and breakwaters) can create intense local vortexes that may signiﬁcantly aﬀect the ships.67 So, the eﬀect of surging on a ship strongly depends on the exact location of the ship, and even on its orientation, in the harbor. In summary, harbor oscillations arise through co-oscillation of sea surface elevations and currents in the harbor with those at the entrance to the harbor. Seichegenerating motions outside the harbor typically have periods of several minutes and most commonly arise from bound and free long waves that are incident on the harbor entrance. 9.3.3. Tsunami Tsunami waves are the main factor creating destructive seiche oscillations in bays, inlets and harbors.30,41,53,54,95 Tsunamis can produce “energies” of 103 –105 cm2 , although such events are relatively rare (depending on the region, from once every 1–2 years to once every 100–200 years). The main generation mechanisms for tsunamis are major underwater earthquakes, submarine landslides and volcanic explosions. Great catastrophic trans-oceanic tsunamis were generated by the 1946 Aleutian (magnitude Mw = 7.8), 1952 Kamchatka (Mw = 9.0), 1960 Chile (Mw = 9.5), and 1964 Alaska (Mw = 9.2) earthquakes. The events induced strong seiche oscillations in bays, inlets, and harbors throughout the Paciﬁc Ocean.90 The magnitude Mw = 9.3 earthquake that occurred oﬀshore of Sumatra in the Indian Ocean on 26 December 2004 generated the most destructive tsunami in recorded history. Waves from this event were recorded by tide gauges around the world, including near-source areas of the Indian Ocean (Fig. 9.7), and remote regions of the North Paciﬁc and North Atlantic, revealing the unmatched global reach of the 2004 tsunami.42,74,86,88 In general, the duration of tsunami “ringing” increased with increasing oﬀ-source distance and lasted from 1.5 to 4 days.74,75 The recorded oscillations were clearly polychromatic, with diﬀerent periods for diﬀerent sites, but with clear dominance of 40–50 min waves at most sites. The analysis of various geophysical data from this event indicates that the initial tsunami source had a broad frequency spectrum, but with most of the energy within the 40–50 min band. Therefore, although tsunami waves at diﬀerent sites induced local eigen modes with a variety of periods, the most intense oscillations were observed at sites having fundamental periods close to 40–50 min. Diﬀerences in spectral peaks among the various tide gauge records are indicative of the inﬂuence of local topography. For example, for the Paciﬁc coast of Vancouver Island (British Columbia), the most prominent peaks in the tsunami spectra were observed for Winter Harbor (period ∼30–46 min) and Toﬁno (∼50 min). In fact, the frequencies of most peaks in the tsunami spectra invariably coincide with

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West and Central Indian Ocean 200 cm

E

Colombo (SL)

Hanimaadhoo (Md)

Sea level

Male (Md)

Salalah (Om)

Pointe La Rue (Sl)

Port Louis (Mt)

26

27 December 2004 (UTC)

28

Fig. 9.7. Tsunami records in the Indian Ocean for the 2004 Sumatra tsunami for six selected sites: Colombo (Sri Lanka); Male and Gan (both Maldives); Salalah (Oman); Pointe La Rue (Seychelles); and Port Louis (Mauritius). Solid vertical line labeled “E” denotes the time of the main earthquake shock (from Ref. 74).

corresponding peak frequencies in the background spectra. This result is in good agreement with the well-known fact that periods of observed tsunami waves are mainly related to the resonant properties of the local/regional topography rather than to the characteristics of the source, and are almost the same as those of ordinary (background) long waves for the same sites. For this reason, the spectra of tsunamis from diﬀerent earthquakes are usually similar at the same location.30,48,69,g It is therefore diﬃcult to reconstruct the source region spectral characteristics based on data from coastal stations. Rabinovich69 suggested a method for separating the eﬀects of the local topography and the source on the resulting tsunami wave spectrum. This method can be used to reconstruct the open-ocean spectral characteristics of tsunami waves. The approach is based on the assumption that the spectrum S(ω) of both the tsunami and background sea level oscillations near the coast can be represented as S(ω) = W (ω)E(ω),

(9.24)

g The resonant characteristics of each location are always the same; however, diﬀerent sources induce diﬀerent resonant mode, speciﬁcally, large seismic sources generate low-frequency modes and small seismic sources generate high-frequency modes.

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where W (ω) = H 2 (ω), H(ω) is the frequency admittance function describing the linear topographic transformation of long waves approaching the coast, and E(ω) is the source spectrum. It is assumed that the site-speciﬁc properties of the observed spectrum Sj (ω) at the jth site are related to the topographic function Hj (ω) for that site, while all mutual properties of the spectra at all sites are associated with the source (assuming that the source is the same for all stations). For typical background oscillations the source spectrum has the form, E(ω) = S0 (ω), where S0 (ω) = Aω −2 , and A = 10−3 −10−4 cm2 .68,69 During tsunami events, sea level oscillations observed near the coast can be represented as ζobs (t) = ζt (t) + ζb (t),

(9.25)

where ζt are the tsunami waves generated by an underwater seismic source and ζb are the background surface oscillations. If the spectra of both tsunami, St (ω), and background oscillations, Sˆb (ω) and Sb (ω) (during and before the tsunami event, respectively) have the form (9.24), and the admittance function, W (ω), is the same for the observed tsunami and the background long waves, then the spectral ratio R(ω), is estimated as

ˆ0 (ω) E(ω) + S ˆ St (ω) + Sb (ω) (9.26) R(ω) = = = A−1 ω 2 E(ω) + 1.0. Sb (ω) S0 (ω) The function R(ω), which is independent of local topographic inﬂuence, is determined solely by the external forcing (i.e., by tsunami waves in the open ocean near the source area) and gives the ampliﬁcation of the longwave spectrum during the tsunami event relative to the background conditions. The close similarity of Rj (ω) for various sites conﬁrms the validity of this approach.69 The topographic admittance function Hj (ω), which is characteristic of the resonant properties of individual sites, can be estimated as

1/2

1/2 1/2 j Sb (ω) Sbj (ω) Sj (ω) = =ω . (9.27) Hj (ω) = E(ω) S0 (ω) A The same characteristic can be also estimated numerically. 9.3.4. Seismic waves There is evidence that seismic surface ground waves can generate seiches in both closed and semi-closed basins. In particular, the Great 1755 Lisbon earthquake triggered remarkable seiches in a number of Scottish lochs, and in rivers and ponds throughout England, western Europe and Scandinavia.95 Similarly, the Alaska earthquake of 27 March 1964 (Mw = 9.2) induced seismic surface waves that took only 14 min to travel from Prince Williams Sound, Alaska, to the Gulf Coast region of Louisiana and Texas where they triggered innumerable seiches in lakes, rivers, bays, harbors, and bayous.15,34 Recently, the 3 November 2002 Denali earthquake (Mw = 7.9) in Alaska generated pronounced seiches in British Columbia and Washington State.2 Sloshing oscillations were also observed in swimming pools during

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these events.2,15,40 The mechanism for seiche generation by seismic waves from distant earthquakes is not clear, especially considering that seismic waves normally have much higher frequencies than seiches in natural basins. McGarr40 concludes that there are two major factors promoting eﬃcient conversion of the energy from distant large-magnitude earthquakes into seiches: (1) A very thick layer of soft sediments that amplify the horizontal seismic ground motions. (2) Deeper depths of natural basins, increasing the frequencies of eigen periods for the respective water oscillations. It should be noted, however, that seismic origins for seiches must be considered as very rare in comparison, for example, with seiches generated by meteorological disturbances.95 9.3.5. Internal ocean waves In some regions of the World Ocean, deﬁnitive correlation has been found between tidal periodicity and the strong seiches observed in these regions. For example, at Palawan Island in the Philippines, periods of maximum seiche activity are associated with periods of high tides.21 Bursts of 75-min seiches in the harbor of Puerto Princesa (Palawan Island) are assumed to be excited by the arrival at the harbor entrance of internal wave trains produced by strong tidal current ﬂow across a shallow sill located about 450 km from the harbor.8 Internal waves can have quite large amplitudes; furthermore, they can travel over long distances without noticeable loss of energy. Internal waves require 2.5 days to travel from their source area in the Sulu Sea to the harbor of Puerto Princessa, resulting in a modulation of the seiche oscillations that are similar to those of the original tidal oscillations. Similarly, large amplitude seiches on the Caribbean coast of Puerto Rico are also related to tidal activity and are usually observed approximately seven days after a new or full moon (syzygy). Highest seiches in this region occur in late summer and early fall, when thermal stratiﬁcation of the water column is at its annual maximum. The seven-day interval between syzygy and maximum seiche activity could be accounted for in terms of internal tidal soliton formation near the southwestern margin of the Caribbean Sea.34 A theoretical model of seiche generation by internal waves, devised by David Chapman (Woods Hole Oceanographic Institution), demonstrated that both periodic and solitary internal waves can generate coastal seiches.8,20 Thus, this mechanism can be responsible for the formation of seiches in highly stratiﬁed regions. 9.3.6. Jet-like currents Harbor oscillations (coastal seiches) can also be produced by strong barotropic tidal and other currents. Such oscillations are observed in Naruto Strait, a narrow channel between the Shikoku and Awaji islands (Japan), connecting the Paciﬁc Ocean and

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the Inland Sea. Here, the semi-diurnal tidal currents move large volumes of water back and forth between the Paciﬁc and the Inland Sea twice per day with typical speed of 13–15 km/h. This region is one of the greatest attractions in Japan because of the famous “Naruto whirlpool”, occurring twice a month during spring tides, when the speed of tidal ﬂow reaches 20 km/h. Honda et al.30 noticed that ﬂood tidal currents generate near both coasts signiﬁcant seiche oscillations with a period of 2.5 min, which begin soon after low tide and cease near high tide; the entire picture repeats with a new tidal cycle. No seiches are observed during ebb tidal currents (i.e., between high and low water) when the water is moving in the opposite direction. Nakano57 explained this phenomenon by assuming that a strong current passing the mouth of a bay could be the source of bay seiches, similar to the way that a jet of air passing the mouth piece of an organ pipe produces a standing oscillation within the air column in the pipe. Special laboratory experiments by Nakano and Abe58 demonstrated that jet-like ﬂow with a speed exceeding a speciﬁc critical number generates a chain of antisymmetric, counter-rotating von Karman vortexes on both sides of the channel. The checker-board pattern of vortexes induce standing oscillations in nearby bays and harbors if their fundamental periods match the typical vortex periods, Tvor =

l , u

(9.28)

where l is the distance between vortexes, and u is the speed of the vortexes (u = 0.4V to 0.6V , where V is the speed of the tidal currents). For the parameters of the Naruto tidal currents, the laboratory study revealed that values of Tvor agreed with the observed seiche period of 2.5 min. Apparently, the same mechanism of seiche generation can also work in other regions of strong jet currents. 9.3.7. Ice cover and seiches It seems clear that ice cannot generate seiches (except for the case of calving icebergs or avalanches that generate tsunami-like waves). However, an ice cover can significantly impact seiche motions, suppressing them and impeding their generation. At the same time, strong seiches can eﬀectively break the ice cover and promote polynya creation. Little is known on the speciﬁc aspects of ice cover interaction with seiche modes. Hamblin26 suggested that the ice cover in Lake Winnipeg inﬂuences the character of seiche activity. Schwab and Rao83 assumed that absence of certain peaks in the sea level spectra for Saginaw Bay (Lake Huron) in winter may have been due to the presence of ice cover. Murty55 examined the possible eﬀect of ice cover on seiche oscillations in Kugmallit Bay and Tuktoyaktuk Harbor (Beaufort Sea) and found that the ice cover reduces the eﬀective water depth in the bay and harbor and in this way diminishes the frequency of the fundamental mode: in Kugmallit Bay from 0.12 cph (ice-free period) to 0.087 cph (ice-covered); and in Tuktoyaktuk Harbor from 1.0 cph to 0.9 cph.

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9.4. Meteorological Tsunamis As discussed in Sec. 9.3.3, tsunamis are the main source of destructive seiches observed in various regions of the World Ocean. However, atmospheric disturbances (atmospheric gravity waves, pressure jumps, frontal passages, squalls) can also be responsible for signiﬁcant, even devastating, long waves, which have the same temporal and spatial scales as typical tsunami waves. These waves are similar to ordinary tsunami waves and can aﬀect coasts in a similar damaging way, although the catastrophic eﬀects are normally observed only in speciﬁc bays and inlets. Nomitsu,61 Defant,10 and Rabinovich and Monserrat 71,72 suggested to use the term “meteorological tsunamis” (“meteotsunami”) for this type of waves. At certain places in the World Ocean, these hazardous atmospherically induced waves occur regularly and have speciﬁc local names: “rissaga” in the Balearic Islands, “ˇs´ciga” on the Croatian coast of the Adriatic Sea, “marubbio” (“marrobio”) in Sicily, “milghuba” in Malta, “abiki” and “yota” in Japan, “Seeb¨ ar” in the Baltic Sea, “death waves” in Western Ireland, “inchas” and “lavadiads” in the Azores and Madeira islands. These waves are also documented in the Yellow and Aegean seas; the Great Lakes; northwestern Atlantic; Argentina and New Zealand coastal areas; and Port Rotterdam.7,9–12,16,27,29,30,34,43,51,64,68,71,72,91,92 Table 9.5 gives a list of destructive harbor oscillations, which apparently have the same atmospheric origin and similar resonances due to similarities in the characteristics of the atmospheric disturbances and local geometry and topography of the corresponding basins. Because of the strong likeness between “meteotsunamis” and seismically generated tsunamis,51,88 it is quite diﬃcult sometimes to recognize one from another. Catalogues of tsunamis normally contain references to numerous “tsunami-like” events of “unknown origin” that are, in fact, atmospherically generated ocean waves. “Rissaga” (a local Catalan word that means “drying”, similar to a Spanish word “resaca”) is probably the best known example of meteorological tsunamis.h These signiﬁcant short-period sea level oscillations regularly occur in many bays and harbors of the Catalan and Valencian coasts of the Iberian Peninsula, and on the coast of the Balearic Islands. The waves in Ciutadella Harbor, Menorca Island [Fig. 9.8(a)] are particularly high and occur more frequently than in any other location.18,22,49–51,71–73,81,85 Ciutadella Inlet is about 1 km long, 100 m wide, and 5 m deep; the harbor is located at the head of the inlet [Fig. 9.8(a)]. The fundamental period of the inlet (Helmholtz mode) is approximately 10.5 min [Figs. 9.8(b,c)]. Due to the particular geometry of Ciutadella Inlet, it has a large Q-factor, which results in signiﬁcant resonant ampliﬁcation of long-wave oscillations arriving from the open sea. Seiche h For

this reason Derek Goring, a wave specialist from New Zealand, suggested to apply the term “rissaga” to all rissaga-like meteorological seiches in other areas of the World Ocean.23 However, if we were to adopt this term, then we would loose information on the cause of the oscillations and the fact that they are part of a family of events that include seismically generated tsunamis, landslide tsunamis, volcanic tsunamis, and meteotsunamis.

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224 Table 9.5.

Region

Extreme coastal seiches in various regions of the World Ocean.

Local name

Typical peroid

Maximum observed height

References

Nagasaki Bay, Japan Pohang Harbor, Korea Longkou Harbor, China Ciutadella Harbor, Menorca I., Spain

Abiki

35 min

4.78 m

—

25 min

>0.8 m

Honda et al.30 Amano [1957], Akamatsu1 , Hibiya and Kajiura29 Chu [1976], Park et al. [1986]

—

2h

2.93 m

Wang et al.93

Rissaga

10.5 min

>4.0 m

Gulf of Trieste, Italy West Sicily, Italy

—

3.2 h

1.6 m

Marrubio (Marrobbio)

∼15 min

>1.5 m

Malta, Mediterranean West Baltic, Finland coast Croatian coast East Adriatic Newfoundland, Canada Western Ireland Azores Is and Madeira Is, East Atlantic Rotterdam Harbor, The Netherlands

Milghuba

∼20 min

∼1.0 m

Fontser´e,17 Tintor´e et al.,88 Monserrat et al.,49−51 Gomis et al.,22 Garcies et al.,18 Rabinovich and Monserrat,71,72 Rabinovich et al.73 Caloi [1938], Greco et al. [1957], Defant,10 Wilson95 Plattania [1907], Oddone [1908], Defant,10 Colucci and Michelato,9 Candela et al.7 Airy [1878], Drago16

Seeb¨ ar

—

∼2.0 m

ˇciga S´

10–30 min 10–40 min ? ?

∼6.0 m

— Death Waves Inchas, Lavadiads

85–100 min

2.0–3.0 m

Doss [1907], Meissner [1924], Defant,10 Credner [1988], Hodˇzi´ c [1979/1980]; Orli´c60 ; Vilibi´ c et al.92 ; Monserrat et al.51 Mercer et al. [2002]

? ?

Berninghausen [1964], Korgen34 Berninghausen [1964], Korgen34

>1.5 m

de Looﬀ and Veldman [1994], de Jong et al.,11 de Jong and Battjes12 [2005]

Comment: Exact references can be found in: Wiegel (1964), Korgen (1995), Rabinovich and Monserrat (1996), de Jong et al. (2003) and Monserrat et al. (2006).

oscillations of duration ranging from a few hours to several days and wave heights exceeding 0.5 m recur in Ciutadella every summer. However, rissaga events (largeamplitude seiches) having wave heights more than 3–4 m, with dramatic consequences for the harbor, usually take place once in 5–6 years. During the rissaga of 21 June 1984 (Fig. 9.9), about 300 boats were destroyed or strongly damaged.71 More recently, on 15 June 2006, Ciutadella Harbor was aﬀected by the most dramatic rissaga event of the last 20 years, when almost 6-m waves were observed in the harbor and the total economic loss was of several tens millions of euros.51 Fontser´e,17 in the ﬁrst scientiﬁc paper on extreme seiches for the Catalan coast, showed that these seiches always occur from June to September and ﬁrst suggested their atmospheric origin. This origin of rissaga waves was supported by

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Fig. 9.8. A map of the Balearic Islands and positions of four tide gauges (M0, M1, M2 and MW3) deployed in Ciutadella and Platja Gran inlets and on the shelf of Menorca Island during the LAST97 experiment.50 The arrow shows the predominant direction of propagation of atmospheric waves during “rissaga” events. (b) The strong “rissaga” event recorded in Ciutadella Inlet on 31 July 1998 by a tide gauge located at position M0. (c) Spectra for “rissaga” of 24 July 1997 (solid line) and background oscillations (dashed line) for four tide gauges indicated in (a). The actual four-day records during this event are shown in the insets.

Ramis and Jans`a81 based on observed oscillations on the Balearic Islands. These authors also deﬁned a number of typical synoptic atmospheric conditions normally associated with rissaga events. The atmospheric source of rissaga is now well established.18,22,49,50,85 During late spring and summer, meteorological conditions in

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Fig. 9.9. Ciutadella Harbor during the rissaga of 21 June 1984. Photo by Josep Gornes (from Ref. 71).

the western Mediterranean are favorable for the formation of high-frequency atmospheric pressure disturbances with parameters promoting the generation of rissaga waves. These conditions include the entrance of warm air from the Sahara at nearsurface levels, and relatively strong middle level winds from the southwest. When this synoptic meteorological situation exists, trains of atmospheric pressure gravity waves (with periods of minutes) are reported traveling from SW to NE.49 If these atmospheric pressure disturbances propagate from SW to NE with a phase speed of about 22–30 m/s, resonant conditions are set up for the southeastern shelf of Mallorca Island (“Proudman resonance”) and dynamic energy associated with the atmospheric waves is eﬃciently transferred into the ocean waves. When these waves reach the coast of Menorca Island, they can generate signiﬁcant (and sometimes even hazardous) seiche oscillations inside Ciutadella and other inlets due to harbor resonance. The Q-factor for the fundamental Helmholtz mode in Ciutadella Inlet (10.5 min), roughly estimated by Eq. (9.20), is about 9. Spectral estimates based on Eq. (9.23) give a similar value, Q ≈ 10.73 As shown in Fig. 9.8(b), rissaga oscillations in Ciutadella Inlet have a very regular monochromatic character. Maximum wave heights occur during the fourth to the sixth oscillations, in good agreement with the criterion by Miles and Munk47 that time of the order of Q/π cycles is necessary for

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the harbor oscillations to adjust themselves to external forcing. The peak period of 10.5 min for the Helmholtz mode strongly dominates the spectra for the M0 and M2 gauges located in Ciutadella Inlet [Fig. 9.8(c)] both for rissaga and background spectra, while in the adjacent inlet Platja Gran (M1), where rissaga waves are also observed but weaker than in Ciutadella, the dominant peak associated with the Helmholtz mode is 5.5 min. In contrast, on the shelf (MW3) both peaks are absent and oscillations are signiﬁcantly weaker. Spectral analysis results [Fig. 9.8(c)] reveal that harbor resonance is a crucial factor in the formation of rissaga waves, as well as “meteorological tsunamis” in other bays, inlets, and harbors of the World Ocean. Barometric data from the Balearic Islands49–51,81 as well as from Japan,29 and Eastern Adriatic Sea64,91,92 demonstrate that generation of these destructive waves is associated with strong atmospheric disturbances, e.g., trains of atmospheric gravity waves, or isolated pressure jumps. These atmospheric disturbances may have diﬀerent origin: dynamic instability, orographic inﬂuence, frontal passages, gales, squalls, storms, tornados, etc.24 However, even during the strongest events, the atmospheric pressure oscillations at the meteotsunami scales (from a few minutes to a few hours) reach only 2–6 hPa, corresponding to only a 2–6 cm change in sea level. Consequently, these atmospheric ﬂuctuations may produce signiﬁcant sea level response only when resonance occurs between the ocean and the atmosphere. During the resonance process, the atmospheric disturbance propagating above the ocean surface generates significant long ocean waves by continuously pumping additional energy into these waves. Possible resonances that are responsible for the formation of meteorological tsunamis are68 : disturbance speed (U ) • Proudman resonance,65 when U = c, i.e., the atmospheric √ equals to the long-wave speed of ocean waves c = gh; • “Greenspan resonance”,25 when Ul = cj , the alongshore component (Ul ) of the atmospheric disturbance velocity equals the phase speed of the jth mode of edge waves (cj ); • “shelf resonance,” when the atmospheric disturbance and associated atmospherically generated ocean wave have a period/wavelength equal to the resonant period/length of the shelf. These resonant eﬀects may signiﬁcantly amplify ocean waves approaching the coast. Nevertheless, even strong resonant ampliﬁcation of atmospherically generated ocean waves normally cannot produce waves with suﬃcient energy to extensively aﬀect the open coast (for example, a 3–4 hPa pressure jump and a factor of 10 resonant ampliﬁcation, will only produce ocean wave heights of 30–40 cm). It is when energetic ocean waves arrive at the entrance of a semi-closed coastal basin (bay, inlet, fjord or harbor) that they can induce hazardous oscillations in the basin due to harbor resonance. On the other hand, intense oscillations inside a harbor (bay or inlet) can only be formed if the external forcing (i.e., the waves arriving from the open sea) are energetic enough. Seismically generated tsunami waves in the open ocean can

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be suﬃciently energetic even in the absence of additional resonant eﬀects (thus, according to satellite altimetry measurements, tsunami waves generated by the 2004 Sumatra earthquake in the open Indian Ocean had trough-to-crest wave heights of approximately 1.0–1.2 m,86 while atmospherically generated tsunami-like can reach such potentially dangerous levels only in the case of some external resonance. This is an important diﬀerence between tsunami waves and meteotsunamis. It follows from expression (9.17) that a large Q-factor is critical but that anomalously pronounced harbor oscillations can only be produced when there is resonant matching between the dominant frequency (f ) of the arriving (external) waves and an eigen frequency f0 of the harbor (normally, the eigen frequency of the fundamental — Helmholtz — harbor mode). This means that catastrophic harbor oscillations are the result of a double resonance eﬀect 51,68 : (a) external resonance between the moving atmospheric disturbances and open-ocean waves; and (b) internal resonance between the arriving open-ocean waves and the fundamental eigen mode of the harbor (bay, inlet). An additional favorable factor is the speciﬁc direction of the propagating atmospheric waves (and corresponding open-ocean waves) toward the entrance of the harbor (bay). Summarizing what has been presented above, we can formulate the particular conditions promoting creation of extreme atmospherically induced oscillations near the coast (meteotsunamis) as follows: • A harbor (bay, inlet or fjord) with deﬁnite resonant properties and high Q-factor. • The occurrence of strong small-scale atmospheric disturbance (a pressure jump or a train of internal atmospheric waves). • A propagation direction that is head-on toward the entrance of the harbor. • The occurrence of an external resonance (Proudman, Greenspan or shelf) between the atmospheric disturbance and ocean waves. • The occurrence of internal resonance between the dominant frequency of the incoming open-ocean waves and the fundamental harbor mode frequency. Due to these necessary levels of matching between the atmospheric disturbance, the open-ocean bathymetry and the shelf-harbor geometries, the direction and speed of the atmospheric disturbance probably are even more important than the actual energy content of the incoming waves. In any case, the necessary coincidence of several factors signiﬁcantly diminishes the possibility of these events occurring, and is the main reason why this phenomenon is relatively rare and restricted to speciﬁc locations.68 Honda et al.30 and Nakano and Unoki59 investigated more than 115 gulfs, bays, inlets, and harbors of the Japanese coast and found that highly destructive seiches (not associated with tsunami waves) occur only in a few of them. Extremely strong seiche oscillations (so-called “abiki” waves) are periodically excited in Nagasaki Bay. In particular, the abiki waves of 31 March 1979 with periods of about 35 min reached wave heights of 478 cm at the northern end of the bay and killed three people.1,29 High meteotsunami risk in certain exceptional locations mainly arises from the combination of shelf topography and coastline geometry coming together to create a multiple resonance eﬀect. The factors (internal and external) of critical importance

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are: (1) well-deﬁned resonant characteristics of the harbor (bay, inlet, etc.); and (2) speciﬁc properties of the shelf favorable for external resonance (between atmospheric and open-ocean waves) and internal resonance (between arriving open-ocean waves and harbor oscillations). The combination of these factors for some particular sites is like a “time-bomb”: sooner or later it will explode (when the atmospheric disturbance is strong enough and the parameters of disturbance coincide with the resonant parameters of the corresponding topography/geometry). Locations with known regular extreme seiches are just the places for these “time-bombs”.51 The catastrophic abiki wave event of 31 March 1979 best illustrates the physical mechanisms responsible for the generation of meteotsunamis [Fig. 9.10(a)]. Hibiya and Kajiura29 (HK in the following text) examined this event in detail and constructed an eﬀective numerical model that agrees well with observational data. Nagasaki Bay is a narrow, elongated bay located on the western coast of Kyushu Island, Japan [Fig. 9.10(b)]; the length of the bay is about 6 km, the width is 1 km, and the mean depth is 20 m. The fundamental period of the bay (Helmholtz mode) is 35 min, and this period prevails in seiche oscillations inside the bay (95% of all observed events) and it was speciﬁcally this period that was observed on 31 March 1979.1 HK noticed that almost all known cases of signiﬁcant abiki waves are associated with pressure jumps. In the case of the 1979 event, there was an abrupt pressure jump (∆Pa ) of 2–6 hPa (according to the observations at several sites) that propagated eastward (more precisely, 5.6◦ north of east) over the East China Sea with an approximate mean speed U = 31 m/s (Fig. 9.5). HK approximated this jump as ∆Pa = 3 hPa over a linear increase distance L1 = 28 km and a linear decrease distance L2 = 169 km. So, the corresponding static inverted barometer response of sea level was ∆ζ¯ ≈ −3 cm [Fig. 9.10(a)]. Moreover, the depth of the East China Sea between mainland China and Kyushu Island is between 50 and 150 m, and the corresponding long-wave speed c ≈ 22–39 m/s. Thus, it was a classical example of Proudman resonance. HK presented a simple expression describing resonant ampliﬁcation of forced open-ocean long waves as: ∆ζ =

∆ζ¯ xf , L1 2

(9.29)

where xf = U t is the distance traveled by the pressure jump during time t. If L1 = 28 km and xf = 300 km [from the source area to the Goto Islands — see Fig. 9.10(b)], then ∆ζ ≈ 16 cm. More precise numerical computation with realistic two-dimensional bathymetry gives the resonant factor ε = ∆ζ/∆ζ¯ = 4.3 and ∆ζ ≈ 12.9 cm in good agreement with observation. Therefore, due to the resonance, the initial disturbance of 3 cm increased in the open sea by four to ﬁve times [Fig. 9.10(a)]. It is interesting to note that the resonant ampliﬁcation is inversely proportional to L1 [see Eq. (9.29)], so the faster the change in atmospheric pressure (the more abrupt is the pressure jump), the stronger is the ampliﬁcation of the generated waves (HK). According to the HK computations, the outer shelf region between the Goto Islands and the mainland of Kyushu (“Goto Nada”) has resonant periods of 64, 36, and 24 min. The second period (36 min) almost coincides with the fundamental period of Nagasaki Bay (35 min). The Goto Nada shelf did not signiﬁcantly amplify

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U = 31 m/s

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air-pressure disturbance

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Fig. 9.10. (a) A sketch illustrating the physical mechanism for formation of the catastrophic meteotsunami at Nagasaki Bay (Japan) on 31 March 1979. Numbers “1”, “2”, and “3” correspond to locations shown in (b). (b) Map of Nagasaki Bay and the initial atmospheric pressure disturbance (shaded rectangular region). (c) Tide records of the catastrophic “abiki waves” of 31 March 1979 at Nezumi (9.1) and Nagasaki (2); positions of the tide gauges are shown in the inset in panel (b).

the incoming wave (the ﬁrst crest height was 16 cm at the shelf depth of 60 m) but it selected and ampliﬁed waves with speciﬁc periods, in particular those with a period of 36 min. Between the outer sea (depth 60 m) and the head of Nagasaki Bay, the arriving waves were ampliﬁed by a factor of 2.4 due to the combined eﬀects of topographic convergence, partial reﬂection, and shoaling inside the bay. Finally, resonant ampliﬁcation in Nagasaki of incoming wave train with a period of about 35 min formed catastrophic oscillations within the bay with a maximum recorded

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wave height of 278 cm [as measured by a tide gauge located in the middle of the bay — see Fig. 9.10(c)] and an estimated wave height in the head of the bay of 478 cm.1 Thus, for this extreme event, we observe the full combination of “hazardous” conditions (factors) responsible for the formation catastrophic oscillations inside Nagasaki Bay: (1) A pronounced atmospheric disturbance (pressure jump of 2–6 hPa), (2) propagating toward the bay with (3) near-resonant phase speed of 31 m/s; this disturbance resonantly generated open-sea long waves with selected (over the shelf) 36 min period that matched (4) the fundamental 35-min period of the bay that has (5) high Q-factor and well-deﬁned resonant properties. As a result, 3 cm ocean waves in the source area resulted in 478 cm waves at the head of the bay (Fig. 9.4). Analysis of destructive meteotsunami events in the Mediterranean18,22,50,51,64, 71,72,91,92 indicate that the physical mechanisms of these events were similar to those for Nagasaki Bay event. Tides in the Mediterranean are small; consequently, harbors are not designed to accommodate large amplitude sea level changes associated with occasional meteotsunamis. Consequently, it is atmospherically generated phenomena (not ordinary tsunamis) that are normally responsible for signiﬁcant ﬂooding and damage in this region. However, the main reason for the damaging nature of meteotsunamis is likely due to the strong currents in the harbor that accompany the sea level oscillations. Seiches with a 10 min period give raise to currents that are 70 times stronger than semi-diurnal tides having the same amplitude. Acknowledgments This work was initiated by Professor Fred Raichlen (CalTech, Pasadena, CA); the author sincerely appreciates his help and friendly support. He is also very grateful to Professor Young Kim (California State University, Los Angeles), the Editor of this Handbook, for his patience and useful comments and to Drs. Sebastian Monserrat (Universitat de les Illes Balears, Palma de Mallorca, Spain) and Ivica Vilibi´c (Institute of Oceanography and Fisheries, Split, Croatia) for their assistance and providing various observational data. Dr. Richard Thomson (Institute of Ocean Sciences, Sidney, BC, Canada) did tremendous work editing this chapter and encouraging the author, Patricia Kimber (Sidney, BC) helped to draft the ﬁgures. Partial ﬁnancial support was provided by the Russian Foundation on Basic Research, grants 06-0565210a, 08-05-13582, 09-05-00599 and 09-05-01125a and by NATO Science for Peace project SfP-981382. References 1. H. Akamatsu, On seiches in Nagasaki Bay, Pap. Meteor. Geophys. 33(2), 95–115 (1982). 2. A. Barberopoulou, A. Qamar, T. L. Pratt and W. P. Steele, Long-period eﬀects of the Denali Earthquake on water bodies in the Puget lowland: Observations and modeling, Bull. Seism. Soc. Amer. 96(2), 519–535 (2006).

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3. J. A. Battjes, Surf-zone dynamics, Ann. Rev. Fluid Mech. 20, 257–293 (1988). 4. W. A. M. Botes, K. S. Russel and P. Huizinga, Modeling harbor seiching compared to prototype data, Proc. 19th Int. Coastal Eng. Conf., Houston (1984), pp. 846–857. 5. A. J. Bowen and D. A. Huntley, Waves, long waves and nearshore morphology, Marine Geol. 60, 1–13 (1984). 6. E. C. Bowers, The modelling of waves and their eﬀects in harbors, in Hydraulic Modelling in Maritime Engineering (Thomas Telford, London, 1982), pp. 121–127. 7. J. Candela, S. Mazzola, C. Sammari, R. Limeburner, C. J. Lozano, B. Patti and A. Bonnano, The “Mad Sea” phenomenon in the Strait of Sicily, J. Phys. Oceanogr. 29, 2210–2231 (1999). 8. D. C. Chapman and G. S. Giese, Seiches, Encyclopedia of Ocean Sciences, eds J. H. Steele, K. K. Turekian and S. A. Thorpe, Vol. 5 (Academic Press, London, 2001) pp. 2724–2731. 9. P. Colucci and A. Michelato, An approach to study of the ‘Marubbio’ phenomenon, Boll. Geofis. Theor. Appl. 13, 3–10 (1976). 10. A. Defant, Physical Oceanography, Vol. 2 (Pergamon Press, Oxford, UK, 1961), 598 pp. 11. M. P. C. de Jong, L. H. Holthuijsen and J. A. Battjes, Generation of seiches by cold fronts over the southern North Sea, J. Geophys. Res. 108, 3117 (2003), doi:10.1029/2002JC001422. 12. M. P. C. de Jong and J. A. Battjes, Low-frequency sea waves generated by atmospheric convection cells, J. Geophys. Res. 109, C01011 (2004), doi:10.1029/2003JC001931. 13. V. A. Djumagaliev, E. A. Kulikov and S. L. Soloviev, Analysis of ocean level oscillations in Malokurilsk Bay caused by tsunami of 16 February 1991, Science Tsunami Hazards, 11, 47–58 (1993). 14. V. A. Djumagaliev, A. B. Rabinovich and I. V. Fine, Theoretical and experimental estimation of transfer peculiarities of the Malokurilsk Bay coast, the Island of Shikotan, Atmosph. Oceanic Phys. 30, 680–686 (1994). 15. W. L. Donn, Alaskan earthquake of 27 March 1964: Remote seiche simulation, Science 145, 261–262 (1964). 16. A. F. Drago, A study on the sea level variations and the ‘Milghuba’ phenomenon in the coastal waters of the Maltese Islands, Ph.D. thesis, University of Southampton (1999). 17. E. Fontser´e, Les ‘seixes’ de la costa catalana. Servei Meteorol` ogic de Catalunya, Notes d’Estudi, 58 (1934) (in Catalan). 18. M. Garcies, D. Gomis and S. Monserrat, Pressure-forced seiches of large amplitude in inlets of the Balearic Islands. Part II: Observational study, J. Geophys. Res. 101, 6453–6467 (1996). 19. G. S. Giese and D. C. Chapman, Coastal seiches, Oceanus 36, 38–46 (1993). 20. G. S. Giese, D. C. Chapman, P. G. Black and J. A. Fornshell, Causations of largeamplitude harbor seiches on the Caribbean coast of Puerto Rico, J. Phys. Oceanogr. 20, 1449–1458 (1990). 21. G. S. Giese and R. B. Hollander, The relation between coastal seiches at Palawan Island and tide-generated internal waves in the Sulu Sea, J. Geophys. 92, 5151–5156 (1987). 22. D. Gomis, S. Monserrat and J. Tintor´e, Pressure-forced seiches of large amplitude in inlets of the Balearic Islands, J. Geophys. Res. 98, 14437–14445 (1993). 23. D. G. Goring, Rissaga (long-wave events) on New Zealand’s eastern seaboard: A hazard for navigation, Proc. 17th Australasian Coastal Ocean Eng. Conf., 20–23 September 2005, Adelaide, Australia (2005), pp. 137–141. 24. E. E. Gossard and B. H. Hooke, Waves in Atmosphere (Elsevier, New York, 1975), 456 pp.

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25. H. P. Greenspan, The generation of edge waves by moving pressure disturbances, J. Fluid Mech. 1, 574–592 (1956). 26. P. F. Hamblin, Seiches, circulation and storm surges of an ice-free Lake Winnipeg, J. Fisheries Res. Board Canada 33, 2377–2391 (1976). 27. D. L. Harris, The eﬀect of a moving pressure disturbance on the water level in a lake, Meteorol. Monogr. 2, 46–57 (1957). 28. R. F. Henry and T. S. Murty, Tsunami ampliﬁcation due to resonance in Alberni Inlet: Normal modes, inTsunami: Progress in Prediction, Disaster Prevention and Warning, eds. Y. Tsuchiya and N. Shuto (Kluwer Acad. Publ., Dordrecht, The Netherlands, 1995), pp. 117–128. 29. T. Hibiya and K. Kajiura, Origin of ‘Abiki’ phenomenon (kind of seiches) in Nagasaki Bay, J. Oceanogr. Soc. Japan 38, 172–182 (1982). 30. K. Honda, T. Terada, Y. Yoshida and D. Isitani, An investigation on the secondary undulations of oceanic tides, J. College Sci., Imper. Univ. Tokyo (1908), 108 pp. 31. G. E. Hutchinson, A Treatise on Limnology. Volume 1. Geography, Physics and Chemistry (J. Wiley, New York, 1957), 1015 pp. 32. G. H. Keulegan and L. H. Carpenter, Forces on cylinders and plates in an oscillating ﬂuid, J. Res. Nat. Bureau Stand. 60, 423–440 (1958). 33. B. Kinsman, Wind Waves: Their Generation and Propagation on the Ocean Surface (Prentice Hall, Englewood Cliﬀs, 1965), 676 pp. 34. B. J. Korgen, Seiches, Amer. Sci. 83, 330–341 (1995). 35. H. Lamb, Hydrodynamics, 6th Edn. (Dover, New York, 1945), 738 pp. 36. J. J. Lee, Wave induced oscillations in harbors of arbitrary geometry, J. Fluid Mech. 45, 375–394 (1971). 37. B. Le M´ehaut´e and B. W. Wilson, Harbor paradox (discussion), J. Waterways Harbor Division, ASCE 88, 173–195 (1962). 38. P. L.-F. Liu, S. Monserrat, M. Marcos and A. B. Rabinovich, Coupling between two inlets: Observation and modeling, J. Geophys. Res. 108, 3069 (2003), doi:10.1029/ 2002JC001478, 14-(1–10). 39. M. S. Longuet-Higgins and R. W. Stewart, Radiation stress and mass transport in gravity waves, with application to ‘surf-beats’, J. Fluid Mech. 13, 481–504 (1962). 40. A. McGarr, Excitation of seiches in channels by seismic waves, J. Geophys. Res. 70, 847–854 (1965). 41. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientiﬁc, London 1992), 740 pp. 42. M. A. Merriﬁeld et al., Tide gage observations of the Indian Ocean tsunami, December 26, 2004, Geophys. Res. Lett. 32, L09603 (2005), doi:10.1029/2005GL022610. 43. M. Metzner, M. Gade, I. Hennings and A. B. Rabinovich, The observation of seiches in the Baltic Sea using a multi data set of water levels, J. Marine Systems 24, 67–84 (2000). 44. G. N. Mikishev and B. I. Rabinovich, Dynamics of a Solid Body with Cavities Partially Filled with Liquid (Mashinostroyeniye Press, Moscow, 1968), 532 pp. (in Russian). 45. J. W. Miles, Ring damping of free surface oscillations in a circular tank, J. Appl. Mech. 25, 26–32 (1958). 46. J. W. Miles, Harbor seiching, Ann. Rev. Fluid Mech. 6, 17–36 (1974). 47. J. Miles and W. Munk, Harbor paradox, J. Waterways Harbor Division, ASCE 87, 111–130 (1961). 48. G. R. Miller, Relative spectra of tsunamis, HIG-72–8 Hawaii Inst. Geophys (1972), 7 pp.

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73. A. B. Rabinovich, S. Monserrat and I. V. Fine, Numerical modeling of extreme seiche oscillations in the region of the Balearic Islands, Oceanology 39, 16–24 (1999). 74. A. B. Rabinovich, R. E. Thomson and F. E. Stephenson, The Sumatra tsunami of 26 December 2004 as observed in the North Paciﬁc and North Atlantic oceans, Surveys Geophys. 27, 647–677 (2006). 75. A. B. Rabinovich and R. E. Thomson, The 26 December 2004 Sumatra tsunami: Analysis of tide gauge data from the World Ocean Part 1. Indian Ocean and South Africa, Pure Appl. Geophys. 164, 261–308 (2007). 76. A. B. Rabinovich, L. I. Lobkovsky, I. V. Fine, R. E. Thomson, T. N. Ivelskaya and E. A. Kulikov, Near-source observations and modeling of the Kuril Islands tsunamis of 15 November 2006 and 13 January 2007, Adv. Geosci. 14, 105–116 (2008). 77. B. I. Rabinovich and Y. V. Tyurin, Numerical Conformal Mapping in TwoDimensional Hydrodynamics (Space Research Institute RAS, Moscow, 2000), 312 pp. 78. F. Raichlen, Harbor resonance, Estuary and Coastline Hydrodynamics, ed. A. T. Ippen (McGraw Hill Book Co., New York, 1966), pp. 281–340. 79. F. Raichlen, The eﬀect of ship-harbor interactions on port operations, Maritime Technol., 11–20 (2002). 80. F. Raichlen and J. J. Lee, Oscillation of bays, harbors and lakes, Handbook of Coastal and Ocean Engineering, ed. J. B. Herbich (Gulf Publishing Company Houston, Texas, 1992), pp. 1073–1113. 81. C. Ramis and A. Jans` a, Condiciones meteorol´ ogicas simult´ aneas a la aparici´ on de oscilaciones del nivel del mar de amplitud extraordinaria en el Mediterr´ aneo occidental, Rev. Geof´ısica 39, 35–42 (1983) (in Spanish). 82. T. Sawaragi and M. Kubo, Long-period motions of a moored ship induced by harbor oscillations, Coast. Eng. Japan 25, 261–275 (1982). 83. D. J. Schwab and D. B. Rao, Gravitational oscillations of Lake Huron, Saginaw Bay, Georgian Bay and the North Channel, J. Geophys. Res. 82, 2105–2116 (1977). 84. R. M. Sorensen and E. F. Thompson, Harbor hydrodynamics, Coastal Engineering Manual, Part II (U.S. Army Corps. of Engineers, Washington, D.C., New York, 2002), Chapter 7, pp. 1–92. 85. J. Tintor´e, D. Gomis, S. Alonso and D. P. Wang, A theoretical study of large sea level oscillations in the Western Mediterranean, J. Geophys. Res. 93, 10797–10803 (1988). 86. V. V. Titov, A. B. Rabinovich, H. Mofjeld, R. E. Thomson and F. I. Gonz´alez, The global reach of the 26 December 2004 Sumatra tsunami, Science 309, 2045–2048 (2005). 87. R. E. Thomson, P. W. Vachon and G. A. Borstad, Airborne synthetic aperture radar imagery of atmospheric gravity waves, J. Geophys. Res. 97, 14249–14257 (1992). 88. R. E. Thomson, A. B. Rabinovich and M. V. Krassovski, Double jeopardy: Concurrent arrival of the 2004 Sumatra tsunami and storm-generated waves on the Atlantic coast of the United States and Canada, Geophys. Res. Lett. 34, L15607 (2007), doi:10.1029/2007GL030685. 89. M. J. Tucker, Surf beats: Sea waves of 1 to 5 minute period, Proc. Roy. Soc. London Ser. A 202, 565–573 (1950). 90. W. G. Van Dorn, Some tsunami characteristics deducible from tide records, J. Phys. Oceanogr. 14, 353–363 (1984). 91. I. Vilibi´c and H. Mihanovi´c, A study of resonant oscillations in the Split harbor (Adriatic Sea), Estuar. Coastal Shelf Sci. 56, 861–867 (2003). 92. I. Vilibi´c, N. Domijan, M. Orli´c, N. Leder and M. Pasari´c, Resonant coupling of a traveling air-pressure disturbance with the east Adriatic coastal waters, J. Geophys. Res. 109, C10001 (2004), doi:10.1029/2004JC002279.

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93. X. Wang, K. Li, Z. Yu and J. Wu, Statistical characteristics of seiches in Longkou harbor, J. Phys. Oceanogr. 17, 1063–1065 (1987). 94. R. L. Wiegel, Tsunamis, storm surges, and harbor oscillations, Oceanographical Engineering (Prentice-Hall, Englewood Cliﬀs, NJ, 1964), Ch. 5, pp. 95–127. 95. B. Wilson, Seiches, Advances in Hydrosciences 8, 1–94 (1972). 96. J.-K. Wu and P. L.-F. Liu, Harbor excitations by incident wave groups, J. Fluid Mech. 217, 595–613 (1990).

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Chapter 10

Finite Diﬀerence Model for Practical Simulation of Distant Tsunamis Sung Bum Yoon Department of Civil and Environmental Engineering Hanyang University Ansan, 426-791, Korea [email protected] A practical numerical model based on the dispersion-correction ﬁnite diﬀerence scheme of Yoon et al. [Terres. Atmos. Oceanic Sci. 18(1), 31–53 (2007)] equipped with the grid nesting scheme of Lim et al. [Natural Hazards 47(1), 95–118 (2008)] is introduced. The model is applied to simulate the propagation of a historical tsunami event that attacked the east coast of Korea. The calculated free surface displacements are compared with the observations at two tidal stations along the east coast of Korea. The comparison shows that the results agree well with the observations. The analyses of the simulated results show that the underwater topography such as submerged rises and ridges plays an important role in the propagation of tsunamis in this region.

10.1. Introduction Tsunamis are the ocean water surface waves generated by an undersea earthquake, a landslide, a volcanic eruption, or even a meteoric impact on water surface. As crust movement of the earth became active, the region along the rim of the Paciﬁc Ocean experienced large-scale tsunamis including the 1946 Alaskan tsunami, the 1960 Chilean tsunami, and the 1964 Alaskan tsunami, for instance. These large-scale tsunamis propagated across the Paciﬁc Ocean, and attacked the coastal area along the Paciﬁc rim. The damage including loss of human lives caused by these catastrophic tsunamis was well documented by Lander and Lockridge.3 More recent tsunamis in this area have been investigated by many scientists, and reported by Satake et al.4 for the 1992 Nicaragua tsunami, and Hokkaido Tsunami Survey Group5 for the 1993 Hokkaido tsunami. On the other hand, large-scale tsunamis including the 1992 and 1994 Indonesia tsunamis, 1995 Philippine tsunami, 1996 Indonesia tsunami, and 1998 Papua New Guinea tsunami were also generated along the rim of the Indian Ocean. In particular, the tsunami of magnitude 9.3 237

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was generated oﬀ the west coast of Sumatra in the Indian Ocean on 26 December 2004. The west coast of northern Sumatra was most severely damaged by this tsunami. The coastal areas along the rim of the Indian Ocean including Malaysia, Thailand, Myanmar, Bangladesh, India, Sri Lanka, Arabian, and African countries were attacked by this tsunami. More than 290,000 people were killed due to this tsunami event. And the property damages and injuries cannot be countable. In order to reduce such damages caused by tsunamis, a proper monitoring, forecasting, and warning system should be established. For this we need to fully understand the mechanism such as generation, propagation, and inundation of tsunamis, and predict the possible quantities such as arrival times and run-up heights of a tsunami event. Thus, the development of an accurate and eﬃcient numerical model for computing all the aspects of tsunami propagation is indispensable. When the tsunamis are generated by an earthquake, they have a shape of solitary wave with a long crest. After they propagate a long distance from the source area over a deep ocean, the tsunamis evolve into a train of waves due to the dispersion eﬀect of waves. When the source area is narrow, the dispersion eﬀect plays an important role in the deformation of tsunamis also in the relatively shallow water area. Boussinesq equations are one of the best choices for governing equations to construct a numerical model for the far-ﬁeld tsunamis, because they can take into account the dispersion eﬀects. The numerical models such as FUNWAVE6 and COULWAVE7,8 were developed based on a fully nonlinear and dispersive Boussinesq equations, and were applied to simulate the 26 December 2004 Indian Ocean tsunami.9,10 However, these Boussinesq models require a small mesh size to suppress the numerical dispersion error. The small mesh system consumes huge computer resources due to the implicit nature of the solution technique to deal with dispersion terms. Thus, the Boussinesq model is not preferred for the simulation of the far-ﬁeld tsunamis, and the shallow-water equations are generally employed instead. Considerable eﬀorts to construct a numerical model for the simulation of tsunamis have been made. For example, ﬁnite diﬀerence models were developed by Hwang and Divoky,11 Goto and Shuto,12 Kowalik and Murty,13 and Mader and Curtis.14 Most of the existing numerical models are based on the shallow-water equations and have been successfully applied to the near-ﬁeld tsunamis which have little dispersion eﬀect of waves. However, distant tsunamis generated far from the region of interest are mainly transformed due to the accumulation of dispersion eﬀects during the propagation. Thus, the numerical models based on the shallowwater equations will suﬀer from the lack of accuracy for the simulation of distant tsunamis. Imamura et al.15 presented a ﬁnite diﬀerence model for the simulation of transoceanic propagation of tsunamis. The model solves the shallow-water equations using a leap-frog scheme. The physical dispersion is compensated by the numerical dispersion introduced by the truncation error of the numerical scheme. This can be done only if the grid size is appropriately selected for the given water depth and the time step satisfying the criterion proposed by Imamura et al.15 Cho16 improved the numerical model of Imamura et al.15 for tsunamis obliquely propagating to

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the principal axes of computational grids. However, strictly speaking, those models have no capability to consider the dispersion eﬀect of waves in the case of practical application. The use of numerical models developed by Imamura et al.15 and Cho16 is limited to the case of constant water depth. Yoon17 developed a ﬁnite-diﬀerence scheme that uses a uniform grid, but the actual computations are made on a hidden grid of variable size determined from the condition proposed by Imamura et al.15 This model satisﬁes local dispersion relationships of waves for a slowly varying topography. However, the accuracy of the model becomes degraded when applied to short waves due to numerical errors associated with the interpolation of quantities at hidden grid points. Yoon et al.1 developed an eﬃcient and relatively accurate ﬁnite-diﬀerence model to simulate the propagation of distant tsunamis over varying topography. This ﬁnite-diﬀerence model solves a linear Boussinesq-type wave equation employing a uniform ﬁnite-diﬀerence grid. A considerable accuracy is achieved by eliminating the numerical dispersion error and by providing the physical dispersion. The dispersioncorrection is performed by controlling the parameters determined from the given time step, grid size, and local water depth. As a result, an accurate dispersioncorrection is possible although the grid size does not meet Imamura’s mesh condition. As the tsunamis approach the coastal area, they experience a shoaling due to decreasing water depth over a continental shelf. The wavelength becomes short, and the grid resolution becomes low near the coastline. Thus, the accuracy of the farﬁeld model proposed by Yoon et al.1 is degraded. This diﬃculty can be solved by nesting a ﬁner grid system to the coarse grid trans-oceanic model. As the tsunamis approach the coastline, nonlinearity and bottom friction play a signiﬁcant role in the transformation of tsunami waves. In this case, a near-ﬁeld model such as full Boussinesq or shallow-water equation model can be nested to the trans-oceanic model as described in Yoon17 and Lim et al.2 Thus, various nesting schemes between systems with diﬀerent grid sizes or governing equations are indispensable to extend the applicability of the far-ﬁeld numerical model. The objective of this chapter is to review a recent development of a practical ﬁnite-diﬀerence model to simulate tsunami propagation and runup for constructing inundation maps. The contents in this chapter are not an original work but a remake of research works presented by Yoon et al.1 and Lim et al.2,18 Section 10.2 describes governing equations and numerical scheme employed in the dispersion-correction ﬁnite-diﬀerence model developed by Yoon et al.1 and Lim et al.2,18 Section 10.3 presents the test of numerical model for uniform and variable depth cases. Section 10.4 describes the near-ﬁeld model. Section 10.5 introduces the grid nesting scheme developed by Lim et al.2,18 to extend the applicability of the numerical model. In order to evaluate the applicability of the present model to real tsunami events, a historical tsunami is simulated and the results are compared with the observations at two harbors along the east coast of Korea in Sec. 10.6. Then, the eﬀect of underwater topography on the propagation of tsunamis in this sea is analyzed based on the simulated results. Finally, this study is summarized in Sec. 10.7.

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10.2. Governing Equations and Numerical Scheme Tsunamis are generally regarded as long waves in comparison with the windgenerated waves. However, tsunamis can be short when the tsunami source is narrow. In this case the dispersion eﬀect can play an important role in the transformation of tsunami waves. The dispersion eﬀect, rather than the nonlinear or the bottom friction eﬀects, dominates the transformation of the far-ﬁeld tsunamis. For the far-ﬁeld tsunami, the linear Bousinessq equations, which neglect the nonlinear eﬀects but include the dispersion eﬀects, can be used as a governing equation. On the other hand, the nonlinear eﬀect and the bottom friction eﬀect become important for the transformation of tsunamis in the near-ﬁeld where the water is shallow near the coastal area. For the near-ﬁeld tsunami, the nonlinear shallowwater equations are generally employed as a governing equation. In the present numerical model, the linear Boussinesq-type wave equation proposed by Yoon et al.1 is employed for the far-ﬁeld tsunami propagation, and the nonlinear shallow-water equations with the bottom friction term are used for the near-ﬁeld tsunamis. For trans-oceanic propagation of tsunamis, the nonlinearity of waves can be neglected because the free surface displacement is much smaller in comparison with the water depth. However, the dispersion eﬀect of tsunami waves should be considered properly for the far-ﬁeld tsunamis. Thus, the following linear Boussinesq equations can be used as governing equations. ∂ζ ∂P ∂Q + + = 0, ∂t ∂x ∂y ∂ζ h2 ∂ ∂ ∂P ∂ ∂Q ∂P + gh = + ∂t ∂x 2 ∂x ∂x ∂t ∂y ∂t 2 ∂ P ∂2 Q h3 ∂ + , − 6 ∂x ∂t∂x h ∂t∂y h ∂Q ∂ζ h2 ∂ ∂ ∂P ∂ ∂Q + gh = + ∂t ∂y 2 ∂y ∂x ∂t ∂y ∂t 2 ∂ P ∂2 Q h3 ∂ + , − 6 ∂y ∂t∂x h ∂t∂y h

(10.1)

(10.2)

(10.3)

where ζ represents the free surface elevation from still water level (m), and P and Q are the depth-integrated volume ﬂuxes (m2 /s) in the x- and y-directions, respectively, g is the acceleration of gravity (m/s2 ), and h is the still water depth (m). The equations of motion (10.2) and (10.3) include the dispersion terms in the right-hand side. These dispersion terms cause numerical diﬃculties in practice because of the mixed form of diﬀerentiations with respect to both time and space. Consequently, it calls for the use of implicit scheme, which solves a matrix system. The implicit scheme requires very ﬁne grids to reduce the numerical dispersion errors inherent in the numerical scheme such as ﬁnite-diﬀerence method. A ﬁne grid

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needs the tremendous computational eﬀort with a huge computer memory space and excessive execution time. To develop an eﬃcient and relatively accurate numerical model for the propagation of dispersive tsunamis over slowly varying topography, Yoon et al.1 derived the following linear Boussinesq-type wave equation (LBTWE) from the linear Boussinesq equations (10.1)–(10.3) as ∂2ζ ∂ ∂ζ ∂ ∂ζ h2 ∂ 2 ∂ζ ∂ ∂ζ ∂ gh + gh − gh + gh − ∂t2 ∂x ∂x ∂y ∂y 3 ∂x2 ∂x ∂x ∂y ∂y ∂ ∂ζ ∂ ∂ζ ∂2 gh + gh = 0, (10.4) + ∂y 2 ∂x ∂x ∂y ∂y where the water depth is assumed to be slowly varying in the computational domain. This equation includes the dispersion terms in the last part. Equation (10.4) can be split into two ﬁrst-order partial diﬀerential equations in time as 2 ∂h ∂ζ ∂h ∂ζ ∂ ζ ∂v ∂ 2ζ =g + + gh + 2 , (10.5) ∂t ∂x ∂x ∂y ∂y ∂x2 ∂y 2 ∂ 2v ∂ v ∂ζ + , (10.6) = v − γ∆x2 ∂t ∂x2 ∂y 2 where v denotes an auxiliary variable introduced for computational convenience, and γ is the dispersion-correction parameter. The solution of (10.5) and (10.6) is identical with that of (10.4) if γ is set to be −h2 /3∆x2 . Since the linear Boussinesqtype wave equations (10.5) and (10.6) can be solved by a pure explicit numerical scheme, the dispersive tsunami propagation over a large domain can be simulated with low computational cost. Yoon et al.1 developed a dispersion-correction ﬁnite-diﬀerence model which solves the equations (10.5) and (10.6) using a relatively simple explicit numerical scheme given by n n n−1/2 n − vi,j hi+1,j − hni−1,j ζi+1,j − ζi−1,j +g ∆t 2∆x 2∆x n n n hi,j+1 − hni,j−1 ζi,j+1 − ζi,j−1 + 2∆y 2∆y n n n n n n ζi,j+1 + ζi−1,j − 2ζi,j + ζi,j−1 ζi+1,j − 2ζi,j + − gh (1 − α) ∆x2 ∆y 2 n n n n n n ζi+1,j+1 − 2ζi,j−1 + ζi−1,j−1 − 2ζi,j+1 + ζi−1,j+1 α ζi+1,j−1 + + 2 ∆x2 ∆x2 n n n n n n ζi+1,j+1 − 2ζi−1,j + ζi−1,j−1 − 2ζi+1,j + ζi+1,j−1 α ζi−1,j+1 + + = 0, 2 ∆y 2 ∆y 2 (10.7)

n+1/2

vi,j

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and n+1 n − ζi,j ζi,j n+1/2 − vi,j + γ∆x2 ∆t n+1/2 n+1/2 n+1/2 n+1/2 n+1/2 n+1/2 + vi−1,j + vi,j−1 vi+1,j − 2vi,j vi,j+1 − 2vi,j = 0, × + ∆x2 ∆y 2

(10.8)

where indices i (and j) and n represent a spatial grid point and a time level, respectively, as shown in Fig. 10.1. ∆x(= ∆y) is the spatial grid size, and ∆t denotes the time step. The weighting factor α to get orientation-free solution is given as 1/6, and the dispersion-correction parameter γ is given by γ=

∆x2 − (4h2 + gh∆t2 ) . 12∆x2

(10.9)

If the dispersion-correction parameter γ calculated by (10.9) for given local water depth, spatial grid size, and time step are used, a considerably accurate solution can be achieved for the linear Boussinesq wave equation (10.4). Thus, the selection of grid size ∆x is free from the condition proposed by Imamura et al.15 This means that the present dispersion-correction scheme is much more ﬂexible in the selection of grid size than the conventional scheme. The stability analysis of the dispersion-correction numerical scheme shows that the stability criterion is Cr ≤ 0.67 and −0.125 ≤ γ ≤ 0.083, where Cr is the Courant number. The lower limit of γ imposes a limitation on the ratio of grid size to water depth as 1.27. This means that the uniform grid size greater than approximately 1.27 times of a local water depth must be employed for the stability of the present dispersion-correction scheme. To satisfy the stability criterion of the dispersion-correction ﬁnite-diﬀerence model, the range of applicable water depth is limited if a uniform grid size is used for varying water depth. For practical purposes, this stability criterion can be solved by imposing intentionally a limitation on the

Fig. 10.1. Sketch of the arrangement of variables and grid points; (a) grid system in space, (b) grid system in time.

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γ value within the range of −0.125 ≤ γ ≤ 0.083. This means that the γ value is set to be −0.125 if the γ calculated by (10.9) is less than −0.125. The γ value is always less than 0.083. Thus, the upper limit of γ is automatically satisﬁed for very large grid size. 10.3. Test of Numerical Scheme 10.3.1. Tsunami propagation over constant depth region In order to test the applicability of the present model, the propagation of tsunamis is ﬁrst simulated with an initial Gaussian shape of water surface for the case of various constant water depths, and the computed free surface displacements are compared with the analytical solutions of the linear Boussinesq equations.19 The initial free surface proﬁle and the velocity of free surface movement are described by (10.10) and (10.11), respectively, as 2

ζ(r, θ) = 2e−(r/a) ,

(10.10)

∂ζ(r, θ) = 0, ∂t

(10.11)

where a is the characteristic radius of Gaussian function, r(= (x2 + y 2 )1/2 ) denotes the distance from the center of the Gaussian hump, and θ is the angle from the x-axis, as shown in Fig. 10.2. The analytical solution of the linear Boussinesq equations is given by Carrier19 as   √ ∞ 2 ghkt  ζ(r, t) = a2 e−(ak) /4 k cos  (10.12) J0 (kr)dk, 2 0 1 + (kh) 3 where J0 is the 0th order Bessel function of the ﬁrst kind and g = 9.81 m/s2 . The parameters used for the solution are a = 7500 m, ∆x = 2086 m, and ∆t = 6 s. The numerical simulations using the present scheme are performed for various constant water depths of h = 500 m, 1000 m, and 1500 m. Figures 10.3 and 10.4 present the comparisons of analytical solutions and numerical results calculated considering or neglecting the dispersion-correction

Fig. 10.2. Coordinate system and initial free surface proﬁle to test the accuracy of the present numerical scheme.

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Fig. 10.3. Comparison of free surface calculated using FDM and analytical solution for the case of h = 500 m (∆xIm = 1085 m, ∆x = 2086 m): (a) γ = 0.0, α = 0.015 ; (b) γ = 0.0, α = 1/616 ; (c) γ = 0.061, α = 1/6 (present).1

parameters γ and the weighting factor α. We remark here that the numerical scheme of Imamura et al.15 is recovered if we set intentionally γ = α = 0.0. The orientation-free scheme of Cho16 is obtained if γ = 0.0 and α = 1/6. These ﬁgures show a time history of free surface displacements at the location of 150∆x(r = 312, 900 m) far from the center of the initial Gaussian hump. The free surface proﬁles calculated by the present ﬁnite-diﬀerence model considering or neglecting dispersion-correction are compared with the analytical solution of Carrier19 for the case of 500 m water depth in Fig. 10.3.

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Fig. 10.4. Comparison of free surface calculated using FDM and analytical solution for the case of h = 1500 m (∆xIm = 3087 m, ∆x = 2086 m): (a) γ = 0.0, α = 0.015 ; (b) γ = 0.0, α = 1/616 ; (c) γ = −0.099, α = 1/6 (present).1

Since the grid size ∆x(= 2086 m) employed in the computationis larger than ∆xIm (= 1085 m) evaluated by Imamura condition, i.e., ∆xIm = 4h2 + gh∆t2 , the free surface proﬁles calculated without dispersion-correction are more dispersive than those of the analytical solutions as shown in Fig. 10.3(a). Especially, the numerical model gives a less dispersive proﬁle along the diagonal direction (θ = 45◦ ) than along principal axes (θ = 0◦ , 90◦ ), while the numerical solution with α = 1/6 shows no directional dependency as shown in Fig. 10.3(b). The present model with the dispersion-correction parameter γ = 0.061 and the weighting factor α = 1/6

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gives a good agreement with the analytical solution in the surface proﬁles in all directions as shown in Fig. 10.3(c). On the other hand, for the case of 1500 m water depth, ∆x(= 2086 m) is smaller than ∆xIm (= 3087 m). Thus, the numerical solutions calculated with γ = 0.0 show less dispersive nature in the surface proﬁles than the analytical solutions based on the linear Boussinesq equations as shown in Figs. 10.4(a) and 10.4(b). The present numerical model using γ = −0.099 and α = 1/6, however, still gives a correct dispersion eﬀect in each direction as shown in Fig. 10.4(c). From the results discussed above, it is concluded that the present numerical model is less sensitive to the choice of grid size than the models of Imamura et al.15 and Cho.16 As a result, the present model gives more ﬂexibility in selecting grid size. 10.3.2. Tsunami propagation over a submerged circular shoal In this section, the validity of the present ﬁnite-diﬀerence model (FDM) in the case of slowly varying topography is tested using a submerged circular shoal. Figure 10.5 shows a computational domain with a submerged circular shoal with a conical frustum shape. The center of the shoal is located at x0 = 500 km and y0 = 250 km. The base radius R1 and the top radius R2 are 150 km and 86 km, respectively. The bathymetry of the computational domain is given by  r ≥ R1 ,  1500 2 2 1500r /R1 R2 < r < R1 , h(r) = (10.13)  500 r ≤ R2 , where r = (x − x0 )2 + (y − y0 )2 is the distance from the center of the submerged circular shoal. Along x = 0, the initial water surface displacement in the form of a solitary wave is prescribed as 2

ζ(x) = 2e−(x/7500) .

(10.14)

Four wave gages are installed to measure the free surface proﬁle. Wave gages ➀ and ➂ are positioned at the front and back slopes of the shoal, respectively, where

Fig. 10.5.

Schematic diagram of a submerged circular shoal, wave gages, and initial condition.

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the water depth is 1000 m. Gage ➁ is located at the center of the shoal where the water depth is 500 m. Gage ➃ is placed behind the shoal. Sponge layers are placed along both ends of the computational domain to absorb the energy of outgoing waves. Since no analytical solution is known for this case, numerical solutions using the present dispersion-correction ﬁnite-diﬀerence model are compared with those of the linearized Nwogu’s Boussinesq equations20 implemented in the FUNWAVE code.6,21 FUNWAVE is a fully nonlinear Boussinesq wave model with improved dispersion relationships for short waves. The accuracy of FUNWAVE has been veriﬁed for various coastal problems such as shoaling, refraction, diﬀraction, and breaking of waves. The numerical simulation using FUNWAVE is performed with a uniform grid size of 500 m, the ﬁnest grid allowable on the personal computer employed in this study, to minimize the numerical dispersion. An additional computation using FUNWAVE is also conducted with 2000 m grid size to test the sensitivity of the grid resolution. On the other hand, the numerical simulation employing the present dispersion-correction model is made with a uniform grid size of 2000 m. The time step ∆t is determined from the stability criteria for each wave model. The algorithms to compute various physical processes, such as nonlinear advection, nonlinear dispersion, and wave dissipation due to bottom friction and breaking, are eliminated from the source code of the FUNWAVE model. Thus, the computational time for FUNWAVE model to calculate only the propagation step of small-amplitude waves can be measured for fair comparison with that of the present model. The numerical simulation is conducted for 9000 s after the initial water surface displacement imposed along x = 0 is released. The computational time elapsed for diﬀerent models is presented in Table 10.1. The FUNWAVE model employing a predictor-corrector scheme consumes a long computational time, while the present fully explicit model takes only 1/10 of the computational time required for FUNWAVE in the case of using the same grid size, i.e., ∆x = 2000 m. The computational eﬃciency of the present model can be realized even more dramatically if the computational time is compared with that of the FUNWAVE using ﬁner grid of ∆x = 500 m. The present model is approximately 2200 times faster than the FUNWAVE model. If the accuracy of the present model is comparable to that of FUNWAVE, it can be concluded that the present model is highly eﬃcient for practical problems. Figure 10.6 presents the comparison of the free surface time history at various gage locations computed by FUNWAVE and the present FDM. Figure 10.6(a) shows

Table 10.1.

Comparison of computational time for each simulation.1

Model

FUNWAVE

FUNWAVE

Present FDM

∆x (m) Number of grids ∆t (s) Number of time steps CPU time (s)

500 3001×1001 2 4500 156,888

2000 751×251 8 1125 770

2000 751×251 4 2250 72

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Fig. 10.6. Comparison of time history computed by FUNWAVE and the present FDM: (a) location ➀ (h = 1000 m); (b) location ➁ (h = 500 m); (c) location ➂ (h = 1000 m); (d) location ➃ (h = 1500 m).1

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the free surface time history recorded at the wave gage ➀ located on the front slope of the shoal. This ﬁgure shows that the initial solitary wave evolves into a group of waves due to dispersion of waves. Figure 10.6(b) presents the time history at the location ➁ on the top of the shoal. The free surface calculated by the present model shows a good agreement with that of the FUNWAVE with ﬁne grid of ∆x = 500 m. The results calculated using FUNWAVE with 2000 m grid size, however, show a considerable phase lag in the tail of wave train. This phase lag is caused by the accumulation of numerical dispersion errors due to poor resolution of computational grid near the top of the shoal. Although the same grid size of 2000 m is employed, the present model still gives more accurate results. This implies that the present model is capable of minimizing the numerical dispersion eﬃciently through the dispersion-correction algorithm. Figure 10.6(c) shows the time history at location ➂ on the back slope of the shoal. Two trains of waves arrive here. The ﬁrst train of waves propagating around the shoulders of the shoal meets at this location at t = 6000 s. The second train of waves traveling over the top of the shoal arrives here at t = 6300 s. A good agreement between the results calculated using the present model and the FUNWAVE with ﬁne grids is achieved. The results calculated using the FUNWAVE with coarse grids, however, suﬀers from the numerical dispersion. Figure 10.6(d) shows the time history at location ➃ behind the shoal where the water depth h is 1500 m. The agreements between the numerical solutions are reasonable. In summary, the present FDM is proven to be suﬃciently accurate in comparison with the FUNWAVE model which can deal with a full coverage of dispersion eﬀects for the varying water depth region. On top of this, the present model is shown to be highly eﬃcient. Thus, the present model can be used as a practical numerical model to simulate the propagation of trans-oceanic tsunamis over slowly varying topography.

10.4. Near-Field Model As the tsunamis propagate over a continental shelf and approach a coastal area, the dispersion eﬀect of waves becomes weak, and the nonlinearity and bottom friction of waves dominantly inﬂuence the transformation of the tsunamis. However, the linear Boussinesq-type wave equation does not include the nonlinear and bottom friction terms. Thus, the nonlinear shallow-water equations (NSWE) are employed for the near-ﬁeld transformation of tsunamis. ∂ζ ∂P ∂Q + + = 0, ∂t ∂x ∂y ∂P ∂ P2 ∂ PQ ∂ζ gn2 + + + gD + 7/3 P P 2 + Q2 = 0, ∂t ∂x D ∂y D ∂x D 2 ∂ PQ ∂ Q ∂ζ gn2 ∂Q + + + gD + 7/3 Q P 2 + Q2 = 0, ∂t ∂x D ∂y D ∂y D

(10.15) (10.16)

(10.17)

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where n denotes the Manning’s roughness coeﬃcient (sec/m1/3 ) for the bottom friction, and D(= h + ζ) is the total water depth. A conventional leap-frog ﬁnite diﬀerence scheme with wet–dry technique17 can be employed.

10.5. Grid Nesting Scheme In the central area of the East Sea of Korea, there exists a huge submerged shoal called Yamato Rise. The depth diﬀerence between shallow and deep areas is significant. To achieve the best performance from the dispersion-correction model, the grid size for deep area of 4 km water depth should be approximately 6 km. This grid size causes a problem of poor resolution for the shallow water area near the top of submerged shoal, because the wavelength becomes shorter due to shoaling. Thus, the accuracy of the simulation will be seriously degraded. To overcome this diﬃculty, a nesting between coarser grid for deep area and ﬁner grid for shallow area is indispensable. Yoon17 developed a dynamic grid nesting scheme for the ﬁnite-diﬀerence model to solve shallow-water equations. This grid nesting scheme uses a second-order interpolation in space and a ﬁrst-order interpolation in time. Numerical tests of the nesting scheme developed by Yoon17 show some reﬂection for short waves along the nesting line. To improve the performance of the nesting scheme, a new dynamic grid nesting scheme has been developed by Lim et al.2,18 for the present ﬁnite-diﬀerence model based on the linear Boussinesq-type wave equations. They employed a third-order interpolation in space and a second-order interpolation in time. The higher order interpolation helps short waves propagate with weak reﬂection from the nesting line. Figures 10.7(a) and 10.7(b) show the layout of variables in space and time, respectively, to nest the coarse and ﬁne grid regions. The

Fig. 10.7. Layout of variables for grid nesting; (a) grid nesting system in space, (b) grid nesting system in time.

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grid size and the time step for the ﬁne grid region are 1/3 of the values of the coarse grid region. Along the boundary between two regions, the information calculated in one region should be transferred to another region for each time step. When the information is transferred from the coarse grid region to the ﬁne grid region, both spatial and temporal interpolations of the variables of the coarse grid region are required. Lim et al.2 tested their grid nesting scheme for the propagation of a linear solitary wave over a submerged circular shoal with a signiﬁcant depth diﬀerence between the top and the base of the shoal. Numerical solutions calculated using their nested model were compared with those of the linearized Nwogu’s Boussinesq equations20 implemented in the FUNWAVE code.6,21 They showed that the present model system is approximately 4000 times faster than the FUNWAVE model, while the accuracy of the present model is comparable to that of FUNWAVE. Thus, it can be concluded that the present model system is highly eﬃcient for practical problems. To link the near-ﬁeld model using nonlinear shallow-water equations to the farﬁeld model employing linear Boussinesq-type wave equations, special care should be taken because the governing equations are diﬀerent between two regions. If the dispersion-correction is neglected along the nesting line, i.e., γ = 0.0, (10.6) can be simpliﬁed to give ∂ζ = v. ∂t

(10.18)

Furthermore, the continuity equation (10.15) of the nonlinear shallow-water equations can be rewritten as ∂ζ ∂P ∂Q =− + . (10.19) ∂t ∂x ∂y The right-hand side of (10.18) and (10.19) should be identical. Thus, the auxiliary variable v can be approximated as ∂Q ∂P + . (10.20) v=− ∂x ∂y The auxiliary variable v calculated in the far-ﬁeld propagation model using LBTWE is transferred to the near-ﬁeld model in the form of the divergence of ﬂow rate as a boundary condition to solve the continuity equation of NSWE. The auxiliary variable v required for the far-ﬁeld model can be computed by (10.20) using the ﬂow rates P and Q calculated in the near-ﬁeld model.

10.6. Numerical Simulations of Historical Tsunami Event A historical tsunami event, i.e., the 1993 Hokkaido Tsunami, is selected to verify the nested dispersion-correction ﬁnite-diﬀerence model (NDCFDM). The calculated free surface displacements are compared with the tide gage records obtained at two harbors: Mukho and Ulsan harbors, along the east coast of Korea.

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10.6.1. Computational information For the simulation of the historical tsunami event, the computational domain bounded by four latitudinal and longitudinal lines, i.e., 127◦ E, 142◦ E, 33◦ N, and ◦ 45 N for west, east, south, and north boundaries, respectively, are selected as shown in Fig. 10.8. This ﬁgure presents the bathymetry of the East Sea. For the ﬁnitediﬀerence model, the computational domain is divided into 900 × 931 square grids, for which the grid spacing is 1 min in the longitudinal direction. To make square grids, the grid spacing in the latitudinal direction is adjusted to give the same grid size of 1 min grid spacing in the longitudinal direction at each grid location.18 This can be done if the following condition is satisﬁed. ∆φ = cos φ∆ψ,

(10.21)

where φ and ψ represent the latitude and longitude of grid point, respectively. ∆φ and ∆ψ are the radian grid size in latitudinal and longitudinal directions, respectively. Since ∆ψ is chosen as 1 min, ∆φ is obtained by multiplying ∆ψ by cos φ, and ∆x is identical with ∆y everywhere. Thus, the actual grid size is given by ∆x = ∆y = R cos φ∆ψ,

(10.22)

where R(= 6.378 × 106 m) is the radius of the earth. Subsequently, the ﬁnitediﬀerence mesh system taken into account the curvature of the earth can be regarded

Fig. 10.8.

Computational domain and bathymetry of the East Sea (depth unit: m).

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as locally uniform, and the actual grid size slowly changes in association with the latitude of grid point. It is approximately 1.4 km around the central part of computational domain. The time step ∆t is chosen as 3 s, and the actual grid size ∆x slowly varies in the range of 1312–1556 m. The dispersion-correction parameters γ calculated by given conditions such as time step ∆t, actual spatial grid ∆x, and local depth h, give the values between −2.93 and 0.08 in the whole computational domain. As mentioned above, the minimum value of γ is restricted to −0.125 to maintain the stability for the deep areas of the East Sea. Due to this limitation, the dispersion eﬀects are underestimated in the numerical solutions. However, the present scheme gives more accurate solutions than the conventional models which use a uniform γ value, i.e., γ = 0.0, everywhere. As they approach the shallow-water region, the wavelength of tsunamis becomes short due to shoaling, and the resolution of grids becomes lower. Thus, a set of ﬁner grid system, B, C, D, and E, is nested dynamically to the coarse grid region A. As shown in Fig. 10.8, two subregions, Mukho (C1) and Ulsan (C2) harbors, are selected for computations with ﬁner grid system, because tide gage records are available at these harbors. The computational information such as grid size and time step employed for the nesting is listed in Table 10.2. The subregions are coupled by the nesting scheme as proposed above. The linear Boussinesq-type wave equation (LBTWE) is employed to compute the regions, A, B, C, and D. For the ﬁnest grid regions, E1 and E2, the nonlinear shallow-water equations (NSWE) are solved because the nonlinearity increases as the water depth decreases. Manning-type energy dissipation for bottom friction is implemented in the nonlinear model. The Manning’s roughness coeﬃcient is set to be equal to 0.05 s/m1/3 . To verify the applicability of the present model to the real topography, the calculated tsunami-free surface elevation is compared with the elevation measured by tide gage records at two harbors, Mukho and Ulsan harbors. The bathymetry of computational subregions and the locations of tide gage at Ulsan harbor are presented in Fig. 10.9. Two kinds of boundary conditions, no-ﬂux boundary and absorbing boundary, are mainly used in the tsunami propagation model. No-ﬂux boundary condition is employed to express the rigid impermeable wall, and an absorbing boundary condition proposed by Larsen and Darcy22 is placed in front of the open-sea boundary. The fault parameters of the historical tsunami event, such as epicenter location ◦ ◦ in latitude ( N) and longitude ( E), depth of upper rim of fault H(km), orientation

Table 10.2.

Computational information for subregions of each harbor.

Region

Grid spacing ∆ψ

Time step ∆t (s)

Remark

A B C1, C2 D1, D2 E1, E2

1 min 20 s 6.67 s 2.22 s 0.74 s

3.0 1.0 0.33 0.11 0.037

LBTWE LBTWE LBTWE LBTWE NSWE

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Fig. 10.9.

Computational subregions and detailed bathymetry near Ulsan harbor (depth unit: m).

Fig. 10.10.

Deﬁnition sketch for fault parameters of three historical tsunami events.

◦

◦

◦

of fault with strike angle θ( ), dip angle δ( ), and slip angle λ( ) from the north, length of fault L(km), width of fault W (km), and dislocation of fault D(m), are deﬁned in Fig. 10.10. It is assumed that the water surface displacement created by the earthquake is identical to the vertical displacement of sea bed induced by the ground motion.

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The displacement ﬁeld can be determined by using the fault model proposed by Mansinha and Smylie.23 10.6.2. Simulation of the 1993 Hokkaido Tsunami The magnitude of the 1993 Hokkaido Tsunami which originated from the deep sea near the Okushiri Island of Hokkaido on 12 July 1993 was recorded as M7.8. The source is composed of three faults. The south fault has the largest dislocation of 12 m. The fault parameters for the tsunami event proposed by Takahashi et al.24 are listed in Table 10.3. Figure 10.11 presents the initial free surface proﬁle of the 1993 Tsunami. The 1993 Tsunami is simulated. Figure 10.12 shows the propagation map with 5 min time interval obtained from the numerical simulation. Figure 10.13 shows the distribution of the highest water levels obtained from the numerical simulation Table 10.3.

Fault parameters for the 1993 Tsunami.24

Fault No.

Lat. ◦ ( N)

Long. ◦ ( E)

H (km)

θ ◦ ( )

δ ◦ ( )

λ ◦ ( )

L (km)

W (km)

D (m)

1 2 3

42.10 42.34 43.13

139.30 139.25 139.40

5 5 10

163 175 188

60 60 35

105 105 80

24.5 30 90

25 25 25

12.00 2.50 5.71

Fig. 10.11.

Initial free surface proﬁle of the 1993 Tsunami (unit: m).2

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Fig. 10.12.

Fig. 10.13.

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Propagation map of the 1993 Tsunami (unit: min).2

Distribution of calculated highest water levels due to the 1993 Tsunami (unit: m).2

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Fig. 10.14. Comparison of calculated and measured time history of free surface displacements of the 1993 Tsunami at (a) Mukho harbor and (b) Ulsan harbor.2

of the 1993 Tsunami. The Shimane Peninsula (JF) of Japan and the Imwon area (KA) of the east coast of Korea are shown to be vulnerable to the tsunami attacks generated in the East Sea because of the large wave heights due to the wave trapping over the Yamato Rise (YM) and the submerged ridges. The calculated time histories of free surfaces at two harbors are compared with the tide gage records of the 1993 Tsunami in Fig. 10.14. The calculated results are generally in good agreement with measured data in many respects even though there are slight phase and height diﬀerences. 10.6.3. Eﬀect of topography on propagation of tsunamis As shown in Fig. 10.15, the underwater topography of the East Sea of Korea is complicated. In the central part of the sea there exists a huge submerged shoal called Yamato Rise (YM). This rise is connected to the Shimane Peninsula (JF) of Japan by a submerged ridge. Two submerged ridges which protrude eastward from the Imwon area (KA) on the east coast of Korea form a K-shaped underwater topography. These underwater rises and ridges play a signiﬁcant role of a waveguide for the tsunamis generated in this sea. Figure 10.15 shows a sketch of paths along which the main tsunami energy is transmitted for the case of the 1993 Tsunami. The tsunami waves are ﬁrst captured by Yamato Rise due to topographical lens

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Fig. 10.15.

Schematic diagram of energy stream path for the 1993 Tsunami.2

eﬀect, and then trapped again by the underwater ridge connecting the Yamato Rise and the Shimane Peninsula of Japan. As a result, the main stream of tsunami energy radiated toward the open sea turns back to the Shimane Peninsula (JF) of Japan. On the other hand, the K-shaped submerged ridges catch the tsunami energy propagating toward the east coast of Korea and carry it along the two branch ridges toward the Imwon area (KA) where two ridges meet. Thus, the tsunami energy is focused at this spot. Even though these regions, i.e., Shimane Peninsula (JF) of Japan and Imwon (KA) of Korea, are located far from the tsunami source area, these two regions are vulnerable to the attack of various tsunamis generated in the East Sea of Korea.

10.7. Conclusions A practical ﬁnite-diﬀerence model developed by Yoon et al.1 and Lim et al.2,18 for the eﬃcient and relatively accurate simulation of tsunami propagation is introduced. This model solves a linear Boussinesq-type wave equation to achieve proper dispersion eﬀect of waves. The present model is applied to simulate the propagation of a historical tsunami event occurred in the East Sea of Korea. The calculated free surface displacements for the 1993 Hokkaido Tsunami are compared with the observations at two tidal stations along the east coast of Korea. The comparison shows

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that the results agree well with the observations. Based on the simulated results of the historical tsunami that occurred in the East Sea of Korea, it is found that the underwater topography, such as submerged rises and ridges, plays an important role in the propagation of tsunamis in this region. The Yamato Rise located in the central part of the sea behaves as a topographic lens to focus the tsunami energy, and the submerged ridge connecting the rise to the Shimane Peninsula of Japan behaves like a waveguide along which the focused energy is trapped and carried to the Shimane Peninsula. The K-shaped submerged ridges exert a similar inﬂuence on the tsunami incident on the Imwon area in the east coast of Korea.

Acknowledgment This research was supported by a grant (No. NEMA-06-NH-06) from the Natural Hazard Mitigation Research Group, National Emergency Management Agency of Korea.

References 1. S. B. Yoon, C. H. Lim and J. Choi, Dispersion-correction ﬁnite diﬀerence model for simulation of transoceanic tsunamis, Terres. Atmos. Oceanic Sci. CGU 18(1), 31–53 (2007). 2. C. H. Lim, J. S. Bae, J. I. Lee and S. B. Yoon, Propagation characteristics of historical tsunamis that attacked the east coast of Korea, Natural Hazards 47(1), 95–118 (2008). 3. J. F. Lander and P. A. Lockridge, United States Tsunamis (including United States Possessions) 1690–1988 (National Geophysical Data Center, Boulder, Colorado, 1989). 4. K. Satake, J. Bourgeois, K. Abe, Y. Tsuji, F. Imamura, Y. Iio, H. Katao, E. Noguera and F. Estrada, Tsunami ﬁeld survey of the 1992 Nicaragua Earthquake, EOS Trans. AGU 74, 156–157 (1993). 5. Hokkaido Tsunami Survey Group, Tsunami devastates Japanese coastal region, EOS Trans. AGU 74, 417, 432 (1993). 6. J. T. Kirby, G. Wei, Q. Chen, A. B. Kennedy and R. A. Dalrymple, FUNWAVE 1.0 Fully nonlinear Boussinesq wave model, Documentation and User’s Manual (CACR98-06) Technical report, Center for Applied Coastal Research, Ocean Engineering Laboratory, University of Delaware (1998). 7. P. Lynett and P. L.-F. Liu, A two-layer approach to water wave modeling, Proc. Roy. Soc. London A 460, 2637–2669 (2004). 8. S.-C. Hsiao, P. Lynett, H.-H Hwung and P. L.-F. Liu, Numerical simulations of nonlinear short waves using the multi-layer model, J. Eng. Mech. ASCE 131(3), 231–243 (2005). 9. M. Ioualalen, J. Asavanant, N. Kaewbanjak, S. T. Grilli, J. T. Kirby and P. Watts, Modeling the 26 December 2004 Indian Ocean tsunami: Case study of impact in Thailand, J. Geophys. Res. 112, C07024 (2007), doi:10.1029/2006JC003850. 10. S. T. Grilli, M. Ioualalen, J. Asavanant, F. Shi, J. T. Kirby, P. Watts, Source constraints and model simulation of the December 26, 2004, Indian Ocean tsunami, J. Waterway, Port, Coastal Ocean Eng. ASCE 133(6), 414–428 (2007).

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11. L. S. Hwang and D. Divoky, Tsunami generation, J. Geophys. Res. 75, 6802–6817 (1970). 12. C. Goto and N. Shuto, Numerical simulation of tsunami propagations and run-up, Tsunami — Their Science and Engineering, eds. K. Iida and T. Iwasaki (Terra Science Publishing Company, Tokyo, 1983), pp. 439–451. 13. Z. Kowalik and T. S. Murty, Computation of tsunami amplitudes resulting from a predicted major earthquake in the Shumagin seismic gap, Geophys. Res. Lett. 11, 1243–1246 (1984). 14. C. L. Mader and G. D. Curtis, Numerical modeling of tsunami inundation of Hilo harbor, JIMAR Contribution No. 91–251, University of Hawaii, Honolulu (1991). 15. F. Imamura, N. Shuto and C. Goto, Numerical simulation of the transoceanic propagation of tsunamis, Proc. 6th Congress Asian and Pacific Regional Division, IAHR, Japan (1988), pp. 265–271. 16. Y. S. Cho, Numerical simulations of tsunami propagation and run-up, PhD thesis, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY (1995). 17. S. B. Yoon, Propagation of distant tsunamis over slowly varying topography, J. Geophys. Res. 107(C10), 3140 (2002), doi:10.1029/2001JC000791. 18. C. H. Lim, J. S. Bae, Y. J. Jeon and S. B. Yoon, Grid nesting for dispersion-correction ﬁnite diﬀerence model for tsunami simulation, Proc. 32nd Congress of the International Association of Hydraulic Engineering and Research, Venice, Italy, Paper No. D2.b-050-O (2007). 19. G. F. Carrier, Tsunami propagation from a ﬁnite source, Proc. 2nd UJNR Tsunami Workshop, NGDC, Hawaii (1991), pp. 101–115. 20. O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, J. Waterway, Port, Coastal Ocean Eng. 119(6), 618–638 (1993). 21. G. Wei and J. T. Kirby, A time-dependent numerical code for the extended Boussinesq equations, J. Waterway, Port, Coastal Ocean Eng. 121(5), 251–261 (1995). 22. J. Larsen and H. Dancy, Open boundaries in short wave simulations — A new approach, Coastal Eng. 7, 285–297 (1983). 23. L. Mansinha and D. E. Smylie, The displacement ﬁelds of inclined faults, Bull. Seismol. Soc. Am. 61, 1433–1440 (1971). 24. T. Takahash, N. Shuto, F. Iammura and M. Ortis, Fault model to describe Hokkaido Nansei oﬀshore earthquake for tsunami, J. Coastal Eng. 41, 251–255 (1994) (in Japanese).

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Chapter 11

Tsunami-Induced Forces on Structures Ioan Nistor∗ , Dan Palermo, Younes Nouri, Tad Murty and Murat Saatcioglu Department of Civil Engineering, University of Ottawa 161 Louis Pasteur, CBY, A115, Ottawa, Canada ∗ [email protected] This chapter deals with the estimation of tsunami-induced hydrodynamic forces on infrastructure located in the vicinity of the shoreline. While extensive research has been conducted on the impact of hydrodynamic forces on classical coastal protection works (breakwaters, seawalls, reefs, etc.), there is limited research on their impact on structures such as buildings and bridges located inland. The devastation brought by the 26 December 2004 Indian Ocean Tsunami on coastal communities in Indonesia, India, Sri Lanka, Thailand, and other countries outlined the urgent need for research on the evaluation of structural resilience of infrastructure located in tsunami-prone areas. This chapter summarizes the state-ofthe-art knowledge with respect to forces generated by tsunami-induced hydraulic bores, including debris impact. Further, sample calculations of tsunami loading on a prototype structure are presented.

11.1. Introduction Tsunami waves represent extreme, often catastrophic events, which signiﬁcantly and adversely impact coastal areas. In spite of the lower frequency of occurrence comparing to storms and storm-induced surges, tsunami-induced coastal ﬂooding often leads to massive casualties and tremendous economic losses.1–3 Hence, tsunamis are rare events, high-impact natural disasters. The devastating eﬀects of the 26 December 2004 Tsunami on many countries bordering the Indian Ocean raised public concern and revealed existing deﬁciencies within the current warning and defense systems against tsunamis. One of the important elements that needs signiﬁcant improvement is the estimation of forces generated by tsunami-induced bores, as well as water-borne debris. Before the 2004 Indian Ocean Tsunami, the design of structures against tsunami-induced forces was considered of minor importance when compared to the attention given to tsunami warning systems. This was due to the assumption that tsunamis are 261

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(a)

(b)

(c)

(d)

Fig. 11.1. Tsunami damage in Thailand and Indonesia (December 2004 Indian Ocean Tsunami): (a) severe structural damage, Khao Lak, Thailand; (b) column failure of a reinforced concrete frame, Phuket, Thailand; (c) column failure due to debris impact, Banda Aceh, Indonesia; (d) punching failure of inﬁll walls, Banda Aceh, Indonesia.4

rare events, with signiﬁcantly high return periods (sometimes more than 500 years). Reconnaissance missions of the December 2004 Indian Ocean Tsunami disaster revealed that tsunami-induced forces can lead to severe damage or collapse of structures as shown in Fig. 11.1.3–11 Therefore, these forces should be properly accounted for in the design of infrastructure built within a certain distance from the shoreline in tsunami-prone areas. The design of coastal structures such as breakwaters, jetties, and groins against waves is typically based on considering the eﬀect of breaking waves and their associated forces, and is well established. Unlike coastal structures, the evaluation of tsunami-induced hydrodynamic forces on structures used for habitation and/or economic activity, received little attention by researchers and engineers. Results of ﬁeld surveys conducted in the aftermath of the December 2004 Indian Ocean Tsunami in Indonesia and Thailand showed that poorly detailed concrete structures experienced severe damage.3,4 This highlighted the fact that the current structural design codes do not account for tsunami-induced forces and the impact of associated debris. Reinforced concrete structures have been observed to withstand tsunamis with acceptable low levels of damage.12 However, as shown in

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: The 2004 Indian Ocean Tsunami : Past Tsunamis Damage Partial Damage Withstand

0

1

2

3

4

5

6

7

Lack of data

20

Inundation Depth (m) Fig. 11.2. Relation between the inundation depth and degree of damage to reinforced concrete buildings.13

Fig. 11.2, inundation depths of more than 5 m can induce partial damage to concrete structures. Currently, there are no clearly established procedures to address the aforementioned forces. Moreover, signiﬁcant disagreement on existing empirical formulae fostered new research interest in an eﬀort to properly address the inclusion of both tsunami-induced forces and the impact of debris into design codes. Aspects related to these forces are discussed in this chapter. Some of the shortcomings and inconsistencies of existing codes are also highlighted.

11.2. Literature Review 11.2.1. Tsunami-induced hydraulic bores As tsunami waves advance toward the shoreline and water depth decreases, their wave height increases while celerity decreases. Tsunami waves may break oﬀshore or at shoreline, inundating low-lying coastal areas in the form of a hydraulic bore. On the other hand, tsunami inundation can also occur as a gradual rise and recession of the sea level for the case of nonbreaking tsunami waves. The width of the continental shelf, the initial tsunami wave shape, the beach slope, and the tsunami wave length are the parameters which govern the breaking pattern of tsunami waves.2,15 Figure 11.3 shows a tsunami wave approaching the Khao Lak Beach in Thailand during the December 2004 Indian Ocean Tsunami. Tsunami waves have a larger horizontal length scale compared to the vertical. Consequently, implementing shallow water wave theory (i.e., depth-integrated equations of momentum and mass conservation with the assumption of a hydrostatic pressure ﬁeld) seems to be a reasonable method for representing tsunami wave propagation. Using these equations, it has been shown that the behavior and runup of nonbreaking tsunami waves can be predicted with acceptable accuracy. However, disagreement has been observed between experiment and prediction, in terms of behavior and runup of breaking waves and the resulting bore.16,17

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(a)

(b)

(c)

(d)

Fig. 11.3. Tsunami wave in Khao Lak, Thailand (December 2004 Indian Ocean Tsunami): (a) water recedes; (b) waves approach the shoreline; (c) tsunami waves break close to the shoreline; (d) tsunami waves inundate the shoreline.14

Signiﬁcant eﬀorts were directed toward the experimental investigation of the mechanisms of tsunami bore runup.2,18,19 Although a two-dimensional dam-break phenomenon was used in the experiments, the bore motion was observed to be fully three-dimensional and highly turbulent. This is in agreement with other observations regarding the irregularity of the bore front in transverse direction and the noticeable ﬂuctuation of the front propagation.19 The onshore propagation of the tsunami wave is similar to the classical dambreak problem. Chanson20 compared the instantaneous free surface ﬂow proﬁles of a tsunami-induced bore with ﬂoating bodies to a dam-break ﬂow on a horizontal bed. A frame-by-frame analysis of a video recording taken during the Indian Ocean Tsunami in Banda Aceh, Indonesia was used to obtain the ﬂow proﬁle of the tsunami-induced bore. The agreement between the tsunami ﬁeld data and the dam-break analytical formulation demonstrated the analogy between propagation of tsunami-induced bores and dam-break ﬂow. Direct estimation of tsunami inundation bore velocities is limited. Tsutsumi et al.21 estimated the nearshore ﬂow velocity of the tsunami caused by the Southwest Hokkaido Earthquake of 12 July 1993. Forces exerted on a bent iron handrail and an iron guardrail were estimated using in situ strength tests. Then, the velocity of the tsunami ﬂow was calculated from the forces using Morison’s equation.22

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Matsutomi et al.13 and Chanson20 estimated inundation ﬂow velocities in terms of inundation depth. Frame-by-frame analysis of video recordings from the 2004 Indian Ocean Tsunami in Banda Aceh was conducted. The videos showed that the inundation ﬂow carried numerous ﬂoating bodies with approximately the same speed as that of the bore. This may have inﬂuenced the bore characteristics. Furthermore, Matsutomi et al.13 developed an empirical model that predicts the bore front velocity. Also, surveyed data were used to improve the criterion for estimating the degree of structural damage.

11.2.2. Tsunami-induced forces on structures Snodgrass et al.23 noticed that broken waves induced larger hydrodynamic horizontal forces on a test pile compared to waves breaking at the pile location. As previously mentioned, broken tsunami waves inundate shoreline in the form of a hydraulic bore, which is a fast moving body of water with an abrupt front. However, mechanisms of impingement of broken tsunami waves on structures located inland are not yet well understood. Pioneering analytical and experimental attempts to quantify forces due to bores date back to Stoker,24 Cumberbatch,25 Fukui,26 Cross,27 and Dames and Moore.28 Comprehensive experimental investigation of the interaction of bores and drybed surges with a vertical wall was performed by Ramsden and Raichlen29 and Ramsden.30,31 In these experiments, three ﬂow conditions were analyzed: (1) turbulent bores (initial still water downstream of the gate); (2) dry-bed surges (no initial water depth downstream of the gate); and (3) solitary waves. Forces and overturning moments due to bores and dry-bed surges were recorded and calculated, respectively. The results of Ramsden’s studies are not applicable to the estimation of impulsive forces.31 It was observed that the pressure distribution during impact is essentially nonhydrostatic. The experiment also demonstrated that the transition from undular to turbulent bores led to a discontinuous increase in water-surface slope, followed by an increase in measured runup, pressure head, and exerted forces and moments. Figure 11.4 shows the diﬀerence between a strong turbulent bore and a dry-bed surge with approximately the same celerity. It was shown that recorded forces gradually increased to an approximately constant value for both the case of a surge and a bore. No impulsive (shock) force exceeding the hydrodynamic force was observed. However, an initial impulsive pressure equal to three times the pressure head, corresponding to the measured runup, was recorded. Ramsden31 further derived empirical formulae for the maximum force and moment exerted on a vertical wall due to the bore impact [Eqs. (11.1) and (11.2)]. 3 H 1 + , 7160 h 2 3 1 H H M H 1 + = 1.923 + 0.454 + , Ml h 8.21 h 808 h F = 1.325 + 0.347 Fl

H h

1 + 58.5

H h

2

(11.1) (11.2)

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Fig. 11.4. Comparison of: (a) wave proﬁle; (b) runup; (c) pressure head; (d) force due to a strong turbulent bore and a dry-bed surge.31

where F is the force on the wall; Fl is the force on the wall due to a runup equal to twice the wave height, assuming hydrostatic pressure; H is the wave height at the wall; h is still water depth; M is the moment on the wall; and Ml is the moment corresponding to Fl . Okada et al.32 conducted a survey of previous studies on tsunami wave forces and pressures, and identiﬁed ﬁve empirical formulae for tsunami-induced forces or pressures. It was found that calculation of tsunami load on structures using these formulae would result in approximately the same magnitude of load. These formulae are as follows: • • • • •

Tsunami wave pressure without soliton breakup,33 Tsunami wave pressure with soliton breakup,33 Tsunami wave pressure without soliton breakup,34 Tsunami-induced wave forces on houses,35 Tsunami force exerted on houses.36

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Measured and nondimensional force history for a square column.37

Arnason37 measured forces exerted on rectangular, rhomboidal, and circular structures due to a hydraulic bore on a dry bed. It was observed that the surge force overshot the hydrodynamic force in the case of a square column for small bore heights (Fig. 11.5). However, no overshoot was recorded for the case of circular and rhomboidal columns. This is in agreement with the results obtained by Nouri et al.,38 where similar experiments with larger bore heights, up to three times those of Arnason,37 were performed (Fig. 11.6). Nouri et al.38 conducted experiments with the objective of estimating boreinduced forces on free-standing structural components. The eﬀect of other parameters such as upstream obstacles, ﬂow constrictions, and debris impact was also investigated. The structural components were subjected to a dam-break ﬂow generated by impoundment depths of 0.5, 0.75, and 0.85 m.

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350

Force (N)

250

150 h0 = 1.00 m h0 = 0.85 m

50

-50 10

Fig. 11.6.

12

14

16 Time (s)

22

60

Height of the structure (cm)

Height of the structure (cm)

20

Time history of exerted forces on a circular structure38 ; h0 is the impoundment depth.

60 50

t=0.000 s t=0.006 s t=0.006 s t=0.009 s

40 30 20 10

50

t=0.160 s t=0.170 s t=0.180 s t=0.190 s

40 30 20 10

0 -2

18

0

3

8

13

18

-2

3

8

13

18

P (kPa)

P (kPa)

Fig. 11.7. Variation of pressure distribution on the front face — circular structure38 ; t = 0.0 s is the instant when the bore impacts the structure.

The variation of the vertical distribution of pressure was measured. Selected snapshots from the variation of pressure distribution due to bore impact generated by an impoundment depth of 1.0 m are shown in Fig. 11.7. 11.2.3. Debris impact force Matsutomi39 performed small- and full-scale experiments on impact forces generated by driftwood on rigid structures. Dam-break waves generated in a small ﬂume carried pieces of lumber to the point of impact on a downstream wall. Also, full-scale experiments in which wooden logs impacted a frame were conducted in open air, and impact forces were measured. An empirical formula for estimating the impact force, F , was derived using regression analysis of collected data [Eq. (11.3)]. F = 1.6CM γw D 2 L

u √ gD

1.2

σf γw L

0.4 ,

(11.3)

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Fm/γ D2L

σf / γL = 2000 1500 120

1000 500 80

200 100 40

20

0 0.0

1.0

Fig. 11.8.

2.0

3.0

0.5

VA0/(gD)

Impact forces of wood logs for bores and surges.39

where γw is the speciﬁc weight of wood, D is the diameter of the log, L is the length of the log, CM is a coeﬃcient which depends on the ﬂow passing around the receiving wall (≈ 1.7 for bore or surge, and 1.9 for steady ﬂow), u is the velocity of the log at impact, and σf is the yield stress of the log. Figure 11.8 shows the design chart based on Eq. (11.3). Currently, three basic models are proposed for estimating the forces due to the impact of debris on structures, which are used by a few design codes. In these models, the impact force is calculated based on the mass and velocity of debris, while ignoring the mass and rigidity of the structure. However, other than the mass and velocity of debris, each model needs an additional parameter. The three models and their corresponding additional parameters are • Contact stiﬀness — stiﬀness between debris and structure, • Impulse–momentum — stopping time of debris after impact and time history of impact, • Work energy — distance traveled from where initial contact occurs, to where debris stops. Haehnel et al.40,41 used a single-degree-of-freedom model with eﬀective collision stiﬀness as an additional parameter. They reviewed the current models discussed above and demonstrated that none of the additional parameters are independent. Hence, the three models are equivalent, provided that additional parameters are appropriately selected. Further, small- and large-scale experiments were performed in order to develop the single-degree-of-freedom model. Small-scale tests were performed in a ﬂume where wooden logs were released into the ﬂow and impacted a load frame located further downstream. Large-scale experiments were performed in

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Fig. 11.9.

Eﬀect of impact orientation on force.40

a large basin where water was stationary and logs were placed on a movable carriage. The eﬀect of parameters such as added mass of the water and the eccentricity and obliqueness of the collision were also considered. Figure 11.9 shows the eﬀect of impact orientation on the measured force. It was found that the maximum impact force, Fi,max , can be calculated using Eq. (11.4). ˆ =u Fi,max = Maxkx

ˆ 1, km

(11.4)

where u is the impact velocity of the log, kˆ is the constant eﬀective stiﬀness between the log and the structure, and m1 is the mass of the log. Based on experiments, Haehnel et al.40 found the value of kˆ = 2.4 MN/m to be the representative for the upper envelope of the collected data.

11.3. Existing Design Codes The design of structures in ﬂood-prone areas has previously been investigated. However, few existing codes speciﬁcally address the design of onshore structures built in tsunami-prone areas. Design codes that speciﬁcally address tsunami-induced forces were introduced in order to suggest provisions for designing infrastructure in tsunami-prone areas. Post-tsunami ﬁeld investigations of the December 2004 Indian Ocean Tsunami are indicative of the extreme loads generated by tsunamiinduced ﬂoods,3,4 and have outlined the need for developing new design guidelines. Recent research work42 indicated that tsunami-induced loads are comparable or can exceed earthquake loads. Tsunami-induced forces and the impact of debris are not properly accounted for in the current codes, and signiﬁcant improvement is needed.

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At present, only four design codes and guidelines speciﬁcally account for tsunamiinduced loads as listed below: • FEMA 55: The code is adopted by the Federal Emergency Management Agency, the United States, and recommends formulae for tsunami-induced ﬂood and wave loads.43 • The City and County of Honolulu Building Code (CCH): The code, developed by the Department of Planning and Permitting of Honolulu, Hawaii, United States, makes provisions for regulations that apply to districts located in ﬂood and tsunami-risk areas.44 • Structural Design Method of Buildings for Tsunami Resistance (SMBTR): The code is proposed by the Building Center of Japan32 and outlines the structural design for tsunami refuge buildings. • Development of Guidelines for Structures that Serve as Tsunami Vertical Evacuation Sites: The guidelines were prepared by Yeh et al.45 for estimating tsunamiinduced forces on structures for the Washington State Department of Natural Resources. There are several other design codes (sometimes country-speciﬁc) which contain prescriptions and design guidelines for ﬂood-induced loads. Examples of widely used codes are indicated below: • 1997 Uniform Building Code, Appendix 33, proposed by the International Conference of Buildings Oﬃcials (ICBO),47 • ASCE 7-05 of the American Society of Civil Engineers,48 • 2006 International Building Code by the International Code Council.49 However, none of the above codes address directly the tsunami-induced forces, which represent the focus of this chapter. The reader is advised to refer to these codes when seeking guidance for the design of structures subjected to ﬂood-induced loads other than tsunamis: coastal ﬂooding due to storm surges, ﬂooding of river banks above bank-full conditions, etc. 11.3.1. Tsunami-induced forces A broken tsunami wave running inland generates forces which aﬀect structures located in its path. Three parameters are essential for deﬁning the magnitude and application of these forces: (1) inundation depth, (2) ﬂow velocity, and (3) ﬂow direction. The parameters mainly depend on: (a) tsunami wave height and wave period; (b) coast topography; and (c) roughness of the coastal inland. The extent of tsunami-induced coastal ﬂooding, and therefore the inundation depth at a speciﬁc location, can be estimated using various tsunami scenarios (magnitude and direction) and modeling coastal inundation accordingly. However, the estimation of ﬂow velocity and direction is generally more diﬃcult. Flow velocities can vary in magnitude from zero to signiﬁcantly high values, while ﬂow direction can also vary due to onshore local topographic features, as well as soil cover and obstacles. Forces associated with tsunami bores consist of: (1) hydrostatic force, (2) hydrodynamic

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(drag) force, (3) buoyant force, (4) surge force, and (5) impact of debris. A brief description of these forces is further presented. 11.3.1.1. Hydrostatic force The hydrostatic force is generated by still or slow-moving water acting perpendicular onto planar surfaces. The hydrostatic force per unit width, FHS , can be calculated using Eq. (11.5), where ρ is the seawater density, g is the gravitational acceleration, ds is the inundation depth, and up is the normal component of ﬂow velocity. Equation (11.5) is proposed by CCH and accounts for the velocity head. Alternatively, FEMA 55 does not include the velocity head in its formulation since it is assumed to be a negligible component of the hydrostatic force:

FHS

2 u2p 1 = ρg dS + . 2 2g

(11.5)

The point of application of the resultant hydrostatic force is located at one-third distance from the base of the triangular hydrostatic pressure distribution. In the case of a broken tsunami wave, the hydrostatic force is signiﬁcantly smaller than the drag and surge forces. Conversely, Dames and Moore28 noted that the hydrostatic force becomes important when tsunami is similar to a rapidly-rising tide. 11.3.1.2. Buoyant force The buoyant force is the vertical force acting through the center of mass of a submerged body. Its magnitude is equal to the weight of the volume of water displaced by the submerged body. The eﬀect of buoyant forces generated by tsunami ﬂooding was clearly evident during post-tsunami ﬁeld observations.1,5,6 Buoyant forces can generate signiﬁcant damage to structural elements, such as ﬂoor slabs, and are calculated as follows: FB = ρgV,

(11.6)

where V is the volume of water displaced by submerged structure. 11.3.1.3. Hydrodynamic (drag) force As the tsunami bore moves inland with moderate to high velocity, structures are subjected to hydrodynamic forces caused by drag. Currently, there are diﬀerences in estimating the magnitude of the hydrodynamic force. The general expression for this force is shown in Eq. (11.7). Existing codes use the same expression, but diﬀerent drag coeﬃcient values (CD ). For example, values of 1.0 and 1.2 are recommended for circular piles by CCH and FEMA 55, respectively. For the case of rectangular

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piles, the drag coeﬃcient recommended by FEMA 55 and CCH is 2.0: ρCD Au2 , (11.7) 2 where FD is the drag force acting in the direction of ﬂow, A is the projected area of the body normal to the direction of ﬂow, and u is the tsunami-bore velocity. The ﬂow is assumed to be uniform, and therefore, the resultant force will act at the centroid of the projected area. As indicated, the hydrodynamic force is directly proportional to the square of the tsunami-bore velocity. The estimation of the bore velocity remains one of the critical elements on which there is signiﬁcant disagreement in literature. A brief discussion on the tsunami-bore velocity is presented below. FD =

Tsunami-bore Velocity. Previous research shows that signiﬁcant diﬀerences in estimating forces exerted on structures by tsunami bores, as well as impact of debris, are due to diﬀerences in estimating bore velocity. Since the hydrodynamic force is proportional to the square of the bore velocity, uncertainties in estimating velocities induce large diﬀerences in the magnitude of the resulting hydrodynamic force. Tsunami-bore velocity and direction can vary signiﬁcantly during a major tsunami inundation. Current estimates of the velocity are crude; a conservatively high ﬂow velocity impacting the structure at a normal angle is usually assumed. Also, the eﬀects of runup, backwash, and direction of velocity are not addressed in the current design codes. Although there is certain consensus in the general form of equation for the hydrodynamic force, several researchers proposed diﬀerent empirical coeﬃcients. The general form of the bore velocity is shown below [Eq. (11.8)]: u = C gds , (11.8) where u is the bore velocity, ds is the inundation depth, and C is a constant coeﬃcient. Various formulations were proposed by FEMA 55 (based on Dames and Moore28 ), Iizuka and Matsutomi,35 CCH,44 Kirkoz,50 Murty,51 Bryant,52 and Camﬁeld53 for estimating the velocity of a tsunami bore in terms of inundation depth (Fig. 11.10). Velocities calculated using CCH and FEMA 55 represent a lower and upper boundary, respectively. 11.3.1.4. Surge force The surge force is generated by the impingement of the advancing water front of a tsunami bore on a structure. Due to lack of detailed experiments speciﬁcally applicable to tsunami bores running up the shoreline, the calculation of the surge force exerted on a structure is subject to substantial uncertainty. Accurate estimation of the impact force in laboratory experiments is a challenging and diﬃcult task. CCH recommends using Eq. (11.9), based on Dames and Moore28 : FS = 4.5ρgh2 , where FS is the surge force per unit width of wall and h is the surge height.

(11.9)

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16

CCH: u=h FEMA 55: u=2(gh)^0.5 Iizuka: u=1.1(gh)^0.5 Kirkoz: u=(2gh)^0.5 Murty: u=1.83(gh)^0.5 Bryant: u=1.67h^(0.7)

V (m/s)

12

8

4

0 0

2

4

6

d S (m) Fig. 11.10.

Comparison of various tsunami-bore velocities as a function of inundation depth.42

building qx Design inundation depth

3h z h

z 3ρgh x Fig. 11.11.

Tsunami wave pressure for structural design recommended by SMBTR.32

The point of application of the resultant surge force is located at a distance h above the base of the wall. This equation is applicable to walls with heights equal to, or greater than 3h. Structural walls with height less than 3h require surge forces to be calculated using an appropriate combination of hydrostatic and drag force for each speciﬁc situation. SMBTR recommends using the equation for tsunami wave pressure without soliton breakup derived by Asakura et al.33 [Eq. (11.10)]. The equivalent static pressure resulting from the tsunami impact is associated with a triangular distribution where water depth equals three times the tsunami inundation depth (Fig. 11.11): qx = ρg(3hmax − z),

(11.10)

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where qx is the tsunami wave pressure for structural design, z is the height of the relevant portion from ground level (0 ≤ z ≤ 3h), ρ is the mass per unit volume of water, and g is the gravitational acceleration. Integration of the wave pressure formula for walls with heights equal to or greater than 3h results in the same equation as the surge force formula recommended by CCH [Eq. (11.9)]. The magnitude of the surge force calculated using Eqs. (11.9) and (11.10) will generate a value equal to nine times the magnitude of the hydrostatic force for the same ﬂow depth. However, a number of experiments31,37 did not capture such diﬀerences in magnitude. Yeh et al.45 commented on the validity of Eq. (11.9) and indicated that this equation gives “excessively overestimated values.” On the other hand, Nakano and Paku46 conducted extensive ﬁeld surveys in order to examine the validity of the proposed tsunami wave pressure formula [Eq. (11.10)]. The coeﬃcient 3.0 in Eq. (11.10) was taken as a variable, α, and was calculated such that it could represent the boundary between damage and no damage in the surveyed data. A value of α equal to 3.0 and 2.0 was found for walls and columns, respectively. The former is in agreement with the proposed formulae by both CCH and SMBTR [Eqs. (11.9) and (11.10)]. The tsunami wave force may be composed of drag, inertia, impulse, and hydraulic gradient components. However, SMBTR does not specify diﬀerent components for the tsunami-induced force, and the proposed formula presumably accounts for other components. 11.3.1.5. Debris impact force A high-speed tsunami bore traveling inland carries debris such as ﬂoating automobiles, ﬂoating pieces of buildings, drift wood, boats, and ships. The impact of ﬂoating debris can induce signiﬁcant forces on a building, leading to structural damage or collapse.5,6 Both FEMA 55 and CCH codes account consistently for debris impact forces, using the same approach and recommend using Eq. (11.11) for the estimation of debris impact force: Fi = mb

dub ui =m , dt ∆t

(11.11)

where Fi is the impact force, mb is the mass of the body impacting the structure, ub is the velocity of the impacting body (assumed equal to the ﬂow velocity), ui is approach velocity of the impacting body (assumed equal to the ﬂow velocity), and ∆t is the impact duration taken equal to the time between the initial contact of the ﬂoating body with the building and the instant of maximum impact force. The only diﬀerence between CCH and FEMA 55 resides in the recommended values for the impact duration which has a noticeable eﬀect on the magnitude of the force. For example, CCH recommends the use of impact duration of 0.1 s for concrete structures, while FEMA 55 provides diﬀerent values for walls and piles for various construction types as shown in Table 11.1. According to FEMA 55, the impact force (a single concentrated load) acts horizontally at the ﬂow surface or at any point below it. Its magnitude is equal to

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Table 11.1. Impact duration of ﬂoating debris (FEMA 55). Impact duration (s) Type of construction

Wall

Pile

Wood Steel Reinforced concrete Concrete masonry

0.7–1.1 N.A. 0.2–0.4 0.3–0.6

0.5–1.0 0.2–0.4 0.3–0.6 0.3–0.6

the force generated by 455 kg (1000-pound) of debris traveling with the bore and acting on a 0.092 m2 (1 ft2 ) surface of the structural element. The impact force is to be applied to the structural element at its most critical location, as determined by the structural designer. It is assumed that the velocity of the ﬂoating body goes from ub to zero over some small ﬁnite time interval (∆t). Finding the most critical location of impact is a trial and error procedure that depends, to a large extent, on the experience and intuition of the engineer. 11.3.1.6. Breaking wave force Tsunami waves tend to break oﬀshore and approach shoreline as a broken hydraulic bore or a soliton, depending on the wave characteristics and coastal bathymetry. Consequently, classic breaking wave force formulae are not directly applicable to the case of tsunami bores. Hence, this chapter does not discuss the estimation of breaking wave forces. 11.3.2. Loading combinations for calculating tsunami-induced forces Based on the location and type of structural elements, appropriate combinations of tsunami-induced force components (hydrostatic, hydrodynamic, surge, buoyant, and debris impact force) should be used in calculating the total tsunami force. This is due to the fact that a certain element may not be subjected to all of these force components simultaneously. Loading combinations can signiﬁcantly inﬂuence the total tsunami force and the subsequent structural design. Unlike the case of tsunami waves, loading combinations for ﬂood-induced surges are well-established and have been included in design codes. The literature review revealed that proposed tsunami loading combinations must be signiﬁcantly improved and incorporated into new design codes. Tsunami-induced loads are diﬀerent from ﬂood-induced loads. Therefore, load combinations based on ﬂood surges are not directly applicable to tsunamis. Loading combinations proposed in the literature are as follows: (i) FEMA 55 does not provide loading combinations speciﬁcally for calculation of tsunami force. However, ﬂood load combinations can be used as guidance. Flood load combinations for piles or open foundations, as well as solid walls (foundation) in ﬂood hazard zones and coastal high hazard zones are presented

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as follows: Pile or open foundation: Fbrkp (on all piles) + Fi (on one corner or critical pile only), or Fbrkp (on front row of piles only) + Fdyn (on all piles but front row) + Fi (on one corner or critical pile only). Solid (wall) foundation: Fbrkw (on walls facing shoreline, including hydrostatic component) + Fdyn (assumes one corner is destroyed by debris), where Fbrkp , Fi , Fdyn , and Fbrkw refer to breaking force on piles, impact force, hydrodynamic force, and breaking force on walls, respectively. The reader can refer to FEMA 5543 for more details. (ii) Yeh et al.45 modiﬁed ﬂood load combinations provided by FEMA 55 and adapted them for tsunami forces as follows: Pile or open foundation: Fbrkp (on column) + Fi (on column), or Fd (on column) + Fi (on column), where Fd is the drag force. Solid (wall ) foundation (perpendicular to ﬂow direction): Fbrkw (on walls facing shoreline) + Fi (on one corner), or Fs (on walls facing shoreline) + Fi (on one corner), or Fd (on walls facing shoreline) + Fi (on one corner), where Fs is the surge force on walls. (iii) Dias et al.54 proposed two load combinations called “point of impact” and “post-submergence/submerged” (Fig. 11.12). These load combinations are based on two conditions: (i) the instant that tsunami bore impacts the structure, and (ii) when the whole structure is inundated. Point of impact: Fd (on walls facing shoreline) + Fs (on walls facing shoreline), where Fs is deﬁned as the hydrostatic force by Dias et al.54 Post-submergence/submerged: Fd (on walls facing shoreline) + Fb (on submerged section of the structure). The net hydrostatic force is zero and Fb (γV) is the buoyant force. (a)

(b)

Fd

Fd

Fs

Fs W

Fs W- V

Fig. 11.12. Loading combinations: (a) point of impact/not submerged; and (b) post-submergence/ submerged.54

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(a)

Fi FS

h

(b)

Fi Fd

dS

FHS Fig. 11.13.

FHS

Proposed loading conditions: (a) point of impact; and (b) post-impact.42

(iv) Nouri et al.42 proposed two new load combinations based on the two conditions considered by Dias et al.,54 as shown in Fig. 11.13. The proposed load combinations by Nouri et al.42 are adapted to follow a consistent format as the above combinations: Columns: Fs (on front row of piles only) + Fi (on one corner or critical column in the front row only), or Fd (on all piles) + Fi (on one corner or critical column only), where Fs is the surge force on walls. Solid (wall) foundation: Fs (on walls facing shoreline) + Fi (on walls facing shoreline), or Fd (on walls facing shoreline) + Fi (on one critical wall facing shoreline) + Fb (on submerged section of the structure). 11.4. Design Example Building codes provide guidance for the design of lateral force resisting systems subjected to wind and seismic excitations. Tsunami-induced loading is normally not considered. The objective of this example is to demonstrate the levels of lateral loading associated with tsunamis for a prototype reinforced concrete building located in a tsunami-prone area. Speciﬁcally, the loads generated by a tsunami bore are addressed. Other researchers have provided comparisons between tsunami loading and other lateral loads (Okada et al.,32 Pacheco and Robertson,55 Nouri et al.,42 and Palermo et al.56 ).

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6.0

6.0 m

279

6.0

6.0

6.0

A

B 6.0

Tsunami

6.0

C

D 1 Fig. 11.14.

2

3

4

5

6

Plan view of structural layout of reinforced concrete moment-resisting frame.56

The following example consists of a moment-resisting frame with simple geometry, as shown in Fig. 11.14. The thickness of the slab is assumed to be 200 mm, and the beams are 450 mm deep (including the thickness of the slab) and 300 mm wide. The exterior and interior columns are 450 mm and 500 mm square sections, respectively. The center-to-center storey heights are 3.65 m, and a 10-storey structure is considered. The components of the tsunami-induced forces are calculated based on CCH, FEMA 55, and SMBTR. The calculation of tsunami-induced loads require a number of assumptions, together with engineering judgment and lessons from reconnaissance missions in regions aﬀected by tsunamis. The authors assume that the net hydrostatic force exerted on the lateral system is zero in calculating the base shear. The surge and drag forces require an eﬀective area for load transfer to the lateral force resisting system. In this example, two scenarios are considered: (1) 100% breakaway walls, which expose the structural elements; and (2) nonbreakaway walls. In the latter, the exterior nonstructural elements remain intact. For breakaway walls, the external elements will be damaged and all the columns of the structure will be subjected to drag force. The forces are calculated for inundation depths of 1–5 m, and the structure is oriented with its short side parallel to the shoreline. When designing structures located in tsunami-prone areas, the inundation depth at a speciﬁc location should be obtained from tsunami inundation hazard maps, when available. Otherwise, numerical modeling based on various tsunami scenarios should be conducted in order to estimate the tsunami inundation depth. 11.4.1. Hydrodynamic (drag) forces For the case where 100% breakaway walls are assumed and the columns are exposed to the hydraulic bore, the drag forces are based on a drag coeﬃcient of 2.0 for square columns, as recommended by CCH and FEMA 55. For the second case, nonbreakaway walls are assumed to remain intact and the hydraulic bore impacts

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the entire surface of the building. In this situation, the drag coeﬃcients are taken as 1.5 and 1.25 for CCH and FEMA 55, respectively. 11.4.2. Debris impact forces To calculate debris impact, a mass of 455 kg is used to represent a ﬂoating object at the water surface. The mass used is consistent with the recommendations of CCH and FEMA 55. In this example, it is assumed that the debris will impact a single reinforced concrete column. CCH assumes a duration of 0.1 s for concrete structures. FEMA 55 recommends an impact duration ranging from 0.3 to 0.6 s for reinforced concrete piles or columns. Hence, impact duration of 0.3 s is assumed in the design example. 11.4.3. Surge forces In this example, the surge force is applied over the full length of the building in the direction of the tsunami for nonbreakaway walls. For the case of 100% breakaway walls, it is assumed that the surge force will develop on the four exterior columns which face the hydraulic bore. Note that the surge force is not applicable for FEMA 55 and that SMBTR assumes a diﬀerent surge force per unit width for columns and walls, as mentioned in Sec. 11.3.1.4. 11.4.4. Sample calculation: Breakaway walls The following is a sample calculation for breakaway walls for the given structure subjected to a tsunami inundation depth of 5 m: g = 9.81

m ; s2

ρ = 1030

kg ; m3

ds = 5 m.

Surge Force: kg m 9.81 2 (5 m)2 FS = 2.0ρgh = 2.0 1030 3 m s 2

N ; SMBTR m 6N (4(0.45 m)) = 909 × 103 N = 909 kN; FS × width = 0.505 × 10 m m kg 2 FS = 4.5 ρgh = 4.5 1030 3 9.81 2 (5 m)2 m s = 0.505 × 106

N ; CCH m 6N (4(0.45 m)) = 2046 × 103 N = 2046 kN. FS × width = 1.14 × 10 m = 1.14 × 106

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Drag Force: ρCD Au2 2

FD = u = C gds = 2

m m 9.81 2 (5 m) = 14 ; s s

u = ds = 5

FEMA 55

m ; s

CCH

CD = 2.0 Rectangular columns; A = (5 m)((16 × 0.45 m) + 8(0.50 m)) = 56 m2 ;

FD =

2 1030 kg/m3 (2)(56 m2 ) (14m/s) = 11,317 × 103 N = 11,317 kN; FEMA 55 2 2 1030 kg/m3 (2)(56 m2 ) (5 m/s) = 1442 × 103 N = 1442 kN. = 2

FD

CCH

Debris Impact Force: ui Fi = m = 455 kg ∆t

14m/s 0.3 s

Fi = 455 kg

5 m/s 0.1 s

= 21.2 × 103 N = 21 kN;

FEMA 55

= 22.8 × 103 N = 23 kN.

CCH

11.4.5. Sample calculation: Nonbreakaway walls The following is a sample calculation for nonbreakaway walls for the given structure subjected to a tsunami inundation depth of 5 m. g = 9.81

m ; s2

ρ = 1030

kg ; m3

ds = 5 m

Surge Force: FS =

1.14 × 106

N m

(18 m + 0.45 m)

= 20,973 × 103 N = 20,973 kN

SMBTR/CCH

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Drag Force: FD =

ρCD Au2 2

CD = 1.25 Walls

FEMA 55

CD = 1.5 Walls

CCH

A = (5 m)(18 m + 0.45 m) = 92 m2 ;

FD

1030 kg/m3 (1.25)(92 m2 ) (14 m/s)2 = 2 = 11,652 × 103 N = 11,652 kN;

FEMA 55

2 1030 kg/m3 (1.5) 92 m2 (5 m/s) = 1782 × 103 N = 1782 kN. = 2

FD

Debris Impact Force: Fi = m

14 m/s = 21.2 × 103 N = 21 kN; 0.3 s 5 m/s = 22.8 × 103 N = 23 kN. Fi = 455 kg 0.1 s

ui = 455 kg ∆t

CCH

FEMA 55 CCH

11.4.6. Results Tables 11.2 through 11.7 provide the results for the calculation of the individual force components for the structure considered using CCH, FEMA 55, and SMBTR, respectively. Given the force components, a loading combination must be speciﬁed in order to evaluate the maximum tsunami load that would be used for either design or analysis purposes. Yeh et al.45 suggested loading combinations that are applicable for tsunami loading. CCH does not speciﬁcally provide guidance to evaluate the maximum tsunami load. Nouri et al.42 proposed a two-part loading combination: Initial impact and Post-impact ﬂow. For this example, these loading combinations are similar to those of Nouri et al.42 Table 11.8 provides the results of the tsunami Table 11.2. Tsunami-induced force components based on CCH for breakaway walls.

Code CCH

Inundation depth (m)

Velocity (m/s)

Surge (kN)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

1 2 3 4 5

82 327 737 1310 2046

12 92 311 738 1442

5 9 14 18 23

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Table 11.3. Tsunami-induced force components based on CCH for nonbreakaway walls.

Code

Inundation depth (m)

Velocity (m/s)

Surge (kN)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

1 2 3 4 5

839 3356 7550 13,423 20,973

14 114 385 912 1782

5 9 14 18 23

CCH

Table 11.4. Tsunami-induced force components based on FEMA 55 for breakaway walls.

Code FEMA 55

Inundation depth (m)

Velocity (m/s)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

6 9 11 13 14

453 1811 4074 7243 11,317

10 13 16 19 21

Table 11.5. Tsunami-induced force components based on FEMA 55 for nonbreakaway walls.

Code FEMA 55

Inundation depth (m)

Velocity (m/s)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

6 9 11 13 14

466 1864 4195 7457 11,652

10 13 16 19 21

load calculation for CCH, FEMA 55 and SMBTR for an inundation depth of 5 m based on loading combinations of Nouri et al.42 For the prototype moment-resisting frame structure with the short side perpendicular to the advancing bore, it is apparent that nonbreakaway walls or rigid exterior nonstructural components can lead to large design base shears. CCH and SMBTR estimate signiﬁcantly larger base shears relative to FEMA 55 due to the omission of a surge component in FEMA 55. It is evident that the width of exposed surfaces aﬀects the magnitude of total forces exerted on a structure. Therefore, it would be prudent to orient buildings such that the short side is placed parallel to the shoreline. Furthermore, using breakaway or ﬂexible walls at the lower level would reduce the lateral force that is transmitted to the lateral force resisting system. Note that although the debris impact force is a negligible component in the base shear,

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Table 11.6. Tsunami-induced force components based on SMBTR for breakaway walls. Code SMBTR

Inundation depth (m)

Surge (kN)

1 2 3 4 5

36 146 327 582 909

Table 11.7. Tsunami-induced force components based on SMBTR for nonbreakaway walls. Code SMBTR

Inundation depth (m)

Surge (kN)

1 2 3 4 5

839 3356 7550 13,423 20,973

Table 11.8. Tsunami-induced load based on CCH, FEMA 55, and SMBTR for 5 m inundation depth.

Standard

Case

CCH CCH FEMA 55 FEMA 55 SMBTR SMBTR

Breakaway Nonbreakaway Breakaway Nonbreakaway Breakaway Nonbreakaway

Surge + Impact (kN)

Drag + Impact (kN)

Tsunami load (kN)

2069 20,995 N.A. N.A. 909 20,973

1465 1804 11,338 11,673 N.A. N.A.

2069 20,995 11,338 11,673 909 20,973

it could be critical in the design of individual structural components subjected to the debris impact.

Acknowledgment Special thanks to Dr Andrew Cornett, Group Leader — Coastal, Ports and Oﬀshore at the Canadian Hydraulics Centre, National Research Council of Canada in Ottawa, Canada, for his valuable advice.

References 1. D. C. Cox and J. F. Mink, Bull. Seism. Soc. Am. 53, 1191–1209 (1963). 2. H. Yeh, Natural Hazards 4, 209–220 (1991).

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3. A. Ghobarah, M. Saatcioglu and I. Nistor, Eng. Struct. 28, 312–326 (2006). 4. I. Nistor, M. Saatcioglu and A. Ghobarah, Ann. Conf. Canadian Society for Civil Eng. (2006). 5. M. Saatcioglu, I. Nistor and A. Ghobarah, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 295–320. 6. M. Saatcioglu, I. Nistor and A. Ghobarah, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 355–375. 7. C. V. R. Murty, D. C. Rai, S. K. Jain, H. B. Kaushik, G. Mondal and S. R. Dash, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 321–354. 8. A. Ruangrassamee, H. Yanagisawa, P. Foytong, P. Lukkunaprasit, S. Koshimura and F. Imamura, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 377–401. 9. B. K. Maheshwari, M. L. Sharma and J. P. Narayan, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 475–493. 10. Y. Yamamoto, H. Takanashi, S. Hettiarachchi and S. Samarawickrama, Coastal Eng. J. 48(2), 117–145 (2006). 11. T. Tomita, F. Imamura, T. Arikawa, T. Yasuda and Y. Kawata, Coastal Eng. J. 48(2), 99–116 (2006). 12. N. Shuto, Tsunami Engineering Technical Report No.11, DCRC, Tohoku University (in Japanese) (1994). 13. H. Matsutomi, T. Sakakiyama, S. Nugroho and M. Matsuyama, Coastal Eng. J. 48(2), 167–195 (2006). 14. CNN, http://www.cnn.com/interactive/world/0502/gallery.tsunami.photos/frameset. exclude.html (2007). 15. H. Yeh, Int. Workshop on Fundamentals of Coastal Eﬀects of Tsunamis, Hilo, Hawaii (2006). 16. S. Hibberd and D. H. Peregrine, J. Fluid Mech. 95, 323–345 (1979). 17. A. R. Packwood and D. H. Peregrine, Report AM-81-07, University of Bristol (1981). 18. H. Yeh, A. Ghazali and I. Marton, J. Fluid Mech. 206, 563–578 (1989). 19. H. Yeh and K. M. Mok, Phys. Fluids A 2, 821–828 (1990). 20. H. Chanson, Coastal Eng. J. 48(4), 355–370 (2006). 21. A. Tsutsumi, T. Shimamoto, E. Kawamoto and J. M. Logan, J. Waterway, Port, Coastal, Ocean Eng., ASCE 126(3), 136–143 (2000). 22. J. R. Morison, M. P. O’Brien, J. W. Johnson and S. A. Schaaf, Petroleum Trans., AIME 189, 149–157 (1950). 23. F. E. Snodgrass, E. K. Rice and M. Hall, University of California, Berkeley, CA, Inst. of Engineering Research, Technical Report Series 35, Issue 4, June (1951). 24. J. J. Stoker, Water Waves (Interscience Publishers, New York, 1957). 25. E. Cumberbatch, J. Fluid Mech. 7(3), 353–373 (1960). 26. Y. Fukui, M. Nakamura, H. Shiraishi and Y. Sasaki, Coastal Eng. Jpn. 6, 67–82 (1963). 27. R. H. Cross, PhD thesis, Dept. Civil Engineering, University of California, Berkeley, CA (1966), p. 106. 28. Dames and Moore, in Design and Construction Standards for Residential Construction in Tsunami-Prone Areas in Hawaii (Prepared for the Federal Emergency Management Agency) (1980). 29. J. D. Ramsden and F. Raichlen, J. Waterway Port, Coastal and Ocean Eng., ASCE 116(5), 592–613 (1990). 30. J. D. Ramsden, Report No. KH-R-54, W. M. Keck Laboratory, California Institute of Technology, Pasadena, California (1993), p. 251.

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31. J. D. Ramsden, J. Waterways, Port, Coastal and Ocean Eng. 122(3), 134–141 (1996). 32. T. Okada, T. Sugano, T. Ishikawa, S. Takai and T. Tateno, The Building Centre of Japan (2005). 33. R. Asakura, K. Iwase and T. Iketani, Proc. Coastal Eng. JSCE 47, 911–915 (2000). 34. M. Ikeno, N. Mori and H. Tanaka, Proc. Coastal Eng. JSCE 48, 846–850 (2001). 35. H. Iizuka and H. Matsutomi, Proc. Conf. Coastal Eng., JSCE 47 (2000) (in Japanese). 36. M. Omori, N. Fujii and O. Kyotani, Proc. Coastal Eng. JSCE 47, 376–380 (2000). 37. H. Arnason, PhD thesis, University of Washington, Seattle (2005), p. 172. 38. Y. Nouri, I. Nistor, D. Palermo and A. Cornett, Coastal Structures 2007, Venice, Italy (2007). 39. H. Matsutomi, J. Hyd. Coastal Environ. Eng. JSCE, No. 621/II-47, 111–127 (1999) (in Japanese, with English abs.). 40. R. B. Haehnel and S. F. Daly, Technical Report: ERDC/CRREL TR-02-2, US Army Corps of Engineers (2002), p. 40. 41. R. B. Haehnel and S. F. Daly, J. Hydraul. Eng. 130(2), 112–120 (2004). 42. Y. Nouri, I. Nistor, D. Palermo and M. Saatcioglu, 9th Canad. Conf. Earthquake Engineering, Ottawa, Canada, June (2007). 43. Federal Emergency Management Agency, Coastal Construction Manual (3 Vols.), 3rd edn. (FEMA 55) (Jessup, MD, 2003). 44. Department of Planning and Permitting of Honolulu Hawai, Chapter 16, City and County of Honolulu Building Code, Article 11 (2000). 45. H. Yeh, I. Robertson and J. Preuss, Report No 2005-4, Washington Dept. of Natural Resources (2005). 46. Y. Nakano and C. Paku, Summaries of technical papers of Annual Meeting Architectural Institute of Japan (Kinki) (2005). 47. UBC, Int. Conf. Building Oﬃcials, 1997 Uniform Building Code, California (1997). 48. ASCE Standard, Minimum design loads for buildings and other structures, SEI/ASCE 7-05, 424 (2006). 49. International Code Council (INC), 2006 International Building Code 2006, Country Club Hills, IL (2006), p. 675. 50. M. S. Kirkoz, 10th IUGG Int. Tsunami Symposium, Sendai-shi/Miyagi-ken, Japan (Terra Scientific Publishing, Tokyo, Japan, 1983). 51. T. S. Murty, Bull. Fisheries Res. Board of Canada, No. 198, Department of Fisheries and the Environment, Fisheries and Marine Service, Scientific Information and Publishing Branch, Ottawa, Canada (1977). 52. E. A. Bryant, Tsunami: The Underrated Hazard (Cambridge University Press, London, UK, 2001), p. 320. 53. F. Camfield, Tsunami Engineering (Coastal Engineering Research Center, US Army Corps of Engineers, 1980), p. 222, Special Report (SR-6). 54. P. Dias, L. Fernando, S. Wathurapatha and Y. De Silva, Int. Conf. Disaster Reduction on Coasts, Melbourne (2005). 55. K. H. Pacheco and I. N. Robertson, Evaluation of Tsunami Loads and Their Eﬀect on Reinforced Concrete Buildings. University of Hawaii Research Report (2005). 56. D. Palermo, I. Nistor, Y. Nouri and A. Cornett, PROTECT 2007, Whistler, Canada (2007).

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Chapter 12

Nonconventional Wave Damping Structures Hocine Oumeraci Leichtweiss-Institute for Hydraulic Engineering and Water Resources Technical University Braunschweig Beethovenstr. 51a, 38106 Braunschweig, Germany [email protected] After a brief discussion of the necessity (i) to consider in addition to the reduction of wave heights in the sheltered area further aspects of the hydraulic performance when developing and designing new coastal structures for the protection against wave action and (ii) to develop a systematic “roadmap” of the existing concepts and types of structures together with their hydraulic performance characteristics as an important tool for practicing engineers and decision-makers, ﬁve examples from selected research studies performed in the last years by the Leichtweiss-Institute are presented in order to illustrate the development of nonconventional structures with substantially improved performance as compared to their conventional counterparts. These examples include (i) a multi-chamber structure to overcome the drawbacks of the perforated JARLAN-type breakwater concept, (ii) an artiﬁcial reef made of successive submerged permeable screens to increase the wave damping performance and to better control both wave reﬂection and wave transmission, (iii) a “High Mound Composite Breakwater” (HMCB) concept to decrease breaking wave loads, wave reﬂection, overtopping, and spray generation at the structure, (iv) an onshore wave damping barrier made of staggered walls, and (v) a rubble mound breakwater with a core made of geotextile sand containers.

12.1. Introduction In the past, the development of wave damping structures used in coastal protection and harbor engineering was rather directed toward achieving a maximum reduction of wave height in the sheltered area, while less attention was generally paid to further performance aspects such as (i) the reduction of wave loads, wave reﬂection, and wave overtopping, (ii) the reduction of environmental impacts (degradation of the marine landscape and of the water quality in the sheltered area, erosion of the foreshore, down coast erosion, etc.), 287

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(iii) the control of wave periods in the sheltered area, (iv) the reduction of spray generation at the structure, (v) the adaptability of the wave damping structure for multi-purpose use (recreational activities, wave power generation, etc.), and (vi) the reversibility of built structures if not performing as expected. In the future, the increasing interest in the practical implementation of the sustainability principles in coastal engineering18 will require that more eﬀort should be put on the development of innovative low impact structures with better hydraulic performance characteristics, lower capital and maintenance costs for the proper sheltering of harbor and other facilities as well as for the proper protection of threatened coasts against storm surges. The main objective of this chapter is to illustrate, by means of three selected example research studies performed in the last years by the Leichtweiss-Institute (LWI), the process of developing and testing nonconventional structures. This process particularly consists of (i) a better understanding of the hydraulic functioning of the existing concepts, (ii) a clear identiﬁcation of their drawbacks with respect to the commonly accepted and new emerging performance characteristics/requirements (see above), and (iii) a better control of the physical processes and structure parameters which contribute to the improvement of the hydraulic performance and to the reduction of wave loads. However, before starting with these example studies, a ﬁrst attempt is made to discuss, why and how a parameterization of the existing concepts and structures for wave damping might help to develop a systematic “roadmap” of these concepts as an important decision tool for practicing engineers and coastal managers. 12.2. Parameterization of Wave Damping Structures: A First Attempt A large variety of concepts and structures for wave damping is available. However, the designer and coastal manager faced with the problem of ﬁnding the concept mostly suited for the intended use have no proper tool to quickly make the right choice before embarking into detailed investigations. For this purpose, a compact overview of the most important concepts/types of structures is needed, including the basic information on their hydraulic functioning (“roadmap”). The roadmap should not only provide the required performance characteristics. Moreover, the unwanted eﬀects and impacts may also be directly provided or easily derived for each type of structure. The latter issue becomes increasingly important, because the choice will not, as in the past, mainly or exclusively be dictated by the wave damping eﬃciency, but also by further criteria associated with the sustainable protection of coastal zones.18 Unfortunately, such a practical decision tool, which is expected to substantially help both engineers and coastal managers, is still missing. Due to the considerable

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number of existing concepts/types of structures, it will certainly be very diﬃcult or even impossible to elaborate any useful compact overview if a parameterization of the existing types of structures is not undertaken. In order to illustrate this ﬁrst requirement, a very tentative parameterization is proposed for a single structure as a ﬁrst starting base in Fig. 12.1. Such a parameterization may then lead to the very tentative compact overview in a parameterized form in Fig. 12.2.

Ei

Er

Et

SWL

H

h1

L ε

hs

h

h2 b

Fig. 12.1.

Tentative parameterization of single wave damping structure.

hs < h

hs > h

1.1

h1 = 0 h2 > 0

h1 > 0 h2 = 0

h1 = 0 h2 = 0 1.2

hs = 0

1.3

h1 > 0 h2 > 0 1.4

h1 + h2 = h 1.5

ε=0 impermeable

no structure

b=0 thin walls

2.1

2.2

2.3

2.4

2.5

ε≠ 0 permeable

no structure 3.1

3.2

3.3

3.4

3.5

4.1

4.2

4.3

4.4

4.5

ε=0 b>0

impermeable

wide structures

ε≠ 0 permeable

Fig. 12.2.

Tentative compact overview of existing wave damping structures in parametrized form.

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It is seen from Figs. 12.1 and 12.2 that only ﬁve parameters (structure width B, porosity ε, structure height dB , submergence depth h1 , and clearance h2 ) together with the water depth h are suﬃcient to achieve a parametric description of most types of existing wave damping structures. But this represents only a ﬁrst step. The next important step is to provide the characteristic curves describing the hydraulic performance for each of the parametrized structure types in Fig. 12.2. For this purpose, the simplest procedure is to use the fact that the incident wave energy Ei is converted into a reﬂected (ER ), a dissipated (ED ), and a transmitted (ET ) energy component: Ei = ER +E D +E T .

(12.1)

Equation (12.1) written in terms of the reﬂection (KR ), dissipation (KD ), and transmission (KT ) coeﬃcients yields 2 2 1 = KR + KD + KT2 .

(12.2)

If only the wave damping performance PW = 1 − KT2

(12.3)

is of interest, then the functioning curves PW = f (T ) such as those qualitatively described in Fig. 12.3 would be useful, because they clearly illustrate how each type of structure will respond over the full range of incident wave periods T . For instance, it is seen that the curtain wall founded on piles (Type 1.3) performs well for shorter wave periods T , but has a very low wave damping performance

Pw = K

2 R

+K

2 D

(

= 1 +KT

2

)

KR = Reflection Coeff.; KD = Dissipation Coeff.; KT = Transmission Coeff.

Wave damping Performance Pw

Vertical Breakwater (3.1)

1.0

Horizontal Breakwater on Piles (3.5) Curtain-Wall on Piles (1.3) Caisson Breakwater on Piles (3.3)

0.5 Submerged Breakwater (3.2)

0.0

0

Wave Period T [s] Fig. 12.3.

Wave damping performance (principle sketch!).

16

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Fig. 12.4. Hydraulic functioning of a single perforated wall under nonbreaking and nonovertopping wave conditions (adapted from Ref. 4).

for longer periods. Widening the wall to a caisson (Type 3.3) will considerably improve the wave damping performance over a wider range of wave periods. Very often, however, the wave damping performance alone is not suﬃcient and a “full” description of the hydraulic functioning as a function of the most relevant parameter governing wave reﬂection, wave transmission, and energy dissipation [see Eq. (12.1)] is required. For instance, in the case of a single surface piercing perforated screen (Type 2.1) and for nonbreaking and nonovertopping wave conditions, a kind of “dynamic porosity” (here reﬂection number RN ) which combines the inﬂuence of incident wave height Hi , water depth h, and wall porosity ε, is found to be the most appropriate parameter to describe the reﬂected, transmitted, and dissipated components of the total incident wave energy (Fig. 12.4). For most of the types of structure shown in Fig. 12.2, curves describing the hydraulic performance in terms of wave reﬂection, transmission, and damping already exist which are derived empirically, analytically, or numerically.2,11,19,21,22

12.3. Selected Example Studies for the Development of Nonconventional Wave Damping Structures In order to illustrate the processes associated with the development of nonconventional wave damping structures, the main results from three selected example research studies performed in the last years by LWI are brieﬂy discussed below, including

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a multi-chamber caisson structure, an artiﬁcial reef made of a submerged progressive ﬁlter, a high mound composite breakwater (HMCB-concept), an onshore wave damping barrier, and a rubble mound breakwater with a core made of geotextile sand containers (GSC).

12.3.1. Multi-chamber caisson structure One of the most useful concepts to cope with the high reﬂection induced by vertical face breakwaters and seawalls is the perforated Jarlan-type breakwater which was introduced in 1960 in Canada.9 It consists of a single dissipating chamber bounded seaward by a perforated front wall (porosity ε ≈ 20%) and shoreward by an impermeable back wall (One Chamber System). The incident wave energy is partly reﬂected at the front wall and partly transmitted through the perforations into the wave chamber, where a certain amount of the incident wave energy is reﬂected by the back wall while a large part is dissipated due to resonance phenomena, vortices, and friction losses. The relative importance of the reﬂected and dissipated part of the total incident wave energy, and thus the hydraulic performance, depends on the porosity of the front wall, but is essentially governed by the ratio of the chamber width B and the wave length L of the incident waves (B/L). Although the JARLAN-type breakwater concept has been used more or less successfully worldwide, it has a basic drawback (see Fig. 12.6) which requires a further development of this concept. For this purpose, it was necessary to investigate ﬁrst the key processes which contribute to the wave damping by friction (local losses and vortices) and by destructive interference of the incident and reﬂected waves over the full range of B/L ratios (i.e., over the full range of incident wave periods). The experimental results in Fig. 12.5 well illustrate how a traditional JARLAN-type caisson works. As shown by the upper curve in Fig. 12.6, the traditional JARLAN-type caisson (OCS) has, at its optimal working point (B/L ≈ 0.2), a much lower reﬂection coeﬃcient (and thus a much larger energy dissipation) than a vertical impermeable wall. However, the response is very selective with respect to the incident wave periods; i.e., it performs satisfactorily only within a very narrow range of the B/Lratios. In order to overcome this drawback, a new Multi-Chamber System (MCS) was developed and tested in the Large Wave Flume of Hannover. As shown by the lower curve in Fig. 12.6, the new MCS concept not only provides a lower reﬂection coeﬃcient; moreover this reﬂection coeﬃcient is kept at its lowest level over the full range of practical B/L ratios (i.e., for B/L > 0.25, where B is deﬁned as the overall width of the Multi-Chamber System. In addition to the substantial improvement of the hydraulic performance, which is achieved by the new multi-chamber concept, the new concept has also the + advantage to strongly reduce the resulting horizontal wave forces Ftotal (obtained by superposition of the forces acting simultaneously on each wall). Because wave forces are directly related to water surface elevation and thus to wave reﬂection,3,6 + + the total force Ftotal (related to force F0% on a single impermeable vertical wall

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impermeable back wall incident wave energy

B

Ei

Wave damping relative to incident wave energy Ei

Er

reflected wave energy

100%

wave chamber

perforated front wall

total wave damping 80%

wave damping by destructive interference

60% wave damping by friction

40% 20% 0% 0

0.1

0.2

0.3

0.4

0.5

relative chamber width B/L [-] Fig. 12.5.

Wave damping friction and destructive interference (Adapted from Ref. 5).

with zero-porosity) exhibits a very similar behavior to the reﬂection coeﬃcient with respect to the B/L ratio (compare Figs. 12.6 and 12.7). This ﬁrst example study has illustrated how a detailed insight into the physical processes responsible for the hydraulic functioning of an existing concept may lead to a clear identiﬁcation of the drawbacks of this concept by indicating how to overcome them and how to achieve substantial improvements through the introduction of additional structure members. Given its potential to substantially reduce and better control wave reﬂection (less risk to navigation and less sea bed scour), wave loads, wave runup and overtopping, spray generation, etc., the new MultiChamber System represents an ideal alternative as a breakwater, jetty, and quay wall as well as a seawall for the protection of reclaimed sea fronts and artiﬁcial islands. Due to the ﬂexibility of caisson structures to allow any shape and size, the seawalls can be adapted to incorporate promenades and any further facility for recreation activities, etc. 12.3.2. Submerged wave absorber as artificial reef for coastal protection An interesting cost eﬀective and soft alternative to conventional seawalls for coastal protection against erosion are artiﬁcial reefs which have the advantage

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Fig. 12.6. Reﬂection coeﬃcient versus relative chamber width for a traditional JARLAN-type caisson and for the new multi-chamber system (Adapted from Ref. 4).

Fig. 12.7. Resulting horizontal wave force versus relative chamber width for a traditional JARLAN-type caisson and for the new multi-chamber system (Adapted from Ref. 4).

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(i) to attenuate the waves before they reach the shoreline, (ii) to be invisible for viewers from the beach and therefore do not aﬀect the marine landscape, (iii) to reduce the morphological impact on the foreshore (erosion) and on the neighboring coast (down-coast erosion), and (iv) to ensure a suﬃcient water exchange between the open sea and the sheltered area. However, the existing artiﬁcial reef concepts have serious drawbacks: the wave damping performance is limited and sometimes not suﬃcient. Moreover, the overall hydraulic performance is diﬃcult to control, due to the limitations associated with the variation of the structure parameters, etc. Therefore, a reef concept made of submerged permeable screens with predetermined porosity and spacing has been experimentally tested in the Large Wave Flume of Hannover (GWK). As schematically shown in Fig. 12.8 for a three-ﬁlter system, this new reef concept is particularly appropriate for the protection of such coastal areas which are frequently used for recreation activities.12−14 Before starting with the systematic study on the hydraulic performance of this new reef concept, it was important to demonstrate ﬁrst its eﬃciency with respect to the protection against beach erosion. For this purpose, a submerged two-ﬁlter system with porosities ε = 11% (front screen) and ε = 5% (back screen), spacing B = 10.3 m and height dB = 4 m was installed in front of a beach proﬁle. The same beach proﬁle was previously tested in the Large Wave Flume of Hannover17 without any protection under the same storm surge conditions (storm surge water level h = 5.0 m, TMA-wave spectrum with signiﬁcant wave height Hs = 1.20 m and peak

Fig. 12.8.

Submerged three-ﬁlter system as an artiﬁcial reef for coastal protection.23

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Fig. 12.9.

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Beach proﬁle development under storm wave attack with and without reef protection.

period TP = 6.6 s, storm duration t = 10 h). The comparison of the results related to the development of the beach proﬁle at the diﬀerent stages of the tests with and without the reef structure as shown in Fig. 12.9 provides a good demonstration of the eﬃciency of the new reef concept as a soft protection alternative. In both cases (i.e., with and without protection), one can indeed observe a beach erosion, a net seaward transport, and a sand bar formation. However, the net seaward transport rate is about twice larger for the unprotected beach than for the protected beach. In addition, the reef causes the seaward transport to occur only within a limited narrow zone, so that this sand bar does not extend further seaward as in the case of an unprotected beach. Moreover, the eroded volume above the storm water level (h = 5.0 m) for the protected beach is only half as much as the eroded volume for the beach without any protection. As a result, the recession of the shoreline of the protected beach is only half as much as the recession of the unprotected beach (Fig. 12.10). Regarding the hydraulic performance of submerged wave absorbers, a brief illustration is given in Fig. 12.11, showing how a two-ﬁlter system damps the waves over the full range of B/L ratios. The contribution of the seaward screen to the total wave damping of the system varies between 30% for B/L ≈ 0.5 and 85% for B/L ≈ 0.3. The maximum wave damping performance of the system also occurs at B/L = 0.3, while the minimum value is at B/L = 0.5. These and further results with a three-ﬁlter system clearly show that for a given submergence depth (Rc /Hi ) the relative spacing B/L represents the most decisive parameter to describe the hydraulic performance of wave absorbers.23

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Fig. 12.10.

2.5

Beach proﬁle after storm with and without reef (Adapted from Ref. 11).

dB/h = 0.98 wave damping at filter 1

2 relative wave height

Hi/ Ht

rel. damping [%]

297

1.5 H

1

wave damping at filter 2

Ht1/ Ht

i

H

0.5

H

r

t1

H

t

h dB =

Hi/Ht (measurement) Ht1/Ht (measurement)

0 100

B

3 .9 4 m

10m 11%

F i l ter 1

5% F i l ter 2

50 wave damping at filter 1/total wave damping

0 0

0.25

0.5

0.75

1

relative spacing B/L Fig. 12.11.

Wave damping of a submerged two-ﬁlter system (Adapted from Ref. 11).

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298

Fig. 12.12.

Hydraulic performance of submerged single screens and ﬁlter systems.

From the comparative analysis of the hydraulic performance shown in Fig. 12.12 for a submerged single screen with diﬀerent porosities (ε = 0%, 5%, and 11%) and submerged two- or three-ﬁlter systems it is seen that (i) using a ﬁlter system instead of a single screen substantially increases the amount of dissipated energy, (ii) unlike a single screen, a ﬁlter system can substantially reduce and simultaneously control both wave reﬂection and wave transmission, (iii) using an optimized three-ﬁlter system, more than 80% of the incident wave energy can be dissipated, (iv) the relative submergence depth Rc /Hi is an important parameter for the wave damping performance of both single screens and ﬁlter systems, and (v) for the range of practical submergence depths (Rc /Hi ≈ −1), the highest improvement of the wave damping performance is achieved when using a twoﬁlter system instead of a single ﬁlter. A further increase of the number of ﬁlters would lead to comparatively less improvement of the wave damping performance. Further results have indeed shown that the relative spacing B/L is much more important than the number and porosity of the ﬁlters to control the reﬂected,

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transmitted, and dissipated wave energy. A more detailed discussion on these issues is provided by Oumeraci and Koether23 and Koether.11 As mentioned in the introduction, there are many cases where the reduction of wave periods in the sheltered area may also be very important. Such situations occur, for instance, when the reef is principally used to control sediment transport or to reduce wave loads and overtopping at a seawall located behind the reef. An example result to illustrate the eﬃciency of the new reef concept by reducing both transmitted wave periods and heights is shown in Fig. 12.13 for two relative heights of a three-ﬁlter system. This second example study has shown again that substantial improvements of the existing concepts can only be achieved through a better insight into the mechanisms and processes governing the hydraulic performance. In fact, using a single submerged screen, only a very limited amount of incident wave energy can be dissipated, i.e., a decrease of wave transmission can only be achieved at the cost of the increase of wave reﬂection (see Ref. 23 for more results). Using a conventional reef made of rubble material would require a very wide structure and a progressive decrease of the porosity in wave direction in order to achieve a satisfactory wave damping performance. This is not only costly and diﬃcult to construct and to maintain, but it is also very diﬃcult to control the hydraulic performance by means of a variation of the structure parameters as it is the case for this new artiﬁcial reef concept.

Fig. 12.13.

Reduction of wave periods by a three-ﬁlter system.

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The understanding of the underlying mechanisms has shown that a new reef concept made of two or three submerged thin ﬁlters is an elegant and costeﬀective alternative to overcome most of the drawbacks of the existing reef concepts, including a substantial reduction and a better control of the reﬂected and transmitted components of the incident wave energy by means of the variation of the structure parameters (submergence depth, porosity, number and spacing of submerged slit walls.) A theoretical model to optimize submerged wave absorbers has been developed by Koether.11,23 The model has been successfully validated by largescale experimental data for regular and irregular waves as well as for submerged single-slit wall and wave absorbers with two and three ﬁlters. Preliminary tests in the Large Wave Flume of Hannover using comparatively a single-wall as well as a two-wall and a three-wall submerged wave absorber subject to 1 m high solitary waves have shown that this reef concept might also be applied for the protection against tsunami. In fact, the part of wave energy of the incident solitary waves dissipated is more than 75% and 85% for a two-wall and a threewall system, respectively. The experimental results showing the incident, reﬂected, and transmitted waves for a single-, two-, and three-wall systems are plotted and discussed by Oumeraci and Koether.23

12.3.3. High mound composite breakwater (HMCB) Based on a historical concept which was ﬁrst used in 1830 in Cherbourg, France, and in 1890 in Alderney, UK, a new HMCB-concept has extensively been investigated within two joint research projects by LWI together with the Port and Harbor Research Institute (PHRI), Yokosuka, Japan and with the Civil Engineering Research Institute (CERI), Sapporo, Japan (Fig. 12.14).

Fig. 12.14.

Old and new high mound breakwater concepts.

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The new HMCB-concept was intended to be used mainly as a seawall for the protection of artiﬁcial islands (oﬀshore airport) and roads with heavy traﬃc along the coast, but also as a breakwater [Fig. 12.14(c), 12.14(d)] in harbors. This concept has recently been implemented at MORI Port in Hokkaido, Japan.14 The governing characteristic of the HMCB-concept is to cause the highest waves in the spectrum to break before reaching the crest structure by means of a relatively ﬂat slope (about 1:3). This concept has the following advantages: (i) the required amount of rubble material is much less than for a conventional rubble mound breakwater, (ii) the required armor units are smaller since they are all placed below still water level, and (iii) the required crest structure is much smaller than a conventional caisson breakwater. In order to further substantially reduce the breaking wave impact loads on the superstructure and to overcome further drawbacks of the old HMCB-concept (excessive wave reﬂection, overtopping, spray generation, etc.), a major innovation was introduced to improve the performance of the concrete superstructure: a front slit wall made of piles (porosity of about 30%) and a relatively short dissipation chamber behind it. If a breaking wave reaches the structure, the total wave force is split spatially and temporarily into the following force components (Fig. 12.15): (i) a force component F1 on the permeable front wall, (ii) a force component F2 on the impermeable back wall, and (iii) a stabilizing downward force F3 on the bottom slab of the dissipation chamber.

Fig. 12.15.

New HMCB-concept.

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In addition to the reduction of wave loads and the subsequent reduction of the required size of the concrete superstructure, a substantial improvement of the overall hydraulic performance characteristics is also achieved by using the new HMCB-concept. A summary of the improvements in comparison to the older concept (vertical impermeable superstructure) as obtained from extensive hydraulic model investigations is given in Table 12.1. With respect to the wave loads, it is seen that a substantial reduction of the horizontal and uplift forces is achieved, which would result in a reduction of about 50% of the required weight of the superstructure to ensure sliding stability. With respect to the hydraulic performance, it is seen that wave reﬂection is reduced by about 25%, and as a result of the reduction of wave overtopping the required crest level above still water level is reduced by about 40%. Even without using any splash reducer at the front and back wall of the concrete superstructure, the splash/spray heights are reduced by half. Further details on the results in Table 12.1 are given by Oumeraci and Muttray,26 Muttray et al.,15 Oumeraci et al.,25 Takahashi et al.,33 Sch¨ uttrumpf et al.,31 Muttray et al.,16 and Oumeraci et al.28 It might be important to stress that the reduction of spray generation at wave damping structures is becoming an increasingly important design requirement, particularly when the structures are used for the protection of oﬀshore airports, the protection of roads with heavy traﬃc along the coastline, etc. The growing importance of this relatively new emerging issue may be explained by the detrimental eﬀects that spray might have on inland and nearshore infrastructures, operations behind the structure as well as on vegetation. In fact, spray can be transported by wind up to 30 km inland and a ﬂux of spray salt up to 400 µg/m2 ·s may result. Large quantities of saltwater dispersed over wide coastal areas may result in the following eﬀects: (i) short-term detrimental eﬀects such as disturbance/stoppage of car traﬃc,10 electric power supply, port, and airport operations and Table 12.1.

Main improvement by the new HMCB-concept.

Improved performance of new HMCB with slit front wall (without splash reducers) Hydraulic performance

rx

CoV [%]

Transmission coeﬃcient Ct Reﬂection coeﬃcient Cr Mean overtopping rate Maximum overtopping rate qmax Number of overtopping waves Now Splash and spray height Rw , Rs Required freeboard Rc

1.0 0.75 0.5 0.6 0.7 0.5 0.6

50 30 30 25 25 60 30

Wave Forces and Stability Horizontal forces Fh Uplift forces Fu Required weight for sliding stability W

0.4 0.6 0.5

50 40 50

CoV = x¯ σx (Coeﬀ. of vacation); τ¯x = mean reduction factor)

x ¯new HMCB x ¯old HMCB

(mean

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(ii) longer-term impacts such as salt corrosion of buildings and other facilities, damage to agriculture and inland vegetation, etc. Therefore, one of the most challenging tasks to cope with salt spray consists of the development of innovative shapes of the structure crest to substantially reduce the splash/spray height induced by the breaking waves at the structure. For this purpose, special tests were performed to analyze the eﬀectiveness of various alternatives to reduce splash/spray height.8,25,28 As shown in Fig. 12.16, the splash/spray height observed in the Large Wave Flume tests for diﬀerent structure alternatives is related to the corresponding value observed for the new HMCB without any splash reducer as a reference structure (Type 1). Even without any splash reducer, the new HMCB-concept is capable of reducing the splash/spray height by half as compared to a conventional impermeable wall (Type 0). Adding a conical splash reducer on each pile of the front slit wall and a splash reducer on the back wall will further reduce the splash/spray height by about half (Type 3). The same improvement was achieved by using a continuous slab on the crest of the front wall (Type 4), but this has the drawback to substantially increase the uplift force.28 Since the eﬃciency of Type 2, both with respect to the reduction of splash heights and overtopping, was found very low, no further analysis was performed. This third example study has highlighted how a very old concept can basically be improved to provide a new solution with a considerable improvement of the wave

Fig. 12.16.

Eﬃciency of various alternatives for the reduction of splash and spray height.

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loading conditions and of the hydraulic performance characteristics such as wave reﬂection, wave overtopping and spray generation. The study also allowed to identify the reduction of spray generation as a new emerging criterion for the development and design of novel wave damping structures. 12.3.4. Onshore wave damping barrier (OWBD) A nonconventional permeable barrier to damp storm waves running up a historical promenade on the North Sea Island of Norderney was developed and tested in the Large Wave Flume (GWK), Hannover.29 The barrier consists of 1.30 m high straight and curved staggered wall elements with a length of 5.5 m and 7.7 m, respectively (Fig. 12.17). The relatively low crest level and the discontinuous nature of the barrier resulted from the requirements that, for the tourists on the promenade the view to the sea and the direct access to the shoreline should not be obstructed. Moreover, the barrier should architecturally and aesthetically ﬁt into the local landscape, so that it will not necessarily be perceived as a coastal protection structure [Fig. 12.18(a)]. The eﬃciency of the OWBD concept in terms of wave overtopping reduction was successfully tested in the Large Wave Flume (GWK) in Hannover [Figs. 12.18(b) and 12.18(c)] and implemented in Norderney [Fig. 12.18(d)], where it withstood without any damage several storm surges for more than ﬁve years.

Fig. 12.17. Ref. 29).

Onshore wave damping barrier for the North Sea Island Norderney (Adapted from

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Fig. 12.18.

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Onshore wave damping barrier (OWBD).

Since broadening the crest of the embankment (Alternative 4) was practically not feasible, the OWBD concept (Alternative 2) proved the most performant in reducing wave overtopping (by a factor of 5) as compared to other conventional alternatives (Fig. 12.19). More details on the results can be found in Refs. 29 and 32. The fourth example study has shown that under some circumstances storm waves can be damped eﬀectively by a discontinuous low barrier made of aesthetically and functionally well-conceived staggered wall elements (Fig. 12.20). This concept can be adapted for the protection against tsunami by using a much wider barrier and more robust wall elements. In fact, the ﬁeld evidence experienced with seawalls and breakwaters during the 2004 Indian Ocean Tsunami clearly suggests that protective structures should not be designed to completely stop the tsunami. Indeed, this is neither economically justiﬁable nor environmentally and socially acceptable. Therefore, protective structures would be preferable which aim at progressively weakening the tsunami power without blocking completely the inundation and which have the overall additional beneﬁt of broadly blocking ﬂoating debris in a rather soft manner. Such a concept would particularly be appropriate for urbanized and touristic coastal areas where forests (bioshields) cannot be planted due to unfavorable local conditions and must therefore be replaced by a man-made barrier ﬁtting architecturally in the local marine landscape.20

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Fig. 12.19. Reduction of wave overtopping by conventional and nonconventional alternatives (Adapted from Refs. 29 and 32).

Fig. 12.20.

Details of wall elements of OWBD in Norderney.

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12.3.5. Rubble mound breakwater with a core made of geotextile sand containers 12.3.5.1. Motivations There are several reasons which might lead the engineer and other decision-makers in practice to use sand instead of conventional quarry run for the core of rubble mound breakwaters and structures, including among others: (i) nonavailability of rock material in suﬃcient quantities and at aﬀordable costs; (ii) sediment inﬁltration through rubble mound structures which may result in the shoaling of navigation channels and harbor basins, and thus in higher maintenance dredging costs; and (iii) reduction of wave transmission through the structure which might particularly be crucial in the case of long waves. On the other hand, the use of sand as a quasi-impervious core instead of quarry stone would result in an increase of (i) wave setup and runup at the structure, (ii) wave overtopping, and (iii) wave reﬂection, which might be detrimental to the stability of the structure, to the operations on and behind the breakwater (due to excessive overtopping) as well as to navigation and seabed stability (due to excessive wave reﬂection). Moreover, serious diﬃculties arise in practice when trying to design and construct the ﬁlter to protect the sand core against washout by wave action. Applying geometrically closed ﬁlter criteria would result in very complex, multiple, and relative thin ﬁlter layers which will not only be very costly and very diﬃcult to build in larger water depths, but also might certainly fail due to the almost unpredictable very complex loading conditions of the sand core under cyclic pulsations by waves and entrained air at the interface with the last ﬁlter layer. Such failures have indeed been observed under both laboratory and ﬁeld conditions in the past. Laboratory evidence has also shown that introducing the so-called “geometrically open ﬁlter” criteria to design a “hydraulic sand-tight ﬁlter” may indeed reduce the number of ﬁlter layers. However, the main practical diﬃculties mentioned above will remain, including those associated with the long-term stability of the sand core due to the high complexity of the loading and its uncontrollability during the entire storm duration and the life cycle of the structure. Geotextile ﬁlters might present themselves as an alternative to the very complex, costly, and uncertain ﬁlter made of multiple layers of granular material. However, geotextile mats are not only diﬃcult to install under waves and currents, but also may introduce a shear surface which might be detrimental to the stability of the armor layer. A more feasible alternative is to use a core made of geotextile sand containers (geocore). This will not only allow to overcome the aforementioned core stability problems, but also to provide (i) a better erosion stability of the core and (ii) an

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Fig. 12.21.

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Class of geocore structures in comparison to conventional rubble mound structures.

increased stability against seismic loads as compared to a core simply made of loose sand. However, many of the drawbacks mentioned above remain with respect to wave setup, runup, overtopping, reﬂection, and armor stability in comparison to a conventional breakwater core. Therefore, an extensive research program has been initiated at LWI to study both the hydraulic performance and armor stability, including the processes involved and the development of prediction formulae for the design of a class of rubble mound structures with a core made of geotextile sand containers (Fig. 12.21). In the following, only the case “rubble mound breakwater with GSC-core” in Fig. 12.17(a) is brieﬂy addressed. Based on comprehensive hydraulic model tests performed in the twin-wave ﬂumes of LWI, a comparative analysis of the two breakwater types was ﬁrst performed with respect to wave reﬂection, wave transmission, wave runup, and wave overtopping performance. As a result, prediction formulae were proposed for the geocore breakwater and compared with those for the conventional rubble mound breakwater alternative.24 12.3.5.2. Wave reflection performance Using the reﬂection model proposed by Oumeraci and Muttray,27 which accounts for the relative depth k0 d as the primary inﬂuencing parameter, the following prediction formula was derived for both breakwaters (Fig. 12.22): tan α Kr = 0.43 · √ , ko d

(12.4)

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Fig. 12.22.

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Reﬂection coeﬃcient Kr versus relative water depth k0 d.

with ko = 2π/L0 (wave number in deepwater); tan α = steepness of the seaward slope of the structure; d = water depth in front of the structure. 12.3.5.3. Wave transmission performance Several models have been proposed in the literature7,24 to predict wave transmission through a rubble mound structure which, however, do not explicitly account for the eﬀect of the core permeability. The parameter which mostly aﬀects the transmission coeﬃcient KT has been found to be a function of the relative freeboard Rc /Hs and the steepness sm = Hs /L0 of the incident waves1: Rc R = Hs ∗

sm . 2π

(12.5)

In fact, the best-ﬁt in analyzing the experimental results for the geocore and conventional breakwater was found by using the modiﬁed freeboard R∗ according to Eq. (12.5). As a result, the following formula was obtained for the transmission coeﬃcient (Fig. 12.23): KT = ar · (Rc∗ )br ,

(12.6)

where ar and br are constants, which depend on the permeability of the structure. The structure parameters ar and br are obtained for the conventional and the geocore breakwater (Fig. 12.23).

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Fig. 12.23.

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Wave transmission coeﬃcient for conventional breakwater and geocore breakwater.

12.3.5.4. Wave runup performance Based on the model proposed in CEM7 for the conventional breakwater type, the following runup formula was determined (CoV = 2.8%): 0.274 Ru2% = 1.217 · ξom Hmo

for ξom = 3.3 − 7.0,

(12.7)

while for the geocore breakwater the following formula was obtained (CoV = 3.4%): 0.274 Hmo Ru2% = 1.415 · ξom

with ξom = surf similarity parameter: ξom = tan α

for ξom = 4.1 − 6.7,

H/

(12.8)

2 gTm−1.0 . 2π

The diﬀerence between the two breakwater types is in the order of 20%; i.e., the required crest level of a geocore breakwater should be increased accordingly if wave runup or/and wave overtopping is an important issue (Fig. 12.24). 12.3.5.5. Wave overtopping performance The best-ﬁt of the experimental data for the conventional breakwater and the geocore breakwater was obtained by using the model proposed by TAW (2002)34 : b · R∗ , (12.9) Q∗ = a · exp − γf

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Fig. 12.24.

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Runup for conventional breakwater versus runup for geocore breakwater.

where Q∗ =

q , 3 )0.5 (g · Hm0

q = average overtopping discharge [m3 /s · m] and R∗ = Rc /Hm0 = relative freeboard [-]. Using a = 0.2 and b = 2.6 in Eq. (12.9) for both breakwater types, the correction factor γf , initially intended to account for the surface roughness eﬀect, is used to distinguish between the eﬀect of permeability on the average overtopping rate q (Fig. 12.25): • γf = 0.52 for the conventional breakwater (CoV = 12%), • γf = 0.60 for the geocore breakwater (CoV = 20%). Based on the results in Fig. 12.25 and the results of further analysis,30 the geocore breakwater is associated with three to four times higher overtopping discharges than the conventional breakwater. 12.3.5.6. Lessons learned from the study The ﬁfth example study has shown that using sand encapsulated in geotextile containers for the core of rubble mound breakwaters, which is about one order of magnitude less permeable than a conventional core made of quarry run, may provide many advantages. This is particularly the case when rock material (quarry run) is

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312

Fig. 12.25.

Average wave overtopping for conventional and geocore breakwater.

not available in suﬃcient quantities and at aﬀordable costs or when the reduction of wave transmission and sediment inﬁltration through the rubble mound structure is an important issue. However, one should keep in mind that using a GSC-core instead of a conventional core will result in larger runup heights and higher wave overtopping rates. Moreover, larger armor units are required to ensure hydraulic stability. More results are given by Oumeraci et al.24 The advantages of the nonconventional “geocore” concept are expected to be particularly revealed when applied to a seawall protecting reclaimed land (Fig. 12.17(d)), due to the diﬃculties to protect the latter from being washed out through a conventional core made of quarry run.

12.4. Concluding Remarks and Perspectives This chapter has attempted to show that one of the challenges of a sustainable protection against wave attack is the development of innovative structures/concepts which are not only capable to further reduce the wave heights in the sheltered area, but also to achieve substantial improvements with respect to the reduction of wave loads, wave reﬂection, and overtopping. Moreover, the novel types of structures should also be able to cope with new emerging requirements such as (i) a better control of the wave periods in the sheltered area, (ii) a reduction of spray generation at the structure, and (iii) a better adaptability to multi-purpose use. Since the importance of these new emerging design criteria is expected to considerably increase in the context of the sustainable development of coastal zones, they might

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represent challenging candidate topics for future research. A further important candidate issue for applied research will certainly consist of the elaboration of a kind of “roadmap” of the existing wave damping concepts (possibly in a parametrized form!), together with the associated characteristic curves which not only summarize the associated hydraulic performance, but also the possible negative impacts and further information required for a fast decision-making.

Acknowledgments Most of the ideas, material, and illustrative examples used herein are based on inputs taken from completed research projects supported by the German Research Council (DFG, Bonn), the Federal Ministry for Education and Research (BMBF, Bonn), the Science and Technology Agency of Japan (STA, Tokyo), and the Fifth Port Construction Bureau of the Ministry of Transport (MOT, Tokyo), the European Community within the EU-MAST research program, NLWK Norden and BBG Bauberatung GmbH & Co. KG, Germany. The author gratefully acknowledges the ﬁnancial support. Since most of these projects were joint projects with Port and Harbor Research Institute (PHRI, Yokosuka), Civil Engineering Research Institute (CERI, Sapporo), the Institute of Naval Architecture and Ocean Engineering, TU Berlin, and many further European Institutes, the author also gratefully acknowledges the fruitful cooperation with these partners.

References 1. N. W. H. Allsop, Low-crest breakwaters, studies in random waves, Proc. Coastal Structures, Specialty Conference on the Design, Construction, Maintenance and Performance of Coastal Structures, Arlington, Virginia, USA (1983). 2. H. Bergmann, Hydraulische Wirksamkeit und Seegangsbelastung senkrechter Wellenschutzbauwerke mit durchl¨ assiger Front, Dissertation, Mitteilungen des LeichtweissInstituts f¨ ur Wasserbau, Heft 147 (2001) (in German). 3. H. Bergmann and H. Oumeraci, Wave loads at perforated caisson structures, Proc. 27th Int. Conf. Coastal Engineering (ICCE’2000), Sydney, Vol. 2 (2000), pp. 1622–1635. 4. H. Bergmann and H. Oumeraci, Digue innovante en caissons multi-chambres — Fonctionnement et sollicitations hydrauliques, Revue fran¸caise de g´enie civil. 5(7), 973–993 (2001) (Sp´ecial issue) (in French). 5. H. Bergmann and H. Oumeraci, Hydraulische Wirksamkeit und Wellenbelastung senkrechter Wellenschutzbauwerke mit durchl¨assiger Front, Hansa (2002), pp. 1–8 (in German). 6. H. Bergmann and H. Oumeraci, Wave-induced water levels and pressure distribution at perforated wall, Proc. COPEDEC VII, Dubai, UAE (2008), Paper No. B-11 (in print). 7. CEM, Coastal Engineering Manual, Engineer Manual 1110-2-1100 (US Army Corps of Engineers, Washington, DC, USA, 2003).

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8. T. Hayakawa, K. Kimura, S. Takahashi, M. Muttray, M. Kudella and H. Oumeraci, Wave splash height on a high-mound composite breakwater, Proc. 4th Conf. Hydrodynamics (ICHD’2000), Yokohama, Japan, Vol. 2 (2000), pp. 671–676. 9. G. E. Jarlan, A perforated vertical wall breakwater, The Dock and Harbour Authority (1961), pp. 394–398. 10. K. Kimura, T. Fujiike, K. Kamikubo, R. Abe and K. Ishimoto, Damages to vehicles on a coastal highway by wave action, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 2 (2000), pp. 1009–1016. 11. G. Koether, Hydraulische Wirksamkeit getauchter Einzelﬁlter und Filtersysteme — Prozessbeschreibung und Modellbildung f¨ ur ein innovatives Riﬀkonzept, Dissertation an der TU Braunschweig, Leichtweiss-Institut f¨ ur Wasserbau (2002), p. 163 (in German). 12. G. Koether, H. Bergmann and H. Oumeraci, Wave attenuation by submerged ﬁlter systems, Proc. 4th Int. Conf. Hydrodynamics (ICHD’2000), Yokosuka, Japan, Vol. 2 (2000), pp. 711–716. 13. G. Koether and H. Oumeraci, Schutzbauwerke f¨ ur Sandstr¨ ande mit touristischer und o ¨kologischer Bedeutung — Unterwasser-Filtersysteme, Hansa, Heft, 11 (2001), pp. 93–98. 14. M. Mori, Y. Yamamoto and K. Kimura, Wave force and stability of upright section of high mound composite seawall, Proc. Abstracts Int. Conf. Coastal Eng., Hamburg (2008), p. 206. 15. M. Muttray, H. Oumeraci, K. Shimosako and S. Takahashi, Hydraulic performance of high mound composite breakwater, Proc. 26th Int. Conf. Coastal Eng. (ICCE‘98) (1998), pp. 2207–2220. 16. M. Muttray, H. Oumeraci, K. Shimosako and S. Takahashi, Wave load on an innovative high mound composite breakwater: Results of large scale experiments and tentative design formulae, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 1 (2000), pp. 353–362. 17. J. Newe, Methodik f¨ ur großmaßst¨ abliche 2D-Experimente zum Strandverhalten unter Sturmﬂutbedingungen, Dissertation an der TU Braunschweig, Leichtweiss-Institut f¨ ur Wasserbau (2002) (in German). 18. H. Oumeraci, The sustainability challenge in coastal engineering, Keynote Lecture, Proc. 4th Int. Conf. Hydrodynamics (ICHD), eds. Y. Goda et al., Yokohama, Japan, Vol. 1 (2000), pp. 57–83. 19. H. Oumeraci, Breakwaters, Part 2, Planing and Design of Ports and Marine Terminals, ed. H. Agerschou (Thomas Telford, London, UK, 2004), pp. 155–262. 20. H. Oumeraci, Nearshore and onshore tsunami eﬀects, Background paper for DFG-Round Table Discussion, Hannover, http://www.fzk.uni-hannover.de/323.html (2006). 21. H. Oumeraci, G. F. Clauss, R. Habel and G. Koether, Unterwasser Filtersysteme zur Wellend¨ ampfung, Final Report of BMBF-Research Project (2001), p. 311 (in German). 22. H. Oumeraci, A. Kortenhaus, N. W. H. Allsop, M. B. De Groot, R. S. Crouch, J. K. Vrijling and H. G. Voortman, Probabilistic Design Tools for Vertical Breakwaters (Balkema, Rotterdam, The Netherlands, 2001), pp. 392. 23. H. Oumeraci and G. Koether, Hydraulic performance of a submerged wave absorber for coastal protection, Advances in Coastal Ocean Engineering (2008), pp. 36 (in print). 24. H. Oumeraci, A. Kortenhaus, H. Breustedt and P. Schley, Hydraulic model investigations on breakwaters with a core made of geotextile sand containers, Research Report No. 933, Report Leichtweiß-Institut f¨ ur Wasserbau, TU Braunschweig (2007), pp. 71 and Annexes.

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25. H. Oumeraci, M. Kudella, M. Muttray, K. Kimura and T. Hayakawa, Wave run-up and wave overtopping on a high mound composite breakwater, Final Report No. 831, Leichtweiss-Institut, TU Braunschweig (1998), pp. 88. 26. H. Oumeraci and M. Muttray, Large scale model tests on a high-mound composite type breakwater, Research Report No. 818, Leichtweiss-Institute for Hydraulics, TU Braunschweig (1997), pp. 115. 27. H. Oumeraci and M. Muttray, Bemessungswellenparameter vor Strukturen mit verschiedenen Reﬂexionseigenschaften, Abschlussbericht DFG-Projekt, OU 1/3-3, Braunschweig, Germany (2001), 93 S (in German). 28. H. Oumeraci, M. M. Muttray, K. Kudella, K. Kimura and S. Takahashi, Wave loading of a high mound composite breakwater (HMCB) with splash reducers, Proc. 4th Int. Conf. Hydrodynamics (ICHD’2000), Yokohama, Japan, Vol. 2 (2000), pp. 659–670. 29. H. Oumeraci, H. Sch¨ uttrumpf, A. Kortenhaus, M. Kudella, J. M¨ oller and M. Muttray, Untersuchungen zur Erweiterung bzw. zum Umbau des Deckwerks am Nordstrand von Norderney, Research Report No. 853, Leichtweiss-Institute for Hydraulics, TU Braunschweig (2000b) (in German). 30. P. Schley, Hydraulische Wirksamkeit von geschuetteten Wellenbrechern mit herkoemmlichen Kern und einem Kern aus geotextilen Sandcontainern — eine experimentelle Vergleichsstudie, Diplomarbeit am Leichtweiss-Institut f¨ ur Wasserbau, Fachbereich Bauingenieurwesen, TU Braunschweig, Germany (2006) 80 S, 8 Anlagen (in German). 31. H. Sch¨ uttrumpf, H. Oumeraci, K. Kimura, T. Hayakawa and J. Moeller, Wave overtopping on a high mound composite breakwater, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 1 (2000), pp. 397–404. 32. H. Sch¨ uttrumpf, H. Oumeraci, F. Thorenz and J. M¨ oller, Reconstruction and rehabilitation of a historical seawall at Norderney, ICE, Proc. Breakwaters, Coastal Structures Coastlines, ed. N. W. H. Allsop (Thomas Telford, London, 2002), pp. 257–268. 33. S. Takahashi, K. Shimosako, H. Oumeraci, M. Muttray and K. Kimura, Reduced wave overtopping characteristics of a new high mound composite breakwater: Results of small-scale experiments, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 1 (2000), pp. 389–396. 34. TAW: Technical Report on wave run-up and wave overtopping at dikes, Techn. Advisory Committee on Flood Defence, Delft, The Netherlands.

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Chapter 13

Wave Interaction with Breakwaters Including Perforated Walls Kyung-Duck Suh Department of Civil and Environmental Engineering Seoul National University, San 56-1, Shillim-dong, Gwanak-gu Seoul 151-744, Republic of Korea [email protected] During the past several decades, breakwaters including perforated walls have been introduced to resolve various problems associated with gravity-type breakwaters. In this chapter, first, the mathematical models are described that predict various hydrodynamic characteristics of single- or multiple-row curtain-wall-pile breakwaters, the upper part of which is a curtain wall and the lower part consisting of an array of vertical piles. Their extension to irregular waves is also described. These models can be used for curtain-wall breakwaters by just removing the piles. They can also be used for pile breakwaters by removing the curtain-wall and extending the piles to the water surface. Second, the mathematical model to predict wave reflection from a fully-perforated-wall caisson mounted on a rubble foundation is described, and its applicability to a partially-perforated-wall caisson and irregular waves is described. Third, a discussion is given for the calculation of the so-called permeability parameter, which represents the energy dissipation and phase-shift of flows passing across a perforated wall.

13.1. Introduction Gravity-type breakwaters using rubble mound or vertical caissons have been widely used to provide a calm basin for ships and to protect harbor facilities from rough seas. In general, the width of these breakwaters increases with water depth, leaving a large footprint and requiring a great amount of construction material, especially when built in deeper water. Often, they block littoral drift and cause severe erosion or accretion in neighboring beaches. In addition, they prevent the circulation of water, thus deteriorating the water quality within the harbor. In some places, they obstruct the passage of ﬁshes and bottom-dwelling organisms. A solid soil foundation is also needed to support such heavy structures. 317

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In order to resolve the above-mentioned problems, perforated wall structures have been introduced especially in small craft harbors. The simplest perforated wall structure may be a curtain-wall breakwater (sometimes called wave screen or skirt breakwater), which consists of a vertical wall extending from the water surface to some distance above the seabed.1−3 Recently, Isaacson et al.4 proposed a slotted curtain-wall breakwater. Another simple perforated wall structure may be an array of vertical piles, which is called a pile breakwater in this chapter. The closely spaced piles induce energy dissipation due to viscous eddies formed by the ﬂow through the gaps. To examine the wave scattering by vertical piles, hydraulic model tests have been used.5−8 Eﬀorts toward developing analytical models to calculate the reﬂection and transmission coeﬃcients have also been made.8−11 Recently, Suh et al.12 introduced a curtain-wall-pile breakwater, the upper part of which is a curtain wall and the lower part consisting of an array of vertical piles. They developed a mathematical model that predicts various hydrodynamic characteristics of a curtain-wallpile breakwater. More recently, Suh and Ji13 extended the model to a multiple-row breakwater. Another type of breakwaters including perforated walls is a perforated-wall caisson breakwater, even though it is still a gravity-type breakwater. It reduces not only wave reﬂection, but also wave transmission due to overtopping. It also reduces wave forces, especially impulsive wave forces, acting on the caisson.14,15 In order to examine the reﬂection characteristics of a perforated-wall caisson breakwater, hydraulic model tests have been used.16−19 Mathematical models for predicting the reﬂection coeﬃcient have also been developed.10,20,21 On the other hand, Fugazza and Natale22 proposed a closed-form solution for wave reﬂection from a perforatedwall caisson. The aforementioned mathematical approaches dealt with the case in which the waves are normally incident to the perforated-wall caisson lying on a ﬂat sea bottom. To resolve these limitations, Suh and Park23 developed a mathematical model that can predict the wave reﬂection from a fully-perforated-wall caisson mounted on a rubble mound foundation when regular waves are obliquely incident on the breakwater at an arbitrary angle. Recently, Suh et al.24 described how to apply this model to a partially-perforated-wall caisson and irregular waves. In this chapter, ﬁrst, the mathematical models of Suh et al.12 and Suh and Ji13 are described. These models were developed to predict the hydrodynamic characteristics of curtain-wall-pile breakwaters. However, they can also be used for curtainwall breakwaters or pile breakwaters by just removing the piles or by removing the curtain wall and extending the piles to the water surface, respectively. Second, the mathematical model of Suh and Park23 and its applicability to a partiallyperforated-wall caisson and irregular waves is described. Third, a discussion is given for the calculation of the so-called permeability parameter, which represents the energy dissipation and phase-shift of ﬂows passing across a perforated wall. 13.2. Curtain-Wall-Pile Breakwaters 13.2.1. Single-row curtain-wall-pile breakwater A curtain-wall-pile breakwater is sketched in Fig. 13.1, in which h = constant water depth in still water; d = height of the curtain wall below the still water level;

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Fig. 13.1.

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A deﬁnition sketch of a curtain-wall-pile breakwater: (a) side view; and (b) front view.

b = thickness of the wall. A Cartesian coordinate system (x, z) is deﬁned with the positive x directing downwave from the crest line of the breakwater and the vertical coordinate z being measured vertically upward from the still water line. The distance between the centers of two neighboring piles is denoted as 2A, and the width of an opening is 2a so that the porosity of the lower part of the breakwater at x = 0 is deﬁned as r0 = a/A. A regular wave train with wave height Hi is incident on the positive x-direction. The ﬂuid domain is divided into region 1 (x ≤ 0) and region 2 (x ≥ 0). Assuming incompressible ﬂuid and irrotational ﬂow motion, the velocity potential exists, which satisﬁes the Laplace equation. Linearizing the free-surface boundary conditions, the following boundary value problem for the velocity potential Φ(x, z, t) is obtained: ∂ 2Φ ∂ 2Φ + = 0, ∂x2 ∂z 2

(13.1)

∂Φ ω 2 − Φ = 0 at z = 0 , ∂z g

(13.2)

∂Φ = 0 at z = −h , ∂z

(13.3)

where ω = wave angular frequency, and g = gravitational acceleration. Assuming periodic motion in time t, we can assume the solution to the above problem as 1 igHi φ(x, z) exp(−iωt) , (13.4) Φ(x, z, t) = Re − 2ω cosh(kh) √ where i = −1; and the symbol Re represents the real part of a complex value. The wave number k must satisfy the dispersion relationship, ω 2 = gk tanh(kh). The spatial variation of the velocity potential φ(x, z) should be determined in each region.

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We assume that the length scale of the ﬂow near the breakwater is of the order of the wall thickness, which is much smaller than the far-ﬁeld length scale of O(k−1 ), so that the wall has no thickness mathematically and the 3D feature near the breakwater does not signiﬁcantly aﬀect the 2D far-ﬁeld solutions. Then, φ1 (x, z) and φ2 (x, z) must satisfy the following matching conditions at x = 0: ∂φ1 ∂φ2 = = 0 for − d ≤ z ≤ 0, x = 0 , ∂x ∂x ∂φ1 ∂φ2 = = iG(φ1 − φ2 ) for − h ≤ z ≤ −d, x = 0 , ∂x ∂x

(13.5) (13.6)

where the subscripts indicate the regions of the ﬂuid domain. The ﬁrst matching condition describes that the horizontal velocities vanish on both sides of the curtain wall of the breakwater. The second one for the lower part of the breakwater describes that the horizontal mass ﬂuxes (or indirectly horizontal velocities) in the two regions must be the same at the breakwater and that the horizontal velocity at the opening is proportional to the diﬀerence of velocity potentials, or the pressure diﬀerence, across the breakwater. The proportional constant G, often called permeability parameter, is in general complex, which will be discussed in detail later. To solve the boundary value problem [Eqs. (13.1) to (13.3)], we use the eigenfunction expansion method of Isaacson et al.4 The velocity potential is expressed in a series of inﬁnite number of solutions: φ1 = φi −

∞

Am cos[µm (h + z)] exp(µm x) ,

(13.7)

Am cos[µm (h + z)] exp(−µm x) ,

(13.8)

m=0

φ2 = φi +

∞ m=0

where φi = cosh[k(h+z)] exp(ikx) is the incident wave potential. The wave numbers µm are the solution of the dispersion relation, ω 2 = −gµm tan(µm h), which has an inﬁnite discrete set of real roots ±µm (m ≥ 1) for nonpropagating evanescent waves and a pair of imaginary roots µ0 = ±ik for propagating waves. We take µ0 = −ik so that the propagating waves in Eqs. (13.7) and (13.8) correspond to the reﬂected and transmitted waves, respectively. We also take the positive roots for m ≥ 1 so that the nonpropagating waves die out exponentially with the distance from the breakwater. Now, Eqs. (13.7) and (13.8) satisfy the free-surface boundary condition in Eq. (13.2) and the bottom boundary condition in Eq. (13.3). Also, they automatically satisfy the requirement that the horizontal velocities must be matched at the breakwater. In order to solve for the unknown coeﬃcients, Am s, we use the matching conditions at the breakwater. First, Eqs. (13.7) and (13.8) are substituted into Eqs. (13.5) and (13.6), respectively. Multiplying each resulting equation by cos[µn (h + z)], integrating with respect to z over the appropriate domain of z (i.e., z = −d to 0, or z = −h to −d), and ﬁnally adding them, we obtain a matrix

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equation for Am : ∞

Cmn Am = bn

for n = 0, 1, 2, . . . , ∞ ,

(13.9)

m=0

where

Cmn = µm fmn (−d, 0) + (µm − 2iG)fmn (−h, −d) ,

(13.10)

bn = −µ0 [f0n (−d, 0) + f0n (−h, −d)] ,

(13.11)

fmn (p, q) =

q p

cos[µm (h + z)] cos[µn (h + z)]dz



q 1 sin[(µm + µn )(h + z)] sin[(µm − µn )(h + z)]    + for m = n , 2 µm + µn µm − µn p =  1   [2µm (h + z) + sin[2µm (h + z)]]qp for m = n .  4µm

(13.12)

Note that the mathematical model developed for a curtain-wall-pile breakwater can be used for a pile breakwater just by setting d = 0. It can also be used for a curtain-wall breakwater by using the permeability parameter G = 1/b, which was derived from the energy dissipation formula for a curtain-wall breakwater of Kriebel.25 Once the wave potentials are calculated, we can obtain various engineering wave properties. The reﬂection and transmission coeﬃcients are given by Cr = |A0 |

(13.13)

Ct = |1 + A0 | ,

(13.14)

and

respectively. The wave runup on the upwave face of the breakwater is given by ∞ Hi 1 Am cos(µm h) . (13.15) Ru = 1 − 2 cosh(kh) m=0 In the limiting case of a full-depth impermeable vertical wall (d → h or r0 → 0), Am = 0 for all m ≥ 1 and A0 = −1, so that Cr = 1.0, Ct = 0.0, and Ru = Hi as expected. Since the vertical distributions of wave pressure on both upwave and downwave sides of the breakwater are known, the wave force can also be calculated. The maximum horizontal wave force Fmax per unit width of the breakwater is given by ∞ A 1 m {sin(µm h) − r0 sin[µm (h − d)]} Fmax = ρgHi cosh(kh) µm m=0

2 ∞ 1 Am cos(µm h) 1− cosh(kh) m=0 2 ∞ 1 , − 1+ Am cos(µm h) cosh(kh) m=0

ρg + Hi2 8

(13.16)

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where ρ = density of ﬂuid. The second term on the right-hand side represents the second-order force contribution of the wave crest regions on the upwave and downwave sides of the breakwater.26 Without this term, in the limiting case of a full-depth impermeable vertical wall (d → h or r0 → 0), Am = 0 for all m ≥ 1 and A0 = −1 so that the preceding equation becomes s = ρgHi Fmax

1 sinh(kh) , k cosh(kh)

(13.17)

where the superscript s stands for the standing wave in front of an impermeable vertical wall. In another limiting case of no breakwater (d → 0 and r0 → 1), Fmax becomes zero, as expected. 13.2.2. Extension to multiple-row breakwaters The single-row curtain-wall-pile breakwater still gives large transmission for longperiod waves. To reduce the wave transmission, the draft of the curtain wall must be increased, or the porosity between the piles must be decreased. Then, however, the wave reﬂection and the wave forces and moments acting on the breakwater may increase. Therefore, it is diﬃcult to increase the draft or decrease the porosity beyond certain limits. A multiple-row breakwater may be a solution for these problems. It may also be advantageous for providing a space on top of the breakwater for ﬁshing, walking, and maintenance of the breakwater, and so on. Suh and Ji13 extended the mathematical model to a multiple-row breakwater, whose deﬁnition sketch is shown in Fig. 13.2, in which dj = height of the jth curtain wall below the still water level; and bj = thickness of the jth wall. The center of the jth wall is located at x = xj . The distance between the centers of two neighboring piles is denoted as 2Aj , and the width of the gap between the piles is 2aj so that the porosity of the lower part of the jth wall is deﬁned as rj = aj /Aj . The ﬂuid domain is divided into J + 1 regions by the J walls. The upwave and downwave regions of the jth wall are deﬁned as Ωj−1 and Ωj , respectively. Using the same assumptions as those made for a single-row breakwater, the same boundary value problem as that given by Eqs. (13.1)–(13.3) is obtained for the velocity potential Φj (x, z, t) in each region, which can be assumed as 1 igHi φj (x, z) exp(−iωt) . Φj (x, z, t) = Re − 2ω cosh(kh)

(13.18)

The spatial variation of the velocity potential φj (x, z) should be determined in each region. The solutions in the regions of Ω0 and ΩJ are given by φ0 = A00 cos[µ0 (h + z)] exp[−µ0 (x − x1 )] +

∞ m=0

B0m cos[µm (h + z)] exp[µm (x − x1 )] ,

(13.19)

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z Hi

Ω1

Ω0

Ωj

Ω2 Ωj-1

ΩJ-1

ΩJ x bj

h

d1

d2

dj

dJ

b1

b2

bj

bJ

r1

r2

rj

rJ

x1

xj

x2

xJ

dj

2aj 2Aj

(a)

(b)

Fig. 13.2. A deﬁnition sketch of a multiple-row curtain-wall-pile breakwater: (a) side view; and (b) bird’s-eye view of jth row.

φJ =

∞

AJm cos[µm (h + z)] exp[−µm (x − xJ )] ,

(13.20)

m=0

respectively. Here, Ajm and Bjm are the coeﬃcients of the component waves propagating forward and backward, respectively. The ﬁrst subscript (j) indicates the row of the wall, while the second one (m) indicates the wave component. The wave numbers µm are the same as those used for a single-row breakwater. Taking A00 = 1, the ﬁrst term on the right-hand side of Eq. (13.19) represents the incident wave potential, and |B00 | and |AJ0 | are the reﬂection and transmission coeﬃcients, respectively. On the other hand, the solutions in the regions from Ω1 to ΩJ−1 are given by φj =

∞

{Ajm exp[−µm (x − xj )] + Bjm exp[µm (x − xj+1 )]} cos[µm (h + z)] ,

m=0

j = 1, 2, . . . , J − 1.

(13.21)

The matching conditions at each row of the breakwater are the same as Eqs. (13.5) and (13.6), i.e., ∂φj ∂φj−1 = = 0 for − dj ≤ z ≤ 0, x = xj , ∂x ∂x ∂φj−1 ∂φj = = iGj (φj−1 − φj ) for − h ≤ z ≤ −dj , x = xj . ∂x ∂x

(13.22) (13.23)

These matching conditions are used to solve for the unknown coeﬃcients Ajm s and Bjm s. First, for x = x1 , Eqs. (13.19) and (13.21) are substituted into Eqs. (13.22) and (13.23). Multiplying each resulting equation by cos[µn (h + z)], integrating with

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respect to z over the appropriate domain of z (i.e., z = −d1 to 0, or z = −h to −d1 ), and ﬁnally adding them, we obtain ∞

[{µm fmn(−d1 , 0) + (µm − iG1 )fmn (−h, −d1 )}B0m

m=0

+ iG1 fmn (−h, −d1 )A1m + iG1 exp(−µm ∆x1 )fmn (−h, −d1 )}B1m ] = µ0 A00 {f0n (−d1 , 0) + f0n (−h, −d1 )} + iG1 A00 f0n (−h, −d1 ) for n = 0, 1, 2, . . . , ∞

(13.24)

and ∞

[µm {fmn (−d1 , 0) + fmn (−h, −d1 )}B0m

m=0

+ µm {fmn (−d1 , 0) + fmn(−h, −d1 )}A1m − µm exp(−µm ∆x1 ){fmn (−d1 , 0) + fmn (−h, −d1 )}B1m ] = µ0 A00 {f0n (−d1 , 0) + f0n (−h, −d1 )}

for n = 0, 1, 2, . . . , ∞ ,

(13.25)

where ∆xj = xj+1 − xj ,

(13.26)

and fmn (p, q) are the same as given in Eq. (13.12). For x = xj (j = 2 to J − 1), Eq. (13.21) is substituted into Eqs. (13.22) and (13.23). Again, multiplying each resulting equation by cos[µn (h + z)], integrating with respect to z over the appropriate domain of z, and ﬁnally adding them, we obtain ∞

[exp(−µm ∆xj−1 ){−µm fmn (−dj , 0)+µm fmn (−h, −d1 )

m=0

+ iGj fmn (−h, −dj )}Aj−1,m + {µm fmn (−dj , 0) − µm fmn (−h, −dj ) + iGj fmn (−h, −dj )}Bj−1,m − iGj fmn (−h, −dj )Ajm − iGj exp(−µm ∆xj )fmn (−h, −dj )Bjm ] = 0 for n = 0, 1, 2, . . . , ∞

(13.27)

and ∞

[µm exp(−µm ∆xj−1 ){fmn(−dj , 0)+fmn(−h, −dj )}Aj−1,m

m=0

− µm {fmn(−dj , 0) + fmn (−h, −dj )}Bj−1,m − µm {fmn(−dj , 0) + fmn (−h, −dj )}Ajm + µm exp(−µm ∆xj ){fmn (−dj , 0) + fmn (−h, −dj )}Bjm ] = 0 for n = 0, 1, 2, . . . , ∞ .

(13.28)

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Finally, for x = xJ , Eqs. (13.20) and (13.21) are substituted into Eqs. (13.22) and (13.23). Using the similar procedure as above, we obtain ∞

[exp(−µm ∆xJ−1 ){−µm fmn(−dJ , 0) + µm fmn (−h, −dJ )

m=0

+ iGJ fmn (−h, −dJ )}AJ−1,m + {µm fmn (−dJ , 0) − µm fmn (−h, −dJ ) + iGJ fmn (−h, −dJ )}BJ−1,m − iGJ fmn (−h, −dJ )AJm ] = 0

for n = 0, 1, 2, . . . , ∞

(13.29)

and ∞

[µm exp(−µm ∆xJ−1 ){fmn (−dJ , 0)+fmn (−h, −dJ )}AJ−1,m

m=0

− µm {fmn (−dJ , 0) + fmn(−h, −dJ )}BJ−1,m − µm {fmn (−dJ , 0) + fmn (−h, −dJ )}AJm ] = 0

for n = 0, 1, 2, . . . , ∞ .

(13.30)

If we take M wave modes (i.e., one progressive and M − 1 evanescent modes), the number of unknown coeﬃcients is 2JM . Equations (13.24), (13.25), and (13.27)– (13.30) give 2JM equations, which can be solved for the unknown coeﬃcients. For a multiple-row breakwater, the reﬂection and transmission coeﬃcients are given by Cr = |B00 | ,

(13.31)

Ct = |AJ0 | ,

(13.32)

and

respectively. The wave runup on the upwave side of the ﬁrst row of the breakwater is given by ∞ 1 Hi Ru = (13.33) B0m cos(µm h) . A00 cos(µ0 h) + 2 cosh(kh) m=0

The wave force on each wall can be calculated by integrating the wave pressure acting on upwave and downwave sides of the wall. The maximum horizontal wave force Fmax per unit width of the ﬁrst wall of breakwater is given by A00 ρgHi 1 sin(µ0 h) Fmax = 2 cosh(kh) µ0 ∞ 1 + {B0m − A1m − B1m exp[µm (x1 − x2 )]} sin(µm h) µ m=0 m A00 −r1 sin[µ0 (h − d1 )] µ0

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∞ 1 + {B0m − A1m − B1m exp[µm (x1 − x2 )]} sin[µm (h − d1 )] µ m=0 m  2 ∞  1 ρgHi2 + A00 cos(µ0 h) + B0m cos(µm h) 8 cosh2 (kh) 

m=0

−

∞

m=0

2   {A1m + B1m exp[µm (x1 − x2 )]} cos(µm h) .  (13.34)

Again, the second term on the right-hand side represents the second-order force contribution of the wave crest regions on the upwave and downwave sides of the wall. Without this term, in the limiting case of a full-depth impermeable vertical wall, B0m = 0 for all m ≥ 1, B00 = A00 = 1, and A1m = B1m = 0 for all m ≥ 0, so that the preceding equation becomes Eq. (13.17). 13.2.3. Extension to irregular waves Using the above regular wave models, the reﬂection and transmission coeﬃcients can be calculated diﬀerently for each frequency component, i.e., Cr (f ) and Ct (f ), where f = wave frequency. In the computation of the permeability parameter G in Eqs. (13.6) and (13.23), the root-mean-squared (rms) wave height is used in place of the incident wave height Hi , because the ﬂow separation due to irregular waves and the resulting energy dissipation are induced not by the individual component waves but by the superposition of the component waves. The rms wave height is √ calculated by Hrms = Hs / 2, where Hs is the signiﬁcant wave height. The spectral densities of the reﬂected and transmitted waves, respectively, are calculated for a particular frequency component by Sη,r (f ) = |Cr (f )|2 Sη,i (f ) , 2

Sη,t (f ) = |Ct (f )| Sη,i (f ) ,

(13.35) (13.36)

where Sη,i (f ) = incident wave energy density. The frequency-averaged reﬂection and transmission coeﬃcients are then calculated as27 m0,r Cr = , (13.37) m0,i m0,t Ct = , (13.38) m0,i where m0,i , m0,r , and m0,t = zeroth moments of the incident, reﬂected, and transmitted wave spectra, respectively. Note that the reﬂection and transmission coefﬁcients calculated by the conventional method, as the ratio of the reﬂected and

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transmitted rms wave heights to the incident one, are equivalent to the frequency√ averaged coeﬃcients, because the rms wave height is proportional to m0 . The force spectrum, SF (f ), is calculated by 2

SF (f ) = |TF (f )| Sη,i (f ) ,

(13.39)

where TF (f ) is the frequency-dependent transfer function between wave amplitude and force amplitude, which can be computed by using the linear term in Eq. (13.16) as TF = 2ρg

∞ 1 Am {sin(µm h) − r0 sin[µm (h − d)]} . cosh(kh) m=0 µm

The zero-moment force, Fm0 , can then be determined as ∞

1/2 Fm0 = 2 SF (f )df .

(13.40)

(13.41)

0

Note that a factor of 2 instead of 4 was used in order to represent the force amplitude.

13.3. Perforated-Wall Caisson Breakwaters 13.3.1. Fully-perforated-wall caisson breakwater Let us consider the perforated-wall caisson breakwater sketched in Fig. 13.3, in which θ1 = incident wave angle; and B = wave chamber width. The x-axis and y-axis are taken to be normal and parallel, respectively, to the breakwater crest line, and the water depth is assumed to be constant in the y-direction. The vertical

Fig. 13.3. A schematic diagram and coordinate system for calculation of wave reﬂection form a perforated-wall caisson breakwater.

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coordinate z is measured vertically upward from the still water line. In region 2 (−b ≤ x ≤ 0), the water depth h(x) is a varying function of x. For x ≤ −b (region 1) and 0 ≤ x ≤ B (region 3), the water depth is constant and equal to h1 and h3 , respectively. The rubble mound foundation is assumed to be impervious. Assuming incompressible ﬂuid and irrotational ﬂow motion, the velocity potential Φ(x, y, z, t) for the monochromatic wave propagating over the water depth h(x) with the angular frequency ω and wave height H can be expressed as igH Φ(x, y, z, t) = Re − φ(x, y, z) exp(−iωt) . (13.42) 2ω Linearizing the free-surface boundary conditions, the following boundary value problem for the potential φ(x, y, z) is obtained: ∇2 φ +

∂ 2φ = 0, ∂z 2

∂φ − λφ = 0 at z = 0 , ∂z ∂φ ∂φ dh + = 0 at z = −h(x) , ∂z ∂x dx

(13.43) (13.44) (13.45)

where ∇ represents the horizontal gradient operator, and λ = ω 2 /g is a wave number in deepwater. The Galerkin-eigenfunction method is used to formulate the problem. The Galerkin approach assumes that the function φ(x, y, z) can be expanded in terms of M + 1 depth-dependent functions Zm (x, z): φ(x, y, z) =

M

ϕm (x, y)Zm (x, z) .

(13.46)

m=0

The functions Zm (x, z) are taken as Zm (x, z) =

cos[µm (z + h)] , cos(µm h)

(13.47)

so as to form a complete orthogonal set of eigenfunctions in the domain [−h(x), 0]. The wave numbers µm are the solution of the dispersion relation, λ + µm tan(µm h) = 0. The function Z0 (x, z) represents the free propagating wave mode, while the functions Zm (x, z) (m ≥ 1) correspond to the nonpropagating evanescent modes. The functions Zm (x, z) satisfy the free surface boundary condition in Eq. (13.44) and do not satisfy the bottom boundary condition in Eq. (13.45) individually. However, the global set of orthogonal functions should satisfy this condition. This is known as a tau method. In the tau method, a suﬃcient number of functions ϕm (x, y) in the approximated solution of Eq. (13.46) is chosen to ensure exact satisfaction of the bottom boundary condition. Turning back to the breakwater problem sketched in Fig. 13.3, the solution of the boundary value problem given by Eqs. (13.43)–(13.45) can be constructed from

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the particular solutions in each region of the ﬂuid domain: φ1 (x, y, z) = {exp[ik1 (x + b) cos θ1 ] − exp[−ik1 (x + b) cos θ1 ]} × exp(iχy) +

cosh k1 (z + h1 ) cosh k1 h1

Rm exp[ν1,m (x + b)] exp(iχy)Z1,m (h1 , z),

(13.48)

µ1,m

φ2 (x, y, z) =

ϕ˜m (x) exp(iχy)Z2,m (h2 , z) ,

(13.49)

Tm exp(−ν3,m x) exp(iχy)Z3,m (h3 , z) ,

(13.50)

µ2,m

φ3 (x, y, z) =

µ3,m

where νj,m =

µ2j,m + χ2 ,

χ = kj sin θj = k1 sin θ1 = constant (j = 1, 2, 3) .

(13.51)

If m = 0 (propagating mode), the following relationships are obtained: λ = kj tanh(kj hj ),

νj,0 = ±ikj cos θj .

(13.52)

For νj,0 , we take the negative sign for the reﬂected wave in region 1, while we need both positive and negative signs for the waves inside the wave chamber (i.e., region 3). The potential φj (x, y, z) must satisfy the matching conditions which provide continuity of pressure and horizontal velocity, normal to the vertical planes separating the ﬂuid regions and no-ﬂux condition at the wall on the downwave side of the wave chamber, i.e., ∂φ1 ∂φ2 (13.53) = (x = −b, −h1 ≤ z ≤ 0) , ∂x ∂x i ∂φ2 ∂φ2 ∂φ3 φ3 = φ2 + (13.54) , = (x = 0, −h3 ≤ z ≤ 0) , G ∂x ∂x ∂x ∂φ3 = 0 (x = B, −h3 ≤ z ≤ 0) . (13.55) ∂x As shown in Eq. (13.46), the Galerkin solution consists of a free propagating wave mode and nonpropagating evanescent modes. The evanescent wave modes would be of importance in the region near the breakwater. However, since we are interested in the reﬂected wave far from the breakwater, we focus on the solution for the propagating mode. Massel28 showed that in region 2, the function ϕ˜0 (x) satisﬁes the following ordinary diﬀerential equation: φ1 = φ2 ,

dϕ˜0 d2 ϕ˜0 + E(x)ϕ˜0 = 0 , + D(x) dx2 dx where

2τ k dh 2 D(x) = 1 − 3τ + , τ + kh(1 − τ 2 ) τ + kh(1 − τ 2 ) dx

2 u1 dh u 2 d2 h 2 − χ2 , + 2 E(x) = k 1 + 2 k u0 dx k u0 dx2

(13.56)

(13.57) (13.58)

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where τ = tanh(kh), and u0 , u1 , and u2 are given by

K 1 tanh(kh) 1 + , u0 = 2k sinh K u1 =

sech2 (kh) (sinh K − K cosh K) , 4(K + sinh K)

u2 =

ksech2 (kh) [K 4 + 4K 3 sinh K − 9 sinh K sinh(2K) 12(K + sinh K)3 + 3K(K + 2 sinh K)(cosh2 K − 2 cosh K + 3)] ,

(13.59) (13.60)

(13.61)

where the abbreviation K = 2kh was used. In order to solve Eq. (13.56) in region 2, we need the boundary conditions at x = −b and x = 0. Suh and Park23 derived these boundary conditions as dϕ˜0 (−b) = i[2 − ϕ˜0 (−b)]k1 cos θ1 , dx

i dϕ˜0 (0) 1 exp(−ν3,0 B) + exp(ν3,0 B) − . ϕ˜0 (0) = ν3,0 exp(−ν3,0 B) − exp(ν3,0 B) G dx

(13.62) (13.63)

The diﬀerential equation (13.56) with the preceding two boundary conditions can be solved using the ﬁnite-diﬀerence method. Using the forward-diﬀerencing for dϕ˜0 (−b)/dx, backward-diﬀerencing for dϕ˜0 (0)/dx, and central-diﬀerencing for the derivatives in Eq. (13.56), the boundary value problem of Eqs. (13.56), (13.62), and (13.63) can be approximated by a system of linear equations, AY = B, where A is a tridiagonal-band-type matrix, Y is a column vector, and B is also a column vector. The subroutines given in the book of Press et al.29 can be used to solve this matrix equation. In particular, we are interested in ϕ˜0 (−b), from which the reﬂection coeﬃcient Cr is calculated as follows: Cr = ϕ˜0 (−b) − 1 .

(13.64)

As stated in the introduction, the advantage of this model is that it can be used for a perforated-wall caisson mounted on a rubble mound foundation when waves are incident at an arbitrary angle. In order to examine the eﬀect of rubble mound foundation upon wave reﬂection, Suh et al.30 calculated the reﬂection coeﬃcients by changing the height and slope of the rubble mound. Total water depth in front of the breakwater h1 = 10 m, wave period T = 8 s, incident wave height H = 2 m, thickness of the perforated wall l = 0.6 m, porosity of the wall r = 0.3, and wave chamber width B = 10 m were used. Figure 13.4 shows the ratio of the reﬂection coeﬃcient of the breakwater with a mound to that of the breakwater without a mound (i.e., lying on a ﬂat sea bottom). When the mound height is approximately less than 0.3 times the total water depth, the reﬂection coeﬃcient of the breakwater with a mound is less than that of the breakwater without a mound. For the mound height greater than this value, however, the wave reﬂection from the mound slope becomes signiﬁcant so that the reﬂection coeﬃcient becomes greater than that of the breakwater without a mound. In addition, the reﬂection coeﬃcient increases with

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1.0 0.9

Mound Slope

0.8 0.7 0.6 0.5 0.4 0.3 0.0

0.1

0.2

0.3

0.4

0.5

(h1-h3)/h1 Fig. 13.4. Contours of the ratio of the reﬂection coeﬃcient of the breakwater with a mound to that of the breakwater without a mound as a function of mound height and slope.

the height and slope of the mound as expected. Within the range of the practical design situations, however, the rubble mound eﬀect on wave reﬂection is not so large. However, it should be noted that the eﬀect of permeability of the rubble mound was not take into account because the rubble mound was assumed to be impervious. Suh et al.30 also examined the eﬀect of angle of incidence upon wave reﬂection. The reﬂection coeﬃcients were calculated by changing the wave angle from 0◦ to 80◦ and B cos θ/L from 0.1 to 0.5. The caisson was situated in a constant depth without a mound so that h1 = h2 = h3 = 30 cm. Wave height H = 3 cm, wave period T = 1.554 s, porosity of the wall r = 0.25 were used, while the perforated wall thickness was taken to be zero (i.e., the inertia eﬀect of the perforated wall was neglected). The calculated results are shown in Fig. 13.5. For normal incidence of waves, the wave reﬂection is minimum at B cos θ/L = 0.25 and increases as it gets away from this value. The same trend is kept for oblique incidence of waves, but the solution exhibits singular behavior (i.e., the reﬂection coeﬃcient shows a sharp peak) at B/L = 0.5, 1.0, 1.5, and so on, as shown in Fig. 13.5. In the practical design of a perforated-wall caisson, the value of B/L is usually less than 0.5. Thus, when the waves are obliquely incident to the breakwater, to minimize the wave reﬂection, the value of B cos θ/L should be around 0.25, but the condition B/L ∼ = 0.5 should be avoided. 13.3.2. Application to partially-perforated-wall caisson breakwater A conventional perforated-wall caisson consists of a front wave chamber and a back wall as shown in Fig. 13.6(a), and the water depth inside the wave chamber is the same as that on the rubble foundation. The weight of the caisson is less than that of a vertical solid caisson with the same width, and moreover, most of this

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0.5 1.0

B/L = 0.5

1.5

Bcos θ / L

0.4

0.3

0.2

0.1 0

20

40

60

80

Wave Angle (degree)

Fig. 13.5. Contours of the reﬂection coeﬃcient as a function of wave incident angle and B cos θ/L; the lines of B/L = 0.5, 1.0, and 1.5 are also included.

weight is concentrated on the rear side of the caisson. Therefore, diﬃculties are sometimes met in the design of a perforated-wall caisson to satisfy the design criteria against sliding and overturning. In addition, particularly in the case where the bearing capacity of the seabed is not large enough, the excessive weight on the rear side of the caisson may have an adverse eﬀect. In order to solve these problems, a partially-perforated-wall caisson as shown in Fig. 13.6(b) is often used, which provides an additional weight to the front side of the caisson. In this case, however, other hydraulic characteristics of the caisson such as wave reﬂection and overtopping may become worse than a fully-perforated-wall caisson. The mathematical model described in Sec. 13.3.1 assumes that the water depth inside the wave chamber is the same as that on the mound berm as in a fullyperforated-wall caisson breakwater shown in Fig. 13.6(a). However, for a partiallyperforated-wall caisson [see Fig. 13.6(b)], these water depths are diﬀerent from each other, having depth discontinuity at the location of the perforated wall. In order to apply the model to the case of a partially-perforated-wall caisson, it can be assumed that the lower part of the front face of the caisson (below the perforated wall) is not vertical but has a very steep slope. As can be seen in Eq. (13.58), the model includes the terms proportional to the square of the bottom slope and to the bottom curvature, so that it can be applied over a bed having substantial variation of water depth. In order to examine the eﬀect of the slope of the lower part of the caisson (which is inﬁnity in reality), Suh et al.24 calculated the reﬂection coeﬃcient by changing the slope from 0.1 to 10. They found that the reﬂection coeﬃcient

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(a)

(b) Fig. 13.6. Bird’s-eye views of (a) a fully-perforated-wall caisson breakwater and (b) a partiallyperforated-wall caisson breakwater.

virtually does not change for slopes greater than 2.0, and they used the slope of 4.0 in their comparison between model prediction and experimental results. 13.3.3. Extension to irregular waves As for irregular waves, as explained in Sec. 13.2.3, the reﬂection coeﬃcient is calculated diﬀerently for each frequency component. The wave period is determined according to the frequency of the component wave, while the rms wave height is used for all the component waves to compute the energy dissipation at the perforated wall. The spectral density of the reﬂected waves is then calculated for a particular frequency component as in Eq. (13.35), and the frequency-averaged reﬂection coeﬃcient is calculated as in Eq. (13.37). 13.4. Calculation of Permeability Parameter Equations (13.6) and (13.23) state that the ﬂuid velocity normal to the perforated wall is proportional to the pressure diﬀerence across the wall with a complex

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constant of proportionality. To elucidate the physical meaning of the permeability parameter G, however, it is more convenient to express the equations in the form of Eq. (13.54). Then, the real part of i/G is associated with the phase diﬀerence between the velocity and the pressure due to inertial eﬀects, and the imaginary part of i/G corresponds to the resistance of the wall. There are several approaches to calculate the permeability parameter G. Mei et al.31 expressed the resistance and the inertial eﬀects in terms of a head-loss coeﬃcient and an eﬀective oriﬁce length, respectively. On the other hand, Sollitt and Cross32 expressed them in terms of a friction coeﬃcient and an added mass coeﬃcient, respectively. 13.4.1. Mei et al.’s method The permeability parameter based on Mei et al.31 can be expressed as G=

1 β ω

− i

,

(13.65)

where β = energy dissipation coeﬃcient derived by linearizing the nonlinear convective acceleration term in the equation of motion; and = eﬀective oriﬁce length, which is the length of the jet ﬂowing through the perforated wall. The linearized dissipation coeﬃcient β for a pile breakwater is given by Kim33 as β=

1 8α 5 + cosh(2kh) Hw ω , 9π (R + 2)2 + P 2 2kh + sinh(2kh)

(13.66)

where Hw = incident wave height at the perforated wall; P = k; R = βk/ω; and α = head-loss coeﬃcient. The preceding equation was derived for a pile breakwater. However, it could be used for a curtain-wall-pile breakwater as well, because the mechanism of energy dissipation between piles must be similar for these two breakwaters. A pile breakwater has energy dissipation associated with the free surface, but it may be small when compared with the energy dissipation between piles. On the other hand, for a perforated-wall caisson (see Fig. 13.3), Fugazza and Natale22 gave β as β=

W 5 + cosh(2k3 h3 ) 8α Hw ω , 2 2 2 9π 2k W (R + 1) + F 3 h3 + sinh(2k3 h3 )

(13.67)

where W = tan(k3 B), R = βk3 /ω, F = 1 − P W , and P = k3 . Rearrangement of Eq. (13.66) or (13.67) gives a quartic polynomial of β, which can be solved by the eigenvalue method (e.g., Press et al.29 ). Suh et al.34 showed that the eﬀective oriﬁce length is related to the blockage coeﬃcient C by = 2C .

(13.68)

For rectangular piles, Flagg and Newman35 proposed the blockage coeﬃcient as

b 1 2A 1 2 281 4 (13.69) C= 1 − log(4r0 ) + r0 + r . −1 + 2 r0 π 3 180 0

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The head-loss coeﬃcient α could be given by the plate oriﬁce formula36 : 2 1 α= −1 , r0 cos θCc

335

(13.70)

where Cc = empirical contraction coeﬃcient, for which Mei et al.31 suggested using the formula: Cc = 0.6 + 0.4r02 ,

(13.71)

and r0 cos θ denotes the eﬀective ratio of the opening of the perforated wall taking into account the oblique incidence of the waves to the wall. Very recently, on the other hand, Yoon et al.37 proposed the head-loss coeﬃcient taking into account the eﬀect of wall thickness as 2 1 −2.5(b/2a) ) −1 , (13.72) α = (0.6 + 0.4e r0 Cc for normally incident waves. For circular piles, Kakuno and Oda38 proposed the blockage coeﬃcient as C=

1 π A(1 − r0 )2 ; 4 1−ξ

ξ=

π2 (1 − r0 )2 . 12

(13.73)

Park et al.11 proposed a formula for the head-loss coeﬃcient for circular piles as 2 1 α= −1 , (13.74) rCc where Cc is the contraction coeﬃcient given by Eq. (13.71), and the ad hoc porosity r is given by D/2 1 1 dx = , (13.75) D −D/2 r(x)2 r2 with the spatially varying porosity D 2 r(x) = 1 −

2 D 2

− x2

+a

,

(13.76)

where D = diameter of piles. In the case of a perforated-wall caisson mounted on a rubble foundation as in Fig. 13.3, the incident wave height at the perforated wall Hw is a priori unknown. In the case where the caisson does not exist and the water depth is constant as h3 for x ≥ 0, Massel28 has shown that the transmitting boundary condition at x = 0 is given by iϕ(0)k ˜ 3 cos θ3 =

dϕ(0) ˜ . dx

(13.77)

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The governing equation (13.56) and the upwave boundary condition in Eq. (13.62) do not change. After solving this problem, the transmission coeﬃcient Ct is given by Ct = Re{ϕ(0)}, ˜ from which Hw is calculated as Ct times the incident wave height on the ﬂat bottom. The above method to compute the permeability parameter is advantageous compared with other methods in that all the related parameters are known, i.e., incident wave height and period and the geometrical parameters of the barrier. However, Suh et al.12 showed that this method gives a wrong result for very long waves. As can be seen in Eq. (13.66), as kh goes to zero, β goes to inﬁnity, indicating complete dissipation of the long waves. This leads to zero transmission for long waves, which is deﬁnitely wrong because in the limit of long waves the breakwater is invisible to the waves so that complete transmission of the waves should occur. This method can be used for irregular waves as well. Using the regular wave model, the reﬂection and transmission coeﬃcients are calculated diﬀerently for each frequency component. In the computation of the energy dissipation coeﬃcient β in Eq. (13.66), however, the rms wave height should be used instead of the height of the component wave, because the ﬂow separation due to irregular waves and the resulting energy dissipation are induced not by the individual component waves but by the superposition of the component waves. 13.4.2. Sollitt and Cross’s method The permeability parameter based on Sollitt and Cross32 can be expressed as G=

r0 , b (fc − is)

(13.78)

where fc = friction coeﬃcient, and s = inertia coeﬃcient given by s = 1 + Cm

1 − r0 , r0

(13.79)

where Cm = added mass coeﬃcient. In the method of Sollitt and Cross,32 the values of friction and added mass coefﬁcients are not known a priori, and they are estimated on the basis of a best ﬁt between measurement and prediction. Many researchers (e.g., Sollitt and Cross32 ; Losada et al.39 ; Yu40 ; Isaacson et al.4,41 ; Zhu and Chwang42; Hossain et al.43 ) suggested to use the value of Cm = 0, i.e., s = 1 by comparing their numerical predictions of reﬂection and transmission coeﬃcients of perforated structures with experimental results. As for the friction coeﬃcient, recently, Li et al.44 proposed the following empirical equation for estimating fc in terms of b/h, along with the use of Cm = 0: fc = −3338.7

2 b b b + 82.769 + 8.711 for 0.0094 ≤ ≤ 0.05 . h h h

They also recommended using fc = 2.0 if b/h ≥ 0.1.

(13.80)

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13.4.3. Inertia eﬀect on wave reﬂection from perforated-wall caisson It is well known that wave reﬂection from a perforated-wall caisson breakwater depends on the width of the wave chamber relative to the wavelength. For regular waves, neglecting the inﬂuence of the inertia term, Fugazza and Natale22 showed analytically that the resonance inside the wave chamber is important so that the reﬂection is minimal when B/L = (2n + 1)/4 (n = 0, 1, 2, . . .) in which L is the wavelength. However, the resonant mode of practical interest is the fundamental mode (n = 0), i.e., B/L = 0.25 because of the width limit of the breakwater. This is also shown in Fig. 13.5. In reality, however, the reﬂection coeﬃcient becomes minimal at the value of B/L somewhat smaller than 0.25 because of the inﬂuence of the inertia term.24,27 In front of a perforated-wall caisson breakwater, a partial standing wave is formed due to the wave reﬂection from the breakwater. If there were no perforated wall, the node would occur at a distance of about L/4 from the back wall of the wave chamber, and hence the largest energy loss may occur at this point because the horizontal velocity becomes a maximum at a node. But, in reality, there exists inertia resistance at the perforated wall, which interrupts the smooth pass of the wave, thus creating pressure (or free surface) diﬀerence between the front and back of the perforated wall. Consequently, the location of the maximum horizontal velocity will move toward the breakwater, and the point where the maximum energy loss is gained becomes smaller than L/4. Thus, the minimum reﬂection occurs at a value of B/L smaller than 0.25. In the methods of Mei et al.31 and Sollitt and Cross,32 the eﬀective oriﬁce length and inertia coeﬃcient s, respectively, represents the inertia eﬀect. Acknowledgments This chapter was prepared while the author was supported by the Korea Sea Grant Program and the Project for Development of Reliability-Based Design Method for Port and Harbor Structures of Korea Ministry of Marine Aﬀairs and Fisheries. References 1. R. L. Wiegel, J. Waterways Harbors Coastal Eng. Div., Am. Soc. Civ. Eng. 86, 1 (1960). 2. D. L. Kriebel and C. A. Bollmann, Proc. 25th Int. Conf. Coastal Eng., ASCE (1997), p. 2470. 3. D. L. Kriebel, C. Sollitt and W. Gerken, Proc. 26th Int. Conf. Coastal Eng., ASCE (1999), p. 2069. 4. M. Isaacson, S. Premasiri and G. Yang, J. Waterways, Port, Coastal, Ocean Eng. 124, 118 (1998). 5. T. Hayashi, M. Hattori, T. Kano and M. Shirai, Proc. 10th Int. Conf. Coastal Eng., ASCE (1966), p. 873. 6. H. Kojima, M. Utsunomiya, T. Ijima, A. Yoshida and T. Kihara, Proc. 35th Jpn Conf. Coastal Eng., ASCE (1988), p. 542.

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7. T. Uda, A. Omata and T. Kawamura, Rep. Public Works Res. Inst., 2891, Japan Ministry of Construction (1990) (in Japanese). 8. S. Kakuno and P. L.-F. Liu, J. Waterways, Port, Coastal, Ocean Eng. 119, 302 (1993). 9. K. Hagiwara, Proc. 19th Int. Conf. Coastal Eng., ASCE (1985), p. 2810. 10. G. S. Bennett, P. McIver and J. V. Smallman, Coastal Eng. 18, 231 (1992). 11. W. S. Park, B. H. Kim, K. D. Suh and K. S. Lee, Coastal Eng. J. 42, 253 (2000). 12. K.-D. Suh, S. Shin and D. T. Cox, J. Waterways, Port, Coastal, Ocean Eng. 132, 83 (2006). 13. K.-D. Suh and C.-H. Ji, Proc. 30th Int. Conf. Coastal Eng., ASCE (2006), p. 4303. 14. S. Takahashi and K. Shimosako, Proc. Hydro-Port ’94, Port and Harbour Res. Inst., Yokosuka, Japan (1994), p. 747. 15. S. Takahashi, K. Tanimoto and K. Shimosako, Proc. Hydro-Port ’94, Port and Harbour Res. Inst., Yokosuka, Japan (1994), p. 489. 16. G. E. Jarlan, The Dock and Harbour Authority, Vol. XII(486) (1961), p. 394. 17. M. Marks and G. E. Jarlan, Proc. 11th Int. Conf. Coastal Eng., ASCE (1968), p. 1121. 18. F. L. Terret, J. D. C. Osorio and G. H. Lean, Proc. 11th Int. Conf. Coastal Eng., ASCE (1968), p. 1104. 19. K. Tanimoto, S. Haranaka, S. Takahashi, K. Komatsu, M. Todoroki and M. Osato, Technical Report, Port and Harbour Res. Inst., Min. of Transport, Japan, No. 246 (1976) (in Japanese). 20. H. Kondo, Proc. Coastal Structures ’79, ASCE (1979), p. 962. 21. S. Kakuno, K. Oda and P. L.-F. Liu, Proc. 23rd Int. Conf. Coastal Eng., ASCE (1992), p. 1258. 22. M. Fugazza and L. Natale, J. Waterways, Port, Coastal, Ocean Eng. 118, 1 (1992). 23. K. D. Suh and W. S. Park, Coastal Eng. 26, 117 (1995). 24. K.-D. Suh, J. K. Park and W. S. Park, Ocean Eng. 33, 264 (2006). 25. D. L. Kriebel, Proc. Coastal Structures ’99, Balkema, Rotterdam (2000), p. 525. 26. R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (World Scientific, River Edge, New Jersey, 1991). 27. Y. Goda, Random Seas and Design of Maritime Structures (World Scientific, Singapore, 2000). 28. S. R. Massel, Coastal Eng. 19, 97 (1993). 29. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. (Cambridge University Press, Cambridge, 1992). 30. K. D. Suh, W. S. Park and K. D. Yum, Proc. Int. Conf. Coastal and Port Eng. Developing Countries (1995), p. 2311. 31. C. C. Mei, P. L.-F. Liu and A. T. Ippen, J. Waterways, Harbors Coastal Eng. Div., Vol. WW3, ASCE (1974), p. 217. 32. C. K. Sollitt and R. H. Cross, Proc. 13th Int. Conf. Coastal Eng., ASCE (1972), p. 1827. 33. B. H. Kim, Interactions of waves, seabed and structures, PhD. dissertation, Seoul National University, Seoul, Korea (1998). 34. K. D. Suh, S. Y. Son, J. I. Lee and T. H. Lee, Proc. 28th Int. Conf. Coastal Eng., ASCE (2002), p. 1709. 35. C. N. Flagg and J. N. Newman, J. Ship Res. 15, 257 (1971). 36. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (Wiley, New York, 1983). 37. S. B. Yoon, J. I. Lee, D. H. Nam and S. H. Kim, J. Korean Soc. Coastal Ocean Engrs. 18, 321 (2006) (in Korean). 38. S. Kakuno and K. Oda, J. Jpn. Soc. Civil Engrs. 369, 213 (1986) (in Japanese).

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39. 40. 41. 42. 43. 44.

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I. J. Losada, M. A. Losada and A. Baquerize, Appl. Ocean Res. 15, 207 (1993). X. Yu, J. Waterway, Port, Coastal, Ocean Eng. 121, 275 (1995). M. Isaacson, J. Baldwin, S. Premasiri and G. Yang, Appl. Ocean Res. 21, 81 (1999). S. Zhu and A. T. Chwang, J. Eng. Mech. 127, 326 (2001). A. Hossain, W. Kioka and T. Kitano, Coastal Eng. J. 43, 83 (2001). Y. Li, Y. Liu and B. Teng, Coastal Eng. J. 48, 309 (2006).

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Chapter 14

Prediction of Overtopping Jentsje van der Meer Van der Meer Consulting P. O. Box 423, 8440 AK Heerenveen, The Netherlands [email protected] Tim Pullen HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] William Allsop HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] Tom Bruce School of Engineering, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JL, UK [email protected] Holger Sch¨ uttrumpf RWTH-Aachen University Institute of Hydraulic Engineering and Water Resources Management Mies-van-der-Rohe-Str. 1, 52065 Aachen, Germany [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics, Technical University of Braunschweig Beethovenstr, 51a, 38106 Braunschweig, Germany [email protected]

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This chapter describes the processes of wave overtopping at sea defense and related coastal or shoreline structures. It introduces a range of methods to calculate mean overtopping discharges, individual and maximum overtopping volumes, and the proportion of waves overtopping a seawall. It describes the principal hazards from wave overtopping and will help engineers by suggesting limiting tolerable discharges for frequent, design, and extreme wave conditions. This chapter is supported by more detailed material in Chaps. 15 and 16 which focus on the methods to predict overtopping for rubble mound structures (with partly sloping embankments), and on vertical structures and battered walls. All of these three chapters have been based closely on the new EurOtop Overtopping Manual.5

14.1. Introduction 14.1.1. Wave overtopping Wave overtopping has always been of principal concern for coastal structures constructed to defend against ﬂooding: often termed sea defenses. Similar structures may also be used to provide protection against coastal erosion: sometimes termed coast protection. Other structures may be built to protect areas of water for ship navigation or mooring within ports, harbours, or marinas: often formed by breakwaters or moles. Within harbours, or along shorelines, reclaimed areas must be defended against both erosion and ﬂooding. Some structures may be detached from the shoreline, often termed oﬀshore, nearshore, or detached, but most structures used for sea defense or similar function form a part of the shoreline. Sloping dikes have been widely used for sea defenses along the coasts of the Netherlands, Denmark, Germany, and the United Kingdom. Dikes or embankment seawalls are also used to defend low-lying areas in the Far East, including China, Korea, and Vietnam. Historically, dikes or embankment seawalls were built along many North Sea coastlines, sometimes subsuming an original sand dune line, protecting the land behind from ﬂooding, and sometimes providing additional amenity value. Similar structures have been formed by clay materials or even from a vegetated shingle ridge, in both instances allowing the side slopes to be steeper. All such embankments need some degree of protection against direct wave erosion, often using a revetment facing on the seaward side (Fig. 14.1). Revetment facing may take many forms, but may commonly include closely-ﬁtted concrete blockwork, cast in situ concrete slabs, or asphaltic materials. Embankment or dike structures are generally most common along rural frontages. A second type of coastal structure consists of a mound or layers of quarried rock ﬁll, protected by rock or concrete armour units (Fig. 14.2). The outer armour layer is designed to resist wave action without signiﬁcant displacement of armour units. Under-layers of quarry or crushed rock support the armour and separate it from ﬁner material in the embankment or mound. These porous and sloping layers dissipate a proportion of the incident wave energy in breaking and friction. Simpliﬁed forms of rubble mounds may be used for rubble seawalls or as protection to vertical walls or revetments. Rubble mound revetments may also be used to protect embankments

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Fig. 14.1.

Fig. 14.2.

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Wave overtopping on a revetment seawall.

Wave overtopping on a rubble mound breakwater.

formed from relict sand dunes or shingle ridges. Rubble mound structures tend to be more common in areas where harder rock is available. Along urban frontages, especially close to ports, erosion or ﬂooding defense structures may include vertical (or battered/steep) walls (Fig. 14.3). Such walls

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Fig. 14.3.

Wave overtopping on a battered/vertical seawall.

may be composed of stone or concrete blocks, mass concrete, or sheet steel piles. Typical vertical seawall structures may also act as retaining walls to material behind. Shaped and recurved wave return walls may be formed as walls in their own right, or smaller versions may be included in sloping structures. Some coastal structures are relatively impermeable to wave action. These include seawalls formed from blockwork or mass concrete, with vertical, near-vertical, or steeply sloping faces. Such structures may be liable to intense local wave impact pressures, may overtop suddenly and severely, and will reﬂect much of the incident wave energy. Reﬂected waves cause additional wave disturbance and/or may initiate or accelerate local bed scour. It is worth noting that developments along waterfronts are highly valued with purchase or rental prices substantially above those for properties not on the waterfront. Yet, direct (or indirect) eﬀects of wave overtopping have the potential to generate signiﬁcant hazards to such developments and their users. Residential and commercial properties along a waterfront will often be used by people who may be unaware of the possibility, of the severity, or of the eﬀects of wave overtopping in storm conditions. Regulatory authorities may therefore wish to impose onerous ﬂood defense requirements on new developments. For instance, protection against ﬂooding (including wave overtopping) for any new developments in the United Kingdom is now required to be 0.5% annual probability, equivalent to 1:200 year return. Exposure to overtopping of many coastal sites will, however, be inﬂuenced by climate change, probably increasing wave heights and periods as well as sea level rise.

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14.1.2. Predicting wave overtopping A number of diﬀerent methods may be available to predict overtopping of particular structures (usually simpliﬁed sections) under given wave conditions and water levels. Each method will have strengths or weaknesses in diﬀerent circumstances. In theory, an analytical method can be used to relate the driving process (waves) and the structure to the response through equations based directly on the knowledge of the physics of the process. It is, however, extremely rare for the structure, the waves, and the overtopping process to all be so simple and well-controlled that an analytical method on its own can give reliable predictions. Analytical methods are not therefore discussed further in this chapter. The primary prediction methods are therefore based on empirical methods that relate the overtopping response to the main wave and structure parameters. These are by far the most commonly used methods to predict overtopping. Two other methods have been derived during the CLASH European project3 based on the use of measured overtopping from model tests and ﬁeld measurements. The ﬁrst of these techniques uses the CLASH database of structures, waves, and overtopping discharges, with each test described by 31 parameters. Using the database is, however, potentially complicated, requiring some familiarity with these type of data. A simpler approach, and much more rapid, is to use the Neural Network tool that has been trained using the test results in the database. The Neural Network tool can be run automatically on a computer as a stand-alone device, or embedded within other simulation methods. For situations for which empirical test data do not already exist, or where the methods above do not give reliable enough results, then two alternative methods may be used, but both are more complicated than the methods above. A range of numerical models can be used to simulate the process of overtopping. All such models involve some simpliﬁcation of the overtopping process and are therefore limited to particular types of structure or types of wave exposure. They may, however, run sequences of waves giving overtopping (or not) on a wave-by-wave basis. Generally, numerical models require more skill and familiarity to run successfully. They will not be described in this chapter. The ﬁnal method to be mentioned is physical modeling in which a scale model is tested with correctly scaled wave conditions. Typically, such models may be built to a geometric scale in the range 1:10 to 1:60. Waves will be generated as random wave trains each conforming to a particular energy spectrum. The model may represent a structure cross section in a 2D model tested in a wave ﬂume. Structures with more complex plan shapes, junctions, transitions, etc., may be tested in a 3D model in a wave basin. Physical models can be used to measure many diﬀerent aspects of overtopping such as wave-by-wave volumes, overtopping velocities and depths, as well as other responses.

14.1.3. Performance requirements Most sea defense structures are constructed primarily to limit overtopping volumes that might cause ﬂooding. For defenses that protect people living, working, or

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enjoying themselves, designers and owners of these defenses must, however, also deal with potential direct hazards from overtopping. This requires that the level of hazard and its probability of occurrence be assessed, allowing appropriate action plans to be devised to ameliorate risks arising from overtopping. Section 14.6 deals with tolerable wave overtopping.

14.2. Empirical Models, Including Comparison of Structures 14.2.1. Mean overtopping discharge Empirical methods use a simpliﬁed representation of the physics of the process presented in (usually dimensionless) equations to relate the main response (overtopping discharge, etc.) to key wave and structure parameters. The form and coeﬃcients of the equations are adjusted to reproduce results from physical model (or ﬁeld) measurements of waves and overtopping. Empirical equations may be solved explicitly, or may occasionally require iterative methods to solve. Historically, some empirical methods have been presented graphically, although this is now very rare. The mean overtopping discharge, q, is the main parameter in the overtopping process. It is, of course, not the only measure of overtopping, but it is relatively easy to measure in a laboratory wave ﬂume or basin (or even in the ﬁeld), and most other parameters are related in some way to this overtopping discharge. The overtopping discharge is generally calculated in m3 /s per m width, but in practical applications it may be quoted as liter/s per m width. Although it is given as a discharge, it is usually very far from a steady discharge as the actual processes of wave overtopping are much more dynamic. For most defenses, only large waves will reach the crest of the structure and will overtop, but they may do so with a lot of water in a few seconds. The individual volumes in wave-by-wave overtopping are more diﬃcult to measure in a laboratory than the mean discharge; so data on wave-by-wave volumes are much rarer. As mean overtopping discharges are relatively easy to measure, many physical model tests have been performed all over the world, both for idealized structures and real applications or designs. The European CLASH project3 collected a large database worldwide with more than 10,000 wave overtopping test results on all kinds of structures (see Sec. 14.5). Some series of tests have been used to develop empirical methods for the prediction of overtopping. Such empirical methods or formulae are, however, only directly applicable to idealized structures, like smooth slopes (dikes, sloping seawalls), simple rubble mound structures or vertical structures (caissons) or walls, and may require extrapolation when applied to many existing structures.

14.2.2. Comparing overtopping performance Chapters 15 and 16 will describe overtopping formulae for diﬀerent kinds of structures, based on the EurOtop Overtopping Manual.5 In this section, an overall view is given to compare the performance of diﬀerent structure types and to give insight

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into how wave overtopping behaves for diﬀerent structures. Those structures considered here are: smooth sloping structures (dikes, seawalls); rubble mound structures (breakwaters, rock armored slopes); and vertical structures (caissons, sheet pile walls). The principal prediction formula for many types of wave overtopping is

q 3 gHm0

= a exp(−bRc /Hm0 ).

(14.1)

It is an exponential function with the dimensionless overtopping discharge 3 )1/2 and the relative crest freeboard Rc /Hm0 . This type of equation shows q/(gHm0 a straight line on a log-linear graph, which makes it easy to compare formulae for diﬀerent structures. Speciﬁc equations are given in Chaps. 15 and 16. Two equations are considered for pulsating waves on a vertical structure. Allsop et al.1 consider relatively shallow water and Franco et al.6 more deep water (caissons). Vertical structures in shallow water, and often with a sloping foreshore in front, may become subject to impulsive forces, i.e., high impacts and water splashing high up into the air. Speciﬁc formulae have been developed for these kinds of situations. For easy comparison of diﬀerent structures, like smooth and rubble mound sloping structures and vertical structures for pulsating and impulsive waves, some simpliﬁcations will be assumed. In order to simplify the smooth structure, no berm is considered (γb = 1), only a normal wave attack is considered (γβ = 1), and the sloping seawall does not feature any wavewall on top (γv = 1). As the slope is smooth and impermeable, γf = 1. This limits the structure to a smooth and straight slope with normal wave attack. The slope angles considered for smooth slopes are cotα = 1−8, which means from very steep to very gentle. If relevant, a wave steepness of sm−1,0 = 0.04 (steep storm waves) and 0.01 (long waves due to swell or wave breaking) will be considered. Detailed deﬁnitions are given in Chap. 15. The same equation as for smooth slopes is applicable for rubble slopes, but now with a roughness factor of γf = 0.5, simulating a rock armoured structure. Rubble mound structures are often steep, but rock armoured slopes may also be gentle. Therefore, slope angles with cotα = 1.5 and 4.0 are considered. For vertical structures under pulsating waves, both formulae of Allsop et al.1 and Franco et al.6 will be compared, together with the formula for impulsive waves. Impulsive waves can only be reached with a relatively steep foreshore in front of the vertical wall. For comparison, values of the breaker ratio (wave height/water depth) of Hm0 /hs = 0.5, 0.7, and 0.9 will be used. These will be discussed further in Chap. 16. Overtopping on smooth slopes can be compared with rubble mound slopes and with vertical structures under pulsating or impulsive conditions. First, the traditional graph is given in Fig. 14.4 with the relative freeboard Rc /Hm0 versus the 3 1/2 logarithmic dimensionless overtopping q/(gHm0 ) . In most cases the steep smooth slope gives the largest overtopping. Steep means cotα < 2, but also a little gentler if long waves (less steepness) are considered. Under these conditions, waves surge up the steep slope. For gentler slopes, waves break as

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Dimensionless overtopping q/(gH

3 0.5 m0 )

1.E+00 steep smooth slopes, cota<2 gentle smooth slope, cota=4, so=0.04 steep rubble mound slope, gf=0.5 gentle rubble mound slope, gf=0.5 vertical structure Allsop (1995) vertical structure Franco et al. (1994) impulsive vertical Hm0/hs=0.5 impulsive vertical Hm0/hs=0.7 impulsive vertical Hm0/hs=0.9

1.E-01

1.E-02

1.E-03

1.E-04

1.E-05

1.E-06 0

0.5

1

1.5

2

2.5

3

3.5

4

Relative freeboard Rc/Hm0 Fig. 14.4.

Comparison of wave overtopping formulae for various structure types.

plunging waves and this reduces wave overtopping. The gentle slope with cotα = 4 gives much lower overtopping than the steep smooth slopes. Both slope angle and wave period have inﬂuence on overtopping for gentle slopes. The high roughness and permeability of a rubble mound can reduce overtopping substantially (Fig. 14.4). A roughness factor of γf = 0.5 was used here although γf = 0.4 (two layers of rock on a permeable under layer) would reduce the overtopping further. A gentle rubble mound slope with cotα = 4 gives very low overtopping. Vertical structures under pulsating waves1,6 give lower overtopping than steep smooth slopes, but more than a rough rubble mound slope. The impulsive conditions give a diﬀerent trend. First of all, the inﬂuence of the relative water depth is fairly small as all curves with diﬀerent Hm0 /hs are quite close. For low vertical structures (Rc /Hm0 < 1.5), there is hardly any diﬀerence between pulsating and impulsive conditions. The large diﬀerence is present for higher vertical structures and certainly for the very high structures. With impulsive conditions, water can be thrown high into the air, which means that overtopping occurs even for very high structures. The vertical distance that the discharge travels is more or less independent of the actual height of the structure. For Rc /Hm0 > 3 the curves are almost horizontal. Another way of comparing the eﬀectiveness of structure types is to show the inﬂuence of slope angle on wave overtopping, as in Fig. 14.5. A vertical structure means cotα = 0. Battered walls have 0 < cotα < 1. Steep slopes are generally described by 1 ≤ cotα ≤ 3. Gentle slopes have roughly cotα ≥ 2 or 3. Overtopping

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1.E+00

Dimensionless overtopping q/(gHm03)0.5

vertical, Allsop (1995) vertical, Franco et al. (1994)

1.E-01

impulsive vertical, Hm0/hs=0.9, so=0.04 smooth slope, so=0.04

1.E-02

smooth slope, so=0.01

Rc/Hm0=1.5

1.E-03

Rc/Hm0=3.0

1.E-04

1.E-05

1.E-06

1.E-07 0

1

2

3

4

5

6

7

8

Slope angle cot α Fig. 14.5.

Comparison of wave overtopping as a function of slope angle.

prediction curves for two relative freeboards: Rc /Hm0 = 1.5 and 3.0 are shown in Fig. 14.5. Of course, similar conclusions can be drawn as for the previous comparison. Steep slopes give the largest overtopping, which reduces for gentler slopes, for a given wave condition and water level. Vertical slopes give less overtopping than steep smooth slopes, except for a high vertical structure under impulsive conditions. This graph gives also the method to calculate for a battered wall: interpolate between a vertical wall and a slope 1:1 with cotα as the parameter to interpolate. Details of all equations used here are described in more detail in Chaps. 15 and 16 (sloping smooth structures, rubble mound structures, and vertical structures). 14.2.3. Overtopping volumes and Vmax Wave overtopping is a dynamic and irregular process and the mean overtopping discharge, q, does not cover this aspect. But by knowing the storm duration, t, and the number of overtopping waves in that period, Now , it is possible to give some description of this irregular and dynamic overtopping, if the overtopping discharge, q, is known. Each overtopping wave gives a certain volume of water, V , and these can be described by a distribution. The two-parameter Weibull distribution can be ﬁtted to many distributions. This equation has a shape parameter, b, and a scale parameter, a. The shape parameter gives a lot of information on the type of distribution. Figure 14.6 gives an overall view of some well-known distributions. The horizontal axis gives the probability of

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14

3

overtopping volume (m /m)

16

12 10 8 6 4 2 0 100

90

70 50

30 20 10

5

2

1 0.5

0.1

probability of exceedance (%) Fig. 14.6. Various distributions on a Rayleigh scale graph. A straight line (b = 2) is a Rayleigh distribution.

exceedance and has been plotted according to the Rayleigh distribution. It is known that wave heights in deep water generally conform to a Rayleigh distribution; so, responses governed by deep water wave conditions will plot on or close to a straight line, whilst shallow water eﬀects will show deviations from the Rayleigh distribution. When waves enter shallow water and the highest waves break, wave heights more closely match a Weibull distribution with b > 2. An example with b = 3 is shown in Fig. 14.6, and this indicates that there are more large waves of similar height. The exponential distribution (often found for extreme wave climates) has b = 1, and shows that extremes become larger compared to most of the data. Such an exponential distribution would give a straight line in a log-linear graph. The distribution of overtopping volumes for all kinds of structures has average values even smaller than b = 1. Such a distribution is even steeper than an exponential distribution. It means that the wave overtopping process can be described by many fairly small overtopping volumes and a few very large volumes. The EAmanual 2 gives values for b and a, based on limited data sets. The b-values are mostly within the range 0.6 < b < 0.9. For comparison, curves with b = 0.65 and b = 0.85 are given in Fig. 14.6. The curves are very similar, except that the extremes diﬀer a little. It is for this reason that an average value of b = 0.75 was chosen for smooth slopes and not diﬀerent values for various subsets. The same average value has been used for rubble mound structures, which makes smooth and rubble mound structures easily comparable. The exceedance probability, PV , of an overtopping volume per wave is then similar to 0.75 V PV = P (V ≤ V ) = 1 − exp − , (14.2) a with a = 0.84 · Tm ·

q = 0.84 · Tm · q · Nw /Now = 0.84 · q · t/Now . Pov

(14.3)

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Equation (14.3) shows that the scale parameter a depends not only on the overtopping discharge, q, but also on the mean period, Tm , and the probability of overtopping, Now /Nw , or (similarly) on storm duration, t, and the actual number of overtopping waves Nw . The maximum overtopping during a certain event is fairly uncertain as most maxima, but depends on the duration of the event. In a 6-h period, individual peak volume is likely to be larger than that in 15 min. The maximum overtopping volume by a single wave during an event depends on the actual number of overtopping waves, Now , and can be calculated by Vmax = a · [ln(Now )]4/3 .

(14.4)

A comparison can be made between the mean overtopping discharge, q, and the maximum overtopping volume in the largest wave. Note that the mean overtopping is given in l/s per m width and that the maximum overtopping volume is given in liters per m width. Again, three example structures are considered: a smooth 1:4 slope; a rubble slope at 1:1.5; and a vertical wall, and again speciﬁc equations are given in Chaps. 15 and 16. The storm duration has been assumed as 2 h (the peak of the tide), and a ﬁxed wave steepness of sm−1,0 = 0.04 has been considered. Figure 14.7 gives the q − Vmax lines for the three structures and for relatively small waves of Hm0 = 1 m and for fairly large waves of Hm0 = 2.5 m.

Maximum volume in overtopping wave (l per m)

100000 Smooth slope Hs=2.5m Rubble mound Hs=2.5m Vertical Hs=2.5m Smooth slope Hs=1m Rubble mound Hs=1m Vertical Hs=1m

10000

1000

100 0.1

1

10

100

1000

Overtopping discharge q (l/s per m) Fig. 14.7. Relationship between mean discharge and maximum overtopping volume in one wave for smooth, rubble mound, and vertical structures for wave heights of 1 m and 2.5 m.

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A few conclusions can be drawn from Fig. 14.7. First of all, the ratio q/Vmax is about 1000 for small q (roughly around 1 l/s per m), and about 100 for large q (roughly around 100 l/s per m). So, the maximum volume (in liters per m width) in the largest wave is about 100–1000 times larger than the mean overtopping discharge (in l/s per m). Secondly, the lines for a 1 m wave height are lower than for a larger 2.5 m wave height, which means that for lower wave heights, but similar mean discharge, q, the maximum overtopping volume is also smaller. For example, a vertical structure with a mean discharge of 10 l/s per m gives a maximum volume of 1000 l/m for a 1 m wave height and a volume of 4000 l/m for a 2.5 m wave height. Finally, the three diﬀerent structures give diﬀerent relationships, depending on the equations to calculate q and the equations to calculate the number of overtopping waves. 14.2.4. Wave transmission by wave overtopping For structures like nearshore breakwaters, large overtopping can be allowed as this overtopping simply falls into the water behind the structure causing new waves behind the structure. This is termed wave transmission and is most easily deﬁned by the wave transmission coeﬃcient Kt = Hm0,t /Hm0,i , with Hm0,t = transmitted signiﬁcant wave height, and Hm0,i = incident signiﬁcant wave height. The limits of wave transmission are Kt = 0 (no transmission) and 1 (no reduction in wave height). If a structure has its crest above water, the transmission coeﬃcient will never be larger than about Kt = 0.4–0.5. Wave transmission was studied in the European DELOS project,4 and the following prediction formulae were derived for smooth sloping structures: Kt = −0.3 ·

Rc 2/3 + 0.75 · (1 − exp (−0.5 · ξop )) · (cos β) , Hm0,i

(14.5)

with a minimum Kt = 0.075 and maximum Kt = 0.8; and limitations 1 < ξop < 3; 0◦ ≤ β ≤ 70, and 1 < B/Hm0,i < 4, where β is the angle of wave attack and B is the crest width (and not berm width). The transmission coeﬃcient Kt is shown in Fig. 14.8 as a function of the relative freeboard Rc /Hm0 and for a smooth structure with slope angle cotα = 4 (a gentle smooth low-crested structure). Three wave steepnesses have been used: sop = 0.01 (long waves), 0.03 and 0.05 (short wind waves). Again, normal wave attack has been assumed. Wave transmission decreases for increasing crest height, but longer waves give more transmission. Wave overtopping can be calculated for the same structure and wave conditions (see Chap. 15 and Fig. 14.9). Wave overtopping and transmission can be related if Figs. 14.8 and 14.9 are combined, and Fig. 14.10 shows this relationship. For convenience, the graphs are not dimensionless, but for a wave height of Hm0,i = 3 m. A small transmitted wave height of Hm0,t = 0.1 m is found if overtopping exceeds q = 30−50 l/s per m. In order to reach a transmitted wave height of Hm0,t = 1 m (Kt ≈ 0.33), the overtopping discharge should exceed q = 500−2500 l/s per m or

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0.6 so=0.01

0.5

so=0.03 so=0.05

Kt

0.4 0.3 0.2 0.1 0.0 0

Fig. 14.8.

0.5

1 Rc /Hm0

1.5

2

Wave transmission for a gentle smooth structure of 1:4 and for diﬀerent wave steepneses.

1.000 so=0.01

q/(gHm0 )

3 0.5

so=0.03 so=0.05

0.100

0.010

0.001 0

0.5

1 Rc /Hm0

1.5

2

Fig. 14.9. Wave overtopping for a gentle smooth structure of 1:4 and for diﬀerent wave steepnesses.

0.5–2.5 m3 /s per m. One may conclude that any signiﬁcant wave transmission is always associated with (very) large wave overtopping. Wave transmission for rubble mound structures has also been investigated in the European DELOS project,4 and the following prediction formulae were derived for wave transmission: Kt = −0.4Rc /Hm0 + 0.64B/Hm0 − 0.31(1 − exp(−0.5ξop )) for 0.075 ≤ Kt ≤ 0.8.

(14.6)

Wave overtopping for a simple trapezoidal rubble mound can be calculated by methods presented in Chap. 15. A typical rubble mound structure has been used as an example, with cotα = 1.5, armour rock of 6–10 ton (Dn50 = 1.5 m), and crest

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3500 so=0.01

3000

so=0.03 so=0.05

q (l/s per m)

2500 2000 1500 1000 500 0 0

0.5

1 Hm0 transmitted (m)

1.5

2

Fig. 14.10. Wave transmission versus wave overtopping for a smooth 1:4 slope and a wave height of Hm0 = 3 m.

3500 3000

so=0.01 so=0.03

q (l/s per m)

2500

so=0.05 2000 1500 1000 500 0 0

0.5

1 Hm0 transmitted (m)

1.5

2

Fig. 14.11. Wave transmission versus wave overtopping discharge for a rubble mound structure, cotα = 1.5, 6–10 ton rock, B = 4.5 m, and Hm0 = 3 m.

width of Bw = 4.5 m (3Dn50 ). A wave height of Hm0,i = 3 m has been assumed with the following wave steepness: sm−1,0 = 0.01 (long waves), 0.03 and 0.05 (short wind waves). In the calculations, the crest height has been changed to calculate wave transmission as well as wave overtopping. These are compared in Fig. 14.11 which shows that longer waves (sm−1,0 = 0.01) give more wave transmission, even for similar overtopping discharge. The reason is probably the eﬀect of wave action penetrating through the permeable upper layers, easier for long waves, thus contributing to the waves behind the structure. So, for permeable mounds, there may still be signiﬁcant wave transmission through the structure even without substantial overtopping discharge. In this

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example, transmitted wave heights between Hm0,t = 0.5 m and 1 m are found for overtopping discharges smaller than q = 100−200 l/s per m. A simple equation for wave transmission at vertical structures has been given by Goda,7 although again more complete methods are given in Chap. 16: Kt = 0.45 − 0.3Rc /Hm0

for 0 < Rc /Hm0 < 1.25.

(14.7)

For vertical walls, transmission is primarily governed by the relative crest height with little or no eﬀect of wave period or steepness. A simple vertical structure has been used as an example with Hm0,i = 3 m. Overtopping and wave transmission are compared in Fig. 14.12, where in the calculations the crest height has been changed to calculate wave transmission as well as wave overtopping. For comparison, the same rubble mound structure has been used as the example in Fig. 14.11, with cotα = 1.5, 6–10 ton rock (Dn50 = 1.5 m) as armour, a crest width of 4.5 m (3Dn50 ), and a wave steepness sop = 0.03. The curve for a smooth structure (Fig. 14.10) and for sop = 0.03 has been given too in Fig. 14.12. A rubble mound structure gives more wave transmission than a smooth structure for similar overtopping discharge, but a vertical structure can give even more transmission. The reason may be that overtopping water over the crest of a vertical breakwater always falls in a mass from a distance into the water, rather than ﬂowing relatively smoothly over or through a sloping structure. One may conclude that even without considerable wave overtopping discharge at the crest of a vertical structure, there still might be signiﬁcant wave transmission. In this example of a vertical structure, transmitted wave heights between 0.5 m and 1 m are found for overtopping discharges smaller than 100–200 l/s per m. Finally, an example is shown of both wave overtopping and wave transmission on a rubble mound breakwater in Fig. 14.13. 3500

q (l/s per m)

vertical 3000

rubble mound (so=0.03)

2500

smooth (so=0.03)

2000 1500 1000 500 0 0

0.5

1 Hm0 transmitted (m)

1.5

2

Fig. 14.12. Comparison of wave overtopping and transmission for a vertical, rubble mound, and smooth structure.

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Fig. 14.13.

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Wave overtopping and transmission at breakwater IJmuiden, the Netherlands.

14.3. PC-OVERTOPPING The computer program pc-overtopping was created using the results of the TAW8 Report “Wave runup and wave overtopping at dikes” and is used for the ﬁve-yearly safety assessment of all water defenses in the Netherlands. The TAW8 Report has now been replaced by Chap. 5 (dikes and embankments) in the EurOtop Overtopping Manual5 and extended for rubble mound and vertical structures in Chaps. 6 and 7 of that manual. pc-overtopping has been translated into English, and is available from the EurOtop Overtopping Manual5 web site. The program was based on dike-type structures. The structure should be sloping, although a small vertical wall on top of the dike may be included. Some eﬀects of roughness and/or permeability can be included, but not a crest with permeable and rough rock or armour units. In such a case, the structure should be modeled up to the transition to the crest, and other formulae should be used to take into account the eﬀect of the crest. pc-overtopping was set up so that almost every sloping structure can be modeled by an unlimited number of sections. Each section is given by x–y coordinates, and each section can have its own roughness factor. The program calculates most relevant overtopping parameters (except ﬂow velocities and ﬂow depths), such as: • • • •

2% runup level; mean overtopping discharge; percentage of overtopping waves; overtopping volumes per wave (maximum and for every percentage deﬁned by the user); • required crest height for given mean overtopping discharges (deﬁned by the user).

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The main uses of pc-overtopping are: • modeling of overtopping on any sloping structure, including diﬀerent roughnesses along the slope; • calculation of most overtopping parameters, not only the mean discharge. A disadvantage of pc-overtopping is that it does not calculate overtopping for vertical structures or for rubble structures with rough/permeable crest. An example illustrates some of its capabilities. An example dike cross section with the design water level at +1 m CD is shown in Fig. 14.14. Diﬀerent materials are used on the slope: rock, basalt, concrete asphalt, open concrete system, and grass on the upper part of the structure. The structure has been schematized in Fig. 14.15 by x–y coordinates and selection of the material of the top layer. The program selects the right roughness factor. The input parameters are wave height, wave period (either spectral period Tm−1,0 or peak period Tp ), wave obliquity, water level (with respect to the same level as used for the structure geometry), and ﬁnally, number of waves (derived from the storm duration and mean period) for the calculation of overtopping volumes, etc. Fig. 14.16 gives the input ﬁle.

Fig. 14.14.

Fig. 14.15.

Example cross section of a dike.

Input of geometry by x–y coordinates and choice of top material.

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Fig. 14.16.

Fig. 14.17.

Input ﬁle.

Output of pc-overtopping.

The output is given in three columns (Fig. 14.17). The left column gives the 2% runup level, the mean overtopping discharge, and the percentage of overtopping waves. If the 2% runup level is higher than the actual dike crest, this level is calculated by extending the highest section in the cross section. The middle column gives the required dike height for the given mean overtopping discharges. Again, the highest section is extended if required. Finally, the right-hand column gives the number of overtopping waves in the given storm duration, together with the maximum overtopping volume, and volumes for speciﬁed overtopping percentages, given as a percentage of the total number of overtopping waves. pc-overtopping also provides a check on whether the results for the 2% runup level and mean overtopping discharge fall within the measured ranges. Results on which the formulae were based are shown in the runup or overtopping graphs (see Figs. 14.18 and 14.19). These show the measured runup or overtopping, including eﬀects of reductions due to roughness, berms, etc. The curve gives the maximum for smooth straight slopes with normal wave attack. The program then plots the calculated point in these graphs (the point within the circle).

14.4. Neural Network Tools Artiﬁcial neural networks are tools that allow meaning to be extracted from very large quantities of data. Neural networks (NN) are organized in the form of layers, within which there are one or more processing elements called “neurons.” The ﬁrst layer is the input layer, and the number of neurons in this layer is equal to the number of input parameters. The last layer is the output layer and the number of neurons in this layer is equal to the number of output parameters to be predicted. The layers between the input and output layers are the hidden layers, consisting of a number of neurons to be deﬁned in conﬁguring the NN. Neurons in each layer receive

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Fig. 14.18.

Fig. 14.19.

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Check on 2% runup level.

Check on mean overtopping discharge.

information from the preceding layer, carry out a standard operation, and produce an output. Each connection has a weight factor assigned from the calibration of the NN. The input of a neuron consists of a weighted sum of the outputs of the preceding layer. This procedure is followed for each neuron; the output neuron generates the ﬁnal prediction of the NN. Artiﬁcial NNs have applications in many ﬁelds including coastal engineering where examples have been applied to predicting armour stability, forces on walls, wave transmission, and wave overtopping. The development of an artiﬁcial NN is useful, where: • the process to be described is complicated with many parameters involved, • there is a large amount of data.

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β

h Hm0,toeTm-1,0,toeht

Bt

f

cot

d

cot

u

Rc

B

hb tan

B

Ac

Gc

INPUT LAYER

HIDDEN LAYER

OUTPUT LAYER q (m3/s/m) Fig. 14.20.

Conﬁguration of the NN for wave overtopping.

It has already been seen that overtopping cannot be predicted by a single formula, but requires a number of diﬀerent formulae. A single NN can, however, cover the full range of structures, provided that suﬃcient data are available to “train” the NN. If too few data are available, predictions in the less-well populated regions will be unreliable, particularly where the prediction is trying to extrapolate out of range. Providentially, international cooperation supported by the European CLASH research project3 collected many test results on wave overtopping for all kinds of coastal structures and embankments. Within the CLASH project, two NNs were developed, one within the main project, and one alongside a PhD project.9 In both cases, the NN conﬁguration was based on Fig. 14.20, where the input layer has 15 input parameters (β, h, Hm0toe , Tm−1,0toe, ht , Bt , γf , cotαd , cotαu , Rc , B, hb , tanαb , Ac , Gc ) and one output parameter in the output layer (i.e., mean overtopping discharge, q). CLASH was focused on a three-layered NN, where a conﬁguration with one single hidden layer was chosen. Development of an artiﬁcial NN requires that all data be checked thoroughly (rubbish in = rubbish out), and that training be done by those with appropriate skills. Using NN as a prediction tool, however, is easy for most practical engineers! It is for this reason that the CLASH NN was adopted as part of the EurOtop Overtopping Manual.5 Applying NN requires an Excel or ASCII input ﬁle with parameters, run the program (push a button), and get a result ﬁle with mean overtopping discharge(s). It is therefore as easy as using a formula programmed in Excel and does not need knowledge about NNs. The advantages of the neural network are: • it works for almost every structure conﬁguration; • it is easy to calculate trends instead of just one calculation with one answer.

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The input exists of 10 structural parameters and four hydraulic parameters. The hydraulic parameters are wave height, wave period, and wave angle and water depth just in front of the structure. The structural parameters describe almost every possible structure conﬁguration by a toe (two parameters), two structure slopes (including vertical and wave return walls), a berm (two parameters), and a crest conﬁguration (three parameters). The tenth structural parameter is the roughness factor for the structure (γf ) and describes the average roughness of the whole structure. Although guidance is given, estimation is not easy if the structure has diﬀerent roughness on various parts of the structure. An overall view of possible structure conﬁgurations is shown in Fig. 14.21. It clearly shows that the NN is able to cope with most structure types.

Fig. 14.21.

Overall view of possible structure conﬁgurations for the NN.

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Gc = 5 m Rc = 4 m

Ac = 5 m Hs = 5 m; Tp = 10 s β = 0 Xbloc 1:1.5 h = 12 m

ht = 9 m Bt = 4 m

Fig. 14.22.

Example cross section with parameters for application of NN.

Very often one is not only interested in one calculation, but in more. As the input ﬁle has no limitations in the number of rows (= number of calculations), it is easy to incrementally increase one or more parameters and to ﬁnd a trend for a certain (design) measure. As an example for the calculation of a trend with the NN tool, an example cross section of a rubble mound embankment with a wave wall has been chosen (Fig. 14.22). If, for example, the cross section in Fig. 14.22 experiences too much overtopping, the following measures could be considered: • • • •

increasing the crest; applying a berm; widening the crest; increasing only the crest wall.

An example input ﬁle with the ﬁrst six calculations is shown in Table 14.1, where incremental increase of the crest will show the eﬀect of raising the crest on the amount of wave overtopping. Calculations will give an output ﬁle with the mean overtopping discharge q (m3 /s per m width) and with conﬁdence limits. Table 14.2 shows as an example the output related to the input in Table 14.1. Assembling the input ﬁle for this example took 1 h and resulted in 1400 rows or calculations. The calculation of the NN took less than 10 s. The results were copied into the Excel input ﬁle, and the resulting graph was plotted within Excel, which took another hour. Figure 14.23 gives the ﬁnal result, where the four trends are Table 14.1.

Example input ﬁle for NN with ﬁrst six calculations.

β

h

Hm0

Tm−1,0

ht

Bt

γf

cotαd

cotαu

Rc

B

hb

tan αB

Ac

Gc

0 0 0 0 0 0

12 12 12 12 12 12

5 5 5 5 5 5

9.1 9.1 9.1 9.1 9.1 9.1

9 9 9 9 9 9

4 4 4 4 4 4

0.49 0.49 0.49 0.49 0.49 0.49

1.5 1.5 1.5 1.5 1.5 1.5

1.5 1.5 1.5 1.5 1.5 1.5

4 4.05 4.1 4.15 4.2 4.25

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

5 5.05 5.1 5.15 5.2 5.25

5 5 5 5 5 5

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Output ﬁle of NN with conﬁdence limits.

2.50%

5.00%

25.00%

50.00%

75.00%

95.00%

97.50%

2.45E−02 2.35E−02 2.26E−02 2.19E−02 2.07E−02 1.99E−02

2.77E−02 2.64E−02 2.49E−02 2.39E−02 2.27E−02 2.18E−02

4.15E−02 3.99E−02 3.82E−02 3.69E−02 3.55E−02 3.38E−02

5.91E−02 5.58E−02 5.33E−02 5.08E−02 4.85E−02 4.62E−02

8.35E−02 7.91E−02 7.52E−02 7.17E−02 6.89E−02 6.60E−02

0.1299 0.1246 0.119 0.1133 0.1079 0.1033

0.1591 0.1516 0.1448 0.1383 0.1324 0.1265

Fig. 14.23.

Results of a trend calculation.

shown. The base situation had an overtopping discharge of 59 l/s per m. The graph clearly shows what measures are required to reduce the overtopping by, for example, a factor 10 (to 5.9 l/s per m) or to only 1 l/s per m. It also shows that increasing the structure height is most eﬀective, followed by increasing only the crest wall. Within the CLASH research project, two diﬀerent NNs exist. One is the oﬃcial NN developed by Delft Hydraulics in the main part of the CLASH project. It runs as executable and can be downloaded from the CLASH web site or the EurOtop Overtopping Manual web site.5 The other NN has also been developed within CLASH, but as part of a PhD R thesis at Gent University.9 The network was developed in MatLab and so it can R only be run if the user has a licence for MatLab . The advantage of this NN is that it ﬁrst decides whether there will be overtopping or not (classiﬁer). If there is no overtopping, it gives q = 0. If there is overtopping, it will quantify the overtopping with a similar network as the CLASH network (quantiﬁer). This use of both “qualiﬁer” and “quantiﬁer” NNs is certainly an advantage over the single-stage CLASH

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NN. The CLASH network was only trained with overtopping data (tests with “no overtopping” were not considered) and, therefore, this network always gives a prediction of overtopping, even in the range where no overtopping should be expected.

14.5. Use of CLASH Database The EU research project CLASH generated an extensive database of overtopping tests from data submitted around the world. Each test was described not only by 31 parameters as hydraulic and structural parameters, but also parameters describing the reliability and complexity of the test and structure. The database includes more than 10,000 tests and was set up as an Excel database. At its simplest, the database is no more than a matrix with 31 columns and more than 10,000 rows. It can be downloaded from the CLASH or Manual web site. If a user has a speciﬁc structure, there is a possibility to look into the database and ﬁnd more or less similar structures with measured overtopping discharges. It may even be possible that the structure has already been tested with the right wave conditions! Finding the right tests can be done by using ﬁlters in the Excel database. Every test of such a selection can then be studied thoroughly. One example will be described here in depth. Suppose one is interested in the improvement of a vertical wall with a large wave return wall. The wave conditions are Hm0 toe = 3 m, the wave steepness sm−1,0 = 0.04 (Tm−1,0 = 6.9 s), and the wave attack is perpendicular to the structure. The design water depth h = 10 m and the wave return wall starts 1 m above design water level and has a height and width of 2 m (the angle is 45◦ seaward). This gives a crest freeboard Rc = 3 m, equal to the wave height. Have tests been performed which are close to this speciﬁc structure and given wave conditions? The ﬁrst ﬁlter selects data with a vertical down slope, i.e., cotαd = 0. The second ﬁlter could select data with a wave return wall overhanging more than about 30◦ seaward. This means cotαu < −0.57. In the ﬁrst instance every large wave return wall can be considered, say, at least 0.5Hm0 wide. This gives the third ﬁlter, selecting data with −cotαu∗ (Ac + hb )/Hm0 ≥ 0.5. With these three ﬁlters, the database gives 212 tests from four independent test series. These data are summarized in Fig. 14.24 with the expression of Franco et al.6 for a vertical wall. There are 22 tests which gave no measurable overtopping. These 3 1/2 results are represented in the ﬁgure with a value of q/(gHm0 = 10−7. The toe ) majority of the data are situated below the simple prediction curve for a vertical wall, indicating that a wave return wall is eﬃcient, but the data are too scattered to be decisive. The next step in the ﬁltering process could be that only wave return walls overhanging more than 45◦ seaward are selected. This means cotαu < −1. The water depth is relatively large for the considered case, and shallow water conditions are excluded if h/Hm0 toe > 3. Figure 14.25 shows this further ﬁltering process. The number of data has been reduced to 78 tests from only two independent series. In total, 12 tests result in no measureable overtopping. The data show the trend that the overtopping discharges are on average about 10 times smaller than for a vertical

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1.E+00 series 508 series 914

1.E-01

series 113

q/(gHm0 toe )

3 0.5

1.E-02

series 033 Franco et al.,(1994)

1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 0

0.5

1

1.5

2

2.5

3

3.5

4

Rc /Hm0 toe Fig. 14.24.

Overtopping for large wave return walls — ﬁrst selection.

1.E+00 series 914 series 113

1.E-01

Franco et al.,(1994)

q/(gHm0 toe )

3 0.5

1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 0

0.5

1

1.5

2

2.5

3

3.5

4

Rc/H m 0 toe Fig. 14.25.

Overtopping for large wave return walls — second selection with more criteria.

wall, given by the dashed line. But for Rc /Hm0 toe > 1, there are quite some tests without any overtopping. As still quite some data are remaining in Fig. 14.25, it is possible to narrow the search area even further. With a wave steepness of sm−1,0 = 0.04 in the considered case, the wave steepness range can be limited to 0.03 < sm−1,0 < 0.05. The width

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1.E+00 series 113

1.E-01

Franco et al.,(1994)

q/(gH m0 toe )

3 0.5

1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 0

0.5

1

1.5

2

2.5

3

3.5

4

R c/Hm 0 toe Fig. 14.26. Overtopping for wave return wall with so = 0.04, seaward angle of 45◦ , a width of 2 m and a crest height of Rc = 3 m. For Hm0 toe = 3 m, the overtopping can be estimated from Rc /Hm0 toe = 1.

of the wave return wall of 2 m, with the wave height of Hm0 = 3 m, gives a relative width of 0.67. The range can be limited to 0.5 < −cotαu∗ (Ac + hb )/Hm0 < 0.75. Finally, the transition from vertical to wave return wall is 1 m above the design water level, giving hb /Hm0 toe = −0.33. The range can be set at −0.5 < hb / Hm0 toe < −0.2. The ﬁnal selection obtained after ﬁltering is given in Fig. 14.26. Only four tests remain from one test series, one of which gave no measurable overtopping. The data now give a clear picture. For a relative freeboard lower than about Rc /Hm0 toe = 0.7, the overtopping will not be much diﬀerent from the overtopping at a vertical wall. The wave return wall, however, becomes very eﬃcient for large freeboards and eﬀectively prevents any measurable overtopping for Rc /Hm0 toe > 1.2. For the structure considered with Rc /Hm0 toe = 1, the wave overtopping will be 20–40 times less than that for a vertical wall and will probably amount to q = 0.5 − 2 l/s per m width. In this particular case, it was possible to ﬁnd four tests in the database with very close similarities to the considered structure and wave conditions. 14.6. Tolerable Discharges 14.6.1. Hazards from overtopping Most sea defense structures are constructed primarily to limit overtopping that might cause ﬂooding. Over a particular storm or tide, the overtopping volumes that can be tolerated will be site-speciﬁc as the overall volume of water that can be accepted will depend on the size and use of the receiving area, extent, and magnitude

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of drainage ditches, damage versus inundation curves, and return period. Guidance on modeling inundation ﬂows is being developed within the Floodsite project (see: http://www.ﬂoodsite.net/html/project overview.htm), but ﬂood volumes, per se, are not distinguished further in this chapter. Instead, advice here focuses on direct hazards from wave overtopping. For sea defenses that protect people living, working, or enjoying themselves, designers and owners of these defenses must deal with potential direct hazards from overtopping. This requires that the level of hazard and its probability of occurrence be assessed, allowing appropriate action plans to be devised to ameliorate risks arising from overtopping. The main hazards on or close to sea defense structures are of death, injury, property damage, or disruption from direct wave impact or by drowning. On average, approximately 2–5 people are killed each year in each of the United Kingdom and Italy through wave action, chieﬂy on seawalls and similar structures (although this rose to 11 in the United Kingdom during 2005). It is often helpful to analyze direct wave and overtopping eﬀects, and their consequences under three general categories: • direct hazard of injury or death to people immediately behind the defense; • damage to property, operation, and/or infrastructure in the area defended, including loss of economic, environmental, or other resource, or disruption to an economic activity or process; • damage to defense structure(s), either short-term or longer-term, with the possibility of breaching and ﬂooding. The character of overtopping ﬂows or jets, and the hazards they cause, also depend upon the geometries of the structure, the hinterland behind the seawall, and the form of overtopping. Rising ground behind the seawall may allow people at potential risk to see incoming waves, and the slope will slow overtopping ﬂows. Conversely, a defense that is elevated signiﬁcantly above the land defended will obscure visibility of incoming waves (Fig. 14.27), and post-overtopping ﬂows may increase in speed rather than decreasing. Hazards caused by overtopping therefore depend upon both the local topography and structures as well as on the direct overtopping characteristics. It is not possible to give unambiguous or precise limits to tolerable overtopping for all conditions. Some guidance is, however, oﬀered here on tolerable mean discharges and maximum overtopping volumes for a range of circumstances or uses, and on inundation ﬂows and depths. These limits may be adopted or modiﬁed depending upon the circumstances and uses of the site. 14.6.1.1. Wave overtopping processes and hazards Overtopping hazards can be linked to a number of simple direct ﬂow parameters (Fig. 14.28): • mean overtopping discharge, q; • individual and maximum overtopping volumes, Vi and Vmax ;

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Fig. 14.27.

Defended area below seawall and foreshore (saltmarsh) level.

Fig. 14.28.

Overtopping on embankment and promenade seawalls.

• overtopping velocities over the crest, horizontally and vertically, vxc and vzc or vxp and vzp ; • overtopping ﬂow depth, again measured on crest or promenade, dxc or dxp . Less direct responses (or similar responses, but farther back from the defense) may be used to assess the eﬀects of overtopping, perhaps categorized by: • overtopping falling distances, xc ; • post-overtopping wave pressures (pulsating or impulsive), pqs or pimp ; • post-overtopping ﬂow depths, dxc or dxp ; and horizontal velocities, vxc or vxp . The main response to direct overtopping hazards has most commonly been the construction of new defenses, but should now always consider three options, in increasing order of intervention: • move human activities away from the area subject to overtopping or ﬂooding hazard, thus modifying the land-use category and/or habitat status;

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• accept hazard at a given probability (acceptable risk) by providing for temporary use and/or short-term evacuation with reliable forecast, warning and evacuation systems, and/or use of temporary/demountable defense systems; • increase defense standard to reduce risk to (permanently) acceptable levels probably by enhancing the defense and/or reducing loadings. For any structure expected to reduce overtopping, the crest level and/or the front-face conﬁguration will be dimensioned to give acceptable levels of overtopping under speciﬁed extreme conditions or combined conditions (e.g., water level and waves). Setting acceptable levels of overtopping depends on: • • • •

use of the defense structure itself; use of the land behind; national and/or local standards and administrative practice; economic and social basis for funding the defense.

Under most forms of wave attack, waves tend to break before or onto sloping embankments with the overtopping process being relatively gentle (see Fig. 14.1). Relatively few water levels and wave conditions may cause “impulsive” breaking where the overtopping ﬂows are sudden and violent. Conversely, steeper, vertical, or compound structures are more likely to experience intense local impulsive breaking, and may overtop violently and with greater velocities (see Fig. 14.3). The form of breaking will therefore inﬂuence the distribution of overtopping volumes and their velocities, both of which will have impact on the hazards that they cause. Additional hazards that are not dealt with here are those that arise from wave reﬂections, often associated with steep-faced defenses. Reﬂected waves increase wave disturbance, which may cause hazards to navigating or moored vessels; may increase waves along neighboring frontages, and/or may initiate or accelerate local bed erosion thus increasing depth-limited wave heights. 14.6.1.2. Form of overtopping hazard Wave overtopping which runs up the face of the seawall and over the crest in (relatively) complete sheets of water is often termed “green water.” In contrast, “white water” or spray overtopping tends to occur when waves break seaward of the defense structure or break onto its seaward face, producing noncontinuous overtopping, and/or signiﬁcant volumes of spray. Overtopping spray may be carried over the wall either under its own momentum, or assisted and/or driven by an onshore wind. Additional spray may also be generated by wind acting directly on wave crests, particularly when reﬂected waves interact with incoming waves to give severe local “clapoti.” This type of spray is not classed as overtopping nor is it predicted by the methods described in this manual. Without a strong onshore wind, spray will seldom contribute signiﬁcantly to overtopping volumes, but may cause local hazards. Light spray may reduce visibility for driving, important on coastal highways, and will extend the spatial extent of salt spray eﬀects such as damage to crops/vegetation, or deterioration of buildings. The eﬀect of spray in reducing visibility on coastal highways (particularly when

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intermittent) can cause sudden loss of visibility in turn leading drivers to veer suddenly. Eﬀects of wind and generation of spray have not often been modeled. Some research studies have suggested that eﬀects of onshore winds on large green water overtopping are small, but that overtopping under q = 1 l/s per m might increase up to four times under strong winds, especially where much of the overtopping is as spray. Discharges between q = 1 and 0.1 l/s per m are, however, already greater than some discharge limits suggested for pedestrians or vehicles, suggesting that wind eﬀects may inﬂuence overtopping at and near acceptable limits for these hazards. This is discussed further in Sec. 14.7. 14.6.1.3. Return periods Return periods at which overtopping hazards are analyzed, and against which a defense might be designed, may be set by national regulation or guidelines. As with any area of risk management, diﬀerent levels of hazard are likely to be tolerated at inverse levels of probability or return period. The risk levels (probability × consequence) that can be tolerated will depend on local circumstances, local and national guidelines, the balance between risk and beneﬁts, and the level of overall exposure. Heavily traﬃcked areas might therefore be designed to experience lower levels of hazard applied to more people than lightly used areas, or perhaps the same hazard level at longer return periods. Guidance on example return periods used in evaluating levels of protection suggest example protection levels versus return periods as shown in Table 14.3. In practice, some of the return periods in Table 14.3 may be too short. National guidelines have recommended lower risk, e.g., a low probability of ﬂooding in the United Kingdom is now taken as <0.1% probability (1:1000 year return), and the medium probability of sea ﬂooding as between 0.5% and 0.1% (1:200 to 1:1000 year return). Many existing defenses, however, oﬀer levels of protection far lower than these. In the Netherlands, where two-thirds of the country lies below the storm surge level, protection was substantially improved after the ﬂood in 1953 where almost 2000 people drowned. Standards of protection for large rural areas are currently 1:10,000 years, less densely populated areas at 1:4000 years, and protection for high river discharge (without threat of storm surge) is given to 1:1250 years. Table 14.3.

Hazard types.

Hazard type and reason Temporary or short-term measures Majority of coast protection or sea defenses Flood defenses protecting large areas Special structure, high capital cost Nuclear power stations, etc. a Total

probability return period.

Design life (years)

Level of protectiona (years)

1–20 30–70 50–100 200 —

5–50 50–100 100–10,000 Up to 10,000 10,000

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The design life for ﬂood defenses like dikes which are fairly easy to upgrade, is taken in the Netherlands as 50 years. In urban areas, where it is more diﬃcult to upgrade a ﬂood defense, the design life is taken as 100 years. This design life increases for very special structures with high capital costs, like the Eastern Scheldt storm surge barrier, Thames barrier, or the Maeslandtkering at the entrance to Rotterdam. A design life of around 200 years is then usual. Variations from simple “acceptable risk” approach may be required for publicly funded defenses based on beneﬁt — cost assessments, or where public aversion to hazards causing death requires greater eﬀorts to ameliorate the risk, either by reducing the probability of the hazard or by reducing its consequence. 14.6.2. Tolerable mean discharges and overtopping simulator Guidance on overtopping discharges that can cause damage to seawalls, buildings, or infrastructure, or danger to pedestrians and vehicles have been related to mean overtopping discharges or (less often) to peak volumes. Guidance quoted previously were derived initially from the analysis in Japan of overtopping perceived by port engineers to be safe. Further guidance from Iceland suggests that equipment or cargo might be damaged for q ≥ 0.4 l/s per m. Signiﬁcantly, diﬀerent limits are discussed for embankment seawalls with back slopes, or for promenade seawalls without back slopes. Some guidance distinguishes between pedestrians or vehicles, and between slow and faster speeds for vehicles. Tests on the eﬀects of overtopping on people suggest that information on mean discharges alone may not give reliable indicators of safety for some circumstances, and that maximum individual volumes may be better indicators of hazard than average discharges. The volume (and velocity) of the largest overtopping event can vary signiﬁcantly with wave condition and structure type, even for a given mean discharge. There remain, however, two diﬃculties in specifying safety levels with reference to maximum volumes rather than to mean discharges. Methods to predict maximum volumes are available for fewer structure types, and are less well-validated. Secondly, data relating individual maximum overtopping volumes to hazard levels are still very rare. In most instances, the discharge (or volumes) discussed here are those at the point of interest, e.g., at the roadway or footpath or building. It is noted that the hazardous eﬀect of overtopping waters reduces with the distance away from the defense line. As a rule of thumb, the hazard eﬀect of an overtopping discharge at a point x m back from the seawall crest will be to reduce the overtopping discharge by a factor of x; and so the eﬀective overtopping discharge at x (over a range of 5–25 m), qeﬀective is given by qeﬀective = qseawall /x.

(14.8)

The overtopping limits suggested in Tables 14.4–14.7 derive from a generally precautionary principle informed by previous guidance and by observations and measurements made by the CLASH partners and other researchers. Limits for pedestrians in Table 14.4 show a logical sequence, with allowable discharges reducing

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372 Table 14.4.

Limits for overtopping for pedestrians.

Hazard type and reason Trained staﬀ, well shod and protected, expecting to get wet, overtopping ﬂows at lower levels only, no falling jet, low danger of fall from walkway. Aware pedestrian, clear view of the sea, not easily upset or frightened, able to tolerate getting wet, wider walkway.b

Mean discharge q (l/s per m)

Max volumea Vmax (l/m)

1–10

500 at low level

0.1

20–50 at high level or velocity

a These limits relate to overtopping velocities well below vc ≤10 m/s. Lower volumes may be required if the overtopping process is violent and/or overtopping velocities are higher. b Not all of these conditions are required, nor should failure of one condition on its own require the use of a more severe limit.

Table 14.5.

Limits for overtopping for vehicles. Mean discharge q (l/s per m)

Max volumea Vmax (l/m)

Driving at low speed, overtopping by pulsating ﬂows at low ﬂow depths, no falling jets, vehicle not immersed

10–50a

100–1000

Driving at moderate or high speed, impulsive overtopping giving falling or high velocity jets

0.01–0.05b

5–50b at high level or velocity

Hazard type and reason

a These

limits probably relate to overtopping deﬁned at highway. limits relate to overtopping deﬁned at the defense, but assumes the highway to be immediately behind the defense. b These

steadily as the recipient’s ability or willingness to anticipate or receive the hazard reduces. A further precautionary limit of q = 0.03 l/s per m might apply for unusual conditions where pedestrians have no clear view of incoming waves; may be easily upset or frightened or are not dressed to get wet; may be on a narrow walkway or in close proximity to a trip or fall hazard. Research studies have, however, shown that this limit is only applicable for the conditions identiﬁed, and should NOT be used as the general limit for which q = 0.1 l/s per m in Table 14.4 is appropriate. For vehicles, the suggested limits are rather more widely spaced as two very different situations are considered. The higher overtopping limit in Table 14.5 applies where wave overtopping generates pulsating ﬂows at roadway level, akin to driving through slowly varying ﬂuvial ﬂow across the road. The lower overtopping limit in Table 14.5 is, however, derived from considering more impulsive ﬂows, overtopping at some height above the roadway, with overtopping volumes being projected at

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speed and with some suddenness. These lower limits are however based on few site data or tests, and may therefore be relatively pessimistic. Rather fewer data are available on the eﬀects of overtopping on structures, buildings, and property. Site-speciﬁc studies suggest that pressures on buildings by overtopping ﬂows will vary signiﬁcantly with the form of wave overtopping, and with the use of sea defense elements intended to disrupt overtopping momentum (not necessarily reducing discharges). Guidance derived from the CLASH research project and previous work suggests limits in Table 14.6 for damage to buildings, equipment, or vessels behind defenses. A set of limits for structures in Table 14.7 has been derived from early work by Goda7 and others in Japan. These give a ﬁrst indication of the need for speciﬁc protection to resist heavy overtopping ﬂows. It is assumed that any structure close to the sea will already be detailed to resist the erosive power of heavy rainfall and/or spray. Two situations are considered: • embankment seawall or dike, elevated above the defended area, and so overtopping ﬂows pass over the crest and down the rear face; • promenade defense in which overtopping ﬂows remain on or behind the seawall crest before returning seaward.

Table 14.6.

Limits for overtopping for property behind the defense.

Hazard type and reason

Mean discharge q (l/s per m)

Max volumea Vmax (l/m)

50

5,000–50,000

10a 1b 0.4a

1,000–10,000 ∼ ∼

Signiﬁcant damage or sinking of larger yachts Sinking small boats set 5–10 m from wall. Damage to larger yachts Building structure elements Damage to equipment set back 5–10 m a These b This

limits relate to overtopping deﬁned at the defense. limit relates to the eﬀective overtopping deﬁned at the building.

Table 14.7.

Limits for overtopping for damage to the defense crest or rear slope.

Hazard type and reason Embankment seawalls/sea dikes No damage if crest and rear slope are well protected No damage to crest and rear face of grass covered embankment of clay No damage to crest and rear face of embankment if not protected Promenade or revetment seawalls Damage to paved or armoured promenade behind seawall Damage to grassed or lightly protected promenade or reclamation cover

Mean discharge q (l/s per m) 50–200 1–10 0.1 200 50

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Fig. 14.29.

Principle of the wave overtopping simulator.

The limits for the latter category cannot be applied where the overtopping ﬂows can fall from the defense crest where the nature of the ﬂow may be more impulsive. The limits in Table 14.7 are precautionary and are generally based on old data. In order to clarify the erosion resistance of grass protection under wave overtopping, tests were performed using the overtopping simulator (Figs. 14.29 and 14.30) in 2007 on a sea dike in the Netherlands. The grass dike had a 1:3 inner slope of fairly good clay, sand content smaller than 30%. The overtopping simulator was used to test the erosion resistance of this inner slope for a simulated 6 h storm for each overtopping condition. These started with overtopping equivalent to a mean discharge of 0.1 l/s per m and increased to 1, 10, 20, 30, and ﬁnally even 50 l/s per m. After all these simulated storms, the slope was still in good condition and showed little erosion. The erosion resistance of this dike was very high. Another test was performed on bare clay by removing the grass sod over the full inner slope to a depth of 0.2 m. Overtopping conditions of 0.1 l/s per m, 1, 5, and ﬁnally, 10 l/s per m were performed, again for 6 h each. Erosion damage started for the ﬁrst condition (two erosion holes) and increased during the other overtopping conditions. After 6 h at a mean discharge of 10 l/s per m (see Figs. 14.30 and 14.31), there were two large erosion holes, about 1 m deep, 1 m wide, and 4 m long. This situation was considered as “not too far from initial breaching.” The overall conclusion of this ﬁrst overtopping test on a real dike is that clay with grass can be highly erosion-resistant. Even without grass the good quality clay also survived extensive overtopping. The conclusions may not yet be generalized to all dikes as

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Fig. 14.30.

Fig. 14.31.

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Overtopping simulator discharging a large volume on the inner slope of a dike.

Wave overtopping test on bare clay; result after 6 h with 10 l/s per m width.

clay quality and type of grass cover still may play a role and, therefore, more testing is required to come to general conclusions. One remark, however, should be made on the strength of the inner slopes of dikes by wave overtopping. Direct erosion of the slope is one possible failure mechanism. A major failure mechanism, especially in the past, was slip failure of the (rear) slope. Slip failures may directly lead to a breach, and such slip failures often occur

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mainly for steep inner slopes like 1:1.5 or 1:2. For this reason, most dike designs in the Netherlands in the past 50 years have used a 1:3 inner slope, where it is unlikely that slip failures will occur due to overtopping. This mechanism might, however, occur for steep inner slopes, and so it should be taken into account in safety analysis. 14.6.3. Tolerable maximum volumes and velocities 14.6.3.1. Overtopping volumes and velocities Guidance on suggested limits for maximum individual overtopping volumes have been given in Tables 14.4–14.7 where data are available. Research studies with volunteers at full scale or ﬁeld observations suggest that danger to people or vehicles might be related to peak overtopping volumes, with “safe” limits for people covering: Vmax = 1000 − 2000 l/m for trained and safety-equipped staﬀ in pulsating ﬂows on a wide-crested dike; Vmax = 750 l/m for untrained people in pulsating ﬂows along a promenade; Vmax = 100 l/m for overtopping at a vertical wall; Vmax = 50 l/m where overtopping could unbalance an individual by striking their upper body without warning. Few data are available on overtopping velocities and their contribution to hazards. Chapter 15 gives guidance on overtopping ﬂow velocities and depths at crest and inner slope for simple sloping embankments as well. Velocities of 5–8 m/s are possible for maximum overtopping waves during overtopping discharges of 10– 30 l/s per m width. Studies of hazards under steady ﬂows suggest that limits on horizontal velocities for people and vehicles will probably need to be set below vx < 2.5–5 m/s. On vertical and battered walls, upward projected velocities (vz ) have been related to inshore wave celerity (see Chap. 16). Relative velocities, vz /ci , have been found to be roughly constant at vz /ci ≈ 2.5 for pulsating and slightly impulsive conditions, but increase signiﬁcantly for impulsive conditions, reaching vz /ci ≈ 3–7. 14.6.3.2. Overtopping loads Post-overtopping wave loads have seldom been measured on defense structures, buildings behind sea defenses, or on people; so little generic guidance is available. If loadings from overtopping ﬂows could be important, they should be quantiﬁed by interpretation of appropriate ﬁeld data or by site-speciﬁc model studies. An example model study during the CLASH research project indicates how important these eﬀects might be. A simple 1 m high vertical secondary wall was set in a horizontal promenade about 7 m back from the primary seawall, itself a concrete recurve fronted by a steep beach and short rock armour slope. Pulsating wave pressures were measured on the secondary wall against the eﬀective overtopping discharge arriving at the secondary wall. (This discharge was deduced by applying Eq. (14.7) to the overtopping measured at the primary wall, 7 m in front.) Whilst

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strongly site-speciﬁc, these results suggest that quite low discharges (0.1–1.0 l/s per m) may lead to loadings up to 5 kPa.

14.6.4. Eﬀects of debris and sediment in overtopping ﬂows There are virtually no data on the eﬀect of debris on hazards caused by overtopping, although anecdotal comments suggest that damage can be substantially increased for a given overtopping discharge or volume if “hard” objects such as rocks, shingle, or timber are included in overtopping. It is known that impact damage can be particularly noticeable for seawalls and promenades where shingle may form the “debris” in heavy or frequent overtopping ﬂows.

14.7. Model Eﬀects, Scale Eﬀects, and Uncertainties 14.7.1. Model and scale eﬀects This section deals with the types of model and scale eﬀects that result from the use of hydraulic models to quantify wave overtopping. Firstly, scale and model eﬀects are deﬁned. Secondly, a methodology based on the current knowledge is introduced on how to account for these eﬀects. Model or laboratory eﬀects originate from the incorrect reproduction of the geometry or materials of the prototype structure, or of the waves and currents, or due to the boundary conditions of a wave ﬂume (sidewalls, wave paddle, etc.). Modeling techniques have developed signiﬁcantly, but model eﬀects may still inﬂuence test results. One noticeable feature is that very few model tests include wind, whilst wind eﬀects may be signiﬁcant for some forms of overtopping, particularly splash or spray. Scale eﬀects result from incorrect (or distorted) reproduction of a prototype water-structure interaction in the model. The ideal model requires that both Froude’s and Reynolds’ laws are met simultaneously. This is not possible without changing the test ﬂuid; so, scale eﬀects cannot be avoided when performing scaled model tests. They can, however, be minimized for the main processes, and/or corrections can be applied where the distortion is understood. Gravity, pressure, and inertial forces are the relevant forces for wave motion; so, physical models of seawalls/breakwaters are scaled according to Froude’s law. Viscosity forces are governed by Reynolds’ law, elasticity by Cauchy’s law, and surface tension forces by Weber’s law, and these forces are generally neglected for most models. Distortions or errors resulting from ignoring these forces are called scale eﬀects, and are generally unquantiﬁed. Measurement eﬀects result from distortions to the process by the use of measurement equipment and/or data sampling methods. These distortions may signiﬁcantly inﬂuence the comparison of results between prototype and model, or between two models. It is therefore essential to quantify the eﬀects and the uncertainty related to the diﬀerent techniques available.

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Whilst these deﬁnitions are reasonably clear, in reality, it is sometimes diﬃcult to assign all the causes of diﬀerences between model and prototype data. During CLASH, the major contributions to model eﬀects were found to be wind since this is seldom included in hydraulic models. Additional diﬀerences were also found and were ascribed to the model eﬀects in addition to those by the wind. Examining repeatability of example tests on armoured slopes showed that wave parameters (Hm0 , Tp , Tm−1,0 ) have a coeﬃcient of variation CoV ∼ 3%. Diﬀerences between overtopping measured in two wave ﬂumes were CoV ∼ 13% and CoV ∼ 10%. Diﬀerent time windows for wave analysis and diﬀerent types of wave generation methods had little or no inﬂuence on the estimated wave parameters (CoV ∼ 3%). The number of waves in a test inﬂuences overtopping, and the use of 200 waves compared to 1000 waves shows diﬀerences in the mean overtopping rates up to 20%. The position of the overtopping tray at the side of the ﬂume also showed diﬀerences in the overtopping rates (CoV ∼ 20%) from the results where the tray was located at the center of the crest. This could be because of the diﬀerent arrangements of the armour units in front of the overtopping tray or due to the inﬂuence of the sidewalls of the ﬂume. Scale eﬀects have been investigated by various authors, and this has led to some generic rules that should be observed for physical model studies. Generally, water depths in the model should be much larger than h = 2.0 cm, wave periods larger than T = 0.35 s, and wave heights larger than Hs = 5.0 cm to avoid the eﬀects of surface tension. For rubble mound structures, the Reynolds number for the stability of the armour layer should exceed Re = 3 × 104 ; for overtopping of coastal dikes Re > 1 × 103 ; and stone sizes in underlayers and core of rubble mounds should be scaled according to the velocities in the core rather than the stone dimensions, especially for small models. This leads to the use of larger material in the core than that demanded by simple Froude scaling. Critical limits for the inﬂuence of viscosity and surface tension are given in Table 14.8. From the observations in the prototype and scaled models, a methodology was derived to account for these diﬀerences without speciﬁcally deﬁning the contribution from the model, scale, or measurement eﬀects. These recommendations are given Table 14.8.

Scale eﬀects and critical limits.

Process

Relevant forces

Similitude law

Critical limits

Wave propagation

Gravity force Friction forces Surface tension Gravity force Friction forces Surface tension Gravity force Friction forces Surface tension Gravity force Friction forces Surface tension

F rW , ReW , We F rW , ReW , We F rA , F rq Req , We F rA , F rq , Req , We

ReW > ReW,crit = 1 × 104 T > 0.35 s; h > 2.0 cm

Wave breaking

Wave runup

Wave overtopping

ReW > ReW,crit = 1 × 104 T > 0.35 s; h > 2.0 cm Req > Req,crit = 103 W e > W ecrit = 10 Req > Req,crit = 103 W e > W ecrit = 10

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in Chaps. 15 and 16 for dikes, for rubble mound slopes, and for vertical walls, respectively.

14.7.2. Uncertainties in predictions Sections 14.2–14.5 have proposed various models or ways to predict wave overtopping of coastal structures. These models will now be discussed with regard to their uncertainties.

14.7.2.1. Empirical models Unless the empirical method has been conﬁgured to give an “upper bound” estimate, model uncertainty can usually be described by using a mean factor of 1.0 and a Gaussian distribution around the mean prediction. The standard deviation is derived by comparing model data and the prediction. This has two implications for design. Probabilistic design values for all empirical models used in this manual describe the mean approach for all underlying data points. This means that, for normally distributed variables, about 50% of the data points exceed the prediction by the model, and 50% are below the predicted values. This value should be used if probabilistic design methods are used. Deterministic design values may be given as the mean value plus one standard deviation, which in general gives a safer approach, and takes into account that model uncertainty for wave overtopping is always signiﬁcant.

14.7.2.2. Neural network When running the NN, the user is provided with overtopping rates based on the CLASH database and the NN prediction. Together with these results the user will also obtain the uncertainties of the prediction through the 5% and 95% conﬁdence intervals. Assuming a normal distribution will allow the standard deviation of overtopping to be estimated from those conﬁdence intervals, and (if required) the whole Gaussian distribution.

14.7.2.3. CLASH database The CLASH database was described earlier. It provides a large data set of model data on wave overtopping of coastal structures. As these are model data, it is noted that corrections for model/scale eﬀects discussed in Sec. 14.7.1 above have not been applied to the database. The user will therefore need to decide to apply any scale/model correction procedure whenever these data are used for prototype predictions. With respect to uncertainties, all model results show variations in measured overtopping. Most of these variations will result from the measurement and model eﬀects as discussed earlier, and will therefore apply to the database.

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14.8. Guidance on Use of Methods The EurOtop Overtopping Manual5 is accompanied by an overall Calculation Tool, which includes the following elements: • Empirical Calculator programmed with the main empirical overtopping equations in this chapter and the next two (limited to those that can be described explicitly, that is without iteration). • PC-Overtopping, which codes all the prediction methods for mean overtopping discharge and other parameters, for (generally shallow sloped) sea dikes, see Sec. 14.3. • Neural Network tool developed in the CLASH research project to calculate mean overtopping for many types of structures (see Sec. 14.4). • CLASH database, a listing of input parameters and mean overtopping discharge from each of approximately 10,000 physical model tests on both idealized (research) test structures, and site-speciﬁc designs. These data can be sifted to identify test results that may apply for conﬁgurations close to the reader’s (see Sec. 14.5). None of these methods give the universally “best” results, and indeed there may still be a need for site-speciﬁc model tests for some defenses. The most reliable method to be used will depend on the type and complexity of the structures, and the closeness with which it conforms to simplifying assumptions used in the previous model testing (on which all of the methods above are inherently based). In selecting which method to use, or which set of results to prefer when using more than one method, the user will need to take account of the origins of each method. It may also be important in some circumstances to use an alternative method to give a check on a particular set of calculations. To assist these judgments, a set of simple rules of thumb are given here, but as ever, these should not be treated as universal truths. • For simple vertical, composite, or battered walls which conform closely to the idealizations in Chap. 16 on vertical walls, the results of the Empirical Calculator are likely to be more reliable than the other methods as test data for these structure types do not feature strongly in the Database or NN, and pcovertopping is not applicable. • For simple sloped dikes with a single roughness, many test data have been used to develop the formulae in the Empirical Calculator; so, this may be the most reliable, and simplest to use/check. For dikes with multiple slopes or roughness, pc-overtopping is likely to be the most reliable, and easiest to use, although independent checking may be more complicated. The Database or NN methods may become more reliable where the structure starts to include further elements. • For armoured slopes and mounds, open mound structures that most closely conform to the simplifying models may best be described by the formulae in the Empirical Calculator. Structures of lower permeability may be modeled using pcovertopping. Mounds and slopes with crown walls may be best represented by the application of the Database or NN methods.

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• For unusual or complex structures with multiple elements, mean overtopping discharge may be most reliably predicted by pc-overtopping (if applicable) or by the Database or NN methods. • For structures that require use of the NN method, it is possible that the use of many data for other conﬁgurations to develop a single NN method may introduce some averaging. It may therefore be appropriate to check in the database to see whether there are already test data close to the conﬁguration being considered. This procedure may require some familiarity with manipulating these data. In almost all instances, the use of any of these methods will involve some degree of simpliﬁcation of the true situation. The further the structure or design (analysis) conditions depart from the idealized conﬁgurations tested to generate the methods/ tools discussed, the wider will be the uncertainties. Where the importance of the assets being defended is high, and/or the uncertainties in using these methods are large, then the design solution may require use of site-speciﬁc physical model tests.

14.9. Conclusions and Outlook It is clear that increased attention to ﬂood risk reduction, and to wave overtopping in particular, have increased interest and research in this area. The EurOtop Overtopping Manual5 is therefore not expected to be the “last word” on the subject; indeed even whilst preparing the ﬁrst version of the manual, the author team expected that there will be later revisions. The reader of this handbook is therefore advised to check whether an improved version of the EurOtop Overtopping Manual has been released. Beyond that manual, it is probable that there will be significant improvements in numerical modeling, although it should be acknowledged that improved numerical models will require substantial measurement data to validate them before their results can be relied upon in detailed analysis or design.

Acknowledgments This chapter, and also Chaps. 15 and 16, are based on the EurOtop Overtopping Manual,5 which was funded in the the United Kingdom by the Environmental Agency, in Germany by the German Coastal Engineering Research Council (KFKI), in the Netherlands by Rijkswaterstaat and Netherlands Expertise Network (ENW) on Flood Protection. The Project Team for the creation, editing, and support of the manual was formed by Prof William Allsop (HRW) Dr Tom Bruce (UoE) Dr Andreas Kortenhaus (LWI) Dr Jentsje van der Meer (Infram) Dr Tim Pullen (HRW) Dr Holger Sch¨ uttrumpf (BAW)

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The project was guided and supervised by a Project Steering Group: Dr Chrissy Mitchell (EA) Dr Daniel Schade (IM Gmbh) Dick Thomas (Faber Maunsell) Hans van der Sande (Waterboard Zeeuwse Eilanden) Mark Klein Breteler (Delft Hydraulics) Dr Mervyn Bramley (EA Advisor) Michael Owen (EA Advisor) The EurOtop authors are grateful to a wide group of contributors including (but not limited) John Alderson, Phillip Besley,2 Laurence Banyard, Karl-Friedrich Daemrich, Leopoldo Franco, Daan Heineke, Hocine Oumeraci, Jon Pearson, Thorsten Piontkowitz, and Piebe van den Berg. References References in this chapter have been kept to a real minimum. An extensive list of relevant references, however, can be found in the EurOtop Overtopping Manual.5 1. N. W. H. Allsop, P. Besley and L. Madurini, Overtopping performance of vertical and composite breakwaters, seawalls and low reflection alternatives, Paper 4.7 in MCS Project Final Report, University of Hannover (1995). 2. P. Besley, Overtopping of Seawalls — Design and Assessment Manual, R & D Technical Report W 178, Environment Agency, Bristol, ISBN 1 85705 069 X (1999). 3. CLASH, Crest level assessment of coastal structures by full scale monitoring, neural network prediction and hazard analysis on permissible wave overtopping, Fifth Framework Program of the EU, Contract no. EVK3-CT-2001-00058. www.clash-eu.org. 4. DELOS, Environmental design of low crested coastal defence structures, Fifth Framework Program of the EU, Contract no. EVK3-CT-00041. www.delos.unibo.it. 5. EurOtop Overtopping Manual, Wave Overtopping of Sea Defences and Related Structures — Assessment Manual, eds. T. Pullen, N. W. H. Allsop, T. Bruce, A. Kortenhaus, H. Sch¨ uttrumpf and J. W. van der Meer, www.overtopping-manual.com. 6. L. Franco, M. de Gerloni and J. W. van der Meer, Wave overtopping on vertical and composite breakwaters, Proc. 24th Int. Conf. Coastal Eng. Kobe (1994), pp. 1030–1044. 7. Y. Goda, Random Seas and Design of Maritime Structures, 2nd edn. (World Scientific Publishing, Singapore, 2000), ISBN 981-02-3256-X. 8. TAW, Technical Report — Wave run-up and wave overtopping at dikes, J. W. van der Meer (author) Technical Advisory Committee for Flood Defence in the Netherlands (TAW), Delft (2002). 9. H. Verhaeghe, Neural network prediction of wave overtopping at coastal structures, PhD. thesis, University Gent, Belgium, ISBN 90-8578-018-7 (2005).

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Chapter 15

Wave Run-Up and Wave Overtopping at Armored Rubble Slopes and Mounds Holger Sch¨ uttrumpf Institute of Hydraulic Engineering and Water Resources Management RWTH-Aachen University, 52065 Aachen, Germany [email protected] Jentsje van der Meer Van der Meer Consulting P. O. Box 423, 8440, AK Heerenveen, The Netherlands [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics Technical University of Braunschweig Beethovenstr 51a, 38106, Braunschweig, Germany [email protected] Tom Bruce School of Engineering, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JL, UK [email protected] Leopoldo Franco Department of Civil Engineering, University of Roma Tre Via V. Volterra 62, 00146 Roma, Italy [email protected]

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Wave overtopping and to a lesser extent wave run-up for armored rubble slopes and mounds have been subject to a number of investigations in the past. The objective of the present chapter is to summarize existing information to be present as a closed guidance on the use of wave run-up and wave overtopping formulae for a wide range of possible applications in practice. Therefore, guidance is given ﬁrst on the use of wave run-up and wave overtopping formulae for simple slopes, excluding the eﬀects of composite slopes, direction of wave attack, roughness, wave walls, etc. Then, formulae are presented to include these parameters in the calculation procedure. Guidance is also given on wave overtopping volumes, overtopping velocities, and the spatial distribution as well as for wave overtopping for shingle beaches. Finally, the eﬀect of model and scale eﬀects on the calculation of average overtopping rates are discussed. This chapter has mainly been composed from Chap. 6 of the EurOtop Overtopping Manual (2007), with some additions from Chap. 5. The present chapter is related to the previous Chap. 14 and the next Chap. 16 of this manual.

15.1. Introduction Armored rubble slopes and mounds are characterized by a mound with some porosity or permeability, covered by a sloping porous armor layer consisting of large rock or concrete units. In contrast to dikes and embankment seawalls, the porosity of the structure and armor layer plays a role in wave run-up and overtopping. The cross section of a rubble mound slope, however, may have great similarities with an embankment seawall and may consist of various slopes. As an example for armored slopes and mounds, a rock-armored embankment is given in Fig. 15.1.

Fig. 15.1.

1:4 rock-armored embankment.

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15.2. Wave Run-Up and Run-Down Levels, Number of Overtopping Waves 15.2.1. Introduction to wave run-up The wave run-up height is deﬁned as the vertical diﬀerence between the highest point of wave run-up and the still water level (SWL) (Fig. 15.2). Due to the stochastic nature of the incoming waves, each wave will give a diﬀerent run-up level. In many design formulae, the wave run-up height Ru2% is used as a characteristic parameter to describe wave run-up. This is the wave run-up height, which is exceeded by 2% of the number of incoming waves at the toe of the structure. The idea behind this was that if only 2% of the waves reach the crest of a coastal structure, the crest and inner slope do not need speciﬁc protection measures. It is for this reason that much research in the past has been focused on the 2%-wave run-up height. In the past decade the design or safety assessment has been changed to allowable overtopping instead of wave run-up. Wave run-up has always been less important for rock-armored slopes and rubble mound structures, and the crest height of these types of structures has mostly been based on allowable overtopping, or even on allowable transmission (low-crested structures). Still an estimation or prediction of wave run-up is valuable as it gives a prediction of the number or percentage of waves which will reach the crest of the structure and eventually give wave overtopping. And this number is needed for a good prediction of individual overtopping volumes per wave, overtopping velocities, and ﬂow depths. The general formula that can be applied for the 2% mean wave run-up height is given by Eq. (15.1): The relative wave run-up height Ru,2% /Hm0 in Eq. (15.1) is related to the breaker parameter ξm−1,0 .

Ru2% = wave run-up height RC = freeboard Hm0 = wave height at the toe of the structure h = water depth at the toe of the structure α = seaward slope steepness Ru2%

RC

Hm0

SWL

α h

Fig. 15.2.

Deﬁnition of the wave run-up height Ru2% on a smooth impermeable slope.

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Ru2% = 1.65 · γb · γf · γβ · ξm−1,0 Hm0 , = 1.00 · γb · γf · γβ 4.0 − √ 1.5 with a maximum of RHu2% m0

(15.1)

ξm−1,0

where Ru2% Hm0 γb γf γβ

= = = = =

wave run-up height exceeded by 2% of the incoming waves [m]; signiﬁcant wave at the toe of the structure [m]; inﬂuence factor for a berm [see 15.3.4(b)] [–]; inﬂuence factor for roughness on the slope [–]; inﬂuence factor for oblique wave attack (see 15.3.3) [–]; √ 2 ξm−1,0 = breaker parameter = tan α/ sm−1,0 = tan α 2πHm0 /(gTm−1,0 ) [–]; tanα = average slope angle (see Fig. 15.2) [–]; Tm−1,0 = spectral moment at the toe of the structure, based on m−1 and m0 [s]. The breaker or surf similarity parameter ξm−1,0 relates the slope steepness tan α 2 (or 1/n) to the ﬁctitious wave steepness sm−1,0 = 2πHm0 /(gTm−1,0 ) and is often used to distinguish diﬀerent breaker types. The combination of structure slope and wave steepness gives a certain type of wave breaking (Fig. 15.3). For ξm−1,0 > 2–3 waves are considered not to be breaking (surging waves), although there may still be some breaking, and for ξm−1,0 < 2–3 waves are breaking. Waves on a gentle foreshore break as spilling waves and more than one breaker line can be found on such a foreshore. Plunging waves break with steep and overhanging fronts and the wave tongue will hit the structure or back washing water. The transition between plunging waves and surging waves is known as collapsing. The wavefront becomes almost vertical, and the water excursion on the slope (wave run-up + run-down) is often largest for this kind of breaking. Values are given for the majority of the larger waves in a sea state. Individual waves may still surge for generally plunging conditions or plunge for generally surging conditions. The relative wave run-up height increases linearly with increasing ξm−1,0 in the range of breaking waves and small breaker parameters less than about 2. For nonbreaking waves and higher breaker parameters, the increase is less and

Fig. 15.3.

Type of breaking on a slope.

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becomes more or less horizontal. The relative wave run-up height Ru,2% /Hm0 is also inﬂuenced by the geometry of the structure, the eﬀect of wind, and the properties of the incoming waves.

15.2.2. Wave run-up on rock-armored slopes Figure 15.4 gives 2% wave run-up heights for various rocks slopes with cot α = 1.5, 2, 3, and 4, and for an impermeable and permeable core of the rubble mound. These run-up measurements were performed during the stability tests on rock slopes of van der Meer.11 First of all, the graph gives values for a large range of the breaker parameter ξm−1,0 , due to the fact that various slope angles were tested, but also with long wave periods (giving large ξm−1,0 -values). Most breakwaters have steep slopes 1:1.5 or 1:2 only and then the range of breaker parameters is often limited to ξm−1,0 = 2–4. The graph gives rock slope information outside this range, which may be useful also for slopes with concrete armor units. The highest curve in Fig. 15.4 gives the prediction for smooth straight slopes (γf = 1). A rubble mound slope dissipates signiﬁcantly more wave energy than an equivalent smooth and impermeable slope. Not only both the roughness and porosity of the armor layer cause this eﬀect, but also the permeability of the under-layer and core contribute to it. Figure 15.4 shows the data for an impermeable core (geotextile on sand or clay underneath a thin under-layer) and for a permeable core (such as most breakwaters). The diﬀerence is most signiﬁcant for large breaker parameters. Equation (15.2) includes the inﬂuence factor for roughness γf . For two layers of rock on an impermeable core, γf = 0.55. This reduces to γf = 0.40 for two layers of rock on a permeable core. This inﬂuence factor is used in the linear part of the runup formula, say, for ξm−1,0 ≤ 1.8. From ξm−1,0 = 1.8, the roughness factor increases linearly up to 1 for ξm−1,0 = 10, and it remains 1 for larger values.

4.0 3.5

Ru2%/Hm0

3.0 2.5 2.0 1.5 1.0

imp. cota=2 imp. cota=3 imp. cota=4

0.5

perm. cota=1.5 perm. cota=2 perm. cota=3

0.0 0

1

2

3

4

5

6

7

8

9

10

Spectral breaker parameter ξ m−1, 0 Fig. 15.4. Relative run-up on straight rock slopes with permeable and impermeable core, compared to smooth impermeable slopes.

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The prediction for the 2% mean wave run-up value for rock or rough slopes can be described by Ru2% = 1.65 · γb · γf · γβ · ξm−1,0 , Hm0 with a maximum of

Ru2% Hm0

= 1.00 · γb · γf surging · γβ 4.0 − √

(15.2) 1.5 ξm−1,0

.

From ξm−1,0 = 1.8, the roughness factor γf surging increases linearly up to 1 for ξm−1,0 = 10, which can be described by γf surging = γf + (ξm−1,0 − 1.8) · (1 − γf )/8.2 γf surging = 1.0 for ξm−1,0 > 10. For a permeable core, however, a maximum is reached for Ru2% /Hm0 = 1.97. The physical explanation for this is that if the slope becomes very steep (large ξm−1,0 -value) and the core is impermeable, the surging waves slowly run up and down the slope, and all the water stays in the armor layer, leading to a fairly high run-up. The surging wave actually does not “feel” the roughness anymore and acts as a wave on a very steep smooth slope. For a permeable core, however, the water can penetrate into the core which decreases the actual run-up to a constant maximum (the horizontal line in Fig. 15.4). Equation (15.2) may also give a good prediction for run-up on slopes armored with concrete armor units, if the right roughness factor is applied. Deterministic design or safety assessment. For design or a safety assessment of the crest height, it is advised not to follow the average trend, but to include the uncertainty of the prediction. As the basic equation is similar for a smooth and a rough slope, the method to include uncertainty is also the same. This means that for a deterministic design or safety assessment, Eq. (15.3) should be used: Ru2% = 1.75 · γb · γf · γβ · ξm−1,0 , Hm0 with a maximum of From ξm−1,0 for ξm−1,0 γf surging γf surging

= = = =

Ru2% Hm0

= 1.00 · γb · γf surging · γβ 4.3 − √

(15.3) 1.6 ξm−1,0

.

1.8, the roughness factor γf surging increases linearly up to 1 10, which can be described by γf + (ξm−1,0 − 1.8) · (1 − γf )/8.2 1.0 for ξm−1,0 > 10.

For a permeable core a maximum is reached for Ru2% /Hm0 = 2.11. Probabilistic design. For probabilistic calculations, Eq. (15.2) should be used together with a normal distribution and variation coeﬃcient of CoV = 0.07. For prediction or comparison of measurements, the same Eq. (15.2) is used, but now for instance with the 5% lower and upper exceedance lines.

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15.2.3. Number of overtopping waves or overtopping percentage Till now only the 2% run-up value has been described. It might be that one is interested in another percentage, for example, on the design of breakwaters where the crest height may be determined by an allowable percentage of overtopping waves, say, 10–15%. A few ways exist to calculate run-up heights for other percentages, or to calculate the number of overtopping waves for a given crest height. van der Meer and Stam10 give two methods. One is an equation like Eq. (15.2) with a table of coeﬃcients for the 0.1%, 1%, 2%, 5%, 10%, and 50% (median). Interpolation is needed for other percentages. The second method gives a formula for the run-up distribution as a function of wave conditions, slope angle, and permeability of the structure. The distribution is a two-parameter Weibull distribution. With this method, the run-up can be calculated for every percentage required. Both methods apply to straight rock slopes only and will not be described here. The given references, however, give all the details. The easiest way to calculate run-up (or overtopping percentage) diﬀerent from 2% is to take the 2%-value and assume a Rayleigh distribution. The probability of overtopping Pov = Now /Nw (the percentage is simply 100 times larger) can be calculated by 2 √ Rc − ln 0.02 , (15.4) Pov = Now /Nw = exp − Ru,2% where Pov Now Nw Rc

= = = =

probability of overtopping [–]; number of overtopping waves in a sea state [–]; number of waves in a sea state [–]; crest freeboard [m].

Equation (15.4) can be used to calculate the probability of overtopping, given a crest freeboard Rc or to calculate the required crest freeboard, given an allowable probability or percentage of overtopping waves. One warning should be given in applying Eqs. (15.2)–(15.4). The equations give the run-up level in percentage or height on a straight (rock-armored) slope. This is not the same as the number of overtopping waves or overtopping percentage. Figure 15.5 gives the diﬀerence. The run-up is always a point on a straight slope, where for a rock-armored slope or mound the overtopping is measured some distance away from the seaward slope and on the crest, often behind a crown wall. This means that Eqs. (15.2)–(15.4) always give an overestimation of the number of overtopping waves. Figure 15.6 shows measured data for rubble mound breakwaters armored with Tetrapods, AccropodeTM , or a single layer of cubes. All tests were performed at Delft Hydraulics. The test setup was more or less similar to Fig. 15.4 with a crown wall height Rc a little lower than the armor freeboard Ac . CLASH-data on speciﬁc overtopping tests for various rock and concrete armored slopes were added to Fig. 15.6. This ﬁgure gives only the percentage of overtopping waves passing the crown wall.

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run-up level (eq. 2 and 3) calculated here

CREST Gc

1.5 Hm0toe

Ac

overtopping measured behind wall

Rc

swl 1.5 Hm0toe

Fig. 15.5.

Run-up level and location for overtopping diﬀer.

Percentage of overtopping waves (%)

100 90

tetrapod DH

80

accropode DH 1 layer cube DH

70

CLASH 60 50 40 30 20 10 0 -0.2

0

0.2

0.4

0.6

0.8

1

2

Dimensionless crest height Ac*Dn/Hm0

Fig. 15.6. Percentage of overtopping waves for rubble mound breakwaters as a function of relative (armor) crest height and armor size (Rc ≤ Ac ).

Analysis showed that the size of the armor unit relative to the wave height had 2 inﬂuence, which gave a combined parameter Ac · Dn /Hm0 , where Dn is the nominal diameter of the armor unit. The ﬁgure covers the whole range of overtopping percentages, from complete overtopping with the crest at or lower than SWL to no overtopping at all. The CLASH-data give maximum overtopping percentages of about 30%. Larger

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percentages mean that overtopping is so large that it can hardly be measured and that wave transmission starts to play a role. Taking 100% overtopping for zero freeboard (the actual data are only a little lower), a Weibull curve can be ﬁtted through the data. Equation (15.5) can be used to predict the number or percentage of overtopping waves or to establish the armor crest level for an allowable percentage of overtopping waves. Pov

1.4 Ac Dn . = Now /Nw = exp − 2 0.19Hm0

(15.5)

It is clear that Eqs. (15.2)–(15.4) will come to more overtopping waves than Eq. (15.5). But both estimations together give a designer enough information to establish the required crest height of a structure given an allowable overtopping percentage. 15.2.4. Wave run-down on rock-armored slopes When a wave on a structure has reached its highest point, it will run down on the slope till the next wave meets this water and run-up starts again. The lowest point to where the water retreats, measured vertically to SWL, is called the run-down level. Run-down often is less or not important compared to wave run-up, but both together they may give an idea of the total water excursion on the slope. Therefore, only a ﬁrst estimate of run-down on straight rock slopes is given here, based on the same tests of van der Meer,11 but re-analyzed with respect to the use of the spectral wave period Tm−1,0 . Figure 15.7 gives an overall view. Breaker parameter ξ m- 1,0 0

1

2

3

4

5

6

7

8

9

Relative run-down R d2%/Hm0

0

0.5

1

1.5

2

imp; cota=2 perm; cota=1.5 hom; cota=2

imp; cota=3 perm; cota=2 imp; Deltaflume

imp; cota=4 perm; cota=3 perm; Deltaflume

2.5

Fig. 15.7. Relative 2% run-down on straight rock slopes with impermeable core (imp), permeable core (perm), and homogeneous structure (hom).

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The graph shows clearly the inﬂuence of the permeability of the structure as the solid data points (impermeable core) generally show larger run-down than the open data symbols of the permeable core. Furthermore, the breaker parameter ξm−1,0 gives a fairly clear trend of run-down for various slope angles and wave periods. Figure 15.7 can be used directly for design purposes, as it also gives a good idea of the scatter.

15.3. Overtopping Discharges 15.3.1. Simple armored slopes The mean overtopping discharge is often used to judge allowable overtopping. It is easy to measure, and an extensive database on mean overtopping discharge has been gathered in CLASH. This mean discharge does not of course describe the real behavior of wave overtopping, where only large waves will reach the top of the structure and give overtopping. Random individual wave overtopping is random in time, and each wave gives a diﬀerent overtopping volume. But the description of individual overtopping is based on the mean overtopping, as the duration of overtopping multiplied with this mean overtopping discharge gives the total volume of water overtopped by a certain number of overtopping waves. The mean overtopping discharge has been described in this section. The individual overtopping volumes is the subject in Sec. 15.4.1. Wave overtopping occurs if the crest level of the coastal structures is lower than the highest wave run-up level. In that case, the freeboard Rc deﬁned as the vertical diﬀerence between the SWL and the crest height becomes important (Fig. 15.2). Wave overtopping depends on the freeboard Rc and increases for decreasing freeboard height Rc . Usually, wave overtopping for rubble slopes and mounds is described by an average wave overtopping discharge q, which is given in m3 /s per m width, or in l/s per m width. An average overtopping discharge q can only be calculated for quasi-stationary wave and water level conditions, a so-called sea state. If the amount of water overtopping for a structure during a storm is required, the average overtopping discharge has to be calculated for each more or less constant storm water level and constant wave conditions. Many model studies were performed to investigate the average overtopping discharge for speciﬁc dike geometries or wave conditions. For practical purposes, empirical formulae were ﬁtted through experimental model data which obey often one of the following expressions: Q∗ = Q0 (1 − R∗ )b

or Q∗ = Q0 exp (−b · R∗ ) .

(15.6)

Q∗ is a dimensionless overtopping discharge, R∗ is a dimensionless freeboard height, Q0 describes wave overtopping for zero freeboard, and b is a coeﬃcient which describes the speciﬁc behavior of wave overtopping for a certain structure. Sch¨ uttrumpf 8 summarized expressions for the dimensionless overtopping discharge ∗ Q and the dimensionless freeboard height R∗ .

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3 The dimensionless overtopping discharge Q∗ = q/ gHm0 is a function of the wave height, originally derived from the Weir formula. Deterministic design or safety assessment. The equation, including a standard deviation of safety, should be used for deterministic design or safety assessment: 0.067 q γb · ξm−1,0 =√ 3 tan α g · Hm0 Rc · exp −4.3 ξm−1,0 · Hm0 · γb · γf · γβ · γv with a maximum of √

q 3 g·Hm0

(15.7)

= 0.2 · exp −2.3 Hm0R·γcf ·γβ ,

where γv = the inﬂuence of a small wall on top of the embankment. Probabilistic design. The mean prediction should be used for probabilistic design, or prediction of or comparison with measurements. This equation is given by

q 3 g · Hm0

with a maximum of √

0.067 γb · ξm−1,0 = √ tan α Rc · exp −4.75 ξm−1,0 · Hm0 · γb · γf · γβ · γv

q 3 g·Hm0

(15.8)

= 0.2 · exp −2.6 Hm0R·γcf ·γβ .

The reliability of Eq. (15.8) is described by taking the coeﬃcients 4.75 and 2.6 as normally distributed stochastic parameters with means of 4.75 and 2.6 and standard deviations σ = 0.5 and 0.35, respectively. For probabilistic calculations, Eq. (15.8) should be taken together with these stochastic coeﬃcients. For predictions of measurements or comparison with measurements also Eq. (15.8) should be taken with, for instance, 5% upper and lower exceedance curves. It has to be mentioned that the ﬁrst part of Eqs. (15.7) and (15.8) is valid for mostly breaking waves. Considering the steep slopes of armored rubble slope and mounds this part has less importance in practice than the second part of the equation, describing the maximum of overtopping. In that case, the relative freeboard does not depend on the breaker parameter ξm−1,0 for nonbreaking waves (Fig. 15.8), as the line is horizontal. This means that a composite slope and even a, not too long, berm leads to the same overtopping discharge as for a simple straight rubble mound slope. Only when the average slope becomes so gentle that the maximum part in Eqs. (15.7) and (15.8) do not apply anymore, then a berm and a composite slope will have eﬀect on the overtopping discharge. Generally, average slopes around 1:2 or steeper do not show inﬂuence of the slope angle, or only to a limited extent, and the maximum part in Eqs. (15.7) and (15.8) are valid.

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.

relative overtopping discharge q/(gHm0 )

3 1/2

[-]

0,016 1:3 slope

0,014

1:4 slope 1:6 slope

0,012 0,01 0,008 0,006 0,004 0,002 0 0

0,01

0,02

0,03

0,04

0,05

0,06

2

wave steepness 2πHm0/(gTm-1,0 ) [-]

Fig. 15.8. Wave overtopping as a function of the ﬁctitious wave steepness sm−1,0 = 2πHm0 / (gTm−1,0 2 ) and a smooth slope.

As part of the EU research program CLASH2 tests were undertaken to derive roughness factors for rock-armored slopes and diﬀerent armor units on sloping permeable structures. Overtopping was measured for a 1:1.5 sloping permeable structure at a reference point 3Dn50 from the crest edge, where Dn50 is the nominal diameter. The wave wall had the same height as the armor crest, so Rc = Ac . As discussed in Sec. 15.2 and Fig. 15.5, the point to where run-up can be measured and the location of overtopping may diﬀer. Normally, a rubble mound structure has a crest width of at least 3Dn50 . Waves rushing up the slope reach the crest with an upward velocity. For this reason, it is assumed that overtopping waves reaching the crest will also reach the location 3Dn50 further. Results of the CLASH-work are shown in Fig. 15.9 and Table 15.1. Figure 15.9 gives all data together in one graph. Two lines are given, one for a smooth slope, Eq. (15.8) with γf = 1.0, and one for rubble mound 1:1.5 slopes, with the same equation, but with γf = 0.45. The lower line only gives a kind of average, but shows clearly the very large inﬂuence of roughness and permeability on wave overtopping. The required crest height for a steep rubble mound structure is at least half of that for a steep smooth structure, for similar overtopping discharge. It is also for this reason that smooth slopes are often more gentle in order to reduce the crest heights. TM R R , Xbloc , and In Fig. 15.9, one-layer systems, like Accropode , CORE-LOC one layer of cubes, have solid symbols. Two-layer systems have been given by open symbols. There is a slight tendency that one-layer systems give a little more overtopping than two-layer systems, which is also clear from Table 15.1. Equation (15.8) can be used with the roughness factors in Table 15.1 for the prediction of mean overtopping discharges for rubble mound breakwaters. Values in italics in Table 15.1 have been estimated/extrapolated, based on the CLASH results.

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Wave overtopping q/(gHm0 )

3 0.5

1.E+00 Smooth Cube rough Anitfer Tetrapod Accropde Xbloc rough gf=0.45

1.E-01

1.E-02

Rock Cube flat Haro Cube-1 layer Coreloc smooth gf=1.0

1.E-03

Eq. 8

f

= 1.0

1.E-04

1.E-05

Eq. 8

f

= 0.45

1.E-06

1.E-07

0

0.5

1

1.5

2

2.5

3

3.5

4

Relative crest height Rc/Hm 0

Fig. 15.9.

Mean overtopping discharge for 1:1.5 smooth and rubble mound slopes. Table 15.1. Values for roughness factors γf for permeable rubble mound structures with slope of 1:1.5. Values in italics are estimated/extrapolated. Type of armor layer

γf

Smooth impermeable surface Rocks (one layer, impermeable core) Rocks (one layer, permeable core) Rocks (two layers, impermeable core) Rocks (two layers, permeable core) Cubes (one layer, random positioning) Cubes (two layers, random positioning) Antifers HARO’s AccropodeTM R Xbloc R CORE-LOC Tetrapods Dolosse

1.00 0.60 0.45 0.55 0.40 0.50 0.47 0.47 0.47 0.46 0.45 0.44 0.38 0.43

15.3.2. Eﬀect of armored crest berm Wave overtopping on simple straight slope include an armored crest berm up to about three nominal diameters. It is clear, however, that wide crests will certainly decrease the overtopping as much more energy will be dissipated in a wider crest. The crest width can be described by Gc (see Fig. 15.5). The EA-Manual (Besley1 ) describes in a simple and eﬀective way the inﬂuence of a wide crest. First, the wave overtopping discharge should be calculated for a simple slope, with a crest width up to 3Dn50 . Then, the following reduction factor on the overtopping discharge can

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be applied: Cr = 3.06 exp(−1.5Gc /Hm0 ),

with maximum Cr = 1.

(15.9)

Equation (15.9) gives no reduction for a crest width smaller than about 0.75 Hm0 . This is fairly close to about 3Dn50 and is, therefore, consistent. A crest width of 1Hm0 reduces the overtopping discharge to 68%, a crest width of 2Hm0 gives a reduction to 15%, and for a wide crest of 3Hm0 , the overtopping reduces to only 3.4%. In all cases, the crest wall has the same height as the armor crest: Rc = Ac . Equation (15.9) was determined for a rock-armored slope and can be considered as conservative, as for a slope with Accropode, more reduction was found. 15.3.3. Eﬀect of oblique waves In the CLASH-project, speciﬁc tests on a rubble mound breakwater were performed with a slope of 1:2 and armored with rock or cubes7 to investigate the eﬀect of oblique waves on wave overtopping. The structure was tested both with long-crested and short-crested waves, but only the results by short-crested waves are given. Results for the eﬀect of oblique waves on smooth slopes, dikes, or embankments are given in the EurOtop Overtopping Manual,4 and in the TAW-report.9 Here, only the results for armored rubble mound slopes will be discussed. For oblique waves, the angle of wave attack β (deg.) is deﬁned as the angle between the direction of propagation of waves and the axis perpendicular to the structure (for perpendicular wave attack, β = 0◦ ). And the direction of wave attack is the angle after any change of direction of the waves on the foreshore due to refraction. Just like for smooth slopes, the inﬂuence of the angle of wave attack is described by the inﬂuence factor γβ . Just as for smooth slopes, there is a linear relationship between the inﬂuence factor and the angle of wave attack, but the reduction in overtopping for rock slopes is faster with increasing angle: γβ = 1 − 0.0063|β|

for 0◦ ≤ |β| ≤ 80◦ for |β| > 80◦ , the result of β = 80◦ can be applied.

(15.10)

The wave height and period are linearly reduced to 0 for 80◦ ≤ |β| ≤ 110◦. For |β| > 110◦ , the wave overtopping is set to q = 0. 15.3.4. Composite slopes and berms, including berm breakwaters In every formula where a cot α or breaker parameter ξm−1,0 is present, a procedure has to be described how a composite slope has to be taken into account. Hardly any speciﬁc research exists for rubble mound structures, and, therefore, the procedure for composite slopes at sloping impermeable structures like dikes and sloping seawalls is assumed to be applicable. (a) Average slopes. A characteristic slope is required to be used in the breaker parameter ξm−1,0 for composite proﬁles or bermed proﬁles to calculate wave runup or wave overtopping. Theoretically, the run-up process is inﬂuenced by a change

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of slope from the breaking point to the maximum wave run-up height. Therefore, often it has been recommended to calculate the characteristic slope from the point of wave breaking to the maximum wave run-up height. This approach needs some calculation eﬀort, because of the iterative solution since the wave run-up height Ru2% is unknown. For the breaking limit, a point on the slope can be chosen which is 1.5Hm0 below the still water line. It is recommended to use also a point on the slope 1.5Hm0 above water as a ﬁrst estimate to calculate the characteristic slope and to exclude a berm (Fig. 15.10). First estimate: tan α =

3 · Hm0 . LSlope − B

(15.11)

As a second estimate, the wave run-up height from the ﬁrst estimate is used to calculate the average slope [LSlope has to be adapted (see Fig. 15.11)]:

1.5 · Hm0 + Ru2%(from 1st estimate) Second estimate: tan α = . (15.12) LSlope − B 1st estimate:

tan α =

3H m0 LSlope − B

B Average slope

SWL

dB 1.5 HS

1.5 HS

B

LSlope

Fig. 15.10.

Determination of the average slope (ﬁrst estimate).

2nd estimate:

tan α =

Ru2% (from 1st estimate)

1.5 H m 0 + Ru 2% ( from 1st estimate) LSlope − B Average slope

SWL

dB 1.5 HS

B

LSlope Fig. 15.11.

Determination of the average slope (second estimate).

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If the run-up height or 1.5Hm0 comes above the crest level, then the crest level must be taken as the characteristic point above SWL. Also, the inﬂuence of a berm in a sloping proﬁle has been adapted from smooth sloping sea dikes for rubble mound structures. There is, however, often a diﬀerence in the eﬀect of composite slopes or berms for rubble mound and smooth gentle slopes. On gentle slopes, the breaker parameter ξm−1,0 has large inﬂuence on wave overtopping (see Eqs. (15.2) and (15.3) as the breaker parameter will be quite small). Rubble mound structures often have a steep slope, leading to the formula for “nonbreaking” waves, the maximum in Eqs. (15.7) and (15.8). In these equations, no slope angle or breaker parameter is present, and the eﬀect of a small berm will be very small and probably negligible. (b) Influence of berms. A berm is a part of a dike or breakwater proﬁle in which the slope varies between horizontal and 1:15. A berm is deﬁned by the width of the berm, B, and by the vertical diﬀerence dB between the middle of the berm and the SWL (Fig. 15.12). The width of the berm, B, may not be greater than 0.25 · Lm−1,0 . If the berm is horizontal, the berm width B is calculated according to Fig. 15.12. The lower and the upper slope are extended to draw a horizontal berm without changing the berm height dB . The horizontal berm width is therefore shorter than the angled berm width. dB is 0 if the berm lies on the still water line. The characteristic parameters of a berm are deﬁned in Fig. 15.12.

(a) Calculation of width B and height db of berm Bhorizontal

SWL

db Bangled

(b) Calculation of berm length Lberm

1.0 H S SWL

db

1.0 HS B LBerm

Fig. 15.12.

Determination of the characteristic berm length LBerm .

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A berm reduces wave run-up or wave overtopping. The inﬂuence factor γb for a berm consists of two parts. γb = 1 − rB (1 − rdb )

for 0.6 ≤ γb ≤ 1.0.

(15.13)

The ﬁrst part (rB ) stands for the width of the berm LBerm and becomes 0 if no berm is present. rB =

B LBerm

.

(15.14)

The second part (rdb ) stands for the vertical diﬀerence dB between the SWL and the middle of the berm and becomes 0 if the berm lies on the still water line. The reduction of wave run-up or wave overtopping is maximum for a berm on the still water line and decreases with increasing dB . Thus, a berm lying on the still water line is most eﬀective. A berm lying below 2 · Hm0 or above Ru2% has no inﬂuence on wave run-up and wave overtopping. Diﬀerent expressions are used for rdB in Europe. Here, an expression using a cosine-function for rdb (Fig. 15.13) is recommended which is also used in pcovertopping (see Chap. 14). db rdb = 0.5 − 0.5 cos π for a berm above SWL , Ru2% db for a berm below SWL , (15.15) rdb = 0.5 − 0.5 cos π 2 · Hm0 rdb = 1 for berms lying outside the area of inﬂuence . Berm lying at Ru,2%

Berm lying at SWL Berm lying above SWL

Berm lying . at 2 Hm0 below SWL Berm lying below SWL

1 0.9 0.8 0.7 rdb [-]

0.6 0.5 0.4 0.3 0.2 0.1 0 -1

-0.5 db/Ru2% [-]

Fig. 15.13.

0

0.5 . db/(2 Hm0) [-]

Inﬂuence of the berm depth on factor rdh .

1

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The maximum inﬂuence of a berm is actually always limited to γB = 0.6. This corresponds to an optimal berm width B on the SWL of B = 0.4 · LBerm . The deﬁnition of a berm is made for a slope smoother than 1:15, while the deﬁnition of a slope is made for slopes steeper than 1:8. If a slope or a part of the slope lies in between 1:8 and 1:15, it is required to interpolate between a bermed proﬁle and a straight proﬁle. For wave run-up, this interpolation is written by

(1/8 − tan α) Ru2% = Ru2%(1:8slope) + Ru2%(Berm) − Ru2%(1:8slope) · . (1/8 − 1/15)

(15.16)

A similar interpolation procedure should be followed for wave overtopping. 15.3.5. Wave overtopping on a berm breakwater A speciﬁc type of rubble mound structure is the berm breakwater. The original idea behind the berm breakwater is that a large berm, consisting of fairly large rock, is constructed into the sea with a steep seaward face. The berm height is higher than the minimum required for construction with land-based equipment. Due to the steep seaward face the ﬁrst storms will reshape the berm and ﬁnally a structure will be present with a fully reshaped S-proﬁle. Such a proﬁle has then a gentle 1:4 or 1:5 slope just below the water level and steep upper and lower slopes (see Fig. 15.14). The idea of the reshaping berm breakwater has evolved in Iceland to a more or less nonreshaping berm breakwater (Fig. 15.15). The main diﬀerence is that during rock production from the quarry, care is taken to gather a few percent of really big rocks. Only a few percent is required to strengthen the corner of the berm and part of the down slope and upper layer of the berm in such a way that reshaping will hardly occur. An example with various rock classes (class I being the largest) is given in Fig. 15.16. Therefore, distinction has been made between conventional reshaping berm breakwaters and the nonreshaping Icelandic type berm breakwater. In order to calculate wave overtopping on reshaped berm breakwaters, the reshaped proﬁle should be known. The basic method of proﬁle reshaping is given in van der Meer,11 and the program breakwat (WL | Delft Hydraulics) is able to calculate the proﬁle. The ﬁrst method described here to calculate wave overtopping at reshaping berm breakwaters is by using Eqs. (15.7) or (15.8) which have been developed for smooth slopes. Equations (15.7) and (15.8) include the eﬀect of an average slope with the roughness factors given in Table 15.1 of γf = 0.40 for

Fig. 15.14.

Conventional reshaping berm breakwater.

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Fig. 15.15.

Fig. 15.16.

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Icelandic berm breakwater.

Nonreshaping Icelandic berm breakwater with various classes of big rocks.

reshaping berm breakwaters and γf = 0.35 for nonreshaping Icelandic berm breakwaters. The method of composite slopes and berms should be applied as described above. The second method is to use the CLASH neural network. As overtopping research at that time on berm breakwaters was limited, this method also gives quite some scatter, but a little less than the ﬁrst method described above. Recent information on berm breakwaters has been described by Andersen.5 Only part of his research was included in the CLASH database and consequently in the Neural Network prediction method. He performed about 600 tests on reshaping berm breakwaters and some 60 on nonreshaping berm breakwaters (ﬁxing the steep slopes by a steel net). The true nonreshaping Icelandic type of berm breakwaters with large rock classes, has not been tested and, therefore, his results might lead to an overestimation. One comment should be made on the application of the results of Andersen.5 3 0.5 The maximum overtopping discharge measured was only q/(gHm0 ) = 10−3. In practical situations with wave heights around 5 m, the overtopping discharge will then be limited to only a few l/s per m width. For berm breakwaters and also for conventional rubble slopes and mounds, allowable overtopping may be much higher than this value.

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The ﬁnal result of the work of Andersen5 is a quite complicated formula, based on multi-parameter ﬁtting. The advantage of such a ﬁtting is that by using a large number of parameters, the data set used will be quite well described by the formula. The disadvantage is that physical understanding of the working of the formula, certainly outside the ranges tested, is limited. But, due to the fact that so many structures were tested, this eﬀect may be neglected. The formula is valid for berm breakwaters with no superstructure and gives the overtopping discharge at the back of the crest (Ac = Rc ). In order to overcome the problem that one encounters when calculating the reshaped proﬁle before any overtopping calculation can be done, the formula is based on the “as built” proﬁle, before reshaping. Instead of calculating the proﬁle, a part of the formula predicts the inﬂuence of waves on recession of the berm. The parameter used is called fHo , which is an indicative measure of the reshaping, and can be deﬁned as a “factor accounting for the inﬂuence of stability numbers.” Note that fHo is a dimensionless factor and not the direct measure of recession and that Ho and To are also dimensionless parameters (see below). ∗ fHo = 19.8s−0.5 om exp(−7.08/Ho ) for To ≥ To ,

fHo = 0.05Ho To + 10.5 for To < To∗ ,

(15.17)

where Ho = Hm0 /∆Dn50 , To = (g/Dn50 )0.5 Tm0,1 , and To∗ = {19.8s−0.5 exp(−7.08/Ho) − 10.5}/(0.05Ho). om The berm level dh is also taken into account as an inﬂuence factor, d∗h . Note that the berm depth is positive if the berm level is below SWL, and therefore, for berm breakwaters often negative. Note also that this inﬂuence factor is diﬀerent from that for a bermed slope. This inﬂuence factor is described by d∗h = (3Hm0 − dh )/(3Hm0 + Rc )

for dh < 3Hm0 ,

d∗h = 0 for dh ≥ 3Hm0 .

(15.18)

The ﬁnal overtopping formula then takes into account the inﬂuence factor on recession, fHo , the inﬂuence factor of the berm level, d∗h , the geometrical parameters Rc , B, and Gc , and the wave conditions Hm0 and the mean period Tm0,1 . It means that the wave overtopping is described by a spectral mean period, and not by Tm−1,0 . 3 1.34 q/(gHm0 )0.5 = 1.79 · 10−5 (fHo + 9.22)s−2.52 ∗, op 1.39 . exp[−5.63(Rc/Hm0 )0.92 − 0.61(Gc /Hm0 )1.39 − 0.55h1.48 b∗ (B/Hm0 )

(15.19)

Equation (15.19) is only valid for a down slope of 1:1.25 and an upper slope of 1:1.25. For other slopes, one has to reshape the slope to a slope of 1:1.25, keeping the volume of material the same and adjusting the berm width B and for the upper slope also the crest width Gc . Note also that in Eq. (15.19), the peak wave period

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Tp has to be used to calculate sop , where the mean period Tm0,1 has to be used in Eq. (15.17). Although no tests were performed on the nonreshaping Icelandic berm breakwaters (see Fig. 15.16), a number of tests were performed on nonreshaping structures by keeping the material in place with a steel net. The diﬀerence may be that Icelandic berm breakwaters show a little less overtopping, due to the presence of larger rocks and, therefore, more permeability. The tests showed that Eq. (15.19) is also valid for nonreshaping berm breakwaters, if the reshaping factor fHo = 0. 15.3.6. Eﬀect of wave walls Most breakwaters have a wave wall, capping wall, or crest unit on the crest, simply to end the armor layer in a good way and to create access to the breakwater. For design, it is advised not to design a wave wall much higher than the armor crest, for the simple reason that wave forces on the wall will increase drastically if directly attacked by waves and not hidden behind the armor crest. For rubble mound slopes as a seawall, design waves might be a little lower than for breakwaters and a wave wall might be one of the solutions to reduce wave overtopping. Nevertheless, one should realize the increase in wave forces in designing a wave wall signiﬁcantly above the armor crest. Equations (15.7) and (15.8) for a simple rubble mound slope include a berm of 3Dn50 wide and a wave wall at the same level as the armor crest: Ac = Rc . A little lower wave wall will hardly give larger overtopping, but no wave wall at all would certainly increase overtopping. Part of the overtopping waves will then penetrate through the crest armor. No formulae are present to cope with such a situation, unless the use of the Neural Network prediction method. Various researchers have investigated wave walls higher than the armor crest. None of them compared their results with a graph like Fig. 15.9 for simple rubble mound slopes. During the writing of the EurOtop Overtopping Manual, 2007, some of the published equations were plotted in Fig. 15.7 and most curves fell within the scatter of the data. Data with a wider crest gave signiﬁcantly lower overtopping, but that was due to the wider crest, not the higher wave wall. In essence, the message is: use the height of the wave wall Rc and not the height of the armored crest Ac in Eqs. (15.7) and (15.8) if the wall is higher than the crest. For a wave wall lower than the crest armor the height of this crest armor should be used. The Neural Network prediction might be able to give more precise predictions. 15.3.7. Scale and model eﬀect corrections Results of the recent CLASH-project suggested signiﬁcant diﬀerences between ﬁeld and model results on wave overtopping. This has been veriﬁed for diﬀerent sloping rubble structures. Results of the comparisons in this project have led to a scaling procedure which is mainly dependent on the roughness of the structure γf [–]; the seaward slope m of the structure [–]; the mean overtopping discharge, upscaled to prototype, qss [m3 /s/m]; and whether wind is considered or not. Data from the ﬁeld are naturally scarce, and hence the method can only be regarded as tentative. Furthermore, it is only relevant if mean overtopping rates are

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Zeebrugge (1:1.4) Ostia (1:4)

30,0

Eq. (20), f_q,max = 7.9 Eq. (20), f_q,max = 30

fq [-]

20,0

10,0

0,0 1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

3

qss [m /s/m]

Fig. 15.17. Proposed adjustment factor applied to data from two ﬁeld sites (Zeebrugge 1:1.4 rubble mound breakwater, and Ostia 1:4 rubble slope).

lower than 1.0 l/s/m but may include signiﬁcant adjustment factors below these rates. Due to the inherent uncertainties, the proposed approach tries to be conservative. It has, however, been applied to pilot cases in CLASH and has proved good corrections with these model data. The adjustment factor fq for the model and scale eﬀects can be determined as follows (Fig. 15.17): 1.0 for qss ≥ 10−3m3 /s/m , (15.20) fq = min{(− log qss − 2)3 ; fq,max } for qss ≥ 10−3m3 /s/m , where fq,max is an upper bound to the adjustment factor fq and can be calculated as follows:  for γf ≤ 0.7 ,  fq,r (15.21) fq,max = 5 · γf · (1 − fq,r ) + 4.5 · (fq,r − 1) + 1 for 0.7 < γf ≤ 0.9 ,   1.0 for γf > 0.9 . In Eq. (15.21), fq,r is the adjustment factor for rough slopes which is mainly dependent on the slope of the structure and whether wind needs to be included or not:  for m ≤ 0.6 ,  1.0 fq,r = fw · (8.5 · m − 4.0) for 0.6 < m ≤ 4.0 , (15.22)   fw · 30 for m > 4.0 , in which m = cot α (slope of structure); fw accounts for the presence of wind and is set to fw = 1.0 if there is wind and fw = 0.67 if there is no wind. This set of equations include the case of smooth dikes which will, due to γf = 0.9 in this case, always lead to an adjustment factor of fq = 1.0. In case of a very rough

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1:4 slope with wind fq,max = fqr = 30.0, which is the maximum, the factor can get to (but only if the mean overtopping rates get below qss = 10−5 m3 /s per m). The latter case and a steep rough slope is illustrated in Fig. 15.14.

15.4. Individual Overtopping Waves 15.4.1. Overtopping volumes per wave The following section is a summary of Sec. 14.2.2 in Chap. 14 of this handbook. Parts of that section are repeated in the following with a focus on rubble mound structures. Wave overtopping is a dynamic and irregular process and the mean overtopping discharge, q, does not cover this aspect. But by knowing the storm duration, t, and the number of overtopping waves in that period, Now , it is easy to describe this irregular and dynamic overtopping, if the overtopping discharge, q, is known. Each overtopping wave gives a certain overtopping volume of water, V , with dimension m3 per m width or l per m width. As many equations in this chapter, the two-parameter Weibull distribution describes the behavior quite well. This equation has a shape parameter, b, and a scale parameter, a. The shape parameter gives a lot of information on the type of distribution. The exceedance probability, PV , of an overtopping volume per wave is similar to Eqs. (15.23) and (15.24). 0.75 V , (15.23) PV = P (V ≤ V ) = 1 − exp − a with a = 0.84 · Tm ·

q = 0.84 · Tm · q · Nw /Now = 0.84 · q · t/Now . Pov

(15.24)

Equation (15.24) shows that the scale parameter depends on the overtopping discharge, but also on the mean period (not the spectral period Tm−1,0 !) and probability of overtopping, or which is similar, on the storm duration and the actual number of overtopping waves. The probability of wave overtopping for rubble mound structures has been described in Sec. 15.2 and Eq. (15.4). Equations for calculating the overtopping volume per wave for a given probability of exceedance is given by Eq. (15.25): V = a · [− ln (1 − PV )]4/3 .

(15.25)

The maximum overtopping during a certain event is fairly uncertain, as most maxima, but depends on the duration of the event. In a 6-h period, one may expect a larger maximum than only during 15 min. The maximum during an event can be calculated by Eq. (15.26): 4/3

Vmax = a [ln (Now )]

.

(15.26)

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15.4.2. Overtopping velocities and spatial distribution The hydraulic behavior of waves on rubble mound slopes and on smooth slopes like dikes, is generally based on similar formulae, as clearly shown in this chapter. This is diﬀerent, however, for overtopping velocities and spatial distribution of the overtopping water. A dike or sloping impermeable seawall generally has an impermeable and more or less horizontal crest. Up-rushing and overtopping waves ﬂow over the crest, and each overtopping wave can be described by a maximum velocity and ﬂow depth. These velocities and ﬂow depths form the description of the hydraulic loads on crest and inner slope and are part of the failure mechanism “failure or erosion of inner slopes by wave overtopping.” This is diﬀerent for rubble mound slopes or breakwaters where wave energy is dissipated in the rough and permeable crest and where often overtopping water falls over a crest wall onto a crest road or even on the rear slope of a breakwater. A lot of overtopping water travels over the crest and through the air before it hits something else. Only recently in CLASH and a few other projects at Aalborg University, attention has been paid to the spatial distribution of overtopping water at breakwaters with a crest wall.6 The spatial distribution was measured by various trays behind the crest wall. Figure 15.18 gives diﬀerent cross sections with a setup of three arrays. Up to six arrays have been used. The spatial distribution depends on the level with respect to the rear side of the crest wall and the distance from this rear wall (Fig. 15.19). The coordinate system (x, y) starts at the rear side and at the top of the crest wall, with the positive y-axis downward. The exceedance probability F of the travel distance is deﬁned as the volume of overtopping water passing a given x- and y-coordinate, divided by the total overtopping volume. The probability, therefore, lies between 0 and 1, with 1 at the crest wall. The spatial distribution can be described with the following equations, which have slightly been rewritten and modiﬁed with respect to the original formulae by Andersen and Burcharth.6 The probability F at a certain location can be

Fig. 15.18.

Deﬁnition of y for various cross sections.

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Fig. 15.19.

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Deﬁnition of x- and y-coordinate for spatial distribution.

described by F (x, y) = exp[−1.3/Hm0 · {max(x/ cos β − 2.7ys0.15 op ; 0)}] .

(15.27)

Equation (15.27) can be rewritten to calculate directly the travel distance x (at a certain level y) by rewriting the above equation: x/ cos β = −0.77Hm0 ln(F ) + 2.7ys0.15 op .

(15.28)

Suppose cos β = 0, then we get F = 1 with x = 0; F = 0.1 with x = 1.77Hs; and F = 0.01 with x = 3.55Hs . It means that 10% of the volume of water travels almost two wave heights through the air and 1% of the volume travels more than 3.5 times the wave height. These percentages will be higher if y = 0, which is often the case with a crest unit. The validity of Eqs. (15.27) and (15.28) is for rubble mound slopes of approximately 1:2 and for angles of wave attack between 0◦ ≤ |β| < 45◦ . It should be noted that the equation is valid for the spatial distribution of the water through the air behind the crest wall. All water falling on the basement of the crest unit will, of course, travel on and will fall into the water behind and/or on the slope behind.

15.5. Overtopping of Shingle Beaches Shingle beaches diﬀer from the armored slopes principally in the size of the beach material, and hence its mobility. The typical stone size is suﬃciently small to permit signiﬁcant changes of beach proﬁle, even under relatively low levels of wave attack. A shingle beach may be expected to adjust its proﬁle to the incident wave conditions, provided that suﬃcient beach material is available. Run-up or overtopping levels on a shingle beach are therefore calculated without reference to any initial slope. The equilibrium proﬁle of shingle beaches under (temporary constant) wave conditions is described by van der Meer.11 The most important proﬁle parameter for run-up and overtopping is the crest height above SWL, hc . For shingle with Dn50 < 0.1 m, this crest height is only a function of the wave height and wave

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steepness. Note that the mean wave period is used, not the spectral wave period Tm−1,0 . hc /Hm0 = 0.3s−0.5 om .

(15.29)

Only the highest waves will overtop the beach crest and most of this water will percolate through the material behind the beach crest. Equation (15.29) gives a run-up or overtopping level which is more or less close to Ru2% . Acknowledgments This chapter was based on the EurOtop Overtopping Manual,4 which was funded in the United Kingdom by the Environmental Agency, in Germany by the German Coastal Engineering Research Council (KFKI), in the Netherlands by Rijkswaterstaat and Netherlands Expertise Network (ENW) on Flood Protection. The Project Team for the creation, editing, and support of the manual; the Project Steering Group for guidance and supervision; and a number of individual persons, have been listed and acknowledged in Chap. 14. References References in this chapter have been kept to a real minimum. An extensive list of relevant references, however, can be found in the EurOtop Overtopping Manual (2007). 1. P. Besley, Overtopping of seawalls — Design and assessment manual, R & D Technical Report W 178, Environment Agency, Bristol (1999), ISBN 1 85705 069 X. 2. T. Bruce, J. W. van der Meer, L. Franco and J. M. Pearson, Overtopping performance of diﬀerent armour units for rubble mound breakwaters, Coastal Eng. 56(2), 166–179 (2009). 3. CLASH, Crest level assessment of coastal structures by full scale monitoring, neural network prediction and hazard analysis on permissible wave overtopping, Fifth Framework Programme of the EU, Contract no. EVK3-CT-2001-00058, www.clash-eu.org. 4. EurOtop Overtopping Manual, Wave Overtopping of Sea Defences and Related Structures — Assessment Manual, eds. T. Pullen, N. W. H. Allsop, T. Bruce, A. Kortenhaus, H. Sch¨ uttrumpf and J. W. van der Meer (2007), www.overtoppingmanual.com. 5. T. L. Andersen, Hydraulic response of rubble mound breakwaters. Scale eﬀects — Berm breakwaters, PhD. Thesis, Aalborg University, Denmark (2006), ISSN 0909 4296 Series Paper No. 27. 6. T. L. Andersen and H. F. Burcharth, Landward distribution of wave overtopping for rubble mound breakwaters, Proc. First Int. Conf. Application of Physical Modelling to Port and Coastal Protection (2006). 7. T. L. Andersen and H. F. Burcharth, Overtopping and rear slope stability of reshaping and non-reshaping berm breakwaters, Proc. 29th Int. Conf. Coastal Engineering, Lisbon (2004).

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8. H. Sch¨ uttrumpf, Wellen¨ uberlaufstr¨ omung bei Seedeichen — Experimentelle und Theoretische Untersuchungen, PhD. thesis (2001), http://www.biblio.tu-bs.de/ediss/ data/20010703a/20010703a.html. 9. J. W. van der Meer, Technical Report — Wave run-up and wave overtopping at dikes, Technical Advisory Committee for Flood Defence in the Netherlands (TAW), Delft (2002). 10. J. W. van der Meer and C. J. M. Stam, Wave runup on smooth and rock slopes, ASCE J. WPC OE 188(5), 534–550 (1992); also, Delft Hydraulics Publication No. 454. 11. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, PhD. thesis, Delft University of Technology (1988); also, Delft Hydraulics Publication No. 396.

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Chapter 16

Wave Overtopping at Vertical and Steep Structures Tom Bruce School of Engineering, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JL, UK [email protected] Jentsje van der Meer Van der Meer Consulting P. O. Box 423, 8440 AK Heerenveen, The Netherlands [email protected] Tim Pullen HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] William Allsop HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] Wave overtopping prediction at vertical structures in earlier days was mainly based on caisson-type structures in relatively deep water. Recent research in many EU-projects has been concentrated on shallower water with waves breaking onto the structure as well. It has led to the deﬁnition of two situations: nonimpulsive and the most severe impulsive condition. This chapter relies on the EurOtop Overtopping Manual, as well as the two previous chapters, 14 and 15, in this handbook. It ﬁrst describes the mean overtopping discharges for many conﬁgurations of vertical and composite vertical structures. Later sections quantify inﬂuences such as oblique wave attack, wind eﬀects, model, scale eﬀects, etc.

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Individual overtopping volumes are then described. Finally, post-overtopping processes and parameters — landward distribution of discharge; velocities and downfall pressures — are described.

16.1. Introduction This chapter presents guidance for the assessment of overtopping and postovertopping processes at vertical and steep-fronted coastal structures such as caisson and blockwork breakwaters and vertical seawalls (see Figs. 16.1 and 16.2). Also included are composite vertical wall structures (where the emergent part of the structure is vertical, fronted by a modest berm) and vertical structures which include a recurve/bull-nose/parapet/wave return wall as the upper part of the defense. Large vertical breakwaters (Fig. 16.1) are almost universally formed of sand-ﬁlled concrete caissons usually resting on a small rock mound. Such caisson breakwaters may reach depths greater than 100 m, under which conditions with no wave breaking at all at the wall would be expected. Conversely, older breakwaters may, out of necessity, have been constructed in shallower water or indeed, built directly on natural rock “skerries.” As such, these structures may ﬁnd themselves exposed to breaking wave, or “impulsive” conditions when the water depth in front of them is

Fig. 16.1. Examples of vertical breakwaters: (left) modern concrete caisson and (right) older structure constructed from concrete blocks.

Fig. 16.2. Examples of vertical seawalls: (left) modern concrete wall and (right) older stone blockwork wall.

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suﬃciently low. Urban seawalls (e.g., Fig. 16.2) are almost universally fronted by shallow water, and are likely to be exposed to breaking or broken wave conditions, especially in areas of signiﬁcant tidal range. There are three principal sources of guidance on this topic preceding this chapter; in the United Kingdom, the Environment Agency’s Overtopping of Seawalls: Design and Assessment Manual 2 ; in the United States, the US Army Corps of Engineers’ Coastal Engineering Manual 6 ; in Japan, Goda’s design charts.11 The guidance presented in this chapter builds upon that of Besley,2 with adjustments to many formulae based upon further testing since 1999. For those familiar with Besley,2 the principal changes/additions are: • minor adjustments to recommended approach for distinguishing impulsive/ non-impulsive conditions (Sec. 16.2); • all formulae are now given in terms of wave period Tm−1,0 resulting in an adjusted deﬁnition of the h∗ , d∗ , and Vbar parameters (Secs. 16.2.2, 16.2.3, and 16.4.2, respectively) in order to maintain comparability with the earlier work; • in line with convergence on the Tm−1,0 measure, formulae using ﬁctitious wave steepness sop have been adjusted to use the new preferred measure sm−1,0 (Secs. 16.3.1 and 16.4.2); • new guidance on mean overtopping under conditions when all waves break before reaching the wall (part of Sec. 16.3.1); • new guidance on overtopping at vertical walls at zero freeboard (Sec. 16.3.1); • minor adjustments to formulae for mean overtopping under impulsive conditions due to the availability of additional data, from e.g., the CLASH database7 (Sec. 16.3.1); • extension of method for mean overtopping to account for steep (but not vertical) “battered” walls (Sec. 16.3.2); • new guidance on prediction of mean and wave-by-wave overtopping for oblique wave attack under impulsive conditions (Sec. 16.3.4); • new guidance on reduction in mean overtopping discharge due to wave return walls/parapets/recurves (Sec. 16.3.5); • new guidance on the eﬀect of wind on mean overtopping discharge (Sec. 16.3.6); • inclusion of summary of new evidence on scale eﬀects for laboratory study of overtopping at vertical and steep walls (Sec. 16.3.7); • new guidance on “post-overtopping” processes, speciﬁcally; velocity of “throw”; landward spatial extent of overtopping (Sec. 16.5); • all formulae are now given explicitly in terms of basic wave and structural parameters without recourse to intermediate deﬁnitions of dimensionless overtopping discharge and freeboard parameters speciﬁc to impulsive conditions. The qualitative form of the physical processes occurring when the waves reach the wall are described in Sec. 16.2. Distinctions drawn between diﬀerent wave/structure “regimes” are reﬂected in the guidance for assessment of mean overtopping discharges given in Sec. 16.3. The basic assessment tools are presented for plain vertical walls (Sec. 16.3.1), followed by subsections giving advice on how these basic tools should be adjusted to account for other commonly-occurring conﬁgurations, such as battered walls (Sec. 16.3.2); vertically composite walls (Sec. 16.3.3);

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the eﬀect of oblique wave attack (Sec. 16.3.4); the eﬀect of recurve/wave-return walls (Sec. 16.3.5). The eﬀect of wind is described in Sec. 16.3.6, with scale and model eﬀects reviewed in Sec. 16.3.7. Methods to assess individual “wave-by-wave” overtopping volumes are presented in Sec. 16.4. The current knowledge and advice on post-overtopping processes including velocities, spatial distributions, and postovertopping loadings are reviewed in Sec. 16.5.

16.2. Wave Processes at Walls 16.2.1. Overall view For assessment of overtopping at steep-fronted and vertical structures the regime of the wave/structure interaction must be identiﬁed ﬁrst, with quite distinct overtopping responses expected for each regime. On steep walls (vertical, battered, or composite), “non-impulsive” or “pulsating” conditions occur when waves are relatively small in relation to the local water depth, and of lower wave steepnesses. These waves are not critically inﬂuenced by the structure toe or approach slope. Overtopping waves run up and over the wall giving rise to (fairly) smoothly-varying loads and “green water” overtopping (Fig. 16.3). In contrast, “impulsive” conditions (Fig. 16.4) occur on vertical or steep walls when waves are larger in relation to local water depths, perhaps shoaling up over the approach bathymetry or structure toe itself. Under these conditions, some waves will break violently against the wall with (short-duration) forces reaching 10–40 times greater than for non-impulsive conditions. Overtopping discharge under these conditions is characterized by a “violent” uprushing jet of (probably highly aerated) water.

Fig. 16.3. A non-impulsive (pulsating) wave condition at a vertical wall, resulting in a nonimpulsive (or “green water”) overtopping.

Fig. 16.4. An impulsive (breaking) wave at a vertical wall, resulting in an impulsive (violent) overtopping condition.

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Broken wave at a vertical wall, resulting in a broken wave overtopping condition.

Lying in a narrow band between non-impulsive and impulsive conditions are “near-breaking” conditions where the overtopping is characterized by suddenness and a high-speed, near-vertical up-rushing jet (like impulsive conditions), but where the wave has not quite broken onto the structure and so has not entrained the amount of air associated with fully impulsive conditions. This “near-breaking” condition is also known as the “ﬂip through” condition. This condition gives overtopping in line with impulsive (breaking) conditions and is thus not treated separately. Many seawalls are constructed at the back of a beach such that breaking waves never reach the seawall, at least not during frequent events where overtopping is of primary importance. For these conditions, particularly for typical shallow beach slopes of less than (say) 1:30, design wave conditions may be given by waves which start breaking (possibly quite some distance) seaward of the wall. These “broken waves” arrive at the wall as a highly-aerated mass of water (Fig. 16.5), giving rise to loadings which show the sort of short-duration peak seen under impulsive conditions (as the leading edge of the mass of water arrives at the wall), but smaller in magnitude due to the high level of aeration. For cases where the depth at the wall hs > 0, overtopping can be assessed using the method for impulsive conditions. For conditions where the toe of the wall is emergent (hs ≤ 0), these methods can no longer be applied and an alternative is required. In order to proceed with the assessment of overtopping, it is therefore necessary ﬁrst to determine which is the dominant overtopping regime (impulsive or nonimpulsive) for a given structure and design sea state. No single method gives a discriminator which is 100% reliable. The suggested procedure for plain and composite vertical structures includes a transition zone in which there is signiﬁcant uncertainty in the prediction of dominant overtopping regime, and thus a “worst-case” is taken.

16.2.2. Overtopping regime discrimination — plain vertical walls A method will be described to distinguish between impulsive and non-impulsive conditions at a vertical wall where the toe of the wall is submerged (hs > 0; Fig. 16.6). When the toe of the wall is emergent (hs < 0), only broken waves reach the wall. For submerged toes (hs > 0), a wave breaking or “impulsiveness” parameter, h∗ is deﬁned based on the depth at the toe of the wall, hs , and incident wave conditions

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Fig. 16.6.

Definition sketch for assessment of overtopping at plain vertical walls.

inshore: h∗ = 1.3

hs 2π hs 2 Hm0 g Tm−1,0

or h∗ =

hs 2π hs , 2 Hm0 g Tm

(16.1)

where Hm0 = the signiﬁcant wave height at the toe of the structure and Tm−1,0 = the spectral wave period based on spectral moments m−1 and m0 , at the same location. Non-impulsive (pulsating) conditions dominate at the wall when h∗ > 0.3, and impulsive conditions occur when h∗ < 0.2. The transition between conditions for which the overtopping response is dominated by breaking and nonbreaking waves lies over 0.2 < h∗ < 0.3. In this region, overtopping should be predicted for both non-impulsive and impulsive conditions, and the larger value assumed. It should be noted that, for certain long-period waves, h∗ < 0.2 can be found under conditions for which no breaking at the vertical wall would be expected. Although the term “impulsive overtopping” may not be accurate in these conditions, the overtopping responses nevertheless follow those described by the “impulsive” formulae.

16.2.3. Overtopping regime discrimination — composite vertical walls For vertical composite walls where a berm or signiﬁcant toe is present in front of the wall, an adjusted version of the method for plain vertical walls should be used (see Fig. 16.7).

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Fig. 16.7.

417

Definition sketch for assessment of overtopping at composite vertical walls.

A modiﬁed “impulsiveness” parameter, d∗ , is deﬁned in a similar manner to the h∗ parameter (for plain vertical walls, Sec. 16.2.2): d∗ = 1.3

d 2π hs 2 Hm0 g Tm−1,0

or d∗ =

d 2π hs , 2 Hm0 g Tm

(16.2)

with parameters deﬁned according to Fig. 16.7. Non-impulsive conditions dominate at the wall when d∗ > 0.3, and impulsive conditions occur when d∗ < 0.2. The transition between conditions for which the overtopping response is dominated by breaking and nonbreaking waves lies over 0.2 < d∗ < 0.3. In this region, overtopping should be predicted for both nonimpulsive and impulsive conditions, and the larger value assumed.

16.3. Mean Overtopping Discharges for Vertical and Battered Walls 16.3.1. Plain vertical walls This section will give design equations for wave overtopping. The use of these equations may be diﬀerent, depending on the design process followed. First, an equation is given, which describes the mean of the prediction and the associated uncertainty. This mean prediction should be used for probabilistic design, or for prediction of, or comparison with, measurements. In the ﬁrst case, the associated uncertainty as given should be considered, and in the latter case, for instance, 5% upper and lower exceedance curves. As prediction of wave overtopping is fairly uncertain, the mean prediction should not be used for deterministic design or safety assessment. In that case a safety margin should be included in order to account for the uncertainty. Often, this will

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be one standard deviation. Each equation will be given as a mean prediction with uncertainty, followed by the application in a deterministic way. For simple vertical breakwaters, the following equations should be used. 16.3.1.1. Non-impulsive conditions (h∗ > 0.3) Equation (16.3) gives the mean prediction for non-impulsive conditions and should be used for probabilistic design, or for comparison with measurements. The coeﬃcient of 2.6 for the mean prediction has an associated normal distribution with a standard deviation of σ = 0.4. This standard deviation can be used to calculate, for instance, 5% exceedance curves. Equation (16.3) has the same shape as equations for smooth and rubble mound slopes (see Chaps. 14 and 15 in this handbook): q Rc valid for 0.1 < Rc /Hm0 < 3.5. = 0.04 exp −2.6 (16.3) 3 Hm0 gHm0 Note that a steep (smooth or rubble mound) slope has the same coeﬃcient of 2.6, which means that curves in a log-linear graph will be parallel. For steep slopes, however, the ﬁrst coeﬃcient is 0.2 and not 0.04, which gives more overtopping. (See also Fig. 16.1 in Chap. 14 of this handbook.) A well-known formula for vertical walls is given by Allsop et al.1 It is similar to Eq. (16.3) with coeﬃcients 0.05 and 2.78. These coeﬃcients are fairly close to those of Eq. (16.3) with 0.04 and 2.6, and, therefore, this known formula can also be used for simple vertical walls and non-impulsive conditions. For deterministic design or safety assessment, Eq. (16.3) should be used with a coeﬃcient of 2.2 instead of 2.6. Figure 16.8 shows measurements taken from the CLASH database and Eq. (16.3) with a mean prediction, the curve for deterministic design and 5% exceedance curves. Zero freeboard: For a vertical wall under non-impulsive conditions only one investigation is available,18 with zero freeboard (Rc = 0 or Rc /Hm0 = 0). Figure 16.9 gives the measurements as a function of the ﬁctitious wave steepness. 3 The mean was q/ gHm0 = 0.062 with a standard deviation of σ = 0.006. The value of 0.062 is slightly larger than the coeﬃcient 0.04 in Eq. (16.3), and even closer to the coeﬃcient of 0.05 of Allsop et al.1 Figure 16.9 shows that the value does not depend on the wave steepness, which is according to Eq. (16.3).For probabilistic 3 = 0.062 for design or comparison with measurements, one can use q/ gHm0 plain vertical walls with zero freeboard. For deterministic design or safety assessment, it is recommended to increase the average overtopping discharge by one standard deviation and use the value of 0.068. No data are available for impulsive overtopping at zero freeboard at vertical walls. 16.3.1.2. Impulsive conditions (h∗ < 0.2) The mean prediction for impulsive conditions is given by Eq. (16.4) (Fig. 16.10). The reliability of this equation is described by considering the scatter in the logarithm of

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1.E+00

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0.5

CLASH database set 028 CLASH database set 106 CLASH database set 224

1.E-01

CLASH database set 225 CLASH database set 351 CLASH database set 402

1.E-02

CLASH database set 502 probabilistic = mean deterministic

1.E-03

5%

1.E-04 5% 1.E-05

1.E-06 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

dimensionless freeboard Rc/Hm0

Fig. 16.8.

Mean overtopping at a plain vertical wall under non-impulsive conditions [Eq. (16.3)].

RC=0

0.09

3

dimensionless discharge q/(gHm0 )0.5

0.10

0.08

3 0.5

q/(g Hm0 )

= 0.062 ± 0.006

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.01

0.02

0.03

0.04

0.05

(fictitious) wave steepness sm-1,0 Fig. 16.9.

Dimensionless overtopping discharge for zero freeboard.18

0.06

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dimensionless discharge q / h *2(ghs3)

0.5

1.E+03 1.E+02 1.E+01

CLASH database set 028

CLASH database set 224

CLASH database set 225

CLASH database set 351

CLASH database set 502

CLASH database set 802

probabilistic

deterministic

1.E+00 1.E-01 1.E-02 1.E-03

5%

1.E-04 5% 1.E-05 1.E-06 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(impulsive) dimensionless freeboard h* Rc/Hm0

Fig. 16.10.

Mean overtopping at a plain vertical wall under impulsive conditions [Eq. (16.4)].

the data about the mean prediction: log10 (qmeasured )−log10 (qpredicted ) is taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation σ = 0.37 (i.e., 68% of predictions lie within the range of ×/÷ 2.3). For probabilistic calculations, Eq. (16.4) should be taken together with these stochastic coeﬃcients. For predictions of measurements or comparison with measurements also Eq. (16.4) should be taken with, for instance, 5% upper and lower exceedance curves. For deterministic design or safety assessment, a coeﬃcient of 2.8 × 10−4 should be used in Eq. (16.4), instead of 1.5 × 10−4 : −3.1 q Rc Rc −4 h∗ = 1.5 × 10 valid over 0.03 < h∗ < 1.0 . (16.4) 2 3 H H h∗ ghs m0 m0 For h∗ Rc /Hm0 < 0.02 arising from hs reducing to very small depths (as opposed to from small relative freeboards), there is evidence supporting an adjustment downward of the predictions of the impulsive formulae due to the observation that only broken waves arrive at the wall.4 The mean prediction is then described by Eq. (16.5). The reliability of this equation is described by considering the scatter in the logarithm of the data about the mean prediction: log10 (qmeasured ) − log10 (qpredicted ) is taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation σ = 0.15 (i.e., 68% of predictions lie within the range of ×/ ÷ 1.4): −2.7 q Rc Rc = 2.7 × 10−4 h∗ valid for h∗ < 0.02, broken waves. Hm0 Hm0 h2∗ gh3s (16.5)

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1.E+04

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0.5

Bruce et al (2003) data broken waves only- probabilistic (Eq. 5) broken waves only - deterministic (Eq. 5)

1.E+03

breaking waves - probabilistic (Eq. 4)

1.E+02 5%

1.E+01 5%

1.E+00 0.000

0.005

0.010

0.015

0.020

0.025

0.030

(impulsive) dimensionless freeboard h* Rc / Hm0

Fig. 16.11. Mean overtopping discharge for lowest h∗ Rc /Hm0 (for broken waves only arriving at wall) with submerged toe (hs > 0). For 0.02 < h∗ Rc /Hm0 < 0.03, overtopping response is illdefined — lines for both impulsive conditions (extrapolated to lower h∗ Rc /Hm0 ) and broken wave only conditions (extrapolated to higher h∗ Rc /Hm0 ) are shown as dashed lines over this region.

For probabilistic calculations or predictions of measurements or comparison with measurements, Eq. (16.5) should be taken together with these stochastic coeﬃcients or, for instance, 5% upper and lower exceedance curves. For deterministic design or safety assessment, a coeﬃcient of 3.8 × 10−4 should be used in Eq. (16.5), instead of 2.7 × 10−4 . For 0.02 < h∗ Rc /Hm0 < 0.03, there appears to be a transition between Eq. (16.4) (for “normal” impulsive conditions) and Eq. (16.5) (for conditions with only broken waves). There is, however, insuﬃcient data upon which to base a ﬁrm recommendation in this range. It is suggested that Eq. (16.4) is used down to h∗ Rc /Hm0 = 0.02 unless it is clear that only broken waves will arrive at the wall, in which case Eq. (16.5) could be used. Formulae for these low h∗ Rc /Hm0 conditions are shown in Fig. 16.11. 16.3.1.3. Impulsive conditions, toe emergent (hs ≤ 0) Data for conﬁgurations where the toe of the wall is emergent (i.e., at or above still water level, hs ≤ 0) is limited. The only available study suggests an adaptation of a prediction equation for plunging waves on a smooth slope may be used, but particular caution should be exercised in any extrapolation beyond the parameter ranges of the study, which only used a relatively steep (m = 10) foreshore slope. This adaptation is given in Eq. (16.6). The reliability of Eq. (16.6) is described by taking the coeﬃcient 2.16 as a normally distributed stochastic parameter with a

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1.E-04

dim'less discharge q/(gHm03)

0.5

.(sm-1,0 x m)

0.5

1.E-03

1.E-05

5%

Bruce et al (2003) data

1.E-06

emergent toe - probabilistic (Eq. 6)

5%

emergent toe - deterministic (Eq. 6)

1.E-07 0.0

1.0

2.0

3.0

dimensionless freeboard Rc/Hm0 x

Fig. 16.12.

4.0 0.33 sm-1,0

5.0

6.0

xm

Mean overtopping discharge with emergent toe (hs < 0).

mean of 2.16 and a standard deviation σ = 0.21: q Rc √ · m sm−1,0 = 0.043 exp −2.16 m s0.33 , m−1,0 Hm0,deep 3 gHm0,deep valid for 2.0 < m s0.33 m−1,0

Rc < 5.0; Hm0,deep

0.55 ≤ Rc /Hm0,deep ≤ 1.6;

sm−1,0 ≥ 0.025 . (16.6)

Note: Data only available for m = 10 (i.e., 1:10 foreshore slope). For deterministic design or safety assessment, Eq. (16.6) should be used with a coeﬃcient 1.95 instead of 2.16. Equation (16.6) for overtopping under emergent toe conditions is illustrated in Fig. 16.12. It is emphasized that this formula is based upon a limited dataset of small-scale tests with 1:10 foreshore only and should not be extrapolated outwith the given limits. 16.3.2. Battered walls Near-vertical walls with 10:1 and 5:1 batters are found commonly for older UK seawalls and breakwaters (e.g., Fig. 16.13). Mean overtopping discharges for battered walls under impulsive conditions are slightly in excess of those for a vertical wall over a wide range of dimensionless

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Fig. 16.13. Battered walls: typical cross-section (left), and Admiralty Breakwater, Alderney Channel Islands (right, courtesy G. M¨ uller).

dimensionless discharge q / h *2(ghs3)0.5

1.E+01

plain vertical (Eq. 3)

1.E+00

10:1 batter (Eq. 7a) 5:1 batter (Eq. 7b)

1.E-01

1.E-02

1.E-03

1.E-04 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(impulsive) dimensionless freeboard, h* Rc/Hm0 Fig. 16.14.

Overtopping for a 10:1 and 5:1 battered walls.

freeboards. Multiplying factors are given in Eqs. (16.17a) and (16.7b) (plotted in Fig. 16.14). 10:1 battered wall:

q10:1batter = qvertical × 1.3 ,

(16.7a)

5:1 battered wall:

q5:1batter = qvertical × 1.9 ,

(16.7b)

where qvertical is arrived at from Eq. (16.4). The uncertainty in the ﬁnal estimated overtopping discharge can be estimated as per the plain vertical cases. No dataset is available to indicate an appropriate adjustment under nonimpulsive conditions. An alternative method, however, may be to calculate

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overtopping for a vertical structure and for a steep smooth slope 1:1. By using the cot α of the battered wall as the parameter (between the limits 0 and 1), a fair guess will be reached (see also Fig. 14.2 in Chap. 14 of this handbook). 16.3.3. Composite vertical walls It is well established that a relatively small toe berm can change wave-breaking characteristics, thus substantially altering the type and magnitude of wave loadings. Many vertical seawalls may be fronted by rock mounds with the intention of protecting the toe of the wall from scour. The toe conﬁguration can vary considerably, potentially modifying the overtopping behavior of the structure. Three types of mounds can be identiﬁed: (i) Small toe mounds which have an insigniﬁcant eﬀect on the waves approaching the wall — here, the toe may be ignored and calculations proceed as for simple vertical (or battered) walls. (ii) Moderate mounds, which signiﬁcantly aﬀect wave-breaking conditions, but are still below water level. Here, a modiﬁed approach is required. (iii) Emergent mounds in which the crest of the armor protrudes above still water level. Prediction methods for these structures may be adapted from those for crown walls on a rubble mound (Chap. 15 of this handbook). For assessment of mean overtopping discharge at a composite vertical seawall or breakwater, the overtopping regime (impulsive/non-impulsive) must be determined (see Sec. 16.2.3). When non-impulsive conditions prevail, overtopping can be predicted by the standard method given previously for non-impulsive conditions at plain vertical structures [Eq. (16.3)]. For conditions determined to be impulsive, a modiﬁed version of the impulsive prediction method for plain vertical walls is recommended, accounting for the presence of the mound by use of d and d∗ . 16.3.3.1. Impulsive conditions (d∗ ≤ 0.2) The mean prediction for impulsive conditions at a composite vertical structure is given by Eq. (16.8), and Fig. 16.15. The reliability of this equation is described by considering the scatter in the logarithm of the data about the mean prediction: log10 (qmeasured) − log10 (qpredicted ) is taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation σ = 0.28 (i.e., 68% of predictions lie within the range of ×/ ÷ 1.9). For probabilistic calculations, Eq. (16.8) should be taken together with these stochastic coeﬃcients:

d2∗

q

gd3

= 4.1 × 10

valid for 0.05 < d∗

−4

−2.9 Rc , d∗ Hm0

Rc < 1.0 and h∗ < 0.3 . Hm0

(16.8)

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dimensionless discharge q / d *2(gd3)0.5

1.E+03 VOWS data

1.E+02 composite vertical - probabilistic (Eq. 8)

1.E+01

composite vertical - deterministic (Eq. 8)

1.E+00

1.E-01

1.E-02 5% 1.E-03 5% 1.E-04 0.0

0.2

0.4

0.6

0.8

1.0

1.2

(impulsive, composite) dimensionless freeboard d* Rc/Hm0

Fig. 16.15.

Overtopping for composite vertical walls.

For deterministic design or safety assessment, Eq. (16.8) should be used with a coeﬃcient of 7.8 × 10−4, instead of 4.1 × 10−4 , and an exponent of −2.6 instead of −2.9. 16.3.4. Eﬀect of oblique waves Seawalls and breakwaters seldom align perfectly with incoming waves. The assessment methods presented thus far are only valid for shore-normal wave attack. In this subsection, advice is given on how the methods for shore-normal wave attack (obliquity β = 0◦ ) should be adjusted for oblique wave attack. As for zero obliquity, overtopping response depends critically upon the physical form (or “regime”) of the wave/wall interaction — non-impulsive, impulsive, or broken. As such, the ﬁrst step is to use the methods given in Sec. 16.2 to determine the form of overtopping for shore-normal (zero obliquity). Based upon the outcome of this, guidance under “non-impulsive conditions” or “impulsive conditions” should be followed. For non-impulsive conditions, an adjusted version of Eq. (16.3) should be used: 2.6 Rc q = 0.04 exp − , (16.9) 3 γβ Hm0 gHm0 where γβ is the reduction factor for angle of attack and is given by γβ = 1 − 0.0062β

for 00 < β < 45◦ ,

γβ = 0.72 for β ≥ 45◦ , and β is the angle of attack relative to the normal, in degrees.

(16.10)

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dimensionless discharge q / h *2(ghs3)0.5

1.E+01 beta = 0 degrees (Eq. 3) beta = 15 degrees (Eq. 11b)

1.E+00

beta = 30 degrees (Eq. 11c)

1.E-01

1.E-02

1.E-03

1.E-04 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(impulsive) dimensionless freeboard, h* Rc/Hm0 Fig. 16.16.

Overtopping of vertical walls under oblique wave attack.

For conditions that would be identiﬁed as impulsive for normal (β = 0◦ ) wave attack, a more complex picture emerges.13 Diminished incidence of impulsive overtopping is observed with increasing obliquity (angle β) of wave attack (Fig. 16.16). This results not only in reductions in mean discharge with increasing β, but also, for β ≥ 60◦ , a switch back over to the functional form observed for non-impulsive conditions [i.e., a move away from a power-law decay such as Eq. (16.4) to an exponential one such as Eq. (16.3)]. The mean predictions are given by Eqs. (16.11a)–(16.11d). Data only exist for the discrete values of obliquity listed: for β = 15◦ , h∗

Rc ≥ 0.2, Hm0

−3.7 q Rc = 5.8 × 10−5 h∗ , Hm0 h2∗ gh3s

(16.11a)

for β = 15◦ , h∗

Rc < 0.2, Hm0

as per impulsive β = 0◦ (Eq. 16.4) ,

(16.11b)

for β = 30◦ , h∗

Rc ≥ 0.07, Hm0

−4.2 q Rc = 8.0 × 10−6 h∗ , Hm0 h2∗ gh3s

(16.11c)

for β = 60◦ , h∗

Rc ≥ 0.07, Hm0

as per non-impulsive β = 60◦ (Eq. 16.10) .

(16.11d)

Signiﬁcant spatial variability of overtopping volumes along the seawall under oblique wave attack are observed/measured in physical model studies. For deterministic design, Eqs. (16.12a)–(16.12c) should be used, as these give estimates of

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the “worst case” conditions at locations along the wall where the discharge is the greatest: for β = 150 , h∗

Rc ≥ 0.2, Hm0

for β = 300 , h∗

Rc ≥ 0.07, Hm0

as per impulsive β = 15◦ (Eq. 16.11b),

(16.12b)

for β = 600 , h∗

Rc ≥ 0.07, Hm0

as per non-impulsive β = 0◦ (Eq. 16.3).

(16.12c)

as per impulsive β = 0◦ (Eq. 16.4),

(16.12a)

16.3.5. Eﬀect of bullnose and recurve walls Designers of vertical seawalls and breakwaters have often included some form of seaward overhang (recurve/parapet/wave return wall/bullnose) as a part of the structure with the design motivation of reducing wave overtopping by deﬂecting back seaward uprushing water (e.g., Figs. 16.17 and 16.18). The mechanisms determining the eﬀectiveness of a recurve are complex and not yet fully described. The guidance presented here is based upon physical model studies.12,15 Parameters for the assessment of overtopping at structures with bullnose/recurve walls are shown in Fig. 16.19. Two conditions are distinguished: • the familiar case of the parapet/bullnose/recurve overhanging seaward (α < 90◦ ), and

Fig. 16.17. An example of a modern, large vertical breakwater with wave return wall (left) and cross-section of an older seawall with recurve (right).

Fig. 16.18. A sequence showing the function of a parapet/wave return wall in reducing overtopping by redirecting the uprushing water seaward (back to right).

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Fig. 16.19. Parameter definitions for assessment of overtopping at structures with parapet/wave return wall.

• the case where a wall is chamfered backward at the crest [normally admitting greater overtopping (α > 90◦ )]. For the latter, chamfered wall case, inﬂuence factors γ should be applied to Franco et al.’s equation10 for non-impulsive mean discharge [Eq. (16.13)] with the value of γ selected as shown8 : 4.3 Rc q = 0.2 exp − , (16.13) 0.5 3 γ Hm0 (gHm0 ) γ = 1.01 for α = 120◦ , γ = 1.13 for α = 135◦ , γ = 1.07 for α = 150◦ . For the familiar case of overhanging parapet/recurve/bullnose, the eﬀectiveness of the recurve/parapet in reducing overtopping is quantiﬁed by a factor k deﬁned as k=

qwith recurve . qwithout recurve

(16.14)

The decision chart in Fig. 16.20 can then be used to arrive at a value of k, which in turn can be applied by multiplication to the mean discharge predicted by the most appropriate method for the plain vertical wall (with the same Rc , hs , etc.). The decision chart shows three levels of decision: • whether the parapet is angled seaward or landward; • if seaward (α < 90◦ ), whether conditions are in the small (left box), intermediate (middle box), or large (right box) reduction regimes;

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Fig. 16.20. “Decision chart” summarizing methodology for tentative guidance. Note that symbols R∗0 , k23 , m, and m∗ used (only) at intermediate stages of the procedure are defined in the lowest boxes in the figure. Please refer to text for further explanation.

• if in the regime of largest reductions (greatest parapet eﬀectiveness; Rc /Hm0 ≥ R0∗ + m∗ ), which of the three further subregimes (for diﬀerent Rc /hs ) is appropriate. Given the level of scatter in the original data and the observation that the methodology is not securely founded on the detailed physical mechanisms/processes, it is suggested that it is impractical to design for k < 0.05, i.e., reductions in mean discharges by factors greater than 20 cannot be predicted with conﬁdence. If such large (or larger) reductions are required, a detailed physical model study should be considered. 16.3.6. Eﬀect of wind Wind may aﬀect overtopping processes and thus discharges by (1) changing the shape of the incident wave crest at the structure resulting in a possible modiﬁcation of the dominant regime of wave interaction with the wall;

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(2) blowing up-rushing water over the crest of the structure (for an onshore wind, with the reverse eﬀect for an oﬀshore wind) resulting in possible modiﬁcation of mean overtopping discharge and wave-by-wave overtopping volumes; (3) modifying the physical form of the overtopping volume or jet, especially in terms of its aeration and breakup resulting in possible modiﬁcation to postovertopping characteristics such as throw speed, landward distribution of discharge, and any resulting post-overtopping loadings (e.g., downfall pressures). The modeling of any of these eﬀects in small-scale laboratory tests presents very great diﬃculties owing to fundamental barriers to the simultaneous scaling of the wave-structure and water–air interaction processes. Very little information is available to oﬀer guidance on eﬀect (1) — the reshaping of the incident waves. Comparisons of laboratory and ﬁeld data (both with and without wind) have enabled some upper (conservative) bounds to be placed upon eﬀect (2) — the intuitive wind-assistance in “pushing” of up-rushing water landward across the crest. These are discussed immediately below. Discussion of eﬀect (3) — modiﬁcation to “postovertopping” processes — is reserved for Secs. 16.5.3 and 16.5.4 (on distributions and downfalling pressures, respectively). Several investigations on vertical structures have suggested diﬀerent adjustment factors fwind ranging from 30% to 40% to up to 300% (Fig. 16.21) either using a paddle wheel or large fans to transport uprushing water over the wall. When these tests were revisited, a simple adjustment factor was proposed for the mean discharge based upon small-scale tests qss , which is already scaled up by

10,0 De Waal et al. (1996) Davey (2004) Pullen and Allsop (2004)

f Wind [-]

Eq. (15)

5,0

0,0 1,0E-07

1,0E-06

1,0E-05

1,0E-04

1,0E-03

qss [m3/s/m]

Fig. 16.21. datasets.

Wind adjustment factor fwind plotted over mean overtopping rates qss for three

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appropriate scaling to full-scale (see also Ref. 9):  for qss ≤ 10−5 m3 /s/m  4.0 fwind = 1.0 + 3 · (− log qss − 4) for 10−5 < qss < 10−4 m3 /s/m   1.0 for qss ≥ 10−4 m3 /s/m .

(16.15)

From Eq. (16.15), it becomes clear that the inﬂuence of wind only gets important for very low overtopping rates below qss = 0.1 l/s/m. Hence, in many practical cases, the inﬂuence of wind may be disregarded. The mean overtopping discharge including wind becomes qwith wind = fwind × qss .

(16.16)

16.3.7. Scale and model eﬀect corrections Tests in a large-scale wave channel (Fig. 16.22) and ﬁeld measurements (Fig. 16.23) have demonstrated that with the exception of wind eﬀect (Sec. 16.3.6), results of overtopping measurements in small-scale laboratory studies may be securely scaled to full-scale under non-impulsive and impulsive overtopping conditions.16,17 No information is yet available on the scaling of small-scale data under conditions where broken wave attack dominates. Comparison of measurements of wave loadings on vertical structures under broken wave attack at small scale and in the ﬁeld suggests that prototype loadings will be over-estimated by small scale tests in the presence of highly-aerated broken waves. Thus, although the methods presented for

dimensionless discharge q / h *2(ghs3)

0.5

1.E+01 Large-scale data (Pearson et al, 2002)

1.E+00

Eq. 7a (10:1 batter)

1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(impulsive) dimensionless freeboard, h* Rc/Hm0

Fig. 16.22. Large-scale laboratory measurements of mean discharge at 10:1 battered wall under impulsive conditions showing agreement with prediction line based upon small-scale tests [Eq. (16.7a)].

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dimensionless discharge q / d *2(gd3)0.5

1.E+04 field data (Pullen et al, 2005)

1.E+03

composite vertical - probabilistic (Eq. 8)

1.E+02 composite vertical - deterministic (Eq. 8)

1.E+01 1.E+00 1.E-01 1.E-02 5% 1.E-03

5%

1.E-04 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(impulsive, composite) dimensionless freeboard d* Rc/Hm0 Fig. 16.23. Results from field measurements of mean discharge at Samphire Hoe, UK, plotted together with Eq. (16.8).

the assessment of overtopping discharges under broken wave conditions given in Sec. 16.3.1 have not been veriﬁed at large scale or in the ﬁeld, any scale correction is expected to give a reduction in the predicted discharge.

16.4. Overtopping Volumes 16.4.1. Introduction While the prediction of mean discharge (Sec. 16.3) oﬀers the information required to assess whether overtopping is slight, moderate, or severe, and make a link to any possible ﬂooding that might result, the prediction of the volumes associated with individual wave events can oﬀer an alternative (and often more appropriate) measure for the assessment of tolerable overtopping levels and possible direct hazard. First, a method is given for the prediction of maximum overtopping volumes expected associated with individual wave events for plain vertical structures under perpendicular wave attack (Sec. 16.4.2). This method is then extended to composite (bermed) structures (Sec. 16.4.3) and to conditions of oblique wave attack (Sec. 16.4.4). Finally, a short section on scale eﬀects is included (Sec. 16.4.5). The methods given for perpendicular wave attack are the same as those given previously in UK guidance,2 but now using the Tm−1,0 period measure (with Weibull parameters, steepness values, etc., adjusted accordingly). The extension to oblique wave attack is new.

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16.4.2. Overtopping volumes at plain vertical walls The ﬁrst step in the estimation of a maximum expected individual wave overtopping volume is to estimate the number of waves overtopping (Now ) in a sequence of Nw incident waves. For non-impulsive conditions, this was found to be well described by Franco et al.,10 as follows: 2

Rc (16.17) (for h∗ > 0.3) . Now = Nw exp −1.21 Hm0 Under impulsive conditions, Now is better described by2 Now = 0.031 Nw ×

Hm0 h∗ Rc

(for h∗ < 0.3) .

(16.18)

The distribution of individual overtopping volumes in a sequence is generally well described by a two-parameter Weibull distribution (see also Chap. 14, Sec. 14.2.2 of this handbook):

b V , (16.19) PV = 1 − exp − a where PV is the probability that an individual event volume will not exceed V ; a and b are Weibull “shape” and “scale” parameters, respectively. Thus, to estimate the largest event in a wave sequence predicted to include (e.g.) Now = 200 overtopping events, Vmax would be found by taking PV = 1/200 = 0.005. Equation (16.19) can then be rearranged to give Vmax = a (ln Now )

1/b

.

(16.20)

The Weibull shape parameter a depends upon the average volume per overtopping wave Vbar , where Vbar = 0.8

qTm−1,0 Nw Now

or

Vbar =

qTm Nw . Now

(16.21)

For non-impulsive conditions, there is a weak steepness-dependency for the scale and shape parameters a and b: 0.66 for sm−1,0 = 0.024 0.74Vbar b= (16.22) a= for h∗ > 0.3 . 0.82 for sm−1,0 = 0.048 0.90Vbar (Note that the ﬁctitious steepness values sm−1,0 = 0.024 and 0.048 in Eq. (16.22) correspond to the values of sop = 0.02 and 0.04 quoted by Besley2 .) For impulsive conditions2,16 : a = 0.92Vbar ,

b = 0.85

for h∗ < 0.3 .

(16.23)

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1.E-01

1.E-02

1.E-03

1.E-04 1.E-04

1.E-03

1.E-02

1.E-01

1.E+00 3

predicted max. individual overtopping volume, Vmax [m /m] Fig. 16.24. Predicted and measured maximum individual overtopping volume — small- and largescale tests.16

The eﬀectiveness of the predictor under impulsive conditions can be gauged from Fig. 16.24. Note that all the a values are fairly similar, leading to similar steep distributions (see also Sec. 14.2.2 in Chap. 14 of this handbook). A further step in development could be to choose a ﬁxed value for a and modify b accordingly. 16.4.3. Overtopping volumes at composite (bermed) structures There is very little information available speciﬁcally addressing wave-by-wave overtopping volumes at composite structures. The guidance oﬀered by Besley2 remains the best available. No new formulae or Weibull a, b values are known; so, for the purposes of maximum overtopping volume prediction, the methods for plain vertical walls (Sec. 16.4.2) are used. The key discriminator is that composite structures whose mound is suﬃciently small to play little role in the overtopping process are treated as plain vertical, non-impulsive, whereas those with large mounds are treated as plain vertical, impulsive. For this purpose, the signiﬁcance of the mound is assessed using the “impulsiveness” parameter for composite structures, d∗ [Eq. (16.2)]. “Small mound” is deﬁned as d∗ > 0.3, with d∗ < 0.3 being “large mound.” 16.4.4. Overtopping volumes at plain vertical walls under oblique wave attack For non-impulsive conditions, an adjusted form of Eq. (16.17) is suggested10 : 2

Rc 1 for h∗ > 0.3 . (16.24) Now = Nw exp − 2 C Hm0

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Table 16.1. Summary of prediction formulae for individual overtopping volumes under oblique wave attack. Oblique cases valid for 0.2 < h∗ Rc /Hm0 < 0.65. For 0.07 < h∗ Rc /Hm0 < 0.2, the β = 00 formulae should be used for all β. β = 15◦ Now = 0.01 Nw × a = 1.06 Vbar b = 1.18

“

β = 30◦ Hm0 h∗ Rc

”−1.6

Now = 0.01 Nw × a = 1.04 Vbar b = 1.27

“

β = 60◦ Hm0 h∗ Rc

”−1.4

Treat as non-impulsive Treat as non-impulsive Treat as non-impulsive

C is given by

 0.91 C = 0.91 − 0.00425β  0.74

for β = 0◦ for 0◦ < β < 40◦ for β ≥ 40◦

for h∗ > 0.3 .

(16.25)

For impulsive conditions (as determined for perpendicular i.e., β = 0◦ wave attack), the procedure is the same as for perpendicular (β = 0◦ ) wave attack, but diﬀerent formulae should be used for estimating the number of overtopping waves (Now ) and Weibull shape and scale parameters (Table 16.1).13 16.4.5. Scale eﬀects for individual overtopping volumes Measurements from large-scale laboratory tests indicate that formulae for overtopping volumes, based largely upon small-scale physical model studies, scale well (Fig. 16.24).16 No data from the ﬁeld is available to support “scale-ability” from large-scale laboratory scales to prototype conditions. 16.5. Overtopping Velocities, Distributions, and Down-Fall Pressures 16.5.1. Introduction to post-overtopping processes There are many design issues for which knowledge of just the mean and/or waveby-wave overtopping discharges/volumes are not suﬃcient, e.g., • assessment of direct hazard to people, vehicles, and buildings in the zone immediately landward of the seawall; • assessment of potential for damage to elements of the structure itself (e.g., crown wall, crown deck, secondary defenses). The appreciation of the importance of being able to predict more than overtopping discharges and volumes has led to signiﬁcant advances in the description and quantiﬁcation of what can be termed “post-overtopping” processes, speciﬁcally, the current state of prediction tools for: • the speed of an overtopping jet (or “throw velocity”), • the spatial extent reached by (impulsive) overtopping volumes, and

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max. dim'less vertical throw speed, u z / ci

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

h* parameter

Fig. 16.25. Speed of upward projection of overtopping jet past structure crest plotted with h∗ parameter (after Bruce et al.5 ).

• the pressures that may arise due to the downfalling overtopped jet impacting on the structure’s crown deck. 16.5.2. Overtopping throw speeds Studies at small scale, based upon video footage suggest that the vertical speed with which the overtopping jet leaves the crest of the structure (uz ) may be estimated as (Fig. 16.25): uz ≈ where ci =

√

2 to 2.5 × ci

for non-impulsive conditions ,

5 to 7 × ci

for impulsive conditions ,

(16.26)

ghs is the inshore wave celerity.5

16.5.3. Spatial extent of overtopped discharge The spatial distribution of overtopped discharge may be of interest in determining zones aﬀected by direct wave overtopping hazard (to people, vehicles, buildings close behind the structure crest, or to elements of the structure itself). Under green water (non-impulsive) conditions, the distribution of overtopped water will depend principally on the form of the area immediately landward of the structure’s crest (slopes, drainage, obstructions, etc.) and no generic guidance can be oﬀered (though, see Sec. 16.5.2 for information on speeds of overtopping jets).

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1 0.9 0.8

c. 95% of discharge falls within 0.25 × wavelength

0.7 0.6 0.5

c. 90% of discharge falls within 0.2 × wavelength

0.4 0.3 0.2

approx. lower (worst case) envolope

0.1 0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

dimensionless landward distance x/L0 Fig. 16.26. Landward distribution of overtopping discharge under impulsive conditions. Curves show proportion of total overtopping discharge which has landed within a particular distance shoreward of seaward crest.

Under violent (impulsive) overtopping conditions, the idea of spatial extent and distribution has a greater physical meaning — where does the airborne overtopping jet come back to the level of the pavement behind the crest? The answer to this question, however, will (in general) depend strongly upon the local wind conditions. Despite the diﬃculty of directly linking a laboratory wind speed to its prototype equivalent (see Sec. 16.3.6), laboratory tests have been used to place an upper bound on the possible wind-driven spatial distribution of the “fall back to ground” footprint of the violently overtopped volumes.3,17 Tests used large fans to blow air at gale-force speeds (up to 28 m/s) in the laboratory. The lower (conservative) envelope of the data (Fig. 16.26) gives the approximate guidance that 95% of the violently-overtopped discharge will land within a distance of 0.25 × Lo , where Lo is the oﬀshore (deepwater) wavelength. No data is available for the case of a shallow foreshore with no clear deﬁnition of Lo . 16.5.4. Pressures resulting from downfalling water mass Wave impact pressures on the crown deck of a breakwater have been measured in small- and large-scale tests.19,20 These impacts are the result of an impacting wave at the front wall of the breakwater generating an upward jet which in turn falls back onto the crown deck of the structure. Small-scale tests suggest that local impact pressure maxima on the crown deck are smaller than but of the same order of magnitude as wave impact pressures on the front face. For high-crested structures (Rc /Hm0 > 0.5), pressure maxima were observed to occur within a distance of ∼ 1.5×Hm0 behind the seaward crest. For lower-crested structures (Rc /Hm0 < 0.5),

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this distance was observed to increase to ∼ 2 × Hm0 . Over all small-scale tests, pressure maxima were measured over the range 2<

p1/250 < 17 with a mean value of 8. ρ gHm0

(16.27)

The largest downfall impact pressure measured in large-scale tests was 220 kPa (with a duration of 0.5 ms). The largest downfall pressures were observed to result from overtopping jets thrown upward by very-nearly breaking waves (the “ﬂip through” condition). Although it might be expected that scaling small-scale impact pressure data would overestimate pressure maxima at large scale, approximate comparisons between small- and large-scale test data suggest that the agreement is good. Acknowledgments This chapter was based on the EurOtop Overtopping Manual,14 which was funded in the United Kingdom by the Environmental Agency, in Germany by the German Coastal Engineering Research Council (KFKI), and in the Netherlands by Rijkswaterstaat and Netherlands Expertise Network (ENW) on Flood Protection. The Project Team for the creation, editing, and support of the manual; the Project Steering Group for guidance and supervision; and a number of individual persons, have been listed and acknowledged in Chap. 14.

References References have been kept to an absolute minimum. An extensive bibliography can be found in the EurOtop Overtopping Manual (2007). 1. N. W. H. Allsop, P. Besley and L. Madurini, Overtopping performance of vertical and composite breakwaters, seawalls and low reﬂection alternatives, Paper 4.7 in MCS Project Final Report, University of Hannover (1995). 2. P. Besley, Overtopping of seawalls — Design and assessment manual, R&D Technical Report W 178, Environment Agency, Bristol (1999). 3. T. Bruce, T. Pullen, W. Allsop and J. Pearson, How far back from a seawall is safe? Spatial distributions of wave overtopping, Proc. Int. Conf. Coastlines, Structures and Breakwaters 2005, ICE London, Thomas Telford (2005), pp. 166–176. 4. T. Bruce, J. Pearson and N. W. H. Allsop, Violent wave overtopping — Extension of prediction method to broken waves, Proc. Coastal Structures 2003, ASCE, Reston, Virginia (2003), pp. 619–630. 5. T. Bruce, N. W. H. Allsop and J. Pearson, Hazards at coast and harbour seawalls — Velocities and trajectories of violent overtopping jets, Proc. 28th Int. Conf. Coastal Eng., Cardiﬀ (2002), pp 2216–2226. 6. CEM/H. F. Burcharth and S. A. Hughes, Fundamentals of design, Coastal Engineering Manual, eds. L. Vincent and Z. Demirbilek, Part VI, Design of Coastal Project Elements. Chapter VI-5-2, Engineer Manual 1110-2-1100 (US Army Corps of Engineers, Washington, DC, 2002).

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7. CLASH, Crest level assessment of coastal structures by full scale monitoring, neural network prediction and hazard analysis on permissible wave overtopping, Fifth Framework Programme of the EU, Contract N. EVK3-CT-2001-00058, www. clash-eu.org. 8. A. Cornett, Y. Li and A. Budvietas, Wave overtopping at chamfered and overhanging vertical structures, Proc. Int. Workshop on Natural Disasters by Storm Waves and their Reproduction in Experimental Basins, Kyoto, Japan (1999). 9. J. De Rouck, J. Geeraerts, P. Troch, A. Kortenhaus, T. Pullen and L. Franco, New results on scale eﬀects for wave overtopping at coastal structures, Proc. Coastlines, Structures and Breakwaters 2005, ICE London, Thomas Telford (2005), pp. 29–43. 10. L. Franco, M. de Gerloni and J. W. van der Meer, Wave overtopping on vertical and composite breakwaters, Proc. 24th Int. Conf. Coastal Eng., Kobe (1994), pp. 1030–1044. 11. Y. Goda, Random Seas and Design of Maritime Structures, 2nd edn. (World Scientiﬁc Publishing, Singapore, 2000). 12. A. Kortenhaus, J. Pearson, T. Bruce, N. W. H. Allsop and J. W. van der Meer, Inﬂuence of parapets and recurves on wave overtopping and wave loading of complex vertical walls, Proc. Coastal Structures 2003, ASCE, Reston, Virginia (2003), pp. 369–381. 13. N. Napp, T. Bruce, J. Pearson and N. W. H. Allsop, Violent overtopping of vertical seawalls under oblique wave conditions, Proc. 29th Int. Conf. Coastal Eng., Cardiﬀ (2002), pp. 4482–4493. 14. Overtopping Manual, Wave Overtopping of Sea Defences and Related Structures — Assessment Manual, eds. T. Pullen, N. W. H. Allsop, T. Bruce, A. Kortenhaus, H. Sch¨ uttrumpf and J. W. van der Meer (2007), www.overtopping-manual.com. 15. J. Pearson, T. Bruce, N. W. H. Allsop, A. Kortenhaus and J. W. van der Meer, Eﬀectiveness of recurve wave walls in reducing wave overtopping on seawalls and breakwaters, Proc. 29th Int. Conf. Coastal Eng., Lisbon (2004), pp. 4404–4416. 16. J. Pearson, T. Bruce, N. W. H. Allsop and X. Gironella, Violent wave overtopping — Measurements at large and small scale, Proc. 28th Int. Conf. Coastal Eng., Cardiﬀ (2002), pp. 2227–2238. 17. T. Pullen, N. W. H. Allsop, T. Bruce, J. Pearson and J. Geeraerts, Violent wave overtopping at Samphire Hoe: Field and laboratory measurements, Proc. 29th Int. Conf. Coastal Eng., Lisbon (2004), pp. 4379–4390. 18. R. Smid, Untersuchungen zur Ermittlung der mittleren Wellen¨ uberlaufrate an einer senkrechten Wand und einer 1:1,5 geneigten B¨ oschung f¨ ur Versuche mit und ohne Freibord, Student study at Leichtweiss-Institute for Hydraulics. Braunschweig (2001) (in German). 19. T. Bruce, L. Franco, P. Alberti, J. Pearson and N. W. H. Allsop, Violent wave overtopping: Discharge throw velocities, trajectories and resulting crown deck loading, Proc. Ocean Wave Measurement and Analysis (“Waves 2001”), ASCE (1999), pp. 1783–1796. 20. G. Wolters, G. M¨ uller, T. Bruce and C. Obhrai, Large scale experiments on wave downfall pressures on vertical and steep coastal structures, Proc. ICE Maritime Engineering 158, 137–145 (2005).

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Chapter 17

Surf Parameters for the Design of Coastal Structures Dong Hoon Yoo Department of Civil Engineering, Ajou University Suwon, 443-749, Korea [email protected] Iribarren number is widely used by coastal engineers for classifying the type of breaking waves and for estimating armor weight, run-up height, and even for estimating wave overtopping. Iribarren number is the ratio of breakwater slope or beach slope to the square root of wave steepness. Another existing surf parameter is the ratio of breakwater slope to the wave steepness. In the present chapter new surf parameters, which are called “ﬁrst order wave action slope” and “second order wave action slope,” are introduced for representing the local wave conditions in shallow waters by employing local values of wavelength as well as wave height. The use of linear wave theory on a ﬂat bed of the depth at the front of breakwater might be considered far better than the simple adoption of the deepwater wavelength for characterizing surﬁng waves at a shoaling depth. The wave action slopes are formed by the product of the breakwater slope and the celerity ratio to the wave height. The run-up height is related to the ﬁrst order wave action slope, and the optimum or minimum weight of armor unit is related to the second order wave action slope.

17.1. Introduction When waves approach a coast, wave particles transform from elliptic movement to horizontal movement and energy transport velocity (group velocity) continues to reduce to near zero. Then wave heights increase up to a certain limit, that is, until they lose their stability of wave formation. The type of breaking waves is one of the major factors for the determination of wave forces aﬀecting beaches and coastal structures. Iribarren1 suggested a wave breaking parameter to determine the type of breaking waves and the Iribarren number is in recent years widely employed for the estimation of run-up height and unit weight of armor block. Iribarren number is the ratio of beach slope to the square root of wave steepness with introducing 441

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Fig. 17.1.

Wave deformation on a beach slope.

deepwater wavelength. The use of deepwater wavelength is reckoned to be largely due to the diﬃculty of measuring or estimating the wavelength at a local point. In practice, the deﬁnition of wavelength does not seem to be clear particularly when waves approach a beach slope or sloping breakwater, as the water depth varies from deep to shallow as shown in Fig. 17.1. The wavelength of oﬀshore direction from the breaking point is certainly longer than the wavelength of onshore direction on a sloping beach. It is, therefore, often practice to use a deepwater wavelength instead of a local value, although a local value of wave height is normally employed in order to represent partly the local condition at the breaking point. It may possibly be very diﬃcult to estimate or measure or even deﬁne the wavelength at a sloping bed in shallow waters. But the wavelength of the local point water depth given by any wave theory, assuming the sloping bed is ﬂat, may be far better than the deepwater wavelength for the estimation of wave steepness at a local point. Iribarren adopted the square root of wave steepness for his nondimensional physical number. Direct use of wave steepness also results in nondimensional physical number, which yields new form of Iribarren number. On the other hand, new surf parameters are formulated by introducing Froude number which is the ratio of wave celerity to group velocity of wave height. On the result four sets of surf parameters are suggested, and it is found that one set is related to the other set. All of them are employed for developing empirical equations of run-up height and unit weight of armor block. 17.2. Iribarren Numbers Using local values of wavelength and wave height, the ﬁrst order Iribarren number I is represented by1 : S I= √ , M

(17.1)

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where M = H/L (wave steepness) and S = tan θ (beach slope or breakwater slope). Oﬀshore Iribarren number Io uses the values of wave height Ho and wavelength Lo at deepwater for the estimation of wave steepness. Inshore Iribarren number Ii uses the local value of wave height but still uses the value of wavelength at deepwater condition. On the other hand, the local Iribarren number I takes the local values of both wave height and wavelength for the estimation of wave steepness. The second order Iribarren number J is also, using local values of wavelength and wave height, represented by: J=

S . M

(17.2)

The second order Iribarren number is directly related to the wave steepness, while the ﬁrst order Iribarren number is related to the square root of the wave steepness. The former is associated with the square of wave celerity, and the latter with the wave celerity with no power. Similarly, oﬀshore second order Iribarren number Jo uses the values of wave height Ho and wavelength Lo at deepwater for the estimation of wave steepness. Inshore second order Iribarren number Ji uses the local value of wave height but still uses the value of wavelength at deepwater condition. On the other hand, the local second order Iribarren number J takes the local values of both wave height and wavelength for the estimation of wave steepness. 17.3. Wave Action Slopes Wave action slope is the product of the beach slope S and the Froude number associated with wave celerity C and wave height H as suggested by Yoo and Na.2 Two types of wave action slopes are suggested; ﬁrst order and second order. The ﬁrst order wave action slope SX is represented by: SX = FH S, (17.3) √ where FH = C/ gH. Similarly, the oﬀshore ﬁrst order wave action slope SXo uses the values of wave height Ho and wave celerity Co at deepwater. And the inshore ﬁrst order wave action slope SXi uses the local value of wave height but still uses the value of wave celerity at deepwater condition. On the other hand, the local ﬁrst order wave action slope takes the local values of both wave height and wave celerity. The second order wave action slope SY (Ref. 3) is represented by: 2 S. SY = FH

(17.4)

Oﬀshore second order wave action slope SY o uses the values of wave height Ho and wave celerity Co at deepwater. And inshore second order wave action slope SY i uses the local value of wave height but still uses the value of wave celerity at deepwater condition. On the other hand, the local second order wave action slope takes the local values of both wave height and wave celerity. Using dispersion relation as follows: σ 2 = gk tanh kh,

(17.5)

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444 Table 17.1.

Relationship of wave action slopes and Iribarren numbers. SXo

SXi

Deﬁnition

√Co S gHo

√Co S gH

Relationship

√1 Io 2π

√1 Ii 2π

SX q

SY o

√C S gH

2 Co gHo

tanh kh I 2π

1 J 2π o

S

SY i 2 Co

gH

S

1 J 2π i

SY C2 S gH tanh kh J 2π

where σ is the angular frequency (σ = 2π/T ) and k is the wave number (k = 2π/L), the local ﬁrst order wave action slope SX is related with the local ﬁrst order Iribarren number I as follows: tanh kh C SX = √ . (17.6) S=I 2π gH √ For√the oﬀshore condition tanh kh ≈ 1, and hence SXo = Io / 2π and SXi = Ii / 2π. That is, the ﬁrst order wave action slopes are found simply proportional to the ﬁrst order Iribarren numbers. Similarly, the local second order wave action slope SY is related with the local second order Iribarren number J as follows: SY =

C2 tanh kh S=J . gH 2π

(17.7)

For the oﬀshore condition tanh kh ≈ 1, and hence SY o = Jo /2π and SY i = Ji /2π. That is, the second order wave action slopes are found simply proportional to the second order Iribarren numbers. The ﬁrst order wave action slope is closely related to the ﬁrst order Iribarren number and the second order wave action slope is closely related to the second order Iribarren number. It is, however, found that the relations do not have simple linear proportionality. The wave action slopes are related to the Iribarren numbers as presented in Table 17.1. As summarized in the table, the oﬀshore numbers and the inshore numbers are linearly proportional to each other, but the local numbers are not directly proportional to each other. Further eﬀect of depth dispersion is reﬂected in the wave action slopes through dispersion relation.

17.4. Run-up Height The run-up height is the vertical length from the mean surface to the top of runup reach, and the run-up height ratio is deﬁned by the ratio of run-up height to wave height. Saville4 conducted laboratory experiments on a concrete slope with various conditions of beach slope and wave steepness for the monochromatic waves, and presented two separate graphs of run-up height ratio to the slope with various inshore wave steepness (Mi = H/L0 ) as shown in Fig. 17.2(a). It is found that the run-up height ratio is almost proportional to the beach slope up to a certain limit, from where reﬂection might be a signiﬁcant factor on the wave particle movement on the slope. Depending on the water depth condition, the run-up height ratio is

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Fig. 17.2.

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Run-up height ratio ηR versus (a) S or (b) SXi using the laboratory data of Saville.4

related to the Iribarren number or ﬁrst order wave action slope as follows: ηR = 2.64SXi ,

(SXi < 0.427Mi−0.25, 1 < h/H < 3),

(17.8a)

ηR = 2.39SXi ,

(SXi < 0.674Mi−0.16, 3 < h/H),

(17.8b)

where ηR = hR /H (run-up height ratio), hR is the run-up height, H is the wave height. As shown in Fig. 17.2(b), the eﬀect of wave period is found well implied in the expression of inshore wave action slope SXi . Further simpliﬁcation can be made by the use of local wave action slope, and the result is given by: ηR = 2.80SX (SX < 0.394M −0.25).

(17.9)

When SX < 0.394M −0.25, the inﬂuence of reﬂection is considered very small. Otherwise, the eﬀect of reﬂection becomes strong and then the run-up height becomes almost irrespective of slope or the wave action slope. Mase5 conducted laboratory experiments of run-up on smooth concrete slope (S = 0.033 − 0.25) for irregular waves, and suggested empirical equation of run-up height related to the oﬀshore Iribarren number as follows: ηR−n = αIoβ ,

(17.10)

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Table 17.2. Coeﬃcients α, β, and δ for the estimation of run-up height ratio.

hR−n hR−max hR−2 hR−1/10 hR−1/3 hR−avr

Eq. (17.10)

Eq. (17.11)

Eq. (17.12)

α 2.32 1.86 1.70 1.38 0.88

α — 0.37 0.34 0.27 0.17

α 0.41 0.44 0.38 0.30 0.20

β 0.77 0.71 0.71 0.70 0.69

β — 1.38 1.27 1.04 0.66

β 2.2 1.64 1.52 1.25 0.79

Eq. (17.13) δ 1 0.742 0.688 0.568 0.359

where ηR = hR /HS , HS is the signiﬁcant wave height, and the subscript n indicates the reaches of diﬀerent probability. The values of constants α and β for diﬀerent probability are presented in Table 17.2. Hedges and Mase6 re-analyzed the laboratory data of run-up height of irregular waves of Mase.4 They deducted the set-up height from the total run-up height, and related the pure run-up height to the inshore Iribarren number as follows: ηR−n = α + βIi ,

(17.11)

where α and β are constants, but the values vary depending on the probability of run-up reaches as presented in Table 17.2. For the case of irregular waves, the run-up height ratio is determined by using signiﬁcant wave height HS , that is, ηR = hR /HS . For the case of 2% reach ηR−2 they suggested α = 0.37 and β = 1.38. New empirical equation of run-up height ratio is developed using the local Iribarren number as follows: ηR−n = α + βI.

(17.12)

Comparison of the equations (17.10), (17.11), and (17.12) are shown in Fig. 17.3 against Mase’s laboratory data.5 Up to the value of Ii = 2, Eqs. (17.10) and (17.11) show very good correlation with the laboratory data. But when the inshore ﬁrst order Iribarren number exceeds approximately 2, the results of Eqs. (17.10) and (17.11) deviate considerably from the laboratory data. On the other hand, when using the local Iribarren number as expressed by Eq. (17.12), good correlations are found over the whole region of I. Another empirical equation of run-up height ratio is developed using the local wave action slope SX instead of the inshore Iribarren number Ii or the local Iribarren number I. Hedges and Mase6 used diﬀerent separate values of constants of Eq. (17.10) or (17.11) for diﬀerent reaches of probability in order to keep the accuracy. For the engineering use the run-up height of diﬀerent reaches of probability ηR−n can be simply related to the maximum run-up height as follows: ηR−n = δηR−max ,

(17.13)

where δ is the reduction factor of probability. The maximum run-up height ratio of irregular waves ηR−max is expressed by: ηR−max = 0.32 + 6.9SX

(17.14)

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Fig. 17.3.

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Run-up height ratio versus Io , Ii , I, and SX using the laboratory data of Mase.5

or ηR−max = 8.1SX .

(17.15)

The degree of accuracy of Eq. (17.11) or (17.12) is slightly better than that of Eq. (17.13), but Eq. (17.13) is also acceptable for an engineering use with less than 5% endurance. The reduction factor of probability δ = 1 for the maximum reach, δ = 0.742 for the 2% reach, δ = 0.688 for the 10% reach, δ = 0.568 for the 1/3 reach, and δ = 0.359 for the average reach as summarized in Table 17.2. The computation results are presented in Fig. 17.4. As shown in the ﬁgure, the correlation of Eq. (17.13) is generally found very good simply by introducing the

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448

Fig. 17.4.

Various run-up height ratios versus SX using the laboratory data of Mase.5

reduction factor. The correlation coeﬃcients are estimated over 0.97 for all cases, and the determination coeﬃcients are about 0.93 for most cases. Comparing both constants of Eqs. (17.9) and (17.15) each other, the maximum run-up reach of irregular waves is found almost tripple times bigger than the run-up reach of monochromatic waves. The reduction factor of average reach of irregular waves is about 0.36, and this is almost the same as the ratio of the two constants of Eqs. (17.9) and (17.15). The run-up height can be signiﬁcantly reduced for the rough slope of breakwaters from the values given for the smooth concrete slope. Ahrens7 suggested to use another reduction factor for the rough wall. The run-up height is, hence, ﬁnally computed by: ηR−n = ξδηR−max ,

(17.16)

where ξ is the reduction of run-up height for rough wall. ηR−max is the maximum run-up height ratio on smooth concrete slope. The values of ξ are given in Table 17.3. 17.5. Armor Weight The equation of Hudson8 is still world-widely employed for the estimation of minimum or optimum weight of armor block. Hudson equation is expressed by: ηW =

S , (s − 1)3 KD

(17.17)

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449

Reduction factor of wall roughness for the estimation of run-up height.

Quality of a slope

Deferment system

Roughness coeﬃcient

— Fitted Fitted Fitted — — Fitted Random Random Random Random

1.00 0.90 0.85∼0.90 0.85∼0.90 0.85∼0.90 0.80 0.75∼0.80 0.60∼0.65 0.60∼0.65 0.50∼0.55 0.45∼0.50

Smooth and impermeable slope Concrete block Basalt block Gobi block Grass One layer of quarry stone Quarry stone Rounded quarry stone Three layer of quarry stone Quarry stone Concrete slope of 50% percolation

where ηW = W/γH 3 , W is the armor weight, γ is the speciﬁc weight of armor block, s is the speciﬁc gravity of armor block, and KD is the empirical constant which is primarily associated with the type of armor block. Hudson equation indicates that the weight ratio is solely dependent on the breakwater slope, but various workers found that the optimum weight of armor block is also signiﬁcantly inﬂuenced by wave period, percolation of slope, and the formation of irregular waves. van der Meer9 employed the inshore ﬁrst order Iribarren number, and suggested a set of two equations depending on the condition determined by the Iribarren number. The equation of van der Meer9 is represented by: ηW =

0.0042NS Ii1.5 (s − 1)3 P 0.54

(0.4 < Ii < 2.5),

(17.18a)

NS P 0.39 S 1.5 (2.5 < Ii ), (17.18b) (s − 1)3 Ii3P √ where P is the percolation, NS = ( N/D)0.6 , N is the number of incident waves, and D is the degree of damage which is deﬁned by the ratio of damage area to the projected area of armor unit. The number of incident waves for a ﬁeld is normally between 1000 and 3000, and the degree of damage is between 1 and 3. The armor weights measured in the laboratory by van der Meer9 are related to the Iribarren number Ii and the second order wave action slope SY . As shown in Fig. 17.5, the use of the second order wave action slope gives the best correlation with the laboratory data. As found by van der Meer,9 the variation of ηW becomes complicated when Ii is bigger than 2.5. New empirical equation has been developed using the second order wave action slope as follows: ηW =

ηW =

0.178KY SY , (s − 1)3

(17.19)

where KY is a constant depending on percolation, degree of damage, the condition of reﬂection, but primarily depending on the type of armor block. The constant 0.178 is determined by adopting the values of N = 3000, D = 3, and P = 0.1. The empirical constant KY = 1 for the armored slope of three-layered round stones.

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Armor block weight ratio versus Ii and SY (laboratory data of van der Meer9 ).

Sealock is one of the most well-devised concrete armor blocks. It is widely used in Japan and recently introduced to Korea. The stability of the block is well tested in the laboratory experiments. The experiments were conducted in the channel of 1.5 m deep, 1.0 m wide, and 30 m long against the breakwater of slope S = 3/4. The number of incident waves was 500. Waves are generated with four diﬀerent wave periods 1.3, 1.5, 1.8, 2.1, and 2.4 s, and six or seven diﬀerent wave heights for each wave period between 0.15 m and 0.24 m. The degree of damage (DS ) is deﬁned by the percentage of failure units, and it varies from 0.0 to the maximum value 4.9. For the zero damage level Hudson parameter of Sealock was estimated 12.0 for T = 2.4 s, 14.7 for T = 2.1 s, 21.3 for T = 1.8 s, 22.0 for T = 1.5 s, and 17.8 and 28.5 for T = 1.3 s as shown in Fig. 17.6(b). The values of Hudson parameter are found to become small for longer wave period, that is, longer waves are found more damageable to the stability of armored breakwaters. The parameter KD of the Hudson equation is found strongly dependent on the wave period. From the results Hudson parameter KD is almost proportional to the degree of damage but strongly dependent on the wave period as follows: KD = 3.874DS + 23.31 (T = 1.3 ∼ 1.8 s),

(17.20a)

KD = 3.529DS + 13.51 (T = 2.1 ∼ 2.4 s).

(17.20b)

For the safety with a conﬁdence Samsung Hydraulic Engineering suggested that KD equals to 10.0 for the armor block Sealock.10 The set of the data is re-analyzed by relating them to the local second order wave action slope SY . The results are demonstrated in Figs. 17.6(c) and 17.6(d). The ﬁgure demonstrates that even the local second order wave action slope does not seem to include suﬃciently well the eﬀect of wave period. There still exists some deviation between the regression lines of diﬀerent wave period. It may be further reﬁned by considering the eﬀect of set-down and set-up for the computation

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Fig. 17.6.

451

Weight ratio of armor block Sealock versus S with KD and SY with KY .

of the local wavelength or wave celerity. The variation of the parameter KY is not ignorable as shown in Fig. 17.6(d). Nevertheless, the variation of the parameter KY becomes almost irrespective of wave period and it is given by: KY = −0.010DS + 0.140.

(17.21)

When the degree of damage deﬁned by Samsung Hydraulic Engineering is zero, KY for the armor unit Sealock is approximately 0.14. The values of the parameter KY for various armor blocks are suggested in Table 17.4. The values of three-layered rounded quarry stones and Sealocks are determined by analyzing the laboratory data, and other values are deduced by relating it with the Hudson parameter KD . More conﬁdent values should be determined by testing each relevant set of laboratory data.

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452 Table 17.4.

KD and KY of various armor blocks. KD

Two-layer rounded quarry stones Three-layer rounded quarry stones Sharp edged quarry stones TTP Dolos T-bar Sealock

KY

Nonbreaking

Breaking

(Nonbreaking)

2.4 3.2 4.0 8.0 31.8 10.0 10.0

1.2 1.6 2.0 7.0 15.8 7.0 10.0

1.33 1.0 0.8 0.4 0.1 0.32 0.14

17.6. Conclusions In the present chapter, several new surf parameters are presented primarily using the local values of wavelength as well as wave height. The ﬁrst one has the similar form to the Iribarren number, and it is called “second order Iribarren number.” The second order Iribarren number is related to the wave steepness, while the original (ﬁrst order) Iribarren number is related to the square root of wave steepness. The second group is called “wave action slope.” √ It is the product of the beach slope S and the Froude number deﬁned by FH = C/ gH, where C is the wave celerity and H is the wave height. The ﬁrst order wave action slope is FH S(= SX ), the second order 2 wave action slope is FH S(= SY ). The ﬁrst order wave action slope is closely related to the ﬁrst order(original) Iribarren number and the second order wave action slope is closely related to the second order Iribarren number. It is, however, found that the relations do not have simple linear proportionality. Run-up height is related to the ﬁrst order wave action slope, and unit weight of armor block is related to the second order wave action slope. Both relations have simple linear forms, and excellent agreements are found in comparison with laboratory data. The eﬀects of wave period or wavelength are found to be well included in the new parameters. The use of the new parameters are considered to be suitable particularly when the wave period is relatively big and the water depth is relatively deep. Acknowledgment This research was supported by Heyin ENC. References 1. R. C. Iribarren, Generalization of the formula for calculation of rock ﬁll dikes and veriﬁcation of its coeﬃcients (Revista de Obras Pulicas, Madrid, 1950 WES Translation N.51-4). 2. D. H. Yoo and D. Y. Na, Hydraulic condition of breaking waves, J. Korean Soc. Civil Eng. 25(2B), 159–163 (2005) (in Korean).

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3. D. H. Yoo, S. K. Koo and I. H. Kim, Minimum weight of breakwater armor unit, Proc. 1st Asian and Pacific Coastal Engineering Conf., Dalian, China (2001), pp. 605–612. 4. T. J. P. Saville, Wave run-up on shore structures, Proc. ASCE, Paper No. 925 82(WW2), 359–373 (1956). 5. H. Mase, Random wave run-up height on gentle slope, J. Waterway, Port, Coastal, and Ocean Eng. 115(5), 649–661 (1989). 6. T. S. Hedges and H. Mase, Modiﬁed Hunt’s equation incorporating wave setup, J. Waterway, Port, Coastal and Ocean Eng. ASCE 130(3), 109–113 (2004). 7. J. P. Ahrens, Approximate Upper Limit of Irregular Waves Run-Up on Riprap, Department of the US Army Corps of Engineers (1988). 8. R. V. Hudson, Laboratory investigation of rubble-mound breakwater, ASCE, Trans., Paper No. 3213, 126 (1961). 9. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Delft Hydraulics Communication No. 336 (1988). 10. Samsung Hydraulic Engineering, Hydraulic experiments of Sealock, Report No. 11 (1978) (in Japanese).

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Chapter 18

Development of Caisson Breakwater Design Based on Failure Experiences Shigeo Takahashi Port and Airport Research Institute 3-1-1, Nagase, Yokosuka 239-0826, Japan takahashi [email protected] The experiences of caisson breakwater failures are explained and the current design of caisson breakwaters is described to prevent such failures. The design method of conventional caisson against wave forces is especially explained in this chapter. The design method of caisson breakwaters is still under development including a new design method of performance design. The future direction will be discussed in the ﬁnal section.

18.1. Introduction Composite type breakwater is one of major types of breakwaters in the world, consisting of vertical wall installed on a rubble mound foundation. To resist the enormous power of storm waves, concrete caissons are usually employed as the vertical wall.1 The composite breakwater with caissons (caisson breakwater) requires a smaller body width and quantities of material compared with conventional rubble mound breakwaters. This is one of the biggest advantages of using a caisson breakwater, which makes the breakwater construction more economical, especially in deep water. The design and construction methods of the caisson breakwater have developed rapidly in the latter half of the 20th century through many bitter experiences of failures in Japan. Especially, the design technique for the caisson breakwaters is nearly established including the calculation method for determining the wave forces acting on the breakwater. In addition, the improvement of the conventional caisson breakwaters has been encouraged. The horizontally composite caisson breakwater is an improved version of the conventional composite caisson breakwater. Such breakwaters are not new, however, since vertical wall breakwaters suﬀering damage to the vertical walls were often strengthened by placing large stones or concrete blocks in front of them so as 455

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to dissipate the wave energy and reduce the wave force, and is now frequently being constructed especially in shallow surf zone in Japan. Various new caisson breakwaters have been invented and commercialized in order to mitigate the drawbacks associated with conventional composite breakwaters. One of typical new caisson breakwater is the perforated wall caisson breakwater invented by Jarlan2 in 1961. The caisson dissipates wave energy by front perforated wall and wave chamber. Many breakwaters of this type were subsequently constructed throughout the world with further improvement. In this chapter, the experiences of caisson breakwater failures are explained and the current design of caisson breakwaters is described to prevent such failures. The design method of conventional caisson breakwaters against wave forces is especially described in this chapter. The design method of caisson breakwaters is still developing including an integrated method of performance design. The future direction will be discussed in the ﬁnal section.

18.2. Failures of Caisson Breakwaters It is very important to learn from the actual failures. The design methods of the breakwaters have been developed signiﬁcantly from such experiences, and the development of the design methods has prevented them from suﬀering failure. It should be noted that the total length of breakwaters in Japan is more than 800 km and the failures of the breakwaters are now very limited compared with the total amount of caissons in the country. Before explaining each failure, it should be noted that the most important cause of failures of caisson breakwaters is the caisson sliding and breakage due to impulsive wave pressures. It should also be noted that the rubble mound/rubble foundation of composite breakwaters is vital to prevent the failure of the upright section by scouring, as well as stabilizing the foundation against the wave force and caisson weight.3

18.2.1. Meandering failures Photo 18.1 shows an oﬀshore breakwater at Sendai Port after Typhoon 9119 hit causing several caissons to slide. The length of the breakwater is 700 m. Although the attacking waves were estimated to be about 20% higher than the design wave height, only caissons at particular locations suﬀered sliding. This is called “Meandering Sliding.” The breakwater consisted of caissons of 11.8 m wide on 6 m thick rubble foundation. The water depth is 21 m, and the estimated incident wave was H1/3 = 6.8 m and T1/3 = 12 s, with an incident wave angle 65◦ . The waves attacking the caisson were not breaking waves but nonbreaking waves. This meandering sliding is a typical sliding phenomenon due to nonbreaking waves. This is caused by diﬀracted waves from breakwater heads in an oblique wave condition as discussed in Sec. 18.3.3.

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Photo 18.1.

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Meandering failure of caisson.

Photo 18.2.

Impulsive wave pressure.

18.2.2. Impulsive wave pressures Photo 18.2 shows a wave hitting the oﬀshore side of a caisson at Minamino-hama Port. The breakwater forms a jetty type breakwater designed to protect small ferryboats, with its rear side to be used as a quay wall. Big splash in the photo is typical, when an impulsive breaking wave force act on the vertical wall. During a typhoon, waves equivalent to the design wave or larger attacked the breakwater head caisson from the breakwater alignment direction. Plunging breakers almost completely destroyed the caisson at the breakwater head. Caisson damage started when the sidewall of the caisson began breaking, then progressed to the whole caisson. Such caisson breakage was caused by impulsive wave pressures acting on a caisson installed on a steep seabed slope. Actually, the breakwater was under construction and the damaged caisson was going to be protected

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by another caisson which was designed to resist such severe wave pressures, but unfortunately not in place before the storm. Similar failure due to impulsive wave pressure occurs when breaking waves acting on a caisson installed on a high/wide rubble foundation. Impulsive wave pressures occur when the vertical wall is attacked by an almost vertical wave front, and therefore larger vertical wave front due to plunging or surging breakers gives larger impact pressures. Such caisson failures due to impulsive pressures had occurred frequently in history but have recently been greatly reduced using accumulated knowledge about impulsive wave pressures including impulsive pressure coeﬃcient as discussed in Sec. 18.3.2. 18.2.3. Scattering of armor for rubble foundation Photo 18.3 shows a typical caisson failure at a breakwater head, where the caisson moved toward the harbor side. It should be noted that the caisson was not moved by wave force, rather by scouring of the rubble foundation. Photo 18.4 shows scouring of the rubble foundation at the breakwater head in a model experiment. It is known that very strong wave-induced current occurs at the corner of the breakwater head caisson. Scattering of armor stones occurs when the weight of armor stones was insuﬃcient against very high water particle velocity around the breakwater head. In such cases scour of the rubble mound and the sandbed under the rubble mound occur. This type of failure can particularly occur during the construction period, although the design method against such high water particle velocity is well established as explained in Sec. 18.3.4. 18.2.4. Scouring of rubble stones and seabed sand due to oblique waves Figure 18.1 shows the inclined caisson in a relatively calm harbor. This is due to the scouring of rubble mound stones and the sandbed under the rubble mound.

Photo 18.3.

Breakwater head.

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Scouring

Photo 18.4.

Fig. 18.1.

Scouring of rubble mound around head.

Scouring of rubble mound along caisson.

Oblique waves caused a strong wave-induced current along the breakwater caissons, although the wave height is not large. An estimation method of the wave-induced current in front of the caisson due to normal as well as oblique waves was already proposed and was included in the current design.4 18.2.5. Erosion of front seabed (scouring of mound toe) Figure 18.2 shows a cross section of a large composite breakwater which suﬀered severe erosion of the mound toe area. Due to high waves, which exceeded 7 m in signiﬁcant wave height, the front area was scoured more than 3 ms. This phenomenon is not seldom when such high waves attack a breakwater. The front erosion of breakwaters comprises two phenomena; large-scale sea bottom change and local front scouring. It is really diﬃcult to protect a breakwater from such front erosion although Irie et al.5 described its fundamental mechanism. Only empirical countermeasures such as a gravel mat or asphalt mat are usually adopted to reduce such scouring.

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Fig. 18.2.

Photo 18.5.

Erosion at rubble mound toe.

Settlement of caisson due to through-wash.

It should be noted that protecting the caisson is essential even though some part of the rubble foundation is scoured. The rubble foundation is usually designed with an enough length considering its deformation due to front erosion.

18.2.6. Seabed through-wash Photo 18.5 shows settlement of a breakwater due to through-wash of the sandbed under it.6 The relatively ﬁne sand under the rubble mound was washed away by severe wave actions. This type of damage is normally prevented by placing a geotextile sheet under the rubble foundation as discussed in Sec. 18.3.4. If, however, high waves hit the breakwater during construction, this may lead to improper placement of the geotextile sheet, which results in settlement as shown here.

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18.2.7. Rubble foundation failure Breakwater failure due to foundation failure is seldom seen since the current design method seems to be a little conservative which evaluates the bearing capacity of rubble mound and seabed foundation using the Bishop method.7 Figure 18.3 shows a special case of that due to rubble mound failure. An asphalt mat was placed under the caisson to increase the friction coeﬃcient between the caisson and the rubble foundation. Due to high waves exceeding the design wave, sliding took place, which did not occur at the caisson bottom, rather in the foundation. It was thought that this sliding occurred between the rubble mound and sandbed.

Fig. 18.3.

Rubble mound failure.

Fig. 18.4.

Failure path diagram.

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Use of an asphalt mat is very eﬀective in reducing the necessary width of the caisson, and has been widely applied in recent designs. Care must be taken to check foundation stability, however. 18.2.8. Failure path diagram Figure 18.4 is a failure path diagram of composite breakwaters. It shows the weaker parts of breakwaters and major causes of failures that were already explained above. For example, due to wave-induced strong current around the breakwater head, scattering of armor blocks around the head occurs, which results in settlement of mound and therefore the settlement of the caisson. Then caisson can easily slide. It should be noted that damage to breakwaters seldom occurs even when storm waves exceed the design wave. Typically, only a part of the breakwater weaker than other parts suﬀers damage due to storm wave heights less than the design wave height. The failure can be reduced by more careful design especially against armor layer scattering and seabed scouring. It will be also eﬀective to reduce failures to include the wave height increase along the breakwater alignment. Most of the design methods against such failures are already established but a new integrated design method should be developed further to increase the reliability and to reduce the total cost of breakwaters.

18.3. Design of Conventional Caisson Breakwaters 18.3.1. Example of a caisson breakwater Figure 18.5 shows a typical cross section of a conventional caisson breakwater. The upright section is a 21.5 m × 27 m caisson installed on a 3.5-m-thick rubble foundation. The depth of the caisson h is 12.5 m and the height of the crown hc is 6.1 m at LWL. The caisson is divided into 5 × 6 chambers by 20-cm-thick inner walls and 40-cm-thick outer walls. The chambers are ﬁlled with sand, capped by concrete, and a concrete superstructure is placed on the caisson.

Fig. 18.5.

Typical cross section of caisson breakwater, Noshiro Port.8

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Foot protection blocks are placed to prevent through-wash of the rubble foundation and the sand below, while shape-designed concrete blocks are installed to act as the armor layer of the rubble foundation. The water depth d above the rubble mound including the armor layer is 10 m at LWL, and the berm width BM of the rubble mound is 12.8 m. For scour protection, additional gravel is placed, being called a “gravel mat.” A vinyl sheet is also used to prevent scouring of the sand under the rubble foundation. This breakwater is designed to withstand a wave of HD = 11.6 m (signiﬁcant wave height H1/3 = 6.66 m), signiﬁcant wave period T1/3 = 13 s, and wave angle θ of 22◦ at a water depth h of 21 m. The design signiﬁcant wave in deepwater H0 is 12.2 m at a return period of 50 years. The design wave at the breakwater site was evaluated using a wave transformation calculation, with wave pressures on the caisson being evaluated by the extended Goda pressure formula. 18.3.2. Wave forces on vertical walls 18.3.2.1. Extended Goda formula (Goda–Takahashi formula) In 1973, Goda9 used his own theoretical and laboratory studies to establish a comprehensive formula to calculate the design wave forces. After a later modiﬁcation to account for the eﬀect of oblique wave incidence, this formula was successfully applied to the design of vertical breakwaters built in Japan. The original Goda formula10,11 has many advantageous features, with the main ones being as follows: (1) It can be employed for all wave conditions, i.e., both for standing and breaking waves. (2) The formula’s design wave is the maximum wave height and can be evaluated by given diagrams and/or equations. (3) It is partially based on nonlinear wave theory and can represent wave pressure characteristics by considering two pressure components: the breaking and slowly varying pressure components. Consequently, it is relatively easy to extend the Goda formula in order to apply it to other vertical wall-type structures. (4) The Goda formula clariﬁes the concept of uplift pressure on the caisson bottom, since the buoyancy of the upright section in still water and its uplift pressure due to the wave action are deﬁned separately. The distribution of the uplift pressure has a triangular shape. The Goda formula was subsequently extended to include the following parameters: (1) The incident wave direction,12 (2) The modiﬁcation factors applicable to other types of vertical walls, (3) The impulsive pressure coeﬃcient.13 In the extended Goda formula (named here the Goda–Takahashi formula), the wave pressure acting along a vertical wall is assumed to have a trapezoidal distribution both above and below the still water level, while the uplift pressure acting

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Fig. 18.6.

Goda pressure formula.

on the bottom of the upright section is assumed to have a triangular distribution as shown in Fig. 18.6. The buoyancy is calculated using the displacement volume of the upright section in still water at the design water level. As indicated, h denotes the water depth in front of the breakwater; d, the depth above the armor layer of the rubble mound foundation; h , the distance from the design water level to the bottom of the upright section; and hc , the elevation of the breakwater above the design water level. The elevation at which the wave pressure is exerted, η ∗ , and the representative wave pressure intensities p1 , p3 , p4 , and pu can be written in a generalized form as η ∗ = 0.75(1 + cos θ)λ1 HD , p1 = 0.5(1 + cos θ)(λ1 α1 + λ2 α∗ cos2 θ)w0 HD , p 3 = α3 p 1 , p 4 = α4 p 1 , pu = 0.5(1 + cos θ)λ3 α1 α3 w0 HD , in which

α1 = 0.6 + 0.5

(4πh/LD ) sinh(4πh/LD )

h∗c , η∗

h∗c = min{η ∗ , hc },

(18.2) (18.3)

2 ,

α∗ = max{α2 , αI }, hb − d (HD /d)2 2d , , α2 = min hb 3 HD 1 h 1− , α3 = 1 − h cosh(2πh/LD ) α4 = 1 −

(18.1)

(18.4) (18.5) (18.6) (18.7) (18.8) (18.9)

where θ is the angle between the direction of wave approach and a line normal to the breakwater;

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λ1 , λ2 , λ3 are the modiﬁcation factors dependent on the structural type; HD , LD are the wave height and wave length applied to calculate design wave forces; αI is the impulsive pressure coeﬃcient; w0 is the speciﬁc weight of sea water (= ρo g); hb is the oﬀshore water depth at a distance ﬁve times the signiﬁcant wave height H1/3 ; min{a, b}, the minimum of a and b; and max{a, b}, the maximum of a and b. 18.3.2.2. Pressure component and pressure coeﬃcients (α1 , α2 , and αI ) Figure 18.7 shows the transition of wave pressure from nonbreaking to impulsive pressure, where the pressure component is indicated by coeﬃcients α1 , α2 , and αI . α1 represents the slowly varying pressure component and α2 the breaking pressure component, while αI represents the impulsive pressure component, which includes the dynamic response eﬀect of the caisson sliding. α1 increases from 0 to 1.1 as the relative depth decreases, and α2 increases as d/hb decreases, though it peaks and then decreases as d/hb decreases, ranging 0–1.0. The value of αI is described next in more detail. 18.3.2.3. Impulsive pressure coeﬃcient αI Takahashi et al.13 obtained the impulsive pressure coeﬃcient αI by reanalyzing the results of comprehensive sliding tests, being a nondimensional value representing the impulsive pressure component, which should be regarded as an additional eﬀect to the slowly varying pressure component. The eﬀect of the dynamic (impulsive) pressure indicated by α2 in Goda’s formula does not under all conditions accurately estimate the eﬀective pressure (equivalent static pressure) due to impulsive pressure, and therefore, αI was introduced. Figure 18.8 shows a calculation diagram for αI , in which it is expressed by the product of αI0 and αI1 , where αI0 represents the eﬀect of wave height on the

Fig. 18.7.

Transition of wave pressure.

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Fig. 18.8.

Calculation diagram of impulsive pressure coeﬃcient.13

mound, i.e., αI = αI0 αI1 , H/d when H ≤ 2d, αI0 = 2.0 when H > 2d,

(18.10) (18.11)

and αI1 represents the eﬀect of the mound shape (shown by the contour lines). This term can be evaluated using   cos δ2 when δ2 ≤ 0, (18.12) αI1 = cosh δ1  0.5 −1 when δ2 > 0, [(cosh δ1 )(cosh δ2 ) ] 20δ11 when δ11 ≤ 0, (18.13) δ1 = 15δ11 when δ11 > 0, 4.9δ22 δ2 = 3δ22

when δ22 ≤ 0, when δ22 > 0,

h−d BM − 0.12 + 0.36 − 0.6 , = 0.93 L h h−d BM − 0.12 + 0.93 − 0.6 . = −0.36 L h

(18.14)

δ11 δ22

(18.15)

The value of αI reaches a maximum of 2 at BM /L = 0.12, d/h = 0.4, and H/d > 2. When d/h > 0.7, αI is always close to zero and is less than α2 . It should be noted that the impulsive pressure signiﬁcantly decreases when the angle of incidence θ is oblique.

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18.3.2.4. Modiﬁcation factors (λ1 , λ2 , and λ3 ) For the ordinary vertical breakwater, λ1 , λ2 , and λ3 are taken as unity since the Goda formula was originally proposed to describe this type of breakwater. The modiﬁcation factor λ1 represents the reduction or increase of the wave’s slowly varying pressure component, λ2 represents changes in the breaking pressure component (dynamic pressure component or impulsive pressure component), while λ3 represents changes in the uplift pressure. These modiﬁcation factors are introduced to express the pressures on other types of caisson breakwaters. 18.3.2.5. Design wave height The wave height and length applied to calculate the design wave forces are those of the highest wave in the design sea state. The height of this wave is taken as HD = Hmax = H1/250 = 1.8H1/3 seaward of the surf zone, or within the surf zone as the largest wave height Hb of random breaking waves at the water depth hb . The term H1/250 is the average height of the highest one-two hundred ﬁftieth waves. 18.3.2.6. Wave direction The wave angle θ is the angle between the direction of wave approach and a line normal to the breakwater alignment. Starting from its principal direction, the wave direction should be rotated toward the line normal to the breakwater alignment by up to 15◦ . This adjustment is made to compensate for both the uncertainty in estimating the wave direction and the waves’ directional spreading. 18.3.3. Other problems related to wave forces 18.3.3.1. Concave section Since the breakwater alignment is usually somewhat complex and a vertical breakwater is so reﬂective, the reﬂection and diﬀraction from it should be taken into account in wave force calculations. Figure 18.9 shows the calculated distribution of the ratio of the wave height to the incident wave, kd , in front of a vertical breakwater which has an alignment forming two lines having a concaved shape with respect to the incident waves. Due to the reﬂection from one of the lines of the breakwater, the wave height along the vertical wall is not simply 2 for standing waves. This eﬀect should be taken into account by considering the ampliﬁcation factor of the incident wave to be equal to kd /2. It should be noted that the ampliﬁcation factor is limited by standing wave breaking and that its maximum value is recommended to be 1.4 based on carrying out a series of experiments. 18.3.3.2. Meandering eﬀect Another problem related to the plane shape of the breakwater is the so-called meandering eﬀect.15 Figure 18.10 shows the calculated value of kd for a singleline detached breakwater with length LB = 200 m and wavelength L1/3 = 92.3 m,

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Fig. 18.9.

Fig. 18.10.

Wave height distribution along concaved breakwater alignment.14

Wave height distribution along detached one-line breakwater.14

where the wave height ﬂuctuates along the breakwater alignment and signiﬁcantly increases near the breakwater head. If waves exceeding the design wave height attack the breakwater, and if the breakwater caissons slide, the shape of these caissons will subsequently form a meandering shape. 18.3.4. Design of rubble mound foundation 18.3.4.1. Armor for rubble foundation 1. Wave force on armors The rubble mound foundation under a vertical wall should be protected to prevent it from scattering due to wave actions. This is accomplished by covering it with armor

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Fig. 18.11.

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Scattering of armor stones of rubble foundation.16

stones or concrete blocks. Figure 18.11 illustrates the movement of the armor stones in observed sections, where in the upper ﬁgure, the stones are scattered from the slope of the mound when incident waves approach perpendicular to the breakwater alignment. Note the occurrence of heavy scattering around the breakwater head. In the middle ﬁgure where θ = 45◦ , the armor stones not only scatter away from the slope, but in the ﬂat berm near the caisson as well. The stones also scatter around the breakwater head on its downstream side and especially at the caisson edge. In the lower ﬁgure with θ = 60◦ , the stones on the slope do not move, though they signiﬁcantly scatter in the ﬂat berm, especially near the caisson. It is postulated that the stability of armor stones or blocks is mainly threatened by the water particle velocity induced by the waves, i.e., by the drag and uplift forces produced by the water particle velocity. In Fig. 18.11, the water particle velocity is high where the stones moved, which supports this hypothesis. Figure 18.12 shows a hodograph of the water particle velocity at the breakwater trunk. When θ = 0, the water particle only moves perpendicular to the breakwater alignment and the velocity is almost zero near the vertical walls, being largest at the node of the clapotis (standing wave). As θ increases, the water velocity component parallel to the breakwater alignment increases, with the velocity near the vertical wall also signiﬁcantly increasing. Figure 18.13 shows the distribution of the peak water particle velocity around the breakwater head, where it is revealed that a high velocity occurs around the edge of the upright section.

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Fig. 18.12.

Fig. 18.13.

Water particle velocity.

Water particle velocity distribution around the breakwater head.

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To withstand velocity-induced forces, the armor stones or blocks should have enough weight, which can be evaluated using Isbash’s equation for stones embedded in the bottom of a sloped channel, i.e., W =

πγr U 6 , 48g 3 y 6 (Sr − 1)3 (cos α − sin α)3

(18.16)

where W is the necessary weight of armor stone; γr , the speciﬁc weight of armor stone (γr = ρr g); y, the Ishbash number (1.2 for embedded stones and 0.86 for stones placed on a ﬂat bottom); Sr , the speciﬁc gravity of stone; U , the water particle velocity on the stone; g, the acceleration of gravity; and α, the bottom slope. 2. Necessary weight of armor Ishbash’s equation relates the stable weight of stones to the water particle velocity. Brebner and Donnelly,16 however, proposed a method to directly determine the necessary weight from the wave height. In their method, the stable weight of armor units W can be expressed as W =

3 γr H1/3

Ns3 (Sr − 1)3

,

(18.17)

where γr denotes the speciﬁc weight of the armor unit; H1/3 , the design signiﬁcant wave height; and Ns , the stability coeﬃcient. This is a kind of Hudson’s equation, which uses Ns instead of KD cot α, such that H1/3 {(Sr − 1)(W/γr )1/3 } = Ns .

(18.18)

Ns depends on variables such as the shape of the armor unit, their manner of placement, the shape of the rubble mound foundation, and wave conditions (height, period, and direction). Tanimoto et al.17 proposed a formula to calculate the stability coeﬃcient for two layers of quarry stones, being based on analytical considerations and the results of random wave experiments. Takahashi et al.18 modiﬁed Tanimoto’s formula so that it can be applied to obliquely incident waves, i.e.,

1−κ h (1 − κ)2 h + 1.8 exp −1.5 Ns = max 1.8, 1.3 H1/3 H1/3 κ1/3 κ1/3 (18.19) in which κ = κ1 (κ2 )B , 2kh , κ1 = sinh 2kh (κ2 )B = max{αs sin2 θ cos2 (kBM cos θ), cos2 θ sin2 (kBM cos θ)},

(18.20) (18.21) (18.22)

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Fig. 18.14.

The calculated necessary weight of armor stones for rubble foundation.18

where max {a, b} denotes the maximum of a and b; h , the water depth above the rubble mound foundation; L , the wavelength corresponding to the signiﬁcant wave period at the depth h ; k, the wave number (= 2πL ); and BM is the berm width as shown in Fig. 18.14. αs is a correction factor obtained using wave tank experiments and is 0.45. Equation (18.19) is also extended to include the stability of the rubble mound armor layer in the breakwater head. In the breakwater head, the term (κ2 )T is used instead of (κ2 )B to represent the water particle velocity at the breakwater head, where κ = κ1 (κ2 )T ,

(18.23)

1 (αs τ 2 ), 4

(18.24)

(κ2 )T =

where τ expresses the ratio of the water particle velocity at the breakwater head to that of the incident wave and is determined to be 1.4 for the wave angle less than 45◦ . Figure 18.14 shows sample calculations to determine the necessary weight of armor stones. The shape of the breakwater is indicated and the necessary weight

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for diﬀerent wave conditions and water depths in front of the vertical wall are calculated. When θ = 60◦ , the weight is the largest, whereas that for θ = 30◦ is the smallest. It is obvious to see that the weight increases as the wave period increases or the water depth in front of the wall decreases. When the weight is large, armor concrete blocks must be used to cover the rubble mound. 18.3.4.2. Foot protection of the upright wall Foot-protection concrete blocks are usually placed in front of the upright section. Figure 18.15 shows the failure modes related to such foot-protection, whereas the foot-protection blocks are removed, erosion of rubble mound takes place near the foot of the upright section. Also, through-wash (rapid current through the rubble mound) will cause scouring of the sand under the rubble mound; thus, the foot-protection concrete blocks must prevent the direct intrusion of wave pressure into the rubble mound and also the subsequent pressure-induced current in the mound. These blocks work as a ﬁlter and also provide weight for the rubble mound. The critical force on a foot-protection block is that due to the pressure diﬀerence between the upper and lower faces of the block. The absolute value of the wave pressure under the block was experimentally found to be 5–40% less than that on the upper side. This pressure diﬀerence can be reduced and the stability increased by making holes in the blocks, although if the holes are too large, the ﬁltering eﬀect is reduced. An opening ratio of 10% is therefore recommended by Tanimoto et al.15 Several ways exist to empirically determine the necessary weight. Figure 18.16 shows a diagram to determine the thickness t of concrete blocks having a 10% opening, with the dimensions of the blocks being subsequently determined and summarized in Table 18.1. Figure 18.17 shows a concrete block with t = 1.2 m. It should be noted that the foot-protection blocks also act as armor blocks for the rubble mound as discussed, especially in oblique seas and at the breakwater head. Therefore, they should be stable against the velocity-induced force. Additionally, care should be taken to prevent the occurrence of scouring underneath the rubble foundation. Too thin a rubble mound may cause this type of scouring to occur due to severe wave actions, and consequently, a vinyl sheet is sometimes placed on the sand bed to prevent it.

Fig. 18.15.

Failure due to damage to toe protection blocks.

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Fig. 18.16.

Thickness of foot-protection blocks.19

Table 18.1. blocks.

Speciﬁcation

t (m)

l(m) × b(m) × t (m)

W (t)

∼0.8 ∼1.0 ∼1.2 ∼1.4 ∼1.6 ∼1.8 ∼2.0 ∼2.2

2.5 × 1.5 × 0.8 3.0 × 2.5 × 1.0 4.0 × 2.5 × 1.2 5.0 × 2.5 × 1.4 5.0 × 2.5 × 1.6 5.0 × 2.5 × 1.8 5.0 × 2.5 × 2.0 5.0 × 2.5 × 2.2

6.2 15.6 24.8 37.0 42.3 47.6 52.9 58.2

Fig. 18.17.

of

foot-protection

Foot-protection block.

18.3.4.3. Toe protection against scouring Figure 18.18 shows scouring damage at the toe of the rubble mound for a composite breakwater situated on a sandy seabed.20 Due to the weight of the caisson, the entire breakwater slightly settles and the toe area is signiﬁcantly scoured, which deepens the toe area by about 2 or 3 m. Even though this toe erosion occurs, the

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Fig. 18.18.

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Scour around rubble mound toe.20

caisson remains intact and the foot-protection blocks and armor blocks of the rubble mound still function as designed. In actuality, one of the primary roles of the rubble mound and the foot-protection blocks is to protect the caisson from such scouring. The cause of toe area scouring is probably due to the strong wave-induced velocity. These velocities are especially increased by reﬂected waves from the vertical wall. However, the scouring mechanism is very complicated and has not yet been properly explained. Generally, two types of toe scouring exist: local scouring and large-scale sand movement. Local scouring was investigated by Xie,21 Irie et al.,5 Oumeraci,22 and others. Irie found the occurrence of N- and L-type scours, in which the former type scour is due to suspension of relatively ﬁne sand which causes accretion at the node, while the later type is due to “bed load” of relatively coarse sand that causes erosion at the node and accretion at the loop. L-type scour appears to be predominant in proto type seas where erosion at the node is usually found. Scour is usually inevitable for vertical breakwaters built on a sandy sea bottom. However, scour is not a fatal problem due to the protection features provided by the rubble mound. Nevertheless, scour protection should be included in breakwaters in which severe scouring is expected. There are several scour protection methods, e.g., the use of gravel, geotextile, or asphalt mats. These methods can prevent scouring to some extent, though no fully suﬃcient method has yet been realized.

18.4. Recent Development of Conventional Caisson Breakwater Design Over the last 15 years improved awareness of wave impact induced failures3,23,24 has focused attention on the need to account for the dynamic response of maritime structures to wave impact loads. Also, the attention was focused on the probabilistic consideration in the design of breakwaters.25–28 The most relevant collaborative research projects carried out in the recent past in the European Union on wave loads on caisson breakwaters and seawalls include: (1) PRObabilistic design tools for VERtical BreakwaterS, see Oumeraci et al.29 and references therein;

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(2) Violent Overtopping of Waves at Seawalls, see Cuomo30 and references therein; (3) Breaking Wave IMpacts on step fronted COastal Structures, see Bullock et al.31 and references therein. Recently, a new innovative design method is being developed for the design of breakwater caissons which is called the performance design method. Performance design can be considered as a design process that systematically and clearly deﬁnes performance requirements and respective performance evaluation methods. In other words, performance design allows the performance of a structure to be explicitly and concretely described.32 A new performance design for the stability of breakwater caissons was being developed which is called deformation-based reliability design.33–37 Sliding distance is selected as the performance evaluation item and the probabilistic nature is fully considered. Performance design requires a reliable performance evaluation method. Thus, in deformation-based reliability design of a breakwater caisson, a calculation method to determine the sliding distance due to wave actions is used, employing Monte Carlo simulation to include the probabilistic nature of waves and response of the breakwater caisson. Since breakwaters are designed to withstand wave heights having a particular return period, such as 50 years, a high probability exists that higher waves than the design wave will attack them and there still remain some failures by heavy storms. Recently, it becomes very important to evaluate the extent of damage/deformation as well as the consequence including restoration and the total economic losses. The performance design is just the design process to consider these items for more reasonable and economical design of breakwaters. Also, the performance design is becoming increasingly important for the design of coastal defenses against storm surges and tsunami.38

References 1. S. Takahashi, Design of vertical breakwaters, Reference Document No. 34, Port and Harbour Research Institute (1996) 85 pp. 2. G. E. Jarlan, A perforated vertical breakwater, The Dock and Harbour Authority 41(488), 394–398 (1961). 3. S. Takahashi, K. Shimosako, K. Kimura and K. Suzuki, Typical failures of composite breakwaters in Japan, Proc. 27th Int. Conf. Coastal Eng., ASCE (2000), pp. 1899–1910. 4. K. Kimura, S. Takahashi and K. Tanimoto, Stability of rubble mound foundations of caisson breakwater under oblique wave attack, Proc. 24th Int. Conf. Coastal Eng. Kobe, ASCE (1994). 5. I. Irie, Y. Kuriyama and H. Asakawa, Study on scour in front of breakwaters by standing waves and protection method, Rep. Port. Harbour Res. Inst. 25(1), 3–86 (1986) (in Japanese). 6. K. Suzuki, S. Takahashi and Y. H. Kang, Experimental analysis of wave-induced liquefaction in a ﬁne sand bed, Proc. 26th Int. Conf. Coastal Eng. (1998), pp. 3643–3654.

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7. M. Kobayashi, M. Terashi and K. Takahashi, Bearing capacity of a rubble mound supporting a gravity structure, Rept. Port and Harbour Res. Inst. 26(5), 215–252 (1987). 8. S. Kataoka and S. Saida, Compilation of breakwater structures, Tech. Note of Port and Harbour Res. Inst., No. 556, 150 pp. (1986) (in Japanese). 9. Y. Goda, Laboratory investigatation of wave pressures exerted upon vertical and composite walls, Rept. Port and Harbour Res. Inst. 11(2), 3–45 (1972) (in Japanese) or Y. Goda, Experiments on the transition from nonbreaking to post breaking wave pressures, Coastal Eng. 15, 81–90 (1972) (in Japanese). 10. Y. Goda, A new method of wave pressure calculation for the design of composite breakwater, Rept. Port and Harbour Res. Inst. 12(3), 31–70 (1973) (in Japanese) or Proc. 14th Conf. Coastal Eng., ASCE, Copenhagen (1974), pp. 1702–1720. 11. Y. Goda, Random Seas and Design of Maritime Structures (Univ. Tokyo Press, Tokyo, 1985). 12. K. Tanimoto, K. Moto, S. Ishizuka and Y. Goda, An investigation on design wave force formulae of composite-type breakwaters, Proc. 23rd Japanese Conf. Coastal Eng. (1976), pp. 11–16 (in Japanese). 13. S. Takahashi, K. Tanimoto and K. Shimosako, A proposal of impulsive pressure coefﬁcient for design of composite breakwaters, Proc. Int. Conf. Hydro-technical Eng. for Port and Harbor Construction, Port and Harbour Res. Inst. (1994). 14. K. Kobune and M. Osato, A study of wave height distribution along a breakwater with a corner, Rept. Port and Harbour Res. Inst. 15(2) (1976) (in Japanese). 15. Y. Ito and K. Tanimoto, Meandering damages of composite type breakwaters, Tech. Note of Port and Harbour Res. Inst. 112 (1972) (in Japanese). 16. A. Brebner and D. Donnelly, Laboratory study of rubble foundations for vertical breakwaters, Proc 8th Int. Conf. Coastal Engineering, New Mexico City, ASCE (1962), pp. 406–429. 17. K. Tanimoto, T. Yagyu and Y. Goda, Irregular wave tests for composite breakwater foundation, Proc. 18th Conf. Coastal Eng., Capetown (1982), pp. 2144–2163. 18. S. Takahashi, K. Kimura and K. Tanimoto, Stability of armor units of composite breakwater mound against oblique waves, Rept. Port and Harbour Res. Inst. 29(2) (1990) (in Japanese). 19. R. Ushijima, R. Mizuno and T. Imoto, Laboratory stability test of foot-protection blocks for upright section of composite breakwaters, Rept. of Civil Engineering Research Institute, Hokkaido Development Bureau, No. 424 (1988), pp. 1–14 (in Japanese). 20. H. Funakoshi, Survey of long-term deformation of composite breakwaters along the Japan sea, Proc. Int. Workshop on Wave Barriers in Deepwaters, PHRI, Yokosuka, Japan, PHRI (1994), pp. 239–266. 21. S. L. Xie, Scouring patterns in front of vertical breakwaters and their inﬂuence on the stability of foundation of breakwaters, Rept. of Department of Civil Engineering, Delft University of Technolgy (1981), 61 pp. 22. H. Oumerachi, Scour in front of vertical breakwaters — Review of problems, Proc. Int. Workshop on Wave Barriers in Deepwaters, PHRI (1994), pp. 281–317. 23. H. Oumeraci, Review and analysis of vertical breakwater failures — Lessons learned, Special issue on vertical breakwaters, Coastal Engineering 22, 3–29 (1994). 24. L. Franco, Vertical breakwaters: The Italian experience. Special Issue on vertical breakwaters, Coastal Engineering 22, 31–55 (1994). 25. H. F. Burcharth, Development of a partial safety factor system for the design of rubble mound breakwaters, PIANC PTII working group 12, subgroup F, Final Report (PIANC, Brussels, 1993).

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26. M. A. Losada and J. Llorca, Breakwaters under the action of sea oscillations. Spanish recommendations ROM 1.1, Coastal Structures (2007). 27. J. W. van der Meer, Deterministic and probabilistic design of breakwater armor layers, Proc. American Society of Civil Engineers, J. Waterways, Coastal Ocean Eng. 114(1), 66–80 (1988). 28. J. K. Vrijling, H. G. Vorrtman, H. F. Burcharth and J. D. Soren-sen, Design philosophy for a vertical breakwa-ter, Proc. Coastal Structures ’99 (1999), pp. 631–635. 29. H. Oumeraci, A. Kortenhaus, N. W. H. Allsop, M. B. De Groot, R. S. Crouch, J. K. Vrijling and H. G. Voortman, Probabilistic Design Tools for Vertical Breakwaters (Balkema, Rotterdam, 2001), 392 pp. 30. G. Cuomo, Dynamics of wave induced loads and their eﬀects on coastal structures, PhD. dissertation, University of Roma TRE, Italy (2005). 31. G. N. Bullock, D. H. Peregrine, H. Bredmose, C. Obhrai, G. Walter and G. Muller, Characteristics and design implication of breaking wave impacts, Proc. 29th Int. Conf. Coastal Engineering (ASCE, 2004), pp. 3966–3978. 32. SEACO, Vision 2000-Performance-based seismic engineering of bridges (1995). 33. K. Shimosako and S. Takahashi, Application of deformation-based reliability design for coastal structures, American Society of Civil Enginners (ASCE), Coastal Structures ’99 (1999), pp. 363–371. 34. T. Takayama, S. Ikesue and K. Shimosako, Eﬀect of directional occurrence distribution of extreme waves on composite breakwater reliability in sliding failure, Proc. 27th Int. Conf. Coastal Engineering (ASCE, 2000), pp. 1738–1750. 35. Y. Goda and H. Takagi, A reliability design method for caisson breakwaters with optimal wave heights, Coastal Engineering 42(4), 357–387 (2000). 36. M. Hanzawa, N. Yamagata, T. Nishihara, Y. Umezawa, T. Takayama and S. Takahashi, Performance of seawall against wave overtopping, American Society of Civil Engineers, Proc. 29th Int. Conf. Coastal Eng. (2004), pp. 4314–4325. 37. J. A. Melby and N. Kobayasi, Damage progression on breakwaters, Proc. 26th Int. Conf. Coastal Eng. (1998), pp. 1884–1897. 38. S. Takahashi, H. Kawai, T. Tomita and T. Takayama, Performance design concept for storm surge defenses, American Society of Civil Engineers, Proc. 29th Int. Conf. Coastal Eng. (2004), pp. 3074–3086.

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Chapter 19

Design of Alternative Revetments Krystian Pilarczyk (Former) Hydraulic Engineering Institute Public Works Department, Delft, The Netherlands HYDROpil Consultancy, 23 Nesciohove, 2726BJ Zoetermeer, The Netherlands [email protected] Revetments are most common structures in coastal engineering. The design of revetments is a complex process and needs proper understanding of loads and structural interactions. The basic information on composition and dimensioning of various types of revetments under wave and current attack are provided. Special attention is given to ﬁlter structures using geotextiles and transitions into splash areas and toe protection. Reference to actual developments and manuals is also provided.

19.1. Introduction There has been an increasing need in recent years for reliable information on design methodology and stability criteria of revetments exposed to wave and current action. The use of revetments, such as riprap, blocks and block mats, various mattresses, and asphalt in civil engineering practice is very common. The granular ﬁlters, and more recently the geotextiles, are more or less standard components of the revetment structure.6,14,20,22,23,26,28−30 However, the proper design rules are not always available. Within the scope of the research on the stability of open slope revetments, much knowledge has been developed about the stability of rock under wave and current load and stability of placed (pitched) stone revetments under wave load.5,7,8 Until recently, no or unsatisfactory design tools were available for a number of other (open) types of revetment and for other stability aspects. This is why the design methodology for placed block revetments has recently been extended in applicability by means of a number of desk-studies for other (open) revetments: • interlock systems and block mats; • concrete mattresses; • gabions; 479

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• geosystems, such as sandbags and sand mattresses; and • other stability aspects, such as ﬂow-load stability, soil-mechanical stability, and residual strength. This chapter aims at giving a summary of the increased knowledge, especially that concerning the design tools that have been made available. The details behind it can be found in Refs. 9, 16, and 26.

19.2. Basics of Hydraulic Loading and Structural Response 19.2.1. Wave-load approach Waves attack on revetments, especially those with low permeability of cover layer, will lead to a complex ﬂow over and through the revetment structure (ﬁlter and cover layer). During wave run-up the resulting forces by the waves will be directed opposite to the gravity forces. Therefore, the run-up is less hazardous than the wave run-down. The wave run-down will lead to two important mechanisms: • The downward ﬂowing water will exert a drag force on the cover layer and the decreasing phreatic level will coincide with a downward ﬂow gradient in the ﬁlter (or in a gabion). The ﬁrst mechanism can be schematized by a free ﬂow in the ﬁlter or gabion with a typical gradient equaling the slope angle. It may result in sliding. • During maximum wave run-down there will be an incoming wave that a moment later will cause a wave impact. Just before the impact, there is a “wall” of water giving a high pressure under the point of maximum run-down. Above the rundown point the surface of the revetment is almost dry and therefore there is a low pressure on the structure. The high pressure front will lead to an upward ﬂow in the ﬁlter or a gabion. This ﬂow will meet the downward ﬂow in the rundown region. The result is an outward ﬂow and uplift pressure near the point of maximum wave run-down (Fig. 19.1). The schematized situation can be quantiﬁed on the basis of the Laplace equation for linear ﬂow. After complicated calculations, the uplift pressure in the ﬁlter or a gabion can be derived.9,26 The uplift pressure is dependent on the steepness and height of the pressure front on the cover layer (which is dependent on the wave height, period, and slope angle), the thickness of the cover layer, and the level of the phreatic line in the ﬁlter or a gabion. In case of riprap or gabions, it is not dependent on the permeability of the cover layer, if the permeability is much larger than the sublayer and/or subsoil. For semi-permeable cover layers the equilibrium of uplift forces and gravity forces (deﬁned by components of a revetment) leads to the following (approximate) design formula26 : 0.67 D Hscr =f ∆D Λξop

with Λ =

bDk k

(19.1a)

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Fig. 19.1.

or

bk Dk

and

or Hscr −0.67 = F ξop ∆D

481

Pressure development in a revetment structure.

Λ/D =

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0.33 D k Hscr −0.67 =f ξop ∆D b k

with maximum

Hs ∆D

(19.1b)

cr

= 8.0 and ctgα ≥ 2,

(19.1c)

where √ Hscr = signiﬁcant wave height at which blocks will be lifted out (m); ξop = tan α/ (Hs /(1.56Tp2)) = breaker parameter; Tp = wave period at the peak of the spectrum (s); Λ = leakage length (m), ∆ = (ρs − ρw )/ρw = relative volumetric mass of cover layer, with: ρs = density of the protection material and ρw = density of water (kg/m3 ); b = thickness of a sublayer/ﬁlter (m), D = thickness of a top (cover) layer (m), k = permeability of a sublayer (m/s), k = permeability of a top layer (m/s), f = stability coeﬃcient, mainly dependent on structure type, tan α and friction; F = total (black-box) stability factor. The leakage length (Λ) and stability coeﬃcient (F ) are explained more in detail for speciﬁc applications in the next sections. There are two practical design methods available: the analytical model based on Eq. (19.1a) and the black-box model based on Eq. (19.1c). In both cases, the ﬁnal form of the design method can be presented as a critical relation of the load (H) compared to strength (∆D), depending on the type of wave attack. More general form (also applicable for riprap and ctgα ≥ 1.5) is provided by Pilarczyk24,26 as: F cos α tan α Hs = , where ξop = (19.2) b ∆D cr ξop Hs /Lop in which F = revetment (stability) factor, Hs = (local) signiﬁcant wave height (m), ∆ = relative density, D = thickness of the top layer (m), ξop = breaker parameter

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(–), Lop = gTp2 /(2π), deep-water wavelength at the peak period (m), and Tp = wave period at the peak of the spectrum (s) and b = exponent; 0.5 ≤ b ≤ 1.0. For porous top layers, such as sand mattresses and gabions, the relative density of the top layer must be determined, including the water-ﬁlled pores: ∆=

ρs − ρw ρw

and ∆t = (1 − n) · ∆

(19.3)

in which ∆t = relative density including pores and n = porosity of the top layer material. D and ∆ are deﬁned for speciﬁc systems such as: • for rock: D = Dn = (M50 /ρs )1/3 (= nominal diameter) and ∆t = ∆ = (ρs − ρw )/ρw , • for blocks: D = thickness of block and ∆t = ∆, • for mattresses: D = d = average thickness of mattress and ∆t = (1 − n)∆, where n = bulk porosity of ﬁll material and ∆ = relative density of ﬁll material. For sand and common quarry stone (1 − n) ∆ ∼ 1. The approximate values of stability factor F are: F = 2.25 for riprap, F = 2.5 for pitched stone of irregular shape and placed geobags, F = 3.0–3.5 for pitched basalt, F = 4.0 for cement geomattresses, 3.5 ≤ F ≤ 5.5 for block revetments (4.5 as an average/usual value), 4.0 ≤ F ≤ 6.0 for block mats (higher value for cabled systems), 6.0 ≤ F ≤ 8.0 for gabions, and 6.0 ≤ F ≤ 10 for (asphalt or concrete) slabs. Exponent b refers to the type of wave–slope interaction and its value is inﬂuenced by the roughness and the porosity of a revetment. The following values of exponent b are recommended: b = 0.5 for permeable cover layers (i.e., riprap, gabions, pattern grouted riprap, open block mats, and geobags), b = 2/3 for semipermeable cover layers (i.e., pitched stone and placed blocks, block mats, concreteor sand-ﬁlled geomattresses, and 2/3 < b ≤ 1.0 for slabs). The advantage of this black-box design formula is its simplicity. The disadvantage, however, is that the value of F is known only very roughly for many types of structures. The analytical model is based on the theory for placed stone revetments on a granular ﬁlter (pitched blocks). In this calculation model, a large number of physical aspects are taken into account. In short, in the analytical model, nearly all physical parameters that √ are relevant to the stability have been incorporated in the “leakage length”: Λ = (bDk/k ). The ﬁnal result of the analytical model may, for that matter, again be presented as a relation such as Eq. (19.1) where F = f (Λ). A system without a ﬁlter layer is (directly on sand or clay and geotextile) not the permeability of the ﬁlter layer, but the permeability of the subsoil (eventually with gullies/surface channels) is ﬁlled in. For the thickness of the ﬁlter layer, it is examined to which depth changes at the surface aﬀect the subsoil. One can ﬁll in 0.5 m for sand and 0.05 m for clay. The values for D and ∆ depend on the type of revetment. In the case of a geotextile situated directly under the cover layer, the permeability of the cover layer decreases drastically. Since the geotextile is pressed against the cover layer by the outﬂowing water, it should be treated as a part of the cover layer. The water ﬂow trough the cover layer is concentrated at the joints between the

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blocks, reaching very high ﬂow velocities and resulting in a large pressure head over the geotextile. The presence of a geotextile may reduce k by a factor 10 or more. To be able to apply the design method for placed stone revetments under wave load to other systems, the following items may be adapted: the the the the the

revetment parameter F ; (representative) strength parameters ∆ and D; design wave height Hs ; (representative) leakage length Λ; increase factor Γ (friction/interlocking between blocks) on the strength.

Only such-like adaptations are presented in this summarizing review. The basic formulae of the analytical model are not repeated here. For these, the reader is referred to Refs. 9 and 26. The wave attack on a slope can be roughly transformed into the maximum velocity component on a slope during run-up and run-down, Umax , by using the following formula: (19.4) Umax = p gHs ξop where p ≈ 1.5 can be assumed as a ﬁrst approximation. 19.2.2. Flow-load stability There are two possible approaches for determining the stability of revetment material under ﬂow attack. The most suitable approach depends on the type of load: • ﬂow velocity: “horizontal” ﬂow, ﬂow parallel to dike; • discharge: downward ﬂow at slopes steeper than 1:10, overﬂow without waves; stable inner slope. The general velocity criterion can be expressed as5,8,25 : U2 = Λh Ψ, 2g∆D

(19.5)

where U = depth-average (mean) velocity, Ψ = Shields parameter, ∆ = relative density, g = acceleration of gravity (g = 9.81 m/s2 ), Λh = depth related ﬂow resistance function; i.e., Λh = C 2 /2g for Chezy approach, C = 18 log(12h/ks ), Λh = 33(h/ks )1/3 for Strickler–Manning, Λh = 33(h/ks )0.2 for Pilarczyk25 (based on Neill’s data19 ). When the ﬂow velocity is known, or can be calculated reasonably accurately, Pilarczyk’s relation is applicable24−26 : ∆D = 0.035

Φ KT Kh u2cr Ψ Ks 2g

(19.6)

in which D = characteristic thickness (m): for riprap D = Dn = nominal diameter as deﬁned previously, Ucr = critical vertically averaged ﬂow velocity (m/s), Φ = stability parameter, Ψ = Ψcr = critical Shields parameter KT = turbulence factor,

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Kh = depth parameter: Kh = 33/Λh, and Ks = slope parameter. These parameters are explained below. Stability parameter Φ The stability parameter Φ depends on the application. Some guide values are: Revetment type

Riprap and placed blocks Sand-ﬁlled units Block mats, gabions, washed-in blocks, Concrete-ﬁlled geobags, and geomattresses

Continuous toplayer

Edges and transitions

1.0

1.5

0.75

1.1

Shields parameter Ψcr With the critical Shields parameter Ψ, the type of material can be taken into account: • riprap, small regular bags Ψ ≈ 0.035 • placed blocks, geobags Ψ ≈ 0.05 • blockmats Ψ ≈ 0.05−0.07 (if washed-in) • gabions Ψ ≈ 0.07 • geomattresses Ψ ≈ 0.07 Turbulence factor K T The degree of turbulence can be taken into account with the turbulence factor KT . Some guide values for KT are: • Normal turbulence of rivers: KT ≈ 1.0 • Increased turbulence: river bends: KT ≈ 1.5 downstream of stilling basins: KT ≈ 1.5 • Heavy turbulence hydraulic jumps: KT ≈ 2.0 strong local disturbances: KT ≈ 2.0 sharp bends: KT ≈ 2.0 to 2.5 • Load due to water (screw) jet: KT ≈ 3.0 to 4.0 Depth parameter Kh With the depth parameter Kh , the water depth is taken into account, which is necessary to translate the depth-averaged ﬂow velocity into the ﬂow velocity just above the revetment. The depth parameter also depends on the development of the ﬂow proﬁle and the roughness of the revetment. The fully developed proﬁle can be found in natural rivers. In case of local civil engineering works as bottom protection or slope protection, the nondeveloped proﬁle is usually present. The following formulae

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Fig. 19.2.

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Kh factor as a function of relative roughness h/ks .

are recommended (see also Fig. 19.2): 2 • fully developed velocity proﬁle: Kh = 2 log 12h ks −0.2 h • nondeveloped proﬁle: Kh = 1 + ks h < 5 : Kh = 1.0 • very rough ﬂow ks

(19.7a)

(19.7b) (19.7c)

in which h = water depth (m) and ks = equivalent roughness according to Nikuradse (m). In the case of dimensioning the revetment on a slope, the water level at the toe of the slope must be used for h. The equivalent roughness depends on the type of revetment/geosystem. For riprap, ks is equal usually to one or twice the nominal diameter of the stones; for bags, it is approximately equal to the thickness (d); for mattresses, it depends on the type of mattress: ks of about 0.05 m for smooth types and about the height of the rib for articulating mats. Note: Usually 12h/ks is applied, however, by using (1 + 12h/ks ) the discontinuity at small values of h can be avoided; the same adjustment is also applied for other velocity distributions. The eﬀect of this additional (imaginary) depth practically vanishes for h/ks > 2.

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Slope parameter Ks The stability of revetment elements also depends on the slope gradient under which the revetment is applied, in relation to the angle of internal friction of the revetment. This eﬀect on the stability is taken into account with the slope parameter Ks , which is deﬁned as follows: 2 2 sin α tan α Ks = 1 − = cos α 1 − (19.8a) sin θ tan θ or Ks = cos αb ,

(19.8b)

where θ = angle of internal friction of the revetment material, α = transversal slope of the bank (◦ ), and αb = slope angle of river bottom (parallel along ﬂow direction) (◦ ). The following values of θ can be assumed as a ﬁrst approximation: 40◦ for riprap, 30◦ –40◦ for sand-ﬁlled systems, and 90◦ for stiﬀ and anchored mortar-ﬁlled mattresses and (cabled) block mats (Ks = cos α). However, for ﬂexible nonanchored mattresses and block mats (units without contact with the neighboring units) this value is much lower, usually about 3/4 of the friction angle of the sublayer. In case of geotextile mattress and block mats connected to geotextile lying on a geotextile ﬁlter, θ is about 15◦ –20◦. Pilarczyk’s formula was veriﬁed using Neill’s data (see Fig. 19.3). The advantage of this general design formula is that it can be applied in numerous situations. The disadvantage is that the scatter in results, as a result of the large margin in parameters, can be rather wide. With a downward ﬂow along a steep slope and small h/ks values, it is diﬃcult to determine or predict the ﬂow velocity because the ﬂow is very irregular. In such case formulae based on the discharge are developed.17,26

Fig. 19.3.

Comparison of Pilarczyk’s equation with the experimental data collected by Neill.19

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It should be noted that Pilarczyk formula can be transformed into similar form as Maynord formula, which is more or less a standard formula in the USA. Maynord formula4,18 :

2.5 2.5 1/2 γw U U ⇒ CM · h D30 = Sf Cs Cv CT h γs − γw (Ks gh) (Ks g∆h) (19.9) where D30 = characteristic riprap size of which 30% is ﬁner by weight, Sf = safety factor (minimum = 1.1), Cs = stability coeﬃcient for incipient failure: = 0.30 for angular rock and = 0.375 for rounded rock, Cv = velocity distribution coeﬃcient: = 1.0 for straight channels, 1 < Cv < 1.28 for river bends = 1.25 downstream of concrete channels, CT = blanket thickness coeﬃcient (CT = 1 for blanket thickness t = 1.5D50 ), Ks = side slope correction factor, and h = local depth (use depth at 20% upslope from toe for side slopes). D30 can be translated (approximately) into D50 using gradation ratio of rock (D85 /D15 ) as: D50 = D30 (D85 /D15 )1/3 . At the ﬁrst approximation one may assume: D30 = 0.75D50 ≈ 0.90Dn50. The minimum value of the total stability coeﬃcent CM is about 0.3. Transformation Pilarczyk formula: 2.5 2.5

1.25

0.035ΦKT U U h = CP · h . Dn50 = 2Ψcr (Ks g∆h) (Ks g∆h) (19.10) The original minimum value of CP is about 0.3; however, for practical applications CP = 0.4 is recommended. Equation (19.10) has the same structure as Eq. (19.9) by Maynord. Comparison of the total stability coeﬃcients CP and CM provides a similar result when using comparable input data and taking into account the diﬀerence between Dn50 and D30 . 19.2.3. Soil-mechanical stability The water motion on a revetment structure can also aﬀect the subsoil, especially when this consists of sand. Geotechnical stability is dependent on the permeability and stiﬀness of the grain skeleton and the compressibility of the pore water (the mixture of water and air in the pores of the grain skeleton). Wave pressures on the top layer are passed on delayed and damped to the subsoil under the revetment structure and to deeper layers (as seen perpendicular to the slope) of the subsoil. This phenomenon takes place over a larger distance or depth as the grain skeleton and the pore water are stiﬀer. If the subsoil is soft or the pore water more compressible (because of the presence of small air bubbles), the compressibility of the system increases and large damping of the water pressures over a short distance may occur. Because of this, alternately, water under-tension and over-tension may develop in the subsoil and corresponding to this an increasing and decreasing grain pressure occurs. It can lead to sliding or slip circle failure; see Fig. 19.4.

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Fig. 19.4.

Schematized development of S-proﬁle and possible local sliding in sand.

Fig. 19.5. Geotechnical stability; design diagram for block mats and wave steepness Hs / Lop = 0.03.

The design method with regard to geotechnical instability is presented in the form of design diagrams. An example is given in Fig. 19.5 (more diagrams and details can be found in Refs. 9, 26, and 27). The maximum admissible wave height is a function of equivalent thickness of cover layer deﬁned as: Deq = D + b/∆t , where D is the cover layer thickness, b is the ﬁlter thickness, and ∆t is the relative system density of the cover layer (inclusive open area of blocks).

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19.2.4. Filters Granular and/or geotextile ﬁlters can protect structures subjected to soil erosion when used in conjunction with revetment armor such as riprap, blocks and block mats, gabions and mattresses, asphalt or concrete slabs, or any other conventional armor material used for erosion control.11,14,21,22 However, there is still a misunderstanding about the function of geotextiles in the total design of these structures, especially in comparison with the granular ﬁlters. In this section, the general principles of designing revetments incorporating granular or geotextiles are reviewed. Attention is paid to the replacing of a granular ﬁlter by a geotextile, which may often lead to geotechnical instability. Furthermore, it appears that a thicker granular ﬁlter gives larger geotechnical stability, but lower cover layer stability (uplift of blocks). The conclusion is therefore that the wave loads must be distributed (balanced) adequately over the sand (shear stress) and the cover layer (uplift pressure). Filters have two functions: erosion prevention and drainage. Traditional design criteria for ﬁlters are that they should be “geometrically tight” and that the ﬁlter permeability should be larger than the base (soil) permeability. However, it results in a large number of layers which are often unnecessary, uneconomical, and diﬃcult to realize. In several cases, a more economical ﬁlter design can be realized using the concept of “geometrically open ﬁlters” (e.g., when the hydraulic loads/gradients are too small to initiate erosion). Recently, some criteria for “geometrically open” ﬁlters including geotextiles were developed (and are still under further development). However, the application of these criteria requires the knowledge/prediction of the hydraulic loads. In the cases when the erosion exceeds an acceptable level, a ﬁlter construction is a proper measure for solving this problem. In revetment structures, geotextiles are mostly used to protect the subsoil from washing away by the hydraulic loads, such as waves and currents. Here, the geotextile replaces a granular ﬁlter. Unfortunately, the mere replacing of a granular ﬁlter by a geotextile can endanger the stability of other components in the bank protection structure. The present section shows that designing a structure is more than just a proper choice of geotextile. Filter structures can be realized by using granular materials (i.e., crushed stone), bonded materials (i.e., sand asphalt, sand cement), and geotextiles, or a combination of these materials. The choice between the granular ﬁlter, a bonded ﬁlter, or geotextile depends on a number of factors. In general, a geotextile is applied because of easier placement and relatively lower cost. For example, the placement of granular ﬁlter underwater is usually a serious problem; the quality control is very diﬃcult, especially when placement of thin layers is required. When designing with geotextiles in ﬁltration applications, the basic concepts are essentially the same as when designing with granular ﬁlters. The geotextile must allow the free passage of water (permeability function) whilst preventing the erosion and migration of soil particles into the armor or drainage system (retention function). In principle, the geotextile must always remain more permeable than the base soil and must have pore sizes small enough to prevent the migration of the larger particles of the base soil. Moreover, concerning the permeability, not only

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the opening size but also the number of openings per unit area (Percent Open Area) is of importance.27 It has to be stressed that geotextiles cannot always replace the granular ﬁlter completely (see Sec. 19.2.3). A granular layer can often be needed to reduce (damp) the hydraulic loadings (internal gradients) to an acceptable level at the soil interface. After that, a geotextile can be applied to fulﬁll the ﬁltration function. In respect to the ﬁlters for erosion control (granular or geotextile), the distinction can be made between: • geometrically tight ﬁlters, • geometrically open ﬁlters, and • transport ﬁlters (when a limited settlement is allowed). Only geometrically tight ﬁlters are discussed. For other type of ﬁlters, the reader is guided to Refs. 6, 26, and 27. 19.2.4.1. Design criteria for geometrically tight granular filters In this case, there will be no transport of soil particles from the base, independent of the level of hydraulic loading. That means that the openings in the granular ﬁlter or geotextile are so small that the soil particles are physically not able to pass the opening. This principle is illustrated in Fig. 19.6 for granular ﬁlters. The main design rules (criteria) for geometrically tight (closed) granular ﬁlters are summarized below. • Interface stability (also called “piping” criterion): Df 15 ≤ 4 to 5, Db85

(19.11)

where Df 15 is the grain size of the ﬁlter layer (or cover layer) which is exceeded by 15% of the material by weight in m; Db85 is the grain size of the base material (soil) which is exceeded by 85% of the material by weight in m.

Fig. 19.6.

Principles of geometrically tight ﬁlters.

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Sometimes, a similar equation is deﬁned as: Df 50 < 6 to 10. Db50

(19.12)

However, Eq. (19.12) is generally less than Eq. (19.11) and can be used for “narrow” gradation only. Therefore, Eq. (19.11) is recommended for general use. • Internal stability In the case of very “wide” gradation, the situation requires an additional check with respect to the internal migration. In this respect, an important parameter is the so-called “uniformity coeﬃcient” Cu , deﬁned by Eq. (19.13) and the shape of the sieve curve: Cu =

Db60 , Db10

(19.13)

where Cu is the coeﬃcient of uniformity. Usually, if this ratio is smaller than 6 (to 10), the soil or ﬁlter structure is considered internally stable. However, internal stability can be judged more carefully by the following rules [Eq. (19.14)]: D10 < 4D5 , D20 < 4D10 , D30 < 4D15 ,

(19.14b) (19.14c)

D40 < 4D20 .

(19.14d)

Df 15 > 5. Db15

(19.15)

(19.14a)

• Permeability criterion

19.2.4.2. Summary of design rules for geotextiles Current definitions for geotextile openings There are a large number of deﬁnitions of the characteristic of geotextile openings. Moreover, there are also diﬀerent test (sieve) methods for the determination of these openings (dry, wet, hydrodynamic, etc.) which depend on national standards. These all make the comparison of test results very diﬃcult or even impossible. That also explains the necessity of international standardization in this ﬁeld. Some of the current deﬁnitions are listed below: O90 corresponds with the average sand diameter of the fraction of which 90% of the weight remains on or in the geotextile (or 10% passes the geotextile) after 5 min of sieving (method: dry sieving with sand). O98 corresponds with the average sand diameter of the fraction of which 98% of the weight remains on or in the geotextile after 5 min of sieving. O98 gives a practical approximation of the maximum ﬁlter opening and therefore plays an important role

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in the sand tightness criterion for a geotextile in strong cyclic loading situations. O98 is also referred to as Omax . Of ﬁltration opening size (FOS). Of is comparable with O95 (hydrodynamic sieve method). AOS is the apparent opening size (acc. to ASTM method), also called EOS (eﬀective opening size). The AOS is determined by sieving spherical glass particles of known size through a geotextile. The AOS, also frequently referred to as O95 (dry sieve method), is deﬁned as a standard sieve size, x, mm, for which 5% or less of the glass particles pass through the geotextile after a speciﬁed period of sieving. Dw eﬀective opening size which corresponds to the sand diameter of the fraction of which 10%, determined by the wet sieve method, passes through the geotextile. Dw is comparable with O95 . The transport of soil particles within a grain structure is possible when there is enough space and a driving force (groundwater pressure, hydraulic gradients within the soil). In most cases, it is the intention to prevent the transport of small-sized soil particles in the subsoil and therefore the term soil tightness is used and not the term space for transport or pore volume (in the case of the transport of water the terms pore volume and water permeability are used). The relation between pore magnitude and grain diameter can be characterized by: pore diameter ≈ 20% of the grain diameter. Just as for the characterization of the performance of a grain structure with regard to the transport of soil particles, for geosynthetics, too, the term soil tightness is used. As was mentioned before (Fig. 19.6), in a theoretical case when the soil is composed of spheres of one-size diameter, all spheres can be retained if all apertures in the geosynthetic are smaller than the diameter of the spheres. Usually, the soil consists of particles with diﬀerent diameters and shapes, which is reﬂected in the particle-size distribution curves. Smaller particles can disappear straight across the geosynthetic by groundwater current. In this case, the retained soil structure can

Fig. 19.7.

Schematic representation of a natural ﬁlter with a soil-retaining layer.

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function as a natural ﬁlter; see Fig. 19.7. The better the soil particles are distributed, the better the soil tightness of the soil structure is eﬀected. Smaller soil particles get stuck into the spaces between larger ones and the soil structure prevents the ﬂow of ﬁne particles. When certain particle-size fractions are lacking, the soil structure is not stacked very well and cavities develop through which erosion can occur. The displacement of soil particles not only depends on the soil tightness but also on the hydraulic gradient in the soil structure. Moreover, the dynamic eﬀects due to heavy wave loading may not allow the forming of a natural ﬁlter, and the process of washing-out may continue. According to some researchers the forming of a natural ﬁlter is only possible for stationary ﬂow.6 However, for small values of the hydraulic gradients, this is also possible for nonstationary ﬂow. For heavy wave attack (i.e., exposed breakwaters, coastal revetments), this is usually not the case. In extreme situations, soil liquefaction is even possible. In such situations the soil particles can still reach the surface of a geotextile and be washed out. In order to judge the risk of wash-out of soil particles through the geosynthetics, some aspects have to be considered. An important factor is the internal stability of the soil structure. In the case of a loose particle stacking of the soil, many small soil particles may pass through the geosynthetic before a stable soil structure is developed near the geosynthetic. Also, a proper compaction of soil is very important for the internal stability of soil. The internal stability is deﬁned by the uniformity coeﬃcient Cu [see Eq. (19.13)]. In case of vibration, for instance, caused by waves or by traﬃc, the stable soil structures can be disturbed. To avoid such situations, the subsoil has to be compacted in advance and a good junction between geosynthetic and subsoil has to be guaranteed and possibly, a smaller opening of geotextile must be chosen. The shape of the sieve curve also inﬂuences the forming of a natural ﬁlter. Especially, when Cu > 6, the shape of the base gradation curve and its internal stability must be taken into account.27 For a self-ﬁltering linearly graded soil, the representative size corresponds to the average grain size, Db50 . For a self-ﬁltering gap graded soil, this size is equal to the lower size of the gap. For internally unstable soils, this size would be equivalent to Db30 in order to optimize the functioning of the ﬁlter system. It is assumed that the involved bridging process would not retrogress beyond some limited distance from the interface. Soil tightness With respect to the soil tightness of geotextiles, many criteria for geometric soil tightness have been developed and published in the past.2,6,10,12,15,20,27 An example of such design criteria, based on Dutch experience, is presented in Table 19.1. An additional requirement is that the soil should be internally stable. The internal stability of a grain structure is expressed in the ratio between Db60 and Db10 . As a rule, this value has to be smaller than 10 to guarantee suﬃcient stability. However, in many situations additional requirements will be necessary, depending on the local situation. Therefore, for design of geometrically tight geotextiles the method applied in Germany can be recommended.2,13,20,27 In this method a distinction is made between the so-called stable and unstable soils.

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Design requirements for geosynthetics with a ﬁlter and separation function.

Description

Filter function/soil tightness

• Stationary loading • Cyclic loading with natural ﬁlter (stable soil structure) • Cyclic loading without a natural ﬁlter (unstable soil structure) • When wash-out eﬀects acceptable • When wash-out eﬀects not acceptable

O90 ≤ 2Db90 O98 ≤ 1(to 2) Db85

Table 19.2.

O98 ≤ 1.5Db15 O98 ≤ Db15

Design criteria for geometrically soil-tight geotextiles. Soil type Db40 < 60µm

Stationary loading Dynamic loading

Db40 < 60µm

Stable soil

Unstable soil

Stable soil

Unstable soil

O90 < 10Db50 and O90 < 2Db90

O90 < 10Db50 and O90 < Db90 O90 < Db90 and O90 < 0.3 mm (300 µm)

O90 < 5Db10 Cu and O90 < 2Db90 O90 < Db90

1/2

O90 and O90 O90 and O90 O90

1/2

< 5Db10 Cu

< Db90 1/2 < 1.5Db10 Cu < Db50 < 0.5 mm

O90 is determined by wet sieve method.

Soils are deﬁned as unstable (susceptible to down-slope migration) when the following speciﬁcations are fulﬁlled: • • • •

a proportion of particles must be smaller than 0.06 mm; ﬁne soil with a plasticity index (Ip ) smaller than 0.15 (it is not a cohesive soil); 50% (by weight) of the grains will lie in the range 0.02 < Db < 0.1 mm; clay or silty soil with Cu < 15.

If Ip is unknown at the preliminary design stage, then the soil may be regarded as a problem soil if the clay size fraction is less than 50% of the silt size fraction. The design criteria are presented in Table 19.2.27 In the case of ﬁne sand or silty subsoils, however, it can be very diﬃcult to meet these requirements. A more advanced requirement is based on hydrodynamic sand tightness, viz. that the ﬂow is not capable of washing out the subsoil material, because of the minor hydrodynamical forces exerted (although the apertures of the geotextile are much larger than the subsoil grains). Requirements concerning water permeability To prevent the formation of water pressure (uplift) in the structure, causing loss of stability, the geotextile has to be water permeable. One has to strive for the increase of water permeability of a construction in the direction of the water current. In

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the case of a riverbank protection, it means that the permeability of the geotextile has to be larger than the permeability of the soil on which the geotextile has to be applied. In the case of a dike slope or dike foundation, the geosynthetic is often applied on an impermeable layer of clay. Proper permeability of geotextiles is very important in respect to the stability of relatively less permeable cover layers as, for example, block and block mats. When a geotextile lies directly under the cover layer it considerably reduces the open area of the cover layer, and as a result the uplift forces increase (see example in Sec. 19.3.2). The water permeability of woven fabrics and nonwovens may decrease in the course of time owing to the fact that ﬁne soil particles, which are transported by the groundwater ﬂow from the subsoil, block the openings in the geotextile, or migrate into the pores of the geosynthetic (clogging). To prevent mineral clogging, the pore size of the geotextiles has to be chosen as large as possible; but, of course, this pore size has still to meet the requirements for soil tightness. The danger of clogging increases when the soil contains more than 20% of silt or in the case of gap-grading of a soil. On the other hand, there usually is no danger of clogging when the total hydraulic gradient (over the subsoil and geotextile together) is less than 3, or when the subsoil is well graded. In all situations it holds that the soil must be internally stable. For less critical situations no clogging can be expected if: • for Cu > 3: O95 /Db15 > 3; • for Cu < 3: criterion of internal stability of soil should be satisﬁed or geotextile with maximum opening size from soil-tightness criteria should be speciﬁed. In respect to the water permeability of geosynthetics/geotextiles, a distinction should be made between “normal to the interface” and “parallel to the interface.” For geotextile ﬁlters the permeability parallel to the interface, is of importance, while for drainage structures the permeability normal to the interface is of most importance. As a general design criterion for ﬂow normal to the interface one can hold that the water permeability of a geosynthetic/geotextile has to be greater than that of the soil at the side from where the water ﬂow comes. As a rule one can keep to: kgeotextile (ﬁlter) = ksoil × factor, where ks and kg are usually (basically) deﬁned as permeability for laminar conditions. For normal (stationary) conditions and applications and clean sands a factor of 2 is suﬃcient to compensate the eﬀect of blocking. If a geotextile is permeable with a factor of 10 more than the (noncohesive) subsoil, overpressure will usually occur, neither below the geosynthetic, nor in the case of reduced permeability caused by clogging or blocking. However, for special applications (i.e., for dam-clay cores with danger of clogging) this factor can be 50 or more.27,32

19.3. Stability Criteria for Placed Blocks and Block Mats 19.3.1. System description Placed block revetments (or stone/block pitching) are a form of protection lying between revetments comprised of elements which are disconnected, such as rubble, and monolithic revetments, such as asphalt/concrete slabs. Individual elements of

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Fig. 19.8.

Examples of block mats.

a pitched block revetment are placed tightly together in a smooth pattern. This ensures that external forces such as waves and currents can exert little drag on the blocks and also that blocks support each other without any loss of ﬂexibility when there are local subsoil irregularities or settlement. A (concrete) block mat is a slope revetment made of (concrete) blocks that are joined together to form a “mat”; see Fig. 19.8. The interconnection may consist of cables from block to block, of hooks connecting the blocks, or of a geotextile on which the blocks are attached with pins, glue, or other means. The spaces between the blocks are usually ﬁlled with rubble, gravel, or slag. The major advantage of block mats is that they can to be laid quickly and eﬃciently and partly underwater. Block mats are more stable than a setting of loose blocks, because a single stone cannot be moved in the direction perpendicular to the slope without moving other nearby stones. It is essential to demand that already with a small movement of an individual stone a signiﬁcant interactive force with the surrounding stones is mobilized. Large movements of individual blocks are not acceptable, because transport of ﬁlter material may occur. After some time, this leads to a serious deformation of the surface of the slope. The block mats are vulnerable at edges and corners. If two adjacent mats are not joined together, then the stability is hardly larger than that of pitched loose stones. 19.3.2. Design rules with regard to wave load The usual requirement that the permeability of the cover layer should be larger than that of the underlayers cannot be met in the case of a closed block revetment and other systems with low permeable cover layer. The low permeable cover layer introduces uplift pressures during wave attack. In this case, the permeability ratio of the cover layer and the ﬁlter, represented in the leakage length, is found to be the most important structural parameter, determining the uplift pressure. This is also the base of analytical model. The analytical model is based on the theory for placed stone revetments on a granular ﬁlter.9 In this calculation model, a large number of physical aspects are taken into account (see Fig. 19.1). In short, in the analytical model, nearly all physical parameters that are relevant to the stability have been incorporated in the “leakage length” factor. The ﬁnal result of the analytical model may, for that matter, again be presented as a relation such as Eq. (19.1) or (19.2) where F = f (Λ). For

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systems on a ﬁlter layer, the leakage length Λ is given by bf Dk bf k or Λ/D = , Λ= k Dk

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(19.16a)

where Λ = leakage length (m), bf = thickness of the ﬁlter layer (m), kf = permeability of the ﬁlter layer or subsoil (m/s), and k = permeability of the cover layer (m/s). With a system without a ﬁlter layer (directly on sand or clay, without gullies being formed under the top layer) not the permeability of the ﬁlter layer, but the permeability of the subsoil (eventually with gullies/surface channels) is ﬁlled in. For the thickness of the ﬁlter layer, it is examined to which depth changes at the surface aﬀect the subsoil. One can ﬁll in 0.5 m for sand and 0.05 m for clay. The values for D and ∆ depend on the type of revetment. When schematically representing a block on a geotextile on a gully in sand, the block should be regarded as the top layer and the combination of the geotextile and the small gully as the ﬁlter layer (Fig. 19.9). The leakage length can be calculated using: (kf dg + kg Tg )D Λ= , (19.16b) k where kf = permeability of the ﬁlter layer (gully) (m/s), dg = gully depth (m), kg = permeability of the geotextile (m/s), Tg = thickness of the geotextile (m), D = thickness of the top layer (m), and k = permeability of the top layer (m/s). In the case of a geotextile situated directly under the cover layer, the permeability of the cover layer decreases drastically. Since the geotextile is pressed against the cover layer by the outﬂowing water, it should be treated as a part of the cover layer. The water ﬂow through the cover layer is concentrated at the joints between the blocks, reaching very high ﬂow velocities and resulting in a large pressure head over the geotextile. The presence of a geotextile may reduce k by a factor 10 or more (see Fig. 19.10). The leakage length clearly takes into account the relationship between kf and k and also the thickness of the cover layer and the ﬁlter layer. For the theory behind this relationship, reference should be made to literature.9,16 The pressure head diﬀerence which develops on the cover layer is larger with a large leakage length

Fig. 19.9.

Schematization of a revetment with gully (cavity).

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Fig. 19.10.

Combined ﬂow resistance determining the permeability of a system.

than with a small leakage length (eﬀect of kf /k in the leakage length formula). The eﬀect of the leakage length on the dimensions of the critical wave for semi-permeable revetments is apparent from the following equations: 0.67 0.33 Hscr Hscr D D k Hscr −0.67 −0.67 =f =f = F ξop ξop = , ∆D Λξop ∆D b k ∆D

(19.17)

where Hscr = √ signiﬁcant wave height at which blocks will be lifted out (m); ξop = tan α/ (Hs /(1.56Tp2)) = breaker parameter; Tp = wave period (s); ∆ = relative volumetric mass of cover layer = (ρs − ρ)/ρ; f = stability coeﬃcient mainly dependent on structure type and with minor inﬂuence of ∆, tan α, and friction, and F = total (black-box) stability factor. These equations indicate the general trends and have been used together with measured data to set up the general calculation model.9,26 This method works properly for placed/pitched block revetments and block mats within the following range: 0.01 < k kf < 1 and 0.1 < D/bf < 10. Moreover, when D/Λ > 1 use D/Λ = 1, and when D/Λ < 0.01 use D/Λ = 0.01. The range of the stability coeﬃcient is: 5 < f < 15; the higher values refer to the presence of high friction among blocks or interlocking systems. The following values are recommended for block revetments: f = 5 for static stability of loose blocks (no friction between the blocks), f = 7.5 for static stability of a system (with friction between the units), f = 10 for tolerable/acceptable movement of a system at design conditions. From these equations, neglecting the minor variations of “f ,” it appears that: • An increase in the volumetric mass, ∆, produces a proportional increase in the critical wave height. If ρb is increased from 2300 to 2600 kg/m3 , Hscr is increased by about 23%;

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• If the slope angle is reduced from 1:3 to 1:4 (tan α from 0.33 to 0.25), Hscr is increased by about 20% (due to the breaker parameter, ξop ); • An increase of 20% in the thickness of the cover layer, D, Hscr increases by about 27%; • A 30% reduction in the leakage length, Λ, Hscr increases by about 20%. This can generally be achieved by halving the thickness of the ﬁlter layer or by doubling the k /kf value. The latter can be achieved by approximation by: — — — —

reducing the grain size of the ﬁlter by about 50%, or by doubling the number of holes in (between) the blocks, or by making hole sizes 1.5 times larger, or by doubling joint width between blocks.

Example: In 1983, the Armorﬂex mat on a slope 1:3 was tested on prototype scale at the Oregon State University: closed blocks with thickness D = 0.12 m and open area 10% on two types of geotextiles and very wide-graded subsoil (d15 = 0.27 mm, d85 = 7 mm). In the case of a sand-tight geotextile, the critical wave height (instability of mat) was only Hscr = 0.30 m. In the case of an open net geotextile (opening size about 1 mm), the critical wave height was more than 0.75 m (maximum capacity of the wave ﬂume). The second geotextile was 20 times more permeable than the ﬁrst one. This means that the stability increased by factor 200.33 = 2.7. In most cases, the permeability of the cover layer and sublayer(s) are not exactly known. However, based on the physical principles as described above, the practical “black-box” method has been established where parameter Λ and coeﬃcient “f ” are combined to one stability factor “F ”. F depends on the type of structure, characterized by the product of the ratios of k /kf and D/bf . With the permeability formulae9 it is concluded that the parameter (k /kf ) × (D/bf ) ranges between 0.01 and 10, leading to a subdivision into three ranges of one decade each. Therefore, the following types are deﬁned: (a) Low stability: (k /kf )(D/bf ) < 0.05–0.1, (b) Normal stability: 0.05–0.1 < (k /kf )(D/bf ) < 0.5–1, (c) High stability: (k /kf )(D/bf ) > 0.5–1. For a cover layer lying on a geotextile on sand or clay, without a granular ﬁlter, the leakage length cannot be determined because the size of bf and k cannot be calculated. The physical description of the ﬂow is diﬀerent for this type of structure. For these structures there is no such a theory as for the blocks on a granular ﬁlter. However, it has been experimentally proved that Eq. (19.1) or (19.17) are also valid for these structures. It can be concluded that the theory has led to a simple stability formula [Eq. (19.17)] and a subdivision into four types of (block) revetment structures: • • • •

(a1) (a2) (a3) (a4)

cover cover cover cover

layer layer layer layer

on on on on

granular ﬁlter possibly including geotextile, low stability; granular ﬁlter possibly including geotextile, normal stability; geotextile on sand; clay or on geotextile on clay.

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Lower and upper value for F .

Type

Description

a1.1

Pitched irregular natural stones on granular ﬁlter Loose closed blocks/basalt on granular ﬁlter, low stability Loose blocks on granular ﬁlter, normal stability Loose blocks on geotextile on compacted sand/clay Linked/interlocked blocks on geotextile on good clay or on ﬁne granular ﬁlter

a1.2 a2 a3 a4

Low F

High F

Usual F

2.0

3.0

2.5

3.0

5.0

3.5

3.5

6.0

4.5

4.0

7.0

5.0

5.0

8.0

6.0

The coeﬃcient, F , is quantiﬁed for each structure type by way of ﬁtting Eq. (19.1c) or (19.17) to the large collection of results of model studies from all over the world. Usually, only large-scale studies are used because both the waves and the wave induced ﬂow in the ﬁlter should be well represented in the model. In the classiﬁcation of structures according to the value of (k D/kf bf ), the upper limit of (k D/kf bf ) is 10 times the lower limit. Therefore, the upper limit of F of each structure type (besides a1.1) is assumed 100.33 = 2.14 times the lower limit, since F = f (k D/kf bf )0.33 . A second curve is drawn with this value of F . In Table 19.3, all available tests are summarized and for each type of structure a lower and upper boundary for the value of F is given (see also an example in Fig. 19.11). The lower boundary gives with [Eq. (19.1c)] a stability curve below which stability is guaranteed. Between the upper and lower boundaries, the stability is uncertain. It depends on various unpredictable inﬂuences whether the structure will be stable or not. The upper boundary gives a curve above which instability is (almost) certain. The results for structure type a3 (blocks on geotextile on sand) may only be applied if the wave load is small [Hs < 1 or 1.5 m (max.)], or to structures with a subsoil of coarse sand (D50 > 0.3 mm) and a gentle slope (tan α < 0.25), because geotechnical failure is assumed to be the dominant failure mechanism (instead of uplift of blocks). A good compaction of sand is essential to avoid sliding or even liquefaction. For loads higher than H = 1.2 m, a well-graded layer of stone on a geotextile is recommended (e.g., layer 0.3–0.5 m for 1.2 m < H < 2.5 m). The results for structure type a4 can be applied on the condition that clay of high quality and with a smooth surface is used. A geotextile is recommended to prevent erosion during (long duration) wave loading. The general design criteria for geotextiles on cohesive soils are given by Pilarczyk.27 In the case of loose blocks, an individual block can be lifted out of the revetment with a force exceeding its own weight and friction. It is not possible with the cover layers with linked or interlocking blocks. Examples of the second type are: block mattresses, ship-lap blocks, and cable mats. However, in this case, high forces will be exerted on the connections between the blocks and/or geotextile. In the case of blocks connected to geotextiles (i.e., by pins), the stability should be treated as

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Fig. 19.11. Example of stability function for type a1.2 (loose closed blocks on granular ﬁlter); low stability. Table 19.4. Recommended values for the revetment parameter F for blockmats (the lower values refer to blocks connected to geotextile while the higher ones refer to cabled blocks). Type of revetment

F (−)

Linked blocks on geotextile on compacted sand or good clay For uncertain conditions/mediocre clay use Linked blocks on a granular ﬁlter:

5–6 4–5

— Favorable construction — Normal construction — Unfavorable construction

5–6 4–5 3–4

for loose blocks in order to avoid the mechanical abrasion of geotextiles by moving blocks. The lower boundary of stability of cabled mats can be increased by a factor of 1.25 (or 1.5, if additionally washed-in with granular material) in comparison with loose blocks. Such an increase of stability is only allowable when special measures are taken with respect to the proper connection between the mats. The upper boundary of stability (F = 8) remains the same for all systems. Application of this higher stability requires optimization of design (including application of geometrically open but stable ﬁlters and geotextiles).6,9 To be able to apply the design method for placed stone revetments under wave load to other semi-permeable systems, the following items may be adapted: the revetment parameter F , the (representative) strength parameters ∆ and D, the

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design wave height Hs , and the (representative) leakage length Λ. Table 19.4 gives an overview of usable values for the revetment stability parameter F in the blackbox model for linked blocks (block mats). The terms “favorable,” “normal,” and “unfavorable” refer to the composition of the granular ﬁlter and the permeability ratio of the top layer and the ﬁlter layer.9 In the case of ﬁne granular ﬁlter and relatively permeable top layer, the total composition can be deﬁned as “favorable.” In a case of very coarse granular layer and less permeable top layer, the composition can be deﬁned as “unfavorable.” In a case of blocks connected to a geotextile and concrete-ﬁlled mattresses on a ﬁlter layer, the construction can be usually deﬁned as between “unfavorable” and “normal,” and the stability factor F = 3.0–3.5 (max. 4.0) can be applied. For blockmats and permeable mattresses on sand F = 5 (max. 6.0) can be applied. The higher values can also be used in cases that the extreme design loading is not very frequent or when the system is (repeatedly) washed in by coarse material providing additional interlocking. This wide range of recommended values for F only gives a ﬁrst indication of a suitable choice. Furthermore, it is essential to check the geotechnical stability with the design diagrams (see Fig. 19.5 and for a full set of diagrams see Refs. 26 and 27). 19.4. Stability Criteria for Concrete-Filled Mattresses 19.4.1. Concrete mattresses Characteristics of concrete mattresses are the two geotextiles with concrete or cement between them. The geotextiles can be connected to each other in many patterns, which results in a variety of mattress systems, each having its own appearance and properties. Some examples are given in Fig. 19.12.

Fig. 19.12.

Examples of concrete-ﬁlled mattresses.

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Fig. 19.13.

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Principles of permeability of Filter Point mattress.

The permeability of the mattress is one of the factors that determine the stability. It is found that the permeability given by the suppliers is often the permeability of the geotextile, or of the so-called Filter Points (Fig. 19.13). In both cases, the permeability of the whole mattress is much smaller. A high permeability of the mattress ensures that any possible pressure buildup under the mattress can ﬂow away, as a result of which the uplift pressures across the mattress remain smaller. In general, with a subsoil of clay and silty sand, the permeability of the mattress will be higher than the permeability of the subsoil. Therefore, the water under the mattress can usually be discharged without excessive lifting pressures on the mattress. The permeability of the mattress will be lower than the permeability of the subsoil or sublayers if a granular ﬁlter is applied, or with sand or clay subsoil having an irregular surface (gullies/cavities between the soil and the mattress). This will result in excessive lifting pressures on the mattress during wave attack. 19.4.2. Design rules with regard to wave load The failure mechanism of the concrete mattress is probably as follows: • First, cavities will form under the mattress as a result of uneven subsidence of the subsoil. The mattress is rigid and spans the cavities.

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• With large spans, wave impacts may cause the concrete to crack and the spans to collapse. This results in a mattress consisting of concrete slabs which are coupled by means of the geotextile. • With suﬃciently high waves, an upward pressure diﬀerence over the mattress will occur during wave run-down, which lifts the mattress (Fig. 19.1). • The pumping action of these movements will cause the subsoil to migrate, as a result of which an S-proﬁle will form and the revetment will collapse completely. It is assumed that local settlement of the subsoil will lead to free spans of the concrete mattress. Then, the wave impact can cause the breaking of these spans, if the ratio of Hs /D is too large for a certain span length. A calculation method is derived on the basis of an empirical formula for the maximum wave impact pressure and the theory of simply supported beams. The collapsing of small spans (less than 1 or 2 m) is not acceptable, since these will lead to too many cracks. The empirical formula for the wave impact is16 : Fimpact = 7.2Hs2 tan α, ρg

(19.18)

where Fimpact = impact force per m revetment (N). Calculation has resulted in an average distance between cracks of only 10–20 cm for a 10-cm thick mattress and wave height of 2 m. This means that at such a ratio of Hs /D the wave impacts will chop the mattress to pieces. For a mattress of 15 cm thick and a wave height of 1.5 m, the crack distance will be in the order of 1 m. Apart from the cracks due to wave impacts, the mattress should also withstand the uplift pressures due to wave attack. These uplift pressures are calculated in the same way as for block revetments. For this damage mechanism the leakage length is important. In most cases, the damage mechanism by uplift pressures is more important than the damage mechanism by impact. The calculated representative values of the leakage length for various mattresses are presented in Table 19.5. The results of calculated stability for various values of leakage length (permeability) are presented in Fig. 19.14.

Table 19.5.

Estimated leakage length for concrete mattresses.

Mattress Leakage length Λ(m) Standard — FP FPM Slab Articulated (Crib) a Good

On sanda

On sandb

On ﬁlter

1.5 1.0 3.0 0.5

3.9 3.9 9.0 1.0

2.3 2.0 4.7 0.5

contact of mattress with sublayer (no gullies/cavities underneath). assumption: poor compaction of subsoil and presence of cavities under the mattress. b Pessimistic

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Fig. 19.14. Calculation results for concrete mattresses (Hs /∆D < 4 because of acceptable crack distance due to impacts on spans).

Taking into consideration the above failure mechanisms, the following design (stability) formula has been derived for the mattresses [similar to Eq. (19.1c)]:

Hs F Hs = 2/3 with = 4, (19.19) ∆D ∆D max ξop where D ∆ ρs F

= = = =

mass per m2 /ρs (which can be called Deﬀective or Daverage ), relative volumetric mass of the mattress (−) = (ρs − ρw )/ρw , volumetric mass of concrete (kg/m3 ), stability factor (see below).

For an exact determination of the leakage length, one is referred to the analytical model.9,16 However, besides the mattresses of a type as, for example, the tube mat (Crib) with relative large permeable areas, the other types are not very sensitive to the exact value of the leakage length. It can be recommended to use the following values of F in design calculations: F = 2.5 or (≤ 3) — for low-permeable mattresses on (ﬁne) granular ﬁlter, F = 3.5 or (≤ 4) — for low-permeable mattress on compacted sand, F = 4.0 or (≤ 5) — for permeable mattress on sand or ﬁne ﬁlter (Df 15 < 2 mm). The higher values can be applied for temporary applications or when the soil is more resistant to erosion (i.e., clay), and the mattresses are properly anchored.

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19.5. Stability of Gabions and Stone-Mattresses 19.5.1. Introduction Gabions are made of rectangular baskets of wire mesh, which are ﬁlled with stones. The idea of the protection system is to hold the rather small stones together with the wire mesh. Waves and currents would have easily washed away the small stones, but the wire mesh prevents this. A typical length of gabions is 3–4 m, a width of 1–3 m, and a thickness of 0.3–1 m. The gabions with small thickness (less then 0.5 m) and large length and width are usually called Reno-mattresses or stone-mattresses. An important problem of this protection system is the durability. Frequent wave or current attack can lead to a failure of the wire mesh because of the continuously moving grains along the wires, ﬁnally cutting through. Another problem is the corrosion of the mesh. Therefore, meshes with plastic coating or corrosion resistant steel are used. On the other hand, the system is less suitable where waves and currents frequently lead to grain motion. 19.5.2. Hydraulic loading and damage mechanisms Wave attack on gabions will lead to a complex ﬂow over the gabions and through the gabions. During wave run-up the resulting forces by the waves will be directed opposite to the gravity forces. Therefore, the run-up is less hazardous than the wave run-down. Wave run-down, as it was already mentioned in Sec. 19.2, will lead to the following mechanisms: The downward ﬂowing water will exert a drag force on top of the gabions and the decreasing phreatic level will coincide with a downward ﬂow gradient in the gabions. During maximum wave run-down, there will be an incoming wave that a moment later will cause a wave impact. Just before the impact, there is a “wall” of water giving a high pressure under the point of maximum run-down. Above the run-down point the surface of the gabions is almost dry and therefore there is a low pressure on the gabions. The interaction of high pressure and low pressure is shown in Fig. 19.1. A simple equilibrium of forces leads to the conclusion that the section from the run-down point to the phreatic line in the ﬁlter will slide down: • if there is insuﬃcient support from gabions below this section, • if the downward forces exceed the friction forces: (roughly) f < 2 tan α with: f = friction of gabion on subsoil and α = slope angle. From this criterion, we see that a steep slope will easily lead to the exceeding of the friction forces, and furthermore, a steep slope is shorter than a gentle slope and will give less support to the section that tends to slide down. Hydrodynamic forces, such as wave attack and current, can lead to various damage mechanisms. The damage mechanisms fall into three categories: (1) Instability of the gabions (a) The gabions can slide downwards, compressing the down slope mattresses. (b) The gabions can slide downwards, leading to upward buckling of the down slope mattresses.

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(c) All gabions can slide downwards. (d) Individual gabions can be lifted out due to uplift pressures. (2) Instability of the subsoil (a) A local slip circle can occur, resulting in an S-proﬁle. (b) The subsoil can wash away through the gabions. (3) Durability problems (a) Moving stones can cut through the mesh. (b) Corrosion of the mesh. (c) Rupture of the mesh by mechanical forces (vandalism, stranding of ship, etc.).

19.5.3. Stability of gabions under wave attack An analytical approach of the development of the uplift pressure in the gabions can be obtained by applying the formulae for the uplift pressure under an ordinary pitched block revetment, with leakage length: Λ = 0.77D. With this relation the stability relations according to the analytical model are also applicable to gabions. Substitution of values, which are reasonable for gabions, in the stability relations provides stability function which indeed match the line through the measured points.16 After complicated calculations the uplift pressure in the gabions can be derived.16 The uplift pressure is dependent on the steepness and height of the pressure front on the gabions (which is dependent on the wave height, period, and slope angle), the thickness of the gabions, and the level of the phreatic line in the gabions. It is not dependent on the permeability of the gabions, if the permeability is larger than the subsoil. The equilibrium of uplift forces and gravity forces leads to the following (approximate) design formula: Hs −2/3 = F · ξop ∆D

with 6 < F < 9 for slope of 1:3 (tan α = 0.33)

(19.20a)

or, using Pilarczyk’s equation (19.2) with b = 2/3 and F = 9 as an upper-limit (Fig. 19.15):

Hs ∆D

= cr

F cos α 9 cos α = 2/3 , b ξop ξop

(19.20b)

where Hs ∆ D F ξop Tp

= = = = = =

signiﬁcant wave height of incoming waves at the toe of the structure (m), relative density of the gabions (usually: ∆ = ∆t ≈ 1), thickness of the gabion (m), stability factor, √ breaker parameter = tan α/ (Hs /(1.56Tp2) wave period at the peak of the spectrum (s).

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Fig. 19.15.

Summary of test results (Ashe1 ) and (Brown3 ) and upper-limit design curve.

For practical applications, F ≤ 7 is recommended in (19.20b) or F = 6 and b = 0.5 in Eq. (19.2) or (19.20b). It is not expected that instability will occur at once if the uplift pressure exceeds the gravity forces. On the other hand, the above result turns out to be in good agreement with the experimental results. The experimental veriﬁcation of stability of gabions is rather limited. Small scale model tests have been performed by Ashe1 and Brown3 ; see Fig. 19.15.

19.5.4. Motion of filling material It is important to know if the ﬁlling material will start to move during frequent environmental conditions, because it can lead to rupture of the wire mesh. Furthermore, the integrity of the system will be aﬀected if large quantities of ﬁlling material are moved. During wave attack the motion of the ﬁlling material usually only occurs if ξop < 3 (plunging waves). Based on the van der Meer’s formula for the stability of loose rock5,7,8 and the assumption that the ﬁlling of the gabion will be more stable than loose rock, the following criterion is derived (van der Meer formula with permeability factor: 0.1 < P < 0.2; number of waves: N < 5000; and damage level: 3 < S < 6): Hs F = , ∆f D f ξop

(19.21)

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where: ∆f = relative density of the stones in the gabions (usually: ∆f ≈ 1.65), Df = diameter of stones in the gabion (m), and F = stability factor, 2 < F < 5. In all situations the stone size must be larger than the size of the wire mesh in the basket; this deﬁnes the minimum size.

19.6. Scour and Toe Protection Toe protection consists of the armoring of the beach or bottom surface in front of a structure which prevents it from scouring and undercutting by waves and currents. Factors that aﬀect the severity of toe scour include wave breaking (when near the toe), wave run-up and backwash, wave reﬂection, and grain size distribution of the beach or bottom materials. Toe stability is essential because failure of the toe will generally lead to failure throughout the entire structure. Toe scour is a complex process. Speciﬁc (generally valid) guidance for scour prediction and toe design based on either prototype or model results have not been developed as yet, but some general (indicative) guidelines for designing toe protection are given in Refs. 4, 5, 22, and 31. The maximum scour force occurs where wave downrush on the structure face extends to the toe and/or the wave breaks near the toe (i.e., shallow water structure). These conditions may take place when the water depth at the toe is less than twice the height of the maximum expected unbroken wave that can exist at that water depth. The width of the apron for shallow water structures with a high reﬂection coeﬃcient, which is generally true for slopes steeper than about 1 on 3, can be planned based on the structure slope and the expected scour depth. The maximum depth of a scour trough due to wave action below the natural bed is about equal to the maximum expected unbroken wave at the site. To protect the stability of the face, the toe soil must be kept in place beneath a surface deﬁned by an extension of the face surface into the bottom to the maximum depth of scour. This can be accomplished by burying the toe, when construction conditions permit, thereby extending the face into an excavated trench the depth of the expected scour. Where an apron must be placed on the existing bottom, or can only be partially buried, its width should not be less than twice the wave height. If the reﬂection coeﬃcient is low (slopes milder than 1 on 3), and/or the water depth is more than twice the wave height, much of the wave force will be dissipated on the structure face and a smaller apron width may be adequate, but it must be at least equal to the wave height (minimum requirement). Since scour aprons generally are placed on very ﬂat slopes, quarrystone of the size (diameter) equal to 1/2 of the primary cover layer probably will be suﬃcient unless the apron is exposed above the water surface during wave action. Quarrystone of primary cover layer size may be extended over the toe apron if the stone will be exposed in the troughs of waves, especially breaking waves. The minimum thickness of cover layer over the toe apron should be two quarrystones. Quarrystone is the most favorable material for toe protection because of its ﬂexibility. If a geotextile is used as a secondary layer it should be folded back at the end, and then buried in cover stone and sand to form a Dutch toe. It is recommended to provide an additional ﬂexible

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Fig. 19.16.

Alternative toe protections.

edge (at least 1 m) consisting of loose material which may easily follow the scour at the toe. Some alternative designs of toe protection are shown in Fig. 19.16. The size of toe protection against waves can be roughly estimated by using the common formulae on slope protection and schematizing the toe by mild slopes (i.e., 1 on 10) or by using formulae developed for breakwaters.

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Toe protection against currents may require smaller protective stone, but wider aprons. The necessary design data can be estimated from site hydrography and/or model studies. Special attention must be given to sections of the structure where scour is intensiﬁed; i.e., to the head, the areas of a section change in alignment, the channel sides of jetties, and the downdrift sides of groynes. Where waves and reasonable currents (>1 m/s) occur together, it is recommended to increase the cover size at least by a factor of 1.3. Note that the conservatism of the apron design (width and size of cover units) depends on the accuracy of the methods used to predict the waves and current action and to predict the maximum depth of scour. For speciﬁc projects, a detailed study of scour of the natural bottom and at nearby similar existing structures should be conducted at a planned site, and/or model studies should be considered before determining a ﬁnal design. In all cases, experience and sound engineering judgment play an important role in applying these design rules.

19.7. Protection Against Overtopping If a structure (revetment) is overtopped, even by minor splash, the stability can be aﬀected. Overtopping can: (a) erode the area above or behind the revetment, negating the structure’s purpose; (b) remove soil supporting the top of the revetment, leading to the unraveling of the structure from the top down; and (c) increase the volume of water in the soil beneath the structure, contributing to drainage problems. The eﬀects of overtopping can be limited by choosing a higher crest level or by armoring the bank above or behind the revetment with a splash apron. For a small amount of overtopping, a grassmat on clay can be adequate. The splash apron can be a ﬁlter blanket covered by a bedding layer and, if necessary, by riprap, concrete units, or asphalt. No deﬁnite method for designing against overtopping is known due to the lack of the proper method on estimating the hydraulic loading. Pilarczyk (1990) proposed the following indicative way of design of the thickness of protection of the splash area (Fig. 19.17): Hs 1.5 cos αi , = Rc ∆Dn ΦT ξ 2b 1 − R n

(19.22)

where Hs = signiﬁcant wave height, ξ = breaker index; ξ = tan α(Hs /Lo )−0.5 , α = slope angle, αi = angle of crest or inner slope, L0 = wavelength, b = coeﬃcient equal to 0.5 for smooth slopes and 0.25 for riprap, Rc = crest height above still water level, Rn = wave run-up on virtual slope with the same geometry (see Fig. 19.17),

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Fig. 19.17.

Deﬁnition of splash area.

D = thickness of protective unit (D = Dn for rock), and ΦT = total stability factor equal to 1.0 for rock, 0.5 for placed blocks, and 0.4 for block mats. The length of protection in the splash area, which is related to the energy decay, depends on the permeability of the splash area. However, it can be roughly assumed as equal to: Ls =

ψ T g(Rn − Rc ) ≥ Lmin , 5

(19.23)

with a practical minimum (Lmin ) equal at least to the total thickness of the revetment (including sublayers) as used on the slope. ψ is an engineering-judgment factor related to the local conditions (importance of structure), ψ ≥ 1, usually 1 < ψ < 2. Stability of rockﬁll protection of the crest and rear slope of an overtopped or overﬂowed dam or dike can also be approached with the Knauss formula.17 The advantage of this approach is that the overtopping discharge, q, can be used directly as an input parameter for calculation. Knauss analyzed steep shute ﬂow hydraulics (highly aerated/turbulent) for the assessment of stone stability in overﬂow rockﬁll dams (impervious barrages with a rockﬁll spillway arrangement). This kind of ﬂow seems to be rather similar to that during high overtopping. His (simpliﬁed) stability relationship can be re-written to the following form: √ q = 0.625 g(∆Dn )1.5 (1.9 + 0.8Φp − 3 sin αi ),

(19.24)

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where q = maximum admissible discharge (m3 /s/m), g = gravitational acceleration (9.81 m/s2 ), Dn = equivalent stone diameter, Dn = (M50 /ρs )1/3 , ∆ = relative density; ∆ = (ρs − ρw )/ρw , αi = inner slope angle, and Φp = stone arrangement packing factor, ranging from 0.6 for natural dumped rockﬁll to 1.1 for optimal manually placed rock; it seems to be reasonable to assume Φp = 1.25 for placed blocks. Note: When using the Knauss formula, the calculated critical (admissible) discharge should be identiﬁed with a momentary overtopping discharge per overtopping fraction of a characteristic wave, i.e., volume of water per characteristic wave divided by overtopping time per wave, roughly (0.3–0.4)T (T = wave period), and not with the time-averaged discharge (q).

19.8. Joints and Transitions Despite a well-designed protective system, the construction is only as strong as the weakest section. Therefore, special care is required when designing transitions. In general, slope protection of dike or seawall consists of a number of structural parts such as: toe protection, main protection in the area of heavy wave and current attack, upper slope protection (very often grass mat), berm for run-up reduction, or as maintenance road. Diﬀerent materials and diﬀerent execution principles are usually applied for these speciﬁc parts. Very often a new slope protection has to be connected to an already existing protective construction which involves another protective system. To obtain a homogeneous strong protection, all parts of protective structures have to be taken under consideration. Experience shows that erosion or damage often starts at joints and transitions. Therefore, important aspects of revetment constructions, which require special attention are the joints and the transitions; joints onto the same material and onto other revetment materials, and transitions onto other structures or revetment parts. A general design guideline is that transitions should be avoided as much as possible, especially in the area with maximum wave attack. If they are inevitable, the discontinuities introduced should be minimized. This holds for diﬀerences in elastic and plastic behavior and in the permeability or the sand tightness. Proper design and execution are essential in order to obtain satisfactory joints and transitions. When these guidelines are not followed, the joints or transitions may inﬂuence loads in terms of forces due to diﬀerences in stiﬀness or settlement, migration of subsoil from one part to another (erosion), or strong pressure gradients due to a concentrated groundwater ﬂow. However, it is diﬃcult to formulate more detailed principles and/or solutions for joints and transitions. The best way is to combine the lessons from practice with some physical understanding of systems involved. Examples to illustrate the problem of transitions are given in Fig. 19.18.

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Fig. 19.18.

Transitions in revetments.

As a general principle, one can state that the transition should be of strength equal to or greater than the adjoining systems. Very often it needs reinforcement in one of the following ways: (a) increase the thickness of the cover layer at the transition, (b) grout riprap or block cover layers with bitumen, and (c) use concrete edge strips or boards to prevent damage progressing along the structure. Top edge and ﬂank protection are needed to limit the vulnerability of the revetment to erosion continuing around its ends. Extension of the revetment beyond the point of active erosion should be considered but is often not feasible. Care should therefore be taken that the discontinuity between the protected and unprotected areas is as small as possible (use a transition roughness) so as to prevent undermining. In some cases, open cell blocks or open block mats (eventually vegetated) can be used as transition (i.e., from hard protection into grass mat). The ﬂank protection between the protected and unprotected areas usually needs a thickened

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Fig. 19.19. Construction aspects of revetments; examples of toe protection and placing of block mats (mattresses), and some methods of anchoring.

or grouted cover layer, or a concrete edge strip with some ﬂexible transition, i.e., riprap. 19.9. General Construction (Execution) Aspects Revetments are constructed in a number of phases, for example: — construction of the bank/dike body, — placement of toe structure,

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— placement of revetment sublayers (clay and/or ﬁlter layers), — laying the blocks or mattress, — anchoring the mattress and, possibly, applying the joint ﬁller. A well-compacted slope is important in order to produce a smooth surface and thus ensure that there is a good connection between the mattress and the subsurface. When laying mattresses on banks, it is strongly recommended that they are laid on undisturbed ground and that areas excavated too deeply are carefully reﬁlled. Before using a geotextile, the slope must be carefully inspected for any projections which could puncture the material. When laying a mattress on a geotextile, care must be taken to ensure that extra pressures are not applied and that the geotextile is not pushed out of place. Geotextile sheets must be overlapped and/or stitched together with an overlap of at least 0.5–1.0 m to prevent subsoil being washed out. This is particularly important if the mattress is laid directly on sand or clay. Block mattresses are laid using a crane and a balancing beam. The mattress must be in the correct position before it is uncoupled because it is diﬃcult to pick up again and also time-consuming. Provided that part of the mattress can be laid above the water line, it can generally be laid very precisely and joints between adjacent mattresses can be limited to 1–2 cm. Laying a mattress completely under water is much more diﬃcult. The spacing between the blocks of adjacent mattresses, nonetheless, should never be more than 3 cm. Once in place, mattresses should be joined so that the edges cannot be lifted/turned up under the action of waves. Loose corners are particularly vulnerable. In addition, the top and bottom edges of the revetment should be anchored, as shown in Fig. 19.19. In such a case, a toe structure is not needed to stop mattresses sliding. More information on execution aspects of revetments can be found in Refs. 4, 5, 7, 8, 9, and 31

19.10. Conclusions The newly derived design methods and stability criteria will be of help in preparing the preliminary alternative designs with various revetment systems. However, there are still many uncertainties in these design methods. Therefore, experimental veriﬁcation and further improvement of design methods are necessary. Also, more practical experience at various loading conditions is still needed.

References 1. G. W. T. Ashe, Beach erosion study, gabion shore protection, Hydraulics Laboratory, Ottawa, Canada (1975). 2. BAW, Code of practice: Use of geotextile ﬁlters on waterway, Bundesanstalt f¨ ur Wasserbau, Karlsruhe, Germany (1993). 3. C. Brown, Some factors aﬀecting the use of Maccaferi gabions, Water Research Lab. Australia, Report 156 (1979). 4. CEM, Coastal Engineering Manual (US Army Corps of Engineers, Vicksburg, 2006).

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5. CIRIA-CUR-CETMEF, The Rock Manual (CIRIA, London, 2007). 6. CUR, Filters in Hydraulic Engineering (Civil Engineering Research and Codes (CUR), Gouda, the Netherlands, 1993) (in Dutch). 7. CUR/CIRIA, Manual on Use of Rock in Coastal Engineering, CUR/CIRIA Report 154 Gouda, the Netherlands (1991). 8. CUR/RWS, Manual on Use of Rock in Hydraulic Engineering, CUR Report 169, Gouda, the Netherlands (1995). 9. CUR/TAW, Design Manual for Pitched Slope Protection, CUR Report 155 (A.A. Balkema, Rotterdam, 1995). 10. DVWK, Guidelines for water management no. 306: Application of geotextile in hydraulic engineering, German Association for Water Resources and Land Improvement (DVWK), Bonn, Germany (1993). 11. Flexible Armoured Revetments, Proceedings of the International Conference, Thomas Telford Ltd., London (1984). 12. FHWA, Geosynthetics Design and Construction Guidelines, Federal Highway Administration, FHWA-HI-95-038, Washington, DC (1995). 13. G. Heerten, Dimensioning the ﬁltration properties of geotextiles considering long-term conditions, Proc. 2nd Int. Conf. on Geotextiles, Las Vegas (1982). 14. G. Heerten, Geotextiles in coastal engineering, 25 years experience, Geotex. Geomemb. 1(2), 119–141 (1984). 15. R. D. Holtz, B. R. Christopher and R. R. Berg, Geosynthetic Engineering (BiTech Publishers Ltb., Richmond, Canada, 1997). 16. M. K. Breteler, K. W. Pilarczyk and T. Stoutjesdijk, Design of alternative revetments, Proc. 26th Int. Conf. on Coastal Engineering, Copenhagen (1998); http://www. wldelft.nl/rnd/publ/search.html (insert for author: Breteler or Pilarczyk). 17. J. Knauss, Computation of maximum discharge at overﬂow rock-ﬁll dams, 13th Congress des Grand Barrages (ICOLD), New Delhi, Q50, R.9 (1979). 18. S. T. Maynord, Corps riprap design guidance for channel protection, in River, Coastal and Shoreline Protection, eds. R. Thorne Colin et al. (John Wiley & Sons, Chichester, UK, 1995). 19. C. R. Neill, Stability of coarse bed material in open channel ﬂow, Edmonton (also, IAHR Congress, Fort Collins) (1967). 20. PIANC, Guidelines for the design and construction of ﬂexible revetments incorporating geotextiles for inland waterways, Report WG 4, PTC I, Supplement to Bulletin No. 57, Brussels, Belgium (1987). 21. PIANC Bulletin, Special issue on propeller jet action, erosion and stability criteria near the harbour quays, Pianc Bulletin no. 58, Brussels, Belgium (1987). 22. PIANC, Guidelines for the design and construction of ﬂexible revetments incorporating geotextiles in marine environment, Report WG 21, PTC II, Supplement to Bulletin No. 78/79, Brussels, Belgium (1992). 23. PIANC, Guidelines for the design of armoured slopes under piled quay walls, Supplement to Bulletin No. 96, Brussels, Belgium (1997). 24. K. W. Pilarczyk (ed.), Coastal Protection (A.A. Balkema, Rotterdam, 1990), www.enwinfo.nl (select English, downloads). 25. K. W. Pilarczyk, Simpliﬁed uniﬁcation of stability formulae for revetments under current and wave attack, in River, Coastal and Shoreline Protection, eds. R. Thorne Colin et al. (John Wiley & Sons, Chichester, UK, 1995). 26. K. W. Pilarczyk (ed.), Dikes and Revetments (A.A Balkema, Rotterdam, 1998). 27. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, 2000), www.balkema.nl.

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28. K. Pilarczyk, International perspectives on coastal structures uses, in Advances in Coastal Structure Design, eds. K. Mohan Ram et al. (ASCE, 2003). 29. EAU, Recommendations of the committee for waterfront structures, German Society for Harbour Engineering (Ernst & Sohn, Berlin, 2000). 30. RWS, The Closure of Tidal Basins, Rijkswaterstaat (The Delft University Press, Delft, 1987). 31. SPM, Shore Protection Manual, U.S. Army Corps of Engineers, Vicksburg (1984). 32. G. Van Santvoort (ed.), Geotextiles and Geomembranes in Civil Engineering, revised edition (A.A. Balkema, Rotterdam, 1994).

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Appendix A. Comparative Stability of Revetments (Pilarczyk, 1990, Coastal Protection) Limits φ(rock) = 2.25 ctgα ≥ 2

Criterion Hs cos α cos α = ψu · φ b = ψu 2.25 b ∆m D ξp ξp

0.5 ≤ b ≤ 1.0 Sublayer

Riprap (2 layers) Riprap (tolerable damage)

Granular Granular

Pitched Stone

1.00 1.33 1.50

Poor quality (irregular-)stone Good quality (regular-)stone Natural basalt

Granular Granular Granular

Loose closed blocks; Hs < 1.5 m Loose (closed-)blocks Blocks connected to geotextile Loose closed blocks Cabled blocks/Open blocks (>10%) Grouted (cabled-) blocks/Interlocked blocks adequately designed Surface grouting (30% of voids) Pattern grouting (60% of voids)

Geotextile on sand Granular Granular Geotextile on clay Granular Granular

Blocks/Blockmats

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Geotexile on clay Sand asphalt

Gabion/mattress as a unit, Hs < 1.5 Stone-ﬁll in a basket, dmin = 1.8 Dn Pm 1 less permeable mattress Pm ≈ 1 (Pm = ratio permeab. top/sublayer Pm 2 Permeable mattress of special design Grass-mat on poor clay; Up < 2 m/s Grass-mat on proper clay; Up < 3 m/s/s

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Chapter 20

Remarks on Coastal Stabilization and Alternative Solutions Krystian Pilarczyk (Former) Hydraulic Engineering Institute Public Works Department, Delft, The Netherlands HYDROpil Consultancy, 23 Nesciohove 2726BJ Zoetermeer, The Netherlands [email protected] Erosion control and coastal stabilization are common problems in coastal engineering. A brief overview of some available alternative systems for shore stabilization and beach erosion control is presented. Special attention is paid to artiﬁcial reefs and geosystems. Geosystems (geotubes, geocontainers, etc.) have gained popularity in recent years because of their simplicity in placement, cost eﬀectiveness, and environmental aspects. However, all these systems have some advantages and disadvantages, which have to be recognized before application.

20.1. Introduction Some coastal environments may be regarded as rather stable (rock and reef coasts) while others are more vulnerable (sand and mud coasts, soft cliﬀs). In this last case, coastal users and managers all over the world are frequently faced with serious erosion of their sandy coasts. Possible causes of erosion include natural processes (i.e., action of waves, tides, currents, sea level rise, etc.) and sediment deﬁcit due to human impact (i.e., river regulations, sand mining, and coastal engineering works). Countermeasures for beach erosion control function depend on local conditions of shore and beach, coastal climate, and sediment transport. Continuous maintenance and improvement of the coastlines, together with monitoring and studies of coastal processes have yielded considerable experience on various coastal protection measures all over the world. This contribution presents an overview of the various available methods for shore stabilization and beach erosion control, with special emphasis on the novel/alternative systems in various design implementations. More detailed information on other coastal protection systems and measures applied nowadays throughout the world can be found in extensive list of references.15,16,51,52,54,80 521

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20.2. General Approach 20.2.1. Types and functions of coastal structures In general, a coastal structure is planned as a practical measure to solve an identiﬁed problem. Starting with identiﬁcation of the problem (e.g., shoreline erosion), a number of stages can be distinguished in the design process for a structure: deﬁnition of functions, determination of boundary conditions, creating alternatives, geometrical design, and the ﬁnal choice of functional solution. After the choice of functional solution has been made the structural design starts including creating structural alternatives (i.e., using diﬀerent materials and various execution methods). The ﬁnal choice will be made after veriﬁcation of various structural solutions in respect to the functional, environmental, and economic criteria. Coastal management, in its broadest sense, must take into account all factors, which bearing on the future of the coastal zone. Politics, policy making, planning, economy, and a multitude of economic, and noneconomic users (e.g., natural reserve areas), environment protection and, by and large, sustainable development, all play signiﬁcant roles and provide both motivation and background for coastal management. The factors of coastal management may well entail many scientiﬁc and engineering disciplines other than coastal engineering and at sites and locations far removed from the costal zone. Some of these factors interact with one another; others are incompatible. The extent to which this applies in any particular region, area, or speciﬁc site needs careful evaluation and compromise solutions. This is one of the major roles of coastal management. The primary objectives of a typical coastal management study are to formulate long-term engineering planning, including ﬁnancial strategies for the future usage, development, and conservation of the coastal zone. In this process, priorities should be deﬁned both for new works and essential maintenance, with estimates for contingency items to cover emergency situations, which inevitably occur (see Fig. 20.1). The key element in any coastal management study is thorough understanding of coastal processes by which is meant the interaction between the hydraulic environment of winds, waves, tides, surges, and currents with the geological conditions in the coastal zone. To be eﬀective, this may require a very broad view to be taken on a regional basis in the ﬁrst instance. A regional cell could then be subdivided into smaller cells once the basic coastal processes had been established, and so on, with decreasing cell sizes until the cell in question becomes the speciﬁc one of the project itself. It is only in this way that the impact of new works in the coastal zone can be evaluated satisfactorily or long-term planning undertaken. The basic tools of the coastal engineering are still fairly limited and comprise cross-shore structures (such as groins, jetties spurs, etc.) shore-parallel structures (oﬀshore breakwaters, seawalls, revetments generally close to shoreline), and dikes, headland structures, and artiﬁcial beach nourishment (see Fig. 20.2). • Groins generate considerable changes in wave and circulation patterns but their basic function — to slow down the rate of littoral drift — is sometimes overlooked. In the absence of beach nourishment, groins can redistribute the existing supply and, in a continuous littoral system, may be expected to create a deﬁciency at

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Analysis of coastal strategies.

the downdrift end where the uncontrolled drift rate is re-established. Without an adequate supply of beach material, groins are of limited value. In addition to controlling the rate of drift, groins are also used extensively to control the distribution of material along a frontage and to limit the temporary eﬀects of drift reversal. There are unfortunately many examples where either bad design or failure to provide for the downdrift consequences has resulted in an adverse eﬀect

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524

Fig. 20.2.

Schematic presentation of various shore protection measures.

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on the coastline. In other instances, failure to maintain groin systems might be more worse than having no groins at all. Oﬀshore breakwaters are usually provided either to reduce wave energy at shoreline structures or to modify the wave climate and enhance sediment transport patterns so as to improve beach levels and create desirable beach features, such as salients.17,22 Oﬀshore breakwaters can be shore-connected or detached, submerged or emerging, alongshore or oblique. Perched beach is a system consisting of a submerged breakwater (“sill” or “reef”), usually located not far away from shoreline, and artiﬁcial beach nourishment providing sand to the area extending between sub-aerial beach and the sill crest. Seawall (wording sometimes used interchangeably with bulkhead) is either a retaining wall intended to hold or prevent sliding of the soil behind it or a massive structure whose primary purpose is to protect the backshore from heavy wave action. Sometimes one speaks of “beach wall” or “shore wall.”15,16,52,53 Revetment is placed on a slope to shelter the adjacent uplands from erosion. Wave reﬂection, a serious disadvantage of vertical-wall bulkheads (seawalls), does not accelerate toe erosion as strongly at revetments as it does at seawalls.51 Dikes are generally intended as means of ﬂood prevention. The crest of a dike is elevated high enough to counteract or conﬁne overtopping in rare storm surge events.53,55,56 Beach nourishment or ﬁll (or recharge), consists of importation of granular material to beach from an external source.17,53 It is not new, and has been used in some countries for decades, but is now being applied to an increasing extent and in a greater variety of ways. The resulting beach provides some protection to the area behind it and also serves as a valuable recreational resource. The beach ﬁll functions as an eroding buﬀer zone, and its useful life will depend on how quickly it erodes. One must be prepared to periodically renourish (add more ﬁll) if erosion continues. Headland control has been devised by analogy to the Nature’s eﬀorts to keep in equilibrium a certain crenulate shape of erosion bays sculptured for thousands or so years. The crenulate shaped bays can be kept in equilibrium by use of a system of headlands. The headland system is claimed to be in feedback with coast and to combine the advantages of groins and detached breakwaters (shore-parallel or oblique).64

All forms of shore protection (i.e., groins, breakwaters, seawalls, revetments, bulkheads, beach-ﬁll, etc.) have certain advantages and disadvantages. A shoreparallel (or oblique) breakwater, placed near the shoreline or oﬀshore and designed either to intercept a portion of moving sediment or to protect a placed beach-ﬁll, has the potential to perform close to the ideal for many types of coastal environments. A number of innovative solutions are given by Silvester and Hsu.64 Recent experience with design of beach stabilization structures is reviewed by Bodge.12 Various low-cost, environment-friendly, emergency and temporary measures, and their combinations provide alternatives to the principal measures. These systems are often appropriate for application only in sheltered waters. Inherent in the concept of environmental friendliness and low cost is the assumption on the equal importance

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of physics, engineering, ecology, and economy. The approach in this contribution is to provide a background for some of these alternatives. 20.2.2. Functional requirements and design It is common knowledge that the realm of coastal processes and the interactions of oﬀshore breakwaters and beaches can be broadly classiﬁed as far-ﬁeld and nearﬁeld phenomena. The scales are somehow arbitrary but can be roughly identiﬁed as those greater and smaller, respectively, than characteristic dimensions of a structure or coastal feature. The design procedures for coastal structures should embody functional/ geometrical design and structural design reﬂecting, respectively, the far-ﬁeld and near-ﬁeld requirements imposed on structures. This corresponds to the division of design procedures into two basic groups concentrating on the overall layout and conﬁguration of a structure as a whole, and its interaction with the coastal environment to produce desirable sedimentation patterns and coastal management eﬀects stability and reliability of the structure and its components, hence dimensioning of structural constituents, associated with possible unavoidable and undesirable hazards due to the loadings exerted by the coastal environment. In other words, the ﬁrst group involves design parameters producing the best environmental eﬀectiveness of a structure in “ideal” conditions, i.e., upon negligence of possible “harmful by-eﬀects,” such as diﬀerent modes of failures and instabilities, both overall and internal. The second group is concerned about these “by-eﬀects” and provides the tools, which secure the integrity and proper operation of the structure and its components. As in many other engineering activities, the design of coastal structures should encompass the following considerations and stages (see also Fig. 20.3): (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

speciﬁcation of the structure’s function(s); description of the physical environment (boundary conditions); selection of envisaged construction technologies; inclusion of design of the structure’s operation and maintenance; conceptual design; preliminary design and selection of alternatives; geometrical dimensioning basing on far-ﬁeld considerations; detailed design basing on near-ﬁeld factors, including structural design; inclusion of possible construction constraints aﬀecting the design; inclusion of some design ﬂexibility allowing for redesign basing on monitoring of the operation and eﬀectiveness of the structure after construction.

Functional requirements and design outline are depicted in Fig. 20.3. It is seen that the design in various stages is veriﬁed through the use of simulation models at diﬀerent levels of complexity. Boundary conditions (bottom) constitute input to both design considerations and the models employed, while the functional requirements (top) ensure evaluation of the suitability of the design and provide design objectives at the same time.

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Design process.

The starting point in the design process consists of the identiﬁcation of the beach erosion problem, followed by the selection of the type of protection measure; the ﬁnal design can incorporate the risk analysis. Attention is drawn to the proper choice of the shore protection measure. The selection is usually aﬀected by the cost. For example, beach nourishment can be cost-eﬀective for low sediment deﬁcit but might be comparable with oﬀshore breakwaters for higher erosion rates. Aside from the cost, many other aspects must also be taken into account upon selection of the shore protection measure. Not shown in the drawing are legal restrictions, regional constraints and priorities, construction, operation and maintenance aspects, etc. Coastal structures are constructed to protect life and property against storm surges, to combat erosion and/or to create (often artiﬁcial) beaches for recreational purposes, and to preserve the natural environment. However, the absolute safety of an area or structure is nearly impossible to achieve. Therefore, it is much better to speak about the probability of failure (or safety) of a certain protection system. To implement this concept, all possible causes and outcomes of failure have to be analyzed. This concept is actually being developed for breakwaters52 and the dike and dune design, mostly in the Netherlands (see, www.enwinfo.nl).18,53,55 The “fault tree” is a handy tool for this aim. In the fault tree all possible modes of failure of elements, which can eventually lead to the failure of a structure section and to inundation are included. They can also badly aﬀect the behavior of the structure, even if the latter is properly designed on the whole. Although all categories of

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events which may cause the inundation of a land or damage of structure are equally important for the overall safety, the engineer’s responsibility is mainly limited to the technical and structural aspects. In the case of coastal structures the following major events can be distinguished: • • • • • •

overﬂow or overtopping of the structure (i.e., instability of the superstructure); erosion or instability of slopes; instability of inner sections leading to progressive failure; scour and instability of toe-protection; instability of the foundation and internal erosion (i.e., piping); instability of the whole structure.

20.3. Alternative Systems for Coastal Protections Various coastal structures, as already discussed, can be applied to solve, or at least, to reduce erosion problems. They can provide direct protection (seawalls, dikes, revetments) or indirect protection (groins and oﬀshore breakwaters of various designs), thus reducing the hydraulic load on the coast (Fig. 20.4). Rock and concrete are usually the construction materials. However, there is a growing interest both in developed and in developing countries in low cost or novel methods of shoreline protection particularly as the capital cost of defense works and their maintenance continues to rise. The shortage of natural rock in certain geographical regions can also be a reason for looking to other materials and systems. Despite this interest there is little published and documented information about the performance of low cost or patented structures especially at more exposed wave climate. Novel systems as geosystems (geotubes, geocontainers, geocurtains) and some other (often patented) systems (Reef Balls, Aquareef, prefabricated units, beach drainage, etc.) have gained popularity in recent years because of (often but not always) their simplicity in placement and constructability, cost eﬀectiveness, and their minimum impact on the environment. These new systems were applied successfully in number of countries and they deserve to be applied on a larger scale. Because of the lower price and easier execution, these systems can be a good alternative for traditional coastal protection/ structures. The main obstacle in their application is, however, the lack of proper design criteria. An overview is given on application and performance of some existing novel systems and reference is made to the actual design criteria. Additional information on these systems can be in references and on websites. 20.3.1. Low-crested structures Low-crested and submerged structures (LCS) as detached breakwaters and artiﬁcial reefs are becoming very common coastal protection measures (used alone or in combination with artiﬁcial sand nourishment).5,9,17,22,57 As an example, a number of systems and typical applications of shore-control structures is shown in Figs. 20.4–20.7.

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Fig. 20.4.

Fig. 20.5.

Examples of shore-control and low-crested structures.

Definitions and objectives of low-crested/reef structures.

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Fig. 20.6.

Fig. 20.7.

Example of Aquareef.33

Example of Reef Balls units.31

The purpose of LCS structures or reefs is to reduce the hydraulic loading to a required level allowing for a dynamic equilibrium of the shoreline. To obtain this goal, they are designed to allow the transmission of a certain amount of wave energy over the structure in terms of overtopping and transmission through the porous structure (emerged breakwaters) or wave breaking and energy dissipation on shallow crest (submerged structures). Due to aesthetical requirements low freeboards are usually preferred (freeboard around SWL or below). However, in tidal environment and frequent storm surges they become less eﬀective when design as a narrow-crested structures. That is also the reason that broad-crested submerged breakwaters (also called, artiﬁcial reefs) became popular, especially in Japan (Figs. 20.5, 20.6 and 20.10).5,33,40,41,48,60−62 They can also be an important measure in combating the eﬀect of sea-level rise; the water level will gradually rise but the wave heights and thus, coastal erosion, run-up and overtopping can be reasonable reduced.

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However, broad-crested structures are much more expensive and their use should be supported by a proper cost-beneﬁt studies. On the other hand, the development in alternative materials and systems, for example, the use of sand-ﬁlled geotubes as a core of such structures, can reduce eﬀectively the cost.24,45,56 The upgrading of (integrated/multidisciplinary) design criteria for LCS structures recently took place in the scope of European project DELOS22 ; (see also: www.delos.unibo.it). The relatively new innovative coastal solution is to use artiﬁcial reef structures called “Reef Balls” as submerged breakwaters, providing both wave attenuation for shoreline erosion abatement, and artiﬁcial reef structures for habitat enhancement.7 An example of this technology using patented Reef BallTM is shown in Fig. 20.7. Reef Balls are mound-shaped concrete artiﬁcial reef modules that mimic natural coral heads. The modules have holes of many diﬀerent sizes in them to provide habitat for many types of marine life. They are engineered to be simple to make and deploy and are unique in that they can be ﬂoated to their drop site behind any boat by utilizing an internal, inﬂatable bladder. Worldwide a large number of projects have already been executed by using this system. The ﬁrst applications were based purely on experience from previous smaller projects. Since recently, more well-documented design criteria are available. Stability criteria for these units were determined based on analytical and experimental studies. For high energetic wave sites the units can be hydraulically anchored with cables to the sea bed. Wave transmission was studied in Canada.4 Technical design aspects are treated by Harris.31 20.3.2. Prefabricated systems There exist a number of other novel and/or low-cost materials and methods for shore protection (gabions and stone mattresses, open stone asphalt, used tire pile breakwaters, sheet pile structures, standing concrete pipes ﬁlled with granular materials, concrete Z-wall (zigzag) as breakwater, geotextiles curtains (screens), natural and mechanical drainage of beaches, and various ﬂoating breakwaters, etc.)54,56 Most of them are extensively evaluated and documented. However, more recently, a new family of prefabricated concrete elements as SURGEBREAKER oﬀshore reef system, BEACHSAVER reef, WAVEblock, T-sill elements, and others have been developed and applied.54 The details on these systems can be found in references and on the websites. However, because of very narrow crest these prefabricated breakwaters are only eﬃcient during mild wave conditions and their eﬀects usually disappear during storm conditions, and because of scour and/or settlement, even losing their stability. The recent evaluation on performance of prefabricated, narrowcrested breakwaters can be found on the US Army website and publications.66 Some of these breakwaters are applied for comparison with other systems in recent US National Shoreline Erosion Control Development and Demonstration Program (Section 227), and more reliable information on the eﬀectiveness of these systems can be expected within a few years (http://chl.erdc.usace.army.mil/CHL.aspx?p = s&a=PROGRAMS;3). The website provides details on sites/systems, and also documentation, if available.

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532

Fig. 20.8.

Principle of distorted ripple mat and application.50

20.3.3. Some other systems 20.3.3.1. Distorted ripple mat A new concept for creating shore accretion is actually developed and applied in Japan. A distorted (precast concrete blocks) ripple mat (DRIM) laid in the surf zone induces a landward bottom current providing accretion of a shore; see Fig. 20.8.50 The strong asymmetry of (artiﬁcial) ripple proﬁle generates current near the bottom to one direction and thus sediment movement, whose concentration is high near the bottom, can be controlled with only very little environmental impact. The hydraulic condition on which the distorted ripple mat can control the sediment transport most eﬀectively is studied experimentally and numerically and its capability to retain beach sand is tested through laboratory experiments and ﬁeld installation. The deﬁnite onshore sediment movement by the control of DRIM is expected if the relative wave height H/h is less than 0.5, where H is the wave height and h is the water depth. The optimum condition for the eﬃcient performance of DRIM is that d0 /λ > 1.7, where d0 is the orbital diameter of water particle and λ is the pitch length of DRIM, and this condition coincides with the condition in which natural sand ripples grow steadily. 20.3.3.2. Beach drainage (dewatering) systems Beach watertable drainage is thought to enhance sand deposition on wave uprush while diminishing erosion on wave backwash (Fig. 20.9). The net result is an increase in subaerial beach volume in the area of the drain. The larger prototype drainage by pumping installations used in Denmark and Florida suggest that beach aggradation

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Principles of beach drainage.

may be artiﬁcially induced by beach watertable drainage. The state of the art of this technique is presented by Vesterby.77 An extensive evaluation of drainage systems is provided by Curtis and Davis18 and actual information can be found on the website: http://shoregro.com/. It is concluded that the drainage system has, in general, a positive eﬀect on diminishing the beach erosion; however, its eﬀectiveness is still diﬃcult to control. The idea to achieve lowering of the water table without pumps by enhancing the beach’s own drainage capacity or hydraulic conductivity through the use of strip drains has been applied in Australia and in Japan.39

20.4. Some Remarks on Wave Transmission and Coastal Response 20.4.1. Wave transmission For shoreline control the ﬁnal morphological response will result from the timeaveraged (i.e., annual average) transmissivity of the applied systems. However, to simulate this in the designing process, for example, in numerical simulation, it is necessary to know the variation in the transmission coeﬃcient for various submergence conditions. Usually, when there is need for reduction in wave attack on structures and properties the wave reduction during extreme conditions (storm surges) is of interest (reduction of wave pressure, run-up and/or overtopping). In both cases the eﬀectiveness of the measures taken will depend on their capability to reduce the waves; the submergence ratio and crest width are important factors (see Fig. 20.10).14,19,21,22,26,57,63,75,76,81 The transmission coeﬃcient, Kt , deﬁned as the ratio of the wave height directly shoreward of the breakwater to the height directly seaward of the breakwater, has the range 0 < K < 1, for which a value of 0 implies no transmission (high, impermeable), and a value of 1 implies complete transmission (no breakwater). Factors that control wave transmission include crest height and width, structure slope, core and armor material (permeability and roughness), tidal and design level, wave height, and period. As wave transmission increases, diﬀraction eﬀects decrease, thus decreasing the size of sand accumulation by the transmitted waves and weakening the diﬀractioncurrent moving sediment into the shadow zone.30 It is obvious that the design rules

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Fig. 20.10.

Prototype measurements for Yugawara reef, Japan.3,49

for submerged structures should include a transmission coeﬃcient as an essential governing parameter. Example of the transmission over the submerged structures (Aquareef) is shown in Fig. 20.11. More detailed descriptions of the functional and technical design of these reefs can be found in Hirose et al.33 The construction of detached breakwaters and, especially, artiﬁcial reefs (= submerged breakwaters with broad crest) is very popular and advanced in Japan. Their application had already started in the 1970s, supported by extensive model studies. The design techniques were gradually improved by using the results of a large number of prototype measurements and by monitoring completed projects (Fig. 20.10). Examples of prototype measurements in Japan and the Japanese design procedure can be found in Refs. 25, 27, 49, 68, and 82. 20.4.2. Layout and morphological response Most commonly an oﬀshore obstruction, such as a reef or island, will cause the shoreline in its lee to protrude in a smooth fashion, forming a salient or a tombolo. This occurs because the reef reduces the wave height in its lee and thereby reduces the capacity of the waves to transport sand. Consequently, sediment moved by longshore currents and waves builds up in the lee of the reef.9–11,35,36,46 The level of protection is governed by the size and oﬀshore position of the reef, so the size

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Fig. 20.11. General transmission characteristic for Aquareef33 (Ht is the transmitted wave height recorded on the landward side, H1/3 and L1/3 are the significant wave height and wavelength at the toe of the rubble mound, B is the crown width, and R is the submergence of the crown).

of the salient or tombolo varies in accordance with reef dimensions. Of course, one can expect this kind of morphological change only if the sediment is available (from natural sources or as sand nourishment). The examples of simple geometrical empirical criteria for the layout and shoreline response of the detached, exposed (emerged) breakwaters can be found in Refs. 20, 32, and 53. To include the eﬀect of submergence (transmission) Pilarczyk57 proposes, at least as a ﬁrst approximation, adding the factor (1− Kt) to the existing rules. Then the rules for low-crested breakwaters can be modiﬁed to (for example): • Tombolo: Ls /X > (1.0–1.5)/(1 − Kt ) or X/Ls < (2/3 to 1)(1 − Kt ),

or

X/(1 − Kt ) < (2/3 to 1)Ls . • Salient: Ls /X < 1/(1 − Kt ) or X/Ls > (1 − Kt ) or X/(1 − Kt ) > Ls . For salients where there are multiple breakwaters: GX/L2s > 0.5(1 − Kt ), where Ls is the length of a breakwater and X is the distance to the shore, G is the gap width, and the transmission coeﬃcient Kt is deﬁned for annual wave conditions. The gap width is usually L ≤ G ≤ 0.8Ls , where L is the wavelength at the structure deﬁned as: L = T (gh)0.5 ; T = wave period, h = local depth at the breakwater. As ﬁrst approximation, Kt = 0 for emerged breakwaters and Kt = 0.5 for submerged breakwaters can be assumed for average annual eﬀects. These criteria can only be used as preliminary design criteria for distinguishing shoreline response to a single, transmissive detached breakwater. However, the range of veriﬁcation data is too small to permit the validity of this approach to be assessed for submerged breakwaters.

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In general, one may conclude that these simple geometrical design rules, both for emerged as well as submerged breakwaters, are of limited value for design calculations because they do not include the eﬀect of the rate of sediment transport, which can be very diﬀerent for a speciﬁc coast. It is supported by the studies on eﬀect of oﬀshore breakwaters along UK coast.6,67 On the other side, it can be stated that numerical models (i.e., Genesis, Delft 2D-3D, Mike 21, etc.) can already be treated as useful design tools for the simulation of morphological shore response to the presence of oﬀshore structures. Examples can be found in Refs. 13, 17, 28–30, 42, and 70. As mentioned above, while considerable research has been done on shoreline response to exposed oﬀshore breakwaters, very little qualitative work has been done on the eﬀect of submerged oﬀshore reefs, particularly outside the laboratory. Thus, within the Artiﬁcial Reefs Program9 (www.asrltd.co.nz), Andrews2 examined aerial photographs seeking cases of shoreline adjustment to oﬀshore reefs and islands. All relevant shoreline features in New Zealand and eastern Australia were scanned and digitized, providing 123 diﬀerent cases. A range of other statistics, particularly reef and island geometry, was also obtained. Some of these results are repeated below. To examine the eﬀects of wave transmission on limiting parameters, data for reefs and islands were considered separately. The data indicated that tombolo formation behind islands occurs with Ls /X ratio of 0.65, and higher and salients form when Ls /X is less than 1.0. Therefore, for islands the Ls /X ratios determining the division between salients and tombolos are similar to those from previously presented breakwater research. Similarly, data resulting from oﬀshore reefs indicate that tombolo formation occurs at Ls /X ratios of 0.6 and higher, and salients most commonly form when Ls /X is less than 2. The data suggests that variation in wave transmission (from zero for oﬀshore islands through to variable transmission for oﬀshore reefs) allows a broader range of tombolo and salient limiting parameters. Thus, a reef that allows a large proportion of wave energy to pass over the obstacle can be (or must be) positioned closer to the shoreline than an emergent feature. Thus, from natural reefs and islands the following general limiting parameters were identiﬁed: • Islands: Tombolos form when Ls /X > 0.65; Salients form when Ls /X < 1.0; • Reefs: Tombolos form when Ls /X > 0.60; Salients form when Ls /X < 2.0; • Nondepositional conditions for both shoreline formations occur at Ls /X < ≈0.1. The choice of the layout of submerged breakwaters can also be aﬀected by the current patterns around the breakwaters. The Japanese Manual (1988) provides (indicative) information on various current patterns for submerged reefs.82 However, for real applications it is recommended to simulate the speciﬁc situation by numerical or hydraulic modeling.

20.5. Geosystems in Coastal Applications 20.5.1. General overview Geotextile systems utilize a high-strength synthetic fabric as a form for casting large units by ﬁlling with sand or mortar. Within these geotextile systems a distinction

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Examples of application of geosystems: (a) Geotube as a breakwater, (b) BEROSIN

can be made between: bags, mattresses, tubes, containers, and inclined curtains.56 All of which can be ﬁlled with sand or mortar. Some examples are shown in Fig. 20.12. Bags are suitable for slope protection and retaining walls or toe protection but the main application is the construction of groynes, perched beaches, and oﬀshore breakwaters. The tubes and containers are mainly applicable for construction of groynes, perched beaches, and (oﬀshore) breakwaters, and as bunds for reclamation works (Fig. 20.13). Geotubes can form an individual structure in accordance with some functional requirements for the project but they can also be used complementary to the artiﬁcial beach nourishment to increase its lifetime. Especially for creating the perched beaches, the sand tubes can be an ideal, low-cost solution for constructing the submerged sill.34,45,56 Geotubes and geocontainers can also be used to store and isolate

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Example of reef structure composed with geotubes24 ; http://geotecassociates.com/.

contaminated materials from harbor dredging, and/or to use these units as bunds for reclamation works. An interesting application for shore erosion control is the geocurtain known under the name BEROSINTM,71 [Fig. 20.12(b)]. The BEROSIN curtain is a ﬂexible structure made of various woven geotextiles, which after placing by divers near the shore and anchoring to the bed catches the sand transported by currents and waves providing accretion on a shore and preventing the erosion. The horizontal curtain (sheet) can be easily spread by a small workboat and two divers. The upper (shoreside) edge, equipped with some depth-compensated ﬂoaters, should be properly anchored at the projected line. The sea-side edge is kept in position by the workboat. By ballasting some of the outside pockets at the lower edge with sand or other materials and with the help of divers, the lower edge is sinking to the required position. The proper choice of permeability of geotextile creates the proper conditions for sedimentation of suspended sediment in front/or under the curtain and at the same time allowing the water to ﬂow out without creating too high forces on the curtain and thus, on the anchors. In case of Pilot project at the coast of Vlieland (NL), some of the horizontal curtains placed in the intertidal zone have provided a growth of a beach/foreshore of 0.5–1.0 m within a week while others within a few weeks.56 These geocurtains can also be applied for the construction of submerged sills and reefs. In the past, the design of geotextile systems for various coastal applications was based mostly on vague experience rather than on the general valid calculation methods. However, the increased demand in recent years for reliable design methods for protective structures have led to the application of new materials and systems (including geotextile systems) and to research concerning the design of these systems. Contrary to research on rock and concrete units, there has been no systematic research on the design and stability of geotextile systems. However, past and recent research in the Netherlands, USA, Germany, and in some other countries on a number of selected geotextile products has provided some useful results which can be of use in preparing a set of preliminary design guidelines for the geotextile systems under current and wave attack.31,56,58 The recent, large scale tests, with large geobags, can be found on the website: http://sun1.rrzn.unihannover.de/fzk/e5/projects/dune prot 0.html.

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When installing geosystems, one should see to it that this does not take place on a rough foundation. Sharp elements may easily damage the casing of the element. Geosystems must usually not be ﬁlled completely. With a ﬁll ratio of approximately 75% an optimum stability of the elements is reached. A sound soil protection is necessary if gravel (sand) sausages are used in circumstances where they are under attack of ﬂow or waves. Stability criteria are discussed by Pilarczyk.56 The main (large) ﬁll-containing geosystems (geobags, geotubes, and geocontainers ﬁlled with sand or mortar) and their design aspects are brieﬂy discussed below. For more detailed information, the reader will be guided to the relevant manuals and publications (see references and websites). 20.5.2. Geobags Geobags can be ﬁlled with sand or gravel (or cement, perhaps). The bags may have diﬀerent shapes and sizes, varying from the well-known sandbags for emergency dikes to large ﬂat shapes or elongated “sausages” (see Fig. 20.14). The most common use for sandbags in hydraulic engineering is for temporary structures. Uses for sandor cement-ﬁlled bags are, among other things: • • • •

repair works; revetments of relatively gentle slopes and toe constructions; temporary or permanent groynes and oﬀshore breakwaters; temporary dikes surrounding dredged material containment areas.

Fig. 20.14.

Application of geobags (sand or cement-filled).

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Because this material is easy to use and cheap, it is extremely suitable for temporary structures. A training groyne is a good example. The working of a groyne is diﬃcult to predict in advance. That is why it is a good procedure to make such a construction using a relatively cheap product ﬁrst, to see how one thing and another works out, and subsequently either make improvements or, after some time, a permanent structure. However, it is recommended not to use sand-ﬁlled geobags for a wave height above 1 m and/or a ﬂow velocity above 2 m/s. The geosystems ﬁlled with sand cannot be used at these more extreme conditions because the sand in the systems is no longer internally stable. Sandbags can be placed as follows: (1) As a blanket: One or two layers of bags placed directly on the slope. An “interlocking” problem arises if the bags are ﬁlled completely. The bags are then too round. A solution is not to ﬁll the bags completely, so that the sides ﬂatten out somewhat, as a result of which the contact area becomes larger. (2) As a stack: Bags stacked up in the shape of a pyramid. The bags lie halfoverlapping with (usually) the long side parallel to the shoreline. The stacking of geotubes increases their stability due to interlocking with the neighboring bags. More recently, prototype experience and large scale tests with large geobags, are described by Restall,58 and more detailed information can be found on the website: http://sun1.rrzn.uni-hannover.de/fzk/e5/projects/dune prot 0.html. 20.5.3. Tube system Geotube is a sand/dredged material ﬁlled geotextile tube made of permeable but soil-tight geotextile.56 The desired diameter and length are project speciﬁc and only limited by installation possibilities and site conditions. The tube is delivered to the site rolled up on a steel pipe. Inlets and outlets are regularly spaced along the length of the tube. The tube is ﬁlled with dredged material pumped as a water–soil mixture (commonly a slurry of 1 on 4) using a suction dredge delivery line (Fig. 20.15). The choice of geotextile depends on characteristics of ﬁll material. The tube will achieve

Fig. 20.15.

Filling procedure of geotube.

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its desired shape when ﬁlled up to about 70–80% (or a height equal to about a half of the ﬂat width of the tube); a higher ﬁlling grade is possible but it diminishes the friction resistance between the tubes. Tube can be ﬁlled on land (e.g., as dikes for land reclamation, bunds, toe protection, or groyns) or in water (e.g., oﬀshore breakwaters, sills of perched beaches, dikes for artiﬁcial islands, etc.). The tube is rolled out along the intended alignment with inlets/outlets centered on top. When a tube is to be placed in water, the eﬀects of buoyancy on the tube geotextile prior to ﬁlling as well as on the dredged material’s settling characteristics must be considered. In order to maximize inlet/outlet spacing, an outlet distant from the inlet may be used to enhance the discharge of dredged slurry and thereby encourage and regulate the ﬂow of ﬁll material through the tube so that suﬃcient ﬁll will ﬂow to distant points. Commonly, the ﬁlter geotextile on both sides of a tube (against scour) and ﬂat main tube are fully deployed by ﬂoating and holding them in position prior to beginning the ﬁlling operation. The geotextile ﬁlter is often furnished with small tubes at the edges which when ﬁlled with sand holds the ﬁlter apron in place. The required length of apron is usually two times the local wave height.

20.5.3.1. Shape and mechanical strength of geotubes For the selection of the strength of the geotextile and calculation of a required number of tubes for a given height of structure, knowledge of the real shape of the tube after ﬁlling and placing is necessary. The change of the cross-section of the tube depends on the static head of the (sand) slurry. Depending on this static head, the laying method, and the behavior of the ﬁll-material inside the tube, it is possible that the cross-sectional shape of the ﬁlled tube will vary from a very ﬂat hump to a nearly fully circular cross-section. More recently, Silvester64 and Leshchinsky43,44 prepared some analytical or numerical solutions and graphs allowing the determination of the shape of sand- or mortar-ﬁlled tubes based on some experiments with water. The Leshchinsky’s method combines all the previous developments and can be treated as a design tool. The design of the shape of the geotube is an iterative process. To obtain a proper stability of the geotube and to fulﬁll the functional requirements (i.e., required reduction of incoming waves/proper transmission coefﬁcient, the width, and the height of the tube must be calculated). If the obtained shape of geotube does not fulﬁll these requirements, a new (larger) size of a geotube must be taken into account or a double-line of tubes can be used.

20.5.4. Container system Geocontainer is a mechanically ﬁlled geotextile and “box” or “pillow” shaped unit made of a soil-tight geotextile.56 The containers are partially prefabricated by sawing mill widths of the appropriate length together and at the ends to form an elongated “box.” The “box” is then closed in the ﬁeld, after ﬁlling, using a sewing machine and specially designed seams (Fig. 20.16). Barge placement of

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Fig. 20.16.

Example of split barge and filling and placing of geocontainer.

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the site-fabricated containers is accomplished using a specially conﬁgured bargemounted crane or by bottom dump hoppers scows, or split barges. The containers are ﬁlled and fabricated on the barge and placed when securely moored in the desired position. Positioning of barge for consistent placement — a critical element of constructing “stacked” underwater structures — is accomplished with the assistance of modern surveying technology. The maximum volume of applied geocontainers was up to now 1000 m3 . The advantages of these large barge-placed geocontainers include: • Containers can be ﬁlled with locally available soil, which may be available from simultaneous dredging activities. • Containers can be relatively accurately placed regardless of weather conditions, current velocities, tides, or water depths. • Contained material is not subject to erosion after placing. • Containers can provide a relatively quick system buildup. • Containers are very cost competitive (for larger works). 20.5.4.1. General design considerations When applying geobags, geotubes, and geocontainers, the major design considerations/problems are related to the integrity of the units during ﬁlling, release and placement impact (impact resistance, seam strength, burst, abrasion, durability), the accuracy of placement on the bottom (especially at large depths), and the stability under current and wave attack. The geotextile fabric used to construct the tubes is designed to: • • • • •

contain suﬃcient permeability to relieve excess water pressure, retained the ﬁll-material, resist the pressures of ﬁlling and the active loads without seams or fabric rapture, resist erosive forces during ﬁlling operations, resist puncture, tearing, and ultraviolet light.

The following design aspects are particularly of importance for the design of containers: (a) change of shape of units in function of perimeter of unit, ﬁll-grade, and opening of split barge, (b) fall-velocity/equilibrium velocity, velocity at bottom impact, (c) description of dumping process and impact forces, (d) stresses in geotextile during impact and reshaping, (e) resulting structural and executional requirements, and (f) hydraulic stability of structure. Some of these aspects are brieﬂy discussed hereafter.

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Fig. 20.17.

Development of forces during dumping of geocontainers.

20.5.4.2. Dumping process of containers and practical uncertainties A summary of various forces during the dumping and placement process is given in Fig. 20.17.56 (1) The required perimeter of geotextile sheet must be suﬃcient enough to release geocontainer through the given split width b0 for a required cross-sectional area of material in the bin of barge Af (or ﬁlling-ratio of ﬁll-material in respect to the max. theoretical cross-section). The derivation of the required minimum length of perimeter of geotextile sheet is given by Pilarczyk.56 After opening of the split of a barge the geocontainer is pulled out by the weight of soil but at the same time the friction forces along the bin side are retarding this process. Due to these forces the tension in geotextile is developing at lower part and both sides of the geocontainer. The upper part is free of tension till the moment of complete releasing of geocontainer. The question is how far we are able to model a friction and the release process of geocontainer. (2) Geocontainer will always contain a certain amount of air in the pores of soil and between the soil and the top of (surplus) geotextile providing additional buoyancy during sinking. The amount and location of air pockets depends on soil consistency (dry, saturated) and uniformity of dumping. The air pockets will exert certain forces on geotextile and will inﬂuence the way of sinking. The question is how to model in a proper way the inﬂuence of soil consistency and air content on shape and stresses in geotubes/geocontainers. (3) The forces due to the impact with the bottom will be inﬂuenced by a number of factors: • consistency of soil inside the geocontainer (dry, semi-dry, saturated, cohesion, etc.) and its physical characteristics (i.e., internal friction);

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• amount of air; • permeability/air tightness of geotextile; • strength characteristics of geotextile (elasticity/elongation versus stresses, etc.); • fall-velocity (inﬂuenced by consistency of soil; saturated soil diminish amount of air but increases fall speed); • shape and catching surface of geocontainer at impact including eﬀect of not horizontal sinking (i.e., catching of bottom with one end); • type of bottom (sand, clay, soft soil, rock, soil covered with rockﬁll mattress, etc.) and/or type of sublayer (i.e., layer of previous placed containers). During the impact the cross-sectional shape of geocontainer will be undergoing a continuous reshaping; from cone shape, ﬁrst probably into a transitional cylindrical shape, and through a certain relaxation, into a semi-oval shape or ﬂat triangular/rectangular shape dictated by soil type, perimeter, and elongation characteristics of geotextile. The question is how far we are able to model this impact phenomena and resulting forces/stresses in geotextile. The impact forces with the bottom are a function of the fall velocity (dump velocity) of a geocontainer. (4) In ﬁnal situation the geocontainers will perform as a core material of various protective structures or as independent structure exposed to loading by currents and waves, and other loadings (ice, debris, ship collision, vandalism, etc.). In most cases the geocontainers will be ﬁlled by ﬁne (loosely packed) soils. The question is how these structures will behave in practice under various types of external and internal loadings. Practical note: The prototype experience indicates that geocontainers with volume up to 200 m3 and dumped in water depth exceeding 10 m have been frequently damaged (collapse of seams) using geotextile with tensile strength lower than 75 kN/m, while nearly no damage was observed when using the geotextile with tensile strength equal to or more than 150 kN/m. This information can be of use for the ﬁrst selection of geocontainers for a speciﬁc project. The placing accuracy for depths larger than 10 m is still a problem.8 20.5.5. Durability of geotextiles Durability of geotextiles is a frequently asked question especially concerning the applications where a long life-span is required. Geotextile is a relatively new product. The ﬁrst applications are from 1960s. Recently, some 30-year-old geotextiles used as a ﬁlter in revetment structures in the Netherlands have been evaluated. In general, these geotextiles were still in a good condition. The technology of geotextiles is improved to such an extent that the durability tests under laboratory condition indicate the life-time of geotextiles at least of 100 years (when not exposed to UV radiation). There is no problem with durability of the geosystems when they are submerged or covered by armor layers. However, in case of exposed geosystems the UV radiation and vandalism are the factors, which must be considered during the design.

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All synthetics are vulnerable to UV. The speed of UV degradation, resulting in the loss of strength, depends on the polymer used and type of additives. Polyesters (PET) are by nature more light stable than, for example, polyamide (PA) and polypropylene (PP). The long-term prototype tests in the Netherlands indicated that the geosynthetics under water and in the intertidal zone show very little degradation in strength in comparison with geosynthetics placed on land; in the inter-tidal zone the geosynthetics are covered very soon by algae which provide very good UV protection. To avoid the problem with light degradation the fabrics must be properly selected (i.e., polyester) and UV stabilized. As the period in which the fabric is exposed is short (in terms of months), no serious problems are to be expected. In case of more or less permanent applications under exposed conditions the fabric must be protected against direct sunlight. There are a number of methods of surface protection for geosystems. To provide additional UV and abrasion protection to the exposed sections of tubes, a coating of elastomeric polyurethane is often used. This coating, however, has a tendency to peel after about a number of months and therefore, has to be re-applied. The permanent surface protection by riprap or block mats is a rather expensive solution and it will normally be applied only when it is dictated by necessity due to a high wave loading or danger of vandalism or other mechanical damage, i.e., boating, anchoring, etc. In other cases it will be probably a cheaper solution to apply a temporary protection of geotextile tubes by an additional layer of a strong geotextile provided with special UV-protection layer. This extra protection can be realized by adding the highly UV-stabilized nonwoven ﬂeece needled onto the main fabric. The function of this felt layer is also to trap the sediment particles and algae, which give again extra UV protection.

20.6. Remarks on Stability Aspects Structural design aspects of low-crested structures are relatively well described in a number of publications.1,15,16,22,23,37,51,52,65,68,69,74,78,79,82 Some useful information on the design of breakwaters on reefs in shallow water can be found in Jensen et al.38 Usually for submerged structures, the stability at the water level close to the crest level will be most critical. Assuming depth limited conditions (Hs = 0.5 h, where h = local depth), the (rule of thumb) stability criterion becomes: Hs /∆Dn50 = 2 or, Dn50 = Hs /3, or Dn50 = h/6, where Dn50 = (M50 /ρs )1/3 ; Dn50 = nominal stone diameter and M50 and ρs = average mass and density of stone. The upgraded stability formulae for LCS structures, including head eﬀect and scour, can be found in Delos report22 (www.delos.unibo.it). It should be noted that some of useful calculation programs (including formula by van der Meer)72,73 are incorporated in a simple expert system CRESS, which is accessible in the public domain (http://ikm.nl/rwscress/ and http://www.ihe.nl/ we/dicea). Useful information on functional design and the preliminary structural design of low crested-structures, including cost eﬀectiveness, can be found in CUR report.17 Alternative solutions, using geotubes (or geotubes as a core of breakwaters), are treated by Pilarczyk.56 An example of this application can be found in Refs. 21, 24, and 57.

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20.7. Conclusions The author does not intend to provide the new design rules for alternative coastal structures. However, it is hoped that this information will be of some aid to designers looking for new sources, which are considering these kinds of structure and improving their designs. Oﬀshore breakwaters and reefs can be permanently submerged, permanently exposed or inter-tidal. In each case, the depth of the structure, its size, and its position relative to the shoreline determine the coastal protection level provided by the structure. To reduce the cost some alternative solutions using geosystems can be considered. The actual understanding of the functional design of these structures needs further improvement but may be just adequate for these structures to be considered as serious alternatives for coastal protection. Continued research, especially on submerged breakwaters, should further explore improved techniques predict shore response and methods to optimize breakwater design. A good step (unfortunately, limited) in this direction was made in a collective research project in the Netherlands.17 Research and practical design in this ﬁeld is also the focus of the “Artiﬁcial Reefs Program” in New Zealand (www.asrltd.co.nz), the International Society for Reef Studies (ISRS) (www.artiﬁcialreefs.org), the European Project DELOS (Environmental Design of Low Crested Coastal Defence Structures; http://www.delos.unibo.it), and recent US National Shoreline Erosion Control Development and Demonstration Program (Section 227). Some useful documents can also be found on the website: http://www.citg.tudelft.nl/live/pagina.jsp?id=4de0d195-5207-4e6784bb-455c5403ae47&lang=en; www.hydraulicengineering.tudelft.nl. Also, the past and recent research in the Netherlands, USA, Germany, and in some other countries on a number of selected geotextile products (geosystems) has provided some valuable results, which can be of use in preparing a set of preliminary design guidelines for the geotextile systems under current and wave attack. The following conclusions can be drawn on application of geosystems based on the actual developments and experience. • Geosystems oﬀer the advantages of simplicity in placement and constructability, cost eﬀectiveness, and minimal impact on the environment. • When applying this technology the manufacturer’s speciﬁcations should be followed. The installation needs an experienced contractor. • When applying geotubes and geocontainers the major design considerations/problems are related to the integrity of the units during release and impact (impact resistance, seam strength, burst, abrasion, durability, etc.), the accuracy of placement on the bottom (especially at large depths), and the stability. • The geotextile systems can be a good and mostly cheaper alternative for more traditional materials/systems. These new systems deserve to be applied on a larger scale. However, there are still many uncertainties in the existing design methods. Therefore, further improvement of design methods and more practical experience under various loading conditions are still needed. • The state of the art of the actual knowledge on the geosystems in hydraulic and coastal engineering can be found in Refs. 8, 34, 45, 56, and 58.

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These new eﬀorts will bring future designers closer to more eﬃcient application and design of these promising coastal solutions. The more intensive monitoring of the existing structures will also help in the veriﬁcation of new design rules. The intention of this literature search is to uncover the technical information on these systems and make them available for the potential users. It will help to make a proper choice for speciﬁc problems/projects and it will stimulate the further developments in this ﬁeld. International cooperation in this ﬁeld should be further stimulated.

References 1. J. Ahrens, Characteristics of Reef Breakwaters, USAE CERC TR 87-17, Vicksburg (1987). 2. C. J. Andrews, Sandy shoreline response to submerged and emerged breakwaters, reefs or islands, unpublished thesis, University of Waikato, New Zealand (1997) (www.asrltd.co.nz). 3. T. Aono and E. C. Cruz, Fundamental characteristics of wave transformation around artiﬁcial reefs, 25th Coastal Eng., Orlando, USA (1996). 4. H. D. Armono and K. R. Hall, Wave transmission on submerged breakwaters made of hollow hemispherical shape artiﬁcial reefs, Canadian Coastal Conf. (2003). 5. T. Asakawa and N. Hamaguchi, Recent developments on shore protection in Japan, Coastal Structures and Breakwaters’91, London (1991). 6. P. Axe, S. Illic and A. Chadwick, Evaluation of beach modelling techniques behind detached breakwaters, ASCE, Proc. 25th ICCE, Orlando, USA (1996). 7. T. Barber, What are Reef Balls, Southwest Florida Fishing News (2000). 8. A. Bezuijen et al., Placing accuracy and stability of geocontainers, 3rd EuroGeo, Munich, Germany (2004), http://www.wldelft.nl/rnd/publ/search.html (insert for Author: Bezuijen). 9. K. Black and S. Mead, Submerged structures for coastal protection, ASR, Marine and Freshwater Consultants, New Zealand (1999) (www.asrltd.co.nz). 10. K. Black and C. J. Andrews, Sandy shoreline response to oﬀshore obstacles, Part I: Salient and tombolo geometry and shape, J. Coastal Res. Special Issue on Surﬁng (2001). 11. K. Black, Artiﬁcial surﬁng reefs for erosion control and amenity: Theory and application, J. Coastal Res. 1–7 (2001). 12. K. R. Bodge, Beach ﬁll stabilization with tuned structures; experience in the Southeastern USA and Caribbean, Coastlines, Structures and Breakwaters’98, London (1998). 13. K. J. Bos, J. A. Roelvink and M. W. Dingemans, Modelling the impact of detached breakwaters on the coast, 25th ICCE, Orlando, USA (1996). 14. M. Buccino and M. Calabrese, Conceptual approach for prediction of wave transmission at low-crested breakwaters, J. Waterways, Port, Coastal Ocean Eng. (May/June) (2007). 15. CEM, Coastal Engineering Manual, US Army Corps of Engineers, Vicksburg (2006). 16. CIRIA-CUR-CETMEF, The Rock Manual, CIRIA, London (2007) (also CUR/ CIRIA, Manual on use of rock in coastal engineering, CUR/CIRIA report 154, Gouda, the Netherlands, 1991). 17. CUR, Beach nourishments and shore parallel structures, R97-2, Centre for Civil Engineering Research and Codes (CUR), P.O.Box 420, Gouda, the Netherlands (1997).

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18. W. R. Curtis and J. E. Davis, Field evaluation/demonstration of a multisegmented dewatering system for accreting beach sand in a high-wave-energy environment, WES/CPAR-CHL-98-1 (1998). 19. K. d’Angremond, J. W. van der Meer and R. J. de Jong, Wave transmission at lowcrested structures, 25th Int. Conf. on Coastal Eng., Orlando, Florida (1996). 20. W. R. Dally and J. Pope, Detached breakwaters for shore protection, Technical report CERC-86-1, U.S. Army Engineer WES, Vicksburg, MS (1986). 21. Delft Hydraulics, AmWaj Island development, Bahrain; physical modelling of submerged breakwaters, Report H4087 (2002). 22. DELOS, Environmental design of low crested coastal defence structures; D 59 DESIGN GUIDELINES, EU 5th Framework Programme 1998–2002, Pitagora Editrice Bologna (2005) (www.delos.unibo.it). 23. Ch. Fleming and B. Hamer, Successful implementation of an oﬀshore reef scheme, 27th Coastal Engineering, Sydney (2000). 24. J. Fowler, T. Stephens, M. Santiago and P. De Bruin, Amwaj Islands constructed with geotubes, Bahrain, CEDA Conf., Denver, USA (2002), http://geotecassociates. com/. 25. H. Funakoshi, T. Siozawa, A. Tadokoro and S. Tsuda, Drifting characteristics of littoral sand around submerged breakwater, Hydro-Port’94, Yokosuka, Japan (1994). 26. Y. Goda, Wave damping characteristics of longitudinal reef system, Advances in Coastal Structures and Breakwaters’95, London (1995); M. D. Groenewoud, J. van de Graaﬀ, E. W. M. Claessen and S. C. van der Biezen (eds.), Eﬀect of submerged breakwater on proﬁle development, 25th ICCE, Orlando, USA (1996). 27. T. Hamaguchi, T. Uda, Ch. Inoue and A. Igarashi, Field experiment on wavedissipating eﬀect of artiﬁcial reefs on the Niigata Coast, Coastal Engineering in Japan, JSCE 34(1) (1991). 28. H. Hanson and N. C. Kraus, GENESIS: Generalised model for simulating shoreline change. Report 1: Technical Reference, Tech. Rep. CERC-89-19, US Army Eng., WES (1989). 29. H. Hanson and N. C. Kraus, Shoreline response to a single transmissive detached breakwater, Proc. 22nd Coastal Eng. Conf., ASCE, The Hague (1990). 30. H. Hanson and N. C. Kraus, Numerical simulation of shoreline change at Lorain, Ohio, J. Waterways, Port, Coastal Ocean Eng. 117(1) (1991). 31. L. E. Harris, www.artiﬁcialreefs.org/ScientiﬁcReports/research.htm; Submerged Reef Structures for Habitat Enhancement and Shoreline Erosion Abatement; FIT Wave Tank and Stability Analysis of Reef Balls, http://www.advancedcoastaltechnology.com/science/DrHarrisWavereduction.htm. 32. M. M. Harris and J. B. Herbich, Eﬀects of breakwater spacing on sand entrapment, J. Hydraulic Res. 24(5) (1986). 33. N. Hirose, A. Watanuki and M. Saito, New type units for artiﬁcial reef development of eco-friendly artiﬁcial reefs and the eﬀectiveness thereof, PIANC Congress, Sydney (2002) [see also 28th ICCE, Cardiﬀ (2002)]. 34. G. Heerten, Geotextiles in coastal engineering, 25 years experience, Geotex. Geomem. 1(2) (1982). 35. J. R. C. Hsu and R. Silvester, Accretion behind single oﬀshore breakwater, J. Waterway Port Coastal Ocean Eng. 116, 362–381 (1990). 36. J. A. Jimenez and A. Sanchez-Arcilla, Preliminary analysis of shoreline evolution in the leeward of low-crested breakwaters using one-line models, EVK3-2000-0041, EU DELOS workshop, Barcelona, 17–19 January 2002.

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37. K. Itoh, T. Toue and D. Katsui, Numerical simulation of submerged breakwater deformation by DEM and VOF, Advanced Design of Maritime Structures in the 21st Century, Yokosuka, Japan (2001). 38. Th. Jensen, P. Sloth and V. Jacobsen, Wave dynamics and revetment design on a natural reef, 26th Coastal Engineering, Copenhagen (1998). 39. K. Katoh, S. Yanagishima, S. Nakamura and M. Fukuta, Stabilization of beach in integrated shore protection system, Hydro-Port’94, Yokosuka, Japan (1994). 40. T. Kono and S. Tsukayama, Wave transformation on reef and some consideration on its application to ﬁeld, Coastal Eng. Jpn. 23 (1980). 41. Y. Kuriyama, K. Katoh and Y. Ozaki, Stability of beaches protected with detached breakwaters, Hydro-Port’94, Yokosuka, Japan (1994). 42. M. Larson, N. C. Kraus and H. Hanson, Analytical solutions of the one-line model of shoreline change near coastal structures, J. Waterway Port Coastal Ocean Eng., ASCE, 123(4) (1997). 43. D. Leshchinsky and O. Leshchinsky, Geosynthetic conﬁned pressurized slurry (GeoCops): Supplement notes for Version 1.0, May (1995) (Nicolon/US Corps). 44. D. Leshchinsky, O. Leschinsky, H. I. Ling and P. A. Gilbert, Geosynthetic tubes for conﬁning pressurized slurry: Some design aspects, J. Geotech. Eng., ASCE, 122(8) (1996). 45. C. R. Lawson, Geotextile containment: International perspectives, Proc. Seventeenth GRI Conf., Geosynthetic Institute, Philadelphia, USA, December (2003), pp. 198–221. 46. D. Ming and Y.-M. Chiew, Shoreline changes behind detached breakwater, J. Waterway Port Coastal Ocean Eng. 126(2) (2000). 47. A. Nakayama, N. Horikosi and H. Kobayashi, The planning and design of multipurpose artiﬁcial barrier reefs, Coastal Zone’93, Coastline of Japan II, New Orleans (1993). 48. A. Nakayama, M. Yamamoto, J. Yamamoto and A. Moriguchi, Development of waterintake works with submerged mound (WWSM), Hydro Port’94, Yokosuka, Japan (1994). 49. S. Ohnaka and T. Yoshizwa, Field observation on wave dissipation and reﬂection by an artiﬁcial reef with varying crown width; Hydro-Port’94, Yokosuka, Japan (1994). 50. N. Ono, J. Irie and H. Yamaguchi, Preserving system of nourished beach by distorted ripple mat, Coastal Engineering, Lisbon (2004), http://www.civil.kyushu-u.ac.jp/ engan/drim.html. 51. PIANC, Guidelines for the design and construction of ﬂexible revetments incorporating geotextiles in marine environment, Brussels, Belgium (1992). 52. PIANC, Analysis of rubble mound breakwaters, PIANC, P.T.C. II/WG. 12, Suppl. To Bulletin 78/79, Brussels (1992). 53. K. W. Pilarczyk (ed.), Coastal Protection (A.A. Balkema, Rotterdam, 1990). 54. K. W. Pilarczyk and R. B. Zeidler, Oﬀshore Breakwaters and Shore Evolution Control (A.A. Balkema, Rotterdam, 1996), www.balkema.nl. 55. K. W. Pilarczyk, Dikes and Revetments (A.A. Balkema, Rotterdam, 1998). 56. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, 2000), www.balkema.nl (select Pilarczyk). 57. K. W. Pilarczyk, Design of low-crested (submerged) structures: An overview, 6th COPEDEC, Sri Lanka (2003), www.enwinfo.nl (select English, downloads). 58. S. Restall et al., Australian and German experiences with geotextile containers for coastal protection, 3rd EuroGeo, Munich, Germany (2004). 59. A. Sanchez-Arcilla, F. Rivero, X. Gironella, D. Verges and M. Tome, Vertical circulation induced by a submerged breakwater, 26th Coastal Engineering, Copenhagen (1998).

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60. T. Sawaragi, I. Deguchi and S. K. Park, Reduction of wave overtopping rate by the use of artiﬁcial reef, Proc. 21st Int. Conf. Coastal Eng., ASCE (1988). 61. T. Sawaragi, Detached breakwaters; short course on design and reliability of coastal structures, Venice, 1–3 October 1992. 62. T. Sawaragi, Coastal Engineering-Waves, Beaches, Wave–Structure Interactions (Elsevier, 1995). 63. S. R. Seabrook and K. R. Hall, Wave transmission at submerged rubble mound breakwaters, 26th Int. Conf. Coastal Eng., Copenhagen (1998). 64. R. Silvester and J. R. Hsu, Coastal Stabilization, (Prentice Hall Inc., Englewood Cliﬀs, 1993), http://www.worldscibooks.com/engineering/3475.html. 65. D. Smit et al., Submerged-crest breakwater design, Advances in Coastal Structures and Breakwaters’95, London (1995). 66. D. K. Stauble and J. R. Tabar, The use of submerged narrow-crested breakwaters for shoreline erosion control, J. Coastal Res. 19(3), 352 (2003), http://chl.erdc.usace. army.mil/CHL.aspx?p=m&a=MEDIA. 67. F. Thomalla and Ch. Vincent, Designing oﬀshore breakwaters using empirical relationships: a case study from Norfolk, UK, J. Coastal Res. 20(4) (2004). 68. T. Uda, Function and design methods of artiﬁcial reef (in Japanese), Ministry of Construction, Japan (1998) (see also, Coastal Zone’93). 69. US Army Corps, Engineering design guidance for detached breakwaters as shoreline stabilization structures, WES, Technical Report CERC–93–19, December 1993. 70. S. C. Van der Biezen, J. A. Roelvink, J. Van de Graaﬀ, J. Schaap and L. Torrini, 2DH morphological modelling of submerged breakwaters, 26th ICCE, Copenhagen (1998). 71. van der Hidde, BEROSIN, Bureau van der Hidde, Harlingen, P.O.B. 299, the Netherlands (1995). 72. J. W. van der Meer, Stability of breakwater armour layers, Coastal Eng. 11 (1987). 73. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Doctoral thesis, Delft University of Technology (1988) (Also: Delft Hydraulics Communication No. 396). 74. J. W. van der Meer, Low-crested and reef breakwaters, Delft Hydraulics Rep. H 986 (1990). 75. J. W. van der Meer, Data on wave transmission due to overtopping, Delft Hydraulics (1990). 76. J. W. van der Meer and K. d’Angremond, Wave transmission at low-crested structures, Coastal Structures and Breakwaters (Thomas Telford, London, UK, 1991), pp. 25–40. 77. H. Vesterby, Beach drainage — state of the art — Seminar on Shoreline Management Techniques, Wallingford, 18 April 1996. 78. C. Vidal, M. A. Losada, R. Medina, E. P. D. Mansard and G. Gomez-Pina, A universal analysis for the stability of both low-crested and submerged breakwaters, 23rd Coastal Engineering, Venice (1992). 79. C. Vidal, I. J. Losada and F. L. Martin, Stability of near-bed rubble-mound structures, 26th Coastal Engineering, Copenhagen (1998). 80. N. Von Lieberman and S. Mai, Analysis of an optimal foreland design, 27th Coastal Engineering 2000, Sydney (2000). 81. T. Wamsley and J. Ahrens, Computation of wave transmission coeﬃcients at detached breakwaters for shoreline response modelling, Coastal Structures’03, Portland, USA (2003). 82. K. Yoshioka, T. Kawakami, S. Tanaka, M. Koarai and T. Uda, Design manual for artiﬁcial reefs, in Coastlines of Japan II, Coastal Zone’93 (ASCE, 1993).

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Chapter 21

Geotextile Sand Containers for Shore Protection Hocine Oumeraci∗ and Juan Recio Leichtweiss-Institute for Hydraulic Engineering and Water Resources, Technical University Braunschweig, Beethovenstr. 51a, 38106 Braunschweig, Germany ∗ [email protected] This chapter aims at providing a brief overview on geotextile sand containers (GSCs) applied in coastal engineering for shore protection. First, the engineering properties required for the geotextile used for sand containers as well as the durability and the lifetime prediction issue are discussed. Second, some example applications are provided to illustrate the versatility of GSCs as an appropriate soft shore prediction alternative to conventional hard coastal structures made of rock and concrete units. However, the major part of the chapter is aimed to address the hydraulic stability of the containers constituting a shore protection structure subject to wave attack. For this purpose, simple formulae are ﬁrst proposed for the stability of the slope and crest containers. The processes which may aﬀect the hydraulic stability are then discussed to highlight the necessity of developing more process-based stability formulae. New stability formulae are ﬁnally proposed which can also account explicitly for the eﬀect of deformation of the containers. Finally, a discussion is provided on the comparative analysis of the stability of the slope and crest containers with and without consideration of the deformation eﬀect.

21.1. Introduction In view of the increasing storminess associated with climate change and its eﬀect on coastal ﬂood and erosion, more versatile materials and solutions are required for the design of new, cost-eﬀective shore protection structures as well as for the reinforcement of existing threatened coastal barriers and structures, including dune reinforcement and scour protection. In search for low-cost, soft, and reversible solutions, the concept of geotextile sand containers (GSCs) as “soft rock” for coastal defense structures was introduced for the ﬁrst time in the 1950s in the form of “sand bags” for the construction of a dike to close the inlet “Pluimpot” in the Netherlands.53 553

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At the same time some test groins were constructed by using “sand bags” at the German North Sea coast.10,57 Later, GSCs have mainly been used as temporary shore protection measures due to the diﬃculties associated with the assessment of the long-term performance. Meanwhile, signiﬁcant advances have been made with respect to the following issues: • improvement of the long-term performance of geotextiles (additives and stabilizers against UV-radiation, coating against abrasion, etc.), • assessment of the durability and lifetime prediction (accelerated testing, standards, etc.),3,14,46 • survey of GSC-built structures and analysis of past experience with respect to the degradation mechanisms, and • understanding of the mechanisms of failure, including hydraulic instability under severe wave action.32,34 These advances, together with numerous advantages of GSCs as “soft rock,” have contributed to extend the use of GSCs to permanent coastal defenses, including a wide range of types of structures such as seawalls, revetments, groins, artiﬁcial reefs, oﬀshore breakwaters, perched beaches, dune reinforcement, core of rubble mound structures, scour protection, etc. (Fig. 21.1). Generally, in coastal engineering any containment of sand encapsulated in geotextile to build ﬂexible and erosion-resistant gravity structures is called “geotextile sand container.” A variety of size and shapes of GSCs have been used, including “geotubes,” “geocontainers,” and “geobags.” The latter have a volume of about 0.05–5 m3 and are generally ﬁlled oﬀsite. Geocontainers are much larger (sausage shape up to 700 m3 ) and generally ﬁlled in split bottom barges. Geotubes have a diameter up to 5.5 m and are ﬁlled directly at the location where they are built (see Fig. 21.2 and Table 21.2). In most cases, it is advantageous to use smaller volume containers because: (i) they are more versatile and can be adapted to build any type of structures; (ii) they can better fulﬁll any requirements with respect to structure slope and geometry (better tolerance); (iii) maintenance and remedial work are much easier in case of vandalism or failure by wave action; (iv) less tensile strength is required and less change of shape will be experienced, thus resulting in longer lifetime; (v) higher density of the sand ﬁll can be achieved; (vi) less risk of liquefaction of the sand ﬁll is expected and thus less GSCdeformations; (vii) simpler mobilization of the required equipment; and (viii) the smaller the containers, the larger the self-healing capacity of the structure. However, larger containers may be required, for instance, in the case of higher waves forces and for temporary structures. A brief illustration of the container sizes used in coastal engineering is given in Fig. 21.2.

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Fig. 21.1.

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Coastal engineering applications of geotextile sand containers.

In the following, some basic information on geotextiles, including a discussion on the durability issue are ﬁrst provided. Second, the versatility of the use of GSCs as a soft shore protection alternative to conventional hard structures made of rock and concrete units is illustrated by some example applications. The major part of the chapter will, however, focus on the hydraulic stability of the GSCs under wave action. For this purpose, simple formulae for the stability of slope and crest GSCs will ﬁrst be proposed which do not take explicitly into account the eﬀect of GSC-deformation and friction between containers. A detailed description of the processes, which may lead to failure under wave action, is then provided to illustrate the necessity of developing more process-based stability formulae. Finally, the new detailed stability formulae and the simple formulae are comparatively analyzed to stress the eﬀect of GSC-deformation on the hydraulic stability.

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Fig. 21.2.

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Range of nonwoven geotextile sand container sizes applied in coastal engineering.

21.2. Geotextile Properties and Durability of Geotextile Sand Containers 21.2.1. Required properties of the geotextile for sand containers Geotextiles and geomembranes, including their related products, are called geosynthetics;19,50 i.e., fabrics that are specially manufactured for civil and environmental engineering applications. While geomembranes are impermeable to water, geotextiles are permeable. The most widely used polymer for geotextiles is polypropylene (PP > 90%), followed by polyester (PET ≈ 5%) and polyethylene (≈ 2%). Based on the manufacturing process two major categories of geotextiles may be distinguished: nonwoven (≈ 60%) and woven geotextiles (≈ 40%). Nonwoven geotextiles are composed of directionally or randomly oriented ﬁbers which are mechanically (needle punching), chemically, or thermally bonded into a loose web. Woven geotextiles are obtained by interlacing two or more sets of yarns (one or several ﬁbers), using conventional weaving processes with a weaving loom. The yarns can be mono-ﬁlament, slit ﬁlm, ﬁbrillated or multi-ﬁlament. Besides these two main groups there are also knitted and stitched geotextiles. The main engineering properties of nonwoven and woven geotextiles are comparatively summarized in Table 21.1.

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Properties of nonwoven and woven geotextiles (After Lawson and Kemplin, 1995)∗ .

Types of geosynthetics Nonwovens Heat-bonded Needle-punched Resin-bonded Wovens Mono-ﬁlament Multi-ﬁlament Flat tape Knitted Weft Warp Stitch-bonded

Tensile strength (kN/m)

Extension at max. load (%)

Apparent opening size (mm)

Water ﬂow rate (volume permeabiltiy) (liters/m2 /s)

Mass per unit area (g/m2 )

3–25 7–90 5–30

20–60 30–80 25–50

0.02–0.35 0.03–0.20 0.01–0.25

10–200 30–300 20–100

60–350 100–3000 130–800

20–80 40–1200 8–90

20–35 10–30 15–25

0.07–4.0 0.05–0.90 0.10–0.30

80–2000 20–80 5–25

150–300 250–1500 90–250

2–5 20–800 30–1000

300–600 12–30 10–30

0.20–2.0 0.40–1.5 0.07–0.50

60–2000 80–300 50–100

150–300 250–1000 250–1000

∗ C. R. Lawson and G. T. Kempton, Geosynthetics and Their Use in Reinforced Soils (Terram Ltd., UIC, 1995).

As shown in Table 21.1, nonwoven and woven fabrics have signiﬁcantly diﬀerent properties which can be exploited to produce the best solution for each speciﬁc need, including composite fabrics to combine the advantages of both types. The lower tensile strength of nonwoven geotextiles as compared to woven geotextiles might represent a drawback, if the containers have to accommodate very large stresses during installation without failure. Generally, however, the stresses on the containers are much lower in service than during installation. The higher capacity for elongation of nonwoven geotextiles can to some extent compensate the disadvantage of lower tensile strength in the sense that it allows to accommodate large strains without failure. This is particularly important when the container is required to reshape during installation and in service (adaptation to scour development and settlement, self-healing eﬀect). Project experiences have shown that the ﬁnal dimensions of sand-ﬁlled containers made of nonwoven geotextile are comparable to those made of woven geotextile. Based on tests with both materials — woven and nonwoven — the ﬁnal height of hydraulically ﬁlled geotextile containers (tubes) was approved to be 80% of the theoretical diameter.17 The hydraulic permeability of geotextile used for sand containers is particularly important when subject to cyclic wetting and drying (e.g., in tidal regime). Water should be drained from the sand container fast enough to avoid excess pressure build up and to ensure overall stability. Therefore, it is generally required that the geotextile should have a much higher permeability than the sand ﬁll without losing the ﬁner fractions. Alternatively, the geotextile can be selected to fulﬁll the commonly used ﬁlter criteria. Nonwoven geotextiles have a higher permeability and a higher ﬁne retention capability than their woven counterparts, but it should be stressed that the permeability is a function of the fabric thickness, and thus depends on its compressibility under normal stresses. The abrasion resistance is particularly important in the surf environment where coarse angular sand, shell fragments, or coral debris are present. The larger thickness of nonwoven geotextiles and the ability of their structure to retain sand material

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make them more resistant. For testing abrasion resistance of diﬀerent geotextile products, a special test setup has been introduced by the Federal Waterways Engineering and Research Institute (BAW, Germany) already about 30 years ago.47 For geotextiles being used in the surf environment abrasion tests, standard test setup (as described in BAW Guidelines for Testing Geotextiles for Navigable Waterways4,47) are recommended. The puncture resistance is important in the case of vandalism, drift ice, drift wood, or dropped rock material during construction. The catalogue for geotextile testing in waterway engineering established by the German Institute for Waterway Engineering (BAW) also includes a puncture test setup which allows a 1:1 test according to the actual stone drop energy expected for the speciﬁc job conditions.23,47 Due to their high elongation capacity and to the retained sand within the fabric structure, nonwoven geotextiles are more susceptible to limit damage from puncture, including vandalism. A higher friction angle between sand containers is desirable to enhance the hydraulic stability against wave and current actions. Due to their structure, nonwoven geotextiles provide a higher friction (see also Sec. 21.4.25). The resistance against UV-radiation and oxidation still represents one of the most critical issues which can limit the service time of exposed GSCs. The resins, ﬁbers, and yarns of the used woven and nonwoven fabrics need special stabilization packages to avoid deterioration already during production and to guarantee longterm stability for more than 50 years. As they are reasonable cost factors, the special stabilization packages need to be described in the speciﬁcations or material certiﬁcates with, e.g., conﬁdential information to the certiﬁcation institute. In general, increasing the thickness of ﬁbers and yarns and increasing weight of the fabric improve UV resistance and long-term stability.49 Dug-up operations from woven and nonwoven geotextiles carried out in the seventies have already shown up to 25 years long-term stability with no indication of further concern to the upcoming years. This is also linked to the fact that the design parameters for surviving handling and installation are giving much reserve for the container simply lying on the sea bed after successful installation. Well-designed and well-installed woven and nonwoven geotextiles for coastal engineering applications have indeed shown that they can be resistant over a long term.9 Despite the signiﬁcant progress in the use of UV-stabilizers, coating or/and armoring of the exposed geotextile containers still remain the sole alternative to achieve a satisfactory lifetime without damage. Simultaneously, the coating/armoring will also protect the geotextile against abrasion, vandalism, drift ice, and drift wood. The resistance against chemical and microbiological attack of polymers (e.g., polypropylene) is very high, so that no signiﬁcant loss of strength is expected during the design lifetime. The ability of geotextiles to enhance marine growth and to attract/support diverse invertebrate communities also becomes an important issue, if maximizing the biodiversity of the recruiting communities is relevant for the choice of the type of geotextile. First, results of investigations comparing woven and nonwoven geotextiles in Australia have shown that the latter are more favorable in this respect.5 The marine growth may represent an enhancement of the resistance against

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UV-radiation and abrasion. However, it is still not clear whether it may cause serious changes or detrimental eﬀects on the mechanical properties of the geotextile over lifetime. 21.2.2. Durability and lifetime prediction Unlike conventional construction materials such as rock and concrete, synthetic geotextiles are, even after 50 years experience, for most coastal engineers relatively new products. Therefore, their degradation and long-term performance are still not well understood. Instead of trying to answer the most frequently asked question: “How long will a geotextile structure last?”, it is more practical from the engineering point of view to ask, how long must a geotextile structure last. The expected lifetime when designing “permanent” shore protection structures is typically 20–100 years. Modern geotextiles are designed to be resistant to degradation from UV-radiation, chemical/biological attack, abrasion, and hydraulic loading. Generally, a lifetime in the order of 20–25 years can be expected if damages during construction and through vandalism are avoided. However, not all applications in coastal engineering require such a level of lifetime, for instance, in the case of temporary protection measures. Although lifetimes up to 100 years have been suggested based on accelerated testing and extrapolation, the following question still remains unanswered: “How to predict/achieve a 100 years or more lifetime for geotextile structures applied for shore protection?” Since much is known about the degradation mechanisms and degradation rates of polymer materials, and how these can be reduced or prevented (Fig. 21.3), it seems reasonable to use this knowledge:

Fig. 21.3.

Degradation mechanisms and reduction procedures.

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• to develop further “index” tests similar to those proposed in Annex B of the European Standards EN 13249–13257 and EN 13265 which are believed to ensure a minimum durability of 25 years and • as a basis for planning and interpreting site monitoring (see ISO 13437), including the development of further procedures and techniques to reduce the degradation rates. Ideally, a degradation curve over the design lifetime for each relevant property of the geotextiles such as tensile strength, speciﬁc mass in g/m2 , elongation capacity, and hydraulic permeability should be determined, together with the associated acceptable degradation limits at which the geotextile cannot perform its primary function (Fig. 21.4). The most authoritative evidence for long-term durability is generally obtained from monitoring of the degradation under real service conditions. In fact, the empirical evidence of long-term durability from retrieved (nonexposed) geotextiles has shown that the reduction of tensile strength and other important engineering properties strongly depends on the prevalent service conditions and therefore signiﬁcantly diﬀers from one site to another. Samples of PP (polypropylene) and PET (polyester) nonwoven geotextiles retrieved from 25 sites in France lost up to 30% of their tensile strength after 10–15 years service time as ﬁlters, separators, and drains, while no chemical/biological eﬀects were identiﬁed.51 A further interesting case was reported by Lefaive,21 showing that the reduction in tensile form strength after 17 years of the same PET straps embedded in a

Fig. 21.4.

Degradation curves (principle sketch).

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concrete facing wall and in the backﬁll was completely diﬀerent: 2% in the backﬁll with pH = 8.5 and up to 40% at the transition between the wall and the backﬁll where pH values of 13–14 and temperature up to 30◦ C prevailed. This case well illustrates the contribution of alkaline surface attack (25%), internal hydrolysis (5–10%), and mechanical damage to degradation of PET ﬁbers or yarns. The eﬀect of UV-radiation is illustrated by a case reported by Troots et al.,52 where woven PET samples were retrieved after 13 years from an earth embankment: while along the section of the slope covered by vegetation and bitumen to provide protection against UV-attack no signiﬁcant changes of the geotextile properties occurred; a reduction up to 50% of the tensile strength was identiﬁed in the nonprotected part. These and further numerous examples from the literature show that the results/ data from retrieved (essentially nonexposed) fabrics, although very valuable, have serious limitations when intended to be used for the prediction of long-term durability and lifetime of permanent structures. Moreover, the obtained data are often incomplete and relate to conditions that are generally far from those for which the prediction/assessment is being made (Fig. 21.5). In order to make the best use of ﬁeld data some brief recommendation for future site monitoring are given in Plate 21.1. These limitations and the urgent necessity for both users and manufacturers to predict lifetime of geotextiles have led to the development of accelerated tests which also have serious limitations (Fig. 21.5).

Fig. 21.5. Limitations of present approach to predict/assess lifetime of permanent geotextile structures for shore protection.

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Plate 21.1.

Fig. 21.6.

Recommendations for future site monitoring.

Principle and limitations of accelerated tests.

The principle of accelerated testing is brieﬂy summarized in Fig. 21.6, showing that (i) generally only one dominant degradation mechanism can be considered, thus ignoring the interaction with other mechanisms and (ii) the approach cannot be applied to all degradation mechanisms.

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Moreover, if two degradation mechanisms occur sequentially (e.g., UV-radiation followed by mechanical degradation), then the two mechanisms are analyzed separately and the predicted lifetimes simply added. Further diﬃculties arise when extrapolating the short lifetime obtained from accelerated tests at increased load frequency/intensity and increased temperature to predict longer lifetime at service conditions. Power laws are often used for extrapolation. For more details, refer to Greenwood and Friday.7 Ideally, the results of both site monitoring and accelerated tests in combination are expected to provide the best basis for the prediction of long-term durability and lifetime. However, a consistent methodology to combine both approaches still needs to be developed (Fig. 21.5). In summary, it can be stated that geotextile applications, although using previous weaker versions of polymer material, performed relatively well over many decades when not exposed to UV-radiation.56 Most failures observed were rather due to faulty design, incorrect choice of material, and poor quality of installation. The knowledge available on the degradation mechanisms, although still limited, allows to predict rationally lifetime up to about 25 years. Since rational prediction cannot foresee problems for which there is no empirical evidence or scientiﬁc basis, the primary goal of future research toward the assessment of durability and lifetime is to improve the understanding of all relevant degradation mechanisms — separately as well as in combination — by making use of both site monitoring and laboratory testing. Future signiﬁcant improvement of UV-resistance and aesthetical aspects as well as the improvement of the long-term performance of GSCs against large wave loads represent further R&D challenges toward avoiding to cover or armor geotextile structures.

21.3. Example Applications of Geotextile Sand Containers for Shore Protection The types of GSCs used in coastal engineering as temporary or permanent structures are generally referred to in the literature as geotubes, geocontainers, or geobags (see Table 21.2 and Sec. 21.1). For permanent coastal structures, small volume containers oﬀer more advantages and are therefore often preferred (see Sec. 21.1). In particular, they are more versatile in application and are used for diﬀerent class of structures, including dune reinforcement, seawalls and revetments, detached breakwaters and artiﬁcial reefs, groins, etc. (see Fig. 21.5). Comprehensive largescale model investigations on GSCs used for the scour protection of monopile foundation of oﬀshore wind turbines have also been conducted by Gr¨ une et al.8 The results are described in more detail in a ﬁnal report by Oumeraci et al.29 Reviews of example applications related to geotextiles in general but, also including geocontainers, can be found in Heerten,11,12 van Santvoort et al.,53 and Pilarczyk.31 Comprehensive reviews on the applications of geosynthetics in hydraulic engineering and for the protection of land ﬁll (including coastal areas) which may also provide valuable information and inspiration for the application

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564 Table 21.2. Type 1. Geotubes

2. Geocontainers

3. Geobags

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Types of geotextile containments used in coastal engineering.

Volume (m3 ) Sand ﬁll Generally >700 m3

On site

Generally 100–700 m3

Split bottom barge Oﬀsite

0.05–5 m3

Shape

Applications

• • • Cylindrical/pillow • (D < 5 m) • Cylindrical (D = 1–55 m)

Pillow, box, mattress

Groins Containment dikes Nonpermanent structures Reef structures (surf zone) Defense structures against tsunami As soft rock units to build any type of coastal structures. Also for scour protection and dune reinforcement

in coastal engineering are given by Heibaum et al.15 and Kavazanjian et al.,18 respectively. More speciﬁc reviews related to geocontainers are provided by Fowler and Trainer,6 Lenze et al.,22 Lawson,20 Saathoﬀ et al.,48 and Jackson et al.16 Rather than trying to duplicate the examples from the aforementioned reviews and to provide a further comprehensive review, it is attempted in the following to select only few examples from three classes of structures and applications: (a) long-shore barriers in the form of seawalls, revetments and dune reinforcement, (b) cross-shore barriers in the form of sea groins, and (c) a new possible application of GSCs as a core of rubble mound structures. The latter type of application may also become particularly important when armoring the GSC-structure is required due to too severe wave attack, abrasion, UV-radiation, and vandalism. 21.3.1. Revetments, seawalls, and reinforcement of beach–dune system Most of the applications of geotextile containment in coastal engineering belong to this type of shore protection; i.e., the containment is built directly along the shoreline to prevent erosion and to stabilize a beach–dune system during storm surge (Fig. 21.7). For this purpose, diﬀerent types of containments have been applied, very often as a last defense line in combination with beach nourishment. An impressive example of the performance of such a last defense line behind a beach nourishment is the wrapped sand containment needle-punched composite geotextile (woven PP slit ﬁlm and nonwoven PET) to reinforce a dune on the island of Sylt (North Sea, Germany) is shown in Fig. 21.8. The stability of this stepped barrier was successfully tested in the Large Wave Flume (GWK) of Hannover. Since 1990, it survived several storm surges with water levels of about 2.5 m above normal and wave heights up to 5 m. Only the sand cover was removed, conﬁrming that the nickname “Bulletproof Vest” commonly given to this type of construction is appropriate. More details on the design and construction of this shore protection are given by Nickels and Heerten26 and Lenze et al.22 Similar geotextile sand containments have also been used successfully at many other sites. Some of them are well documented by ACT.1

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Fig. 21.7. Seawall made of geotextile sand containers (principle sketches): (a) plan view and (b) cross-section.

Fig. 21.8. Geotextile containment for dune reinforcement, Sylt/Germany (extended and modiﬁed from Ref. 26).

More ﬂexible and much simpler in both engineering design and installation, but equally eﬃcient are smaller volume containers (Fig. 21.9). Moreover, smaller containers have many advantages over larger containers and tubes (see Sec. 21.1 and Ref. 2). A comparative analysis of containers with V = 0.75 m3 and containers with a 30 times larger volume for dune reinforcement which confronted the pros and cons of both methods from the client and contractor view point clearly resulted in the selection of the smaller containers.2

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(a)

(b)

(c) Fig. 21.9. Beach and dune reinforcement with geotextile sand containers. (a) Beach reinforcement in Australia, (b) Stockton Beach Revetment,48 and (c) Dune Reinforcement in Wangerooge, North Sea/Germany.54

Using geotextile containment for a long-shore barrier in the form of seawalls, revetment or beach-dune reinforcement has several advantages over hard barrier such as rock structures (see Sec. 21.1), but there are also some drawbacks. The most important drawback is that the sand cover has to be fully or partially rebuilt after each important storm surge because a natural recovery is not always possible. Due to their lower permeability and larger slope steepness, GSC-structures have generally higher reﬂection coeﬃcients and higher wave overtopping rates than rubble mound structures (Fig. 21.10).

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(a)

(b) Fig. 21.10. Wave reﬂection and overtopping performance of GSC-seawalls.28 (a) Overtopping performance and (b) reﬂection performance.

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Fig. 21.11.

Down drift erosion induced by seawalls.

Moreover, it should be kept in mind that a long-shore GSC-barrier also induces a down drift erosion similar to those induced by hard structures (Fig. 21.11), including erosion of the foreshore. 21.3.2. Sea groins Geotextile sand containers, including geotubes, have often been used for emerged and submerged groins (Fig. 21.12). Generally, the containers are uncovered and directly exposed to wave impact, abrasion, and UV-radiation. Therefore, the fabric should be heavily treated for UV-stability and should consist of an inner layer for strength and an outer layer for robustness, durability, and abrasion resistance (geocomposite). Sometimes, the

Fig. 21.12.

Sea groins (principle sketches).

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fabric is made in a color that blends naturally with the beach environment.24 Heavy-duty UV stabilized nonwoven needle-punched geotextiles with high tenacity polyester thread in all seams (min. 80% of basic fabric strength) have also been successfully used.44 Despite large displacement that occurred during storms, the groin continued to provide protection and even withstood abrasion and UV attack over 10 years.48 To reduce the costs in the case of larger groins and larger projects, the groin core can be made of smaller container with lower requirements while strengthening the outer protective layer.24 Further example applications are provided by Saathoﬀ et al.48 and Restall et al.44,45 21.3.3. Core of rubble mound structures There are several reasons which might lead the engineer and other decision-makers in practice to use sand instead of conventional quarry run for the core of rubble mound breakwaters and structures, including among others: (i) Nonavailability of rock material in suﬃcient quantities and at aﬀordable costs. (ii) Avoiding sediment inﬁltration through rubble mound structures which may result in the shoaling of navigation channels and harbor basins, and thus in higher maintenance dredging costs. (iii) Reduction of wave transmission through the structure which might particularly be crucial in the case of long waves. On the other hand, the use of sand as a quasi-impervious core instead of quarry stone would result in an increase of (i) wave setup and runup at the structure, (ii) wave overtopping, (iii) wave reﬂection, which might be detrimental to the stability of the structure, to the operation on and behind the breakwater (due to excessive overtopping) as well as to navigation and seabed stability. Moreover, serious diﬃculties arise in practice when trying to design and construct the ﬁlter to protect the sand core against wash out by wave action. Applying geometrically closed ﬁlter criteria would result in very complex, multiple, and relative thin ﬁlter layers which will not only be very costly and very diﬃcult to build in larger water depths, but also might certainly fail due to the almost unpredictable very complex loading conditions of the sand core under cyclic pulsations by waves and entrained air at the interface with the last ﬁlter layer. Such failures have indeed been observed under both laboratory and ﬁeld conditions in the past. Laboratory evidence has also shown that introducing the so-called “geometrically open ﬁlter” criteria to design a “hydraulic sand-tight ﬁlter” may reduce the number of ﬁlter layers. However, the main practical diﬃculties mentioned above will remain, including those associated with the long-term stability of the sand-core due to the high complexity of the loading and its uncontrollability during the entire storm duration and over the life cycle of the structure.

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Geotextile ﬁlters might present themselves as an alternative to the very complex, costly, and uncertain ﬁlter made of multiple layers of granular material. However, geotextile mats are not only diﬃcult to install under waves and currents, but also may introduce a shear surface which might be detrimental to the stability of the armor layer. Therefore, geotextile ﬁlters need careful installation by experienced contractors with special equipment, and the design should address the shear plane between the geotextile ﬁlter and the layers below and above. Also, “sandmats” as composite products of nonwoven geotextiles and sand needle-punched together may be a suitable solution.13,14 A more feasible alternative is to use a core made of GSCs. This will not only allow to overcome the aforementioned core stability problems, but also to provide (i) a better erosion stability of the core and (ii) an increased stability against seismic loads as compared to a core simply made of loose sand. However, many of the drawbacks mentioned above remain with respect to wave setup, runup, overtopping, reﬂection, and armor stability in comparison to a conventional core. Therefore, an extensive research program has been initiated at LeichtweissInstitute to study both hydraulic performance and armor stability, including the processes involved and the development of prediction formulae for the design of a class of rubble mound structures with a core made of geotextile sand containers (Fig. 21.13). The ﬁrst phase of this research program which is concerned with hydraulic model tests to study in the twin-wave ﬂumes of LWI the hydraulic performance and the armor stability of a rubble mound breakwater made of GSCs as compared to its conventional counterpart with a core made of quarry stones, has been completed

Fig. 21.13.

Class of geocore structures in comparison to conventional rubble mound structures.

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and the results are published by Oumeraci et al.30 Prediction formulae for wave reﬂection, wave runup and overtopping, wave transmission, and armor stability have been determined for the nonconventional breakwater in comparison with the conventional rubble mound breakwater.30 21.4. Hydraulic Stability of Geotextile Sand Containers Depending on the prevailing wave loads and degradation mechanisms, GSCs may experience diﬀerent types of failure modes: (i) hydraulic failure modes, (ii) geotechnical failure modes, and (iii) failure modes related to the geotextile itself (Fig. 21.14). In the following, only the ﬁrst type which is related to hydraulic stability under wave loads will be addressed. First, simple stability formulae without explicit account of the eﬀect of deformation will be proposed separately for the containers on the slope and the containers on the crest of the structure (called hereafter “slope containers” and “crest containers”). Second, the necessity of a better understanding of the processes responsible for the deformation of the sand containers under wave loads as well as their eﬀect on the hydraulic stability is illustrated by some selected results from recent research. Finally, more detailed stability formulae are proposed which can also explicitly account for the deformation eﬀects, including a comparison with the simple stability formulae.

Fig. 21.14.

Failure modes for geotextile sand containers.

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21.4.1. Simple stability formulae Due to the diﬀerent wave loads and boundary conditions which prevails on the slope and on the crest of a coastal structure, a diﬀerent stability behavior and thus diﬀerent stability formulae are expected for the containers on the slope and those on the crest. The following results are extracted from the research report27 of two comprehensive laboratory studies: small-scale model tests performed in the wave ﬂume of Leichtweiss-Institute, using 1 liter sand containers subject to random waves up to 20 cm height and large-scale model tests in the large wave ﬂume of the Joint Coastal Research Centre (FZK) of both Universities of Hannover and Braunschweig, using 150 liter sand containers subject to random waves up to 1.6 m height.27,28 21.4.1.1. Stability of slope containers The sand containers on the slope which are located around the still water level are repeatedly moved up and down by the waves rushing up and down the slope, leading to an incremental seaward displacement of the containers. This dislodgement/pull out eﬀect is illustrated by Fig. 21.15 as observed in the wave ﬂume and in the ﬁeld. Based on the HUDSON-formula for the hydraulic stability of rock armor units (nondeformable) and similarly to WOUTERS,56 a stability number Ns is formulated and postulated to be a function of the surf similarity parameter ξ0 , which includes both the slope steepness tan α as well as the signiﬁcant wave height Hs and the wavelength Lop (Plate 21.2): Ns =

Cw =√ . ξ0 −1 ·D Hs

ρE ρW

(21.1)

With the surf similarity parameter ξ0 = tan α/ Hs /Lop expressed in terms of the deepwater length Lop = gTp2 /2π (Tp = peak period of wave spectrum) the following stability formula is obtained in terms of the characteristic size D of the container: 3/4

D=

Hs

1/2

· Tp · (tan α)1/2 . 2π 1/4 ρE Cw · d ρW − 1

(21.2)

Deﬁning the characteristic size D as D = lc sin α according to Wouters56 and according to the principle sketch in Plate 21.2, Eq. (21.2) can be reformulated in terms of the length lc of the slope containers as: 3/4

lc = Cw ·

2π g

Hs · 1/4

Tp

ρE ρW

. sin 2α −1 2

(21.3)

The empirical parameter Cw was determined by stability tests in a large and a small wave ﬂume to Cw = 2.75. In Fig. 21.16 only the results of the large-scale model tests are plotted to illustrate that the threshold curve between stable and unstable containers is obtained for Cw = 2.75.

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(a)

(b)

(c)

Fig. 21.15. Hydraulic failure modes of slope containers. (a) Wave uprush and downrush on slope containers, (b) pull-out eﬀect in the FZK-large wave ﬂume, and (c) pull-out eﬀect in a dune reinforcement (Courtesy of Heerten).

For Cw = 2.75 and g = 9.81 m/s2 , Eq. (21.3) reduces to: Tp − 1 · sin(2α)

3/4

lc =

1.74 ·

with Hs = Tp = α= ρE =

signiﬁcant wave height (m), peak period of waves (s), slope angle of structure (◦ ), bulk density of GSC (kg/m3 ),

Hs ρE ρW

·

(21.4)

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Plate 21.2.

Fig. 21.16.

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Stability of slope containers based on Hudson formula.

Stability of slope containers from large-scale model tests (modiﬁed from Ref. 28).

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ρW = ρE = n= ρS =

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density of water (kg/m3 ), (1 − n) · ρs + ρW · n (with ρE ≈ 1800 kg/m3 for sand), porosity of ﬁll material (–), density of grain material (kg/m3 ).

The stability formula in Eq. (21.4) expressed in terms of the required container length lc is plotted in Fig. 21.17 for a structure slope angle α = 45◦ in order to illustrate the sensitivity to the signiﬁcant wave height Hs and to the peak period Tp . To compare the results with those of the Hudson-formula for rock armor, the volume of the tested slope containers (V = lc ·0.46lc ·0.14lc = 0.065lc3) is considered. For Hs = 1.5 m and Tp = 4 s, about the same required weight for slope containers with ρE = 18,000 kg/m3 is obtained as for a rock armor with ρE = 2700 kg/m3 and Kd = 2.0. Using a 10 times higher Kd -value (Kd = 20) and applying Hudsonformula in this speciﬁc case for slope containers would indeed provide the same result as the proposed formula. However, such an approach is not applicable as the stability of slope container is very sensitive to the wave period and the Kd value is expected to be a function of both wave height and wave period, due to the deformation eﬀect caused by wave action. In fact, if for the same wave height Hs = 1.5 m the wave period is increased from Tp = 4 s to Tp = 6 s the required length lc will increase by more than 20% (Fig. 21.17). Moreover, due to the eﬀect of deformation of the slope containers on the long-term stability, it is not advisable to use unprotected slope containers for design wave heights of Hs > 1.5–2.0 m.

Fig. 21.17.

Required length of slope containers for hydraulic stability.

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Fig. 21.18.

Hydraulic failure modes of crest containers.

21.4.1.2. Stability of crest containers The sand containers on the crest of the structure may fail due to two possible mechanisms (Fig. 21.18): (i) uplifting during the wave uprush process and shoreward displacement by the wave overtopping ﬂow, (ii) dislodgement and pull-out eﬀect similar to the mechanism observed for the slope containers. Due to the boundary conditions of the crest containers which are more critical than those of the slope containers (no overburden from upper layers), it is expected that the stability of the crest containers will be more critical than that of the slope containers if the crest level of the structure is not high enough. The relative freeboard Rc /Hs , therefore, represents the most important inﬂuencing parameter. In fact, it was diﬃcult to identify a noticeable eﬀect of the surf similarity parameter

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Fig. 21.19.

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Stability of crest containers form large-scale model tests.28

ξ0 on the stability number Ns as clearly observed for slope containers (Fig. 21.19). However, plotting at the top left of Fig. 21.19, the stability number Ns against the relative freeboard Rc /Hs shows that the stability number of the crest containers increases with increasing relative freeboard Rc /Hs according to the following linear relationship (Fig. 21.19): Ns =

Rc = 0.79 + 0.09 , Hs −1 ·D Hs

ρE ρW

(21.5)

(see Plate 21.2) with D = lc sin α substituted in Eq. (21.5), a stability formula for crest containers is obtained in terms of the required container length lc : lc =

ρE ρW

−1

Hs

. Rc 0.79 + 0.09 H sin α s

(21.6)

In a similar way as in Fig. 21.17 for slope containers, the required container length lc in Eq. (21.6) is plotted against the design wave height Hs for diﬀerent relative freeboard Rc /Hs = 0 − 2.0 (Fig. 21.20). Considering exemplarily a typical value Rc /Hs = 1.2 and Hs = 1.5 m, a container length lc = 3.15 m and a container weight W = 36.6 kN are obtained which are much larger than those obtained for the slope containers in Fig. 21.17.

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Fig. 21.20.

Required length of crest containers for the hydraulic stability.

21.4.2. Processes aﬀecting the hydraulic stability 21.4.2.1. Position of the problem and need to improve process understanding As observed in the experiments and shown in Fig. 21.21, the dislodgment and pull out of the slope containers by wave action, including the sliding and overturning of crest containers are strongly aﬀected by the deformation of the sand containers. Simple stability formulae like those proposed in Sec. 21.4.1 cannot explicitly account for the deformation and other mechanisms aﬀecting the hydraulic stability. An improved understanding of the processes and mechanisms is needed in order to (i) possibly avoid failure (engineering judgment), (ii) develop more process-based stability formulae (see Sec. 21.4.3). For this purpose it is necessary to address the following aspects: (i) hydraulic permeability of GSC-structures and its eﬀect on the stability, (ii) wave loads and identiﬁcation of the most critical loading case and location of the containers, (iii) internal movement of sand ﬁll and its eﬀect on the stability, (iv) eﬀect of the friction angle between geotextile containers on the stability, and (v) eﬀect of the container deformations on the stability. The content of this section and next section represents a brief summary of selected key results which have essentially been obtained from comprehensive laboratory studies, including several types of experiments in combination with numerical

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Fig. 21.21.

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Observed failure mechanisms under wave loads.

studies using a CFD-code (RANS-VOF model COBRAS developed at Cornell University) partially coupled with a CSD-code (FEM-DEM code “UDEC” developed by Itasca). These studies were performed in the framework of the PhD-research work by Recio37 and the results are described in more detail in several research reports33,35,36,38–40 and in a PhD thesis.37 21.4.2.2. Hydraulic permeability of GSC-structures Surprisingly, no information on the assessment of the hydraulic permeability of structures made of GSCs could be found in the published literature, although permeability represents an important parameter for both hydraulic stability and hydraulic performance (wave transmission, reﬂection runup, and overtopping). Moreover, reliable permeability values of GSC-structures are also needed for numerical simulations. Comprehensive laboratory experiments supplemented by numerical simulations using the COBRAS-code were performed for the ﬁrst time to investigate the hydraulic permeability of several types of GSC-structures for diﬀerent modes of placement and sizes of the containers, and based on the achieved results then to develop a conceptual model for the assessment of the hydraulic permeability.37,38,41 The key results may be summarized as follows: (i) The permeability of a GSC-structure is essentially governed by the size of the gaps between the containers. The ﬂow through the sand ﬁll can therefore be neglected in the computation. (ii) The hydraulic permeability of a GSC-structure is generally more than 10 times higher and 10 times lower than the permeability of sand (k ≈ 10−3 m/s) and gravel (k = 10−1 m/s), respectively (see Fig. 21.22).

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Fig. 21.22. Hydraulic geocontainers.38,41

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permeability

coeﬃcients

for

diﬀerent

mode

of

placement

of

(iii) The mode of placement of GSCs generally aﬀects the permeability of the GSC-structure (Fig. 21.22). However, randomly placed GSCs and longitudinally placed GSCs have approximately the same permeability coeﬃcient: k = 2.4 cm/s and k = 2.3 cm/s, respectively. To achieve a lower permeability an “interlaid” placement (gaps of the longitudinally placed containers are blocked by transversally placed containers) is required, allowing to reduce the permeability up to k = 1 cm/s and even to k = 0.5 cm/s. (iv) A conceptual model and a systematic procedure to predict the hydraulic permeability is proposed by Recio and Oumeraci.38,41 Validation by experimental data show that uncertainties less than 30% would be expected when applied to prototype containers. 21.4.2.3. Wave-induced loads and critical location of slope containers Based on the pressure recorded by load sensors around an instrumented container placed at diﬀerent locations along the slope of the GSC-structure subject to both breaking and nonbreaking waves, the horizontal and vertical components of the total wave force on the instrumented containers is obtained for each time during a wave cycle. A typical result of the measurement is exemplarily provided in Fig. 21.23

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Fig. 21.23.

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Wave-induced force: Identiﬁcation of critical container location.38– 40,42

to illustrate that the highest horizontal force in seaward direction and the highest vertical force in upward direction occur on the container located just below still water level. Using pressure measurements within the gaps between slope containers for both breaking wave impacts and nonbreaking waves, it is found that the former are less critical for the hydraulic stability than the nonbreaking waves rushing up and down the slope and causing more damage. In fact, the impact pressure induced by breaking waves is of much shorter duration. Moreover, it is strongly damped when propagating inside the gap (Fig. 21.24). 21.4.2.4. Movement and redistribution of sand inside the containers induced by wave action The analysis of the video records of the movement of colored sand inside a transparent permeable container built in the slope of the GSC-structure (Fig. 21.25) and subject to wave action has shown that (Fig. 21.26): (i) Noticeable sand movements occur only for larger waves that are capable to substantially move the front part of the container up and downward during the wave runup and rundown process [Fig. 21.26(b)]. After about 30 wave cycles the internal sand movement decreases signiﬁcantly due to the accumulated sand at the seaward front of the container [Fig. 21.26(c)]. (ii) Due to the sand ﬁll redistribution at the seaward front of the container, the latter deforms, thus oﬀering a larger impact surface area to the mobilizing wave forces and reducing the contact area with neighboring containers [Fig. 21.26(c)].

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Fig. 21.24.

Fig. 21.25.

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Impact pressure propagation inside a gap between geotextile sand containers.42

Permeable transparent container for the investigation of internal movement of sand.38

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Fig. 21.26.

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Internal movement and redistribution of sand ﬁll in the transparent container.42

With the increased mobilizing forces and the decreased resisting forces, an incremental lateral seaward displacement of the deformed container occurs (pull-out eﬀect) which causes again the start of the internal sand movement in a similar way as during the ﬁrst wave cycles [Fig. 21.26(a)]. These results have considerable practical implications as the internal sand movement is responsible for the deformations of the container which aﬀect • the hydraulic stability by reducing the contact area between GSCs and by increasing the drag and lift forces due to the increased exposed areas, • the crest level of a GSC-structure. Even in the laboratory it is observed that the height of sand containers is reduced by 4% after placement under water and by further 6% due to wave action. As a result, a total reduction of the height of the GSC-structure of about 10% was observed.38,41 Since these eﬀects are strongly dependent on the adopted sand ﬁll ratio, future research and design guidance should be directed toward the deﬁnition of an optimal sand ﬁll ratio by accounting for the deformation properties of the geotextile and

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Table 21.3.

Results of friction tests in a shear box25 (30 cm × 30 cm).25 Measured peak friction stress τ (kN/m2 ) at normal stress σ (kN/m2 ) σ = 50 kN/m2

σ = 100 kN/m2

Nonwoven versus nonwoven geotextiles

27 29

48 47

Woven versus woven geotextiles

16 14

29 26

Interface

by balancing the advantages and drawbacks of high sand ﬁll ratio. Moreover, this issue should also be explicitly addressed in future standards and guidelines due to the considerable eﬀect of the sand ﬁll ratio on stability and long-term performance. 21.4.2.5. Friction between geotextile sand containers Friction angles between woven geotextiles vary from about 12◦ (e.g., MiraﬁGT500) and 20◦ (e.g., Geolon PP120S), while for nonwoven geotextiles values of 20–26◦ (mechanical bound) or even 20–30◦ (thermal bound) are more common. Generally, the friction behavior between geotextiles increases with the surface roughness of the geotextiles. The higher the transmittable shear stress τ in relation to the induced normal σ stress in an interface (τ /σ), the higher is the friction behavior. Table 21.3 shows shear box test results for diﬀerent interfaces at normal stresses of 50 kN/m2 and 100 kN/m2 .25 The results of numerical simulations using the partially coupled CFD and CSD codes (COBRAS and UDEC) previously validated by laboratory data, as shown in Fig. 21.27, highlight the signiﬁcant eﬀect of friction between the GSCs on the hydraulic stability. The eﬀect of the friction angle on the stability is particularly important within the range of the practically relevant values (15–30◦), implying that the friction between the containers should be explicitly considered in future stability formulae. 21.4.2.6. Eﬀect of container deformations on the stability It was shown in Sec. 21.4.2.4 that the sand ﬁll is redistributed, resulting in deformations of the containers (Fig. 21.26). These deformations aﬀect the stability of the containers in the following manner: (i) Reduction of the stability against sliding caused (a) by the increase of the drag and uplift forces as a result of the increased exposed areas AS and AT (Fig. 21.28) as well as (b) by the decrease of the resisting force as a result of the decreased friction area between the containers (Figs. 21.28 and 21.29). (ii) Reduction of the stability against overturning caused by the increase of the mobilizing drag and uplift forces as mentioned above, but also by the seaward

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Fig. 21.27.

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Eﬀect of friction between GSCs on hydraulic stabilization.42

Fig. 21.28. Increase of exposed areas of drag and uplift forces and decrease of friction areas due to container deformation.

shift of the center of gravity of the deformed container leading to a reduction of the resisting moment (Fig. 21.30). A closer experimental and numerical examination of the variation of the “eﬀective” contact areas between containers during wave action (i.e., those areas

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Fig. 21.29.

Fig. 21.30.

Eﬀect of container deformation on sliding stability.38,42

Eﬀect of container deformation on overturning stability.42

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Fig. 21.31.

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Reduction of eﬀective contact areas between containers during wave action.37,42

which contribute to the resistance against dislodgment by friction) has shown that the “eﬀective” contact areas (i) decrease due to the upward movement of the front part of the containers (Fig. 21.31). (ii) increase with increasing slope angle of the GSC-structure. 21.4.3. Process-based stability formulae The insight in the processes aﬀecting the hydraulic stability as described in Sec. 21.4.2 has clearly highlighted the necessity of an explicit account of these processes in future stability formulae, at least those processes which mostly aﬀect the stability such as the eﬀect of deformations of the container and the friction between the containers. In order to examine more closely the eﬀect of deformation on the stability, new process-based stability formulae are proposed: ﬁrst, without any account of

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Fig. 21.32.

Sliding stability formulae without deformation eﬀect.

the deformation eﬀect, and then correction factors are introduced to account for the deformation eﬀect. Finally, a comparative analysis between the results of the stability formulae with and without deformation eﬀect is provided for both slope and crest containers. It should be underlined that the new formulae proposed below were derived analytically for the geometry of the containers commonly used in coastal structures, i.e., with a container length lc which is twice the container width (lc /2) and ﬁve times the container height (lc /5) resulting in the following relationships for the volume of the container V and the application area of the drag and uplift forces AS and AT , respectively [Fig. 21.32(a)]. V = 0.1lc3 , AS = 0.1lc2 ,

(21.7)

AT = 0.5lc2 . These relationships provide the geometrical parameters that govern the resisting forces (weight) and the mobilizing forces (drag, inertia, and uplift forces), thus allowing to express the stability formulae in terms of the container length lc (see also Sec. 21.4.1). If, however, other container geometries, and thus other relationships which diﬀer from those in Eq. (21.7) are adopted, the stability formulae can be modiﬁed accordingly. Further indications on how to proceed with such modiﬁcations and on the limitations of the proposed stability formulae will also be given below.

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21.4.3.1. Stability formulae without deformation eﬀect (a) Stability against sliding: A sand container is stable as long as the resisting force FR = µ(FGSC − FL ) generated by the resulting normal force (FGSC − FL ) due to friction remains larger than the drag force FD and the inertia force FM (Fig. 21.32): µ (FGSC − FL ) ≥ (FD + FM ) .

(21.8)

And with the relative density parameter ∆ = (ρE /ρW ) − 1 (see Plate 21.2):

µ ∆gV − 0.5CL AT · u

2

∂u 2 . ≥ 0.5CD AS u + CM V ∂t

(21.9)

Given the considered container geometry, relationships similar to those in Eq. (21.7) which provide V , AT , and AS as a function of the container length lc can be obtained and substituted in Eq. (21.9) which is then solved to obtain either the required length lc or the required mass WGSC of the container. Using, for instance, the geometry described by Eq. (21.7), the stability formulae are given in Fig. 21.32 in terms of the required length lc or mass WGSC of the container. (b) Stability against overturning: A sand container is stable as long as the stabilizing moment induced by the weight of the container under buoyancy FGSC remains larger than the mobilizing moment induced by the drag, inertia, and uplift forces FC , FM , and FL (Fig. 21.33): FGSC · rs ≥ FD · rh + FM · rh + Fc · rs

(21.10)

(ρE − ρw )gV · rs ≥ 0.5ρw CD u2 As · rh + ρw CM

∂u V · rh + 0.5ρW CL u2 AT · rs . ∂t

(21.11)

Given the considered container geometry, relationships similar to those in Eq. (21.7) which provide V , As , and AT , but also the lever arms rs and rh as a function of the container length lc can be obtained and substituted in Eq. (21.11). The latter is then solved to obtain either the required length lc or the required mass WGSC of the container. Using, for instance, the geometry described by Eq. (21.7), the overturning stability formulae are obtained in terms of the required length lc or mass WGSC of the container (Fig. 21.33). More details on the force coeﬃcients CD , CM , and CL as well as on further input parameters required in the stability formulae summarized in Figs. 21.32 and 21.33 will be given in Sec. 21.4.3.2. 21.4.3.2. Stability formulae including deformation eﬀect The eﬀect of the container deformations on the stability is explicitly accounted for by introducing analytically derived deformation factors into the formulae for the drag force, lift force, inertia force, and resisting forces for both hydraulic failure

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Fig. 21.33.

Overturning stability formulae without deformation eﬀect.

modes37,43 : sliding and overturning. The deformation factors are obtained as correction factors, describing the changes of (Plate 21.3): The resisting force FR against sliding. The correction factor KSR is obtained as the ratio of the eﬀective weight contribution to the frictional force FR with and without deformation. The resisting moment against overturning. The correction factor KOR is obtained as the ratio of lever arm rs of the container weight FGSC under buoyancy with and without deformation eﬀect. The mobilizing drag, lift, and inertia forces contributing to sliding. The correction factors of the drag force FD and lift force FL (KSCD and KSCL ) are obtained as the ratios of the areas AS and AT with and without deformation eﬀect. The correction factor KSCM is assumed to be 1.0 since the container volume V remains constant. The mobilizing moment induced by the drag, lift, and inertia forces. The correction factor for the moments induced by the drag and lift forces (KOCD and KOCL ) area is obtained as the ratio of describing the changes of both surface areas (AS and AT ) and lever arms (rm and rs ) of the drag and lift forces FD and FL . The correction factor KOCM is obtained as the ratio of lever arms rsn of the inertia force with and without deformation eﬀect. The values of the correction factors suggested in Table 21.4 are derived on the basis of a number of simplifying assumptions (see Ref. 37 for more details).

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Plate 21.3.

Stability formulae including the eﬀect of deformation.

Table 21.4.

Deformation factors and force coeﬃcients.

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Deﬁning of parameters and typical values to be used in the stability formulae.

Among the most important assumptions the following are noteworthy: sand ﬁll ratio of 80% and a slope angle of the GSC-structure of 45◦ . Indications to account for other slope angles and recommendations on further research to overcome most of the simplifying assumptions are given by Recio.37 Moreover, the force coeﬃcients CD , CL , and CM are also given as a function of the Reynolds number for diﬀerent locations and boundary conditions which may represent diﬀerent practical applications (scour protection on the sea bed, artiﬁcial reef, slope containers, and crest container of a surface piercing structure such as revetments, seawalls, groins, etc.). The proposed values of CD , CM , and CL have been determined on the basis of systematic laboratory experiments.35 In Plate 21.4 the parameters used in the stability formulae described in Figs. 21.32–21.35 are deﬁned and typical values are also given, including some remarks on the limitations of the suggested values. 21.4.3.3. Comparative analysis of stability formulae with and without including deformation eﬀect In order to illustrate the eﬀect of deformation the results of the new more processbased stability formulae with and without consideration of the deformations of the container as proposed in Secs. 21.4.3.2 and 21.4.3.1 are compared in Figs. 21.35 and 21.34 for a sloping revetment with an angle of 45◦ , a water depth d = 4 m at the structure, and a peak period Tp = 6 s of the waves. Moreover, the simple stability formulae for slope containers (Fig. 21.34) and crest containers (Fig. 21.35) proposed

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Fig. 21.34.

Stability formulae for slope containers: Comparative analysis.

Fig. 21.35.

Stability formulae for crest containers: Comparative analysis.

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in Sec. 21.4.1 are also considered to illustrate the diﬀerence with the results from the more process-based formulae. Depending on the range of the design wave height Hs , the following remarks may be drawn from the comparison of the formulae for the slope containers (Fig. 21.34): (i) For smaller design waves (Hs ≤ 1.5 m): the simple stability formulae proposed in Sec. 21.4.2 are too conservative and the deformation eﬀect on the stability obtained from the comparison of the new formulae proposed on Sec. 21.4.3 are relatively small. (ii) For larger design waves (Hs ≥ 2.5 m): the simple stability formulae become more unsafe with increasing wave height. The eﬀect of deformation also increases with the increase of the design wave height. Depending on the range of the design wave height Hs the following remarks may be drawn from the comparison of the stability formulae for the crest containers (Fig. 21.35): (i) For smaller design waves (Hs ≤ 1.5 m): the simple stability formula proposed in Sec. 21.4.2 is slightly conservative and becomes unsafe as soon as the signiﬁcant wave height exceed 1.5 m. It can therefore be used instead of the more processbased formula only for Hs ≤ 1.5 m. However, the eﬀect of deformation is higher than in the case of slope containers. (ii) For larger design waves (Hs ≥ 2.5 m): the simple stability formula by Oumeraci et al. (2003) becomes more unsafe with increasing wave height Hs . The eﬀect of deformation on the stability also increases with increasing wave height. Comparing Figs. 21.34 and 21.35 also conﬁrms that for commonly used relative freeboards Rc /Hs in the order of 1.2, much larger containers are required for the crest than for the slope of the structure. Moreover, it also shows that the eﬀect of deformation on the stability dramatically increases with increasing design wave height and is much more pronounced for the crest containers than for the slope containers.

21.5. Concluding Remarks After about 50 years of successful experience of geotextile applications in coastal engineering, applications for shore protection are well established. Most failures which have yet been experienced are rather due to bad design, bad choice of material, and/or bad installation. Geotextile sand containers (GSCs) represent nowadays a soft and low cost alternative to conventional hard structures made of rock and concrete. Moreover, GSCmade structures are environmentally more appropriate and more easily reversible as they need essentially sand as construction material which is generally available at any coastal site. As “soft rock” GSCs can be manufactured at any size and used to build any type of shore protection structure, including scour protection, dune reinforcement, and repair of undermined structures.

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However, several problems still need to be solved in order to make use of the full potential of GSCs, particularly including (i) the long-term durability and lifetime prediction and (ii) the hydraulic stability under severe wave action. The facts, limitations and research needs related to the durability and lifetime issue may be summarized as follows: (i) Although ﬁeld evidence (mostly nonexposed geotextile) is available over about 50 years, useful information extracted from nonretrieved samples is often very incomplete or very limited as the results can hardly be transferred to other sites, to present geotextile products and to time durations and conditions beyond the experienced service time/conditions. (ii) Results of accelerated tests — even in combination with those from site monitoring and retrieved samples — are still very limited when trying to predict lifetime of more than about 25 years. In fact, a systematic methodology to combine both laboratory and ﬁeld monitoring approaches for this purpose is still missing. (iii) Recommendations for future research in the mid-term and long-term should focus on two directions: (a) improvement of the understanding of the degradation mechanisms, including physical, biological, and chemical processes, both isolated and in combination, and (b) development of a consistent framework for the assessment of long-term durability and lifetime (up to 100 years and more) based on the results of the above and including site monitoring, laboratory testing, and theoretical/numerical modeling. (iv) Meanwhile, in order to contribute to solve the present problems in practice, the following two recommendations might be helpful. (a) Apply engineering judgment based on the present knowledge of degradation mechanisms rather than relying on “extrapolation approaches” to predict lifetime and (b) apply where feasible and necessary well-established measures to enhance long-term performance, including, for instance, appropriate stabilizers and additives (e.g., against UV-radiation), more robust geotextiles (e.g., multi-layer), and geotextile coating (e.g., against abrasion and vandalism), sand covering (e.g., against UV-radiation and vandalism and to enhance aesthetical aspect), rock covering (e.g., against ice loads, debris, very high waves, UV, and vandalism) and setup of a consistent maintenance plan. Moreover, the collaboration with experienced companies with regard to geotextile production, manufacturing of sand containers, ﬁlling, handling, and placing is also strongly recommended. Regarding the hydraulic stability under severe wave action, the present state of knowledge, the limitations, and the needs for future research may be summarized as follows: (i) The large experience and stability formulae available for rock and concrete armor units cannot simply be transferred to GSCs, essentially due to the deformation of the GSCs under very severe wave attack. (ii) The deformations of the GSCs are essentially induced by the internal movement of the sand ﬁll of the containers.

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(iii) The eﬀect of the deformations of the GSCs on the hydraulic stability rapidly increases with the severity of wave attack, depending on the size and sand ﬁll ratio of the containers as well as on the degree of exceedance of the wave loads required for the inception of the internal movement of the sand ﬁll. (iv) Besides, the eﬀect on the hydraulic stability, the internal sand movement in submerged geotextile containers may lead to a substantial reduction of the height of GSC-structures (up to about 10%) when subject to severe wave attack. (v) The deformation of GSCs aﬀects the hydraulic stability, essentially due to two mechanisms: (a) reduction of the contact areas between GSCs caused by the uplift of the containers by wave action, thus decreasing the stabilizing forces/moments and (b) increase of the surface areas exposed to drag and lift forces which represent the main destabilizing forces/moments. Therefore, the deformation eﬀect should be accounted for explicitly in the stability formulae. (vi) Friction between GSCs aﬀects the hydraulic stability much more than commonly assumed in past and present design practice. Therefore, and due to the possible changes in service life, friction should be incorporated explicitly in stability formulae. (vii) The hydraulic permeability of structures made of GSCs is not only important for the prediction of the hydraulic performance (e.g., wave transmission, runup, overtopping, etc.), but also slightly aﬀects the hydraulic stability. However, no clear correlation could be found between stability and permeability for the range of practical permeability coeﬃcients of GSC-structures in the order of k = 1–3 cm/s. The permeability of a GSC-structure is essentially determined by the gaps between the containers, so that the ﬂow through the sand ﬁll itself can be neglected. Therefore, the hydraulic permeability essentially depends on the mode of placement of the containers. For randomly placed containers and longitudinally (in wave direction) placed containers, the permeability coeﬃcient is in the order of k = 2.5 cm/s. (viii) The eﬀect of breaking wave impact on sliding and overturning stability of slope containers has been found much less than expected, due to the potential of the GSCs to eﬀectively damp impact pressure propagation inside the gaps. More eﬃcient to destabilize the slope containers are the uprush and downrush of the longer nonbreaking waves and partially breaking waves. (ix) The proposed simple stability formulae derived in Sec. 21.4.1 on the basis of the Hudson-formula take additionally into account the eﬀect of the wave period for the slope containers and that of the relative freeboard (Rc /Hs ) for the crest containers. In both cases, the eﬀect of container deformations is taken into account implicitly through the empirical parameters determined from laboratory experiments. These formulae are conservative for waves up to about Hs = 1.5 m and can thus be used for design wave heights not larger than about 2 m. For higher waves (Hs > 2 m) the eﬀect of deformation on the stability becomes more important and must therefore be considered more explicitly in order to ensure long-term stability performance.

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(x) The more detailed stability formulae proposed in Sec. 21.4.3 do not only take into account the most relevant processes, but also allow us to better quantify the eﬀect of deformation on the required size of the container as a function of the incident wave height for both slope containers (Fig. 21.34) and crest containers (Fig. 21.35). In fact, these new process-based formulae clearly highlight the eﬀect of container deformation which dramatically increases with the design wave height and is much more pronounced for the stability of the crest containers than for that of the slope containers. Moreover, they also show that even for the commonly used relative freeboard of about Rc /Hs ≈ 1.2 much larger containers are required for the crest than for the slope of a GSCstructure to ensure hydraulic stability. (xi) Although signiﬁcant advancement has been achieved in the understanding of the processes and mechanisms responsible for the hydraulic failure of GSCs, more systematic research is further needed to investigate and better control the friction between GSCs, the eﬀect of the sand ﬁll ratio, the eﬀect of the slope steepness of the GSC-structure, the internal movement of the sand ﬁll, the container deformations and their more explicit consideration in both stability formulae and numerical simulations. A fully coupled CFD and CSD model system well-validated by experimental data will be needed as a necessary tool in combination with new laboratory experiments to achieve these goals.

Acknowledgments The ﬁnancial support by StAUN Rostock for the experimental investigations described in Sec. 21.4.1 and by NAUE GmbH & Co. KG for those described in Sec. 21.4.3.2 are gratefully acknowledged. Sections 21.4.2 and 21.4.3 are essentially the part of the PhD-Thesis of the second author which has been supported by the Deutsche Akademische Austausch Dienst (DAAD). The ﬁrst author would like to thank Prof. Heerten, Mr. Pilarczyk, and Mrs. Werth for their valuable comments and for reading the manuscript. Thank is also due to Mrs. Werth for providing the ﬁrst author with valuable information and references on the properties and durability of geotextiles. References 1. ACT, Innovative Technology for Coastal Erosion Control — Subsurface Dune Stabilization, Advanced Coastal Technologies, LLC (2006), http://bcs.dep.state.ﬂ.us/ innovative/report/appedic c/06 Subsurface Dune Protect.pdf. 2. J. Buckley and W. Hornsey, Woorin beach protection–chasing the tide sand ﬁll tubes versus sand ﬁlled containers, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 761–764. 3. CEN, Guide to Durability of Geotextiles and Geotextile Related Products (European Normalization Committee (CEN/TC189/WGS/N210/Paris), 1998). 4. EAG-CON, German recommendations on geotextile containers in hydraulic engineering, WG-UG5 “Geotextile Containers of AK 5.1” Geosynthetics, Geotechnics and Hydraulic Engineering (German Geotechnic. Society (DGGT), 2008).

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5. R. A. Edward and S. D. A. Smith, Optimising biodiversity of macro invertebrates on artiﬁcial reefs, Poster presentation AMSA National Conf., Brisbane, Australia (2003). 6. J. Fowler and E. Trainer, Overview of geocontainers projects in the United States, Proc. Western, Dredging Conf., USA (1998). 7. H. H. Greenwood and A. Friday, How to predict hundred year lifetimes for geosynthetics, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 1539–1542. 8. J. Gr¨ une, U. Sparboom, R. Schmidt-Koppenhagen, Z. Wang and H. Oumeraci, Stability test of geotextile sand containers for monopile scour protection, ASCE Proc. Int. Conf. Coastal Eng., San Diego, USA (2006), pp. 5093–5105. 9. G. Heerten, Long-term experience with the use of synthetic ﬁlter fabrics in coastal engineering, Proc. 17th Int. Coastal Engineering Conf., Vol. 3, Sydney, Australia (1980), pp. 2174–2193. 10. G. Heerten and F. F. Zitscher, 25 Jahre Geotextilien im K¨ ustenschutz — Ein Erfahrungsbericht. 1. Nationales Symposium Geotextilien im Erd-und Grundbau, Mainz, Germany, Forschungsgesellschaft f¨ ur Straßen- und Verkehrswesen (Hrsg.), K¨ oln, Germany (1984), pp. 7–15 (in German). 11. G. Heerten, Geotextiles in coastal engineering — 25 years experience, Geotextiles and Geomembranes, Vol. 1 (Elsevier, 1984), pp. 119–141. 12. G. Heerten, A. Jackson, S. J. Restall and F. Saathoﬀ, New development with mega sand containers of non-woven needle-punched geotextiles for the construction of coastal structures, ASCE Int. Conf. Coastal. Eng., Sydney, Australia (2000). 13. G. Heerten, The challenge for the use of geosynthetic construction materials in environmental, coastal and oﬀshore engineering applications, Oﬀshore Arabia 2006, Conf. and Exhibition, Dubai, United Arab Emirates (2006). 14. M. Heibaum, Geosynthetic containers — A new ﬁeld of application with nearly no limits, Proc. 7th Int. Conf. Geosynthetics, Nice, France (2002). 15. M. Heibaum, A. Fourie, H. Girard, G. B. Karunararne, J. Laﬂeur and E. M. Palmeira, Hydraulic applications of geotextiles, Proc. Int. Conf. Geosynthetics, Yokohama, Japan, Millpress, Rotterdam (2006), pp. 79–120. 16. L. A. Jackson, R. Tomlinson, I. Turner, B. Corbett, M. D’Agatha and J. McGrath, Narrow artiﬁcial reef — Results of 4 years of monitoring and modiﬁcations, Proc. 4th Int. Surﬁng Reef Symposium, Manhattan Beach, California, USA (2005). 17. Th. Jansen, Erprobung und Verwendung von sandgef¨ ullten Kunststoﬀschl¨ auchen aus Gewebe und Vlies beim Seedeichbau in der Leybucht, Ostfriesland, 1. Kongress “Kunststoﬀe in der Geotechnik” K-Geo, deutsche Gesellschaft f¨ ur Erd- und Grundbau e.V., Hamburg (1988) (in German). 18. E. Kavazanjian, Jr., N. Dixion, T. Katsumi, A. Kortegast, P. Legg and H. Zanziger Geosynthetic barriers for environmental protection of landﬁll, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 121–152. 19. R. M. Koerner, Designing with Geosynthetics, 5th edn. (Pearson Prentice Hall, Ltd., London, 2005), 796 pp. 20. C. R. Lawson, Geotextile containment for hydraulic and environmental engineering, Proc. Geosynthetics Int. Conf. (Mitchpress, Rotterdam, 2006), pp. 1–48. 21. E. Lefaive, Durability of geotextiles: The French experience, Geotextiles and Geomembranes 7, 553–558 (1988). 22. B. Lenze, G. Heerten, F. Saathoﬀ and K. Stelljes, Geotextile sand containers — Successful solutions against beach erosion at sandy coasts and scour problems under hydrodynamic loads, Proc. EUROCOAST, Littoral 2, Porto, Portugal (2002), pp. 375–381.

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23. MAG, Code of Practice — Use of Geotextile Filters on Waterways (Federal Waterways Engineering and Research Institute (BAW), Karlsruhe, 1993), www.baw.de. 24. A. McClarty, J. Cross, L. Gilbert and O. M. James, Design and construction of coastal erosion protection groyne using geocontainers, Langebaan, South Africa. Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006). 25. NAUE, Scherversuche im Labor der Firma NAUE f¨ ur Soil Filters Australia, NAUE Fasertechnik (2002). 26. H. Nickels and G. Heerten, Objektschutz Haus Kliﬀende, HANSA, Band 137(3), 72–75 (2000) (in German). 27. H. Oumeraci, M. Bleck, M. Hinz and S. K¨ ubler, Large-scale model test for hydraulic stability of geotextile sand containers under wave attack, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 878 (2002) 62 and Annexes. 28. H. Oumeraci, M. Hinz, M. Bleck and A. Kortenhaus, Sand-ﬁlled geotextile containers for shore protection, Proc. COPEDEC, Colombo, Sri Lanka (2003). 29. H. Oumeraci, J. Gr¨ une, H. Sparboom, R. Schmidt-Koppenhagen and Z. Wang, Investigations on scour and scour protection for monopile foundation of wind oﬀshore turbines, Forschungszentrum K¨ uste (FZK), Res. Report (2007) 79 and Annexes (in German). 30. H. Oumeraci, A. Kortenhaus and K. Werth, Hydraulic performance and armour stability of rubble mound breakwaters with core made of geotextile sand containers — Comparison with conventional breakwaters, Proc. Int. Conf. Coastal Structures, Venice, ASCE (2007). 31. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, The Netherlands, 2000). 32. J. Recio and H. Oumeraci, Eﬀect of the deformation on the hydraulic stability of revetments made of geotextile sand containers, Proc. Int. Symp. “Tsunami Reconstruction with Geosynthetics”, Bangkok, Thailand (2005), pp. 53–68. 33. J. Recio and H. Oumeraci, A numerical study on the hydraulic processes associated with the instability of GSC-structures using a VOF-RANS model, LeichtweissInstitute for Hydraulic Engineering and Water Resources, LWI Report No. 941 (2006). 34. J. Recio and H. Oumeraci, Processes aﬀecting the stability of revetments made with geotextile sand containers, Proc. Int. Conf. Coastal Engineering, ICCE 2006, San Diego, USA (2006). 35. J. Recio and H. Oumeraci, Geotextile sand containers for coastal structures, hydraulic stability formulae and tests for drag, inertia and lift coeﬃcients, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 936 (2006). 36. J. Recio and H. Oumeraci, Preliminary experiments and numerical simulations of solitary wave acting on a submerged-ﬁlter-reef, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 945 (2006). 37. J. Recio, Hydraulic stability of geotextile sand containers for coastal structures — Eﬀect of deformations and stability formulae, PhD-Thesis, Leichtweiss-Institute for Hydraulic Eng., Technical University Braunschweig (2007). 38. J. Recio and H. Oumeraci, Permeability of GSC-structures, model tests and analyses, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 943 (2007). 39. J. Recio and H. Oumeraci, “Coupled” numerical simulations on the stability of coastal structures made of geotextile sand containers, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 942 (2007).

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40. J. Recio and H. Oumeraci, Eﬀect of the deformations on hydraulic stability of coastal structures made of geotextile sand containers, Geotex. Geomemb. J. 25, 278–292 (2007). 41. J. Recio and H. Oumeraci, Hydraulic permeability of GSC-structures: Laboratory results and conceptual model, Geotex. Geomemb. 26, 473–487 (2008). 42. J. Recio and H. Oumeraci, Processes aﬀecting the hydraulic stability of GSC structures: Experimental and numerical investigations, Coastal Eng. 56, 632–658 (2009). 43. J. Recio and H. Oumeraci, New stability formulae for coastal structures made of geotextile sand containers, Geotex. Geomemb. 56, 260–284 (2009). 44. S. J. Restall, L. A. Jackson, G. Heerten and W. P. Hornsey, Case studies showing the growth and development of geotextile sand containers — An Australian perspective, Geotex. Geomem. 20(5), 321–342 (2002). 45. S. J. Restall, W. P. Hornsey, H. Oumeraci, M. Hinz, F. Saathoﬀ and K. Werth, Australian and German experiences with geotextile containers for coastal protection, Proc. Euro-Geo 3, Munich, Germany (2005), pp. 141–146. 46. A. L. Rollin, Long term performance of geotextiles, 57th Canadian Geotechnical Conf., Session 4D (2004), pp. 15–20. 47. RPG, Guidelines for Testing Geotextiles for Navigable Waterways (Federal Waterways Engineering and Research Institute (BAW), Karlsruhe, 1994), www.baw.de. 48. F. Saathoﬀ, H. Oumeraci and S. Restall, Australian and German experiences on the use of geotextile containers, Geotex. Geomem 25, 251–263 (2007). 49. H. F. Schroeder, H. Bahr, P. Herrmann, G. Kneip, E. Lorenz and I. Schmuecking, Durability of polyﬁne geosynthetics under elevated oxygen pressure in aqueous liquids, Proc. Second European Geosynthetics Conf. EUROGEO, Bologna, Italy (2000). 50. S. K. Shukla and J. H. Yin, Fundamentals of Geosynthetics Engineering (Routledge, 2006). 51. M. Solton, B. Leclerq, J. L. Paute and D. Fayoux, Some answer’s components on durability problems of geotextiles, Proc. 2nd Conf. Geotechnics 2, 553–558 (1982). 52. G. H. Troost, G. Den Hoedt, P. Risseeuw, W. Voskamp and H. M. Schmidt, Durability of a 13 year old embankment reinforced with polyester woven fabric, 5th Int. Conf. Geotextiles, Geomembranes and Related Products, Singapore, IGS (1994) pp. 1185–1190. 53. G. van Santvoort (ed.), Geotextiles and Geomembranes in Civil Engineering (A.A. Balkema, Rotterdam, The Netherlands, 1994). 54. J. Vohlken, H. Lind and J. Witte, Dune reinforcement with geotextile sand containers, HANSA 4, 60–62 (2003) (in German). 55. J. D. M. Wisse, C. J. M. Broos and W. H. Boels, Evaluation of the life expectancy of polypropylene geotextiles used in bottom protection structures around the Ooster Schelde storm surge barrier — A case study, Proc. 4th Conf. Geosynthetics (1990), pp. 697–702. 56. J. Wouters, Slope Revetment — Stability of Geosystems, Delft Hydraulics Report H. 1939 (1998) Annex 7 (in Dutch). 57. F. F. Zitscher, Kunststoﬀe f¨ ur den Wasserbau, Bauingenieur-Praxis, Heft 125, Verlag von Wilhelm Ernst & Sohn, Berlin, M¨ unchen, D¨ usseldorf (1971).

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Chapter 22

Low Crested Breakwaters Alberto Lamberti∗ and Barbara Zanuttigh Department of Civil Engineering (DISTART), University of Bologna Viale Risorgimento, 2; 40136 Bologna, Italy ∗ [email protected] The chapter describes typical features of low crested breakwaters, their hydraulic stability, eﬀects on waves, induced circulation, erosion, and problems related to construction and maintenance, providing basic tools for design.

22.1. Introduction As low crested breakwaters (LCBs) we mean breakwaters that are frequently overtopped or weakly submerged. They are in any case breakwaters, i.e., the height over the bed of these structures is suﬃcient to cause systematic wave breaking on the crest (submergence less than one signiﬁcant wave height), but they are also low, i.e., not so high to make overtopping rare (freeboard less than one signiﬁcant wave height). In this sense they are distinguished from artiﬁcial reefs (deployed mainly to enhance ﬁsheries) or bed protections on one hand, that are deeply submerged and do not cause breaking, and by usual harbor breakwater on the other hand that are high enough not to cause signiﬁcant overtopping. LCBs mimic in some manner the functioning of coral reefs, that however have usually a much wider crest and a narrower range of submergence/emergence due to their generation mechanism. As a consequence they are sometimes named also “reefs.” LCBs are usually built with natural rock for the defence of beaches (Fig. 22.1), even if some examples of LCBs protecting the main breakwater from highest waves or protecting an external harbor from excessive agitation and currents (Fig. 22.2) can also be found. Beach defence breakwaters are usually built in a few meter water depth within the surf zone at design conditions. The freeboard (positive or negative) is usually of the same order of magnitude as the tidal range, i.e., in contrast with most harbor breakwaters, freeboard variability in lifetime is substantial. 601

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Fig. 22.1. Detached LCBs protecting the beach of Skagen (DK, 57◦ 44 07 N 10◦ 37 50 E; coordinates are given to let the reader obtaining a plan view from Google-Earth).

Fig. 22.2. Jetties protecting Ijmuiden entrance (NL, 52◦ 27 55 N 4◦ 32 18 E) during a storm surge event. From http://www.vandermeerconsulting.nl/.

As it will be shown later, the height of the structure in this case is not greater than a few armor stones, posing serious problems in building the traditional ﬁlter structure separating the armor from the sandy bed. LCBs are usually located in the surf zone where design wave height is depth limited and, therefore, lasts for long periods. It is therefore unsafe to accept any,

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even small, permanent damaging rate in design conditions, since the damage will cumulate in a few years at a signiﬁcant level. For all the reasons sketched above, LCBs cannot be treated as a small scale version of rubble mound harbor breakwaters, even if they share some evident common features. Due to their wide use in beach defence, however, they deserve some special attention. This contribution is dedicated to highlight diﬀerences between high and low breakwaters. We summarize the experience accumulated in the last two decades on the subject, making use of several results focused within DELOS research project, that we had the opportunity to coordinate; we acknowledge at the same time the contribution of project partners, evident to everyone will read one at least of the two major deliverables of the project: the special issue of Coastal Engineering8,24,28,29,35,46,57,68 and the Design Guidelines.5 The sections are arranged as follows: Section 22.2 — typical features, Section 22.3 — hydraulic stability, Section 22.4 — wave reﬂection and transmission, Section 22.5 — overtopping, Section 22.6 — induced circulation, Section 22.7 — scour and erosion, Section 22.8 — construction and maintenance problems.

22.2. Typical Layout and Sections LCBs were used since long time as detached emergent breakwaters for the defence of beaches. The low crest was motivated by economy reason in conditions where some agitation in the protected area could be accepted and sometimes required in order to avoid silt sedimentation. The problem of visual impact was not posed at that time. Later during the last two decades of the 20th century, in Europe particularly along the Mediterranean coasts subject to intensive tourism exploitation, the aim of reducing visual impact of defence works promoted the use of submerged breakwaters. Figure 22.3 shows one of the most deeply submerged breakwater constructed in Pellestrina (near Venice, IT) and completed in 1997.29 It is a 9 km long defence work, where a 4.5 × 106 m3 capital nourishment of 0.2 mm sand was protected by a LCB system. The oﬀshore part of the LCB is a submerged breakwater defending the beach against oﬀshore sand losses. The groins limit the long-shore displacement of sand and are characterized by an emerged part in the aerial beach zone and a submerged continuation to reach the submerged longitudinal breakwater. Average water depth at structure is 4.5 m. Design submergence is −1.5 m a.m.s.l. Spring tidal range near Venice is 1.0 m, but high tide combined with storm surge caused by Scirocco wind may reach 1.5 m a.m.s.l. Incident signiﬁcant wave height with return period 10 years is 3.70 m, therefore, extreme waves at the structure are depth limited. Since the construction of Venice outlet jetties (end of the 19th century), the beach, deprived of any natural sand nourishment, suﬀered a slow but continuous erosion. The beach, after the construction of the defence system, showed to be rather stable with low southerly directed littoral drift.

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Fig. 22.3. Pellestrina (IT, 45◦ 15 57 N 12◦ 18 10 E) oﬀshore submerged breakwater and groins. Courtesy of Consorzio Venezia Nuova.

Fig. 22.4.

Lønstrup defense section (DK, 57◦ 28 37 N 9◦ 47 44 E). From Burcharth et al.6

Figure 22.4 shows the section of a Danish beach protection example at Lønstrup in northern Jutland.29 Mean tidal range at the site is 0.3 m; exceptional high tide may reach 1.5 m; wave are usually depth limited. Longshore net transport is great, approx. 600 × 103 m3 /year. Beach sand size is 0.2 mm. A 1.1 km long segmented breakwater was constructed to protect the cliﬀ and the village from an ongoing 1.5 m/year erosion. To maintain the shoreline position, a 20–30 × 103 m3 /year nourishment was required even after the breakwater construction.

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Fig. 22.5. Sea Palling defence section (UK, 52◦ 47 32 N 1◦ 36 11 E). Adapted from Halcrow;21 see also Sea Palling site at http://argus-data.wldelft.nl/index.html, reefs located north of the video station.

Another example of a typical emerged breakwater for coastal defence is represented by detached breakwater built near Sea Palling in North Norfolk, UK, Thomalla and Vincent.48 The defence system was motivated by severe ﬂooding in 1953 (causing seven casualties in the area) and consisted of a continuous seawall protecting dune foot, requiring for its stability an adequate beach at the toe. In order to reduce wave intensity on the wall and required nourishment volumes in the most critical part of the defence, nine breakwater (or reefs) were constructed since 1993. Figure 22.5 shows the section of the ﬁrst built reefs 5–8.21 The site is characterized by 3.0 m spring tidal range (from −1.3 to +1.7 m over datum -ODN-); extreme level reach 3.7–4.0 m ODN with return period 50–100 years. Oﬀshore signiﬁcant wave height reaches 7.9 m with return period 100 years, therefore, extreme waves are depth limited at the structure. Longshore tidal currents are relevant with a peak south-easterly ﬂood of 0.8 ms−1 and a peak spring ebb to the northwest of 0.6 ms−1 . The net longshore sediment drift has been estimated to be 100–150×103 m3 per year in a southerly direction. The sandy beach shows to be highly volatile, with temporary spatial average erosion under severe storms reaching up to 70 m3 /m. Finally, an example of low crested harbor breakwater is worth to be presented. The outer breakwater prolongation in Leix˜ oes harbor (near Porto, PT) is selected, detailed, described in Vera-Cruz and Reis de Carvalho.58 The two long jetties enclosing the 100 ha outer harbor were built in 1884–1892. The outer harbor was not designed to provide load/unload capacities, but the limited extension of the initial dock area forced ships to stay anchored or moored in it. A storm in January 1917 caused severe damages to several of these ships. The construction of an outer vertical wall breakwater was therefore initiated in 1934, but during the ﬁrst winter the last 120 m of the 400 m long built breakwater were destroyed. Based on economy considerations, a submerged breakwater was proposed. The design was veriﬁed with a hydraulic model carried out in Lausanne (CH), showing that the structure safety and harbor tranquility resulted satisfactory. The main environmental conditions are: water depth at roundhead −15.0 m, tidal excursion 0.0–4.0 m, signiﬁcant wave height 9.5 m, wave period 13–19 s. The construction of the submerged extension was carried out in years 1937–1942 and its behavior was satisfactory, requiring only minor maintenance works. In 1966, in order to provide berthing facility to 100,000 t oil-carriers, the construction of berth A along the outer breakwater and of a wave wall with crest elevation 15 m a.s.l. was decided. The designed wall, apparently tested with regular waves, was protected by a steep (3:4) 40 t tetrapod slope resting on the crest of the 90 t cube submerged

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breakwater. In 1972, two years after the construction ended, a replacement of more than one-third of the placed tetrapods was necessary, and a similar replacement was required in 1974. In 1979, the last 90 m of the wave wall were destroyed. Breakage of tetrapods was observed in prototype that, together with the insuﬃcient characterization of design waves (regular rather than irregular waves with equal signiﬁcant wave height), were recognized as the main causes of the failure. The construction of some protection works of the roundhead was decided and, after comparison of several alternative solutions in a hydraulic model, the construction of a submerged breakwater with section similar to the previous one but shifted 52 m oﬀshore was selected, aiming to reduce through breaking the wave height incident on the wave wall. The construction of the selected design (see Fig. 22.6) ended in 1982; the crest was constructed 1 m higher than shown in the drawing anticipating a 2 m settlement. The behavior of the submerged breakwater was evaluated satisfactory in 1993 either regarding its stability under wave action or regarding the protective eﬀect on tetrapod stability and wave wall overtopping. The prototype veriﬁcation appears signiﬁcant since during the 11 year monitoring period a 9.0 m peak significant wave height event occurred (January 1985), reaching almost design intensity. PERFIL 6 - 6 0

L = 65,0 m - Solução 1 L = 40,0 m - Solução 2 L = 52,0 m - Solução 3 (ADOPTADA)

25

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MACIÇO DE PROTECÇÃO

+ 4,0 (PMAV)↓

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PARALELEPÍPEDOS DE BETÃO

~ 90 tt.

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1 - 5 tt.

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,5) (+ 15,0)

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POSTO A

Fig. 22.6. Terminal section and layout of the Leix˜ oes harbor breakwater (PT, 41◦ 10 22 N 8◦ 42 30 W). From Vera-Cruz and Reis De Carvalho.58

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22.3. Armor Stability in Shallow Water and Signiﬁcant Tide Excursion A low breakwater compared to a high one is less exposed to breaker impact and to wave downrush, since the breaker falls on the crest often covered with water and the impact ﬁnds vent shorewards in the absence of a wave wall. This section is dedicated to describe this eﬀect and to provide proper tools for armor design. Systematic laboratory experiments regarding rock armor stability in low crested breakwaters were carried out by several authors. Givler and Sørensen20 provided results of 2D experiments with regular waves, van der Meer51 and Burger7 with irregular waves. Vidal et al.59,60 Kramer and Burcharth,27 and Kramer et al.28 performed experiments with long crested and short crested irregular waves in a wave basin observing also damages at roundheads. A synthetic presentation of these results with a critical analysis is presented in Burcharth et al.6 on which this section is essentially based. Experiments are described in detail in the referred papers. All experiments are carried out on a ﬁxed bed, where no settlement or erosion can take place, and therefore they do not provide information on these failure modes. Results refer to where the damage takes place and how severe the damage is. Assuming equal armor stone size at trunk and roundhead, all authors agree on where ﬁrst and how damage occurs, see Fig. 22.7. • For submerged or zero freeboard breakwaters (Rc ≤ 0), hydraulic damage occurs ﬁrst at trunk crest and results in a shorewards displacement of stones; roundhead in not more critical than the trunk and damage is distributed all around the head. • For slightly emerged breakwaters (0 < Rc < Dn50 ), the crest remains the most critical part of the trunk with some stones thrown shorewards and other dragged Seaward and middle head Leeward head

Stability number Hs/∆Dn50

5

4

3

X D

Trunk seaward slope Trunk crest

º

Trunk leeward slope

2

1

Least stable section, Eq. (22.1) 0 -3

-2

-1

0

1

Normalized freeboard Rc/Dn50 Fig. 22.7.

Partial and global start of damage curve, from Burcharth et al.5

2

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down the oﬀshore slope; leeward roundhead becomes the ﬁrst damaged part of a breakwater. • For emerged LCBs (Dn50 < Rc ), the seaward slope is the most critical part of the trunk with most stones dragged down the slope and the leeward roundhead is the ﬁrst damaged part of a breakwater. • The crest width, built with stones of the same size as the armor slope, has no evident eﬀect on hydraulic stability. • In all the mentioned tests, slope was 2:3 or 1:2 and no signiﬁcant eﬀect of the slope was observed. Burger7 showed that gradation and stone shape within reasonable limits (D85 /D15 < 2.5, L/D < 3) have not an evident eﬀect on stability. Based on all the available information Burcharth et al.6 proposed the following formula representing the initiation of damage in some part of the breakwater 2 Hs Rc Rc = 0.06 − 0.23 + 1.36 (22.1) ∆Dn50 Dn50 Dn50 for relative crest freeboard less than 1.9. Above this value the stability number remains constant and equals to 1.14, the minimum of Eq. (22.1). This value is signiﬁcantly lower than the one that can be derived from Hudson formula (∼ =1.5). This is due partially to the fact that no damage is assumed in this formula against a moderate damage (5%) accounted for in Hudson formula and partially to the deliberate use of Hs as the characteristic wave height (Hudson formula is used sometimes with H1/10 and some other with Hmax ). Most LCBs are placed in the littoral zone and in design conditions are subjected to depth breaking waves. For many of these the mean water level varies in a relatively wide range, regularly due to tide and occasionally due to storm surge. If waves are depth limited and the breaking limit is represented as a limit ratio γ between signiﬁcant wave height and the local water depth, among the incident wave height Hi , the water depth at structure toe hs , the structure height hc , and structure freeboard Rc the relation holds Hi = γ · hs = γ · (hc − Rc ),

(22.2)

where the breaker index γ depends on seabed slope, wave steepness, and the characteristic wave height in use (signiﬁcant, extreme . . . ). Figure 22.8 shows a typical situation at failure boundary. The shape of the two domains (failure domain and possible sea conditions) is such that stability is assured if breaking limit does not cross the start of damage curve or is tangent to it as a limit. This leads to the hydraulic stability condition under variable water level and depth limited waves characterized by the signiﬁcant wave height: γs /∆ Dn50 ≥ . hc 1.36 − (γs /∆ − 0.23)2 /(4 · 0.06)

(22.3)

If the tangency point does not lie within the possible sea level range or breaking conditions are unlikely, the condition contains an implicit safety margin.

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Fig. 22.8.

609

Failure representation under variable water level, from Burcharth et al.6

Table 22.1. Stability condition and critical degree of submergence for diﬀerent foreshore slope and breaker index. Wave solicitation Foreshore slope 1/∞ 1/200 1/100 1/40 1/20 — —

Tangency point

Hsi /hs

Rc /hc

Dn50 /hc

Hsi /∆Dn50

0.40 0.45 0.50 0.55 0.60 0.65 0.70

−0.02 −0.08 −0.16 −0.25 −0.36 −0.48 −0.64

0.18 0.21 0.23 0.26 0.29 0.33 0.37

1.39 1.46 1.5 1.6 1.7 1.8 1.9

Results for the minimum stone size according to Eq. (22.3) for relative stone density ∆ = 1.6 and diﬀerent values for signiﬁcant breaker index γs are reported in Table 22.1. The breaker index values are evaluated according to van der Meer52 with sop = 0.03 and hs /Lop = 0.05. For a gentle foreshore slope the following simple and cautious rule of thumb is found Dn50 ≥ 0.3 · hc .

(22.4)

The ratio Dn50 /hc imposes serious restrictions to the structure design. A double armor layer with Dn50 /hc > 0.4 require that at least part of the ﬁlter is placed below the natural bed level; this could be beneﬁcial in case a bed erosion might occur. A value Dn50 /hc > 0.3 implies that there is no suﬃcient space for a core and the two layer armor stones rest directly on a coarse ﬁlter. Only when Dn50 /hc < 0.2 there is space for a conventional core and ﬁlter layer.

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It should be noted that Eq. (22.1) is based on model tests with sea-side slope 1:2 (some 1:1.5). For milder slopes, smaller armor stones can presumably be stable. According to Hudson formula, for a slope 1:5, an increase of the stability number for emerged breakwaters by a factor 1.36 is expected; a smaller increase in stability is argued in the case of submerged crest because the crest (including edges) is the most damaged part in this case. Also, some laboratory test cases support this conclusion (Ostia IT, Ferrante et al.18 ; Sea Palling UK, Leasowe Bay UK, Barber and Davies1 ). Oﬀshore and inshore the armor slopes, a toe berm is normally constructed with the function of supporting armor and preventing the propagation of damage resulting from scour or settlement. Regarding berm stability no speciﬁc analysis was carried out on low breakwaters. Design rules suggested by van der Meer et al.56 were evaluated as reasonably safe for the oﬀshore berm of low breakwaters in Burcharth et al.6 For a suﬃciently wide crest, the breaker will fall on the crest and the inshore toe may be designed with the same material as the oﬀshore one. For narrow and submerged breakwaters, the breaker caused by the oﬀshore slope can fall beyond the crest causing a severe solicitation of the inshore toe berm; care should be paid in this case. It is worth noting that the above formulae are derived from ﬁxed bed models, and do not represent any eﬀect of bed mobility and/or erosion, whereas in most prototype conditions the ﬁne sand seabed is subjected to intense mobilization and transport. Indeed, the observed damage of prototype rubble mound (as described by Lamberti et al.29 ) is often the consequence of geotechnical or morphodynamic instabilities. First of all, when proper bed protection is not used, dumping of stones onto bed cause their rapid sinking already during construction followed by a slower settlement. Moreover, breaking waves and related strong currents produce local scour (Sumer et al.46 ) which aﬀects toe stability either directly (stone sliding in scour hole) or indirectly (increase of water depth and breaker height). To avoid sinking of the rubble stone material into a sandy bed it is necessary to separate the two materials with an appropriate ﬁlter: widely dispersed small stone layers, geotextiles, or mattresses. If sinking of stones stops within construction period, the rubble mound volume used for construction must be oversized and that is all; if sinking last longer, the structure must be designed and initially built higher (see, e.g., the Leixoes example). The overheight will correspond to the anticipated settlement, and depends on seabed characteristics, construction method, and structure height. For instance, a large rubble mound built on muddy seabed by means of ﬂoating equipment demands a large overheight, whereas a low mound structure placed on a coarse sandy seabed by landbased equipment running on already placed material demands much less overheight (but great overvolume) as settlement will be almost completed during construction. Burcharth et al.6 describes in detail some settlement cases. Statistics from a much wider set of cases from Japan is presented in Uda.49 Depending on the nature of the bed material the statistics in Table 22.2 was observed. The fraction of settling breakwater increases from 10% on rock (probably degradation of armor units), up to 43% on gravel, 63% on sand and 100% on sylt. The settlement height is highly variable in the range 0–2.80 m, with average and modal value around 1.00–1.20 m.

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Table 22.2. Settlement distribution of detached breakwater as function of bed material. Settlement Bed material

Exist

Nil

Unknown

Total

Rock Gravel Sand Silt

10%(19) 43%(139) 64%(678) 100%(1)

73%(141) 56%(183) 33%(347)

17%(34) 1%(5) 3%(42)

100%(194) 100%(327) 100%(1067) 100%(1)

Total

53%(837)

42%(671)

5%(81)

100%(1589)

Results are also given for scattering (displacement, hydraulic damage) of armor units. Globally scattering was observed in 14.2% of the 1,552 cases, but the frequency varies signiﬁcantly between breakwaters showing evident settlement, for which the scattering frequency was 17.9%, and those for which no settlement was observed, for which scattering frequency was 9.4%. The frequency diﬀerence is highly signiﬁcant, proving that scattering and settlement are related. In our experience hydraulic damage, i.e., the removal of stones from the breakwater crest, is able to justify only a minor part of crest settlement. On the other side, bed erosion around the structure and stone sinking preferably along the structure perimeter are the cause of block dilation on the slopes and therefore of reduced interlocking and instability. The mutual inﬂuence of settlement and block scattering is therefore bilateral and can explain the evident correlation observed.

22.4. Eﬀect on Waves: Transmission and Reﬂection The main eﬀect of a LCB compared to a high one is that a signiﬁcant fraction of incident energy can pass over its crest and generate waves behind the structure. The increase of wave transmission leads, on one hand, to lower beach protection from erosion or to less calm areas for swimming purposes inshore the LCBs. On the other hand, it guarantees a high degree of water recirculation and thus a water quality improvement within the protected cell. Transmission, as well as reﬂection, can be characterized by a global intensity factor, normally the transmission and reﬂection coeﬃcients, and by a description of energy distribution within the energy containing frequency range. The EU-projects DELOS (www.delos.unibo.it) and CLASH (www.clash-eu.org) collected and generated wide datasets from tests on all kind of structures, where the transmission and reﬂection coeﬃcients were available in almost all cases. An extensive and homogeneous database on wave transmission (van der Meer et al.57 ) and on wave reﬂection (Zanuttigh and van der Meer64 ) were prepared, whose analyses produced for LCBs the major outcomes synthesized below. For rubble mound LCBs, to be able to estimate the transmission coeﬃcient Kt also in presence of submerged structures with very wide crests, the following two formulae are given for design conditions (Briganti et al.3 ; van der Meer et al.57 ).

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Equation (22.5a), which is the formula by d’Angremond et al.,12 is used for Bc /Hsi < 8; Eq. (22.5b) for Bc /Hsi > 12: Rc Kt = −0.40 + 0.64 Hsi

Bc Hsi

−0.31

(1 − e−0.50ξop )

(22.5a)

with imposed lower and upper limit values Ktmin = 0.075, Ktmax = 0.8 and Kt = −0.35

Rc + 0.51 Hsi

Bc Hsi

−0.65

(1 − e−0.41ξop )

(22.5b)

with Ktmin = 0.05, Ktmax = 0.93 − 0.006(Bc /Hsi ). A linear interpolation between the two is performed in the range 8 < Bc / Hsi < 12. In Eqs. (22.5) Rc is the structure freeboard (negative for submerged structures), Bc is the crest width, and ξop is the surf similarity parameter based on structure oﬀshore slope and peak wave period. Equations (22.5) were applied in the range −5 ≤ Rc /Hsi ≤ 5. Not all LCBs are of the rubble mound type. Sometimes smooth and impermeable structures exist, for example, LCBs covered with asphalt or armored with a block revetment. Often the slope angles of these structures are gentler (1:3 or 1:4) than for rubble mound structures, mainly for construction reasons. Wave transmission over smooth LCBs is completely diﬀerent from rubble mound structures. First of all, the wave transmission is larger for the same crest height, simply because there is less energy dissipation by friction and structure permeability. Furthermore, the crest width has less or even no inﬂuence on transmission, as also on the crest there is lower energy dissipation. Only for very wide (submerged) structures there could be some inﬂuence on the crest width, but this is not a case that will often be present in reality as asphalt and block revetments are mainly constructed in the dry and not under water. The presence of tide makes it possible to construct these structures above water. For smooth LCBs the prediction formula by van der Meer et al.57 is Rc −0.5ξop + 0.75(1 − e ) (22.6) Kt = −0.3 Hsi with imposed lower and upper limit values Ktmin = 0.075, Ktmax = 0.8, and limitations: 1 < ξop < 3, 1 < Bc /Hsi < 4. Transmitted spectra are often diﬀerent from incident spectra. Waves breaking over a LCB may generate two or more transmitted crests on the lee side per incident crest. The eﬀect is that more energy is present at higher frequencies than in the incident spectrum. Usually the peak period is quite close to the incident one, but the mean period may decrease considerably. If the reduction of wave energy is mainly led by the dissipations due to the ﬂow through the armor layer, however, higher frequencies may be cut. A ﬁrst analysis on this topic can be found in van der Meer et al.,53 who developed a simple model where in average the 60% of the transmitted energy is present in the range f < 1.5fp and the other 40% of the energy is evenly distributed between 1.5fp and 3.5fp. This scheme was analyzed little further by

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Briganti et al.3 and by van der Meer et al.,57 concluding that the redistribution is applicable only to submerged rubble mound structures. Particular care to the representation of the shape of the transmitted wave spectra was paid by Lamberti et al.30 The transmitted wave is reconstructed from the incident wave as the sum of the overtopped and ﬁltered components. Incident waves transform seawards the structure into overtopping events and then regenerate, leeward the structure, due to perturbations caused by impulsive overtopping volumes. The overtopping and ﬁltration discharges are added and converted into displacements of an ideal wave maker placed leeward the structure. A sample of the reconstructed transmitted wave spectrum with comparison of an experimental one is given in Fig. 22.9. The approach describes fairly the spectrum modiﬁcation. Related to spectral change is the eﬀect on obliquity described below. Similarly, for traditional structures several formulae exist that predict the reﬂection coeﬃcient Kr as function of the Iribarren–Battjes surf similarity parameter ξ and are empirically determined for diﬀerent types of armor units, of tested conditions and layouts. The only formula that can be applied, with varying coeﬃcients (a, b) for diﬀerent kinds of structures is the one by Seelig and Ahrens42 Kr =

a1 · ξ 2 . ξ 2 + b1

(22.7)

The coeﬃcients (a1 , b1 ) are, respectively, equal to (0.6, 6.6) for rubble mound structures, and to (1.0, 6.2) for smooth slopes.

Fig. 22.9. Measured transmitted spectra (solid line with +), transmitted spectra components due to ﬁltration (dot line) and to overtopping (dash line), in case of abundant overtopping. From Lamberti et al.30

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A more recent formula by Zanuttigh and van der Meer64 exists that • can be applied to all kinds of structures or revetments in design conditions, provided only that the roughness factor γf is known, • represents physical bounds, • was validated against a wide database. This formula reads: Kr = tanh(a · ξob ),

(22.8)

where a = 0.167 · [1 − exp(−3.2 · γf )]; b = 1.49 · (γf − 0.38)2 + 0.86 and ξo is the Iribarren–Battjes parameter calculated with the mean (−1, 0) spectral period at the structure toe. Based on CLASH results, the parameter ξo is used in this formula since it allows to represent also bi-modal spectra or shallow water with a ﬂat spectrum where a peak period is not well deﬁned. For the most typical cases, the coeﬃcients in Eq. (22.8) are: γf = 0.40, a = 0.12, b = 0.87 for rock slopes; γf = 0.55, a = 0.14, b = 0.90 for rock slopes with impermeable core; γf = 1.0, a = 0.16, b = 1.43 for smooth slopes. Based on the DELOS database, van der Meer et al.,57 proved that as expected low crest breakwaters have smaller reﬂection than high ones, due to the fact that more energy passes over the structure. Moreover, the key parameter to describe the reduction of Kr is the relative crest height Rc /Hsi . Based on these observations, van der Meer et al.57 provided a ﬁrst rough reduction factor for Kr as function of Rc /Hsi to be generally applied to formulae for well-emerged structures. The dependence of Kr on Rc /Hsi can be appreciated for permeable structures in Fig. 22.10, where the measured value of Kr is a dimensionalized by the value 1.20

1.00

K rm/K rc

0.80

0.60

UPC UCA UPD UFI Seabrook

0.40

0.20

0.00 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Rc/Hsi Fig. 22.10. Measured to computed values based on Eq. (22.10) of the reﬂection coeﬃcient with varying degree of overtopping.

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obtained from Eq. (22.6). Similarly to van der Meer et al.,57 Zanuttigh and van der Meer64 proposed an extension of their formula to rubble mound LCBs by including a simple linear regression on Rc /Hsi . Reﬂection is not inﬂuenced if the relative crest height is > 0.5 and decreases to 0 when submergence is twice the wave height. The formulae (22.5), (22.6), and (22.7) are given for perpendicular waves only. Based on a limited set of data, van der Meer et al.57 and Wang et al.61 analyzed the eﬀects of oblique wave attack on wave transmission. With increasing incident wave angle βi , Kt tends to decrease and the obliquity of the transmitted wave βt is a little smaller than the incident βi . The inﬂuence of wave obliquity on transmission is more evident for smooth structures, where it results proportional to cos2/3 β, and hardly observable for rubble LCBs. The deviation of transmitted waves (obliquity reduced to about 80%) can be interpreted as refraction of the average wave due to spectral change and reduced nonlinearity of transmitted waves. Explanation of eﬀects is provided in Wang et al.63 Analysis of the eﬀect due to wave obliquity on Kr can be found in Wang et al.,62 who derived for smooth LCBs a Kr reduction proportional to cos2/3 β.

22.5. Wave Overtopping Overtopping refers to the process of water passing over the structure crest. If the structure is not continuously submerged the process can be naturally interpreted and made up of several events strictly related to the passage of wave crests and characterized by a certain mass and momentum (or volume and velocity). The consequences of overtopping are dependent on speciﬁc statistics of the process related to the considered eﬀect: for instance, the extreme momentum of one overtopping crest may be responsible for damaging exposed elements, whereas the mean volume per unit time (or mean overtopping discharge) is responsible for ﬂooding. Nevertheless, the last is the standard parameter for characterizing overtopping intensity, probably because it is the easiest to measure. The total elapsed time and the total volume of water passing over the structure are simply the sum of elapsed time and passed volume within each wave. This trivial observation is the basis of the relation among mean overtopping discharge qot , overtopping probability Pot , mean overtopping volume ·Vot , and mean period of well formed waves Tm : qot

Volumes Vot Not · Vot = waves = Pot , = Periods N · T Tm w m waves

(22.9)

where the overtopping probability Pot is the ratio between the number of overtopping waves Not and the number of waves Nw . Since every wave whose runup exceeds the structure crest (and only these waves) cause overtopping, the overtopping probability at the oﬀshore edge of the crest is equal to the probability that a single wave runup is higher than the crest. For irregular waves, van der Meer and Stam54 suggested for runup at rubble mound structures a Weibull probability distribution, with parameters k1 and k2 related to

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incident waves and slope characteristics, from which the overtopping probability can be derived: Prob(Ru ≤ z) = 1 − exp(−(z/k1 )k2 ) ⇔ Pot = exp(−(Rc /k1 )k2 ),

(22.10)

where k1 = 0.4Hsi s−0.25 cot gα−0.2 , som is the mean wave steepness and α the om −0.75 structure oﬀshore for plunging waves (ξm < 2.5) and k2 = √ slope; k2 = 3.0ξm −0.3 P 0.52·P ξm cot α for surging waves (ξm > 2.5), ξm is the Iribarren number based on mean wave period and P is the notional structure permeability. Later, van der Meer and Janssen55 provided a simpler expression for the runup distribution at dikes: the shape parameter is constant k2 = 2, i.e., the distribution is Rayleighian, and the location parameter is proportional to incident wave height k1 = 0.81 · γh · γf · γβ · min(γb · ξop , 2) · Hsi . The Rayleigh distribution ﬁts in any case rather well the distribution of runup. In Eq. (22.9) Pot is the factor controlling the order of magnitude of mean overtopping discharge and is the expression of the main eﬀect of relative crest elevation Rc /Hsi , as it can easily vary by orders of magnitude as a consequence of modest variations of structure crest elevation, see the CLASH database (Steendam et al.44 ; wwwclash-eu.org). The data obtained in wave basin by Zanuttigh and Lamberti67 for perpendicular and oblique layouts of LCBs show that the pattern and values of Pot are in good agreement with predictions from Eq. (22.10) and that the overtopping volumes can be well approximated by a Weibull distribution, as in van der Meer and Janssen.55 The mean value of such distribution Vot shows a more or less parabolic relation with the runup Ruot , that is the median potential runup of waves causing overtopping (Fig. 22.11). Ruot is derived from Eq. (22.10) as the value giving exceeding probability just half the overtopping probability. As suggested by Pilarczyk,38 when overtopping is rare, Vot is almost proportional to (Ruot − Rc )2 ; this implies a ﬁxed shape of the overtopping crests. The scaled volumes, however, increase signiﬁcantly as soon as Pot exceeds 0.4–0.5. The location parameter k1 of the volume distribution (typical overtopping volume) can be obtained from the mean value as k1 = Vot /Γ(1 + k2−1 ). The shape parameter k2 of the distribution increases with increasing Pot (Fig. 22.12). Its values range from something below 1 up to 3; the lowest values refer to the case of “rare” overtopping and the highest values to very frequent overtopping. If due attention is paid to diﬀerence in overtopping frequency and parameter uncertainties, the lowest ﬁgures are not substantially diﬀerent from the value 3/4 suggested by van der Meer and Janssen.55 In conclusion, for LCBs that are overtopped by most waves, not just by the highest (Pot > 0.4 or Rc < k1 ), the crest level is so low in the approximate Rayleigh distribution of runup that the shape of all overtopping characteristics distributions become more symmetric and less variable (the shape parameter in a Weibull distribution becomes greater) and the overtopping events become longer. The combined model of a Weibull (or Rayleigh) distribution of runup, providing the overtopping probability, and a shape model for overtopping crests, as the one suggested by Pilarczyk,38 represents properly the process for high and low freeboards if the shape parameters are assumed variable with relative freeboard or overtopping probability.

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8 F=0, Layout 1, Narrow berm F=0, Layout 1, Wide berm F=3, Layout 1, Narrow berm F=3, Layout 1, Wide berm F=0, Layout 2, Narrow berm F=3, Layout 2, Narrow berm F=0, Layout 2, Wide berm F=3, Layout 2, Wide berm

7

/(Ruot-Rc)2

6 5

F

Ru

4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pot Fig. 22.11. Mean overtopping volumes Vot versus overtopping probability Pot , emergent and eres et al.8 zero-freeboard structures. In the plot, F = Rc . From Cac´

3.0

F=0, Layout 1, Narrow berm

F=0, Layout 1, Wide berm 2.5

F=3, Layout 1, Narrow berm

F=3, Layout 1, Wide berm Rayleigh 2.0

F=0, Layout 2, Narrow berm

F=3, Layout 2, Narrow berm

k2

F=0, Layout 2, Wide berm F=3, Layout 2, Wide berm 1.5

Exponential 1.0

0.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pot Fig. 22.12. Shape parameter k2 of the Weibull volume distribution versus overtopping probability Pot , emergent and zero-freeboard structures. From Cac´eres et al.8

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The extrapolation below the aforementioned limits of formulae adapted to high structures has two contrasting eﬀects: the intrinsic limitation Pot < 1 is not recognized and the factor is overestimated; the increasing duration of overtopping events is not recognized and the mean volume is underestimated. The total result on the discharge is weak and may be confused with formulae uncertainty. No speciﬁc formula exists so far to estimate the overtopping discharge at LCBs. The widely adopted formula for surging waves by van der Meer qot Rc , (22.11) = 0.2 · exp −2.6 3 Hsi γf γβ gHsi where the symbol γ denotes the runup reduction factors accounting for slope roughness (γf ) and wave obliquity (γβ ), whose values and expressions can be found in TAW,47 tends to overestimate the overtopping discharge for LCBs, even with the indeed small correction by Kofoed and Burcharth.26 Equation (22.11) can be thus used only when overtopping is rare, i.e., well-emerged LCBs or weak wave attack with respect to the crest freeboard. 22.6. Induced Setup and Planimetric Circulation Unlike high barriers, the ﬂow rate over LCBs is abundant and related to the rear piling-up. Overtopped water accumulates behind the structure, establishing a higher mean water level, or piling-up, which forces return ﬂows along diﬀerent paths: water may return oﬀshore through gaps, originating the so-called rip currents, and, if the crest is submerged, also over the structure itself. In this latter case, the ﬂux over the crest during the wave cycle is alternately directed inshore and oﬀshore, driven by waves and piling-up. Since LCBs are typically made of permeable rubble mound, return ﬂow occurs also through the structure, due to the unbalance between the steady hydraulic gradient induced by piling-up and the breaking wave thrust. Gaps are normally shorter but proportionate to breakwater length (∼ = 1/3), but breakwaters, even when permeable, are far less permeable than the open gap, therefore, when the breakwater is emerged the overtopping discharge returns to the sea almost only through the gaps. The ﬂux through gaps qG globally equals the net ﬂux across the structure qIS , which is the algebraic sum of the inshore directed ﬂux over the structure crest qD and the return ﬂow across the structure qU , see Fig. 22.13. The return ﬂow through gaps qG is frequently named also recirculation ﬂow, since it is usually simulated in a wave ﬂume by a recirculating system. The process can be schematized in analogy to a pump system (Lamberti et al.31 ; Zanuttigh et al.66 ). The head losses associated to return ﬂow through gaps are represented by a characteristic curve and the relation between piling-up and net mass ﬂux across the structure is similarly described by a barrier pumping curve. The system operational point at equilibrium is the intersection between the two curves and is the solution of the equation. The pumping curve was experimentally investigated in wave ﬂumes equipped with a recirculation system and it was found to be approximately linear for emerged conditions (Ruol et al.40,41 ; Cappietti et al.10 ). Piling-up reaches its maximum p0 in

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zO

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qD

zI

qU qG

p ≡ zI − zo qG ⋅ LG LB

Fig. 22.13. At the left-hand side scheme of planimetric circulation; at the right-hand side, scheme of the model approach. From Lamberti et al.31

absence of recirculation and decreases to zero when qIS is recirculated without any ﬂow resistance. For given wave conditions, p in conﬁned conditions is quite greater than for a structure with gaps: indeed, the overall return ﬂow resistance decreases with increasing the gap to structure length ratio LG /LB , and consequently the piling-up required to drive all the return ﬂows is smaller (Martinelli et al.36 ). Waves and currents around LCBs can be estimated by means of physical or numerical modeling. Laboratory experiments are usually dedicated to the analysis of a speciﬁc layout, are time consuming and expensive, and may be aﬀected by scale eﬀects. It is thus common practice for design purposes to adopt numerical models for simulating waves and current ﬁelds. When the structure crest is fully submerged, standard wave averaged models (as) can be used with the assumption of negligible ﬁltration through the structure. If the breakwater is deeply submerged, its permeability has secondary eﬀects when compared to the return ﬂow over breakwater crest. For this case (reef barrier) induced circulation and setup can be evaluated with a 2DH wave-current model. Depending on the relative depth (depth to wavelength ratio) the user is addressed to the following models: • Nonlinear Shallow Water Equations numerical integrator (Brocchini et al.4 ; Hu et al.23 ). It is a rather simple and robust phase resolving model that describes both nondispersive waves and currents at the same time, used since long time and expressly adapted to represent discontinuous bore propagation over an existing body of water or a dry beach. Its use is recommended if everywhere in the computation domain h/L < 1/20 or, that is equivalent, if h/gT 2 < 1/120 and the computation domain is not so wide to make the small dispersive eﬀects relevant during propagation. • Boussinesq equations numerical integrator (Karambas and Koutitas25 ). It is a phase resolving model representing both wave and currents at the same time,

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recommended for greater relative water depth and/or wider space. The Boussinesq model is only a little more time consuming than the previous one and it is potentially less robust, but it is speciﬁcally designed to represent wave dispersion and it can be used with conﬁdence almost up to the deepwater limit if it uses higher order approximations. • Phase averaged models (as MIKE 21 or LIMCIR, and many others, see Johnson et al.24 ). These models represent wave statistics (spectra) and current separately, based essentially on mild slope approximation. They are to be used when the relative water depth variation in a wavelength is small and the computational domain is much wider than the wavelength. When they are applied on a breakwater, which is invariably rather steep, they tend to overestimate wave height decay rate due to breaking, wave setup, and induced currents. For both submerged and emerged LCBs, accurate predictions with high computational eﬀorts can be obtained from the 2DV RANS-VOF code COBRAS by Losada et al.,35 which allows to represent the structure porosity. Some analytical models are also available under simplifying assumptions: both Calabrese et al.9 and Bellotti2 provided the ﬂuxes over an impermeable structure, the ﬁrst including friction eﬀects in a 2D context and the second one excluding friction eﬀects in 3D conditions. A rapid and reasonable assessment of the 3D ﬂuxes around and within LCBs can be derived from Zanuttigh et al.66 for all crest freeboards and wave attacks.

22.7. Scour and Erosion Wave breaking and currents generated by the presence of the LCBs induce in turn morphological changes in their vicinity within a timescale of the order of a storm, i.e., within a time frame of hours to days. A detailed description of seabed dynamics and timescales, together with a classiﬁcation of morphological changes, can be found in De Vriend and Ribberink.13 In describing morphological eﬀects of waves and currents, it is usual to distinguish between near-ﬁeld scour and far-ﬁeld erosion. First of all, scour occurs in the immediate vicinity of the structure (less than 1/4 the wavelength), and it is caused by quite diﬀerent mechanisms from those responsible for far-ﬁeld erosion. These mechanisms include wave streaming, horse-shoe vortices, local turbulence, and possible liquefaction due to vertical pressure gradients. Furthermore, diﬀerent timescales are associated with the development of scour and far-ﬁeld erosion. Similar considerations apply to the shoreline response induced by LCBs, i.e., the development of salient or tombolos, especially with regards to the timescale involved, which is larger than that associated with far-ﬁeld erosion and typically ranges from months to years. Far-ﬁeld erosion in the vicinity of LCBs is intimately linked to the onshore ﬂow over the crest of the structures, which returns oﬀshore mainly through gaps, and to the deﬂection and acceleration of longshore and return currents at the roundheads. Currents ﬂowing oﬀshore through gaps produce irregularly shaped erosion

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Fig. 22.14.

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Field surveys performed in 2004 in Lido di Dante. From Zyserman et al.68

areas resembling a tongue of ﬁre. Similarly, currents bending in oﬀshore direction at roundheads produce crescent-shaped erosion patterns. At some distance from the gaps, these oﬀshore-directed currents bend back toward the structure and merge with the onshore-directed ﬂow over the crest of the LCBs. An example of these sediment transport patterns can be seen in Fig. 22.14, which consists of a bottom survey carried out in Lido di Dante (Ravenna, Italy). The coastal defence in this site is composed of three groynes, two detached zero freeboard LCBs with a protected gap in between, and two low-crested connectors from the northern and southern groynes to the LCBs (Zyserman et al.68 ; Zanuttigh65 ). Deep crescent-shaped erosion areas at about 70 m from the two barrier roundheads can be recognized, which have a maximum depth of 1.5 m, a length of 150 m, and a width of 50 m. A ﬂame-shaped erosion hole, 1.0 m deep, 120 m long, 50 m wide, is present seaward of the central gap, due to the rip current concentration and intensity. Deposition occurs seaward the detached LCBs, due to wave breaking at the oﬀshore LCB slope and to return currents from the roundheads. Far-ﬁeld processes can be best investigated through the analysis of detailed bathymetric surveys coupled with concurrent monitoring of hydrodynamic

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conditions at the site and information about bed sediment characteristics. Besides economical and technical diﬃculties, prototype observations are diﬃcult to generalize due to the particular conditions of each site (Lamberti et al.29 ). Physical modeling is another valid alternative approach, and can also provide data for calibration of numerical models. Few physical model tests of LCS have been carried out in wave basins (Kramer et al.28 ) and most of them adopted a ﬁxed bed and dealt only with hydrodynamics around impermeable submerged structures (Haller et al.22 ; Drønen et al.17 ). So far only two sets of movable bed experiments were performed in a wave basin with the aim of analyzing morphodynamic changes induced by LCBs. Both cases included detached rubble-mound breakwaters separated by gaps of diﬀerent length, under regular (Van der Biezen et al.50 ) and irregular (Martinelli et al.36 ) waves. However, restrictions imposed by the limited extension and boundary conditions of laboratory facilities, and scale eﬀects associated with movable bed models must be kept in mind when selecting this method to investigate far-ﬁeld erosion. Wave and ﬂow ﬁelds in the vicinity of coastal structures and the associated morphological response are thus usually assessed on the basis of coastal area numerical models, see, e.g., De Vriend et al.15 and De Vriend14 for a description. Several references can be found in the literature regarding applications of two-dimensional, depth-integrated (2DH), or quasi three-dimensional (Q3D) models to study sites or simpliﬁed cases. A nonexhaustive list of cases includes: large-scale and local eﬀects of groynes (De Vriend and Ribberink13 ), rip channels and bars evolution (Damgaard et al.11 ; Ranasinghe et al.39 , where numerical results are compared with video imaging analysis), bed changes in presence of breakwaters (Leont’yev32 ; Nicholson et al.37 , which also includes a comparison of diﬀerent model performances), intermediate beach sedimentation inside and close to harbors (Lesser et al.33 ), morphological response induced by LCBs (Zyserman et al.68 ; Martinelli et al.36 ; Zanuttigh65 ). Main advantages of numerical models are the lack of scale eﬀects and the limited amount of work involved in modifying the model setup, which in turn allows testing a wide range of layouts at limited cost. Near-ﬁeld erosion close to LCBs is related to two processes: steady streaming, due to the superposition of incident and reﬂected waves at the oﬀshore structure slope, and plunging breakers inshore the structure. Steady streaming occurs both at the structure trunk and roundheads. The plunging breaker is basically the key element of the scour process inshore the roundhead. After its formation, the plunger travels some distance along the crest width of the breakwater, eventually descends toward the bottom, impinges on the bed, and mobilizes the sand leading to a scour hole. Whilst far-ﬁeld erosion can be generally predicted based on numerical or physical modeling results, near-ﬁeld erosion has been extensively analyzed by means of laboratory tests deriving some analytical expressions for the estimate of its depth and extension. Figures 22.15 and 22.16 show the scour depth and extension, respectively, due to steady streaming and plunging breakers. The plotted data are obtained under regular and irregular waves by Sumer et al.,46 Fredsøe and Sumer,19 Lillycrop and Hughes34 and Dixen.16 In all these works α = 1:1.5, with the exception of Lillycrop and Hughes34 where α = 1:2.

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Fig. 22.15. Depth S (to the left) and width Ly (to the right) of the scour hole induced by steady streaming. In the plots, F = Rc and H = Hsi . From Sumer et al.46

Figures 22.15 and 22.16 are based on a dimensional analysis, which allows to represent the characteristics of the steady streaming [Eq. (22.12)] and of the plungingbreaker [Eq. (22.13)] induced scour by the following nondimensional equations h L Rc , , , Re, ϑ, α, L dn50 h √ S Tw gH Rc = fp , ϑ, α, , Hsi h h

S =f Hsi

(22.12) (22.13)

where S is the scour depth, H is the signiﬁcant incident wave height, h is the bottom depth in front of the structure, L is the incident wavelength, ϑ is the Shields’

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Fig. 22.16. Depth S (to the left) and width Ly (to the right) of the scour hole induced by plunging breakers. In the plots, F = Rc and H = Hsi . From Sumer et al.46

parameter, α is the oﬀshore structure slope, dn50 is the nominal sediment diameter, √ Tw gH/h is the plunger parameter (Fredsøe and Sumer19 ) and represents the ratio between the amount of water in the plunging breaker entering in the main body of water and the distance to be penetrated, Rc is the crest height with respect to mean sea level (positive if the structure is emerged), Re = aUm /ν (being a the amplitude of the orbital motion of water particles, Um the maximum value of the orbital velocity at the bed, Tw is the wave period, ν the kinematic viscosity).

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Field surveys highlight appreciable erosions along the trunk of LCBs, oﬀshore and inshore, and severe erosions at LCB roundheads and gaps. The erosion induced at the oﬀshore slope appears less pronounced and dangerous than in the case of traditional emerged structures. This can be explained because of the lower reﬂection and of sediment transport toward the structure induced by wave overtopping, with consequent deposition oﬀshore the LCB and ﬂattening or canceling erosion holes. Erosion may induce displacements of the structure toe and sliding of the armor layer into the scour holes. The prolongation of such conditions in time can bring to the structure destabilization. It is therefore necessary to protect the LCBs with a suﬃciently wide toe berm. Toe protection layer may be constructed in the form of a protection apron. The apron must be designed so that it will remain intact under wave and current forces, and it should be “ﬂexible” enough to conform to an initially uneven seabed. With this countermeasure, scour can be minimized, but not entirely avoided. Some scour will occur at the edge of the protection layer, and consequently, armor stones will slump down into the scour hole. This latter process, however, will lead to the formation of a protective slope, a desirable eﬀect for “ﬁxing” the scour. The determination of the width of the protection layer is an important design concern. The width should be suﬃciently large to ensure that some portion of the protection apron remain intact, providing adequate protection for the stability of the breakwater. On the basis of the works by Sumer and Fredsøe45 and Sumer et al.,46 it is recommended to estimate from the following empirical equation the width of the protection apron at the trunk section W (from the oﬀshore structure edge) L (22.14) − αhs , 4 where hs is the structure slope. The validity range of Eq. (22.14) is h/L = 0.1 ÷ 0.2. On the inshore side, the same W can be cautiously assumed, in order to prevent also damages induced by wave overtopping. At the roundhead, the toe protection width can be calculated as Rc Rc Rc W = We , if > +0.9; W = 0.29 + 0.74 We , if < +0.9, Hsi Hsi Hsi (22.15) where We is the value of W for emerged breakwaters, which is evaluated following Fredsøe and Sumer19 as W =

We = A · Um · Tw ,

(22.16)

where A is 1.5 for complete scour protection and 1.1 for a scour protection which allows a scour depth of 1% of the breakwater width at the toe. The above equation (22.15) is based on experiments where the breakwater slope was 1:1.5. Therefore, for slopes milder than 1:1.5, the width necessary for protection might be reduced (and for steeper slopes increased). Furthermore, Eq. (22.15) is for scour protection against the local scour caused by the combined eﬀect of steady streaming and stirring up of sediment by waves. Due considerations must be given to global scour caused by the far-ﬁeld ﬂow circulations around the breakwater. Finally, the recommended width is for protection at the oﬀshore side of the head.

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Experiments show that berm width designed according to Eq. (22.16) is also able to protect the sand bed against the breaker-induced scour at the inshore side of the head. However, scour might occur in the protection layer itself due to wave plunging. Therefore, over-sizing of the protection material is recommended at the inshore side compared to the usual rule provided, for instance, by formula (22.17), van der Meer et al.,56 0.15 Hsi /(∆Dn50 ) = (0.24hb /Dn50 + 1.6) · Nod ,

(22.17)

where hb is the minimum water depth over the berm and Nod is the number of stones displaced from a one stone wide strip of the berm and for standard toe size (about ﬁve stones wide and two to three stones high at trunk) Nod = 0.5, 2, 4, respectively, for no damage, acceptable, and severe damage. The volume of the toe berm shall be such that its material is suﬃcient to protect the scour/erosion hole from further erosion without destabilizing the armor layer slope, i.e., its width should be around three times the erosion depth and its thickness at least four times its maximum stone size (SPM43 ; Burcharth et al.6 ). In this way, slide berm stones can form although dispersed a stable and continuous slope covering the sand bed. The erosion induced at gaps can both cause serious problem of structure stability and act as sink for sediments inside the protected area, making them ﬁrst slide into the hole and then favoring their exit from the gap transported by rip currents. It is therefore necessary to adequately protect the gaps with a stable and ﬂexible plateau that may follow bottom movements, usually consisting of the same material at the barrier toe. The objective must be to shift erosion from the structure at such a distance not to compromise structure stability. Gap protection shall be extended more in oﬀshore than in inshore direction, although it is not realistic an oﬀshore protection to the limit of the eroded area. The amount of material must exceed the strictly necessary quantity in order to ﬁll the holes that invariably form at the protection boundaries. Maintenance works for restoring toe protection before structure damage occur should be planned. 22.8. Structure Construction and Maintenance As already mentioned, LCBs can be constructed by means of either ﬂoating or landbased equipment. The selection of the construction method depends on environmental conditions, like the water depth, tidal range, and wave climate. The selection has relevant eﬀects on rapidity and accuracy of construction, as well as risk for the contractor. Land-based equipment (dumpers, front loaders, dozers, cranes including backhoes) is preferred when materials arrive to the construction yard by road and the breakwater is either placed in really shallow water and/or close to the shore, or constructed in a site where tidal range is large enough to make the site dry in each cycle for a few hours. Floating equipment (barges and crane on them) are preferred in frequently calm waters more than 3–4 m deep, and when materials are transported to the yard on barges.

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Land-based construction is more accurate and can be carried out usually under sea conditions that do not allow construction with the aid of barges. Moreover, the risk of damaging concrete units is lower. When both type are possible land-based is normally preferred. In many sites, however, depending on local conditions, diﬀerent combinations of land and ﬂoating equipment are actually used. Where the tidal range is large, one can rather safely assume that for half of the daylight period water level is lower than the structure basement level (higher than m.s.l.); during the summer period this allows 8 hrs working period per day, that is normally considered quite suﬃcient to carry out safely a work on the upper beach, where land-based equipment are normally used. In beaches exposed to frequent wave activity, so that a barge cannot operate safely for long periods, land equipment are used also in the lower part of the intertidal beach and even below. LCBs are constructed by dumping rocks from lorries moving on the mound crest and placing armor with land-based cranes. If the structure is to be submerged, the mound is constructed emergent and advancing from land, with equipment moving on it; the emergent crest is lowered in a second phase, when the crane is retreating and crest material is dumped at the sides of the mound. In some other case the access causeway is temporary, and is removed at the end of works. Barge-mounted equipment are normally used in calm waters. Barges are rather insensitive to the short period waves (2–4 s) that are generated by local winds blowing in bays or among islands. The direct dumping from barges with the assistance of a ﬂoating crane for armor placement is the most common construction method in this environment. Material shuttle barges and crane barge for placement are usually separate, allowing crane barge to remain on station. Self unloading split, bottom-door, tilting, or side unloading barges are usually used to dump core material. For bedding layers, scour protection and berms ﬂat deck barges will a bulldozer for discharge can also be used. For the placement of ﬁlter layers only side unloading or ﬂat deck barges can oﬀer a good placement precision. Thin layers (0.50 m) can be placed only with barges operating with a high precision positioning system in one or multiple passage. LCBs pose a challenging separation problem between bed, foundation mound, and armor, requiring a good placement precision. For operations the following site conditions shall be considered: current, wind, waves, tidal level variation, maneuvring space, water depth, and visibility, including seasonal variation. Positioning of barges is obtained by a roundabout anchoring system (usually six anchors) or by a dynamic and computerized positioning system only for large structures. Generally, in sheltered water (no severe currents and waves), a horizontal accuracy of 1 m can be achieved. In exposed conditions the accuracy will be less and will decrease with increasing water depth. Down-time caused by waves and wind is often determined by the inﬂuence on the positioning accuracy and on crane loading, rather than by operational limitation of the barge. Critical are the barge movements; short wind waves have less impact than long swell. Generally, wind waves should not exceed 1.0–1.5 m height, whereas a 0.5 m high swell might be critical for positioning and accurate dumping. More

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severe are the limits for placing heavy armor units with cranes: cranes are normally not designed to sustain lateral forces caused by swinging loads due to barge motions, and heavy concrete units must be laid on bed with a minimal velocity. For these reasons maximum allowable tilts during placement should not exceed a few degrees. Often it is not possible to build a perfect ﬁlter separating the mound breakwater from the bed, particularly under water. Some movement of sand in the mound pores under heavy waves can therefore be anticipated in these cases. Moreover, if the bed is composed of ﬁne sand some liquefaction or partial liquefaction of the upper layers are possible in storm conditions. The sand body sustaining the breakwater weight should not be aﬀected by, not even partial, liquefaction. For all the mentioned reasons wide toe berms are useful: they reduce ﬁltration velocity below the breakwater body, they rest on and let sand at rest; if bed erosion occurs, material sliding from the berm protects the bed from further erosion. The state of berms is therefore an indicator of structure safety that should be monitored. If scour holes of the order of twice the stone diameter are shown by bathymetric surveys, or the berm width is reduced below 2–4 stones diameter, toe berm stability may be compromised and toe protection should be reinforced and widened. Since the functionality of a LCB is very sensitive to crest elevation, the crest settlement should be also monitored. As soon as the crest is lowered on the average more than half the armor stone size or anywhere more than one stone size, the armor layer should be recharged. The design should not foresee a too frequent maintenance (below 10 years) since this will probably induce risk for structure stability and cause disturbance to the surrounding environment due to repeated working activity. Nevertheless, a limited settlement of 3–5 cm/year is present also in properly designed and constructed breakwaters, as those described in Section 22.2. A settlement rate greater than 10–15 cm/year should be considered instead worth of reassessing the stability of the breakwater and of analyzing a reinforcement feasibility. Acknowledgments Most of the research on LCBs synthesized and discussed in this study has been carried out within the framework of the DELOS project (EVK3-CT-2000-00041), through which the support of EC is gratefully acknowledged. References 1. P. C. Barber and C. D. Davies, Oﬀshore breakwaters — Leasowe Bay, Proc. Institution of Civil Eng. (1985), pp. 85–109. 2. G. Bellotti, A simpliﬁed model of rip currents systems around discontinuous submerged barriers, Coastal Eng. 51(4), 323–335 (2004). 3. R. Briganti, J. W. van der Meer, M. Buccino and M. Calabrese, Wave transmission behind low-crested structures, Proc. Coastal Structures 2003, ASCE (2003), pp. 580– 592 (2003).

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4. M. Brocchini, A. Mancinelli, L. Soldini and R. Bernetti, Structure-generated macrovortices and their evolution in very shallow depths, Proc. 28th Int. Conf. Coastal Eng. ASCE (2002), pp. 772–783. 5. H. F. Burcharth, S. Hawkins, B. Zanuttigh and A. Lamberti (eds.), Environmental Design Guidelines for Low Crested Coastal Structures (Elsevier, UK, 2007), 448 pp. 6. H. F. Burcharth, M. Kramer, A. Lamberti and B. Zanuttigh, Structural stability of detaced low crested breakwaters, Coastal Eng. 53(4), 381–394 (2006). 7. G. Burger, Stability of low-crested breakwaters, Delft Hydraulics Rep. H1878/H2415 (1995). 8. I. Cac´eres, A. S. Arcilla, B. Zanuttigh, A. Lamberti and L. Franco, Wave overtapping and inducing currents at emergent low crested structures, Coastal Eng. 52(10–11), 931–947 (2005). 9. M. Calabrese, D. Vicinanza and M. Buccino, Veriﬁcation and recalibration of an engineering method for predicting 2D wave setup behind submerged breakwaters, Proc. Int. Coastal Symposium ’05, Hofn, Iceland (2005). 10. L. Cappietti, E. Clementi, P. Aminti and A. Lamberti, Piling-up and ﬁltration at low crested breakwaters of diﬀerent permeability, Proc. 30th Int. Conf. Coastal Eng., S. Diego, USA, Vol. 5 (2006), pp. 4957–4969. 11. J. Damgaard, N. Dodd, L. Hall and T. Chesher, Morphodynamic modelling of rip channel growth, Coastal Eng. 45, 199–221 (2002). 12. K. d’Angremond, J. W. van der Meer and R. J. de Jong, Wave transmission at low crested structures, Proc. 25th Int. Conf. Coastal Eng., ASCE (1996), pp. 3305–3318. 13. H. J. De Vriend and J. S. Ribberink, Mathematical modeling of meso-tidal barrier island coasts, Part II: process-based simulation models, Adv. Coastal and Ocean Eng., ed. P. L.-F. Liu, Vol. 2 (World Scientiﬁc, 1996), pp. 151–197. 14. H. J. De Vriend, Mathematical modelling of meso-tidal barrier island coasts, Part I: empirical and semi-empirical models, Adv. Coastal and Ocean Eng., ed. P. L.-F. Liu, Vol. 2 (World Scientiﬁc, 1996), pp. 115–149. 15. H. J. De Vriend, J. Zyserman, J. Nicholson, J. A. Roelvink, P. P´echon and H. N. Southgate, Medium-term 2DH coastal area modelling, Coastal Eng. 21(1–3), 193–224 (1993). 16. M. Dixen, Scour around the roundhead of a submerged rubble mound breakwater, Master’s thesis, Tech. Univ. Denmark (2003). 17. N. Drønen, H. Karunarathna, J. Fredsøe, M. Sumer and R. Deigaard, An experimental study of rip channel ﬂow, Coastal Eng. 45, 223–238 (2002). 18. A. Ferrante, L. Franco and S. Boer, Modelling and monitoring of a perched beach at Lido Di Ostia (Rome), Proc. Int. Conf. Coastal Eng. 1992, Vol. 3 (1992), pp. 3305– 3318. 19. J. Fredsøe and B. M. Sumer, Scour at the round head of a rubble-mound breakwater, Coastal Eng. 29, 231–262 (1997). 20. L. D. Givler and R. M. Sørensen, An investigation of the stability of submerged homogeneous rubble mound structures under wave attack, H. R. IMBT Hydraulics Report IHL-110-86. Lehigh Univ. (1986). 21. Halcrow, Anglian coastal management atlas, Anglian water and the sea defence management study for the Anglian region, Supplementary studies report, Anglian water (1988). 22. M. Haller, R. A. Dalrymple and I. A. Svendsen, Experimental study of nearshore dynamics on a barred beach with rip channels, J. Geophysical Res. 107(C6), 1–21 (2002).

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23. K. Hu, C. G. Mingham and D. M. Causon, Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations, Coastal Eng. 41, 433–465 (2000). 24. H. K. Johnson, Th. Karambas, J. Avgeris, B. Zanuttigh, D. Gonzalez and I. Caceres, Modelling of wave and currents around submerged breakwaters, Coastal Eng. 52(10–11), 949–969 (2005). 25. Th. V. Karambas and C. Koutitas, Surf and swash zone morphology evolution induced by nonlinear waves, J. Wat., Port, Coastal and Ocean Eng. 128(3), 102–113 (2002). 26. J. P. Kofoed and H. F. Burcharth, Estimation of overtopping rates on slopes in wave power deviced and other low crested structures, Proc. 28th Int. Conf. Coastal Eng., Cardiﬀ, UK (2002), pp. 2191–2202. 27. M. Kramer and H. F. Burcharth, Stability of low-crested breakwaters in shallow water short crested waves, Proc. Coastal Structures ’03, Portland, USA (2003), pp. 139–149. 28. M. Kramer, B. Zanuttigh, J. W. van der Meer, C. Vidal and X. Gironella, 2D and 3D experiments on low-crested structures, Coastal Eng. 52(10–11), 867–885 (2005). 29. A. Lamberti, R. Archetti, M. Kramer, D. Paphitis, C. Mosso and M. Di Risio, European experience of low crested structures for coastal management, Coastal Eng. 52(10–11), 841–866 (2005). 30. A. Lamberti, B. Zanuttigh and L. Martinelli, Overtopping and wave transmission: An interpretation of spectral change at low-crested rubble mound structures, Proc. 30th Int. Conf. Coastal Eng., Vol. 5 (2006), pp. 4628–4640. 31. A. Lamberti, L. Martinelli and B. Zanuttigh, Piling-up and rip-currents induced by low-crested structures in laboratory and prototype, Proc. Coastal Structures 2007 (2007). 32. I. O. Leont’yev, Modelling of morphological changes due to coastal structures, Coastal Eng. 38, 143–166 (1999). 33. G. R. Lesser, J. A. Roelvink, J. A. T. M. van Kester and G. S. Stelling, Development and validation of a three-dimensional morphological model, Coastal Eng. 51, 883–915 (2004). 34. W. J. Lillycrop and S. A. Hughes, Scour hole problems experienced by the Corps of Engineers; data presentation and summary, Miscellaneous papers, CERC-93-2, US Army Engineer Waterways Experiment Station, Vicksburg, MS (1993). 35. I. J. Losada, J. L. Lara, E. D. Christensen and N. Garcia, Modelling of velocity and turbulence ﬁelds around and within low-crested rubble-mound breakwaters, Coastal Eng. 52(10–11), 887–913 (2005). 36. L. Martinelli, B. Zanuttigh and A. Lamberti, Hydrodynamic and morphodynamic response of isolated and multiple low crested structures: Experiments and simulations, Coastal Eng. 53(4), 363–379 (2006). 37. J. Nicholson, I. Broker, J. A. Roelvink, D. Price, J. M. Tanguy and L. Moreno, Intercomparison of coastal area morphodynamic models, Coastal Eng. 31, 97–123 (1997). 38. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A. A. Balkema, Rotterdam, The Netherlands, 2000). 39. R. Ranasinghe, R. Symondsa, K. Black and R. Holman, Morphodynamics of intermediate beaches: A video imaging and numerical modelling study, Coastal Eng. 51, 629–655 (2004). 40. P. Ruol and A. Faedo, Physical model study on low-crested structures under breaking wave conditions, Proc. Int. MEDCOAST Workshop on Beaches of the Mediterranean and the Black Sea, Kusadasi, Turkey (2002), pp. 83–96.

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41. P. Ruol, A. Faedo and A. Paris, Physical model study of water piling-up behind low crested structures, Proc. 29th Int. Conf. Coastal Eng. (2004), pp. 4165–4177. 42. W. N. Seelig and J. P. Ahrens, Estimation of wave reﬂection and energy dissipation coeﬃcients for beaches, revetments and breakwaters, CERC technical paper 81-1, Fort Belvoir, U.S.A.C.E., Vicksburg, MS (1981). 43. SPM, Shore protection manual, 4th edn., US Army Corps of Engrs (Coastal Engng. Res. Center, US Govt Printing Oﬃce, Washington DC, 1984). 44. G. J. Steendam, J. W. van der Meer, H. Verhaeghe, P. Besley, L. Franco and M. Van Gent, The international database on wave overtopping, Proc. 29th Int. Conf. Coastal Eng., Lisbon (2004), pp. 4301–4313. 45. B. M. Sumer and J. Fredsøe, Experimental study of 2D scour and its protection at a rubble-mound breakwater, Coastal Eng. 40, 59–87 (2000). 46. M. Sumer, J. Fredsøe, A. Lamberti, B. Zanuttigh, M. Dixen, K. Gislason and A. Di Penta, Local scour and erosion around low-crested structures, Coastal Eng. 52(10–11), 995–1025 (2005). 47. TAW, Wave run-up and wave overtopping at dikes, J. W. van der Meer (author), Tech. Report of the Tech. Advisory Committee on Water Defences in the Netherlands (2002). 48. F. Thomalla and C. E. Vincent, Designing oﬀshore breakwaters using empirical relationships: A case study from Norfolk, UK, J. Coastal Res. 20(4), 1224–1230 (2004). 49. T. Uda, Statistical analysis of detached breakwaters in Japan, Proc. 21st Int. Conf. Coastal Eng. (1988), pp. 2028–2042. 50. S. C. Van der Biezen, J. A. Roelvink, J. Van de Graaﬀ, J. Schaap and L. Torrini, 2DH morphological modelling of submerged breakwaters, Proc. 26th Int. Conf. Coast. Eng., Copenhagen, DK (1998), pp. 2028–2041. 51. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Delft Hydraulics Commun. 396 (1988), 214 pp. 52. J. W. van der Meer, Extreme shallow water wave conditions, Report H198. Delft Hydraulics Laboratory, The Netherlands (1990). 53. J. W. van der Meer, H. J. Regeling and J. P. de Waal, Wave transmission: Spectral changes and its eﬀects on run up and overtopping, Proc 27th Int. Conf. Coastal Eng. (2000), pp. 2156–2168. 54. J. W. van der Meer and C. J. M. Stam, Wave runup on smooth and rock slopes of coastal structures, J. Waterway, Port, Coastal and Ocean Eng. 118(5), 534–550 (1992). 55. J. W. van der Meer and J. P. F. M. Janssen, Wave run-up and wave overtopping at dikes, Wave Forces on Inclined and Vertical Structures, eds. N. Kobayashi and Z. Demirbilek (1995), Ch. 1, pp. 1–27. 56. J. W. van der Meer, K. D. Angremond and E. Gerding, Toe structure stability of rubble mound breakwaters, Proc. Advances in Coastal Structures and Breakwaters Conf., Institution of Civil Engineers (Thomas Telford Publishing, London, UK, 1995), pp. 308–321. 57. J. W. van der Meer, R. Briganti, B. Zanuttigh and B. Wang, Wave transmission and reﬂection at low crested structures: Design formulae, oblique wave attack and spectral change, Coastal Eng. 52(10–11), 915–929 (2005). 58. D. Vera-Cruz and J. Reis de Carvalho, Macico submerso de pre-rebentacao das ondas como meio de proteccao de obras maritimas: O caso do quebra-mar de Leixoes, LNEC Memoria no 796, Lisboa (1993), p. 36.

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59. C. Vidal, M. A. Losada, R. Medina, E. P. D. Mansard and G. Gomes-Pina, An universal analysis for the stability of both low-crested and submerged breakwaters, Proc. 23rd Int. Conf. Coastal Eng., Venice (1992), pp. 1679–1697. 60. C. Vidal, M. A. Losada and E. P. D. Mansard, Stability of low-crested rubble mound breakwater heads, J. Wat., Port, Coastal, and Ocean Eng. 121(2), 114–122 (1995). 61. B. Wang, A. K. Otta and A. J. Chadwick, Analysis of oblique wave transmission at smooth and rubble mound structures, ICE Proc. Coastlines, Structures and Breakwaters, London, UK (2005). 62. B. Wang, J. W. van der Meer, A. K. Otta, A. J. Chadwick and J. HorrilloCaraballo, Reﬂection of obliquely incident waves at low-crested structures, Proc. Coastal Dynamics ’05 (2005). 63. B. Wang, A. K. Otta and A. J. Chadwick, Transmission of obliquely incident waves at low-crested breakwaters: Theoretical interpretation of experimental observations, Coastal Eng. 54(4), 333–344 (2007). 64. B. Zanuttigh and J. W. van der Meer, Wave reﬂection from coastal structures, Proc. 30th Int. Conf. Coastal Eng. (World Scientiﬁc Publising Co., 2006), Vol. 5, pp. 4337–4349. 65. B. Zanuttigh, Numerical modeling of the morphological response induced by lowcrested structures in Lido di Dante, Italy, Coastal Eng. 54(1), 31–47 (2007). 66. B. Zanuttigh, L. Martinelli and A. Lamberti, Wave overtopping and piling-up at permeable low-crested structures, Coastal Eng. 55(6), 484–498 (2008). 67. B. Zanuttigh and A. Lamberti, Experimental analysis and numerical simulations of waves and current ﬂows around low-crested coastal defence structures, J. Wat., Port, Coastal and Ocean Eng. 132(1), 10–27 (2006). 68. J. Zyserman, H. K. Johnson, B. Zanuttigh and L. Martinelli, Analysis of far-ﬁeld erosion induced by low-crested rubble-mound structures, Coastal Eng. 52(10–11), 977–994 (2005).

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Chapter 23

Hydrodynamic Behavior of Net Cages in the Open Sea Yu-Cheng Li School of Civil Engineering Dalian University of Technology, Dalian, China [email protected] Knowledge of the hydrodynamic behavior of net cage under the action of waves and currents is the basis of the design and management of net cages in the open sea. Techniques used to investigate the net cage have typically included the use of scaled physical and numerical models, and, where possible, ﬁeld measurements. In this chapter, information on the hydrodynamic behavior of net cages in the open sea is focused on gravity cages. The main methods used for research into hydrodynamic behavior are introduced: physical tests and numerical computation.

23.1. Introduction Exposed net cages in the open sea are subject to wave and current action. Thus, knowledge of their hydrodynamic behavior under the action of waves and currents is important for the design and management of net cages in the open sea. There are many types of net cages, among which gravity cages are the most popular. Thus, in this chapter, information on the hydrodynamic behavior of net cages in the open sea is focused on gravity cages. The main methods used for research into hydrodynamic behavior are introduced: physical tests and numerical computation.

23.2. Experimental Methods 23.2.1. Modeling criteria for ﬁshing nets in experiments Deepwater sea cage engineering has developed rapidly, but research into hydrodynamic characteristics is relatively weak. Model experiments are very useful methods for such studies, but the main diﬃculty is in obtaining reasonable simulation criteria for model tests of ﬁshing nets. For such model tests, the geometric scale λ is usually 633

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greater than 20:1. If the model net is designed strictly according to geometric similarity principles, two diﬃcult problems arise. First, the yarn used for the model net will be very thin, which is diﬃcult to manufacture. Second, because the model yarn is so thin, the Reynolds number (Re) for the model net will change signiﬁcantly relative to the prototype net, leading to a large diﬀerence in hydrodynamic behavior between the model and the prototype nets. Thus, special simulation methods are needed. Tauti’s criteria1,2 were developed in the 1930s during research into ﬁshing net problems and are used only for conditions where a current is present. 23.2.1.1. Tauti’s simulation criteria1,2 Tauti’s simulation criteria involve two geometric scales for nets, λ and λ , which are the global model scale and the model mesh scale, respectively: λ=

Lp Lm

and λ =

dp ap = , dm am

(23.1)

where d is the ﬁlament diameter, a is half the mesh size, and subscripts p and m denote the model and prototype sizes, respectively. Based on geometric similarity and dynamic similarity, the following expressions can be obtained according to Fig. 23.1: rp Sp Tp Lp Fp Wp Sp = = = , Wm Sm rm Sm Tm L m Fm

(23.2)

where W and r are the gravity and the hydrodynamic force per unit area on the net, respectively, and T is the force per unit length on the net edge. When the net material is the same for the model and the prototype, the equation can be replaced by: λ2 λ = λ2

Vp2 Tp Fp =λ = . 2 Vm Tm Fm

(23.3)

In trawl experiments, Tauti’s simulation criteria are suitable and used widely. However, in model tests of net cages, two limitations arise for practical experiments. First, in model tests the model mesh scale cannot be too large, so a greater current velocity is required. Second, Tauti’s simulation criteria cannot be applied to model tests under wave conditions because Eq. (23.3) is not satisﬁed if geometric similarity LT LT S rS WS rS WS Fig. 23.1.

Calculation model of forces acting on a microsegment of a ﬁshing net.

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is applied in the presence of waves. Thus, for model tests of net cages in wave and current conditions, the following gravity simulation criteria are recommended. 23.2.1.2. Extended gravity simulation criteria3,4 The main principle for extended gravity simulation is as follows: (1) Two geometric scales for the nets are still used as in Tauti’s simulation criteria: λ is the global scale and λ is the scale for the yarn diameter and net mesh size. (2) Since the porosity ratio for the model net is the same as for the prototype, external forces acting on the net will follow the gravity simulation criteria by a scale of λ3 . (3) The model net weight simulated by a scale of λ3 should be modiﬁed as follows: 1 1 πd21 4 − × 10 (23.4) ∆W = · (ρ − ρw ) · q · S, · λ λ 4a1 µ1 µ2 where ρ is the density of the net material, ρw is the water density, q is the packing fraction of the ﬁlament, S is the area of the net, and µ1 , µ2 are the hanging ratios (Fig. 23.2) as deﬁned in Eqs. (23.5) and (23.6). In the cross-wise direction of the net, the hanging ratio is deﬁned as: µ1 = a/2L

(23.5)

and in the longitudinal direction as µ2 = b/2L.

(23.6)

Based on gravity simulation criteria, the following equation can be written: λ3 = λ2

Vp2 Tp Fp =λ = . Vm2 Tm Fm

(23.7)

23.2.1.3. Validation of extended gravity simulation criteria3,5 To examine the validity of extended gravity simulation criteria,3,5 special model tests are carried out according to Tauti’s simulation1,2 and extended gravity simulation criteria.3,5 In the model tests, λ = 20 and λ = 2. In the model arrangement shown in Fig. 23.3, the net made of PE has 68×28 diamond meshes with knots. The a

b

L

Fig. 23.2.

Deﬁnition of mesh properties.

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0.4m

transducer

lower bar sinker

1m Fig. 23.3.

16

Sketch of the net model.

Gravity Tauti

Force (KN)

12 8 4 0 0

20

40

60

80

100

120

140

Velocity (cm/s) Fig. 23.4.

Comparison of force values between the gravity similarity and Tauti similarity criteria.

yarn diameter is 1.18 mm and the half-mesh opening is 10 mm. The sinker system consists of a lower bar and sinkers. The prototype mass of the weight system is 240 kg. To record the net conﬁguration, two bulbs were ﬁxed at the middle and bottom of the net. As shown in Figs. 23.4 and 23.5, both the hydrodynamic force and deﬂection of the model net are in agreement under the two similarity criteria, but the weakness in Tauti’s simulation criteria does not exist in the extended gravity simulation criteria. The √ latter can be applied in both current and wave conditions, and since Vm = Vp / λ, the requirement to create a current in the test facility is lower than for Tauti’s simulation criteria. 23.2.2. Scanning method for sea cage motion6 When analyzing cage motion responses in model tests, the data obtained for tracing points include motion trajectory, displacement, inclination angle, velocity, and acceleration. Data processing involves: (1) image acquisition; (2) scanning of tracing points; (3) coordinate transformation; and (4) data analysis, with (2) and (3) requiring the greatest focus. Detailed information on scanning methods for sea cage motion can be found in the literature.6

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Projected area(cm 2 )

800 600 400 200

Gravity Tauti

0 0

20

40

60

80

100

120

140

Velocity (cm/s) Fig. 23.5. criteria.

Comparison of projected area between the gravity similarity and Tauti similarity

23.3. Numerical Methods This section introduces simulation methods for the main parts of a gravity cage, including the ﬂoat collar and ﬁshing net, then the whole gravity cage can be simulated by connecting the main parts of cage.7–15 23.3.1. Model of the ﬂoat collar14,15 23.3.1.1. Forces on the ﬂoat collar In general, the ﬂoat collar system of a gravity cage is at the water surface and double ﬂoating pipes bear the wave-induced loads. For simplicity, the ﬂoat collar system is simpliﬁed to a double-column pipe system, as shown in Fig. 23.6. When calculating wave-induced forces on the ﬂoat collar, the collar can be divided into many mini-segments. The forces acting on the whole collar can be

Fig. 23.6. Sketch of the ﬂoat system for a gravity cage. (a) Simpliﬁed ﬂoat system and (b) section of double pipes.

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u v

b

n

c

a d

u

ο Fig. 23.7.

Schematic diagram of a mini-segment.

obtained by summing the forces acting on each mini-segment. Figure 23.7 is a sketch of a mini-segment of ﬂoat collar with a local coordinate system n, τ , v deﬁned for each mini-segment. As to coordinate system, n and τ are in the normal and tangential direction of the mini-segment, respectively. Thus, v is normal to the mini-plane (Fig. 23.7). According to Brebbia and Walker,16 the n component of wave-induced forces on a mini-segment in local coordinates can be obtained as follows: Fn =

1 CDn ρAn · |(un − Un )| · (un − Un ) + ρV0 an + Cmτ ρV0 (an − U˙ n ), 2

(23.8)

where CDn and Cmn represent the drag and inertial coeﬃcients of the n component, respectively; An is the eﬀective projected area in the direction of the n component; and an and U˙ n represent the acceleration vectors for water particles and minisegments of the n components, respectively. Other parameters are as described above. The same expression can be applied to other wave-induced forces (Fτ , Fν ) of v components. Figure 23.8 is a sketch of a simpliﬁed pipe model for calculation. v n

e

urfac

es Wav

η

n

γ

v

Fig. 23.8.

Sketch of a simpliﬁed pipe model.

O

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The projected areas in diﬀerent directions are as follows:  An = dn · li    1 Aτ = · r · φi · li  π   Av = dv · li ,

(23.9)

where li is the length of the mini-segment. Aτ is related to the arc area of the minisegment (r · φi · li ) in the water. φi is the corresponding central angle of the projected chord-length, which is calculated from cos(φi /2) = 1 − dn /r. For the normal projected area Av , when dn ≥ r, let dv = 2r. In addition to wave-induced forces, the ﬂoat collar is also subjected to gravity, buoyancy, and mooring line forces. The gravity acting on the mini-segment can be written as: Gi = G/N,

(23.10)

where G is the total gravity acting on the ﬂoating system and N is the number of mini-segments. The buoyancy acting on a mini-segment can be calculated as: Ffi = ρg · Vi .

(23.11)

The relationship between mooring line forces and elongation can be obtained directly by experimental measurement. 23.3.1.2. Motion equation for the ﬂoat collar The three-dimensional motions of the ﬂoat collar include surge–sway–heave translation and roll–pitch–yaw rotation. In this section, two coordinate systems are adopted, the ﬁxed coordinate system Oxyz and the body coordinate system G123, as shown in Fig. 23.9. The body coordinate system G123 is rigidly attached to the ﬂoat collar and the coordinate axes 1, 2, 3 are principal axes with origin at the center of mass G. Initially, axes x, y, and z are parallel to axes 1, 2, and 3.

3 p

r G z

2

rp

rG 1 o

y

x Fig. 23.9.

Schematic diagram of the moving coordinate system for a rigid body.

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Six degrees of freedom are required to describe the motion of a ﬂoat collar in general spatial motion, resulting in six equations of motion. According to Newton’s second law, under ﬁxed coordinates, the three translational equations of motion are given by: x ¨G =

n 1 Fxi , mG

y¨G =

i=1

n 1 Fyi , mG

z¨G =

i=1

n 1 Fzi , mG

(23.12)

i=1

where Fxi , Fyi , Fzi are the components of the external force vector Fi (i = 1, n) along the ﬁxed coordinate axes xyz, n is the number of external forces, and mG is the mass of the rigid body. Axes 1, 2, 3 are principal axes with origin at the center of mass G, and thus Euler’s equations of motion for a rigid body17 can be applied. Under the body coordinate system, the three rotational equations of motion are given by: I1 ω˙ 1 + (I3 − I2 )ω2 ω3 =

n

M1i ,

I2 ω˙ 2 + (I1 − I3 )ω3 ω1 =

i=1

I3 ω˙ 3 + (I2 − I1 )ω1 ω2 =

n

n i=1

M2i , (23.13)

M3i ,

i=1

where subscripts 1, 2, 3 represent the body coordinate axes 1, 2, 3. ω1 , ω2 , and ω3 are the components of the angular velocity vector ω along the principle axes, M1i , M2i , M3i are the components of the moment vector Mi (i = 1, n) along the principle axes, n is the number of moment vectors, and I1 , I2 , I3 are the principal moments of inertia. Although six equations of motion have been set up, it is necessary to know the transformation relationship between the ﬁxed coordinates and body coordinates before solving the equations. If Bryant angles φ1 , φ2 , φ3 (Ref. 18) are obtained, the transformation matrix [R] between the ﬁxed coordinates and body coordinates can be expressed as:   cos φ2 cos φ3 cos φ1 sin φ3 sin φ1 sin φ3   + sin φ1 sin φ2 cos φ3 − cos φ1 sin φ2 cos φ3     sin φ1 cos φ3 [R] =  − cos φ2 sin φ3 cos φ1 cos φ3 ,    − sin φ1 sin φ2 sin φ3 + cos φ1 sin φ2 sin φ3  sin φ2 − sin φ1 cos φ2 cos φ1 cos φ2 (23.14) where [R]−1 = [R]T . The relationship between the ﬁxed coordinates and body coordinates is given by:     x 1  2  = [R]  y  . (23.15) z 3

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The kinematic diﬀerential equations for the Bryant angles are given by: ω1 = φ˙ 1 cos φ3 cos φ2 + φ˙ 2 sin φ3 ,

ω2 = −φ˙ 1 sin φ3 cos φ2 + φ˙ 2 cos φ3 ,

ω3 = φ˙ 1 sin φ2 + φ˙ 3 .

(23.16)

The motion of the ﬂoat collar can be described by three translational displacements of the centroid and three Bryant angles, which can be obtained by solving the simultaneous ordinary diﬀerential Eqs. (23.12), (23.13), and (23.16). The equations can be solved using the Runge–Kutta–Verner ﬁfth-order and sixth-order method. During calculation, transformation between the ﬁxed coordinates and body coordinates is carried out by matrix [R]. It should be noted that the forces and motions of the collar are dependent on the net through the mutual mass points attached to both the net and the collar. 23.3.1.3. Hydrodynamic coeﬃcients for the ﬂoat collar in waves Numerical research5,14 has shown that the tangential coeﬃcient Cτ (0.4–0.8) for a pipe in waves is much greater than that of a circular cylinder fully submerged under the water surface. The eﬀects of surface tension may account for this diﬀerence, since the pipe is ﬂoating on the water surface. Results for the normal coeﬃcient Cn (0.6–1.0) are within the range reported by other researchers.19,20 Hydrodynamic coeﬃcients calculated here for a ﬂoat system under wave action will be favorable for the design of and further research into the behavior of net cages in the open sea. It is interesting to note that in wave-only conditions the ﬂoat collar exerts approximately 90% of the total load on the gravity cage, whereas in current-only conditions the ﬂoat collar accounts for <10% of the total load. Thus, it is very important to select reasonable values for the hydrodynamic coeﬃcients of the ﬂoat collar in wave conditions, but is less important for current-only conditions. 23.3.2. Net model 23.3.2.1. Lumped-mass model By applying a lumped-mass model,7 a ﬁshing net is assumed to be a connected structure with limited masses and springs. Lumped point masses are set at each knot and at the center of each mesh bar (Fig. 23.10). Depending on the use of knots, two diﬀerent types of nets are commonly used: nets with knotted meshes and nets with knotless meshes. In the model, the diameter of the sphere d needs to reﬂect the physical properties of the net. For a mesh with knots, d is taken to be 3.14-fold greater than the diameter of the mesh bar. For a knotless mesh, d is taken to be 1.5-fold greater than the diameter of the mesh bar.21 According to Newton’s second law, the equation of motion for a lumped mass i in waves can be expressed as Mi a =

n j=1

+ B, Tij + FD + FI + W

(23.17)

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mesh knot

d

mesh bar point

Fig. 23.10.

Schematic diagram of the mass-spring model.

τ ξ η

Fig. 23.11.

Schematic diagram of the local coordinates for mesh yarns.

where a is the acceleration of mass point i, Tij is the vector of the tension in bar ij (j is the code for knots at the other end of bar ij), n is the number of knots adjacent is the gravity force, and to point i, FD is the drag force, FI is the inertial force, W is the buoyancy force. According to our research,9,10 the inertial force FI on a B ﬁshing net is rather small under wave conditions compared to other external forces, so it can be omitted here. The mass point at a mesh knot is assumed to be a spherical point at which the ﬂuid force coeﬃcient is constant in the direction of motion, so the motion equation is easily set up according to Eq. (23.17). For points at mesh bars, the forces and motion equation are described in the following. Since the points at mesh bars are assumed to be cylindrical elements, the direction of the ﬂuid forces acting on the point masses at each mesh bar should be considered. Therefore, the motion of point mass i is set at the center of a mesh bar, and the local coordinates (τ, η, ξ) passing point i are deﬁned to simplify the procedure (Fig. 23.11). The η axis lies on the plane including the τ axis and V , and the velocity vector V of the water particles at point i can be divided into τ (tangential) and η (normal) components, so the local ﬂuid force on the plane can be estimated by deﬁning the τ and η components of the ﬂuid force. Under global coordinates, the vectors drawn from i to 1 and 2 are ei1 and ei2 , respectively. The unit vectors of the τ , η, ξ axes are eτ = (xτ , yτ , zτ ), eη = (xη , yη , zη ), and eξ = (xξ , yξ , zξ ).

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The relationship between tension and elongation based on Wilson22 is given by: T = C1 εC2 , d2

ε=

l − l0 , l0

(23.18)

where l0 is the undeformed length, l is the deformed length, d is the diameter of the mesh bar, and C1 and C2 are constants that deﬁne the elasticity of the element. Therefore, the τ component of tension at point i can be expressed as: Tτ = Ti1 ei1 · eτ + Ti2 ei2 · eτ ,

(23.19)

where the mid-dot [·] represents the inner product. The same expression can be applied to the other tension forces (Tη , Tξ ) of η, ξ components. The τ, η, and ξ components of the relative velocity of point i are τ˙ − eτ · V , ˙ respectively. Thus, the τ component of the drag force of point i η˙ − eη · V , and ξ, can be represented as: 1 Fdτ = − ρCdτ Dl|τ˙ − eτ · V |(τ˙ − eτ · V ), 2

(23.20)

where Cdτ represents the drag coeﬃcient of the τ component, D is the diameter of the cylindrical element, and l is its length. The same expression can be applied to the other drag forces (Fdη , Fdξ ) of η, ξ components. Dividing the other external forces B and W along the local coordinate axes, the τ component of the motion equation for point i can be expressed as: Mi τ¨i = Tτ − FDτ + Wτ + Bτ .

(23.21)

Thus, the motion equations for point i in the η, ξ directions can be expressed in the same manner. As the displacements are expressed using local coordinate variables in Eq. (23.21), we can transform these into the global coordinate system by: ˙ T, (x˙ i , y˙i , z˙i )T = [C](τ˙ , η, ˙ ξ)

(23.22)

where [C] is given by: 

xτ [C] =  yτ zτ

xη yη zη

 xξ yξ  . zξ

(23.23)

Thus, the motion can be represented by the following system of ordinary diﬀerential equations: dx˙ i = f (xi , yi , zi , x˙ i , y˙ i , z˙i , x1 , y1 , z1 , x2 , y2 , z2 ; t) dt dy˙ i = g(xi , yi , zi , x˙ i , y˙i , z˙i , x1 , y1 , z1 , x2 , y2 , z2 ; t) dt dz˙i = h(xi , yi , zi , x˙ i , y˙ i , z˙i , x1 , y1 , z1 , x2 , y2 , z2 ; t). dt

(23.24)

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The net shape at each time step can be calculated numerically by solving these ordinary diﬀerential equations for a given initial condition. 23.3.2.2. Mesh grouping method When all the knots in the mesh and at the bar are used as mass points, there are more than 10,000 mass points for a common plane net, which requires too long a calculation time for practical simulations. To reduce the computational eﬀort, a mesh grouping method can be employed in calculations. The method consists of modeling a given number of actual meshes as a ﬁctitious equivalent mesh that has the same physical qualities as the actual meshes, such as the projected area of the net, speciﬁc mass, weight, etc. Figure 23.12 shows an example of a 4 × 4 mesh with 65 lumped mass points that is approximated to a 2 × 2 mesh with 21 lumped mass points. According to calculations by Zhao,12 using 1 × 1 (G0), 2 × 2 (G4), and 4 × 4 (G16) meshes to model the same panel yields similar results for predicted hydrodynamic forces for current-only loading, but the time reduction is obvious, as shown in Fig. 23.13.

Fig. 23.12.

Sketch of the mesh grouping method.

30

25

25 Time(s)

Force(N)

20 15 G0 G4

10 5

24

20 15 10 4

5

G16

0

1

0 0

15

30 V(cm/s)

(a)

45

60

G0

G4

G16

(b)

Fig. 23.13. Comparison of the (a) total drag force and (b) calculation time for diﬀerent mesh grouping methods.

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23.3.3. Hydrodynamic coeﬃcients for nets in the numerical model Our previous study8 showed that, even in wave conditions, the hydrodynamic coeﬃcients of ﬁshing nets are related to the Reynolds number only, and are not inﬂuenced by the Kenlengan–Carpenter number. Thus, in both current and wave conditions, we may assume that the hydrodynamic coeﬃcients of a net are related to the Reynolds number only. For each mesh bar, the numerical procedure calculates the drag coeﬃcient Cd using a method described by Choo and Casarella23 that updates the drag coeﬃcients based on the Reynolds number, as follows:  8π  (1 − 0.87s−2 )(0 < Ren ≤ 1)    Ren s (23.25) Cn = 1.45 + 8.55Re−0.90 (0 < Re ≤ 30) n  n    (0 < Ren ≤ 105 ) 1.1 + 4Re−0.50 n 2/3 Ct = πµ(0.55 Re1/2 n + 0.084 Ren ),

(23.26)

where Ren = ρVRn D/µ, s = −0.07721565 + ln(8/Ren ), and Cn and Ct are the normal and tangential drag coeﬃcients. VRn is the normal component of the ﬂuid velocity relative to the bar, ρ is the density of water, and µ is the water viscosity. Fredheim and Faltinsen21 suggested that it is reasonable to use a drag coeﬃcient in the range 1.0–2.0 when modeling the knot part as a sphere.

23.4. Comparison of the Hydrodynamic Behavior of Gravity and Sea Station Cages24 In this section, we compare the force and movement characteristics of diﬀerent cage models. Three models were selected: a gravity cage, a sea station cage, and a quasisea station cage. The sea station cage model has that same enclosed volume as the gravity cage model, but has a greater cage diameter and height. The quasi-sea station cage model has the same diameter as the sea station cage model, but is 1.4fold greater in cage height. This increases its eﬀective aquaculture volume to 2.2fold that of the latter. Model tests to determine the hydrodynamic characteristics of sea cages were carried out under ﬂoating and submerged conditions. Current-only, wave-only, and wave plus current conditions were set during the tests. Table 23.1 shows the model settings in detail. Table 23.1.

Model test conditions.

Model

Cage style

Diameter (m)

Height (m)

Volume (m3 )

Weight (N)

Weight style

Submersion method

I II III

Gravity Sea station Quasi-sea station

0.637 1.0 1.0

0.4 0.5 0.7

0.127 0.131 0.288

3.92 2.26 2.26

Sinker Ballast Ballast

Two-pipe perfusion Center-spar perfusion Center-spar perfusion

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1.0m

0.4m

0.637m

z Gravity Cage

sinker

α

1.09m

0.25m

Sea Station mooring line

α

1.04m

z

0.25m

Quasi-Sea Station

α

x Fig. 23.14. points).

1.71m

1.0m

1.0m

0.25m

1.71m

0.75m

x

0.7m

z

1.0m

1.0m

0.1m

1.71m

0.15m

x

1.02m

Sketch of cage models under ﬂoating conditions (black dots on cages are the tracing

Figure 23.14 shows the model settings for diﬀerent cages under ﬂoating conditions. All cage models were moved to 2/5 of the water depth when submerged. The mooring line forces were acquired by transducers attached to the bottom of each mooring line. Cage movement data were gathered by tracing diodes ﬁxed on the rigid ring or ﬂoat system of each cage model using a CCD camera. Cages of diﬀerent styles will exhibit diﬀerent force and motion characteristics under the same current and wave conditions. For oﬀshore aquaculture facilities, security, management, and investment are the main factors to be considered. Table 23.2 provides a general comparison of the cage models under ﬂoating and submerged conditions, for which the gravity cage was selected as the reference. Values for the mooring line force, aquaculture volume and cost were all set to 1.0, and the data are reported as ratios compared to values for the gravity cage. From the viewpoint of forces, the gravity cage has advantages when it is ﬁxed with heavier sinkers. Management of gravity cages is more convenient and the investment costs are much lower compared to the other two cages. The sea station and quasi-sea station systems are semi-rigid cages, which are beneﬁcial in controlling volume reduction and preventing predator attack. However, the maximum volume of the gravity cage is nearly three-fold greater than that of the sea station cage

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Current only Wave only (H = 35 cm) Wave + currenta Cage model 11.5 cm/s 15.5 cm/s 17.2 cm/s T = 1.6 s T = 1.8 s T = 2.0 s T = 1.6 s T = 1.8 s T = 2.0 s Diameter Volume

Cost

Floating

I II III

1.0 0.97 2.11

1.0 1.02 2.08

1.0 1.04 2.06

1.0 1.16 1.27

1.0 1.0 1.06

1.0 1.04 1.14

1.0 1.24 1.84

1.0 1.04 1.58

1.0 1.18 1.71

1.0 1.57 1.57

1.0 1.03 2.27

1.0 3∼4 >4

Submerged

I II III

1.0 0.84 1.96

1.0 1.12 1.92

1.0 1.11 1.9

1.0 1.43 1.4

1.0 1.63 1.97

1.0 1.16 1.5

1.0 0.64 1.41

1.0 0.63 1.4

1.0 0.81 1.51

1.0 1.57 1.57

1.0 1.03 2.27

1.0 3∼4 >4

a For

wave + current, the wave height was H = 35 cm and the current velocity was u = 17.29 cm/s.

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General comparison of the cage models under ﬂoating and submerged conditions (ratios).

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Table 23.2.

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for the same cage diameter and height. In other words, the advantages of the sea station cage can be attributed to largely losing the original aquaculture volume. Even for the same original volume as the gravity cage, management of sea station and quasi-sea station cages is inconvenient. From the viewpoint of cage motion, the sea station and quasi-sea station cages exhibit better stability, which is considered beneﬁcial for ﬁsh in the cages. However, it should be noted that even if cages can endure the action of tidal currents and waves, there is no guarantee that ﬁsh in such cages will survive.

23.5. Calculation of Velocity Reduction Behind a Plane Fishing Net25 Part of the ﬂuid will ﬂow through adjacent areas of the ﬁshing net owing to its blockage eﬀect. However, it is diﬃcult to determine the ﬂux passing through adjacent areas because of the complexity of the interaction between the ﬂuid and the ﬁshing net. In general, the ﬂuid velocity in adjacent areas decreases as the distance between the ﬂuid and the ﬁshing net increases. The eﬀective adjacent area is related to the fabric property of the ﬁshing net. Here, the area of the ﬁshing net is discretized into two parts: the solid projected area of the twines and knots, and the area of the holes. The latter is treated the same as the eﬀective adjacent area, which means that the average velocity in holes is assumed to be the same as that in the eﬀective adjacent area. Therefore, the eﬀective adjacent area is a key factor for proper evaluation of the velocity reduction behind a ﬁshing net. Figure 23.15 shows the deﬁnition of the ﬂowing areas. It is assumed that the cross-sectional area of the ﬁshing net is A1 with solidity Sn and the eﬀective adjacent area is A2 . Here, the variation in area introduced by changes in water level in diﬀerent sections is neglected. The solid projected area of the ﬁshing net is Sn A1 and the area of holes is (1−Sn )A1 . Therefore, considering the unit: cm

100

50

Area of net (A1, u 1)

70

50

39.6

200

Hypothetic effective adjacent area (A2,u 2)

Fig. 23.15.

Deﬁnition of the ﬂowing areas.

FA

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eﬀective adjacent area, the total area aﬀected by ﬂuid passing through the ﬁshing net is A = (1 − Sn )A1 + A2 .

(23.27)

According to mass conservation, we have: (A1 + A2 ) · u0 = [(1 − Sn )A1 + A2 ] · u2 ,

(23.28)

where u0 is the velocity of the far-ﬁeld approach ﬂuid, which is assumed to be undisturbed by the ﬁshing net; u2 is the velocity in the adjacent area, which is the same as that in the holes of the ﬁshing net. A coeﬃcient k is introduced here to denote the ratio between the adjacent area to the solid projected area. It can be written as: k = A2 /(Sn · A1 ).

(23.29)

Substituting Eq. (23.29) into Eq. (23.28), we have: u2 =

1 + k · Sn · u0 , 1 − (1 − k) · Sn

(23.30)

As described above, the velocity of the ﬂuid passing through the holes will decrease due to ampliﬁcation of the sectional area behind the ﬁshing net. According to mass conservation within the section of ﬁshing net (excluding the adjacent area), we have: u1 = (1 − Sn ) · u2 =

(1 + k · Sn ) · (1 − Sn ) · u0 , 1 − (1 − k)Sn

(23.31)

i.e., u1 (1 + k · Sn ) · (1 − Sn ) = . u0 1 − (1 − k)Sn

(23.32)

Studies on the hydrodynamics of pile groups or double-circle cylinders reveal that the inﬂuence of adjacent piles can be neglected when the gap between them is four-fold greater than the pile diameter, i.e., the hydrodynamics of each pile can be treated separately. Therefore, the range inﬂuenced by one pile is three-fold the pile diameter. For the ﬁshing net, we assume that the eﬀective area inﬂuenced is in proportion to the solid projected area of the net. Here, the ratio k is supposed to be the value of Eq. (23.32), which can be written as: u1 (1 + 3Sn ) · (1 − Sn ) = . u0 1 + 2Sn

(23.33)

According to Eq. (23.32), if Sn = 0, i.e., there is no ﬁshing net in the ﬂow ﬁeld, we have u1 = u0 . If Sn = 1, the ﬁshing net is equivalent to a solid plate and the velocity behind the ﬁshing net is equal to zero. Results under these two cases are reasonable. However, when Sn = 0 and k = 0, the ﬁshing net exists in the whole section, which is similar to a trash rack. It is known that under this case, the velocity behind it will increase to some extent. However, the result calculated with

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Eq. (23.32) is u1 = u0 . Therefore, calibration should be made to Eq. (23.32). It is assumed that ∆u is the calibration velocity, which is the average velocity diﬀerence between the section behind the ﬁshing net and that at in the far ﬁeld in front of the ﬁshing net when k = 0. ∆u can be determined from the equations for energy conservation and mass conservation as follows: H0 +

u20 u2 = H + + hj , 2g 2g

H0 · u0 = H · u ,

(23.34)

(23.35)

where H0 is the water level in the far-ﬁeld section in front of the ﬁshing net; H is the water level behind the ﬁshing net; u is the ﬂow velocity behind the ﬁshing net; and hj is the local energy loss in this section of ﬁshing net, which can be calculated according to the trash rack analogy. The general form of the local energy loss is as follows: 4/3 b u2 u2 (23.36) hj = ξ · 0 = β · · sin α · 0 , 2g t 2g where ξ denotes the local head loss coeﬃcient and β is the shape factor of the trash rack. For the ﬁshing net, β is equal to 1.79. b and t refer to the grid width and the gap between grids, respectively. Thus, b/t is the ratio between the solid projected area and the gap area. For the ﬁshing net, we have b/t = Sn /(1 − Sn ). α in Eq. (23.36) refers to the angle between the cross plane of the trash rack and the horizontal plane, which is 90◦ here. Substituting the value of each parameter into Eq. (23.36) yields: 4/3 2 Sn u (23.37) hj = 1.79 × · 0. 1 − Sn 2g By combining Eqs. (23.34), (23.35), and (23.37), the ﬂow velocity (u ) behind the ﬁshing net can be determined. Then the value of ∆u is obtained according to: ∆u = u − u0 .

(23.38)

The calibration velocity ∆u is then distributed according to the ﬂowing area. Equation (23.32) is then written as: (1 + k · Sn ) · (1 − Sn ) 1 − Sn ∆u u1 = + · . u0 1 − (1 − k)Sn 1 − (1 − k)Sn u0

(23.39)

As shown in Eq. (23.39), the velocity reduction is related to the calibration velocity ∆u and the solid projected area of the ﬁshing net. For a given ﬁshing net, its fabric property is determined. The velocity reduction behind the ﬁshing net can then be calculated using Eq. (23.39). A model experiment of velocity reduction behind a plane net was carried out in a ﬂume tank (60 m long, 2 m wide, and 1.8 m high) equipped with a currentproducing system at one end.

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50

39.6

C1

A2

70

651

20

A1

Flow direction

FA

C2

A3

C3

A4

C4 Plane net Measuring Point

Fig. 23.16.

Measuring point

Model setting and measurement points during experiments.

Table 23.3. Comparison of the velocity reduction between calculated results and experimental data. Measured velocity and velocity reduction

Calculated result (1 − u /u0 )

Position A u0 (cm/s)

Position C u (cm/s)

1 − u /u0

Eq. (23.39)

15.9 31.8 44.21 60.4

13.85 28.22 38.92 52.25

12.89% 11.26% 11.97% 13.49%

12.32% 12.11% 11.84% 11.31%

In Eq. (23.39), u0 is the far-ﬁeld velocity, so velocities at position A in front of the plane net were selected for calculation. Velocities at position C were selected for validation of the calculation model proposed. The model setting and measurement points during experiments are shown in Fig. 23.16. As shown in Table 23.3, an average velocity reduction of 12.4% was observed behind the plane net. Results calculated according to Eq. (23.39) agree well with the experimental data. The calculated results are also consistent with experimental results presented by Fredriksson,26,27 in which an approximate 10% reduction in velocity was found for a sea-station cage. 23.6. Hydrodynamic Characteristics of a Single Gravity Cage Model tests were carried out to investigate the hydrodynamic behavior of a gravity cage of a given structure size. The geometric scale λ and model mesh scale λ were 20:1 and 2:1, respectively. The gravity simulation criteria described in Sec. 23.2.1.2 were applied in designing the model net cage. Under current-only, wave-only, and wave + current conditions, the mooring line force and motion of the net cage were measured when the net cage was ﬂoating or submerged under water. In model tests,

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Table 23.4. Design of experimental conditions (H and T denote the wave height and period, respectively).

Wave H (cm) T (s)

1

2

3

4

5

6

7

8

9

I

J

K

20 1.2

20 1.4

20 1.6

25 1.4

25 1.6

25 1.8

30 1.4

30 1.6

30 1.8

35 1.6

35 1.8

35 2

Velocity (cm/s)

8

12

16

Transducer

z

0.43m

0.796m

20

Tracing point

Bottom collar Sinker

h=1.0m

No.

Sinking ring

x

2.92m

(a)

Transducer

z

Bottom collar Sinking ring

x

h=1.0m

H/3

0.796m

Sinker

2.92m

(b) Fig. 23.17. Sketch of the setup for the gravity cage model. (a) Floating condition and (b) submergence condition.

submergence depths of U1 and U2 , corresponding to h/3 and h/2 (h is the water depth), respectively, were used. The experimental conditions are shown in Table 23.4 and a sketch of the experimental model is shown in Fig. 23.17. The ﬂoat collar with a diameter of 0.796 m was made of HDPE material. The cage net was made of PE with a mass density of 953 kg/m3 . The netting was knotless, with a mesh size of 20 mm and a yarn thickness of 1.18 mm. Mounted as diamond meshes, the net then formed an open vertical cylinder with a diameter of 0.796 m and a height of 0.43 m. The full-scale diameter of the net cylinder is assumed to be 16 m. The weight system of the gravity cage comprises a bottom collar, sinkers, and a sinking ring. The bottom collar is of steel, with a mass of 25 g in water, which is attached to the bottom of the net cylinder by a thin line with a length of 5 cm. There are 10 spherical sinkers, each with a mass of 3.75 g in water and a diameter of 3 cm. When the net cage is ﬂoating, the mass of the sinking ring is zero; when submerged, the mass is 248.2 g. Outer pipe perfusion was applied to submerge the

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cage, and the sinking ring was used to control the submergence depth of the cage. The water mass that was perfused into the pipe was 334.1 g. Using the data measured, empirical formulae for the mooring line force and motion were obtained by the least square method.

23.6.1. Mooring line force 23.6.1.1. Current only Under ﬂoating conditions, the empirical formula for the force of a single mooring line on the net cage is V2 F = 0.149 (correlation coeﬃcient R = 1.000) . ρgD 2 B gh

(23.40)

At submergence depths of U 1 and U 2, the corresponding empirical formulae are F V2 = 0.237 (correlation coeﬃcient R = 1.000) 2 ρgD B gh

(23.41)

V2 F = 0.217 (correlation coeﬃcient R = 0.990) , ρgD 2 B gh

(23.42)

where F is the mooring line force (N ), V is the current velocity (m/s), ρ is the water density (kg/m3 ), D is the diameter of the ﬂoat collar (m), B is the depth of the net cage (m), h is the water depth (m), and g is acceleration due to gravity (kg m/s2 ). 23.6.1.2. Wave only Under ﬂoating conditions, the empirical formula for the maximum force of a single mooring line of net cage is (correlation coeﬃcient R = 0.979): 2 H H F −3 −3 + 1.73 × 10 · = 1.83 × 10 · . (23.43) ρgD 2 B h h At submergence depth U 1, the corresponding empirical formula is (correlation coefﬁcient R = 0.982): 2 H H F −3 −3 = 3.33 × 10 · , (23.44) − 0.28 × 10 · ρgD 2 B h h where H is the wave height and other symbols are as previously described.

23.6.2. Motion of the net cage 23.6.2.1. Current only As shown in Fig. 23.18, the inclination of the ﬂoating collar system is not signiﬁcant under current conditions when the cage is ﬂoating. Even for a prototype current velocity of 0.86 m/s, the inclination is only 1.17◦ . When the cage is submerged, the inclination of ﬂoating collar is signiﬁcant. It can be concluded that current has a destabilizing eﬀect when the net cage is submerged.

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20 Floating 15 Inclination/Deg

Submergence 10

5

0 0

0.2

0.4

0.6

0.8

1

Velocity/ms-1 Fig. 23.18.

Inclination angle for the cage under current conditions.

23.6.2.2. Wave only When a net cage is subject to wave action, the amplitude of the horizontal motion of tracing points is greater on the lee side than on the wave side of the ﬂoating collar. The empirical formula for the maximum amplitude of the horizontal motion of tracing points on the lee side of the ﬂoating collar is (correlation coeﬃcient R = 0.986): ξF = 0.614 × D

H D

2

+ 4.56 × 10−3 ×

gT 2 h

− 0.175 ×

Uh h

,

(23.45)

where ξF denotes the maximum amplitude of the horizontal motion of tracing points on the lee side of the ﬂoating collar and Uh is the submergence depth. In contrast, the amplitude of the vertical motion of tracing points is greater on the wave side than on the lee side of the ﬂoating collar under wave action. The empirical formula for the maximum amplitude of the vertical motion of tracing points on the wave side of the ﬂoating collar is (correlation coeﬃcient R = 0.991): ηF = 0.321 × D

H D

+ 3.05 × 10−3 ×

gT 2 h

− 0.217 ×

Uh h

,

(23.46)

where ηF denotes the maximum amplitude of the vertical motion of tracing points on the wave side of the ﬂoating collar.

23.7. Eﬀects of Structural Arrangement on the Hydrodynamic Behavior of a Gravity Cage9,28,29 A change in the structural arrangement of a gravity cage may have eﬀects on its hydrodynamic behavior. Structural arrangements include the weight system, structure size ratio, mesh type, etc. In this section, researches9,28,29 including model

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test and numerical simulation are introduced, which will give references of the design of net cage. 23.7.1. Eﬀects of the weight system For gravity cage, the holding of available volume for ﬁshing net may mainly depend on the function of weight system. Adding mass of weight system may be a good way to reduce the loss of netting volume, but increase in mass will lead to the tension increase in ﬁshing net, and then the manufacture cost and risk of netting breakage will be higher. In practical application, bottom collar has been proposed and oﬀer promise in maintaining net shape. In this section, the eﬀects of weight system on hydrodynamic behaviors of gravity cage are introduced, according to our recent research.28,29 In our model tests, two types of weight system were applied: (1) sinker, and (2) bottom-collar sinker. Details of the weight systems are shown in Table 23.5. Other settings for the net cage are the same as in Sec. 23.6. A sketch of the model setup is shown in Fig. 23.19.

Weight mode Bottom collar (g) Sinker (g) Total mass (g) Prototype mass (kg)

Weight system models.

A1

A2

A3

B2

B3

0 38.8 38.8 310.4

37.5 38.8 76.3 610.4

100 38.8 138.8 1110.4

— 76.6 76.3 610.4

— 138.8 138.8 1110.4

0.43m

0.796m

Transducer

z

B4 — 176.3 176.3 1410.4

Tracing point

H=1.0m

Table 23.5.

Bottom collar Sinker

x

2.92m

(a) Tracing point

Bottom of net Sinker

x

H=1.0m

Transducer

z

0.43m

0.796m

2.92m

(b) Fig. 23.19. Sketch of the model setup. (a) Weight system comprising a bottom-collar sinker and (b) weight system comprising sinkers.

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60

60

B2

A1

B3

40

Force /(kN)

A2 Force/(kN)

FA

A3

20

0

40

B4

20

0 0.36

0.54

0.72

Velocity /m s

0.89

1.07

0.36

-1

0.54

(a) Fig. 23.20. mode B.

0.72

Velocity /ms

0.89

1.07

-1

(b)

Resultant forces in the two upstream mooring lines. (a) Weight mode A and (b) weight

1.0

Ratio

0.8

0.6 B2/A2

0.4

B3/A3 0.2 0.72

0.89 Velocity /cms

Fig. 23.21.

1.07 -1

Comparison of the resultant mooring-line forces for two diﬀerent weight systems.

23.7.1.1. Mooring line force Values for the resultant forces in upstream mooring lines are shown in Fig. 23.20. When a net cage is subject to current action, the mooring line forces increase with increasing mass of the weight system for both the sinker system and bottom collarsinker system. For the same mass, the mooring line force is greater for the bottomcollar sinker system compared to the sinker system, as shown in Fig. 23.21. When a net cage is subject to wave action, the mass of weight system has little eﬀect on the mooring line force, as shown in Fig. 23.22. For the same mass, the mooring line force is smaller for the bottom-collar sinker system than for the sinker system, as shown in Fig. 23.23. 23.7.1.2. Deformation Net deformation is an important consideration. A sea cage is a 3D structure and thus deformation of a ﬁshing net is also a three-dimensional problem under current

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Force /(kN)

Force /(kN)

60

657

40

20

A1

40

20

B2 B3 B4

A2 A3 0

0 7.2 4m

8.0

8.0

8.9

5m

6m

7m

7.2

T/s H/m

8.0

4m

(a) Fig. 23.22. mode B.

8.0

5m

6m

8.9 7m

T/s H/m

(b)

Resultant forces in the two wave-side mooring lines. (a) Weight mode A and (b) weight

1.5

Ratio

1.2

0.9

B2/A2 B3/A3

0.6

Fig. 23.23.

7.2

8.0

4m

5m

8.0 6m

8.9 7m

T/s H/m

Comparison of the resultant mooring-line forces for two diﬀerent weight systems.

and/or wave conditions. At present, there is no eﬀective method for measuring the net deformation of sea cages. During the experiment, six diodes were arranged in two columns on the ﬁshing net, as shown in Fig. 23.19. Images were gathered with a CCD camera set outside the glass wall of the wave-current ﬂume. The rate of volume loss was obtained by comparing the dynamic areas enclosed by the six tracing points on the ﬁshing net with that under static conditions. Under current conditions, nets with either the sinker system or bottom collarsinker system exhibit a decrease in the rate of area loss with increasing mass of the weight system, as shown in Fig. 23.24. For the same mass, the bottom-collar sinker system can improve net deformation compared to the sinker system, as shown in Fig. 23.25. When a net cage is subject to wave action, the mass of the weight system has little eﬀect on the maximum rate of area loss, as shown in Fig. 23.26. With the same mass of weight system, the maximum rate of area loss is lower for the bottom-collar sinker system than for the sinker system, as shown in Fig. 23.27.

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90%

90%

A1 A2

60%

Rate of area loss

Rate of area loss

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A3

30%

0%

B2 B3

60%

B4

30%

0% 0.36

0.54

0.72 Velocity/ms

0.89

1.07

0.36

-1

0.54

0.72

(a) Fig. 23.24.

0.89

Velocity/ms

1.07

-1

(b)

Inﬂuence of weight size on net deformation. (a) Weight mode A and (b) weight mode B.

Rate of area loss

90%

B2 A2 B3 A3

60%

30%

0% 0.36

0.54

0.72 Velocity/m·s

40%

A1 A2

30%

A3 20% 10%

40% 30% 20% B2 10%

B3 B4

0%

0% 7.2 4m

8.0

8.0

8.9

5m

6m

7m

(a) Fig. 23.26. mode B.

1.07

Inﬂuence of weight style on net deformation.

Maximum Rate of area loss

Maximum Rate of area loss

Fig. 23.25.

0.89 -1

T/s H/m

7.2 4m

8.0

8.0

8.9

5m

6m

7m

T/s H/m

(b)

Inﬂuence of weight system on net deformation. (a) Weight mode A and (b) weight

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3

Ratio

2

1

B2/A2 B3/A3

0 7.2 4m Fig. 23.27.

8.0

8.0

8.9

5m

6m

7m

T/s H/m

Inﬂuence of weight style on net deformation.

23.7.2. Eﬀects of RDH and mesh type on net deformation in current8 In this section, the numerical model described in Sec. 23.3 is used to analyze the eﬀects of RDH (RDH = D/H, where D and H are the diameter and height of the gravity cage) and mesh type on net deformation in a current. 23.7.2.1. Model description In the numerical model, the net cage is designed at a model scale of 1:7.1, as shown in Fig. 23.28. The net cage model comprises a hoop, a net, and a number of weights attached to the bottom of the net. The top of the net is mounted on the hoop, which is kept in a ﬁxed position during each test. The weights are suspended around the bottom of the net to stretch it and maintain its shape under the inﬂuence of a current. The hoop is made of stainless steel and has an overall diameter of 1.435 m and a rod diameter of 0.025 m. The hoop itself is designed to have no deformation. The full-scale diameter of the net cylinder is assumed to be 10 m. The net is made of nylon with a mass density of 1,130 kg/m3 . The total number of meshes in the circumferential direction is 252. The netting is knotless, with a Top of net cage fixed

Net

Current

Sinker

Fig. 23.28.

Sketch of the gravity cage model with a bottom-collar sinker-weight system.

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Properties of the net cages used for numerical simulation. No. of meshes

Cage no.

RDH

Mesh style

Diameter (m)

Height (m)

Circumference

Height

1 2 3 4 5 6

2.0 2.0 1.43 1.43 1.0 1.0

Square Diamond Square Diamond Square Diamond

1.435 1.435 1.435 1.435 1.435 1.435

0.717 0.717 1.0 1.0 1.435 1.435

248 210 248 210 248 210

40 25 56 35 80 50

Table 23.7.

Weight-mode setting.

Mass (g)

Bottom collar

One sinker (g)

Scaled weight system (g)

WA WB

0 931.2

87.3 29.1

1396.8 1396.8

mesh size of 32 mm and a yarn thickness of 1.8 mm. The netting itself is not scaled, and ordinary full-scale netting is used. Mounted as square meshes, the solidity ratio (S) of the netting is 0.225. The net then forms an open vertical cylinder with a diameter of 1.435 m. Keeping the cage diameter constant, three RDH values are applied: 2.0, 1.43, and 1.0. In each RDH mode, two types of knotless net are used and simulated, a square mesh and a diamond mesh. The yarn diameter and mesh size are set to 1.8 and 32 mm, respectively, for both net types. The properties of the net cages are presented in Table 23.6. The bottom of the net cage is not modeled. In this section, two types of weight system are applied: (1) a sinker (WA), and (2) a bottom-collar sinker (WB), as shown in Table 23.7. The gravity cage system is subjected to ﬁve diﬀerent current velocity cases (Usteady = 0.11, 0.19, 0.26, 0.34, and 0.37 m/s) corresponding to the full-scale cases (Usteady = 0.3, 0.5, 0.7, 0.9, and 1.0 m/s) according to the gravity simulation criteria. After entering the net cage, the velocity of a ﬂuid particle will decrease slightly. 23.7.3. Eﬀect of RDH on net deformation Under diﬀerent current velocities, the volume-holding coeﬃcient (Cvh ) of the net cylinder can be calculated as: Cvh =

Vc , Vc0

(23.47)

where Vc is the volume of the net cylinder exposed to the current and Vc0 is the initial volume of the net cylinder not exposed to the current.

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23.7.3.1. Fishing net with square mesh For both weight systems the eﬀect of RDH on net deformation is signiﬁcant, as shown in Fig. 23.29. An increase in RDH helps to decrease net deformation. According to the simulation results, the bottom-collar sinker system holds the volume of the gravity cage better than the sinker system. As shown in Fig. 23.30, at low current velocity the increase in volume-holding coeﬃcient is greater at lower RDH, but when the velocity increases to approximately 0.3 m/s, the opposite trend is apparent.

Volume holding coefficient [-]

Volume holding coefficient [-]

1.0 0.9 0.8 0.7 0.6

RDH=2.0 RDH=1.43 RDH=1.0

0.5 0.4

1.0 0.9 0.8 0.7 0.6

RDH=2.0 RDH=1.43 RDH=1.0

0.5 0.4

0.3 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.10

0.15

0.20

0.25

0.30

Velocity [m/s]

Velocity [m/s]

(a)

(b)

0.35

0.40

Fig. 23.29. Volume-holding coeﬃcient under (a) the sinker system and (b) the bottom-collar sinker system at diﬀerent RDH as a function of current velocity.

Increase in the volume holding coefficient[-]

0.09

RDH=2.0 RDH=1.43 RDH=1.0

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.10

0.15

0.20

0.25

0.30

0.35

0.40

Velocity [m/s] Fig. 23.30. Increase in the volume-holding coeﬃcient with the bottom-collar sinker system for diﬀerent RDH relative to the sinker system.

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Fig. 23.31.

Shapes of net cylinders with diamond and square meshes.

(a) Fig. 23.32.

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(b)

Initial net images (a) with a sinker system and (b) a bottom-collar sinker system.

23.7.3.2. Fishing net with diamond mesh For the same RDH, a net cylinder with a diamond mesh has the same volume as a net with a square mesh when not exposed to the current and not attached to the weight system, as shown in Fig. 23.31. A net cylinder with a diamond mesh exhibits signiﬁcant initial deformation when attached to a weight system but not exposed to the current. Figure 23.32 shows the initial net deformation for the sinker system and the bottom-collar sinker system. With a decrease in RDH, the initial deformation increases. When RDH decreases to 1.0, the initial volume reduction is approximately 50% for weight system WA and 20% for weight system WB, so an RDH of 1.0 is not recommended for use in practice (simulated in this section). For a net cylinder with a diamond mesh, the volume-holding coeﬃcient (Cvh ) is calculated on the basis of the simulation results, but here the symbol Vc0 in Eq. (23.47) is deﬁned as the volume of the net cylinder that is not exposed to the current and not attached to the weight system. It can be concluded from the numerical results that the eﬀect of RDH on net deformation is important with either type of weight system (Fig. 23.33). An increase in RDH is helpful for decreasing net deformation. In comparison with the bottomcollar sinker system, the eﬀect of RDH on net deformation is more signiﬁcant for the cage with the sinker system.

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Volume holding coefficient [-]

1.00

Volume holding coefficient [-]

0.80 0.75 0.70 0.65 0.60 0.55

RDH=2.0 RDH=1.43

0.50 0.45

0.95 0.90 0.85 0.80 0.75 0.70 0.65

RDH=2.0 RDH=1.43

0.60 0.55 0.50 0.45

0.40 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.10

0.15

Velocity [m/s]

0.20

0.25

0.30

0.35

0.40

Velocity [m/s]

(a)

(b)

Fig. 23.33. Volume-holding coeﬃcient under (a) the sinker system and (b) the bottom-collar sinker system at diﬀerent RDH as a function of current velocity.

Increase in the volume holding coefficient [-]

0.26 0.24

RDH=2.0 RDH=1.43

0.22 0.20 0.18 0.16 0.14 0.10

0.15

0.20

0.25

0.30

0.35

0.40

Velocity [m/s] Fig. 23.34. Increase in the volume-holding coeﬃcient with the bottom-collar system at diﬀerent RDH relative to the sinker system.

As shown in Fig. 23.34, the bottom-collar sinker system exhibits lower deformation than the sinker system with a lower RDH. For a gravity cage with a diamond mesh, the structure of the bottom collar is critical when the RDH is low.

23.7.4. Eﬀect of mesh type on net deformation Diamond and square are two mesh types that are commonly used in ﬁshing nets. To investigate the eﬀect of mesh type on netting shape, net deformations of the two mesh types are compared in this section. Figures 23.35 and 23.36 show comparison

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Fig. 23.35.

Comparison image of net deformation with the sinker system (V = 0.37 m/s).

Fig. 23.36. Comparison image of net deformation with the bottom-collar sinker system (V = 0.37 m/s).

0.95

Volume holding coefficient [-]

Volume holding coefficient [-]

1.00

0.90 0.85 0.80 0.75 0.70

Square mesh Diamond mesh

0.65 0.60 0.55 0.10

0.15

0.20

0.25

0.30

0.35

0.40

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30

Square mesh Diamond mesh

0.10

0.15

0.20

0.25

Velocity [m/s]

Velocity [m/s]

(a)

(b)

0.30

0.35

0.4

Fig. 23.37. Comparison of the volume-holding coeﬃcient with the sinker-weight system. (a) RDH = 2.0 and (b) RDH = 1.43.

images of net deformation for the sinker and bottom-collar sinker systems with RDH = 2.0, in which the dashed lines denote net deformation for the square mesh. More quantitative comparisons are shown in Figs. 23.37 and 23.38. According to Fig. 23.37, when the sinker system (WA) is applied, net deformation is greater for a diamond mesh than for a square mesh, but the discrepancy decreases with increasing current velocity. This discrepancy may be induced by the initial net deformation. Even if the net is not exposed to the current, net deformation for the diamond mesh is signiﬁcant, whereas deformation for the square mesh is small. According to Fig. 23.38, when the bottom-collar sinker system (WB) is applied, net deformation for the diamond mesh is less than that for the square mesh with increasing current velocity. This is the reason why rotation of the bottom of the net is reversed for the two diﬀerent mesh types, and with increasing current velocity

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Volume holding coefficient [-]

Volume holding coefficient [-]

1.00 0.95 0.90 0.85 0.80 0.75 0.70

Square mesh Diamond mesh

0.65 0.60 0.55 0.10

0.15

0.20

0.25

0.30

0.35

0.40

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30

665

Square mesh Diamond mesh

0.10

0.15

0.20

0.25

Velocity [m/s]

Velocity [m/s]

(a)

(b)

0.30

0.35

0.4

Fig. 23.38. Comparison of the volume-holding coeﬃcient with the bottom-collar sinker system. (a) RDH = 2.0 and (b) RDH = 1.43.

the orientation of the bottom of the diamond-mesh net is more helpful in reducing net deformation.

References 1. M. Tauti, A relation between experiments on model and on full scale of ﬁshing net, Bull. Jpn. Soc. Scientiﬁc Fisheries 3(4), 171–177 (1934). 2. M. Tauti, The force acting on the plane net in motion through the water, Bull. Jpn. Soc. Scientiﬁc Fisheries 3(1), 1–4 (1934). 3. Y. C. Li, F. K. Gui, H. H. Zhang and C. T. Guan, Simulation criterica of ﬁshing net in aquaculture sea cage experiments, J. Fishery Sciences of China 12(2), 179–187 (2005) (in Chinese). 4. F. K. Gui, Y. C. Li and H. H. Zhang, The proportional criteria for model testing of force acting on ﬁshing cage net, China Oﬀshore Platform 17(5), 22–25 (2002) (in Chinese). 5. F. K. Gui, Hydrodynamic behaviors of deep-water gravity cage, Doctoral dissertation, Dalian University of Technology (2006) (in Chinese). 6. F. K. Gui, Y. C. Li, G. H. Dong and C. T. Guan, Application of CCD image scanning to sea-cage motion response analysis, Aquacultural Eng. 35, 179–190 (2006). 7. Y. C. Li, Y. P. Zhao, F. K. Gui and B. Teng, Numerical simulation of the hydrodynamic behavior of submerged plane nets in current, Ocean Eng. 33(17–18), 2352–2368 (2006). 8. Y. P. Zhao, Y. C. Li, G. H. Dong, F. K. Gui and B. Teng, Numerical simulation of the eﬀects of structures ratio and mesh style on the 3D net deformation of gravity cage in current, Aquacultural Eng. 36(3), 285–301 (2007). 9. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, Numerical and experimental study of submerged ﬂexible plane nets in waves, Aquacultural Eng. 38, 16–25 (2008). 10. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, Numerical simulation of the hydrodynamic behaviour of gravity cage in waves, China Ocean Eng. 21(2), 225–238 (2007).

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11. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, A numerical study on hydrodynamic properties of gravity cage in combined wave-current ﬂow, Ocean Eng. 34, 2350–2363 (2007). 12. Y. P. Zhao, Numerical investigation on hydrodynamic behavior of deep-water gravity cage, Doctoral dissertation, Dalian University of Technology (2007) (in Chinese). 13. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, Wave theory selection in the simulation of gravity cage, Proc. 17th (2007) Int. Oﬀshore and Polar Engineering Conf., Lisbon, Portugal, 1–6 July 2007, pp. 2222–2228. 14. Y. C. Li, F. K. Gui and B. Teng, Hydrodynamic behavior of a straight ﬂoating pipe under wave conditons, Ocean Eng. 34, 552–559 (2006). 15. Y. N. Zheng, G. H. Dong, F. K. Gui and Y. C. Li, Movement response of ﬂoating circle collars of gravity cages subjected to waves, Eng. Mech. 23(Sup I), 222–228 (2006) (in Chinese). 16. C. A. Brebbia and S. Walker, Dynamic Analysis of Oﬀshore Stuctures (NewnesButterworths, 1979), pp. 109–143. 17. R. B. Bhatt and R. V. Dukkipati, Advanced Dynamics (Alpha Science International Ltd., UK, 2001), pp. 213–219. 18. J. Wittenburg, Dynamics of Systems of Rigid Bodied (B.G. Teubner, Stuttgart, 1977). 19. Y. C. Li and B. Teng, Wave Action on the Maritime Structures, 2nd edn. (The Ocean Press, Beijing, 2002), pp. 250–265 (in Chinese). 20. E. H. Hou and Q. L. Gao, Theory and Design of Fishing Gear (The Ocean Press, Beijing, 1998), pp. 41–47 (in Chinese). 21. A. Fredheim and O. M. Faltinsen, Hydroelastic anslysis of a ﬁshing net in steady inﬂow conditions, in 3rd Int. Conf. Hydroelasticity in Marine Technology, Oxford, Great Britain, University of Oxford (2003). 22. B. W. Wilson, Elastic characteristics of moorings, ASCE J. Waterways and Harbors Division 93(WW4), 27–56 (1967). 23. Y. I. Choo and M. J. Casarella, Hydrodynamic resistance of towed cables, J. Hydronautics 5(4), 126–131 (1971). 24. Y. C. Li, F. K. Gui and F. Song, Comparison on the mooring line force and cage movement characteristics of gravity and sea staion cages, in Proc. 15th Int. Oﬀshore and Polar Engineering Conf., Seoul, Korea (2005), pp. 187–193. 25. F. K. Gui, Y. C. Li, Y. P. Zhao and G. H. Dong, A model for the calculation of velocity reduction behind a ﬁshing net, China Ocean Eng. 20(4), 615–622 (2006). 26. D. W. Fredriksson, M. R. Swift, J. D. Irish, I. Tsukrov and B. Celikkol, Fish cage and mooring system dynamics using physical numerical models with ﬁeld measurements, Aquaculture Eng. 27(2), 217–270 (2003). 27. D. Fredriksson, Open Ocean Fish Cage and Mooring System Dynamics (UMI, USA, 2001). 28. Y. C. Li, Y. C. Mao and F. K. Gui, The inﬂuence of sinker forms and weight on mooring line force of graviy sea-cage, China Oﬀshore Platform 1(1), 6–16 (2006) (in Chinese). 29. Y. C. Li, Y. P. Zhao, F. K. Gui and B. Teng, Numerical simulation of the inﬂuences of sinker weight on the deformation and load of net of gravity sea cage in uniform ﬂow, Acta Oceanologica Sinica 25(3), 125–137 (2006).

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Chapter 24

State of Oﬀshore Structure Development and Design Challenges Subrata Chakrabarti Civil and Mechanical Engineering Department University of Illinois at Chicago, 842 West Taylor Chicago, IL 60607, USA [email protected] This chapter will review and highlight the research being carried out today to meet the challenges in the design, and operation of oﬀshore structures. The subject matter, while general in nature, will focus on one of the most unique areas in the oﬀshore structure design, namely, the ﬂuid-induced responses of oﬀshore structures and the associated structural design consequences. Due to the rapid growth in the oﬀshore ﬁeld, particularly in deepwaters, this area is seeing a phenomenological advancement. The chapter will begin with an overview of the historical development of ﬁxed and ﬂoating structures. It will state the design status for these systems. The ﬁxed structure design is more mature today, even though many aspects of it still remain empirical. For ﬂoating structures the design procedure is still advancing and more research is ongoing in this area. These will be highlighted. The treatment of the individual components of the ﬂoating structure, namely, the ﬂoater, the mooring system, and the riser system including their interactive coupling eﬀects with ﬂuid will be discussed. The state-of-the-art in the treatment of the individual components of the ﬂoating structure, namely, the ﬂoater, the mooring system, and the riser system will be brieﬂy described. The design methods for these oﬀshore components will be included. The basic differences between the coupled and uncoupled systems and the complexity of the later method will be discussed. The chapter will conclude with a discussion of the present-day deepwater design challenges that remain and the research that is needed to meet these challenges.

24.1. Introduction of Oﬀshore Structures Oﬀshore structures are located isolated in waters of the ocean with no continuous access to dry land. Their design life-span ranges from a few years to as many as 25 years. In most cases, they may be required to stay in position in all weather 667

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conditions. Oﬀshore structures are deﬁned by either their function or their conﬁguration. The functions of an oﬀshore structure may be one of the following: • Exploratory Drilling Structures: A Mobile Oﬀshore Drilling Unit [MODU] conﬁguration is largely determined by the variable deck payload and transit speed requirements. • Production Structures: A production unit can have several functions, e.g., processing, drilling, workover, accommodation, oil storage, and riser support. • Storage Structures: These structures are used in storing the produced crude oil temporarily at the oﬀshore site before its transportation to the shore for processing. Sometimes a structure may be used for multiple functions. The conﬁguration of oﬀshore structures may be classiﬁed by whether the structure is a ﬁxed structure, either piled or gravity, a compliant or articulated structure, or a ﬂoating structure. The requirements of a ﬂoating structure are that it be moored in place and that the facility remains under the environment within a speciﬁed distance from a desired location achieved either by mooring lines, or by dynamic positioning system using thrusters, or a combination of the two. First, a short description of these structures and their applications will be discussed. Then the current state-of-the-art in the general hydrodynamic analysis associated with the design of these structures, and the future problems that need to be considered in these areas will be addressed. 24.2. Fixed Structures Fixed oﬀshore structures composed of small tubular members are mainly used for the production of oil and gas. These structures may be composed entirely of

Fig. 24.1.

Historical development of ﬁxed jacket structures (Courtesy Shell Oil Co.).

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small battered members which are piled in place. These are called jacket structures. The evolution of ﬁxed jacket structures with water depth is shown in Fig. 24.1. Note that, within a short span of 40 years, the ﬁxed structure extended from a water depth of 6 m to over 400 m. Exploratory drilling operations may be performed with ﬁxed structures, called jack-ups, which are mobile. The jack-ups are usually buoyant during transit and are towed from station to station. Once they reach their destination at the drilling site, the small-membered legs of the jack-up are set on the ocean bottom and the deck is jacked up above the water level on these legs for the drilling operation. The gravity-type ﬁxed structures, on the other hand, are mostly composed of large steel shell-type members, which may also be used for storage in addition to production of oil and gas. There have also been many concrete gravity structures in existence, most of which are located in the North Sea. 24.2.1. Design of fixed jacket structures The ﬁrst step in the design of ﬁxed structures is the computation of the forces on its members due to the extreme environment present at the site. Forces on the submerged members of the structure arise from currents, and waves. The fatigue life of the members is also of concern in the design of these structures. In fact, several structures have failed due to the fatigue failure of critical members in the structure. It is not uncommon to repair or replace the underwater members of a ﬁxed jacket structure during its lifetime. Today’s design of ﬁxed oﬀshore structures with small tubular members is still based on empirical formulae. The forces on the member are the inline (direction of current and wave) and lift or transverse force. For example, for a streamline shape of an airfoil (a) the ﬂow remains attached to the body giving rise to an inline mean drag force, but very little ﬂow separation and lift (or transverse) force [Fig. 24.2(a)]. On the other hand, a bluﬀ body, e.g., a circular cylinder, generates a large separation of ﬂow in current behind the cylinder forming vortices, which remain attached to the body [Fig. 24.2(b)] at low Reynolds number (Re) and separate from it and move away with the ﬂow at higher Re. These vortices are generally alternating in nature, at least at moderate Re. Because of this ﬂow separation

U

(a)

(b)

Fig. 24.2. Flow eﬀects on body shapes. (a) Steady ﬂow past airfoil and (b) steady ﬂow past bluﬀ body.

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Fig. 24.3.

Lift and drag coeﬃcients for Re = 200.3

and the presence of vortices, the pressure behind the body in steady ﬂow is low compared to the forward of the body ﬂuctuating asymmetrically. These forces are diﬃcult to compute numerically specially at high Re and research in this area is continuing. The ﬂow past the body, as shown in Fig. 24.2(b) for a cylinder in steady ﬂow, produces an asymmetric (fore-aft) pressure distribution around the cylinder, which traverses the ﬂow axis at a given (Strouhal) frequency at a low or intermediate Re. This pressure distribution, in turn, generates an oscillating lift (or transverse) force with near-zero mean, in addition to an inline mean drag force, and an oscillating inline force of smaller magnitude over the steady force. Since the vortex shedding behind the cylinder is alternating in nature, the transverse force will be periodic and its frequency (at low Re) will depend on the frequency of vortex shedding. This is demonstrated in Fig. 24.3 computed in a 2D numerical simulation. The result presented applies to a Reynolds number of 200. At higher Reynolds numbers, this shedding of vortices is more random and complex causing a more irregular large transverse force on the body, which, at the present time, causes signiﬁcant computational diﬃculty. 24.2.1.1. Empirical formulae As shown in Fig. 24.3, the mean inline load is large and the transverse force has a zero mean. The ﬂuctuating current load in the inline direction is generally ignored as being small in a ﬁxed-structure design. In the empirical form, the mean inline current load per unit cylinder length, known as the drag force, is given by f=

1 ρCD AU 2 2

(24.1)

and the transverse current load per unit cylinder length, sometimes called the lift force, is written as fL =

1 ρCL AU 2 , 2

(24.2)

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where ρ = ﬂuid density; A = projected cross-sectional area; U = steady ﬂow velocity; D = cylinder diameter; CD = drag coeﬃcient; and CL = lift coeﬃcient. The Reynolds number: Re = UD/ν, and ν = kinematic viscosity of ﬂuid. The values of the coeﬃcients needed to compute these forces have been established through model tests. The drag coeﬃcients for a smooth cylinder have been found to be functions of Re only as shown in Fig. 24.4. If the member surface is considered rough, then the roughness parameter of the member is considered in the choice of the coeﬃcients (generally resulting in an increase in the values of the coefﬁcient depending on the extent of surface roughness). The ﬂuctuating nature of the lift force makes the values of CL in Eq. (24.2) a function of time and it is a common practice to represent CL by its maximum (or rms) value. Experimental data on CL versus Re have shown considerable scatter in the value of CL . Figure 24.5 shows their upper and lower range. Figures 24.4 and 24.5 (with appropriate roughness correction) are normally used for computing the forces on a small member of an oﬀshore structure. Since waves are oscillatory, the inline wave force on a cylinder will also be oscillatory and will depend on both the water particle velocity and the water particle

Fig. 24.4.

Fig. 24.5.

Drag coeﬃcient for a smooth circular cylinder in steady ﬂow.

Lift coeﬃcient for a smooth circular cylinder in steady ﬂow.

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acceleration. No analytical or numerical solution exists for the forces which (per unit length) are computed from the well-accepted empirical formula, commonly known as the Morison equation,29 given as f = CM AI u˙ + CD AD |u|u,

(24.3)

where AI = ρ(π/4)D2 ; AD = 0.5ρD and u = wave particle velocity; u˙ = wave particle acceleration; CM = wave inertia coeﬃcient; and CD = wave drag coeﬃcient. Using linear wave theory the horizontal particle velocity and acceleration are obtained from gkH 2ω gkH u˙ = 2 u=

cosh ks cos(kx − ωt) cosh kd cosh ks sin(kx − ωt), cosh kd

(24.4) (24.5)

where g = gravitational acceleration; k = wave number; ω = wave frequency; H = wave height; s = elevation from the ocean ﬂoor to the mean water level (s = 0 to d); and d = water depth. Total force on a structure member requires integration over its length. These formulae using appropriate hydrodynamic coeﬃcients (see Sarpkaya32 for experimental values) may be applied in the design of the members of ﬁxed structure. The API, DNV, and other similar design guidelines provide suitable data to use for cylindrical structures. For example, the API RP 2A Section C3 Recommended values are given below: Coeﬃcient values

Smooth cylinders

Rough cylinders

CM CD

1.05 0.65

1.2 1.6

When current is present in the direction of wave (negative sign indicates opposing current), the Morison equation is modiﬁed in the presence of current in terms of a relative velocity model: f = CM AI u˙ + CD AD |u ± U |(u ± U ).

(24.6)

In this case the values of the hydrodynamic coeﬃcients will be diﬀerent than those for the wave alone. Similarly for inclined cylinders, e.g., those found in the cylindrical bracings of a jacket structure, the modiﬁed Morison equation considers force normal to the cylindrical member based on the normal component of velocity and acceleration of the water particle at the point. 24.2.1.2. Blockage factor in steady ﬂow Often the structural members (e.g., vertical production riser bundle) appear in close proximity in an array or a group as shown in Fig. 24.6. In these cases the ﬂow is

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S i D

Fig. 24.6.

Group of cylinders in a matrix array.

blocked by the forward cylinders. API Guidelines2 give a composite blockage factor for a dense cylinder group in steady current as follows: CBF = 1 +

N

i=1 (CD A)i 4A

−1 (24.7)

in which CBF = composite current blockage factor, N = no. of cylinders (risers) in the bundle, A = individual cylinder projected area, and A = overall structure area normal to the ﬂow. 24.2.2. Design of large fixed structures Unlike small-membered oﬀshore structures, wave forces on large structures may be computed by an elegant numerical method on the assumption that the ﬂow past the structure remains essentially potential and the irrotationality assumption for the ﬂow is valid. 24.2.2.1. Linear diﬀraction/radiation forces The general analytical approach based on linear theory (commonly called the linear diﬀraction/radiation theory) includes the diﬀraction and radiation eﬀect from the submerged portion of a structure due to linear progressive waves. Several numerical methods may be used in solving the problem mathematically e.g., panel method, ﬂuid ﬁnite element method, hybrid region method, etc. The most common method is the boundary element panel method. It makes use of Green’s mathematical function and Green’s theorem. The spatial part of the total wave potential is written as φ = φo + φs +

6

φRn

(24.8)

n=1

in which φo = incident potential, φs = scattered potential, and φRn = radiated potential due to forced oscillation of unit amplitude in the nth mode. The spatial

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Fig. 24.7.

Illustrative sketch of the linear force calculation on a vertical cylinder.

incident potential at a point (x, y, z) is obtained from gH cosh ks exp(ikx) (24.9) 2ω cosh kd in which i = imaginary quantity. In the numerical computation, the submerged surface of the structure is discretized into small ﬂat panels (Fig. 24.7). The scattered part of the velocity potential in the ﬂuid due to the continuous source distribution on the structure surface is given as 1 φs (x, y, z) = σ(a, b, c)G(x, y, z; a, b, c)ds (24.10) 4π S φo (x, y, z) = i

in which σ represents the source strength, (a, b, c) represents the source point on the surface of the structure and (x, y, z) the ﬁeld point in the ﬂuid, ds is the ﬂat area of the panel on the submerged surface. The function G represents the near-ﬁeld Green’s function given in a series or an integral form. The source strength function σ, is computed from ∂G (x, y, z; a, b, c)ds = −4πun (x, y, z), σs (a, b, c) (24.11) 2πσs (x, y, z) − ∂n S where un = known normal ﬂuid velocity at (x, y, z) due to the incident wave. Equation (24.11) is solved numerically by assuming the ﬁeld point to be on the structure surface and setting it up in a matrix form in terms of the centers of the panels. An N × N complex matrix is formed to describe Eq. (24.11), where N is the total number of panels. The solution for the source strengths σ at the center of each panel is obtained by the inversion of the complex matrix. For a large value of N , this computation is time consuming. For the radiated potential the right-hand side of Eq. (24.11) is replaced by the normal displacement of the body at one of six degrees of freedom. Since the only diﬀerence is the right-hand side, this equation may be solved at the same time once the inverted matrix is obtained. The radiated potential provides the added mass and damping coeﬃcients of the structure in six degrees of freedom. Once the diﬀraction/radiation potentials are known at the center of each panel, the external forces on the submerged body due to the total diﬀraction and radiation potential, are obtained respectively from the integrals FkD = −iρω (φo + φs )nk ds (24.12) S

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and FkR = −iρω

S

φRk nk ds,

(24.13)

where k = 1, 2, . . . , 6 and S = submerged surface area up to SWL (Fig. 24.7). Thus, the exciting wave forces (and moments) by the linear wave theory are obtained from Eq. (24.12), while the hydrodynamic added mass and radiation damping are derived from Eq. (24.13) (see, Chakrabarti8 for details). 24.2.2.2. Time domain fully nonlinear forces Numerous articles are available on nonlinear wave-body interaction with oﬀshore structures.5,23,34 Many of these not only considers the nonlinear forces on the ﬂoating structure, but the response of the structure as well. The consistent nonlinear numerical solutions are quite elaborate and extremely time-consuming. The fully nonlinear wave-structure interaction boundary-value problem may be solved by the mixed Eulerian–Lagragian (MEL) method4,23 without any analytical approximations. This method of solution requires prohibitively large computational eﬀorts and is not yet practical for routine industry use. Moreover, several technical issues are yet to be satisfactorily resolved before this approach can be successfully applied for complex 3D oﬀshore structures.33 To address the need of the industry, several timedomain solution methods have been proposed which minimize this excessive use of computational eﬀorts, while accounting for the so-called essential nonlinearities by some approximate means.11,14,33 In most cases, the hydrodynamic interaction due to radiation and diﬀraction eﬀects is linearized. This allows the use of the usual 2D or 3D linear diﬀraction/radiation theory. In this chapter, a 2D nonlinear wave-structure interaction problem is formulated using a potential-based fully nonlinear Numerical Wave Tank (NWT). Figure 24.8 shows the deﬁnition sketch for a vertical cylinder assumed frozen at an instant in wave. The theory is based (closely following the work of Kim and Koo24 ) on modedecomposition method, and MEL material-node time marching scheme,28 and uses the boundary element method (BEM). As in the case of linear diﬀraction/radiation theory, an ideal ﬂuid is assumed so that the ﬂuid velocity can be described by the gradient of velocity potential

Fig. 24.8.

Illustrative sketch of the fully nonlinear force calculation on a vertical cylinder.

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Φ. A Cartesian coordinate system is chosen such that z is positive upwards corresponding to the still water level. Then the governing equation of the velocity potential is given by ∇2 Φ = 0.

(24.14)

The boundary conditions consist of • Fully nonlinear dynamic free surface condition, satisﬁed on the exact free surface (thick line in Fig. 24.8): ∂Φ 1 = −gη − − |∇φ|2 . ∂t 2

(24.15)

• Fully nonlinear kinematic free surface condition, satisﬁed on the exact free surface: ∂η ∂Φ = − ∇φ · ∇η. ∂t ∂z

(24.16)

• Body boundary condition on the structure: ∂Φ = 0. ∂n

(24.17)

• Input boundary condition: At the inﬂow boundary, a theoretical particle velocity proﬁle is fed along the vertical input boundary. The exact velocity proﬁle of a truly nonlinear wave under the given condition is not known a priori. Therefore, the best theoretical wave proﬁle is input along the input boundary. Since the fully nonlinear free-surface condition is applied in the computational domain, the input wave immediately takes the feature of fully nonlinear waves. Any unnecessary spurious waves inside the domain is accounted for and corrected. An integral equation in terms of two-dimensional Green function satisfying Laplace equation is adopted. To update the fully nonlinear kinematic and dynamic free-surface conditions at each time, Runge–Kutta fourth-order time-integration scheme7 and the MEL approach are adopted. Lagrangian approach for which the free-surface nodes move with water particle motion is used. At each time step, (i) the Laplace equation is solved in the Eulerian frame, and (ii) the moving boundary points and values are updated in Lagrangian manner. To avoid nonphysical sawtooth instability on the free surface in time marching, smoothing, and regriding schemes are used. In the case of fully Lagrangian approach, the free-surface nodes need to be updated and rearranged at every time step. The regriding scheme prevents the free-surface nodes from crossing or piling up on the free surface, and thus makes the integration scheme more stable. Near the end of the computational domain, an artiﬁcial damping zone is applied on the free surface so that the wave energy is gradually dissipated in the direction of wave propagation. The proﬁle and magnitude of the artiﬁcial damping is designed to minimize possible wave reﬂection at the entrance of the damping zone, while maximizing wave energy dissipation. The most physically plausible open boundary condition is Sommerfeld/Orlanski outgoing wave condition.30 The Orlanski radiation condition, for example, was used

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by Boo et al.5 for the simulation of nonlinear regular and irregular waves and by Isaacson and Cheung21 for wave–current–body interaction problems. There exist other open-boundary conditions, such as absorbing beaches by artiﬁcial damping on the free surface6,19,27 or matching with linear time-domain solutions at far ﬁeld.17 It is well known that a properly designed artiﬁcial damping on the free surface does not have to be far from the body and can damp out most of the wave energy, if its length is greater than two wave lengths. Therefore, it is more eﬀective than the matching technique and ideal to damp out relatively short waves.24 When the simulation starts, a ramp-function7 at the input boundary may be applied. The ramp function prevents the impulse-like behavior at the start and consequently reduces the corresponding unnecessary transient waves, which usually wastes computational time to die out. As a result, the simulation is more stable and soon reaches the steady state. Accurate calculation of the time derivative of velocity potential is very crucial in obtaining correct pressure and force on the body surface at each time step. There are several ways to obtain this velocity potential. Backward diﬀerence is the simplest way using the potential values of previous time steps. In case of a stationary structure, more accurate ﬁnite-diﬀerence formulae7 can also be used. The wave force on the body surface is calculated by integrating Bernoulli’s pressure over the instantaneous wetted surface from the nonlinear wave. While the above development is shown for a wave force computation, the method can be easily extended to include moving structures as well.24

24.3. Floating Oﬀshore Structures Floating oﬀshore system (a variety of which is illustrated in Fig. 24.9) consists of three principal structural components: • Floating hull: facilitating the space for the operation of the production work, and storage for supplies,

TLP

Sem i Fig. 24.9.

SPA R Floating oﬀshore structures.

FPSO

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• Mooring system: providing a connection between the structure and the seaﬂoor for the purposes of securing the structure (generally called station-keeping) against environmental loads, and • Riser system: achieving drilling operation or product transport. The station-keeping may also be achieved by dynamic positioning system solely using thrusters, or in combination with mooring lines. The mooring lines and risers provide restoring forces to the ﬂoater. 24.3.1. Frequency domain approach The motion responses of most of the large ﬂoating structures (Fig. 24.9) may be obtained in the frequency domain, which is considered satisfactory for design purposes. The ﬂoating oﬀshore structure is almost always taken as a rigid body in its response calculations. In the simplest analysis, the attachments to the body from the moorings and risers are treated as linear or nonlinear springs externally attached to the body. The computation of wave forces on the ﬂoating structure is carried out the same way as a large ﬁxed structure shown in the earlier section. The environmental forces are determined at the equilibrium position of the body for a linear analysis (Fig. 24.7). Moreover, the free motions of the body generate the linear hydrodynamic added mass and damping eﬀect. In particular, the radiated potential from the body produces a 6 × 6 force matrix. When nondimensionalized by the oscillation frequency (or frequency squared) and structure displacements, the radiated forces give rise to a 6 × 6 added mass coeﬃcient and a 6 × 6 damping coeﬃcient matrix. The motions are obtained by solving a coupled set of equations of motion. In a linear analysis, it is even possible to introduce a nonlinear damping in an approximate manner. If a Morison type nonlinear damping is included in a linearized form, the equations of motion become mk x ¨k +

6 8 Mlk x Dlk |x˙ l0 |x˙ l + Clk xl = fk ; ¨l + Nlk x˙ i + 3π

k = 1, 2, . . . , 6,

l=1

(24.18) where mk = mass or moment of inertia in the kth mode, xk = displacement in the kth direction, dots are time-derivatives, and subscript 0 denotes amplitude, while the variables Mlk , Nlk , Dlk , Clk , = added mass, linear and nonlinear damping and restoring force coeﬃcients, respectively due to l degree of freedom in the kth direction. The factor 8/(3π) in the nonlinear Morison damping term arises from the linearization. The restoring force includes the stiﬀness arising from the structure as well as the mooring lines. The right-hand side, fk , represents the six forces (moments) on the ﬂoater by the linear diﬀraction theory. The stiﬀness due to risers are generally ignored in this analysis, but can be easily accommodated in Eq. (24.18). The stiﬀness is assumed linear (or linearized) for a frequency-domain solution. The solutions for 6 DOF motion xm are obtained by the inversion of the 6 × 6 matrix on the left-hand side by assuming the motions to be harmonic.

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admass+1% damping

Heave, ft/ft

admass + 15%damp

6

no plate+1%damp

4 2 0 10

11

12

13

14 15 16 Period, sec

17

18

19

20

Fig. 24.10. Example heave response of a Truss Pontoon Semisubmersible including additional heave plate added mass and damping.

Note that Eq. (24.18) is still nonlinear in the motion amplitude due to the third term. A simple iterative technique is chosen to solve the above linearized equation. In the ﬁrst step, a linear harmonic solution for xl0 = 0 (l = 1, . . . , 6) on the lefthand side is obtained. This ﬁrst iteration value is introduced on the left-hand side of solution and the process repeated. Generally, between two and three iterations produce convergence in the results. An example of the computed heave motion from Eq. (24.18) of a ﬂoater (pictured on the left of Fig. 24.10) is given in Fig. 24.10. Diﬀerent responses refer to with and without additional added mass and nonlinear damping generated from the heave plates at the bottom of the ﬂoater. 24.3.2. Time domain approach While the above simple linear (or linearized) solution is a useful design tool for a variety of ﬂoating oﬀshore structures, it is limited by the linear restoring force, linear damping, and linear waves. Some of the nonlinear aspects of the motion analysis are well-established, including steady drift force, and second-order low frequency (slow drift) and high frequency (TLP tendon) loads. The second-order steady drift force is derived directly from the ﬁrst-order potential. Therefore, the steady drift force is computed by the pressure-area method within the linear diﬀraction/radiation program. The low or high frequency force calculations are much more complex in terms of a quadratic transfer function, which is extremely time-consuming. In order to reduce the computation eﬀort for routine application in a design, simpliﬁed assumptions are often applied using fewer frequency pairs around the resonance frequency. This type of approximation is a common design practice. In view of several nonlinearities in the oﬀshore system, a frequency domain solution is not always suﬃciently accurate and time domain solution is sought in these cases. In the time domain the analysis still assumes the ﬂoater to be a rigid

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body, and the solution is generated by a forward integration scheme. The analysis can easily accommodate the nonlinearities due to the excitation forces (e.g., wind, current, and waves), as well as damping and restoring forces typically encountered in a moored ﬂoating system. The waves can be single frequency or composed of multiple frequency components following a given wave energy spectral model (e.g., PM or JONSWAP). For the series representation of long crested waves for a given wave energy spectrum, the wave proﬁle η may be decomposed into N components as: η(x, t) =

N

an cos[kn x − ωn t + εn ],

(24.19)

n=1

where an represents the wave amplitude having frequency ωn and a randomly chosen phase angle εn . The corresponding linear and nonlinear wave excitation force, and structure velocity dependent radiation force for each of these wave components are computed from the linear diﬀraction/radiation theory at a given time step. At each time step, a set of second-order diﬀerential equations (similar to Eq. (24.18), but retaining all the nonlinearities present in the system) is solved to obtain the accelerations of the system. For a single degree of freedom system, the current value of acceleration is computed from the equation of motion for the total random force at time t, F (t): x ¨c = [F (t) − cx˙ p − kxp ]/m,

(24.20)

where the subscripts c and p stand for the current and previous values, respectively. The force time history F (t) may be composed of nonlinear wind and current forces including wind spectrum, the linear diﬀraction forces, second-order steady and oscillating forces, and Morison and lift forces. The solution is initiated with prescribed values for the displacements and velocities, and these values are calculated for the next time step from the derived accelerations by the forward integration scheme. e.g., ﬁnite-diﬀerence: xc + x ¨p ) ∗ dt x˙ c = x˙ p + 0.5 ∗ (¨

(24.21)

xc = xp + 0.5 ∗ (x˙ c + x˙ p ) ∗ dt.

(24.22)

The extension of this approach to include multiple ﬂoating structures, e.g., an FPSO and a shuttle tanker, interconnected by mooring lines is straightforward. For example, an example of a time history of responses for a single point moored tanker-tower system7 is shown in Fig. 24.11. The biggest diﬃculty in these computations of the vessel response is an accurate determination of the hydrodynamic damping, specially in slow drift oscillation of soft-moored vessels or high frequency load on the stiﬀ vertically moored vessels, e.g., TLP. For example, the typical percent damping factor of a TLP in heave is found from model test to be about 0.05 for both round and square vertical columns, while the same for a horizontal pontoon are 0.176 and 0.278.20 The evidence in the literature of appropriate and accurate values of damping in real structures is generally rare.

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20 10

Wave profile

0 400 -10

500

600

700

WAVE -20

2

Tower Motion

1

0 400

500

600

700

OMEGA

-1

20 15

Tanker Surge

10 SURGE

5 400

500

600

700

4.E+04 LINE

Line Tension

2.E+04

0.E+00 400

Fig. 24.11.

500

600

700

Time history of motions and load due to a random wave on an SPM.

24.4. Mooring System During the earlier ﬂoating structure response analysis, the mooring lines are succinctly assumed to be merely a nonlinear spring and the eﬀects of the environment on the lines themselves are ignored having small overall eﬀect on the ﬂoater response. In the design of these lines themselves for a ﬂoating system, however, the environmental forces may be an important criterion. In the process of design of an oﬀshore system, one ﬁrst selects their layout, geometry, and the initial mechanical and structural properties. Diﬀerent types of mooring and anchoring system for ﬂoating oﬀshore structures of today are shown in Fig. 24.12. The lines (typically 8 to 12) are arranged in a symmetric arrangement except to make room for risers if needed. There are principally two types of mooring system in use today — catenary mooring and taut mooring system. The catenary lines often consists of chain-wire-chain combination. The taut lines could be steel wires or synthetic polyester. It is customary to use drag anchors for the catenary lines. For the taut polyester lines, a vertically loaded plate or suction anchor may be suitable.

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Fig. 24.12.

Schematic layout of mooring and anchoring system.

For the taut mooring system the restoring force comes primarily from the stretch in the line. The restoring force for the conventional catenary system results principally from the submerged weight and changes in the catenary shape.

24.4.1. Design of mooring lines The design of a mooring line is performed in several ways depending on the sophistication desired in the analysis. Quite often the design is static in which the loadelongation characteristics for a single line and a mooring spread are established from the horizontal excursion of the line, ignoring ﬂuid forces from the environment on the line itself. For a composite catenary line, the customary catenary equation is used. For the taut lines, the stiﬀness under loads appears from the stretch in the lines from their elastic behavior. The static mooring design analysis is accomplished by the following steps. The load-elongation characteristics for each line are ﬁrst computed, given the line end-point coordinates, line length, and submerged weight. The forces for all lines in the mooring spread are then summed based on their orientation, which yield the resultant horizontal and vertical restoring forces as a function of the displacement of the vessel. The overall mean oﬀset of the vessel caused by the steady loads from wind, current, and wave drift is then estimated. Based on this steady load, the tension in the most loaded line is computed by displacing the vessel through this oﬀset. If line length is insuﬃcient, calculations are repeated with increased length. The safety factor for the most heavily loaded line is computed based on the breaking strength of the line and compared with the allowable (generally 2, Ref. 1). If it is too low, the design parameters, e.g., line pre-tension, material speciﬁcations, their end co-ordinates or number of lines, are adjusted and the calculations are repeated. The safety margin is checked again allowing one (generally the most loaded) broken line. In a quasi-static design, the line dynamics are still ignored, but the dynamic loads on the ﬂoater are included in the analysis. Thus, in addition to vessel oﬀset from the mean wind, current, and wave drift forces on the vessel, the maximum excursion

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from the wave-induced vessel motion at wave and slow-drift frequency are computed. Then the maximum line tensions resulting from the total vessel oﬀset for each possible environmental direction are determined. The line tensions are compared with the minimum breaking load given the safety factor. DNV16 currently suggests separate safety factors for the mean and dynamic loads to avoid excessive conservatism, since mean loads are much higher and more reliably predictable. As before, the maximum peak line loads with 1 line broken are recalculated. If the proposed mooring speciﬁcation fails the safety factor test, then a new speciﬁcation is tried. The dynamic design of the mooring lines includes additional loads on the mooring lines themselves. In this case the analysis is much more involved and a numerical method is needed. The dynamic mooring system design is a two-step procedure as follows. First, the motions of the ﬂoater independent of the line dynamics are calculated to determine the top-end oscillation of lines. Loads on the ﬂoater are the steady current, and steady and ﬂuctuating wind, wave and wave drift. For the ﬂoating structure, mooring system is an external nonlinear stiﬀness term (step 1, Fig. 24.13) as shown earlier. Once the ﬂoater dynamics are known, the motions of the mooring line attachment (fairlead) points are determined. For the mooring lines, motions of the ﬂoating structure are included at the attachment point as an externally deﬁned oscillation (step 2). To determine the mooring line dynamics, a time domain analysis is required here. External eﬀects on the mooring lines include the following: topend motion, current load, wave load, seabed lateral friction, and soil spring and damping eﬀect on the portion of the line on bottom. A lumped mass, ﬁnite element or ﬁnite diﬀerence scheme may be used to model the lines. The line is decomposed into a number of straight elements (bars) with linear shape function. The distributed mass plus hydrodynamic added mass is lumped at the element nodes. The hydrodynamic damping is included for the relative motion between the line and the ﬂuid. Damping levels vary signiﬁcantly depending on water depth, line makeup, oﬀsets, and top-end excitation. Quite often a modiﬁed Morison equation is used to represent the environmental eﬀect. At each time step, a standard set of matrix equation is developed composed of the inertia, damping, and stiﬀness matrix.

Fig. 24.13.

Schematic of two-step uncoupled analysis method.31

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It is often important to check the fatigue life of the mooring line. A mooring line fatigue analysis may be based on long-term cycles of the dynamic tension on the line due to time varying loads. In fatigue, the factor of safety (FOS) is recommended1 to be between 3 and 10 depending on the lines being inspectable or not. Because of the low level of experience of lighter polyester lines in deepwater applicable, the FOS is taken to be as high as 60.

24.5. Riser System As noted earlier, the riser is an important component of an oﬀshore structure. For a ﬂoating structure there are a variety of risers in use today, which are schematically identiﬁed in Fig. 24.14. The vertical risers are still the most common and always used for the drilling operation. These risers for drilling and production operation are pretensioned (Top Tension Risers, TTR). The production risers may have a catenary shape (Steel Catenary Riser, SCR) or may be ﬂexible as well. Flexible risers may have additional buoyancy elements giving it an S-shape. 24.5.1. Dynamics of risers In order to implement more advanced design of risers, the current research and development on riser technology follows the following paths: numerical analysis, laboratory testing, and in situ testing. The numerical simulation includes coupled structural ﬁnite element and computational ﬂuid dynamics (FE/CFD) analysis. In addition, several commercially available large CFD programs are currently updated for oﬀshore applications. Limitations of this analysis today are large computational time, limitations in ﬂow solver, including convergence and high Reynolds number. They also need systematic validity with reliable benchmark tests. The small-scale

Flexible w/buoyancy

Free standing

Fig. 24.14.

Float Tensioned

SCR

Types of risers in use today.

TTR

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testing is generally done at the University laboratory water channels, and wave tanks. They are limited by small scales, modeling problems, and measurements. Their main purpose is to increase the understanding of the ﬂow problems past risers and generating empirical coeﬃcients. Field measurements often include insitu testing in full scale and face the constraints of cost, practicality, environmental limitation, and client conﬁdentiality. In spite of these ongoing developments, today almost all riser design analyses are performed by empirical methods. Static analysis is generally based on steady current loads (uniform or shear) and provides the riser deﬂected shape, stresses at various points on the riser, top and bottom angles, and its structural mode shape under load. Dynamic analysis considers both the inline and transverse unsteady loads. In addition to the riser shapes and stresses, it provides the vortex induced vibration, and fatigue life of the riser. 24.5.1.1. Mathematical model In this section, a basic mathematical model for the riser analysis is developed. It should, however, be recognized that there are many approaches in the dynamic analysis of risers and their further details are outside the scope of this chapter. Therefore, the development below should be taken as an exercise to introduce the basic parameters and governing equations for the riser dynamics. 24.5.1.1.1. Governing equations Consider a vertical top-tensioned riser exposed to ocean waves and currents, as illustrated in Fig. 24.15. The waves and current are shown as collinear. The deﬁnition of the co-ordinate system is shown in the ﬁgure. Considering only the current loads, the motion of the risers is governed by the following equations. For inline static analysis, the right-hand side is the current force: d2 d2 x d 1 dx EI(y) − F + m(y)¨ x + c(y)x˙ = ρCD (y)D(y)U 2 (y). (y) e 2 2 dy dy dy dy 2 (24.23) For transverse riser analysis, the right-hand side is the transverse (or lift) force: d2 z d 1 dz d2 EI(y) 2 − Fe (y) + m(y)¨ z + c(y)z˙ = ρCL (y)D(y)U 2 (y), 2 dy dy dy dy 2 (24.24) where x = direction of current, y = vertical direction, z = transverse direction, m = total mass per unit length of riser section, c = damping coeﬃcient of riser section, ρ = mass density of water, Fe (y) = eﬀective tension due to axial tension and pressure force, U (y) = current velocity as a function of the vertical coordinate y, CD (y) = drag coeﬃcient for the riser, and CL (y) = maximum lift coeﬃcient for the riser. Note that the mass m includes the hydrodynamic added mass. It is assumed that the riser bottom end is connected to a frictionless ball joint. The upper end is constrained to the ﬂoater through the top tensioner, but free to move with the ﬂoater both in the horizontal and vertical directions.

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Fig. 24.15.

Schematic of vertical top-tension riser in current.

For a mathematical solution of the system governed by the above equations, suitable initial conditions are also needed. For example, the initial conditions may be given by the static solutions with zero initial velocities.9 The standard modal superposition method may be used to solve the problem. For the dynamic riser solution, the right-hand sides of Eqs. (24.23) and (24.24) are replaced by the forces due to wave. In this case the modiﬁed Morison equation may be used for the inline direction [Eq. (24.23)] and the appropriate form of the lift force is used in the transverse direction [Eq. (24.24)]. The static and dynamic loads may also be combined in a single analysis, even though it is rarely found in the literature. There are several empirical codes to analyze the VIV problems based on 2D measurements of cylinder models most of which analyze cross-ﬂow response only. A few of these application tools are SHEAR7 (MIT), VIVA (MIT), VIVANA (MARINTEK/ NTNU), VICoMo (NTNU), and ABAVIV (Technip). Some of them also perform the in-line static deﬂection using ampliﬁed CD values from the crossﬂow oscillation. 24.5.2. CFD numerical model While the empirical method of riser design is the current accepted method, considerable eﬀorts are being spent in the more pleasant numerical approach. In this case,

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a CFD analysis incorporating the ﬂuid and the structure is desirable. Due to high Reynolds number the problem is quite involved and time-consuming. However, the recent results look promising.12 The numerical analysis applies the turbulent incompressible Navier–Stokes equations in describing the conservation of mass and momentum in three-dimensions. Among the numerical methods, most of the CFD codes are 2D on parallel planes. They include ﬁnite element, ﬁnite diﬀerence, discrete vortex, vortex-in-cell, and vortex tracking method. A few examples are FSI-Navsim (Norsk Hydro), USP, DeepFlow (IFP), VIVIC (Imperial College), TACC, Orcina VT, and Orcina WO of OrcaFlex. There are also a few large 3D Commercial Codes being adopted for oﬀshore applications, e.g., Fluent and Acusolve. A fundamental problem in the simulation of long marine risers is the large scale of the computational ﬂuid dynamics problem itself.12 The risers may have lengths of thousands of meters so that using three-dimensional CFD solutions seems impossible. A practical approach to riser VIV predictions that has been proposed is to combine a series of two-dimensional ﬂuid ﬂow solutions along the riser axis. These “strips” are pieced together with a structural model of the riser to obtain a prediction of the ﬂuid–structure response. This method reduces the large 3D CFD problem to a large number of small 2D problems. However, this “strip” method has some serious shortcomings. In particular, the ﬂow around bluﬀ bodies is inherently three-dimensional so that the 2D strip solutions can only be expected to give approximate answers. In addition, the riser may be at a steep angle of attack to the ﬂow as is often the case with steel catenary risers (SCRs) creating strong axial ﬂow components. Also, VIV suppression devices like helical strakes that have a very strong three-dimensional wake cannot be modeled. Finally, the strip method requires that some kind of interpolation method be used to estimate the forces between the strips. There is no general rule available to make such interpolations. In addition to the sheer scale of the CFD problem, several other diﬃcult problems remain. Although the structural response of the riser usually does not involve nonlinear material response, it usually includes nonlinear geometric eﬀects, such as large displacements and rotations. The presence of these eﬀects means that a nonlinear structural model must eventually be incorporated in the solution of riser problems. In addition, large motions of the generated meshes must be accommodated in the ﬂuid ﬂow solution. Finally, it cannot be expected that the ﬂow environment remains constant over the entire length of the riser. Both ﬂow speed and direction may change. Any general solution must be able to treat these eﬀects. A simple linear structural model for the solutions is described here.12 The riser axis is oriented in the z-direction and the eigenmodes are assumed sinusoidal. Then the eigenvectors have the form:

nπy , (24.25) ζin = sin L where the ζin is the eigenvector associated with the nth mode and the ith index indicates the x or z directions, L is the riser length, y is the distance along the riser axis, and n is the mode number. It should be noted that the risers modeled here are

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tension dominated in that the bending stiﬀness of the riser is negligible compared to the stiﬀness due to tension. Also, the use of a sine shape in Eq. (24.25) implies that the riser tension is constant along the riser length. With this approach, the motion of the riser is assumed to be a linear summation of the vibration modes. The response is found by solving the equation: yin } + [cni ]{y˙ in } + [kin ]{yin } = fin , [mni ]{¨

(24.26)

where yin are the modal amplitudes, and mni mni kin are the associated mass, linear damping and stiﬀness for each mode. The forces fin are computed at each time step using the corresponding eigenvector. The displacement of the riser for the particular mode is then computed before moving to the next time step. Modeling a full scale deepwater riser was possible with this analysis [Eq. (24.26)] using a medium size computational cluster. Satisfactory experimental validation was obtained from the solution within the limitations, uncertainties, and assumptions of the available data and CFD model.12 However, some reﬁning of data is still needed for a successful correlation. This is illustrated by the following case. A test was performed (Fig. 24.16) by towing a long riser model in water and measuring the detailed inline and cross-ﬂow response of the riser model.10 The test setup and input data were supplied to the available software, both research type and commercial codes, for a blind correlation check. The results from this analysis are given in Fig. 24.16. It shows the measured inline and cross-ﬂow response compared with results from these codes. The names of the codes used in this blind test are shown on the left (see the original paper for their identiﬁcation). It is clear that the correlation is quite varied with diﬀerent degrees of success without a consistent trends in them. After this test, the results were made available to the users of the codes for further veriﬁcation. The second correlation turned out much better than the ﬁrst. Thus this type of analysis tool appears to have the capability to predict the results reasonably well, but needs further experience and validation. In order that such validation can be achieved more precisely, more accurate benchmark results should be available. Only then it is possible to make use of such tools in the design of deepwater risers with conﬁdence. Because of the numerous uncertainties in the riser analysis, the factor of safety in the fatigue life in a riser design is considered large. The guidelines, e.g., API, suggest values of FS = 3, for inspectable riser applications, and FS = 10–20, for un-inspectable applications with increased uncertainty (e.g., VIV).

24.6. Uncoupled Analysis The uncoupled analysis is still the traditional design practice for ﬂoating production systems. As already explained, in the uncoupled formulation the numerical analysis tool is based on the hydrodynamic behavior of the ﬂoater, not inﬂuenced by the nonlinear dynamic behavior of the lines. Two distinct stages of design may be identiﬁed in an uncoupled methodology:

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Mode shapes of riser response in steady current.

(1) The ﬁrst stage is the design of the ﬂoating moored system, employing programs based on an uncoupled model, to analyze the hydrodynamic model of the ﬂoating hull and determine its motions. The contribution of the risers is ignored in this analysis, and the behavior of the mooring lines is approximated by scalar coeﬃcients inserted in the 6-DOF equations of motion of the hull, to represent the contribution of the lines (in terms of mass, damping, stiﬀness, and current loading). In this methodology, the stiﬀness of the risers is generally not taken into account.

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(2) The second stage consists of the design of the risers. The ﬂoater motions that result from the ﬁrst stage (expressed either as RAOs or time series) are applied to the top of each individual riser, and the structural response of each riser (subjected also to the wave and current loads) is assessed employing a ﬁniteelement based program. In the analyses of the ﬁrst stage, the values for the coeﬃcients that approximate the behavior of the lines should be estimated by the engineer, determined via simpliﬁed analytical models, or calibrated using results from experimental tests. In this analysis procedure, generally few or no integration between the moored system and the risers take place. However, some level of integration between mooring lines and risers can be achieved even when only uncoupled analysis is used.35 Such reﬁnements consist of employing enhanced procedures for the determination of the scalar coeﬃcients introduced in the vessel equations of motion to represent the behavior of the mooring lines. For example, as a reﬁnement to this two-step procedure, part of the current loads on the mooring lines may be transferred to the ﬂoater, as an excitation load at the fairlead point on the ﬂoater. Traditionally, “basic” design methodologies have considered only the hull damping, ignoring the contribution not only of the risers but also of the mooring lines to the damping of the system. More reﬁned methodologies consider this contribution by, for instance, introducing a scalar coeﬃcient calibrated from the results of an experimental decay test. The contribution of the mooring lines to the total mass of the system may be roughly approximated by introducing a scalar mass coeﬃcient.

24.7. Coupled Analysis The eﬀects of the appendages, namely mooring lines and risers, become increasingly important as the structure moves into deeper waters. Thus their eﬀects in the total system may not be ignored in these cases. In a coupled system analysis, the mooring lines and risers are included as an integral part in the numerical model simultaneously with the ﬂoating structure. It has recently been recognized13,18,22,23,26,33,34 that the most accurate design methodology for ﬂoating oﬀshore systems should employ analysis programs based on coupled formulations. Ideally such programs should incorporate, in a single code and data structure, a hydrodynamic model for the representation of the vessel, coupled to a 3D ﬁnite-element model for the representation of the hydrodynamic and nonlinear structural dynamic behavior of the mooring lines and risers. The characteristics and advantages of this so-called “fullycoupled” methodology are described by Correa et al.13 One characteristic of this methodology is its excessive computer costs; therefore, “hybrid” methodologies and analysis procedures are presented in order to circumvent this problem and to gradually advance toward a fully coupled and integrated design methodology. In a complete coupled analysis (Fig. 24.17) the 6 DOF motions of the ﬂoating vessel is solved at a given time step. The loads and motions of the top of each of the riser/mooring line are determined. A ﬁnite element (or lumped mass) method for

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riser and mooring line is applied. For example, to compute the dynamic position of the entire mooring line at each time step in the simulation, a lumped mass method is used; see, for example, Derksen and Wichers.15 In this lumped mass method, the mass of the mooring line is lumped to a ﬁnite number of nodes that are connected by linear springs, corresponding to the axial stiﬀness of the line. Bending stiﬀness is not taken into account. The current (and wave) loads on the mooring line elements are computed. In the simulation, the mooring forces in the equation of motion are kept constant over a time step (typically taken as 0.1 seconds). During this interval, the equation of motion is integrated with a Runga–Kutta method and the position and velocity of the buoy at the new time step is obtained. The new position of the buoy is then used in the lumped mass method to compute the position of and the tension in the mooring lines at this new time step, after which the procedure is repeated until the simulation reaches its desired convergence. The new values of mooring line top loads and motions are imposed to the vessel and the 6 DOF of the vessel is solved again. Iterations continue until a convergence is reached and proceeded to the next time step.

24.8. Conclusions and Outlook While deepwater development is continuing at a steady pace and considerable progress has been made in the design of oﬀshore structures, several challenges in the analysis of deepwater ﬂoating and ﬁxed structures remain. For the ﬁxed structure of small members the empirical Morison equation has been well-established and the values of the hydrodynamic coeﬃcients have been prescribed in various design codes. While it is interesting to explore a more complete solution that does not depend on empiricism, it is unlikely that considerable eﬀort will be made in this area as the deepwater developments do not call for ﬁxed jacket-type structures. For large structures the theory is well-established and several commercially available codes are routinely used in the design of such structures. One area where further development is sought is in the application of shallow water LNG facilities. In this case the high wave overtopping on the deck may be a serious design consideration

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for which more complete reliable and practical CFD or NWT type solutions are desirable. The design of large ﬂoating structures has traditionally been carried out using the uncoupled analysis in which the dynamics of the ﬂoater are considered along with the mooring lines and risers treated as external stiﬀness terms. Once the ﬂoater dynamics are known, they are applied to the individual mooring line or riser in the design of these components. Today’s design of risers and mooring line is generally empirical in nature. Progress is being made in the development of more complete and sophisticated numerical analysis for their design. This area needs further development for improved eﬃciency and accuracy with less dependence on empiricism. They need careful validation before their practical application. Moreover, ultradeep water and relatively smaller size of ﬂoaters necessitated the interaction of the appendages and the environmental eﬀects on themselves to be incorporated in a coupled analysis with the ﬂoater. This is mostly in the development stage, even though some limited design veriﬁcation is currently being carried out by the oﬀshore industry with this type of tools.

References 1. American Petroleum Institute, Recommended practice for design and analysis of station-keeping systems for ﬂoating structures, API-RP2SK, Washington, DC (1996). 2. American Petroleum Institute, Recommended practice for planning, designing and constructing ﬁxed oﬀshore platforms, API-RP2A, 21st edn, Washington, DC (2000). 3. P. W. Bearman, J. M. R. Graham, X. W. Lin and J. R. Meneghini, Numerical simulation of ﬂow-induced vibration of a circular cylinder in uniform and oscillatory ﬂow, Flow-Induced Vibration, ed. P. W. Bearman (Rotterdan, Balkema, 1995), pp. 231–240. 4. R. F. Beck, Time-domain computation of ﬂoating bodies, Appl. Ocean Res. 16, 267–282 (1994). 5. S. Y. Boo, C. H. Kim and M. H. Kim, A numerical wave tank for nonlinear irregular waves by 3D high-order boundary element method, Int. J. Oﬀshore and Polar Eng. 4, 265–272 (1994). 6. M. S. Celebi, M. H. Kim and R. F. Beck, Fully nonlinear 3-D numerical wave tank simulation, J. Ship Res. 42(1), 33–45 (1998). 7. S. K. Chakrabarti, Numerical simulation of multiple ﬂoating structures with nonlinear constraints, J. Oﬀshore Mechanics and Arctic Eng. ASME (May 2002). 8. S. K. Chakrabarti, Hydrodynamics of Oﬀshore Structures (Computational Mechanics Publication, Southampton, UK, 1987). 9. S. K. Chakrabarti, Nonlinear Methods in Oﬀshore Engineering (Elsevier Publishers, UK, 1990). 10. J. Chaplin, Vortex- and wake-induced vibrations of deep water risers, Fourth Int. Conf. Fluid Structure Interaction, Ashurst, Southampton, UK (2007). 11. A. S. Chitrapu and R. C. Erketin, Time-domain simulation of large amplitude response of ﬂoating platforms, Ocean Eng. 22(4), 367–385 (1995). 12. Y. Constantinides, O. H. Oakley Jr. and S. Holmes, CFD high L/D riser modeling study, Proc. 26th Int. Conf. Oﬀshore Mechanics and Arctic Engineering, San Diego, CA, OMAE2007-29151, June 2007.

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13. F. N. Correa, S. F. Senra, B. P. Jacob, I. Q. Masetti and M. M. Mourelle, Towards the integration of analysis and design of mooring systems and risers, parts I and II: Studies on a dicas system, Proc. 26th Int. Conf. Oﬀshore Mechanics and Arctic Engineering (2002). 14. J. O. De Kat and R. J. Pauling, The simulation of ship motions and capsizing in severe seas, Trans. of the Society of Naval Arch. and Marine Eng. 97, 139–168 (1989). 15. A. Derksen and J. E. W. Wichers, A discrete element method on a chain turret tanker exposed to survival conditions, Boss Conf. (1992). 16. Det Norske Veritas, Oﬀshore standard position mooring, DNV-OS-E301, June 2001. 17. D. G. Dommermuth and D. K. P. Yue, Numerical simulations of nonlinear axisymmetric ﬂows with a free surface, J. Fluid Mech. 178, 195–219 (1987). 18. J. M. Heurtier, P. Le. Buhan, E. Fontaine, C. Le Cunﬀ, F. Biolley and C. Berhault, Coupled dynamic response of moored FPSO with risers, ISOPE, June 2001. 19. S. Y. Hong and M. H. Kim, Nonlinear wave forces on a stationary vertical cylinder by HOBEM-NWT, Proc. 10th Int. Oﬀshore and Polar Engineering Conf., ISOPE Seattle, WA, Vol. 3, (2000), pp. 214–220. 20. E. Huse, Resonant heave damping of tension leg platforms, Proc. 22nd Oﬀshore Technology Conf. Paper 6317 (1990), pp. 431–436. 21. M. Isaacson and K. F. Cheung, Time-domain solution for wave-current interactions with a two-dimensional body, Appl. Ocean Res. 15, 39–52 (1993). 22. C. H. Kim, A. Clement and K. Tanizawa, Recent research and development of numerical wave tanks-a review, Int. J. Oﬀshore and Polar Eng. 9, 241–256 (1999). 23. C. H. Kim, Recent progress in numerical wave tank research: A review, 4th Int. Conf. of the Society of Oﬀshore and Polar Eng., Osaka, Japan (1995), 9 pp. 24. M. H. Kim and W. C. Koo, 2D fully nonlinear numerical wave tanks, Numerical Modeling in Fluid-Structure Interaction, ed. S. K. Chakrabarti (WIT Press, Great Britain, 2005), Chap. 2. 25. M. H. Kim, 2D fully nonlinear numerical wave tanks, Numerical Modeling in FluidStructure Interaction, ed. S. K. Chakrabarti (WIT Press, UK, 2005), Chap. 2. 26. M. H. Kim, E. G. Ward and R. Haring, Comparison of numerical models for the capability of hull/mooring/riser coupled dynamic analysis for spars and TLPs in deep and ultra-deep waters, ISOPE, June 2001. 27. W. C. Koo and M. H. Kim, Fully nonlinear waves and their kinematics: NWT simulation VS experiment, Proc. 4th Int. Symp. Ocean Wave Measurement and Analysis, WAVES 2001, ASCE 2, 1092–1101 (2001). 28. M. Longuet-Higgins and E. D. Cokelet, The deformation of steep surface waves on water: I. A numerical method of computation, Proc. Royal Soc. London A 350, 1–26 (1976). 29. J. R. Morison, M. P. O’Brien, J. W. Johnson and A. S. Schaaf, The force exerted by surface waves on piles, Petroleum Transactions, American Institute of Mining and Metal Eng. 4, 11–22 (1950). 30. J. E. Orlanski, A simple boundary condition for unbounded hyperbolic ﬂows, J. Comput. Phy. 21, 251–269 (1976). 31. H. Ormberg and K. Larsen, Coupled analysis of ﬂoater motion and mooring dynamics for a turret-moored ship, Appl. Ocean Res. 20(1–2), 55–67 (1998). 32. T. Sarpkaya, In-line and transverse forces on cylinder in oscillating ﬂow at high reynolds number, Proc. Oﬀshore Technology Conf., Houston, Texas, OTC 2533, (1976), pp. 95–108.

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33. D. Sen, Time-domain simulation of motions of large structures in nonlinear waves, Proc. 21st Int. Conf. Oﬀshore Mechanics and Arctic Eng., OMAE2002–28033, Oslo, Norway (2002). 34. A. Tavassoli and M. H. Kim, Interactions of fully nonlinear waves with submerged bodies by a 2D viscous NWT, Proc. 11th Int. Oﬀshore and Polar Eng. Conf., Vol. 3, ISOPE, Stavanger, Norway (2001), pp. 348–354. 35. S. F. Senra et al., Towards the integration of analysis and design of mooring systems and risers, Part I: Studies on a semisubmersible platform, Proc. Oﬀshore Mechanics and Arctic Engineering Conf., OMAE2002-28046 (2002).

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Chapter 25

Computer Modeling for Harbor Planning and Design Jiin-Jen Lee Sonny Astani Department of Civil and Environmental Engineering University of Southern California 3620 S. Vermont Avenue, Kaprielian Hall 200 Los Angeles, CA 90089-2531, USA [email protected] Xiuying Xing Sonny Astani Department of Civil and Environmental Engineering University of Southern California 3620 S. Vermont Avenue, Kaprielian Hall 229 Los Angeles, CA 90089-2531, USA [email protected] Harbors are built to provide a sheltered environment for the mooring of ships and vessels. For some wave periods the semi-enclosed harbor basin acts as a resonator to amplify the wave motions in the harbor due to the combined eﬀects of wave diﬀractions, refractions, and multiple reﬂections from the boundaries. This undesirable wave motion could induce signiﬁcant ship motions, damage ships and dock facilities, and delay loading and unloading activities if the resonant wave periods are close to that of the ship mooring system. Harbor planners and engineers need to model the wave induced oscillations as new harbor layouts are contemplated. This chapter presents a computer model to be used for predicting the response characteristics of arbitrary shape harbors with variable depth. The model incorporates the eﬀects of wave reﬂection, refraction, diﬀraction, and dissipation losses due to boundary absorption, bottom friction, and energy losses due to the ﬂow separation at the entrances. The model is applied to four real harbors and the model results have been shown to agree surprisingly well with the ﬁeld data obtained from tsunami-genic events as well as hurricane induced wave motions. The computer model is shown to be an eﬀective engineering tool for harbor

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planning and design to derive ways of eliminating or altering the harbor response so that the harbor may indeed provide a sheltered environment for moored ships and vessels.

25.1. Introduction Harbors are built to provide a sheltered environment for the mooring of ships and vessels. In order to accomplish this goal, marine structures such as breakwaters and jetties are constructed (either detached or undetached to the shore) to reduce the wave energy incident from the open sea. The eﬀectiveness of the breakwaters and jetties in reducing the incident wave energy must be ascertained in the planning and design of harbors. Normally, the breakwaters and jetties are eﬀective in reducing the incident wave amplitude for waves with shorter wave periods (in the order of 16 s or less). As the wave period increases, the eﬀectiveness of the breakwaters or jetties in reducing the incident wave amplitude progressively decreases. For waves of shorter wave period the eﬀect of wave reﬂection from the harbor boundaries is quite small, thus only wave refraction due to changing water depth and wave diﬀraction around the breakwaters and jetties need to be considered. When the wave period increases (thus the wavelength increases), the combined eﬀect of wave diﬀraction, wave refraction, and wave reﬂection from the harbor boundaries is very signiﬁcant. In fact, it is possible that for certain semi-enclosed harbor the combined eﬀect of wave diﬀraction, refraction, and multiple reﬂections from the boundaries can cause signiﬁcant increase in the wave amplitude compared with the incident wave amplitude. This is commonly referred to as “harbor resonance” due to long waves.36,37 Resonances or oscillations due to long period waves in bays and harbors have often been observed. For example, in response to the daily ﬂooding and ebbing tide, the Bay of Fundy (at the border of eastern Canada and northeast of the United States) has produced the largest tidal range in the world (approximately 50 ft). At the Crescent City harbor region in northern California, larger than usual water surface elevations have been observed in response to a tsunami-genic event whether it is distant tsunami or local tsunami. The records at the tide gauge station located in Crescent City harbor will be discussed later in this chapter. The resonance in a harbor could result in large ﬂuctuation of water level in certain areas and produce strong currents throughout the harbor. In addition, it may also induce large ship motions for moored ships, especially if the fundamental resonant period of the harbor is close to that of the ship mooring system. Such oscillations could last as long as several days, delaying the cargo loading and unloading activities, breaking the mooring lines, and even damaging the moored boats and dock facilities. As a result, some ships or boats may have to move out to the open sea to avoid those large oscillations within the harbor, resulting in signiﬁcant economic loss.

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It is important for harbor planners and engineers to have the answers to important questions such as: (a) (b) (c) (d)

What are the important wave periods that the harbor would respond to? Are these wave periods close to the resonant periods of the ship mooring system? What ampliﬁcation of wave and current can be expected? How will the harbor respond to diﬀerent forcing functions such as tsunamis or hurricanes (called typhoons in the Paciﬁc region) as opposed to long period swells? (e) Will the response characteristics change when the harbor layout is changed or modiﬁed?

In this chapter, we have attempted to show how one can use the computer model to arrive at the answers to those questions listed above. Even though no one can completely eliminate the need for physical models, the use of computer models can be shown to provide fast and reliable answers to the questions in such a way that the physical models, if absolutely needed, can be performed more eﬃciently and eﬀectively to save time and money.

25.2. Comments on Computer Model Techniques and Previous Studies 25.2.1. General comments on computer modeling techniques As discussed in Ref. 37, there were three numerical methods to be employed for the computer models: (1) ﬁnite diﬀerence method; (2) boundary element method; (3) ﬁnite element method. The model could either be run in time domain with the results expressed as a function of time or in frequency domain with the results expressed as a function of wave frequency. When the time sequence of the dependant parameters (such as wave amplitude and water particle velocity ﬁeld) is important, time domain simulations will be appropriate. For harbor planning and design purpose, usually the exact incident wave form is not known or not yet occurred, frequency domain computations would appear to be more appropriate, so that one can explore all possible scenarios in the model more eﬀectively. We will ﬁrst discuss the pros and cons of the three numerical methods used in the computer models. We must realize at the outset that no one particular computer modeling technique is superior in all applications. Each has its own pros and cons. Selection of a modeling technique will depend on the experience and background of the individuals, or the scope and objective of the modeling eﬀorts. 25.2.1.1. Finite diﬀerence method Finite diﬀerence method discretizes the function on rectangular grids with equal or varying spacing. The function on a node is related to the values on the neighboring nodes which deﬁne the grid system. Spatial or time derivatives are approximated by the diﬀerence of the values on the neighboring notes or the successive time steps. The

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functions on each node of the whole system are calculated by solving a series of difference equations with certain boundary conditions. The ﬁnite diﬀerence method is not restricted to any governing equations and the approximation is straightforward. But the calculation might not always be stable for time dependent problems and the curved boundary can only be approximated by rectangular grids. The earliest use of the method was conducted by Leendertse.25 The two-dimensional depth-averaged governing equations containing nonlinear terms were solved to study the long wave propagation mechanism. Other examples can be found in Refs. 8 and 40. 25.2.1.2. Boundary element method Boundary element method formulates the solution inside a harbor domain by an integral equation along the boundary, and the boundary integral equation is converted into a matrix equation with approximation from each of the boundary elements. The method was ﬁrst used for harbor resonance study by Lee18 and Hwang and Tuck13 independently. Physically, this method can be considered as distributing sources, sinks, and doublets with proper strength along the boundary to satisfy the governing equation and the boundary conditions. The unique virtue of this method is that it can reduce the domain of calculation by one dimension. A three-dimensional problem can be formulated to solve equations on the boundary surface and a two-dimensional problem can be reduced to integration along the line boundary. The limitation of this method is that it can only be used for wellknown diﬀerential equations such as the Laplace’s equation and the Helmholtz equation with known fundamental solutions. Examples of computer models based on boundary element method can be found in Lee,19 Lee and Raichlen,23 Grilli et al.,10 and Lee et al.20 25.2.1.3. Finite element method Finite element method divides the simulation domain into ﬁnite polygons or polyhedrons, called elements. A shape function is needed to approximate the solution inside each element. The boundary value problem is presented by a variational principle with a functional that contains a bilinear form for the unknown coeﬃcients of the shape function. The functional is then minimized to obtain a series of linear algebraic equations about the unknown coeﬃcients for the shape function, from which the sought unknowns at the nodal points can be obtained. If the functional is not attainable, weak form formulation or Galerkin method can be used. This makes the method applicable to a wide range of physical and engineering problems since almost any governing partial diﬀerential equation can be approximated by the Galerkin procedure. Another advantage of the method is that the boundary of the simulation domain can be more precisely approximated and ﬁner grids can be easily applied for special local regions. One of the ﬁrst applications of this method in this ﬁeld was conducted by Chen and Mei.7 Mild slope equation was solved and an eigenfunction expansion was used at a common boundary in the exterior region of the harbor. It is also called hybrid ﬁnite element method. Examples of ﬁnite element model can be found in Refs. 9, 12, 21, 22, 26, and 30.

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25.2.2. Governing model equations Several diﬀerent partial diﬀerential equations are available for implementation in the computer models with varying degree of success. The classic depth averaged shallow water equation originally used by Leendertse25 has been used by many to solve for the water surface elevation and velocity ﬁeld. The classic Laplace’s equation or Helmholtz equation was used by many to solve for the velocity potential.10,19,20 Higher order Boussinesq type of equations for velocity potential have also been successfully applied to harbor modeling.26,44 An eﬀective and convenient model equation which has been found to be well suited for combining the eﬀects of wave refraction and diﬀraction in the coastal region is the two-dimensional elliptic mild slope wave equation ﬁrst derived by Berkhoﬀ.2 The mild slope equation can be written in the form as ∇ · ccg ∇φ + k 2 ccg φ = 0, where φ is the horizontal variation in velocity potential Φ, cosh k(z + h) Φ = R φ(x, y) exp(−iωt) cosh kh

(25.1)

(25.2)

in which ω and k are wave frequency and wave number. c = ω/k is the wave celerity, and cg = ∂ω/∂k is group velocity. The theory is restricted to irrotational and inviscid linear harmonic waves, and a slowly varying bathymetry condition. The loss of energy due to friction or breaking is not taken into account. It was found that the results of regular mild slope equation tend to overestimate the ampliﬁcation factor of the harbor resonance because of ignoring the separation and dissipation losses.34 The eﬀects of bottom friction and boundary absorption on wave scattering were examined by Chen.6 The regular mild slope equation (25.1) was modiﬁed by including a parameter λ: ∂ ∂φ λccg + k2 ccg φ = 0. ∂xj ∂xj

(25.3)

Similar approaches were developed by Lejeune et al.24 and Yu.43 The regular mild slope equation was modiﬁed through a parameter to account for the bottom friction. Shorling and breaking eﬀects were included in many works such as by Balas and Inan1 and Massel.28 The nonlinear eﬀects of higher order were investigated by Mei.29 Various extended approaches (so-called extended mild slope equation) were generated to improve the performance of the regular mild slope equation for abrupt and undulating topography. Terms of bottom curvature and slope were included into the regular mild slope equation by groups of researchers.4,5,31 Although different approaches were used to derive the equations, they all obtained equivalent formulae as ∇ · ccg ∇φ + (k 2 ccg + f1 ∇2 hg + f2 (∇h)2 gk)φ = 0

(25.4)

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in which f1 and f2 are functions of kh, and h is the water depth. The eﬀects of curvature and slope terms were examined by Chandrasekera and Cheung5 and Lee and Yoon.17 Although the mild slope equation was obtained with the assumption of “mild slope,” the work by Booji3 showed that the regular mild slope equation is applicable for bottom slope as large as 1/3. Since the mild slope equation can be conveniently implemented in a ﬁnite element model, we will apply it to real harbors in this presentation. The basics of the model will be shown in the following section.

25.3. A Computer Model Using the Mild Slope Equation A hybrid ﬁnite element model using the mild slope equation will be brieﬂy described herein. The model solves the mild slope equation over arbitrary shape harbor of variable water depth. It incorporated the eﬀects of wave reﬂection, refraction, diﬀraction, and dissipation losses due to boundary absorption, bottom friction, and energy losses due to the ﬂow separation at the entrances. An earlier version of the model has been applied successfully for the modeling of harbor responses of Los Angeles and Long Beach harbors by Lee et al.22 It has also been applied to model a large coastal region for assessing the potential coastal impacts of tsunamis surrounding the island of Taiwan.21 25.3.1. Governing equation The governing equation is the mild slope equation ﬁrst derived by Berkhoﬀ2 : ∇ · (CCg ∇φ) +

Cg ω 2 φ = 0. C

(25.5)

It can also be written in the form of (25.1). In which φ = φ(x, y) is the horizontal variation of the velocity potential Φ as shown in Eq. (25.2), C = ω/k is the wave celerity, and the group velocity is, C 2kh C Cg = (1 + G) = 1+ (25.6) 2 2 sinh 2kh in which G = 2kh/sinh 2kh. 25.3.2. Boundary conditions The modeling area is divided into two parts, the ﬁnite inner area (region A) and the inﬁnite outer area (region R), as shown in Fig. 25.1. The inner area includes the harbor and a connected semi-circular area. The half ring-shaped outer area has a radius of inﬁnity. Solutions of both regions are matched at the connecting boundary ∂A. The water depth is variable in the inner area but assumed to be constant in the outer area. The energy dissipation is considered in the inner area but is negligible in the outer area.

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Fig. 25.1.

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Simulation domain of the hybrid FEM model.

25.3.2.1. Far-ﬁeld boundary In the simulation domain, wave ﬁeld includes incident waves, reﬂected waves, and scattered waves. φR = φI + φS ,

(25.7)

where φI , φR , and φS are incident wave potential (including incident wave and reﬂected wave), outer region wave potential, and scattered wave potential, respectively. For the far-ﬁeld boundary, the Sommerfeld radiation condition is applied. √ ∂ − ik φS = 0. (25.8) lim r r→∞ ∂r The scattered wave potential function satisfying the Helmholtz equation and the radiation condition can be expressed as φS =

∞

Hn (kr)(αn cos nθ + βn sin nθ),

(25.9)

n=0

where αn and βn are unknown coeﬃcients and Hn is the Hankel function. The wave potential in the inner region is then obtained by solving the mild slope equation and by matching the solutions with outer region at ∂A. 25.3.2.2. Partially absorbing boundary The solid boundary such as a vertical wall or a natural beach and the energy releasing boundary such as a river outlet can be treated as that energy is partially

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absorbed and partially reﬂected. The energy ﬂux out through the boundary is formulated using a second-order scheme as iα ∂ 2 φ ∂φ , = −iαkφ − ∂n 2k ∂s2

(25.10)

where α is the absorption coeﬃcient with a range of 0 ≤ α ≤ 1. It should be mentioned that for partially absorbing boundaries, the changing of absorption coeﬃcient α may represent diﬀerent boundary conditions. When α = 0, it represents fully reﬂecting condition, i.e., ∂φ/∂n = 0. When α = 1, it represents fully absorbing condition. The relation between absorption coeﬃcient and reﬂection coeﬃcient R for an incident wave angel θi is R=−

(α cos θ i )2 − 2α cos θi + α2 . (α cos θ i )2 + 2α cos θi + α2

(25.11)

25.3.2.3. Wave transmission through breakwater Wave energy is considered as partially transmitted and partially absorbed when a wave passes through a porous breakwater. The transmitted wave potential φT is assumed to be proportional to the incoming wave potential φi . ∂φT ∂φi = KT = ikKT φi , ∂n ∂n

(25.12)

where KT is the transmission coeﬃcient through the breakwater. 25.3.2.4. Entrance loss Quadratic law for head loss has been widely used for energy dissipation at harbor entrances and bottom friction.11,14,32,39,42 The quadratic entrance head loss at the harbor entrance is also applied in this model as follows: ∆H = fe

U2 U = fe |U0 | = Ke U 2g 2g

(25.13)

in which, Ke =

fe |U0 |, 2g

(25.14)

where fe represents the dimensionless entrance loss coeﬃcient. |U0 | is the averaged velocity at the harbor entrance computed considering no entrance loss and U is the new entrance velocity to be computed considering the entrance loss in the model. Thus the relationship of the complex velocity potentials at the entrance can be written as φ1 = φ2 + ∆φ = φ2 +

g U fe |U | . iω 2g

(25.15)

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25.3.2.5. Bottom friction The energy dissipation due to bottom friction is described as an instantaneous energy ﬂux throughout the bottom: Ef = τb Ub ,

(25.16)

where τb is the instantaneous complex shear stress at the bottom, which can be formulated by the water particle velocity near the bed, τb =

1 ρKb |Ub |Ub , 2

(25.17)

where Kb is a dimensionless friction coeﬃcient. The particle velocity near the bed is ∂φ 1 = ∇φ Ub = exp(−2iωt). (25.18) ∂s b cosh kh By introducing a bottom friction coeﬃcient, fω = 1/2gKb |Ub |, the energy ﬂux through bottom becomes 2 1 ρ Ef = fω (∇φ)2 exp(−2iωt). (25.19) g cosh kh The bottom friction coeﬃcient can be obtained based on the bottom roughness study by Jonsson and Carlsen.16 The resulting formula is 0.75 0.25 αb fω = 0.2gωCbf

fω = 0.15gωαb

for 1.6 < for

αb < 100 Cbf

αb < 1.6, Cbf

(25.20)

where αb = a/sinh kh, Cbf is the Nikuradse roughness height, a is wave amplitude. 25.3.3. FEM scheme The variational principle method and shape function are used to derive the FEM matrix. With the shape functions, the velocity potential and its variation can be transformed as φ = N i φi

(25.21)

∇φ = ∇N i φi = B i φi

(25.22)

φS = NSi φiS

(25.23)

∂φS ∂NSi i = φ = PSi φiS . ∂r ∂r S

(25.24)

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The matrix equations can be written as [K][Ψ] + [Q] = 0, where [Ψ] is the unknown matrix, and

[M ] [M2 ]T

K=

[M2 ] [M1 ]

(25.25) (25.26)

in which, Cg ω 2 T T N N dxdy − iωCg αN T N ds CCg B B − M = C A ∂B 2 1 i 2 T fω + iωCg Kt N N ds − B T Bdxdy cosh kh ∂T A ω

g − iωCCg (1 − Kj )N T B − Ke B T B ds iω ∂E M1 =

CCg PST NS ds

∂A

M2 = − Q1 = − Q2 =

∂A

CCg N T

∂A

∂A

CCg N T PS ds ∂φI ds ∂nA

CCg PST φI ds.

(25.27)

(25.28) (25.29) (25.30) (25.31)

The quadratic Lagrange elements are used for the region shoreward of the imaginary common boundary ∂A. For the open sea region an eigenfunction expansion is used to represent its solution. Solutions for the two regions are matched at the common boundary ∂A. A substructure technique is used in the model for which the whole calculation domain is divided in several small domains. The matrix equations are generated in each subdomain. The solutions in each subdomain are solved separately based on their own boundary values. The solution for the entire domain is obtained by matching the results at the imaginary boundaries between those subdomains. This technique largely decreases the matrix size, thus eﬀectively accelerating the calculation. 25.4. Application to Real Harbors As mentioned before, many harbors or bays have encountered large oscillations due to long waves. For some harbors, the problem was mainly induced by the

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earthquake generated tsunamis, such as the Crescent City harbor. Some of them were caused by the typhoons and winter storms, such as the Hualien harbor in Taiwan and the Pohang harbor in Korea. The oscillation modeling in the four basins were examined by the numerical model discussed in the previous section. The results will be presented one by one, and then conclusions will be drawn at the end of this section.

25.4.1. Crescent City harbor, California Crescent City harbor, located in northern California, is one of the oldest harbors in California. The surrounding area is well known for its tsunami vulnerability. Due to its location and topography along the Paciﬁc coast (Fig. 25.2), many have described it as a “sitting duck” for tsunami waves originated from the Paciﬁc Ocean. It was severely damaged by tsunamis in the past such as the one generated by Alaskan earthquake in 1964, in which 11 people were killed and the property loss was estimated to be tens of million US dollars.27,35 Most recently, another heavy loss

Fig. 25.2.

Location of Crescent City.

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Fig. 25.3. Simulation domain for Crescent City harbor (left) and locations of special interest as A, B, C, D, E, and the tide gauge (right).

in the harbor was caused by the tsunami produced by the Kuril Islands earthquake on November 15, 2006. The model region for the computer model is shown in Fig. 25.3 (left). Only the major grid blocks are shown, the model contains 9,709 ﬁnite elements and 39,688 nodes with eight incident wave directions (indicated as 1 to 8). The Crescent City harbor is shown in Fig. 25.3 (right) with ﬁve locations of special interest as A, B, C, D, E, and the tide gage noted. Figure 25.4 presents the computer simulation response curves with the two distinct resonant periods at the tide gauge station of Crescent City harbor under different scenarios of wave directions. The ordinate is the ampliﬁcation factor deﬁned as the wave height at the tide gauge station divided by the incident wave height. The abscissa is the dimensionless wave number kl (where k is the wave number, 2π divided by wavelength, and l is the characteristic length of the harbor which is the length from the outer harbor entrance to the facing coastal line about 4,363 feet in the present model). It clearly shows that 22.0 min and 10.3 min resonant periods existed at the tide gauge station. Figure 25.5 shows the response curves at locations A–E and tide gauge station with incident wave coming from direction 2 (oriented toward north). It provides a clear indication that the waves are ampliﬁed as the inner harbor region is approached for both the 22.0 min and 10.3 min resonant wave periods. Thus, it appears that the more inner the location is, the more vulnerable for tsunami hazards. It explained the damage occurred in the small inner harbor on November 15, 2006 during which strong currents were invoked by the large oscillations. This large oscillation pushed the boats against the berth facilities and colliding with other neighboring boats. An examination of the tide record associated with several diﬀerent tsunami events indicates that water levels at the recording station at Crescent City harbor have been ampliﬁed from waves originated from near-ﬁeld as well as that from

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Fig. 25.4.

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Response curves at tide gauge location for diﬀerent incoming wave directions.

Fig. 25.5. Response curves at locations noted by A–E and tide gauge station with incident wave coming from direction 2.

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Fig. 25.6. Historical records at the Crescent City harbor for events on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) August 15, 2007.

far-ﬁeld. Figures 25.6(a)–25.6(d) show tide gauge records of water surface elevations during the near-ﬁeld earthquake generated in oﬀshore of northern California (June 15, 2005, Magnitude 7.2), and the three far-ﬁeld earthquakes. Two generated in Kuril Islands, Japan (November 15, 2006 and January 13, 2007, with Magnitude 8.3 and 8.1), and one generated in oﬀshore of Peru (August 15, 2007, Magnitude 7.9). The spectral analysis of the records shown in Fig. 25.6 has been performed, and the spectral density distributions are correspondingly shown in Fig. 25.7. It can be seen that the dominant waves in those events, which have the highest energy density, are all around 21 to 22 min. The incoming wave directions in the four events are roughly from southwest for the northern California earthquake, from west for the two Kuril Islands earthquakes, and from south for the Peru earthquake. The simulated response curves for these incoming wave directions superimposed with the observed resonant waves corresponding to those events are plotted in Fig. 25.8. The results indicate that the actual resonant periods are very close to those computed, especially the fundamental mode at 22 min. Water depth aﬀects the value of wavelength for a certain wave period. Since the actual water depth may be varied due to the tide condition, the wave period may change for a certain wavelength in diﬀerent events. The slight variation of

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Fig. 25.7. Corresponding spectral density of the historical records at the Crescent City harbor on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) August 15, 2007.

the resonant wave period is reasonable. The results presented clearly indicate the eﬀect of local topography in amplifying the incident tsunami waves whether they are near-ﬁeld or far-ﬁeld tsunamis.

25.4.2. San Pedro bay, California San Pedro bay in southern California is one of the most important economic zones in the United States. It includes the ports of Los Angeles and Long Beach (LA/LB port), which are one of the largest and busiest ports in the world. The simulation for San Pedro bay covers a very large area including the LA/LB harbor to capture the basin resonance characteristics. An air photo of the San Pedro bay is shown in Fig. 25.9 with the simulated region indicated in Fig. 25.9(a). The radius of the semicircle of the simulation domain is about 12 miles. The model grids contain 13,263 elements and 56,134 nodes. The location of tide gauge station LA Berth60 is also shown. The computed response curves at the tide gauge LA Berth60 for diﬀerent incoming wave directions are plotted in Fig. 25.10. It can be seen that the most signiﬁcant resonant wave period is about 60.0 min. Generally, the ampliﬁcation factor increases for incident waves coming from south. Since waves can go inside the harbor

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Fig. 25.8. Simulated response curves with ﬁeld observed dominant waves superimposed at Crescent City harbor for events on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) August 15, 2007.

Fig. 25.9.

Air photos of San Pedro bay with simulation domain imposed (a) and LA/LB port (b).

directly otherwise the reﬂection and diﬀraction due to the coast and breakwaters will weaken the wave ﬁeld. Similar to the modeling for Crescent City harbor, the simulation results were compared with the tide gauge (LA Berth60) records. The ports of Los Angeles and Long Beach have experienced modiﬁcations and expansion over the years. The harbor layout is now diﬀerent from that which existed several decades ago. Thus

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Fig. 25.10.

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Response curve at LA Berth60 for diﬀerent incoming wave directions.

we focused on the recorded data for recent tsunamis generated by: the earthquake in oﬀshore of northern California on June 15, 2005; two major earthquakes near the Kuril Islands, Japan on November 15, 2006 and January 13, 2007; the Solomon Islands earthquake on April 1, 2007, with a magnitude of 8.1. Both local and distant tsunamis generated by those earthquakes reached San Pedro bay, causing water level oscillations. The tide records at station LA Berth60 were analyzed to obtain the periods of the dominant modes of oscillation at that location during those events. The tide gauge records and the corresponding spectral density distributions at LA Berth60 are shown in Figs. 25.11 and 25.12. One or two dominant waves occurred during those events, all of them are around 60 min. The recorded data on April 1, 2007 was not used in the modeling examination for Crescent City harbor because the tide record was not available during that period for Crescent City harbor. For further comparison, the observed dominant waves are superimposed on the response curves simulated with the corresponding incoming wave direction for each event. This is shown in Fig. 25.13. Roughly, the incident wave came from the northwest direction for the northern California oﬀshore earthquake, from west for the Kuril Islands earthquakes, and from southwest for the Solomon Islands earthquake. Apparently, all the observed dominant waves have all been found to correspond to the ﬁrst mode of the response curves, indicating that the computer model catches the resonance characteristics of the basin. It is interesting to note that the four tide records for Crescent City harbor shown in Figs. 25.6 and 25.7 had the dominant wave period at 22 min which was predicted by the computed response curves. However, the same tide records at San Pedro bay LA/LB harbor presented in Figs. 25.11 and 25.12 showed that the dominate

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Fig. 25.11. Historical records at the LA Berth60 for events on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) April 1, 2007.

Fig. 25.12. Corresponding spectral density of the historical records at LA Berth60 on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) April 1, 2007.

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Fig. 25.13. Simulated response curves with ﬁeld observed dominant waves superimposed at LA Berth60 for events on (a) June 15, 2005, (b) November 16, 2006, (c) January 13–14, 2007, and (d) April 1–2, 2007.

resonant modes are at 60 min. This again implies that the local topography and the plan form strongly aﬀect the resonant response. Thus there is a need for performing the computer modeling whenever modiﬁcations of harbor layouts are planned. 25.4.3. Hualien harbor, Taiwan Hualien harbor is located in the eastern coast of Taiwan and facing the Paciﬁc Ocean (shown in Fig. 25.14). It has a long history of harbor resonance problem induced by typhoons during the typhoon seasons. Excessive water surface oscillations in the harbor induce large ship motion, delay cargo loading or unloading activities, and damage ships and marine structures. In the last 20 years, on several occasions, harbor resonances have resulted in mooring lines being snapped, ships and dock facilities being severely damaged. To avoid damages as typhoons approached, the ships have been ordered to move out of the harbor! The simulation region is shown in Fig. 25.15 (left). The radius of the outside semi-circle of the domain is about 4 km. To provide clarity only the major grid blocks are shown. The present grid system contains 72,955 nodes and 17,823 elements. The bathymetry was obtained from the ﬁeld survey data conducted by HMTC and other organization commissioned for this region.

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Fig. 25.14.

Location of Hualien harbor, Taiwan.

The model for the present condition is simulated and compared with the ﬁeld data measured by HMTC during typhoon Tim in July 1994.38 The model results and ﬁeld data at station #22 and #8 are presented in Fig. 25.16. It shows the ampliﬁcation factor as a function of the incident wave period. It is seen that the comparison is quite good with respect to resonant periods, resonant bandwidth, and peak ampliﬁcation factors. The results clearly indicate that there exists a broadband of resonant response for wave periods between 100 and 160 s and the computer model results appear to have captured the resonant modes correctly. 25.4.4. Pohang harbor, Korea Pohang new harbor, situated in the Yongil bay in the southeast of Korea, is the largest industrial harbor in Korea and is one of the largest industrial harbors in the world. The harbor handles cargos of steel company POSCO and other industrial complex in the region. The pier structure and the loading and unloading facilities are capable of handling 36 ships concurrently which handle 47 million tons yearly including 250,000DWT size ship. Figure 25.17 presents an air photo of the harbor region with the model grid layout superimposed (left). A close-up picture of the Pohang new harbor indicated by the rectangle in the left panel is also attached on the right. The ﬁnite element grid used in the computer model contains 5,950 elements and 24,853 nodes.

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Fig. 25.15. Air photos of Hualien harbor, Taiwan with simulation domain imposed (left) and stations #8 and #22 indicated (right).

Fig. 25.16. (right).

Simulated response curves and the observed ﬁeld data at stations #22 (left) and #8

Fig. 25.17. Layout of simulation domain with mesh and incoming wave directions superimposed (left) and the close up photo of the Pohang new harbor (right).

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Due to its location and the coastline orientation, Pohang new harbor has been found to provide fairly good protection against typhoons coming from the south. However, seiche motion due to long period waves have occurred frequently which produced undesirable wave and ship oscillations in the harbor, especially during the season with waves coming from the northeast direction. Field measurement program has shown the existence of the long period wave oscillation within the harbor.15 Figure 25.18(a) presents the simulated response curve at Station T1 in Pohang new harbor at various wave frequencies for the existing harbor conﬁguration. It covers the wave periods from 100 min to 60 s. The ordinate is the ampliﬁcation factor deﬁned as the wave height at Station T1 divided by the incident wave height. The abscissa is the wave frequency with unit at 10 cycles per minute (the right-hand limit corresponds to 60 s wave). Several resonant modes are clearly seen from Fig. 25.18. Figure 25.18(b) presents the spectral density curve for Station T1 covering the same frequency range based on the data obtained in the ﬁeld measurement program.33 The ﬁrst four resonant periods indicated by the four vertical lines in Fig. 25.18(a) are the resonant periods obtained from the ﬁeld data shown in Fig. 25.18(b), i.e., 4,800 s, 1,650 s, 490 s, and 260 s. It is seen that the present computer results compare very well with the ﬁeld data, especially for the ﬁrst few resonant modes. It is also noted

Fig. 25.18.

Results of (a) numerical simulation and (b) ﬁeld data at Station T1.

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Fig. 25.19.

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Results of (a) numerical simulation and (b) ﬁeld data at Station T2.

that several resonant modes also exist for wave coming from the north direction for wave period between 60 and 120 s. Similar results for Station T2 are shown in Figs. 25.19(a) and 25.19(b) for computer results and ﬁeld measurements. Again, the computer model appears to capture the major resonant response of the harbor region very well. 25.4.5. Summary The good agreements between the simulated results and the ﬁeld observations in these harbor resonance studies prove that the numerical model is a very useful tool to investigate the resonance mechanism for long waves. The diﬀerences of the ampliﬁcation factor caused by various incident wave directions are the results of wave reﬂection from the coast or outside jetties, as well as wave diﬀraction due to the natural obstacles or manmade breakwaters, both of which weaken the wave ﬁeld inside the harbors or bays. Tide gauge data in three earthquake events (northern California earthquake on June 15, 2005, Kuril Islands earthquakes on November 15, 2006 and January 13,

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2007) was used in the result examinations for both San Pedro bay and Crescent City harbor. Although the incoming waves in the same events may be similar before they reached the west coast of the USA, the dominant waves in LA/LB harbor (about 1 h) are totally unrelated with those in Crescent City harbor (about 22 min). This provides proofs that harbor resonance is locally induced due to the harbor layout and the local geographical conﬁguration.

25.5. Application for Harbor Improvements As shown in the previous section, resonant response in a bay or harbor is mainly inﬂuenced by the local geometrical layout and the bathymetry and less dependent on the origin of the incident waves. The question now is what we can do to alter the resonant response characteristic to minimize the negative eﬀect of long period oscillations. An example of harbor resonance improvement is the modiﬁcation of Pier J, which is located in Long Beach harbor in San Pedro bay. Figure 25.20 shows the air photo of Pier J and its surrounding area. As reported in Lee et al.,22 the original Pier J (left photo in Fig. 25.20) was modiﬁed to include the outside breakwater (right photo in Fig. 25.20). Thus the planforms of the two layouts are quite diﬀerent. The response curves obtained from the computer model before and after the construction of the breakwater at the chosen locations #81 and #82 (as noted in Fig. 25.20) are plotted in Fig. 25.21. As can be seen from the results, the ampliﬁcation factors for wave periods between 40 and 140 s have been greatly reduced due to the introduction of the breakwater. The major resonant mode at 130 s has been shifted to 170 s or longer period in the hope that the resonant periods associated with moored ships are avoided. Another example for harbor modiﬁcation is the Pohang harbor in Korea. The results of Pohang harbor discussed in the previous section focus on the harbor layout condition about two decades ago. The ﬁeld data used for the examination was obtained in 1987. Later on another breakwater was constructed inside the harbor and the entrance area was also modiﬁed. The old and new layouts are illustrated in Fig. 25.22. The computer model generated response curves before and after the modiﬁcation at station T1 and T9 shown in Fig. 25.22 are plotted in Fig. 25.23. It can be seen

Fig. 25.20.

Air photos of Pier J with the surrounding area included.

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Fig. 25.21. Response curves before and after the construction of the breakwater at location #81 (left) and #82 (right), by Lee et al.22

Fig. 25.22.

Layouts of Pohang harbor with the old one on the left and new one on the right.

Fig. 25.23. Response curves before and after the modiﬁcation at Station T1 (left) and T9 (right) in Pohang harbor.

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that the ampliﬁcation factor is largely diminished for the wave period ranging from 100 to 200 s, a region in which severe ship motions were found to occur. Thus, the modiﬁcation of harbor layout will be helpful in reducing the seiche problem in Pohang harbor. Diﬀerent types of ships may be oscillated by diﬀerent wave periods. Generally, the “troublesome” waves for ships have periods ranging from 100 to 300 s. The ship motion study, such as the modeling done by Tzong et al.41 will help determine the resonant waves for ships inside a particular harbor.

25.6. Concluding Remark As shown in this presentation, wave induced oscillation in complicated harbors can be obtained conveniently using the computer model. The good agreements, which are shown, between the prototype measurement and the computer generated results further reinforce the validity of the computer modeling technique. As mentioned earlier, once the “troublesome wave periods” are determined either by the observation or by computer simulation, they can be diminished or shifted by modifying the harbor conﬁguration to avoid potential troubles. To investigate such problem in a harbor, using physical models will be very costly and time consuming. However, modiﬁcations in a numerical simulation can be done easily especially when the original grid is already generated. The computer model can serve as a very powerful and cost-eﬀective engineering tool for harbor planning and design to provide a sheltered environment for moored ships and vessels. Even though we may not be able to completely eliminate the need of physical model for all applications, computer model can at least help plan the physical model in a more cost-eﬀective way to save time and money.

References 1. L. Balas and A. I. Inan, A numerical model of wave propagation on mild slopes, J. Coastal Res. SI36, 16–21 (2002). 2. J. C. W. Berkhoﬀ, Computation of combined refraction-diﬀraction, Proc. 13th Coastal Eng. Conf., ASCE, New York, NY (1972), pp. 471–490. 3. N. Booji, A note on the accuracy of the mild slope equation, Coastal Eng. 7, 191–203 (1983). 4. P. G. Chamberlain and D. Porter, The modiﬁed mild slope equation, J. Fluid Mech. 291, 393–407 (1995). 5. C. N. Chandrasekera and K. F. Cheung, Extended linear refraction-diﬀraction model, J. Waterway Port Coastal Ocean Eng. 123, 280–286 (1997). 6. H. S. Chen, Eﬀects of bottom friction and boundary absorption on water wave scattering, Appl. Ocean Res. 8(2), 99–104 (1986). 7. H. S. Chen and C. C. Mei, Oscillations and waves forces in an oﬀshore harbor, Report No. 190, Parsons Laboratory, MIT (1974). 8. W. J. Chiang and J. J. Lee, Simulation of large scale circulation in harbors, J. Waterway, Port, Coastal and Ocean Div. 108, WWI (1982).

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9. M. Ganaba, L. C. Wellford and J. J. Lee, Finite element methods for boundary layer modeling with application to dissipative harbor resonance problem, Chapter 15 of Finite Elements in Fluid, Vol. 5 (John Wiley & Sons, 1984), pp. 325–346. 10. S. T. Grilli, J. Skourup and I. A. Svendsen, An eﬃcient boundary element method for nonlinear water waves, Eng. Analysis with Boundary Elements 6(2), 97–107 (1989). 11. K. Horikawa and H. Nishimura, On the function of tsunami breakwaters, Coastal Eng. J. 13, 103–112 (1970). 12. J. R. Houston, Interaction of tsunamis with the Hawaiian Islands calculated by a ﬁnite element numerical model, J. Physical Oceanography 8(4), 93–102 (1978). 13. L. S. Hwang and E. O. Tuck, On the oscillations of harbors of arbitrary shape, J. Fluid Mech. 42, 447–464 (1970). 14. Y. Ito, Head loss at tsunami-breakwater opening, Proc. 12th ASCE Conf. Coastal Eng. (1970), pp. 2123–2131. 15. W. M. Jeong, S. B. Oh, J. W. Chae and S. I. Kim, Analysis of the wave induced downtime at Pohang new harbor, J. Korean Soc. Coastal and Ocean Engineers 9(1), 24–34 (1997). 16. I. G. Jonsson and N. A. Carlsen, Experimental and theoretical investigations in an oscillatory turbulent boundary layer, J. Hydraul. Res. 14, 45–60 (1976). 17. C. Lee and S. B. Yoon, Eﬀect of higher-order bottom variation terms on the refraction of water waves in the extended mild slope equation, Ocean Eng. 31, 865–882 (2004). 18. J. J. Lee, Wave induced oscillations in harbors of arbitrary shape, Report No. KH-R20, W. M. Keck Laboratory, Caltech (1969), p. 266. 19. J. J. Lee, Wave-induced oscillations in harbors of arbitrary geometry, J. Fluid Mech. 45(2), 375–394 (1971). 20. J. J. Lee, C. Chang and F. Zhuang, Interactions of transient nonlinear waves with coastal structures, 23rd Conf. Coastal Eng. (1992). 21. J. J. Lee and C. P. Lai, Assessing impacts of tsunamis on Taiwan’s and China’s southeast coastlines, Proc. ICCE 2006, San Diego (2006). 22. J. J. Lee, C. P. Lai and Y. Li, Application of computer modeling for harbor resonance studies of Long Beach and Los Angeles harbor basins, Proc. 26th Int. Conf. Coastal Eng. (1998), pp. 1196–1209. 23. J. J. Lee and F. Raichlen, Oscillations in harbors with connected basins, J. Waterways, Ports, Coastal and Ocean Eng. Div. 98(WW3), 311–332 (1972). 24. A. Lejeune, M. Lejeune and M. Sahloul, Wave plan computation method in study of the Calvi Bay erosion in Corsica, France, Int. J. Numer. Meth. Eng. 27, 71–85 (1989). 25. J. J. Leendertse, Aspects of a computational model for long-period wave propagation, Memo KM-5294-PR RAND Corp., Santa Monica, CA (1967). 26. T. G. Lepelletier and F. Raichlen, Harbor oscillations induced by non-linear transient long waves, J. Waterway, Port, Coastal and Ocean Eng. 113(4), 381–400 (1987). 27. O. T. Magoon, Structural damage by tsunamis, Coastal Engineering, Santa Barbara Specialty Conf., ASCE (1965), pp. 35–68. 28. S. R. Massel, Ocean Surface Waves: Their Physics and Prediction (World Scientiﬁc Publication, Singapore, 1995). 29. C. C. Mei, Mild-slope approximation for long waves generated by short waves, J. Eng. Math. 35, 43–57 (1999). 30. C. C. Mei, M. Stiassnie and D. K. Yue, Theory and Applications of Ocean Surface Waves (World Scientiﬁc Publishing Company, 2005).

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31. S. R. Massel, Expanded refraction-diﬀraction equation for surface waves, Coastal Eng. 19, 97–126 (1993). 32. J. W. Miles and Y. K. Lee, Helmholtz resonance of harbors, J. Fluid Mech. 67(3), 445–464 (1975). 33. Ministry of Construction, The Report of Wave Measurement and Numerical Experiment in Pohang New Harbor (1987), pp. 108–116. 34. M. Okihiro, R. T. Guza and R. J. Seymour, Excitation of seiche observed in a small harbor, J. Geophys. Res. 98(C10), 18201–18211 (1993). 35. D. M. Powers, The Raging Sea: The Powerful Account of the Worst Tsunami in U.S. History (Kensington Publishing Corporation, 2005). 36. F. Raichlen, Long period oscillations in basins of arbitrary shapes, Chapter 7, Coastal Engineering, Santa Barbara Speciality Conf. (1965), pp. 115–145. 37. F. Raichlen and J. J. Lee, Oscillation of bays, harbors, and lakes, Chapter 13, Handbook of Coastal and Ocean Engineering, ed. J. B. Herbich (Gulf Publishing Company, 1992). 38. C. H. Su and T. K. Tsay, Numerical simulation on harbor oscillations in Haw-Lien harbor, Report NO. MOTC-IOT-IHMT-NB9001-1, Institute of Harbor and Marine Technology Institute of Transportation, Tai-Chung, Taiwan (2002). 39. G. M. Terra, W. J. Berg and L. R. M. Maas, Experimental veriﬁcation of Lorentz’ linearization procedure for quadratic friction, Fluid Dynamics Res. 36, 175–188 (2005). 40. V. V. Titov and C. E. Synolakis, Numerical modeling of tidal wave runup, J. Waterways, Port, Coastal and Ocean Eng. 124(4), 157–171 (1998). 41. T. J. Tzong, C. P. Lai, J. J. Lee and F. Zhuang, Ship motion modeling in Los Angeles and Long Beach harbors, Proc. ICCE 2006, San Diego (2006). ¨ Unl¨ ¨ uata and C. C. Mei, Eﬀects of entrance loss on harbor oscillations, J. Waterways 42. U. Harbors and Coastal Eng. Div. 101(WW2), 161–179 (1975). 43. X. Yu, Finite analytic method for mild-slope wave equation, J. Eng. Mech. 122(2), 109–115 (1996). 44. J. A. Zelt and F. Raichlen, A Lagrangian model for wave-induced harbour oscillations, J. Fluid Mech. 213, 203–225 (1990).

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Chapter 26

Prediction of Squat for Underkeel Clearance Michael J. Briggs Coastal and Hydraulics Laboratory US Army Engineer Research and Development Center 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA [email protected] Marc Vantorre Ghent University, IR04, Division of Maritime Technology Technologiepark Zwijnaarde 904, B 9052 Gent, Belgium [email protected] Klemens Uliczka Federal Waterways Engineering and Research Institute Hamburg Oﬃce, Wedeler Landstrasse 157 D-22559 Hamburg, Germany [email protected] Pierre Debaillon Centre d’Etudes Techniques Maritimes Et Fluviales 2 bd Gambetta, BP60039, 60321 Compiegne, France [email protected] This chapter presents a summary of ship squat and its eﬀect on vessel underkeel clearance. An overview of squat research and its importance in safe and eﬃcient design of entrance channels is presented. Representative PIANC empirical formulas for predicting squat in canals and in restricted and open channels are discussed and illustrated with examples. Most of these formulas are based on hard bottoms and single ships. Ongoing research on passing and overtaking ships in conﬁned channels, and oﬀset distances and drift angles is presented. The eﬀect of ﬂuid bottoms or mud is described. Numerical modeling of squat is an area of future research and some comparisons are presented and discussed.

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26.1. Introduction When a ship travels through shallow water it undergoes changes in its vertical position due to hydrodynamic forces from the ﬂow of water and wave-induced motions of heave, pitch, and roll. The focus of this chapter is on the former mechanism of ship squat. Squat is the reduction in underkeel clearance (UKC) between a vessel at-rest and underway due to the increased ﬂow of water past the moving body. The forward motion of the ship pushes water ahead of it that must return around the sides and under the keel. This water motion induces a relative velocity between the ship and the surrounding water that causes a water-level depression in which the ship sinks. The eﬀect of shallow water and channel banks only exacerbates these conditions. The velocity ﬁeld produces a hydrodynamic pressure change along the ship similar to the Bernoulli eﬀect in that kinetic and potential energy must be in balance.1 This phenomenon produces a downward vertical force (sinkage, positive downward) and a moment about the transverse axis (trim, positive bow up) that can result in diﬀerent values of squat at the bow and stern (Fig. 26.1). This combination of sinkage and change in trim is called ship squat. Most of the time squat at the bow, Sb , represents the maximum value, especially for full-form ships, such as supertankers. In very narrow channels or canals and for high-speed (ﬁne-form) ships, such as passenger liners and containerships, the maximum squat can occur at the stern Ss . The initial trim of the ship also inﬂuences the location of the maximum squat. The ship will always experience maximum squat in the same direction as the static trim.2 If trimmed by the bow (stern), maximum squat will occur at the bow (stern). A ship trimmed by the bow or stern when static will remain that way and will not level out when underway to oﬀset the sinkage at the bow or stern due to squat. So why do we care about ship squat? For one thing, ship squat has always existed, but was less of a concern with smaller vessels and with relatively deeper channels. The new supertankers and supercontainerships have smaller static UKC and higher service speeds. Secondly, the goal of all ports is to provide safe and eﬃcient navigation for waterborne commerce. Since operation and maintenance costs continue to escalate and can easily exceed $3M per vertical meter, it is imperative to minimize required channel depths and associated dredging costs. Finally, even though we have a pretty good handle on squat predictions, accidents continue to occur. Barrass3 noted that there have been 12 major incidents between 1987 and 2004. In 2007, this number of ship incidents had increased to as many as 82 that are partially attributable to ship squat.4 The luxury passenger liner QEII grounded

Fig. 26.1.

Schematic of ship squat at bow and stern.

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oﬀ Massachusetts in 1992 with a repair cost of $13M and another $50M for lost passenger bookings. In the early 1990s, the Maritime Commission (MarCom) of the Permanent International Association of Navigation Congresses (PIANC) formed a working group (WG30) to provide information and recommendations on the design of approach channels.5 In the past 10 years since the WG30 report, research in squat predictions was a dynamic area in naval architecture with new experiments to study the eﬀects of ﬂuid bottoms and passing and overtaking vessels, especially with the increasing size of the shipping ﬂeet. Time domain Reynolds Average Navier–Stokes Equation (RANSE) numerical models are being developed to predict squat, but these models are still being validated. In 2005, the PIANC MarCom formed a new working group Horizontal and Vertical Dimensions of Fairways (WG49) to update the WG30 report on design of deep draft navigation channels.6 A summary of ship squat is presented in this chapter. In the second section, factors governing squat including ship characteristics, channel conﬁgurations, and combined factors are discussed. Some empirical formulas from the PIANC WG30 report are presented and compared in the third section. The fourth section presents some recent research on the eﬀect of squat on passing and overtaking ships in conﬁned channels by the Federal Waterways Engineering and Research Institute (BAW) in Hamburg, Germany, and the Flanders Hydraulic Research (FHR) Laboratory in Antwerp, Belgium. It also includes numerical modeling by Delft University of Technology and laboratory modeling by FHR on the eﬀect of ship oﬀset and drift on squat. The ﬁfth section summarizes the recent studies at FHR on the eﬀect of ﬂuid bottoms (i.e., mud) on squat. The development of numerical models to predict ship squat is an ongoing research area. The current status of this development at Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF), France, is discussed in the sixth section. Finally, a summary and conclusions of ship squat issues is presented in the last section.

26.2. Factors Governing Squat Prediction of ship squat depends on ship characteristics and channel conﬁgurations. These factors are often combined to create new normalized parameters to describe the squat phenomenon. 26.2.1. Ship characteristics The main ship parameters include ship draft, T , hull shape as represented by the block coeﬃcient, CB , and ship speed, VS (m/s) or VK (knots). Other ship parameters include the length between forward and aft perpendiculars Lpp and the beam, B. The CB is a measure of the “ﬁneness” of the vessel’s shape relative to an equivalent rectangular volume with the same dimensions. The range of values of CB is typically between 0.45 for high-speed vessels and 0.85 for slow, full-size tankers and bulk carriers. The most important ship parameter is its speed VS . This is the relative speed of the ship in water, so ﬂuvial and tidal currents must be included. In general,

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squat varies as the square of the speed. Therefore, doubling the speed quadruples the squat and vice versa. There are two calculated ship parameters that are based on the basic ship dimensions. The ship’s displacement volume ∇ (m3 ) is deﬁned as ∇ = CB Lpp BT.

(26.1)

The CB can be determined from the ∇ if the other ship dimensions are known. The underwater midship cross-sectional area AS is generally deﬁned as AS = 0.98BT.

(26.2)

The “0.98” constant accounts for reduction in area due to the keel radius.7 Some researchers ignore this and use a constant of “1.00” since the error is small relative to other uncertainties in the squat calculations. Finally, the bulbous bow and stern-transom are two other characteristics of a ship that aﬀect squat. Many of the early squat measurements were made before bulbous bows were in use. Newer designs of bulbous bows, although mainly to reduce drag and increase fuel eﬃciency, also have an eﬀect on squat. The newer “sterntransoms” on some ships are “blockier” (i.e., wider and less streamlined) than earlier ship designs and aﬀect squat as they become more fully submerged with increases in draft.8 26.2.2. Channel conﬁgurations The main channel considerations are proximity of the channel sides and bottom, as represented by the channel depth h and cross-sectional conﬁguration. If the ship is not in relatively shallow water with a small UKC, squat is usually negligible. Ratios of water depth to ship draft h/T greater than 1.5–2.0 (i.e., relatively deepwater) are usually considered safe from the inﬂuences of squat. The main types of “idealized” channel conﬁguration are (a) open or unrestricted (U), (b) conﬁned or restricted (R), and (c) canal (C). Figure 26.2 is a schematic of these three types of entrance channels for ocean-going or deep draft ships. Unrestricted channels are in relatively larger open bodies of water and usually toward the oﬀshore end of entrance channels. Analytically and numerically, they are easier to describe and were some of the ﬁrst types studied. Sections of rivers may even

Fig. 26.2. canal.

Schematic of three channel types: unrestricted or open, restricted or conﬁned, and

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be classiﬁed as unrestricted channels if they are wide enough. The second type of channel is the restricted channel with an underwater trench that is typical of dredged channels. The restricted channel is a cross between the canal and unrestricted channel type. The trench acts as a canal by containing and inﬂuencing the ﬂow around the ship, and the water column above the hT allows the ﬂow to act as if the ship is in an unrestricted channel. The last type of channel is the canal. These channels are representative of channels in rivers with emergent banks. The sides are idealized as one slope when in reality they may have compound slopes with revetment to protect against ship waves and erosion. The canal may or may not be exposed to tidal ﬂuctuations. For instance, the Panama and Suez Canals have a constant water depth. Many channels can be characterized by two or three of these channel types as the diﬀerent segments or reaches of the channel have diﬀerent cross-sections. Finally, many real-world channels look like combinations of these three types as one side may look like an open unrestricted channel and the other side like a canal or restricted channel with side walls. Most of the PIANC empirical formulas are based on ships in the center of symmetrical channels, so the user has to use “engineering judgment” when selecting the most appropriate formulas. New data are being collected for some of these more realistic channel shapes, so future formulas may account for these diﬀerences in channel shapes. Other important parameters necessary to describe restricted channels and canals are the channel width at the bottom of the channel W , trench height hT from the bottom of the channel to the top of the trench, and inverse bank slope n (i.e., run/rise = 1/ tan θ). The value of n, although not necessarily an integer, typically has a value such as 1, 2, or 3 representing side slopes of 1:1, 1:2, and 1:3, respectively. How does one deﬁne the width of an unrestricted or an open channel since there are no banks or sides? In 2004, Barrass had deﬁned an eﬀective width Weﬀ for the unrestricted channel as the artiﬁcial side boundary on both sides of a moving ship where the ship will experience changes in performance and resistance that aﬀect squat, propeller RPMs, and speed.3 His width of inﬂuence FB is deﬁned for h/T values from 1.10 to 1.40 as 7.04 B. (26.3) FB = Weﬀ = CB0.85 Mean values of FB are of the order of 8B to 8.3B for supertankers (CB range from 0.81 to 0.87), 9B to 9.5B for general cargo ships (CB range from 0.68 to 0.80), and 10B to 11.5B for containerships (CB range from 0.57 to 0.71). The calculated cross-sectional area AC is the wetted cross-section of the canal or the equivalent wetted area of the restricted channel by projecting the slope to the water surface. It is given by AC = W h + nh2 .

(26.4)

For an unrestricted channel, use Barrass’s eﬀective width Weﬀ for channel width W and set n = 0 in the equation for AC .

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26.2.3. Combined ship and channel factors Several dimensionless parameters are required in the PIANC squat prediction formulas that are ratios of both ship and channel parameters. They include the depth Froude number Fnh and the blockage factor S. The most important dimensionless parameter is Fnh , which is a measure of the ship’s resistance to motion in shallow water. Most ships have insuﬃcient power to overcome Fnh values greater than 0.6 for tankers and 0.7 for containerships. Most of the empirical equations require that Fnh be less than 0.7. For all cases, the value of Fnh should satisfy Fnh < 1, an eﬀective speed barrier and the deﬁning level for the subcritical speed range. The Fnh is deﬁned as Vs Fnh = √ gh

(26.5)

with gravitational acceleration g (m/s2 ). The blockage factor S is the fraction of the cross-sectional area of the waterway AC that is occupied by the ship’s underwater midships cross-section AS deﬁned as S=

AS . AC

(26.6)

Typical S values can vary from 0.03 to 0.25 or larger for restricted channels and canals, and to 0.10 or less for unrestricted channels.3,9 Higher values may occur, for example, the canal from Terneuzen (The Netherlands) to Ghent (Belgium) is operated with a blockage factor S = 0.275, and higher values will be evaluated in the near future.10 The value of S is a factor in the calculation of the ship’s critical speed in canals and restricted channels (see Sec. 26.3.3.3 and Appendix 26.A). 26.3. PIANC Squat Formulas 26.3.1. Background In 1997 the PIANC WG30 report included 11 empirical formulas and one graphical method from nine diﬀerent authors for the prediction of ship squat.5 They were based on physical model experiments and ﬁeld measurements for diﬀerent ships, channels, and loading characteristics. The formulas included the pioneering work of Tuck11 , Tuck and Taylor,12 and Beck et al.,13 and the early research by Hooft,14 Dand,15 Eryuzlu and Hausser,16 R¨omisch,17 and Millward.18,19 The PIANC recommends that channels be designed in two stages. The ﬁrst is the “Concept” Design where a “quick” or “ballpark” answer is desired. The WG30 report recommended the International Commission for the Reception of Large Ships (ICORELS) formula20 in this phase. The second stage is the “Detailed” Design phase where more accurate and thorough predictions and comparisons are required. The WG30 recommended the formulas by ICORELS, Huuska,7 Barrass,21,22 and Eryuzlu et al.23 in this second stage. All of these formulas give predictions of bow squat Sb , but only the R¨ omisch formula gives predictions for stern squat Ss for all channel types. The Barrass

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formula gives Ss for unrestricted channels, and for canals and restricted channels depending on the value of CB . Each formula has certain constraints that it should satisfy before being applied, usually based on the ship and channel conditions under which it was developed. Caution should be exercised if these empirical formulas are used for conditions outside those for which they were developed. In 2005 the PIANC MarCom formed WG49,6 which is in the process of reviewing and revising these formulas for an updated report on channel design (expected to be completed in 2010). There have been some new formulations since the WG30 report that are being evaluated. Barrass has continued to develop and reﬁne his formulas and now has predictions for both Sb and Ss . Ankudinov et al.24 proposed the Maritime Simulation and Ship Maneuverability (MARSIM) 2000 formula for maximum squat based on a midpoint sinkage and vessel trim in shallow water. It is one of the most thorough and the most complicated formulas for predicting ship squat. The St. Lawrence Seaway (SLS) Trial and Very Large Crude Carriers (VLCC) formulas are based on the prototype measurements in the SLS by Stocks et al.25 Briggs26 developed a FORTRAN program to calculate squat using most of these formulas. It is not possible to include all the formulas in this chapter. We have selected a representative sample of formulas that can be used for both phases of design. Some are the “old tried and true” formulas and some are based on new research. The Concept Design phase is by deﬁnition the simplest, of course this does not necessarily mean that these formulas are any less accurate than some of the more complicated formulas. In the Detailed Design phase, it is usually a good practice to evaluate the squat with several of the formulas and calculate some statistics such as average and range of values. In some cases, the maximum squat values might be used in design for the case of dangerous cargo and/or hard channel bottoms. The user should always be mindful for the original constraints. Some of these constraints are very restrictive (especially for the newer vessels coming on line) as they are based on the limited set of conditions tested in physical models by the individual researchers. This does not mean that the particular formula would not be applicable if the constraints are exceeded by a reasonable amount. Therefore, the user should exercise Engineering Judgment when deciding the applicability of those predictions. Table 26.1 summarizes the applicable channel conﬁgurations and Table 26.1.

Channel conﬁgurations and parameter constraints for PIANC squat formulas. Conﬁguration Code ID

Constraint Code ID

Formulas

U

R

C

CB

Barrass27

Y

Y

Y

0.5–0.85

1.1–1.4

Eryuzlu et al.23 Huuska7 ICORELS20 Yoshimura28 R¨ omisch17

Y Y Y Y Y

Y Y

Y

≥ 0.8 2.4–2.9 0.6–≥0.8 2.19–3.5

1.1–2.5 1.1–2.0

Y Y

Y Y

0.55–0.8

B/T

2.5–5.5 2.6

h/T

≥1.2 1.19–2.25

Notes: 1. Huuska/Guliev originally for Fnh ≤ 0.7.

hT /h

L/B

6.7–6.8 0.22–0.81 5.5–8.5 3.7–6.0 8.7

L/h

L/T

16.1–20.2

22.9

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parameter constraints according to the individual testing conditions for the formulas in this chapter. 26.3.2. Concept design 26.3.2.1. ICORELS The ICORELS formula20 for bow squat Sb is one of the original formulas from the PIANC WG30 report.5 It was developed for unrestricted or open channels only, so it should be used with caution if applied for restricted and canal channels. It is similar to Hooft’s14 and Huuska’s7 equations and is deﬁned as S b = CS

2 ∇ Fnh 2 L2pp 1 − Fnh

(26.7)

where CS = 2.4 and the other factors have been previously deﬁned. The Finnish Maritime Administration (FMA) uses this formula with diﬀerent values of CS depending on the ship’s CB .29,30    1.7 CB < 0.70 CS = 2.0 0.70 ≤ CB < 0.80 . (26.8)   2.4 C ≥ 0.80 B The BAW, however, recommends a value of CS = 2.0 for the larger containerships of today which may have a CB < 0.70. Their research is based on many measurements along the restricted channel (side slope n varies from 15 to 40), 100-km long, River Elbe.31 The wider stern-transom ships (see Sec. 4.3) require CS = 3 because of the increased bow squat. The FHR has found CS ≥ 2.0 for modern containerships. They typically travel at much higher speeds than the ICORELS formula was originally developed, even in shallow and restricted waters. The Fnh are higher and in this speed range the eﬀect of blockage S on the critical ship speed is considerable. For example, a very small S = 0.01 results in an important decrease in critical speed.10 26.3.2.2. Barrass The Barrass4,27 formula is one of the simplest and “user friendly” and can be applied for all channel conﬁgurations. Based on his earlier work in 1979,21 1981,22 and 2004,3 the maximum squat SMax at the bow or stern is determined by the value of ship’s CB and Vk as SMax =

KCB Vk2 . 100

(26.9)

According to Barrass,2 the value of CB determines whether SMax is at the bow Sb or stern SS (requires even keel when static). He notes that full-form ships with CB > 0.7 tend to squat by the bow and ﬁne-form ships with CB < 0.7 tend to squat by the stern. The CB = 0.7 is an “even keel” situation with squat the same

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at both bow and stern. Of course, for channel design, one is mainly interested in the maximum squat and not necessarily whether it is at the bow or stern. This formula is based on a regression analysis of more than 600 laboratory and prototype measurements. Stocks et al.25 found that the Barrass formulas gave the best results for New and Traditional Lakers in the Lake St. Francis area (unrestricted channel) of the SLS. The BAW feels that the Barrass restricted formula is conservative for their restricted channel applications in the Elbe River. The coeﬃcient K 4 is deﬁned in terms of blockage factor S as K = 5.74S 0.76.

(26.10)

A value of S = 0.10 is equivalent to a “wide” river (unrestricted or open water conditions). The value of K = 1 and the denominator in the equation for SMax remains 100. If S < 0.10, the value of K should be set to 1. For restricted channels, a value of the order of S = 0.25 gives a value of K = 2, and the denominator becomes 50. Thus, the eﬀect of K is to modify the denominator constant between values of 50 to 100. Constraints on these equations are 1.10 ≤ h/T ≤ 1.40 and 0.10 ≤ S ≤ 0.25. This equation can accommodate a medium width river with a value of S between the limits of S above. For ships in unrestricted channels that are at even keel when in a static condition (i.e., moored), one can estimate the squat at the other end of the ship (either bow or stern) based on SMax . Thus, if CB indicates the ship will squat by the bow, then this formula will give the squat at the stern, and vice versa: Sb CB ≤ 0.7 2 [1 − 40(0.7 − CB ) ]SMax = . (26.11) SS CB > 0.7 26.3.2.3. Yoshimura The Overseas Coastal Area Development Institute of Japan32 and Ohtsu et al.33 proposed the following formula for Sb as part of their new Design Standard for Fairways in Japan. This formula was originally developed by Yoshimura28 for open or unrestricted channels typical of Japan. The range of parameters for which this formula is applicable is shown in Table 26.1. In 2007, Ohtsu34 proposed a small change to the ship velocity term Vs (last factor in the equation is now Ve ) to include S to improve its predictions in restricted channels and canals: Ve =

  Vs

Unrestricted



Restricted, canal

Vs (1 − S)

.

(26.12)

Their Sb predictions generally fall near the average for most of the other PIANC bow squat predictions, regardless of ship type:

3 2 CB 1 CB 1 Ve + 15 0.7 + 1.5 . (26.13) Sb = h/T Lpp /B h/T Lpp /B g

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26.3.3. Detailed design 26.3.3.1. Eryuzlu One of the more recent series of physical model tests and ﬁeld measurements was conducted by Eryuzlu et al.23 for cargo ships and bulk carriers with bulbous bows in unrestricted and restricted channels. Their tests used self-propelled models with bulbous bows. Many of the early PIANC formulas did not have ships with bulbous bows. The range of ship parameters was somewhat limited with CB ≥ 0.8, B/T from 2.4 to 2.9, and Lpp /B from 6.7 to 6.8. The Eryuzlu formula should not be used for containerships unless they meet this CB criteria. They conducted some supplemental physical model tests with an hT /h = 0.5 and n = 2 to investigate the eﬀect of channel width in restricted channels. The Canadian Coast Guard35 is using the Eryuzlu et al.23 formula exclusively. Stocks et al.25 recommended the Eryuzlu formula for the chemical tankers in the Lake St. Louis section (unrestricted channel) of the SLS. The Eryuzlu formula for Sb is deﬁned as 2.289 −2.972

Vs h h2 √ Sb = 0.298 Kb . (26.14) T T gT Note that the Ship Froude number rather than Fnh is used in their equation since the ship draft T is used in the denominator instead of the channel depth h. The Kb is a correction factor for channel width W relative to ship’s B given by  W 3.1   < 9.61  W/B B . (26.15) Kb =  W  1 ≥ 9.61 B One should use the second value of Kb = 1 for unrestricted channels regardless of eﬀective width Weﬀ since the channel has no boundary eﬀects on the ﬂow and pressures on the ship. 26.3.3.2. Huuska/Guliev The next empirical formula in the Detailed Design phase is by Huuska.7 This Finnish professor extended Hooft’s work for unrestricted channels to include restricted channels and canals by adding a correction factor for channel width Ks that Guliev36 had developed. The Spanish ROM 3.1-99 (Recommendations for Designing Maritime Conﬁguration of Ports, Approach Channels, and Floatation Areas37 ) and the FMA recommend the Huuska/Guliev formula for all three channel conﬁgurations. In general, this formula should not be used for Fnh > 0.7. The FMA29 also includes some additional constraints for lower and upper limits as follows (Table 26.1): • • • •

CB 0.60 to 0.80 B/T 2.19 to 3.50 Lpp /B 5.50 to 8.50 hT /h 0.22 to 0.81

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Fig. 26.3.

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Huuska/Guliev K1 versus S.

The Huuska/Guliev formula is deﬁned as Sb = CS

2 ∇ Fnh Ks . 2 L2pp 1 − Fnh

(26.16)

The squat constant CS = 2.40 is typically used as an average value in this formula. The value for Ks for restricted channels and canals is determined from

Ks =

7.45s1 + 0.76 s1 > 0.03 s1 ≤ 0.03

1.0

(26.17)

with a corrected blockage factor s1 deﬁned as s1 =

S . K1

(26.18)

The correction factor K1 is given by Huuska’s plot of K1 versus S for diﬀerent trench height ratios hT /h shown in Fig. 26.3. One should use a value of hT = 0 for unrestricted channels and hT = h for canals. Appendix 26.A contains a set of least square ﬁt coeﬃcients for Fig. 26.3 if one wants to program these curves.26 26.3.3.3. R¨ omisch R¨omisch17 developed formulas for both bow and stern squat from physical model experiments for all three channel conﬁgurations. His empirical formulas are some of

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734

the most diﬃcult to use, but seem to give good predictions for bow Sb and stern squat Ss given by Sb = CV CF K∆T T Ss = CV K∆T T

(26.19)

where CV is a correction factor for ship speed, CF is a correction factor for ship shape, and K∆T is a correction factor for squat at ship critical speed. The value for CF is equal to 1.0 for the stern squat. The values for these coeﬃcients are deﬁned as

CV = 8

V Vcr

2

V − 0.5 Vcr

4 + 0.0625

2 10CB CF = Lpp /B K∆T = 0.155 h/T .

(26.20)

(26.21) (26.22)

The ship critical or Schijf-limiting speed Vcr is the speed that ships cannot exceed due to the balance between the continuity equation and Bernoulli’s law.9,38,39 For economic reasons, maximum ship speeds are typically only 80% of Vcr . The Vcr (m/s) varies as a function of the channel conﬁguration given by    CKU Vcr = Cm KC   CmT KR

Unrestricted Canal Restricted

.

(26.23)

The three-wave celerity parameters C, Cm , and CmT (m/s) are deﬁned as C=

gh;

Cm =

ghm ;

CmT =

ghmT .

(26.24)

The mean water depth hm (m) is a standard hydraulic parameter that is used for canals and restricted channels. It is deﬁned as hm =

AC WTop

(26.25)

where WTop (m) is the projected channel width at the top of the channel equal to WTop = W + 2nh.

(26.26)

The relevant water depth hmT (m) is for restricted channels and is deﬁned as hmT = h −

hT (h − hm ). h

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R¨ omisch’s KC versus 1/S.

Table 26.2. 1/S

1

6

10

20

30

∞

KC

0.0

0.52

0.62

0.73

0.78

1.0

R¨omisch’s correction factors KU , KC , and KR for unrestricted, canal, and restricted channels, respectively, are deﬁned as

0.125 h Lpp T B 1.5

Arc sin(1 − S) KC = 2 sin 3

KU = 0.58

KR = KU (1 − hT /h) + KC (hT /h).

(26.27) (26.28) (26.29)

Note that the KR for the restricted channel is a function of both KU and KC . Table 26.2 lists R¨omisch’s limited dataset for KC as a function of 1/S (i.e., AC /AS ). Appendix 26.A contains more detailed descriptions of KC and some additional equations for deﬁning it relative to Schijf’s limiting speed and his limiting Froude number FHL . 26.3.4. Example problems Three example problems are presented in this section to illustrate the diﬀerent formulas for several channel and ship types. All are for bow squat Sb . Comparisons of the diﬀerent formulas with the measured laboratory values are shown for each example in Figs. 26.4–26.6, respectively. Appendix 26.B contains worked examples for at least one Concept and one Detailed Design application for each example problem. 26.3.4.1. Example 1: BAW Post-Panamax containership in unrestricted channel The ﬁrst example is for a Post-Panamax containership traveling at Vk = 13.3 kt (Vs = 6.84 m/s) in an unrestricted channel. This speed matches laboratory data ugge and Uliczka.40,41 This vessel is similar (Sb = 0.70 m) obtained at BAW by Fl¨ to the last generation Emma Maersk containership (launched in August 2006), but with a larger CB . The larger CB is not realistic for the newer containerships (most have CB < 0.7), but was tested by BAW by “lengthening” an existing model during design experiments. A comparable CB is of the order of 0.62 for a ship of this size. The dimensions of the ship and channel are listed in Table 26.3. Figure 26.4 shows comparisons among the Barrass, Eryuzlu, Huuska, ICORELS, R¨omisch, and Yoshimura formulas and the measured BAW laboratory values. The numerical values are from a numerical model described in Sec. 26.6. In general, the best formulas are the Yoshimura (Concept) and Eryuzlu (Detail) as they are slightly

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0.0

Bow Squat for BAW Hansa Container Ship - Unrestricted R N

R R N

-0.5

R R

R

N

R N

N

Sb , m

R

Barrass Eryuzlu Huuska ICORELS Romisch Yoshimura Numerical BAW

N

-1.0

R N R N

-1.5 Example

R N

Unrestricted Channel Bottom

-2.0 9

11

13

15

17

19

Vk, knots Fig. 26.4. Comparison of BAW’s experimental measurements, empirical formulas, and numerical model of bow squat for a Post-Panamax containership in an unrestricted channel (open water).

0.0

Bow Squat for FHR Tanker G, Condition C - Canal

R N

N

-0.5

Barrass Huuska Romisch Yoshimura Numerical FHR

R N

Sb, m

R R

Example R

N R

-1.0 R N

R R

Canal Bottom

-1.5

R

7

8

9

10

11

12

Vk, knots Fig. 26.5. Comparison of FHR’s experimental measurements, empirical formulas, and numerical model of bow squat for a Tanker “G”, in Condition C in a canal with vertical sides.

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Bow Squat for Tothil Canadian Laker - Canal R N

R R R N

R R

N

-0.5

Sb , m

R N

R

-1.0

N

R

Barrass Huuska Romisch Yoshimura Numerical Tothil

N

Example

R

Canal Bottom

-1.5 4

5

6

7

8

Vk, knots Fig. 26.6. Comparison of Tothil’s experimental measurements, empirical formulas, and numerical model of bow squat for a Canadian Laker in a canal. Table 26.3. BAW’s Post-Panamax containership in unrestricted channel. Lpp (m)

B (m)

T (m)

CB

h (m)

400

50

17

0.84

19

conservative (i.e., larger than measured). The R¨omisch is slightly smaller than the measured values, but follows the trend very well. Appendix 26.B contains worked examples for the Concept Design formulas of Yoshimura and ICORELS and the Detail Design formulas of Eryuzlu and R¨omisch. 26.3.4.2. Example 2: FHR “G” Tanker in a canal with vertical side, Condition C The second example is for the “G” Tanker, Condition C in a canal with vertical sides (similar to a restricted channel) from FHR and Ghent University.42 The 1:50 scale laboratory experiments were performed in a 7.0-m-wide (350-m prototype) towing tank. The measured Sb = 1.18 m for the ship sailing at Vk = 10 kt (Vs = 5.14 m/s). The ship and channel characteristics are listed in Table 26.4. Figure 26.5 shows comparisons among the Barrass, Huuska, R¨ omisch, and Yoshimura formulas and the measured FHR laboratory values for the canal with vertical sides. The numerical values are from a numerical model that is described in Sec. 26.6. In general, the best formulas are the Yoshimura (Concept) and R¨omisch (Detail) as they are nearly exact or slightly conservative for the smaller ship speeds

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738 Table 26.4.

FHR “G” Tanker in restricted channel, Condition C.

Lpp (m)

B (m)

T (m)

CB

h (m)

hT (m)

W (m)

WTop (m)

n (deg)

180

33

13

0.85

14.5

14.5

350

350

0.0

(i.e., larger than measured). Appendix 26.B contains worked examples for the Yoshimura, Barrass (Concept), and Huuska (Detail). The R¨omisch is not included in the worked examples for this case as it has already been demonstrated. The Barrass is a little small, especially for higher ship speeds. The Huuska formula is conservative for all ship speeds. 26.3.4.3. Example 3: Tothil’s Canadian Laker in a canal The third example is for a Canadian Laker in a canal with sloping sides (typical canal). These data are from Tothil’s 1:48 scale model experiments.43 The measured Sb = 0.93 m for the ship traveling at 6.98 kt (Vs = 3.59 m/s). Ship and channel features are listed in Table 26.5. Figure 26.6 shows comparisons among the Barrass, Huuska, R¨ omisch, and Yoshimura formulas and the measured Tothil laboratory values for the canal case. The numerical values are from a numerical model that is described in Sec. 26.6. In general, the best formulas are the Barrass (Concept), Huuska (Detail), and R¨omisch (Detail). The Barrass is a good match for ship speeds less than 6.54 kt, but does not follow the measured values for increasing speeds. The Huuska is on the low side, but matches reasonably well until Vk exceeds 6.54 kt. The R¨omisch is on the low side, but follows the measured trend of the data for all speeds. The Barrass and R¨omisch formulas are included in worked examples in Appendix 26.B. 26.4. Recent Investigations of Ship Squat So far we have discussed the PIANC empirical formulas for predicting ship squat. These are based on “idealized” conditions with single vessels that are sailing along the centerline of symmetrical channels. Unfortunately, real-world channels and ship transits are seldom this simple. This section discusses some recent research in laboratory and ﬁeld measurements of ship head-on passing encounters and overtaking maneuvers in two-way traﬃc, stern-transom eﬀects, abrupt sills, and oﬀset and drift angle eﬀects for ships sailing oﬀ the centerline with drift angles. When two ships pass or overtake each other, the water ﬂow and corresponding squat is aﬀected as a function of the other ship’s size, speed, and direction of travel, and the channels conﬁguration. Dand44 was one of the ﬁrst to study this Table 26.5.

Tothil’s Canadian Laker in a canal.

Lpp (m)

B (m)

T (m)

CB

h (m)

W (m)

WTop (m)

n

215.6

22.9

7.77

0.86

9.33

72.3

105.9

1.8 (29 deg)

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phenomenon. He found increases in bow squat of 50–100% during passing and overtaking encounters. During the past 10 years, the BAW has conducted many ﬁeld and laboratory studies to investigate ship–waterway interactions, especially head-on passing encounters and overtaking maneuvers of ships in restricted channels within German federal waterways. Preliminary studies of the dynamic response of large containerships in laboratory models have shown tendencies of reduced squat.40,41,45 These results were conﬁrmed by additional model tests in restricted and unrestricted channels and ﬁeld measurements along the Elbe River.31 The FHR (in cooperation with the Ghent University) has conducted laboratory experiments to study passing and overtaking in their automated towing tank as part of a larger study to improve their ship simulator for traﬃc in Flemish waterways.46 Finally, the Delft University of Technology47 had conducted some numerical modeling of the eﬀects of ship oﬀset and drift angles on ship squat. Thus, this section presents a summary of recent laboratory, ﬁeld, and numerical investigations of ship squat in real-world situations including head-on passing encounters, overtaking maneuvers, wider stern-transoms, and ships with oﬀset and drift angles. 26.4.1. Head-on passing ship encounters 26.4.1.1. BAW laboratory experiments Laboratory experiments were conducted at the BAW-DH shallow water basin to study squat as a function of ship size, hull form, draft, speed, direction of travel, and channel water level. This facility has approximate dimensions of 100-m length, 35-m width, and 0.7-m maximum water depth. Geometric and dynamic conditions were accurately scaled according to dimensional analysis at a scale of 1:40. A section of the River Elbe (i.e., restricted or conﬁned channel) with a width of 1.0 km and length of 1.5 km was modeled. The cross-section had a channel depth h = 18.5 m, channel width of 265 m, and river width of 850 m. The results of a Panamax (PM) containership (PM32) and a Post-Panamax (PPM) bulk carrier (MG58) during head-on passing were investigated (Table 26.6). Note that the MG58 is the larger vessel. The two ships passed each other at a passing distance of 156 m (between course lines). The range of ship speeds for the two ships was approximately 7–14 kt for the PM32 and 7–12 kt for the MG58. A laser measuring system was installed on the self-propelled, cable-guided model ships to record their vertical behavior. Measurements were recorded over a distance of approximately 90 m, including acceleration and braking phases of each run. The velocity-independent precision of the laser system was ∆S <1 mm model, Table 26.6.

BAW ship head-on passing characteristics.

Code

Description

PM32 MG58

Panamax containership Bulk carrier

Lpp (m)

B (m)

T (m)

CB

280 349

32.2 58

12.8 14.5

0.68 0.80

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Fig. 26.7. Laboratory measurement of the eﬀect of head-on passing on bow and stern squat for a PM containership passing a large bulk carrier in the River Elbe. The dark blue curves represent the single runs of the containership; the light blue curves the encounters with the large bulk carrier.

corresponding to <4 cm in the prototype. Additional measurements with a point, laser-geometric method allowed for correlation of squat as a function of ship speed, with an accuracy of ∆S < 1 mm (model) for speeds up to 14 kt prototype. Figure 26.7 illustrates the eﬀect of passing on bow and stern squat for the smaller PM32 containership. The measured squat for the single PM32 sailing by itself is shown in dark blue. The eﬀect of the larger MG58 bulk carrier on the PM32 squat is shown in light blue. An additional increase in maximum bow squat for the PM32 (Vk = 14 kt) of ∆S ≈ 0.6 m was recorded due to the passing encounter with the MG58 (Vk = 12 kt). The trim of the PM32 changed from even keel for single runs in the channel and low speeds to bow trim at higher passing speeds during the encounter situation. Figure 26.8 is the analogous ﬁgure for the larger MG58 bulk carrier, but shown in red colors for ease of readability. The larger and slower MG58 experienced an additional squat of ∆S ≈ +0.2 m at the stern. The trim of the MG58 changed only slightly at the stern from its original trim as a single ship in the channel. 26.4.1.2. BAW ﬁeld measurements Field measurements of 12 transits on PPM containerships along the River Elbe were made between April 2003 and June 2004. Meteorological conditions included very calm to stormy (up to Beaufort Wind Scale 9). The shipping company Hapag Lloyd Container Line GmbH (HLCL) supported eight journeys (transits) of Hamburg Express Class ships (7506 TEU), and Yang Ming Marine Transport Corporation

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741

Fig. 26.8. Laboratory measurement of the eﬀect of head-on passing on bow and stern squat for a large bulk carrier passing a PM containership in the River Elbe. The dark red curves represent the single runs of the large bulk carrier; the light red curves the encounters with the containership.

Table 26.7.

Characteristics of the PPM containerships in the BAW ﬁeld measurements.

Vessel type Hamburg Express Class 5500 TEU YM Class

Lpp (m)

B (m)

T (m)

CB

Capacity TEU

320.4 274.7

42.8 40

10.8–12.6 11.4–13.2

0.62–0.65 0.56–0.59

7506 5500

(YM) supported four journeys on ships of the 5500 TEU Class with Tollerort Container Terminals (TCT) acting as the intermediary.8 Table 26.7 presents selected characteristics of the vessel types as well as the range of mean draft T and draftdependent CB during these journeys. In Hamburg Harbor, the containerships were equipped with four autonomous digital global positioning systems (DGPS) on the bow and the bridge and one data collection system on the bridge. Vessel dynamics data were collected from Container Terminal Altenwerder (CTA) or from TCT until just north of Scharh¨ orn (about 120 km from Hamburg Harbor). The width of the channel in this section ranged from 250 to 400 m. Current, temperature, and conductivity were measured by a very fast, small ship at six cross-sections of the lower River Elbe just before the passing encounters. Head-on and passing situations were recorded and documented.8 Vessel movement, nautical maneuvers, local squat, trim, heel, and net maneuvering lane were calculated using special water gage evaluations, precise DGPS measurements, and calculations of virtual reference positions. Vessel data included propeller speed, rudder position, etc. Maximum diﬀerences between water level interpolation and DGPS zero measurements of <1 cm were obtained.48

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Fig. 26.9. Cumulative distribution of the increase in squat for 125 head-on passing encounters of large PPM containerships (HLCL and YM) at the channel of the lower and outer River Elbe.

Squat measurement errors were estimated to be ∆S = ±0.05 m for UKC determination. Given the quality of the digital terrain model from area and traﬃc soundings, a precision of ∆UKC < ±0.2 m was estimated. Precisions of ∆VS = ±0.08 kt for ship velocity (over ground) and ∆Φ = ±0.07 deg for ship heel were also estimated.48 During the seven-hour transits of the 12 ships from Hamburg to the sea, 125 head-on passing encounters were recorded. Figure 26.9 shows the increase in bow (blue diamond) and stern (red square) squat due to these head-on passing encounters and the corresponding cumulative distribution curve. The increase in squat was about the same at both the bow and stern. For this limited set of large containerships in the River Elbe, the maximum increase in squat was 0.44 m; 50% of the cases experienced bow or stern squat less than 0.16 m, while 90% were less than 0.33 m. 26.4.1.3. FHR laboratory experiments The FHR conducted comprehensive laboratory studies of head-on passing encounters to improve the quality of their ship simulator.42 Figure 26.10 illustrates the eﬀect on the squat of a containership (LOA = 291.3 m, B = 40.3 m, T = 13.5 m) sailing at a forward speed of 12 kt, caused by a head-on passing encounter with a bulk carrier (LOA = 310.6 m, B = 37.8 m, and T = 13.5 m). The lateral distance dy between the two centerlines was 114.5 m and the water depth h was 17.1 m. The triangles indicate the sinkage fore and aft of the containership as a function of the relative longitudinal position of both vessels if the bulk carrier approaches at a speed of 8 kt, while the squares refer to an approach speed of 12 kt. The abscissa takes values of −1 and +1, respectively, when the bows and the sterns are located at the same longitudinal position. When the two bows meet, the ship’s bow sinkage increases, whereas the stern is lifted, resulting in trim by the bow. The trim changes

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743

RELATIVE LONGITUDINAL POSITION (-) -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0

0.2

SINKAGE (m)

0.4

0.6

0.8

1

sinkage aft - VT = 12 knots sinkage fore - VT = 12 knots

1.2

sinkage aft - VT = 8 knots sinkage fore - VT = 8 knots

1.4

Fig. 26.10. Eﬀect of passing encounter on ship bow and stern squat as a function of ship speed in FHR tow tank for containership and bulk carrier.49

sign when the midship sections of both ships are at the same position. During the second part of the meeting, the sinkage aft is increased while the bow is lifted. In the given examples, the sinkage aft of the containership increases from an initial value of 0.6 to about 0.9 m, if the bulk carrier has a speed of 8 kt and to about 1.2 m when both ships have a speed of 12 kt. This corresponds to an increase in squat of 50% for the 8-kt case and 100% for the 12-kt case. 26.4.2. Overtaking ship maneuvers 26.4.2.1. BAW laboratory experiments The squat interaction between overtaking Feeder (VG3) and General Cargo (VG4) vessels (traveling in the same direction) were simulated in a 1:33.3 scale laboratory model. A schematized trapezoidal cross-section of a portion of the western Kiel Canal (approximate length of 100 km, channel width of 70–90 m, and depth of 11 m) was modeled. Maximum overtaking squat values were measured only during the time when both ships were aligned parallel to each other. The lateral passing distance during the time when the ships were parallel was 54 m (between course lines). The two ships in this experiment had the properties listed in Table 26.8. Table 26.8.

BAW model ships during overtaking maneuvers.

Code

Description

VG3 VG4

Feeder Containership General Cargo ship

Lpp (m)

B (m)

T (m)

CB

158 127

23 19

7.5 6.1

0.66 0.725

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Fig. 26.11. Laboratory measurements of the eﬀect of overtaking on bow and stern squat for General Cargo (VG3) and Feeder containership (VG4) at the western Kiel Canal. The bow and stern squat values for the VG3 are shown in red, and the VG4 are shown in blue.

Figure 26.11 is similar to Figs. 26.7 and 26.8 for head-on passing ships. It shows bow and stern squat for both ships as single ships and during the overtaking maneuver. Squat during overtaking is larger than for single ships. The increase in stern squat for the VG3 was ∆S ≈ 0.6 m and ∆S ≈ 0.8 m for the VG4 at a speed of Vk = 8.1 kt (15 km/h). This increase in squat is caused by the eﬀect of the additional hydrodynamic mass and channel blockage of each ship. Since both ships experienced a common speed-dependent long wave, they had the same order of magnitude of total stern squat Ss = 1.0 m at Vk = 8.1 kt (15 km/h) (light blue and light red curves at left side of Fig. 26.11). The shorter VG3 squatted with even keel in the long wave of the larger VG4 (light red curves).

26.4.2.2. FHR laboratory experiments Results from the FHR laboratory experiments on overtaking maneuvers are shown in Fig. 26.12. It shows sinkage fore and aft of the same containership from the ship passing experiments, sailing at a speed of 12 kt, while overtaking a bulk carrier (LOA = 301.5 m, B = 46.7 m, and T = 15.5 m) sailing at 8 kt. The water depth h was 18.6 m. Three lateral distances dy between centerlines were investigated: 84, 124, and 205 m. Squat increased up to 0.3 m as the lateral distance decreased between vessels during these overtaking experiments. This is equivalent to an increase in ship squat of over 40%.

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745

Fig. 26.12. Eﬀect of overtaking maneuver on bow and stern squat as a function of lateral distance between ship centerlines for a containership and bulk carrier in the FHR tow tank.

26.4.3. Wide stern-transom eﬀects The BAW collected ﬁeld measurements of bow and stern squat for the 12 large containerships (HLCL and YM) during transits in the River Elbe from Hamburg to the North Sea. Flow and density conditions covering the entire channel navigation were obtained for nine of these runs. Fig. 26.13 shows the maximum bow squat (HLCL = squares; YM = triangles) for these nine transits along the River Elbe. They represent ranges of vessel types, channel conﬁgurations, and UKC for the two shipping companies. In general, the YM vessels experience a larger bow squat due to the design of the hull with a wider transom-stern. Above a speed of 11 kt, the ship starts to squat and trim strongly. However, once the transom-stern submerges below a draft of 12 m, the ship experiences greater buoyancy, which causes it to trim by the bow. This produces larger bow squat then ships without the wider transomstern. The HLCL ships exhibited a much weaker trim, as the narrower stern does not immerse even with larger drafts. The buoyancy remains approximately equally distributed along the hull compared to the wider transom-stern of the YM ships. Therefore, in extremely shallow water, the trim behavior and the deepest point of a vessel (here the bow squat) clearly depends on the overall design of the underwater hull and especially on the buoyancy distribution in the longitudinal direction. This result indicates that, for these wider transom-stern ships, the use of the CB may not be as reliable an indicator of squat as has traditionally been observed.8

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746

0

2

4

6

8

10

12

14

16

v d.W. [kn] [Kn] vSS c.W. 18 20

0,00

0,50

1,00

1,50

BERLIN EXPRESS 22.06.03 SHANGHAI EXPRESS 01.06.03 SHANGHAI EXPRESS 27.07.03 HAMBURG EXPRESS 15.06.03

2,00

BERLIN EXPRESS 01.02.04 MING COSMOS 22.02.04 MING PLUM 28.03.04

SMAX [m]

MING COSMOS 17.04.04 MING COSMOS 12.06.04

2,50

Fig. 26.13. Field measurements of bow squat for nine container vessels of HLCL (blue/light blue squares) and YM (red/yellow triangles) in the channel of the lower and outer River Elbe.8

26.4.4. Vertical variations in the channel In real channels, the bathymetry is not constant, especially where the entrance channel meets the oﬀshore contours or enters more sheltered waters. Some channels are characterized by undulating ripples along the channel bottom that can have signiﬁcant vertical rise above the bottom. An abrupt change in depth or sill due to dredging can induce a signiﬁcant transient squat that can be critical if the ship is entering at deep water speeds. There has been little new research on these eﬀects, but the designer should be aware of their potential impact. The BAW has conducted laboratory experiments on the eﬀects of these ripples on ship squat. These results should be available in the future, but additional research is recommended. 26.4.5. Ship oﬀset and drift angle Ships in the PIANC formulas are assumed to be sailing on the centerline of the channel. When ships are oﬀset from the centerline, they experience increased squat because the hydrodynamic pressure is aﬀected by the bank. The National Ports Council50 showed that squat increases as the UKC and distance D between the ship and the toe of the bank decrease relative to beam B. Squat increased in a restricted channel from 16% to 47% for 1.1 ≤ h/T ≤ 1.2, 0.5B ≤ D ≤ B, and CB from 0.70 to 0.85. Squat increased even more in a canal due to the larger bank eﬀect. The bank eﬀect became insigniﬁcant for D > 3B. Similarly, a ship with a drift angle to the channel centerline experiences increased water ﬂow past the hull due to the increased blockage factor and a smaller gap

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between the ship and the channel. The ship acts as a lifting surface as it moves asymmetrically through the water. Drift angles are usually the result of trying to compensate for large wind forces, especially on containerships. 26.4.5.1. Delft numerical model The Delft University of Technology (Delft) has recently completed a limited set of numerical modeling of ship squat for ships sailing with an oﬀset and drift angle to the channel centerline.47 A panel method was used in the tests for a 6500 TEU Post-Panamax container ship with one draft but a range of oﬀsets and drift angles. The potential ﬂow model includes inertial eﬀects, but no viscosity that will cause vortices and increased squat if included. The modeled ship had Lpp = 302 m, B = 42.9 m, T = 14 m, and CB = 0.67. The canal had W = 300 m and h = 16 m. The UKC = 2 m with oﬀsets of 0 and ±20 m, and drift angles of 0, ±7.5 deg and ±15 deg. They found that both oﬀsets and drift angles increase squat, in a quadratic manner. High drift angles should be avoided by using tugs if available. They recommended additional research for a range of ships, channels, UKC, oﬀsets, and drift angles. 26.4.5.2. FHR laboratory experiments The FHR has conducted towing tank experiments with containerships to study ship oﬀset and drift angle eﬀects on squat. Figure 26.14 shows that moving the ship laterally from the center of the channel (red) to the toe of the bank (blue) results in an increase in squat of about 20%. At higher ship speeds, however, this eﬀect is ampliﬁed. A slight bow squat turns into a signiﬁcant stern squat, and it is clear that the ship sailing oﬀ-center will reach its critical speed much sooner.

Fig. 26.14. Inﬂuence of oﬀset on squat of a containership (Lpp = 331.3 m, B = 42.8 m, and T = 14.5 m) sailing at constant speed in a channel with h = 19.6 m. Scale 1:80 towing tank tests, no propeller action. Open symbols: stern; closed symbols: bow.49

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Fig. 26.15. Inﬂuence of drift angle on squat of a containership at constant speed in a rectangular channel of 565-m width with h = 16 m. Ship test speciﬁcations same as Fig. 26.14.49

Figure 26.15 shows the bow and stern sinkage (squat) of a containership as a function of speed for several drift angles (0, 5, and 10 deg). The bow sinkage increases signiﬁcantly with the drift angle, whereas the stern sinkage decreases slightly.

26.5. Muddy Navigation Areas 26.5.1. Governing eﬀects The presence of a ﬂuid mud layer on the bottom of a channel has a signiﬁcant inﬂuence on ship behavior in general, and sinkage and trim in particular. Two eﬀects play a dominant role: (1) The pressure ﬁeld around the moving hull causes undulations of the water– mud interface that themselves modify the distribution of vertical forces over the length of the ship and, therefore, sinkage and trim. (2) If ship’s keel penetrates into the mud layer, the hydrostatic (buoyancy) force acting on the submerged hull increases due to the higher density of the mud. The interface deformation is a function of many parameters, such as ship speed, layer thickness, mud density and rheology, and (initial) UKC referred to the mud– water interface. Contact between the ship’s keel and the mud layer depends mainly on the UKC, but is also inﬂuenced by the interface undulations and the ship’s sinkage. As a result, both eﬀects are not independent. A general description of the vertical interface motions on squat is presented. Most of the information available on this subject is based on experimental research, mostly at model scale.

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26.5.2. Mud-water interface undulations A ship navigating above ﬂuid mud layers will cause vertical interface motions (internal waves, internal undulations) that are inﬂuenced by the ship’s forward speed (Fig. 26.16): • At very low speed (ﬁrst speed range), the interface remains practically undisturbed. • At intermediate speed (second speed range), an interface sinkage is observed under the ship’s bow if the ﬂuid mud layer is relatively thick. At a certain time, an internal hydraulic jump, perpendicular to the ship’s longitudinal axis, is observed. The front of this internal jump moves aft with increasing speed. • At higher speeds, the internal or interface jump occurs behind the stern (third speed range). It can be shown by means of a simpliﬁed theory that the critical speed separating the second and third speed ranges is a function of the mud ρ2 to water density ρ1 ratio and the water depth h1 (Fig. 26.17). 8 ρ1 (26.30) (1 − S1 )3 gh1 1 − Ucrit = 27 ρ2 where S1 = AS /Ac1 is similar to blockage factor S except that the Ac1 is the crosssectional area of the channel to the top of the mud layer. This equation is based on ideal ﬂuid assumptions, and appears to underestimate the critical speed for mud layers of higher viscosity.

Fig. 26.16. Mud–water interface undulations for second speed range (top) and third speed range (bottom).51

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10

Ucrit (knots)

8 6

h1 = 25 m h1 = 20 m h1 = 15 m

4

h1 = 10 m

2 0 1

1.05

1.1

1.15

1.2

1.25 1.3 ρ2/ρ1

Fig. 26.17. Critical speed separating second and third speed ranges as a function of mud–water density ρ2 /ρ1 ratio for diﬀerent water depths h1 .51

The description above is typical for motions of the mud–water interface occurring when a ship moves with a positive UKC above a ﬂuid mud layer of low viscosity (black water). In case of a negative UKC (i.e., when the keel penetrates the mud layer), a second internal wave system, comparable to the Kelvin wave system in the water–air interface, interferes with the hydraulic jump. This may result in either an interface rising amidships or a double-peaked rising along the hull. Figure 26.18 illustrates the eﬀect of speed (5 and 10 kt), UKC (−12% to +10%), and mud density (1100–1250 kg/m3 ) on the interface undulation pattern. Due to the vertical motion of the interface and the ship, contact between the ship’s keel and the mud layer can occur even if, initially at rest, the UKC of the ship is positive relative to the mud–water interface. Figure 26.19 shows the initial UKC required to avoid contact between mud and keel as a function of Depth Froude number Fn (speed) for diﬀerent mud characteristics. 26.5.3. Eﬀect of mud layers on sinkage and trim The eﬀect of the presence of a ﬂuid mud layer covering the bottom on the ship’s vertical motions is closely related to the interface deformation. If no contact between the ship’s keel and the mud layer occurs [Figs. 26.20(a) and 26.20(c)], a rising interface yields an increased velocity of the ship relative to the water and, as a result, a pressure drop and a local water depression. A mud–water interface sinkage, on the other hand, leads to a local decrease of the relative velocity and an increased pressure, at least compared to the solid bottom case. In case of contact between keel and a rising mud interface (Fig. 26.20(b)), the velocity of the mud relative to the ship’s surface decreases. Contact with a lowered interface with negative UKC (Fig. 26.20(d)) leads to an increased relative ﬂuid velocity, with associated local pressure ﬂuctuations acting on the ship’s keel. Figure 26.21 illustrates the eﬀect of the presence of a mud layer on the sinkage and trim of a containership for the case in which the initial UKC is suﬃciently large so that the interface undulations do not cause any contact between the keel and the mud layer. The sinkage for a ship sailing in a muddy bottom condition is decreased

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-5 -4.5 -4 -3.5 -3 -2.5 -2

Layer thickness:

-1.5

3.0 m

Density:

-1 -0.5

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Ship’s speed:

UKC to interface:

5 knots

-12% -7% +4% +10%

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interfaceinterface position above bottom (m) position abovesolid solid bottom

Interface motions -5 -4.5 -4 -3.5 -3 -2.5 -2

3.0 m

Layer thickness:

-1.5

Density:

-1 -0.5

1100 kg/m³

Ship’s speed:

UKC to interface:

10 knots

-12% -7% +4% +10%

0 -1

-0.5

0

0.5

1

Interfaceinterface position above solid bottom (m) position above solid bottom (m)

Interfaceinterface position above bottom position abovesolid solid bottom (m) (m)

-5

Fig. 26.18.

-4.5 -4 -3.5 -3 -2.5 -2

Layer thickness:

-1.5

3

UKC to interface:

-1 -0.5

Ship speed:

3.0 .0 m 10 %

5

Density (kg/m³):

5 knots

1100 1150 1180 1210 1250

0

-1

-0.5

0

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1

-5 -4.5 -4 -3.5 -3 -2.5 -2

Layer thickness:

-1.5

3

UKC to interface:

-1 -0.5

Ship's speed:

.0 m 3.0 10 %

10

Density (kg/m³):

10 knots

1100 1150 1180 1210 1250

0

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Inﬂuence of speed, UKC, and mud density on undulations of the interface.52

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(a)

(b) Fig. 26.19. Critical UKC h1,crit /T for diﬀerent mud layers with stopped propeller for model ship D, where mud layer thickness to draft ratio h2 /T (a) 0.11 and (b) 0.22 as a function of Depth Froude Number Fn .52

relative to the condition in which the mud layer is replaced by a solid bottom. This is because the ship can “feel” the hard bottom more than the softer, less dense, mud layer. If the mud layer is replaced by water (normal conditions without a mud layer), however, the sinkage would decrease relative to the condition with the mud layer. However, this does not take into account the eﬀect of extra buoyancy (i.e., mud is denser than water), but this is only important in very dense mud layers and/or important penetration. In general, the inﬂuence on trim is more important than sinkage since the mud layer causes the ship to be dynamically trimmed by the stern over its complete speed range. Thus, the eﬀect of mud layers on average sinkage is only marginal as trim is much more important.

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Fig. 26.20. Eﬀect of mud layers on sinkage and trim (a) no interface contact, (b) contact with mud interface, (c) no contact with interface, and (d) negative UKC. The blue line represents the water surface, the brown line the mud layer interface, and black, the solid bottom.53

The eﬀect of the decrease of UKC is shown in Fig. 26.22. In a range of small positive to negative UKC, the trim is mostly aﬀected in a moderate speed range (second speed range, as deﬁned above). A large negative UKC (keel into the bottom mud–water interface) causes trim by the stern in the complete speed range. The eﬀect of mud on the average sinkage is less important, but the combination of trim and sinkage results in an increase of the sinkage aft in some conditions. Figures 26.21 and 26.22 are valid for slender ships (CB < 0.7) that tend to trim by the stern above a solid bottom. Full-formed ships, on the other hand, usually trim by the bow. In muddy navigation areas, such vessels will experience a reduced trim by the bow — or even trim by the stern — when they have suﬃcient UKC in the second speed range. In the third speed range, this eﬀect will be reduced again. Figure 26.23 shows this eﬀect of midships sinkage and trim as a function of UKC for a full-form trailing suction hopper dredge.

26.6. Numerical Models Many diﬀerent numerical methods can be used to calculate the ship squat. Their only common point is that they calculate the velocity components and the pressure of the ﬂow surrounding the ship. Depending on whether the ﬂuid is modeled as viscous, a potential velocity function can be used or a more sophisticated ﬂow model has to be applied. Some models are based on slender body theory, whereas others use the boundary elements method (BEM) or the ﬁnite element method (FEM).

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(a) sinkage FP (m)

0.6 0.5 0.4 0.3 0.2 0.1 0

(b)

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sinkage AP (m)

0.6 0.5 0.4 0.3 0.2 0.1 0

(c) sinkage midships (m)

0.6 0.5 0.4 0.3 0.2 0.1 0

(d)

0.2 trim (mm/m)

0 -0.2 -0.4 -0.6 -0.8 -1 0

solid 15% mud E

2

4

mud F mud C

6 8 V (knots) mud G mud D

mud H solid 26%

Fig. 26.21. Sinkage (a) fore, (b) aft, (c) and midships, and (d) trim as a function of ship speed for Containership D (LOA = 300 m, B = 40.3 m, h = 13.5 m) sailing above a mud layer of 1.5 m thickness with 15% clearance referenced to mud-water interface (26% to solid bottom). Note the legends are the same for all plots.52

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Fig. 26.22. Sinkage (a) fore, (b) aft, (c) and midships, and (d) trim as a function of ship speed for Containership D (LOA = 300 m, B = 40.3 m, h = 13.5 m) sailing above a mud layer of 3.0 m thickness, ρB = 1,180 kg/m3 , ρD = 1,100 kg/m3 .52

26.6.1. Numerical modeling approaches Most numerical models use the continuity equation to calculate the velocity components and Bernoulli’s equation to obtain pressure that is integrated on the hull for hydrodynamic forces. Then applying Archimedes’ principle, a vertical displacement and a trim angle are calculated. 26.6.1.1. Slender body theory Tuck11 established a mathematical expression for squat with a slender body theory. The slender body theory assumes that the beam, draft, and water depth are very small relative to ship length. This theory uses potential ﬂow where the continuity equation becomes Laplace’s equation. The ﬂow is taken to be inviscid and incompressible and is steady and irrotational. In restricted water, the problem is divided into the inner and the outer problems, following a technique of matched asymptotic expansions to construct an approximate solution. The inner problem deals with ﬂow very close to the ship. The potential is only a function of y and z in the Cartesian coordinate system. In the cross-ﬂow sections, the potential function

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0

1

SHIP SPEED (knots) 3

2

4

5

6

0 SOLID BOTTOM - UKC 10%

MEAN SINKAGE (% of draft)

MUDDY BOTTOM - UKC 10% MUDDY BOTTOM - UKC 0%

1

MUDDY BOTTOM - UKC -10%

2

3

TRAILING SUCTION HOPPER DREDGER LPP = 115.6 m; B = 23.0 m; T = 8.0 m MUD LAYER: ρ2/ρ1 = 1.11 - h2/T = 0.175

4 SHIP SPEED (knots)

0

1

2

3

4

5

6

1 SOLID BOTTOM - UKC 10%

0.8 0.6

MUDDY BOTTOM - UKC 10% MUDDY BOTTOM - UKC -4%

TRIM (mm/m)

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

TRAILING SUCTION HOPPER DREDGER LPP = 115.6 m; B = 23.0 m; T = 8.0 m MUD LAYER: ρ2/ρ1 = 1.225 - h2/T = 0.175

-1 Fig. 26.23. (a) Mean midships sinkage and (b) trim as a function of UKC for a full-form trailing suction hopper dredger (115.6 × 23.0 × 8.0 m3 , scale 1/40) above a simulated mud layer (ρ2 /ρ1 = 1.22, h2 /T = 0.175). Positive trim is equivalent to increased stern squat.54

satisﬁes the two-dimensional (2D) Laplace’s equation with impermeable conditions at the boundaries. The outer problem looks at the ﬂow far from the ship where it mainly depends on x and y directions. Dand and Ferguson55 and Beck56 used slender body theory and found good agreement with squat measurements for ratios of water depth to ship draft h/T > 2. Dand used the cross-sectional strip theory of Korvin-Kroukovsky.57 The slender

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body method works by vertical cross-sections of the ﬂow, so it is also called the one-dimensional (1D) theory of squat. Gourlay58 extended the slender body theory of Tuck with the unsteady slender body theory. This improvement allows one to consider a ship moving in a nonuniform depth since the coordinate system is now earth-ﬁxed, whereas it is shipﬁxed for classic numerical methods. The 1D system still uses vertical cross-sections and decomposition into an inner and outer expansion. The pressure integration is only made on the ship length based on the ship section B(x) at each x along the hull. Resolution of the 1D equation is made with the ﬁnite diﬀerence method. Comparison with experimental results for soft squat situations (h/T > 4) showed good agreement with numerical results. No tests were made for hard squat conditions (i.e., shallow depths) where ﬂow around the ship is aﬀected. 26.6.1.2. Boundary element method The BEM is really based on a particular numerical resolution. It is commonly applied for wave-resistance calculations using Green’s function to calculate the potential velocity function. Derivatives of the potential velocity function give the velocity components in Cartesian coordinates. B¨ uhring59 made a squat model called fast boundary elements method (FBEM) based on this boundary element method. The reliability of the model has to be veriﬁed, however, as no comparisons with ship squat measurements was found. 26.6.1.3. Computational ﬂuid dynamics models A number of commercially available computational ﬂuid dynamics (CFD) models could be used for the prediction of squat. At the core of any CFD problem is a computational grid or mesh where the solution is divided into thousands of elements. These elements are usually 2D quadrilaterals or triangles; and three-dimensional (3D) hexahedral, tetrahedral, or prisms. Mathematical equations are solved for each element by the numerical model. For hydrodynamics the Navier–Stokes equations (NSEs) can be solved to include viscosity and turbulence. The NSEs provide detailed prediction (vortices) of the ﬂow ﬁeld, but require very thin meshes, high central processing unit (CPU) time, and memory storage. Its resolution is also quite diﬃcult with numerical instabilities. Examples of commercial CFD models include Fluent and Fidap. Nowadays, CFD models can solve 3D problems, such as ship squat, but the computation domain has to be relatively narrow using NSE. To extend the width of the computation domain, some models solve the problem by zones. Far from the ship, the model solves a potential function with a nonviscous ﬂuid and, in the vicinity of the ship, the model solves using the NSEs. The advantage of the potential ﬂow solution is that it requires low CPU time and less memory storage. The boundary conditions for the NSE model are extracted from the potential ﬂow solution. One example of this kind of commercial model is ShipFlow. In very restricted water, squat can substantially reduce the vertical cross-section around the ship and can subsequently increase the ﬂow velocity below the hull.

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According to Bernoulli’s principle, the pressure will decrease which will make the ship sink more. Numerical models have to take into account this “over squat” to precisely calculate ship squat in all channel conﬁgurations. So when a ﬁrst squat result has been found, the model has to check that this squat is not disturbing the hydrodynamics in such a way that squat could increase more. This checking is important to ensure a reliable result from the numerical model. As these commercial numerical models do not perform squat checking, they may not be very eﬃcient in restricted water. The user has to be very careful and take the result with reservations since the numerical model could in these conditions underestimate ship squat. 26.6.2. New modeling system to predict ship squat In an unrestricted channel (i.e., open sea or large channel), “over squat” is negligible, and the previous numerical models work well unless the UKC is relatively small (h/T < 1.1). Under a 3-m UKC or in a very restricted channel, the numerical model has to check that the hydrodynamics are not being modiﬁed by the squat. Some empirical formulas try to make such a correction with a restriction factor that multiplies the squat calculated for unrestricted water. For instance, the K coeﬃcient in Barrass and Ks for Huuska are examples of these types of correction factors. Such a modeling system with squat checking was developed by Debaillon.60 The basic principle is to reproduce the physical process of ship squat using a numerical model coupling. As the ship is moving, a return ﬂow is generated around the hull. This induced velocity reduces the pressure under the hull. The ship sinks until pressure forces balance the ship weight. As the ship position changes, ﬂow around it might be diﬀerent, and it has to be updated with a new cycle of hydrodynamic and equilibrium computations. The modeling system (Fig. 26.24) is thus composed of (a) a hydrodynamic model to calculate the ﬂow around the hull, (b) an equilibrium model to move the ship with balanced force and momentum equations, and (c) a mesh updating model to take into account the ship and the free surface displacements. 26.6.2.1. Coupling principle As shown on Fig. 26.24, the system starts from the rest position of the ship with a sinkage equal to its draft T . A 3D mesh of ﬁnite elements (tetrahedral) is constructed with the ship features (i.e., Lpp , B, T , and CB ) and the ﬂuid domain (i.e., h, channel shape, boundary conditions, etc.). A ﬁrst run of the model is done with null velocity of the ship. The equilibrium model is then calibrated with the ship weight (Wb ) and the position of the center of gravity (XG , YG ), as all hull nodes must have no displacements with the hydrodynamic model results. Once these ship features are set up, the system is ready to start. A small ship velocity ∆V is imposed in the hydrodynamic model, which gives hull pressure to the equilibrium model. The latter displaces the hull, so the mesh has to be updated by the third model. The system checks the hull displacement. If it is negligible, the ship velocity is increased by ∆V or the same velocity is retained and a new cycle is begun. The system stops when the velocity has reached the velocity speciﬁed by the user or if the ship has grounded.

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Position at rest

V= ∆ V

Hull Hydrodynamic

Hull node

Fluid mesh

Displacement

updating

Equilibrium pressure

no

759

Displacements < ε yes

V=V+ ∆ V

Fig. 26.24.

Coupling for the ship squat numerical modeling system.

Fig. 26.25.

Example of 3D mesh of a ship in a channel.

26.6.2.2. Finite element method The hydrodynamic model is based on the FEM. It requires a numerical mesh that is a subset of the liquid volume around the ship. An example of the 3D mesh, with the imprint of the ship, is shown in Fig. 26.25. It is important to note that numerical results depend on the resolution of the mesh. If it is not suﬃciently reﬁned, the model results will not be as accurate as possible. The hydrodynamic model solves Laplace’s equation in 3D to obtain the velocity potential function. The moving body is ﬁxed, and an incoming ﬂow is imposed with the same velocity in the opposite direction at the far upstream boundary. A matrix system is assembled with Laplace’s equation for each tetrahedron and the diﬀerent boundary conditions. As the problem is linear, the potential function Ψ is solved by a matrix inversion represented by {Ψ} = [K]−1 {F }

(26.31)

where K is a matrix of Laplace’s equation for each tetrahedral and F is a vector composed of the boundary conditions on the triangles.

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26.6.2.3. Three numerical models In Fig. 26.24, the three numerical models are represented by shaded boxes. The ﬁrst model calculates the velocity components and pressure at each node of the mesh. The second receives pressures at all nodes of the hull. The corresponding vertical force FPZ is obtained by integration of pressure over the hull. Squat ∆Sb is estimated as ∆Sb = α

Wb − FPZ ρgSf

(26.32)

where Sf is the ﬂoating surface of the ship, and α (ﬁxed at 0.9) is a relaxation coeﬃcient to limit the variation of squat in a calculation cycle for the mesh updating model. Pitching θ1 and rolling θ2 angles are estimated in the second model from the x and y components of the moment equations: tan θ1 = α tan θ2 = α

Wb (xG − xP ) x∈Sf y∈Sf x(x − x0 )dx dy

(26.33)

Wb (yG − yP ) . ρg x∈Sf y∈Sf y(y − y0 )dx dy

(26.34)

ρg

The equilibrium (second) model has to calculate the position of the center of vertical thrust P (xP , yP ) by integration of the pressure over the hull, and O(x0 , y0 ) the center of the ﬂoating surface Sf . All nodes of the ship are then vertically translated by ∆z and rotated by θ1 around the y-axis and by θ2 around the x-axis. The third model moves all nodes of the free surface per the results of Bernoulli’s relation, and the hull node displacements in x and y directions. Then the inner nodes of the mesh are moved proportionally according to boundary modiﬁcations and the distance from those boundaries. Since 2006 the University of Compi`egne has been working to improve Debaillon’s squat system. As in Gourlay’s model, the coordinate system will be earth-ﬁxed to allow ship passing or crossing and bridge pile crossings. The system will also be able to take into account a nonuniform water depth along the channel.

26.6.3. Numerical modeling examples Figures 26.4–26.6 showed comparisons of the PIANC empirical formulas with the measured laboratory measurements. These ﬁgures also included comparisons with the numerical model predictions for each example. These examples included BAW’s PPM containership in an unrestricted channel, FHR’s tanker in restricted water, and Tothil’s Canadian Laker in a canal. In general, the numerical model matched the measured values from the laboratory measurements very well. Details of the individual examples are given in the following sections.

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26.6.3.1. BAW PPM containership in an unrestricted channel The mesh contains 57,602 nodes, 255,447 tetrahedra, and 58,458 triangles. A general comparison between squat measurements, empirical formulas, and the numerical model was shown in Fig. 26.4. The numerical modeling match is very good, with a maximum error smaller than 0.12 m. 26.6.3.2. FHR “G” Tanker in restricted water, Condition “C” The mesh contains 35,149 nodes, 158,022 tetrahedra, and 33,532 triangles. Figure 26.5 showed the comparisons. The numerical model had to rotate the hull (trim) around the center of gravity. The agreement with the measured values was reasonable. 26.6.3.3. Tothil’s Canadian Laker in canal The mesh contains 27,191 nodes, 118,513 tetrahedra, and 28,300 triangles. A general comparison between squat measurements, empirical formulas, and the numerical model was shown in Fig. 26.6. The numerical model experienced some numerical instability problems (rotate the hull (trim) around the center of gravity), but still gave good predictions.

26.7. Conclusions and Outlook This chapter has focused on some of the latest advances in predicting ship squat and its eﬀect on underkeel clearance for channel design. Several of the more popular PIANC empirical formulas were presented for Concept and Detailed Design phases. In general, the simpler and more “user friendly” formulas were recommended for the Concept Design phase, but this does not preclude them being used in the Detail Design phase and vice versa. Ultimately, the designer wants the maximum squat value possible (bow or stern) in the Concept Design phase and a more realistic value in the Detail Design phase. All empirical formulas have certain constraints based on the ﬁeld and laboratory data used in their development. It is up to the user to exercise Engineering Judgment when applying these formulas as they give a range of squat values. The PIANC formulas were developed for “idealized” channel and sailing conditions for single ships. Recent research has been conducted to investigate more “real-world” conditions for the latest generation of larger ships. The BAW and FHR have conducted extensive laboratory and ﬁeld investigations of head-on passing and overtaking maneuvers, where squat is a function of ship speed and lateral separation distance. Their results indicate that squat can increase 50–100% relative to a single ship. The BAW measured maximum additional squat during passing encounters of 0.6 m in the laboratory and 0.4 m in the ﬁeld. Similarly, the FHR measured maximum additional squat of 0.3–0.6 m in the laboratory. For overtaking maneuvers in the laboratory, the BAW recorded additional squat of 0.6–0.8 m and

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the FHR measured 0.2–0.3 m. These additional squat values represent the eﬀect of these more realistic ship and channel interactions for a range of conditions. The BAW noted that ships with wider “transom” sterns experience more bow squat than most ships with more streamlined transoms due to the additional buoyancy of the stern. They measured additional bow squat of 0.2–0.5 m as a function of ship speed. The Delft and FHR have conducted research on ships sailing with oﬀsets and drift angle from the channel centerline. Delft found that both oﬀsets and drift angles increase squat, in a quadratic manner. The FHR found that oﬀsets can increase squat by 20% and drift angles produce signiﬁcant increase in bow squat and slight decrease in stern squat. The average sinkage of a vessel navigating in muddy channels is generally reduced by the presence of mud layers. The dynamic trim is aﬀected signiﬁcantly by the generation of interface undulations. For ships navigating above mud layers, the maximum sinkage is comparable to or slightly less than the values occurring if the mud layer were replaced by a solid bottom. Compared to the situation in which the mud layer is not present (i.e., replaced by water), the muddy bottom interface always increases the maximum sinkage, even in case of contact with the mud layer. This means that the mud layer will increase sinkage even if the ship “plows” through it. The maximum sinkage (bow or stern) always increases when the lower part of the water column is replaced by ﬂuid mud. The designer and harbor pilots should be aware that all of these special inﬂuences can increase squat. Sometimes, it is not economically feasible to design a channel for all of these eventualities, but it is always possible to slow down as the conditions warrant. Numerical modeling of ship squat is just beginning to be developed. Historically, ship modeling has been concentrated in the areas of wave resistance models. Squat modeling requires time domain models that are very computer intensive. The increasing cost of dredging and the larger ships coming on line have motivated many institutions around the world (such as USACE, BAW, FHR, CETMEF, and FMA) to begin a more active development of ship squat models. The improvements in computer speed and storage have made these types of models much more promising. The CETMEF numerical model matched measured laboratory values very well in the three examples presented in this chapter. Numerical models will continue to be improved, but ﬁeld measurements and laboratory models will still be necessary to investigate the highly nonlinear dynamic behavior of the newer and larger ships. They will respond diﬀerently than existing ships and continued study of passing and overtaking, bank and bottom eﬀects, and sailing alignment will insure optimum and safe navigation design of channels and waterways.

Acknowledgments The authors wish to acknowledge the Headquarters, US Army Corps of Engineers, Ghent University, Flanders Hydraulics Research, Federal Waterways Engineering and Research Institute, Centre d’Etudes Techniques Maritimes Et Fluviales, and the

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PIANC WG49 for authorizing publication of this chapter. Particular thanks goes to Bryan Barrass, the WG49 Chair Mark McBride, original WG30 and WG49 member Werner Dietze, and WG49 vertical subcommittee members Martin Boll, Hans Moes, Terry O’Brien, and Kohei Ohtsu who assisted with the ship squat and UKC research. Other members of the WG49 making contributions to this eﬀort included Larry Cao, Don Cockrill, Rink Groenveld, Jarmo Hartikainen, Jose Iribarren, Susumu Naruse, Sahil Patel, Carlos Sanchidrian, Esa Sirki¨ a, and Jos Van Doorn.

References 1. J. N. Newman, Marine Hydrodynamics (MIT Press, Cambridge, Massachusetts, 1977), pp. 329–386. 2. C. B. Barrass, Ship Squat Seminar, The Nautical Institute, Humberside Branch, September 1995, pp. 21–33. 3. C. B. Barrass, Thirty-Two Years of Research into Ship Squat, 2nd Squat Workshop 2004, Elsﬂeth, Germany, 3–4 March 2004. 4. C. B. Barrass, Ship Squat and Interaction for Masters, Private report (2007), www. ship-squat.com. 5. PIANC, Approach Channels: A Guide for Design, Final Report of the Joint PIANCIAPH Working Group II-30 in cooperation with IMPA and IALA, Supplement to Bulletin No. 95 (1997). 6. PIANC, Horizontal and Vertical Dimensions of Fairways, Maritime Navigation Commission Working Group 49 (MarCom WG 49), July 2005. 7. O. Huuska, On the Evaluation of Underkeel Clearances in Finnish Waterways, Helsinki University of Technology, Ship Hydrodynamics Laboratory, Otaniemi, Report No. 9 (1976). 8. K. Uliczka and B. Kondziella, Dynamic response of very large containerships in extremely shallow water, Proc. 31st PIANC Cong., Estoril, Spain (2006). 9. U.S. Army Corps of Engineers (USACE), Engineering and Design: Hydraulic Design Guidance for Deep-draft Navigation Projects, Engineer Manual 1110-2-1613, Headquarters, Washington, DC (2004). 10. M. Vantorre, Personal communication (2007). 11. E. O. Tuck, Shallow-water ﬂows past slender bodies, JFM 26(Part 1), 81–95 (1966). 12. E. O. Tuck and P. J. Taylor, Shallow water problems in ship hydrodynamics, Proc. 8th Symp. Nav. Hydrod., Pasadena, CA (1970). 13. R. F. Beck, J. N. Newman and E. O. Tuck, Hydrodynamic forces on ships in dredged channels, J. Ship Res. 19(3), 166–171 (1975). 14. J. P. Hooft, The behavior of a ship in head waves at restricted water depth, Int. Shipbuild. Prog. 21(244), 367–378 (1974). 15. I. W. Dand, Squat Estimation: A Graphical Method for Full Form Ships, National Physical Laboratory, Report No. TM 348 (1975). 16. N. E. Eryuzlu and R. Hausser, Experimental investigation into some aspects of large vessel navigation in restricted waterways, Proc. Symp. Asp. Navi. Const. Waterway. Incl. Harb. Ent. 2, 1–15 (1978). 17. K. R¨ omisch and Empfehlungen zur Bemessung von Hafeneinfahrten, Wasserbauliche Mitteilungen der Technischen Universit¨ at Dresden, Heft 1, 39–63 (1989). 18. A. Millward, A preliminary design method for the prediction of squat in shallow water, Mar. Tech. 27(1), 10–19 (1990).

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19. A. Millward, A comparison of the theoretical and empirical prediction of squat in shallow water, Int. Shipbuilding Prog. 39(417), 69–78 (1992). 20. ICORELS (International Commission for the Reception of Large Ships), Report of Working Group IV, PIANC Bulletin No. 35, Supplement (1980). 21. C. B. Barrass, The phenomenon of ship squat, Int. Shipbuilding Prog. 26(294), 44–47 (1979). 22. C. B. Barrass, Ship Squat — A Reply, The Naval Architect, November 1981, pp. 268–272. 23. N. E. Eryuzlu, Y. L. Cao and F. D’Agnolo, Underkeel requirements for large vessels in shallow waterways, Proc. 28th Int. Navi. Cong., PIANC, Paper S II-2, Sevilla, Spain (1994), pp. 17–25. 24. V. Ankudinov, L. L. Daggett, J. C. Hewlett and B. K. Jakobsen, Prototype measurement of ship sinkage in conﬁned water, Proc. Int. Conf. Mar. Sim. Ship Maneuv. (MARSIM 2000), Orlando, FL, 8–12 May 2002. 25. D. T. Stocks, L. L. Dagget and Y. Page, Maximization of Ship Draft in the St. Lawrence Seaway Volume I: Squat Study, prepared for Transportation Development Centre, Transport Canada, June 2002. 26. M. J. Briggs, Ship Squat Predictions for Ship/Tow Simulator, Coastal and Hydraulics Engineering Technical Note CHETN-I-72, U.S. Army Engineer Research and Development Center, Vicksburg, MS (2006), http://chl.wes.army.mil/library/publications/ chetn/. 27. C. B. Barrass, Ship Squat — A Guide for Masters, Private report (2002), www.shipsquat.com. 28. Y. Yoshimura, Mathematical model for the maneuvering ship motion in shallow water, J. Kansai Soc. Nav. Arch. Japan 200, 41–51 (1986). 29. FMA (Finnish Maritime Administration), The Channel Depth Practice in Finland, Bulletin, Waterways Division, Helsinki, Finland, 12 July 2005. 30. E. Sirkia, Economical eﬃciency to be achieved with a regulatory change only with consideration for navigational risks, PIANC Magazine 129, 23–34 (2007). 31. K. Uliczka, B. Kondziella and G. Fl¨ ugge, Dynamisches Fahrverhalten sehr großer Containerschiﬀe in seitlich begrenztem extremen Flachwasser, HANSA, 141, Jahrgang, Nr. 1 (2004) (in German). 32. Overseas Coastal Area Development Institute of Japan, Technical Standards and Commentaries for Port and Harbor Facilities in Japan (2002). 33. K. Ohtsu, Y. Yoshimura, M. Hirano, M. Tsugane and H. Takahashi, Design standard for fairway in next generation, Asia Navigation Conf. 26 (2006). 34. K. Ohtsu, Personal communication (2007). 35. Canadian Coast Guard, Safe Waterways (A Users Guide to the Design, Maintenance and Safe Use of Waterways), Part 1(a) Guidelines for the Safe Design of Commercial Shipping Channels, Software User Manual Version 3.0, Waterways Development Division, Fisheries and Oceans Canada, December 2001. 36. U. M. Guliev, On squat calculations for vessels going in shallow water and through channels, PIANC Bulletin 1(7), 17–20 (1971). 37. Puertos Del Estado, Recommendations for Maritime Works (Spain) ROM 3.1-99: Designing Maritime Conﬁguration of Ports, Approach Channels and Floatation Areas, CEDEX, Spain (1999). 38. C. J. Huval, Lock Approach Canal Surge and Tow Squat at Lock and Dam 17, Arkansas River Project; Mathematical Model Investigation, Technical Report HL-80-17, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS (1980).

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39. V. Balanin et al., Peculiarities of navigation on canals and restricted channels, originating hydraulic phenomena associated with them and their eﬀect on the canal bed; measurements preventing slope deterioration, Paper S.I-3, 24th Int. Nav. Cong., Leningrad, Russia (1977). 40. G. Fl¨ ugge and K. Uliczka, Dynamisches Fahrverhalten und Wechselwirkungen mit der Fahrrinnensohle von sehr großen Containerschiﬀen unter extremen Flachwasserbedingungen, Proceedings HTG-Kongress 2001, Hamburg (2001) (in German). 41. G. Fl¨ ugge and K. Uliczka, Dynamisches Fahrverhalten mit der Fahrinnensohle von sehr grossen Containershiﬀen unter extremen Flachwasserbedingungen. Hansa (2001). 42. M. Vantorre, E. Verzhbitskaya and E. Laforce, Model Test Based Formulations of Ship-Ship Interaction Forces, Ship Technology Research/Schiﬀstechnik, Band 49, Heft, 3 August 2002. 43. J. T. Tothil, Ships in Restricted Channels, A Correlation of Model Tests, Field Measurements and Theory, National Research Council of Canada Mechanical Engineering Report MB264, January 1966. 44. I. W. Dand, Some Measurements in Interaction between Ship Models Passing on Parallel Courses, NMI R108, August 1981. 45. K. Uliczka and G. Fl¨ ugge, Squat-Untersuchungen f¨ ur sehr große Post-PanamaxContainerschiﬀe, HTG/STG-Sprechtag FA Seeschiﬀfahrststraßen, Hafen und Schiﬀ, Hamburg (2001) (in German). 46. M. Vantorre, E. Laforce, G. Dumon and W. Wackenier, Development of a probabilistic admittance policy for the ﬂemish harbours, 30th PIANC Cong., Sydney, September 2002. 47. H. J. de Koning Gans and H. Boonstra, Squat eﬀects of very large container ships with drift in a harbor environment, MTEC2007 Conference, Singapore, 26–28 September 2007. 48. Ch. Maushake and S. Joswig, Messung von Squat, Trimm und Kr¨angung sehr großer Containerschiﬀe im Rahmen von Grundsatzuntersuchungen auf der Elbe, Hydrographische Nachrichten Nr. 072, Deutsche Hydrographische Gesellschaft (2004) (in German). 49. M. Vantorre and G. Dumon, Model test based requirements for the under keel clearance in the access channels to the ﬂemish harbours, 2nd Squat Workshop Aspects of Under Keel Clearance in Analysis and Application, Elsﬂeth, March 2004. 50. National Ports Council, Ship Behavior in Ports and their Approaches — Part 2: Additional Sinkage Caused by Sailing in the Proximity of Channel Bank, Research Transport Headquarters, London, U.K. (1980). 51. M. Vantorre, Nautical Bottom Approach – Application to the Access to the Harbour of Zeebrugge, HANSA — Schiﬀahrt — Schiﬀbau — Hafen, 138. Jahrgang, Nr. 6 (2001), pp. 93–97. 52. G. Delefortrie, Maneuvering behavior of container vessels in muddy navigation areas, Ph.D. thesis, Ghent University (2007). 53. K. Van Craenenbroeck, M. Vantorre and P. De Wolf, Navigation in Muddy Areas: Establishing the Navigable Depth in the Port of Zeebrugge, Proceedings CEDA/PIANC Conference 1991 (incorporating CEDA Dredging Days): Accessible Harbours, Paper No. E4, Amsterdam (1991). 54. M. Vantorre, Systematische proevenreeksen met het zelfaangedreven schaalmodel van een sleephopperzuiger boven een mengsel petroleum¬trichloorethaan als slibsimulatiemateriaal experimentele waarnemingen en theoretische interpretaties. Rijksuniversiteit Gent/Waterbouwkundig Laboratorium Borgerhout. Gent/Antwerpen (1990).

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55. I. W. Dand and A. M. Ferguson, The squat of hull ships in shallow water, Roy. Inst. Nav. Arch. 115, 237–247 (1973). 56. R. F. Beck, Hydrodynamic forces caused by ship in conﬁned waters, Proc. Am. Soc. Civ. Eng. 107, ASCE, EM3, June 1981, pp. 523–546. 57. B. V. Korvin-Kroukovsky, Investigations of ship motions in regular waves, Trans. SNAME 63, 386–435 (1955). 58. T. Gourlay, Mathematical and computational techniques for predicting the squat of ships. Thesis of the University of Adelaide, Australia (2000). 59. H. B¨ uhring, Prediction of Squat by Fast Boundary Elements, 2nd Squat Workshop, Oldenburg, Germany (2004). 60. P. Debaillon, Syst`eme de mod´elisation de l’enfoncement dynamique des bateaux, Thesis of University of Compi`egne, France (2005).

26.A. Appendix 26.A.1. Least square coeﬃcients for Huuska K1 versus S A least squares polynomial ﬁt of Huuska’s Fig. 26.3 curves for K1 as a function of S was calculated according to the formula: K1 = a0 + a1 S + a2 S 2 + a3 S 3 .

(26.A1)

Table 26.A1 lists the correlation coeﬃcient R2 and the polynomial coeﬃcients for each of the hT /h curves. This allows one to program the value for K1 without having to manually read a plot. 26.A.2. R¨ omisch Kc for canals and restricted channels In Sec. 26.3.3.3, we presented an equation for R¨ omisch’s Kc for canals and restricted channels as a function of the blockage factor S. The Kc given in Sec. 26.3.3.3 is equivalent to

KC = 2 cos

Table 26.A1.

π Arc cos(1 − S) + 3 3

1.5 .

(26.A2)

Huuska K1 versus S least square ﬁt coeﬃcients. Polynomial coeﬃcients

hT /h

R2

a0

a1

a2

a3

0.2 0.4 0.6 0.8 1.0

0.9988 0.9983 0.9963 0.9991 1.0000

1.0 1.0 1.0 1.0 1.0

2.7704 8.0885 −1.9528 1.9453 0.0

214.87 89.87 137.6 45.325 0.0

−569.42 −214.88 −347.93 −129.48 0.0

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Both equations are the explicit solution to 1.5

V2 VCr 2 √ 1 − S + Cr = 3 2ghm ghm

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(26.A3)

which has the critical or limiting ship speed VCr on both sides of the equation. The KC can also be deﬁned several other ways as it is related to Schijf’s limiting speed VL and Schijf’s limiting froude number FHL .9 Schijf’s limiting Froude Number is deﬁned as

VL π Arc cos FHL = √ = 8 cos3 (26.A4) + (1 − S) = KC . 3 3 gh Note that the right-hand side is equivalent to KC for canals and restricted channels, the same as Eqs. (26.28) and (26.A2). Finally, Briggs26 determined the formula for KC from a least square ﬁt of R¨omisch’s limited set of discrete data points in Table 26.2 with an R2 = 0.97. It gives the same results as the other formulas for KC : KC = 0.2472 ln(1/S) + 0.02411.

(26.A5)

26.B. Appendix: Worked Example Problems Several worked examples are contained in this appendix. They are the same examples described in Sec. 26.3. The input ship and channel characteristics were described in the text and listed in Tables 26.3–26.5. 26.B.1. Example 1: BAW’s Post-Panamax Containership in unrestricted channel 26.B.1.1. Constraint check The ﬁrst step is to examine the constraints for this case as listed in Table 26.1 and the input values from Table 26.3. The Fnh is given by 6.84 Vs = 0.50. Fnh = √ = gh 9.81(19)

(26.B1)

Since Fnh ≤ 0.70, it is acceptable for all methods. The full form CB = 0.84 is acceptable for all methods, but slightly exceeds Huuska’s upper limit of CB < 0.8. The ratio of B/T = 50/17 = 2.94, slightly exceeds Eryuzlu and R¨ omisch’s criterion. Next, h/T = 19/17 = 1.12 is acceptable, but minimal UKC. The ratio Lpp /B = 400/50 = 8.0 is larger than Eryuzlu’s upper limit of 6.8. Finally, Lpp /T = 400/17 = 23.53 is slightly larger than the upper limits of Huuska and R¨ omisch. Bottom line: probably acceptable to use the diﬀerent formulas for this case, but remember that this is a much larger vessel than the criteria for which most of these formulas were developed.

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26.B.1.2. ICORELS application This is a logical ﬁrst choice for an unrestricted channel as it is one of the ﬁrst formulas developed as a guide. There are two steps involved in the ICORELS estimate of bow squat Sb . The ﬁrst step is to calculate the ship’s displacement volume Λ using values from Table 26.3 ∇ = CB Lpp BT = 0.84(400)(50)(17) = 285, 600 m3.

(26.B2)

The second step is to substitute this value into the equation for bow squat Sb from Table 26.3 and Sec. 26.B.1.1. Sb = CS

∇ F2 (0.50)2 285, 600 nh = 2.4 = 1.24 m. 2 2 2 Lpp 1 − Fnh (400) 1 − (0.50)2

(26.B3)

This value is too large compared to the measured value of Sb = 0.70 m. Of course, some institutions use a smaller value of the constant CS . Thus, this value could be reduced by using a smaller CS , but it would need to be of the order of 1.4 which is much smaller than commonly recommended. 26.B.1.3. Yoshimura application The Yoshimura formula is a good Concept Design application for this case. It is very straightforward and easy to apply. Substituting values from Table 26.3 and Sec. 26.B.1.1 into Yoshimura’s equation for Sb

3 2 1 (6.84) 0.84 1 0.84 Sb = 0.7 + 1.5 = 1.10 m. (26.B4) + 15 1.12 8.0 1.12 8.0 9.81 Thus, this Concept Design application is on the high side, but at least it is a conservative estimate relative to the measured Sb = 0.70 m. 26.B.1.4. Eryuzlu application This is an example of a Detailed Design application using Eryuzlu’s formula. The steps are described in the following paragraphs. The ﬁrst step is to calculate the correction factor for channel width Kb . Since this is an unrestricted channel with no side boundaries, a value of Kb = 1 was selected since it is assumed that Eryuzlu meant the ﬁrst part of his equation to be for restricted channels only. The second step is to substitute values from Table 26.3 and Sec. 26.B.1.1 into Eryuzlu’s equation for bow squat Sb (19)2 Sb = 0.298 17

6.84

9.81(17)

2.289 (1.12)−2.972 (1) = 1.06 m.

(26.B5)

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This value is also larger than the measured value of Sb = 0.70 m, but it is conservative and similar to Yoshimura’s value. 26.B.1.5. R¨ omisch application This is an example of a Detailed Design application using R¨omisch’s formula. It is probably the most complicated of all the PIANC empirical formulas. The ﬁve steps are described below using the values from Table 26.3 and Sec. 26.B.1.1. The ﬁrst step is to calculate the coeﬃcients required to estimate the critical speed Vcr in an unrestricted channel. They are phase speed or celerity C and the correction factor KU . C = gh = 9.81(19) = 13.65 m/s (26.B6)

0.125 h Lpp = 0.58[(1.12)(8.0)]0.125 = 0.76 (26.B7) KU = 0.58 T B Vcr = CKU = 13.65(0.76) = 10.41 m/s.

(26.B8)

The second step is to calculate the correction factor for ship speed CV 2

4

6.84 6.84 − 0.5 + 0.0625 = 0.22. CV = 8 (26.B9) 10.41 10.41 The third step is the calculation of the correction factor for ship shape CF

2

2 10CB 10(0.84) CF = = = 1.10. (26.B10) Lpp /B 8.0 The fourth step is the calculation of the correction factor for squat at the critical speed K∆T √ K∆T = 0.155 h/T = 0.155 1.12 = 0.16. (26.B11) The last step is to substitute these values into the equation for bow squat Sb Sb = CV CF K∆T T = 0.22(1.10)(0.16)(17) = 0.67 m.

(26.B12)

This value, although slightly small, is in excellent agreement with the BAW measured value of Sb = 0.70 m. 26.B.2. Example 2: FHR “G” Tanker in a canal with vertical sides, Condition C 26.B.2.1. Constraint check As in the ﬁrst example, the ﬁrst step is to determine the constraints for this case using Table 26.1 and input values from Table 26.4. The Fnh is given by 5.14 Vs Fnh = √ = = 0.43. gh 9.81(14.5)

(26.B13)

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Since Fnh ≤ 0.70, it is acceptable for all methods. The full form CB = 0.85 is acceptable for all the methods, but is greater than Huuska’s upper limit of CB < 0.8. The ratio of B/T = 33/13 = 2.54 is good for all methods. Next, h/T = 14.5/13 = 1.12 is acceptable. The ratio Lpp /B = 180/33 = 5.45 is acceptable, although a little low for most methods. Finally, Lpp /T = 180/13 = 13.85 is on the low side, but acceptable. Bottom line: probably acceptable to use the diﬀerent formulas for this case, but remember that some of the original constraints are exceeded. 26.B.2.2. Barrass application The Barrass formula is a good Concept Design formula as it is relatively easy to apply and gives reasonable estimates. The ﬁrst step is to calculate the mid-ship cross-sectional area AS using Table 26.4. AS = 0.98 B T = 0.98(33)(13) = 420.42 m2.

(26.B14)

The second step is to calculate the channel cross-sectional area AC . The zero value of slope n is to account for the vertical sides of the ﬂume. AC = W h + nh2 = 350(14.5) + 0.0(14.5)2 = 5075 m2.

(26.B15)

The third step is to calculate the blockage factor S given by S=

420.42 AS = 0.083. = AC 5075

(26.B16)

This relatively small value of S indicates that the channel should be considered as an unrestricted channel for Barrass application. The fourth step is to calculate the correction coeﬃcient K given by K = 5.74S 0.76 = 5.74(0.083)0.76 = 0.87 => 1.00.

(26.B17)

The value of K = 0.87 is replaced by K = 1.00 since this is the minimum value that Barrass intended for relatively wide channels. The last step is to substitute the values above into the equation for Sb SMax =

1.0(0.85)(10)2 KCB Vk2 = = 0.85 m. 100 100

(26.B18)

This value is a little low compared to the measured value of Sb = 1.18 m, but is a good ﬁrst estimate. 26.B.2.3. Yoshimura application The ﬁrst step in the Yoshimura application is to calculate the equivalent ship speed Ve because it is a canal and not an unrestricted channel. The blockage factor S has

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already been calculated for the Barrass application above. Therefore, Ve is given by Ve =

5.14 Vs = = 5.61 m/s. (1 − S) (1 − 0.083)

(26.B19)

The second step is to substitute values from above, Table 26.4, and Sec. 26.B.2.1 into the Yoshimura equation for Sb

3 1 (5.61) 2 1 0.85 0.85 Sb = 0.7 + 1.5 = 1.19 m. (26.B20) + 15 1.12 5.45 1.12 5.45 9.81 Thus, Yoshimura’s prediction is an excellent match to the measured Sb = 1.18 m. 26.B.2.4. Huuska application The last application will be using Huuska’s Detailed Design formula. R¨omisch’s formula gives very good agreement, but the Huuska formula will be demonstrated here since R¨omisch’s was illustrated in the previous example. The Huuska formula is more complicated to use than some, but not as diﬃcult as the R¨ omisch. It is very similar to the ICORELS formula, but includes a correction factor for restricted channels and canals. The ﬁrst step is to calculate the correction factor K1 that is used in the corrected blockage factor s1 . Since hT = h for this case, which is similar to a canal, one can use the graph from Fig. 26.3 or the least square coeﬃcients in Appendix 26.A to get the value of K1 = 1.0. The second step is to calculate the corrected blockage factor s1 s1 =

S 0.083 = 0.083. = K1 1.0

(26.B21)

The third step is to calculate the correction factor for channel width Ks , which depends on the value of s1 . The ﬁrst equation for Ks is used since s1 > 0.03. Ks = 7.45s1 + 0.76 = 7.45(0.083) + 0.76 = 1.38.

(26.B22)

The fourth step is to calculate the ship’s displacement volume Λ ∇ = CB Lpp BT = 0.85(180)(33)(13) = 65, 637 m3.

(26.B23)

The last step is to substitute these values above into Huuska’s equation for Sb Sb = CS

65, 637 ∇ F2 (0.43)2 nh K = 2.4 (1.38) = 1.38 m. s 2 2 2 Lpp 1 − Fnh (180) 1 − (0.43)2

(26.B24)

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This value is a little large compared to the measured value. The values of both Yoshimura and R¨ omisch are Sb = 1.19 m, which are excellent matches to the measured value of Sb = 1.18 m. 26.B.3. Example 3: Tothil’s Canadian Laker in a canal 26.B.3.1. Constraint check The ﬁrst step is to determine the constraints for this case by comparing calculated values with those in Table 26.1. Using values in Table 26.5, the Fnh is given by 3.59 Vs = 0.38. Fnh = √ = gh 9.81(9.33)

(26.B25)

Since Fnh ≤ 0.70, it is acceptable for all methods. The full form CB = 0.86 is acceptable for all the methods, although it is slightly larger than the upper CB limit of both Barrass and Huuska. The ratio of B/T = 22.9/7.77 = 2.95, slightly exceeds Eryuzlu’s and R¨ omisch’s upper limits. Next, h/T = 9.33/7.77 = 1.2 is acceptable. The ratio Lpp /B = 215.6/22.9 = 9.41 is larger than most upper limits. Finally, Lpp /T = 215.6/7.77 = 27.75 is slightly larger than all upper limits. Bottom line: probably acceptable to use the diﬀerent formulas for this case, but remember that some of the original constraints are exceeded. 26.B.3.2. Barrass application The Barrass is a good Concept Design formula for the canal example. It is one of the best ﬁts to the measured data, but is too low for higher ship speeds above 6.4 kt. It is simple to apply, but there are several steps to follow. Again, the values are substituted from Table 26.5 and Sec. 26.B.3.1. The ﬁrst step is to calculate the mid-ship cross-sectional area AS AS = 0.98 B T = 0.98(22.9)(7.77) = 174.37 m2.

(26.B26)

The second step is to calculate the channel cross-sectional area AC . The slope n (i.e., run/rise = 1.8 = (105.9 − 72.3)/(2 ∗ 9.33)) is equivalent to an angle of θ = 29 deg (i.e., θ = arctan(1/n)). AC = W h + nh2 = 72.3(9.33) + 1.8(9.33)2 = 831.25 m2.

(26.B27)

The third step is to calculate the blockage factor S given by S=

AS 174.37 = 0.21. = AC 831.25

(26.B28)

The fourth step is to calculate the correction coeﬃcient K given by K = 5.74 S 0.76 = 5.74(0.21)0.76 = 1.75.

(26.B29)

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The last step is to substitute the values above into the equation for Sb KCB Vk2 1.75(0.86)(6.98)2 = = 0.73 m. 100 100

SMax =

(26.B30)

This value is a little low compared to the measured value of Sb = 0.93 m, but is a good ﬁrst estimate. 26.B.3.3. R¨ omisch application The R¨omisch formula is a good Detail Design formula, and this application will illustrate how it is used for a canal. The R¨omisch results are slightly low, but the trend follows the measured values reasonably well. The main diﬀerence with the unrestricted case from before is that the correction factor KC replaces KU for the canal application. Input values from Table 26.5 and Secs. 26.B.3.1 and 26.B.3.2 are used in this application. The ﬁrst step is to calculate the coeﬃcients required to estimate the critical speed VCr in a canal. They are phase speed or celerity Cm that is based on the mean water depth hm and the correction factor KC . Although Table 26.2 could be used, the formula for KC [from Eq. (26.28)] is used in this example. AC 831.25 = 7.85 m = WTop 105.9 Cm = ghm = 9.81(7.85) = 8.78 m/s

1.5 Arc sin(1 − S) Kc = 2 sin 3 1.5

Arc sin(1 − 0.21) = 2 sin = 0.46 3 hm =

Vcr = Cm Kc = 8.78(0.46) = 4.06 m/s.

(26.B31) (26.B32)

(26.B33) (26.B34)

The second step is to calculate the correction factor for ship speed CV

CV = 8

3.59 4.06

2

3.59 − 0.5 4.06

4

+ 0.0625 = 0.53.

(26.B35)

The third step is the calculation of the correction factor for ship shape CF

CF =

10CB Lpp /B

2

=

10(0.86) 9.41

2 = 0.83.

(26.B36)

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The fourth step is the calculation of the correction factor for squat at the critical speed K∆T √ K∆T = 0.155 h/T = 0.155 1.2 = 0.17. (26.B37) The last step is to substitute these values into the equation for bow squat Sb Sb = CV CF K∆T T = 0.53(0.83)(0.17)(7.77) = 0.58 m.

(26.B38)

This value is 0.35 m too small compared to the measured value of Sb = 0.93 m. The R¨omisch trend is pretty good, however.

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Chapter 27

Wave-Induced Resuspension of Fine Sediment Mamta Jain Halcrow Inc., 4010 Boy Scout Blvd. Tampa, FL 33607, USA Ashish J. Mehta Department of Civil and Coastal Engineering University of Florida, Gainesville, FL 32611, USA Prediction of turbidity due to ﬁne sediment resuspension in wave-driven environments is a subject of considerable interest. In that regard, experience from shallow lakes, in which forcing is mainly by waves, has been used extensively in this presentation. Turbidity prediction is contingent upon an understanding of two processes, namely the eﬀect of the bottom on wave height and period, and also the rate at which bottom sediment is locally entrained by the wave. Simple approaches pertinent to these processes have been reviewed. Attention is devoted to the role of rheological models in the determination of wave attenuation. This is followed by a brief description of simple modeling of suspended sediment proﬁles subject to sediment erosion, settling, and deposition. The role of sediment-induced stratiﬁcation in governing resuspension is highlighted. Overall, the mainly analytic treatment is meant to help the engineer in developing ﬁrst-order estimates of turbidity levels in wave-driven environments.

27.1. Introduction Resuspension of ﬁne-grained sediment in coastal and inland waters is a subject of extensive interest arising from its signiﬁcance in the maintenance of navigational routes and control of water quality. Enclosed bodies of water such as lakes and reservoirs lend themselves to investigation in this regard, because the eﬀect of astronomical tide is practically nonexistent, and currents due to tributary inﬂows or due to seiching modes are usually (although not always; see, e.g., Refs. 1–3) too weak to resuspend sediment in the absence of wind-generated waves. Such is the case, for example, in Newnans Lake in north-central Florida (Fig. 27.1), which has a surface area of 27 km2 and a maximum depth of 3.9 m.4 Under such conditions, lakes of 775

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Fig. 27.1. (a) Newnans Lake in north-central Florida with location of shallow and deep sites. (b) Typical surface current pattern in the lake (due to two tributary inﬂows and one outﬂow) simulated by the three-dimensional numerical model EFDC (Environmental Fluid Dynamics Code) of Ref. 11. Current magnitudes are generally a few millimeters per second (smaller than the arrowheads) (adapted from Ref. 12).

interest are those that are shallow enough to allow waves to “touch” the bottom and thereby aﬀect its physical state. Sedimentary matter in many lakes tends to be rich in organic matter. A signiﬁcant example of accumulation of such “black mud” is the large but shallow Lake Okeechobee in south-central Florida. This lake has an area of 1,750 km2 and a maximum depth of about 4.5 m below mean sea level. Figure 27.2 shows the thickness contours of the mud deposit. The volume of mud is estimated to be about 193 × 106 m3 .5,6 Such muds can be the primary carriers of sorbed chemicals that play a critical role in governing water quality. In these lakes, suspended ﬁne sediment dynamics is inﬂuenced by pycnoclines arising from temperature and associated hypolimnatic water layer, dissolved substances, or sediment itself. We will be concerned with sediment-induced pycnoclines, since they are usually the most important density gradients that inﬂuence turbidity in such environments. When surface waves propagate over a muddy bottom (Fig. 27.3), their height tends to decrease with distance quite noticeably if the bottom is compliant. Thus, at a given site two issues must be dealt with, namely prediction of the local wave amplitude and period based on the wind ﬁeld, and simulation of sediment resuspension and its signiﬁcance to the turbidity level. We will consider wave damping ﬁrst, followed by resuspension.

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Fig. 27.2. Lake Okeechobee in south-central Florida and contours of black mud thickness in cm (adapted from Ref. 5).

Fig. 27.3.

Wave propagation in a stratiﬁed water–mud system.

27.2. Wave Generation, Growth, and Damping To simulate the (design) maximum eﬀect of waves due to wind from a given direction, duration is assumed to be long enough to allow the generated wave train to reach the downwind shoreline. Also, given the spatial variability in wave amplitude and frequency, it is convenient to examine the eﬀect of waves at a downwind site where the waves are high. Prediction of (nonbreaking) waves at a given wind speed, water depth, and linear distance along the fetch is done in its simplest form via semi-analytic treatments that rely mainly on natural observations. For a spatial representation of wave ﬁeld, use is presently made of “third-generation” models such as SWAN (Simulating WAves Nearshore). This model is based on the wave energy balance equation in the absence of current.7 SWAN has the advantage of enabling the inclusion of the actual bathymetry of the lake. In Fig. 27.4, an application of

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Fig. 27.4. Predicted wave heights in Lake Monroe in central Florida using SWAN (adapted from Ref. 8).

SWAN is illustrated for Lake Monroe in central Florida subject to an easterly wind of 10 m s−1 . The lake area is 32 km2 and the maximum depth is 3.1 m.8 The semi-analytic approach of Young and Verhagen9 is based on extensive observations on waves in a lake. However, this and similar treatments for wave prediction are applicable to noncomplaint, or static, beds only, i.e., there is no allowance for energy dissipation due to mud. On the other hand, in SWAN the inclusion of an appropriate dissipation function for mud is relatively straightforward (e.g., Ref. 10). Given either approach, it is a matter of interest to assess the damping eﬀect of mud on wave amplitude relative to dissipation over a static bottom. For that purpose, we will review simple physical descriptions of bed dynamics for the estimation of wave damping. The wave energy balance equation (as used in SWAN, for example) is based on the consideration that the wave timescale, on the order of seconds, is much shorter than the wind duration over the desired fetch, which is much longer than the wavelength. That duration can be typically on the order of minutes to tens of minutes in the fetch of initial wave growth. The steady-state energy conservation equation is ∂(ECg cos θ) ∂(ECg sin θ) + = −εD ∂y ∂x

(27.1)

in which x and y are coordinates normal and parallel to the upwind shoreline, respectively, θ is the angle between the direction of wave approach relative to the x-coordinate, E is the wave energy per unit water surface area, Cg is the wave group velocity, and εD is the wave-mean energy dissipation per unit bottom area per unit time. Wave propagation in the two directions enables the simulation of wave refraction and damping of the wave crest (e.g., Ref. 7). Inasmuch as the angle between the wave crest and the shoreline is reduced by refraction in shallow water, the eﬀect of the y-gradient in Eq. (27.1) tends to be small compared to the x-gradient. Accordingly, for present purposes this equation may be simpliﬁed by considering waves normally incident to a straight shoreline

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(and shore-parallel isobaths), yielding d (ECg ) = −εD dx

(27.2)

in which Cg is now the wave group velocity in the x-direction. This velocity is related to the wave angular frequency σ (= 2π/T , where T is the wave period) and the wave number kr via Cg = dσ/dkr . The wave energy is obtained from E = ρw ga2 /2, where ρw is the water density, g is the acceleration due to gravity, and a is the wave amplitude (equal to one-half the height). The quantities εD , kr (hence Cg ), and E depend on the mechanism for wave energy dissipation. Wave energy dissipation mechanisms broadly fall within two categories. The ﬁrst includes static-bed mechanisms in which the bed particle matrix remains stationary even as water may move through the pores. The second category includes mechanisms associated with compliant bottoms. In these mechanisms, heaving motion of the matrix (which may be accompanied by pore water ﬂow relative to the matrix) occurs in response to wave-induced variation of dynamic pressure along the bottom surface. Static-bed losses occur over the entire grain size range from sand to clay, but as the grain diameter decreases, losses within the compliant bottom take over and become paramount in soft clayey substrates. As a result, mechanisms pertaining to the latter category are of primary interest here. For the simple case of constant water depth, Eq. (27.2) reduces to Cg

dE = −εD . dx

(27.3)

To proceed we will take the surface wave amplitude decay relationship to be of the form ax = a0 e−ki x

(27.4)

where ki is the wave damping coeﬃcient. By combining Eqs. (27.3) and (27.4), ki is obtained from εD . (27.5) ki = 2Cg E The magnitude of ki , which in turn determines εD , depends on the rheology (stress–strain behavior) of the substrate. 27.3. Basic Rheological Models Loss of energy in water due to frictional resistance at the static-bed surface arises within the (laminar or the turbulent) boundary layer. For nonbreaking waves, this is essentially the only mechanism that operates at the bed surface, and its magnitude depends on how rough the surface is. All other mechanisms are eﬀective within the bed, and therefore require descriptions of dissipation that depend on the state (continuum or two-phased particle–water mixture, solid, or ﬂuid) of the bottom. Several basic rheological models are available to predict ki (Table 27.1).

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780 Table 27.1.

Bottom sediment constitutive properties and models. Property or medium

Constitutive behavior

Surface roughness

Frictional resistance

Two-phase

Percolation

Continuum

Viscousa

Compliant solid

Continuum

Viscoelastica

Fluid Compliant solid and ﬂuid

Continuum Continuum

Viscoelastica Viscoplastic

Compliant porous solid

Two-phase

Poroelastica

Bed type Noncompliant (static) solid Noncompliant (static) porous solid Fluid

a These

Constitutive models Laminar or turbulent boundary-layer Darcy viscous loss Newtonian or nonNewtonian viscous Kelvin–Voigt or higher order Maxwell or higher order Bingham plastic or higher order Coulomb friction and Darcy viscous loss

behaviors are addressed further in this paper.

Fig. 27.5. Rheological domains based on combined eﬀects of solids volume fraction and interparticle interaction (adapted from Ref. 17).

In Fig. 27.5 the domains of diﬀerent (nonturbulent ﬂow) behaviors are deﬁned entirely in the qualitative sense in terms of the combined eﬀect of the solids volume fraction ϕ (or porosity np = 1 − ϕ) and interparticle interaction I. Newtonian behavior is conﬁned to the lower left-hand corner characterized by low values of ϕ and very low (almost negligible) values of I. The zone of particular interest (dashed rectangle) is in the range of moderately low to medium ϕ and low to medium I. This is part of a larger zone within which the stress-versus-strain rate response of mud shows thixotropy, i.e., viscosity depends on the strain rate and on ϕ. In other words, the stress-history of mud plays a role in governing its response to wave action. As a result, for instance, pre- and post-storm response of marine mud of the same density and composition to the same wave action may be quite diﬀerent.

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With increasing interparticle collisions the probability of formation of ﬂocs from dispersed (nonﬂocculated) particles increases. Thus the horizontal axis can also be interpreted to mean a change from weakly ﬂocculated particles on the left to increasingly ﬂocculated particles toward the right. An outcome is that, irrespective of the degree of interparticle interaction, at low values of ϕ the viscosity rises slowly, but tends to increase rapidly when particle packing becomes dense.13 For randomly packed spheres this change occurs at about ϕ = 0.60. A simple viscoplastic model is the Herschel–Bulkley equation τ = τy + K γ˙ n

(27.6)

where τy is the yield stress, γ˙ is the ﬂow shear (strain) rate, K and n are the constants for a given sediment, and n = 1. For ﬂuid muds n typically ranges from 0.3 to 0.9. This variation incorporates eﬀects of mud density, composition, and stresshistory.14 The Casson15 equation is more nonlinear than Eq. (27.6) √ √ τ = τy + η∞ γ˙

(27.7)

where η∞ is the viscosity at (theoretically) inﬁnite shear rate. Dade and Nowell16 found that their data on dense clay suspensions seemed to better agree with Eq. (27.7) than Eq. (27.6). This equation can be written as τ = τy + (K γ˙ n−1 )γ˙ = ηm γ˙

(27.8)

ηm = K γ˙ n−1 + η∞

(27.9)

where

denotes a non-Newtonian viscosity, in which η∞ is the value of ηm when theoretically γ˙ → ∞. This relationship between mud viscosity ηm and shear rate γ˙ is also called the Sisko18 model, whose performance depends on the value of n. When n < 1 a pseudoplastic response results, in which the viscosity decreases with increasing shear rate (shear-thinning) and eventually attains a constant value (η∞ ). When n > 1 the material has a dilatant (shear-thickening) behavior, in which the viscosity increases with increasing shear rate. When n = 1, Eq. (27.6) describes the Bingham model τ = τB + ηm γ˙

(27.10)

in which the (Bingham) yield stress τy = τB is equal to the threshold stress at and below which the material is a solid, and above which behaves as a Newtonian ﬂuid. Thus the Bingham model is a special case of the Herschel–Bulkley equation. Since the rheological behavior of mud is critical to prediction of wave damping, the relevant deﬁnition of mud must be based on a rheological classiﬁcation. Accordingly, mud may be deﬁned as, “A sediment–water mixture which consists of particles that are predominantly less than 63 µm in size, exhibits poroelastic or viscoelastic rheological behavior when the mixture is particle-supported, and is highly viscous and non-Newtonian when it is in a fluid-like state”.19

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This deﬁnition allows for the inclusion of some ﬁne sand (>63 µm) as part of mud, which is often the case. On the other hand, there is no reference to material composition, i.e., the deﬁnition relies solely on the rheological behavior. Thus any sediment that meets the particle size criterion, including organic matter, is deﬁned as mud provided it conforms to the stated rheological criteria. Percolation occurs below the bed surface, as pore water ﬂows (percolates) through the interstitial space in response to the wave-induced dynamic pressure gradient. Energy loss is expressed in terms of the well-known Darcy’s law of viscous ﬂow of pore water (e.g., Ref. 20). This law in the vertical (z) direction is ww = −

Ksp dpt ηw dz

(27.11)

where ww is the pore water velocity, pt is the total pressure, i.e., the sum of dynamic and hydrostatic pressures, and Ksp is the speciﬁc permeability. The bed matrix is static and, also, the bed is two-phased. The poroelastic response of such a bed involves energy loss due to interparticle friction, which is additional to Darcy viscous (percolation) loss. When the bed heaves, the resulting Coulomb friction due to interparticle abrasion becomes the dominant cause of energy loss (compared to percolation). The poroelastic stress (τ )–strain (γ) relationship is given by τ = G(1 + iδc )γ

(27.12)

in which G is the shear modulus of elasticity of the bed material and δc is a characteristic energy loss parameter. For compliant continua, a simple (linear) viscoelastic medium subject to oscillatory forcing is represented by two equivalent representations, one between shear stress τ and shear strain γ (for a solid), and another between τ and time-rate of shear strain γ˙ (for a ﬂuid) τ = G∗ γ = (G − iG )γ ∗

τ = η γ˙ = (η + iη )γ˙

(27.13) (27.14)

in which the equivalent shear modulus G∗ (a complex number characterized by √ i = −1) comprises of the storage modulus G and the loss modulus G , and the equivalent dynamic viscosity η ∗ consists of viscous coeﬃcient η , and a component η incorporating the material’s elastic response. The two basic viscoelastic models include the Kelvin–Voigt (K–V) and the Maxwell elements. The K–V element behaves as a solid when sheared, since the deformed material regains its initial state after the applied stress is relaxed. The components of equivalent shear modulus and equivalent viscosity, respectively, are G = G; η = ηm ;

G = ησ G η = . σ

(27.15) (27.16)

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Kelvin–Voigt

τ

τ

Maxwell ηm

G τ

ηm G τ

τ

Jeffreys

783

ηa ηb

G τ Fig. 27.6.

Mechanical analogs of simple viscoelastic constitutive elements for mud.

The Maxwell element, which represents a ﬂuid because it does not resist deformation under an applied stress, is deﬁned as 2 2 ηm σ G ; 2 2 (ηm σ + G2 ) ηm G2 ; η = 2 2 (ηm σ + G2 )

G =

ηm σG2 2 σ 2 + G2 ) (ηm η 2 σG η = 2 m2 . (ηm σ + G2 ) G =

(27.17) (27.18)

Mechanical analogs of K–V and Maxwell elements are shown in Fig. 27.6. Higher-order analogs are combinations of these two types. For example, the Jeﬀreys element combines, in series, a Maxwell element (with viscous coeﬃcient ηa ) and a K–V element (viscous coeﬃcient ηb ).

27.4. Change in Bed State During wave action, the rheological state of soft mud almost always changes. This change depends on the relative magnitudes of the applied shear stress τ , the shear strength with respect to erosion (or critical shear stress) τs , and the plastic yield stress τy (or the Bingham yield stress τB ). As schematized in Fig. 27.7, ﬂuid mud is formed in diﬀerent ways depending on the stresses and accompanying processes. Case 1: τ ≤ τs < τy . Wave action is too weak to erode the bed surface. However, wave orbits can penetrate the bed, and mainly by pressure work buildup excess pore pressure. Interparticle contacts become impermanent and the eﬀective normal stress, representing the solid-supported part of total normal stress in bed, eventually vanishes. Mud rigidity decreases even as the particle packing density (i.e., wet bulk density) may remain largely unaﬀected. However, the bottom is now in a ﬂuidlike transitory state, which reverts to bed almost as soon as wave action ceases and interparticle bonds are reestablished. For transitory ﬂuid mud to change to a

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Fig. 27.7. Flow chart illustrating the modes of formation of ﬂuid mud, turbidity generation, and reversion of ﬂuid mud to bed.

quasi-steady or semi-permanent ﬂuid mud, transfer of turbulent kinetic energy to the bottom must occur. This transfer increases the potential energy of ﬂuid mud, which rises and is diluted. Dilute ﬂuid mud is sustained until dewatering causes it to revert to bed. Case 2: τs < τ < τy . The bed is subject to pressure work and shear work with the result that pore pressure buildup eventually destabilizes the particle matrix. The bed is liqueﬁed and its surface erodes, thus causing water to become turbid. However, there may be no signiﬁcant change in bed density. Fluid-like mud reverts to bed when wave action ceases, unless its dilution follows, in which case its reversion to bed depends on the rate of dewatering. Case 3: τs < τy < τ . The bed yields rapidly and also erodes quite signiﬁcantly, as a result of which water becomes signiﬁcantly turbid. However, beneath the eroding bed surface the density may remain practically unchanged. When wave action ceases ﬂuid mud develops at depths at which hindered settling occurs. This mud is initially signiﬁcantly lower in density than the original bed. In due course, thixotropic gelling and bed consolidation cause the density increase to its original value. In order to account for a change in bed state, some investigators have adopted empirical stress–strain models based wholly on rheometric tests, because simple mechanical analogs cannot account for such a change. An illustrative case is the model of Isobe et al.,21 which was developed for a mud tested in a concentric-cylinder rheometer. A sample stress–shear (strain) rate relationship is shown in Fig. 27.8. The model (Fig. 27.9) attempts to mimic the characteristic hysteresis loop arising out of a phase lag between the applied stress and the resulting shear rate. Based

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Shear stress (Pa)

200 150 100 50 0 -50

-100 -150 -200 -1.2

Fig. 27.8.

-0.9 -0.6 -0.3 0 0.3 Shear rate (Hz)

0.6

0.9

1.2

Applied stress versus ﬂow shear rate relationship showing hysteresis.

Fig. 27.9.

Stress versus shear rate model of Ref. 21.

on the geometric idealization shown in Fig. 27.9, the relevant model equations for stress τ versus strain rate γ˙ are as given in the following paragraph. Backbone curve (representing the basic ﬂow curve in the absence of hysteresis): τ = τp

|γ| ˙ γ˙ c

n · sign(γ); ˙

τ = τy · sign(γ) ˙ + η1 γ; ˙

|γ| ˙ ≤ γ˙ c

(27.19)

|γ| ˙ > γ˙ c .

When γ˙ > γ˙ c : ˙ + η1 γ; ˙ τ = τy · sign(γ) τ = −τy · sign(γ) ˙ + η2 γ; ˙

|γ| ˙ ≤ γ˙ c ; |γ| ˙ ≤ γ˙ c ;

γ¨ ˙ γ ≤ 0; γ¨ ˙ γ > 0.

or |γ| ˙ > γ˙ c (27.20)

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When γ˙ ≤ γ˙ c : ˙ + η1 γ; ˙ τ = τm0 · sign(γ)

τm1

γ¨ ˙γ ≤ 0

˙ + ηm2 γ; ˙ γ¨ ˙γ > 0 τ = −τy · sign(γ) n γ˙ mc τm0 + τm1 = τp ; τm0 = τm1 − η1 γ˙ mc ; ηm2 = γ˙ c γmc

(27.21) (27.22)

˙ and η1 and η2 are viscosities when the magnitude where γ˙ c is a threshold value of γ, of γ˙ is increasing and decreasing, respectively. Calibration of these sets of equations requires ﬁve mud-speciﬁc parameters: i.e., τp , τy , τm0 , γ˙ c , and γ˙ mc . 27.5. Wave Damping Coeﬃcient Due to their simplicity, basic rheological models merely approximate the actual stress–strain (or strain rate) behavior of muds. Nevertheless, with proper calibration they have been found useful in engineering applications requiring knowledge of wave attenuation. A few illustrative equations for calculating the wave-damping coeﬃcient are given in Table 27.2. Consider a 6-s wave of 1-m amplitude propagating in seawater (density 1,030 kg m−3 ) of 4.5 m depth, giving a wave group velocity of 5.14 m s−1 . The bed consists of mud having a density of 1,200 kg m−3 . Bed surface resistance is characterized by the wave friction factor fw = 0.035, mud dynamic viscosity is 2.5 Pa.s, and the rigidity modulus is 500 Pa. Assume speciﬁc permeability of 1.27 × 10−13 m2 , speciﬁc energy loss parameter equal to 0.5 and bed porosity 0.5. Calculated wave damping coeﬃcients ki (given below) using equations in Table 27.3 indicate that for the selected parameters the viscoelastic K–V model is the only important contributor to wave damping, while bed friction and poroelastic contributions are the least important. Percolation is limited by low permeability. Poroelastic contribution is low under given wave conditions, but can be high when the bed is silty. For soft muds in general, wave damping due to viscoelastic and poroelastic (silty) bed responses deserves greater consideration than damping due to other mechanisms. From dimensional analysis it can be shown that for the viscoelastic ﬂuid 2 σ d δm ∆ρ k˜i = ki h = f n , , , (27.23) gkr h d ρw 2νm /σ is the Stokes where d is mud layer thickness, h is water depth, δm = boundary layer thickness in mud, vm is the kinematic viscosity of mud and ∆ρ = ρm − ρw is the density diﬀerence between mud and water.26 The dimensionless wave damping coeﬃcient k˜i = ki h depends on σ 2 /gkr (i.e., the surface wave dispersion relationship), the relative mud thickness d/h, the relative thickness of the Stokes boundary layer δm /d in mud, and the buoyancy of the lower layer represented by ∆ρ/ρw . For a poroelastic bed 2 σ d Vm n ˜ ki = f , , ,Pe (27.24) gkr h Cs

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Wave damping coeﬃcient p kr2 νw /2σ ki = 2kr h + sinh 2kr h

Viscous ﬂuid over rigid bed

ki =

Inviscid ﬂuid over viscous ﬂuid of indeﬁnite thickness in nonshallow water ki =

Percolation + viscous eﬀects ki =

Viscoelastic Kelvin– Voigt bed

787

Wave damping coeﬃcients over static and compliant beds.

Physical basis

Poroelastic bed

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Investigator(s) Dean and Dalrymple22

√ 1 ρm σνw 2kh 2 e (σ − gkr )2 4 ρw gσ2 Cg

2kr 2kr h + sinh 2kr h

ρw gδc “ 2G cosh2 k0 h 1 +

ki =

„

Ksp σ + kr νw

2k0 h sinh 2k0 h

”» 1−

ρw gηm “ ” 2 + G2 4σ ηm σ2

“

r

σ σ0

Dean and Dalrymple22

νw 2σ

«

1 ”2 –2

Liu23

+ δc2

Yamamoto and Takahashi24 MacPherson25

kr = wave number (= 2π/L, where L is the wavelength), σ = wave angular frequency, h = water depth, vw = ﬂuid (water) viscosity, ρw = upper ﬂuid (water) density, ρm = mud density, ηm = mud dynamic viscosity, Cg = wave group velocity (simply calculated from the Airy linear wave theory, e.g., Ref. 7), Ksp = speciﬁc permeability, a = wave amplitude, g = acceleration due to gravity, G = shear modulus of elasticity (of mud),p δc = Coulomb speciﬁc energy loss, k0 = wave number for the static (rigid) bed case, σ0 (= k0 2G/ρm ) = poroelastic resonance frequency, ρs = particle density, and np = bed porosity.

Table 27.3.

Wave damping coeﬃcient for various physical system or process.

Physical system or process

ki (m−1 )

Viscous ﬂuid over rigid bed Inviscid ﬂuid over viscous ﬂuid of indeﬁnite thickness in nonshallow water Percolation Poroelastic bed Viscoelastic Kelvin–Voigt bed

5.4 × 10−6 8.9 × 10−6 1.1 × 10−5 9.2 × 10−4 2.6 × 10−2

where Vm /Cs (= M aG ) is the shear Mach number in mud, in which Vm = √G/ρm is the shear wave velocity, G is the shear modulus of elasticity and Cs = gh is the shallow water wave celerity. The quantity Pe is a characteristic P´eclet number representing the ratio of rate of advection of ﬂow to its rate of diﬀusion within the

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bed. For a given wave condition, a high value of Pe implies low rate of pore water diﬀusion, which is conducive to undrained bed condition.27,28 These bed states are relevant because under drained condition, excess pore pressure generated by wave action is dissipated via pore water ﬂow, while under undrained condition pore pressure does not dissipate. This distinction in bed behavior is related to particle size. Large size correlates with high permeability and easy drainage, while small size is associated with low ﬂow permeability, i.e., poor drainage. Thus the modality of wave energy dissipation is dependent on Pe. This number, Pe = σdp /Kp, is proportional to a characteristic velocity scale (which is deﬁned in terms of angular wave frequency σ and particle diameter dp ), and inversely proportional to the isotropic coeﬃcient of consolidation, which varies with ﬂow permeability Kp . The relation between the ﬂow permeability and speciﬁc permeability is given by Kp = Ksp gρw /ηw . This deﬁnition of P´eclet number will be apparent in subsequent reference to mechanisms for wave energy dissipation. The quantity M aG varies with the solids volume fraction ϕ = 1 − np , where the porosity np = (ρs − ρm )/(ρs − ρw ) and ρs is the particle density, nominally equal to 2,650 kg m−3 for typical inorganic sediments. When ϕ ≤ ϕsp , where ϕsp is the space-ﬁlling value of ϕ below which the particles do not form a continuous matrix (i.e., mud is a ﬂuid, and therefore cannot transmit a shear wave) Vm and therefore M aG are nil. In the range of ϕ > ϕsp , Vm rapidly increases with ϕ. When waves begin to liquefy mud, Vm decreases even when ϕ (> ϕsp ) remains constant. In a ﬂume experiment on a clayey mud bed at ϕ = 0.10, the onset of steady wave action caused Vm to decrease from 2 to 1.4 m s−1 in one hour.29 The mean range of ρm in which ﬂuid mud occurs is 1,050 to 1,200 kg m−3 , i.e., ϕ in the range of 0.03 to 0.12 (taking ρw = 1,000 kg m−3 ). The upper limit represents the mean value of ϕsp , which has been found to vary between 0.07 and 0.18, depending on mud composition.27 The presence of organic matter can drastically reduce ϕsp . For mud from Lake Okeechobee containing 40% organic matter by weight, ϕsp was reported to be 0.04.30 Such diﬀerences in ϕsp indicate that mud can be a ﬂuid or a solid depending on its composition, and also its stress history.

27.6. Domains of Dissipation Mechanisms Notwithstanding evident constraints against imposing strict limits of applicability of diﬀerent mechanisms for energy dissipation, it is helpful to the modeler to have recourse to a qualitative representation of the domains of applicability of these mechanisms. To that end the bed may be characterized nominally in terms of the solids volume fraction ϕ (= 1 − np ) and the P´eclet number P e. Even though the use of ϕ is appropriate for nonﬂocculating sediment (>∼10–20 µm), it is a less unambiguous choice for ﬁner bed material, because in reality the bed’s constitutive behavior varies with the floc volume fraction. Unfortunately, since the latter quantity is not readily available in most cases, we have eschewed its use in favor of ϕ to represent ﬂocculated sediment beds. The P´eclet number increases with increasing wave frequency σ and grain size dp , and decreases with increasing permeability Kp . Accordingly, Pe depends on whether the bed is sandy, silty or clayey, and for a given bed varies

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with the frequency of wave forcing. These eﬀects on Pe are exempliﬁed by the order-of-magnitude representations in Table 27.4. In the laboratory or in small lakes (σ = 6 rad s−1 ) Pe values are generally high and increase ten-fold (from 0.6 to 6) as the substrate changes from coarse sand to clay. At the other extreme case of open sea or the ocean (σ = 0.5 rad s−1 ) the Pe values are comparatively low, even as they increase ten-fold when the substrate changes from coarse to ﬁne. An important inference to be derived from these eﬀects is that the magnitude of Pe serves to deﬁne the bed as a drained, two-phased medium at one end to an undrained continuum at the other. An ensuing outcome would be that the viscoelastic model is appropriate for high values of Pe and poroelastic model for low values. In Fig. 27.10, the compactness of the bed of given sediment diameter has been deﬁned by ϕ (or np ). The abscissa is speciﬁed by the P´eclet number, with values corresponding to Table 27.4. Since density ρ has been used in the analysis, its scale is also included based on particle density of 2,650 kg m−3 and water density of 1,000 kg m−3 . The boundaries are approximate not only due to the paucity of available information, Table 27.4.

P´eclet number for diﬀerent beds and wave frequencies. P´ eclet number, Pe

Bed Coarse sand Fine sand Clay

Nominal diameter, dp (m)

Permeability, Kp (ms−1 )

σ=6 (rad s−1 )

σ=1 (rad s−1 )

σ = 0.5 (rad s−1 )

1 × 10−3 1 × 10−4 1 × 10−6

1 × 10−2 5 × 10−4 1 × 10−6

0.6 1.2 6

0.1 0.2 1

0.05 0.1 0.5

Fig. 27.10. Semi-quantitative schematization of the domains of applicability of rheological models for predicting wave attenuation.33

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but also because porosity and permeability depend on particle composition, shape and texture, in addition to size. This dependence leads to overlaps in the zones of applicability of the various models. Another factor contributing to uncertainty in the domain boundaries is due to the characteristically thixotropic response of mud to stress. As mud microstructure formed during rising stress diﬀers from that under falling stress, energy dissipation varies with stress history. Viscoelastic solid and viscoplastic models occupy the lower left-hand region, because they are important when the bed is dense and cohesive. If modeling is done within the ﬂuid-ﬂow framework, the viscoelastic model can yield acceptable results only for short-term response of mud to wave forcing. Similarly, viscoplastics are modeled correctly only within the framework of the lower layer as a solid at low stresses.31,32 When the bed is static (noncompliant) and the porosity is low, energy loss occurs due to friction at the surface. The domain boundary shown for this mechanism can vary substantially depending on surface roughness. Porous static beds experience loss due to percolation, whose contribution to overall dissipation is increasingly subsumed by Coulomb friction as the material becomes ﬁner and causes the bottom to heave. Although the choice of the rheological model varies quite signiﬁcantly with ϕ and Pe, some broad albeit approximate trends are also evident in Fig. 27.10. Most importantly, it appears that the poroelastic response is bracketed between Pe = ∼0.4 and ∼2. At higher values of Pe one of the viscoplastic models (including purely viscous) applies depending on the value of ϕ. At lower values of Pe bed friction and percolation are the two important dissipation mechanisms.

27.7. Sediment Resuspension 27.7.1. Suspended sediment mass balance Resuspension may be conveniently deﬁned as the net transfer of sediment from bottom sediment into the water column.34 Since the gross transfer of sediment is called erosion, resuspension is the net result of bottom erosion and deposition of suspended sediment. Transfer of sediment changes the wave-mean-suspended sediment concentration with elevation z (= z + h, conveniently chosen as the coordinate with respect to the rigid bed as datum) and time t. This choice of datum means that, referring to Fig. 27.3 the rigid-bed level, the static-bed level and the mean position of the ﬂuid mud–water interface are one and same. Fluid mud, if it occurs above this level is taken to be within the water layer (above z = 0). The concentration C(z , t) is obtained from the simple sediment mass balance equation in the vertical direction: ∂C εm + εs ∂C ∂ ∂ (27.25) = (w C) + s ∂t ∂z ∂z σT ∂z where ws is the settling velocity, εm is the molecular diﬀusivity, εs is the turbulent momentum diﬀusivity, and σT is the Prandtl–Schmidt number, i.e., the ratio of momentum to mass diﬀusivity. The inequality between momentum and mass diffusion processes is due to the eﬀect of inertia, as sediment particles do not follow

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the turbulent movement of water parcels. In other words, particle motion is not “iso-kinetic”. This discrepancy, which increases with increasing frequency of turbulence, can be usually ignored for particles less than about 100 µm. In Eq. (27.25) it is assumed that σT , which depends on particle size and whose value is determined from empirical evidence, is the same for molecular and turbulent diﬀusion processes in the z -direction. Molecular diﬀusivity is obtained from εm =

κB Tw 6πηw dp

(27.26)

in which κB is the Boltzmann constant (1.38026 × 10−16 erg deg−1 molecule−1 ), and Tw is the (absolute) temperature of the ﬂuid. Since εm εs , we will ignore molecular diﬀusion in the present treatment. In Eq. (27.25) the product ws C is the settling ﬂux and Ds ∂C/∂z is the diﬀusive ﬂux, where Ds = εs /σT denotes turbulent sediment mass diﬀusivity. Thus the time-rate of change of concentration ∂C(z , t)/∂t is governed by the net eﬀect of the settling and diﬀusion. The behavior of C(z , t) is entirely dependent on the surface and bottom boundary conditions required to solve Eq. (27.25). At the free surface the usual boundary condition is that there is no loss or gain of sediment, so the settling and upward diﬀusive ﬂuxes are equal, i.e., ∂C (27.27) ws C + Ds = 0 @z = h. ∂z At the bottom the condition to be satisﬁed is ∂C (27.28) ws C + Ds = S @z = 0 ∂z where S = Fe − Fs characterizes resuspension (when S > 0), Fe is the erosion ﬂux, and Fs is the deposition ﬂux. All three quantities have units of mass (of sediment) per unit volume of suspension and unit time. Dividing each by the bed area gives the respective ﬂuxes Sf , Fef , and Fsf . Three situations can arise with respect to Sf (or S): (1) Sf = 0, i.e., there is no change in the concentration proﬁle C(z ), which represents an equilibrium condition, (2) Sf > 0, i.e., erosion dominates, and C(z , t) changes in such a way that the depth-mean concentration (or turbidity) increases, and (3) Sf < 0, i.e., deposition dominates and turbidity decreases. The solution of Eq. (27.25) (together with Eqs. (27.27) and (27.28)) depends on the choice of the functions for Fef , Fsf (= ws C at z = 0) and Ds . 27.7.2. Erosion flux The functional form of Fef depends on whether the bottom is a bed or ﬂuid mud. The density ρmf at the boundary between these two states varies with inorganic sediment composition and is further inﬂuenced by the amount of organic matter. A density range as wide as 1,002 to 1,300 kg m−3 has been reported.35 The lower limit arose from sediment rich (40% by weight) in organic matter.30 However, as a rule of thumb, the range for ﬂuid mud may be taken as 1,050 to 1,200 kg m−3 . Expressions for Fef are as follows. For a bed (ρm > ρmf kg m−3 ): τb − τs (27.29) Fef = Feo τs

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where τb is the bed shear stress, τs is the bed shear strength with respect to erosion and Feo is the erosion ﬂux constant, equal to value of Fef when τb = 2τs . Typical range of τs values is 0.1 to 0.3 Pa. The constant Feo can be as high as 1 × 10−1 kg m−2 s−1 in the shallow littoral zone. In general, τs and Feo are highly sensitive to sediment composition and bed density (e.g., Ref. 36). For example, if mud were signiﬁcantly palletized and highly cohesive, τs could be higher than the quoted values. For ﬂuid mud (ρm ≤ ρmf kg m−3 ):  2 Rc  αe ρm uwb − Rg − ws Ca ; Rg < Rc (27.30) Fef = Rg  0; Rg ≥ Rc where uwb is the near-bottom velocity amplitude, αe is a sediment-dependent coefﬁcient, Ca is a characteristic near-bottom suspended sediment concentration, and Rg is a global Richardson number. The quantity Rc is the critical value of Rg below which no entrainment ﬂuid mud occurs. From experimental data on a natural and a clayey sediment, Li37 estimated αe = 2 × 10−6 and Rc = 0.043. The Richardson number is deﬁned as Rg =

2 ∆ρgδw ρw u2wb

(27.31)

where δw = 2νw /σ is the thickness of the Stokes boundary layer in water. We will return to the method for calculation of uwb and the signiﬁcance of Ca . 27.7.3. Settling flux In general, the settling velocity function is of the form ws = wso f n (C, γ), ˙ where wso is a characteristic value of ws and f n (C, γ) ˙ denotes a function of C and a representative value of ﬂow shear rate γ. ˙ Substituting these relations in Eq. (27.25) (and assuming εs = 0) yields ∂C ∂C ∂ n w (27.32) = Cf (C, γ) ˙ + D s0 s ∂t ∂z ∂z

where f n (C, γ) ˙ is another function of C and γ. ˙ Based on experimental studies, Teeter38 reported the following relationship relating the settling velocity to the ¯˙ turbulence-mean shear rate γ: 1 + λa γ¯˙ ws|γ=0 ws = (27.33) ˙ 1 + λb γ¯˙ 2 in which λa and λb are sediment-speciﬁc coeﬃcients. From laboratory work on natural sediment, values of these two coeﬃcients were found to be 320 and 75, respectively. In general, bothcoeﬃcients are found to be sensitive to sediment composition. The quantity γ¯˙ = εD /νw , where εD is the rate of energy dissipation in the ﬂuid. Typical range of γ¯˙ is 1 to 30 s−1 . The velocity ws|γ=0 depends on concentration. When C < ∼ 0.1 m s−1 , the ˙ (denoted by the frequency of inter-particle collisions is so low that ws|γ=0 ˙

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Wave-Induced Resuspension of Fine Sediment Table 27.5.

Wolanski et Jiang41 Marv´ an42 Ganju43

Values of coeﬃcients af , nf , bf , and mf . Sediment source

af

bf

mf

nf

Cleveland Bay (Australia); ﬁeld test Cleveland Bay; laboratory tests Jiaojiang (China); neap tide Jiaojiang; spring tide Ortega River, Florida Loxahatchee River, Florida

0.200 0.070 0.045 0.230 0.160 0.190

1.40 1.30 6.00 10.00 4.50 5.80

2.25 2.50 1.51 1.80 1.95 1.80

2.45 2.80 1.50 1.50 1.70 1.80

Investigator(s) al.40

793

symbol wo and termed free-settling velocity) can be taken as constant. For higher concentrations, Wolanski et al.39 proposed the empiric equation = af ws|γ=0 ˙

(C 2

C nf + b2f )mf

(27.34)

where af is a settling velocity scaling coeﬃcient, nf is a ﬂoc-settling exponent, bf is a hindered settling coeﬃcient, and mf is a hindered settling exponent. Selective values of these coeﬃcients are given in Table 27.5. When the concentration exceeds ∼ 75 kg m−3 settling is practically arrested and the deposit begins to gel and consolidate. A representative ﬂow shear rate is uwb uwb γ˙ = = . (27.35) δw 2νw /σ By assuming that the wave boundary layer δw h, the velocity amplitude uw (z ) can be estimated from the Airy theory as uw = aCw kr

cosh kr z sinh kr h

in which the wave celerity Cw is given by g Cw = tanh kr h. kr

(27.36)

(27.37)

At the bottom, the velocity amplitude uwb is obtained from Eq. (27.36) by setting z = 0 to yield uwb =

aσ aCw kr = . sinh kr h sinh kr h

(27.38)

27.8. Mass Diﬀusivity The mass diﬀusivity Ds is obtained from Ds = Dso Φs

(27.39)

where Dso is called neutral mass diﬀusivity applicable to homogeneous ﬂows, and Φs is a correction term for stratiﬁcation induced by suspended sediment.

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In general,

z Φs = f n 1 + Lso

(27.40)

in which Lso is known as the Monin–Obukov length scale speciﬁed from the steadystate form of the turbulent kinetic energy balance for the ﬂow system. This length scale is accordingly deﬁned as ρu3∗ Lso = (27.41) κgρ w where ρ is the ﬂuid density, u∗ is the friction velocity, κ is the Karman constant (= 0.4 in clear water turbulent ﬂow), ρ w is the component of sediment mass ﬂux due to turbulent diﬀusion, ρ is the turbulent ﬂuctuation in density ρ, and w is the turbulent ﬂuctuation in the ﬂuid vertical velocity. Based on Eq. (27.41), Lso is dependent on the ratio of kinetic energy associated with ﬂow shear to the potential energy of the buoyant suspension. When inertia dominates, Lso is large and z/Lso in Eq. (27.40) approaches zero with the result that Φs = 1 and Ds = Dso . The higher is the stratiﬁcation the smaller is the value of Lso , and the greater the eﬀect of buoyancy on diﬀusion. Thus when the ﬂow consists of water stratiﬁed by ﬂuid mud, Ds can be signiﬁcantly diﬀerent from Dso . Expressions relating Φs to parameters that can be easily obtained from measurements of velocity and density proﬁles are commonly used. Munk and Anderson44 proposed a well-known equation of the type 1 Φs = (27.42) 1 + αs Rig where αs is a sediment-dependent coeﬃcient, whose value typically ranges between 0.5 and 2.30,45 The gradient Richardson number Rig is obtained from Rig = −

g(∂ρ/∂z) . ρ(∂uw /∂z)2

(27.43)

The related ﬂux Richardson number Rif is deﬁned as Rif = −

gρ w w ρu w du dz

(27.44)

which indicates a close relationship between Rif and Lso . In the strict sense, in Eq. (27.42) Rig must be replaced by Rif . However, Rig is often used because it is easier to deduce from measurements than Rif . The relationship between Rif and Rig is Rig = σT Rif

(27.45)

where σT is the Prandtl–Schmidt number. Thus in Eq. (27.42) the coeﬃcient αs includes the eﬀect of σT , which is therefore empirically accounted for in that relationship. The neutral mass diﬀusivity Dso is related to the neutral momentum diﬀusivity εmo by εmo Dso = (27.46) σT which makes it convenient to use expressions for εmo derived for the wave boundary layer, provided σT is known or is estimated. Costa46 reported σT in the range of

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about 1 to 2.5 at Hangzhou Bay in China. In the absence of data, σT is often taken to be equal to unity. Hwang30 used the following expression for the neutral momentum diﬀusivity: εmo = 2αw a2 σ

sin h2 kr z 2 sin h2 kr h

(27.47)

where αw is an empiric proportionality coeﬃcient taken as unity. 27.9. Equilibrium Concentration Proﬁle While true sedimentary equilibrium is rare in nature, the assumption of equilibrium over short durations, e.g., minutes, enables one to use measured proﬁles C(z ) to estimate the settling velocity of sediment. In this case we set ∂C/∂t in Eq. (27.32) to zero, which results in ∂ ∂C n Cf (C, γ) ˙ + D Φ w = 0. (27.48) s0 so s ∂z ∂z Thus,

˙ + Dso Φs ws0 Cf n (C, γ)

∂C = C . ∂z

(27.49)

Since there is no net input or removal of sediment from the water column, the constant of integration C = 0. Therefore, Dso Φs

∂C = −ws0 Cf n (C, γ) ˙ ∂z

(27.50)

which can be integrated from a near-bed reference elevation z = za , where the corresponding concentration is Ca , to give

z

C dC dz = −w . (27.51) so n ˙ Ca Cf (C, γ) za Dso Φs Setting Φs = 1 in the absence of stratiﬁcation, this equation reduces to

z

C dC dz = −w . (27.52) so n ˙ Ca Cf (C, γ) za Dso Let us consider a simple case in which the settling velocity is independent of concentration and shear rate, i.e., f n (C, γ) ˙ = 1. Then, from Eq. (27.52)

z C dz = −wso . (27.53) ln Ca za Dso We will further consider Dso to be independent of height in the water column. Thus, the solution of Eq. (27.53) is C wso = exp − (z − za ) . (27.54) Ca Dso

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According to this expression, a plot of ln(C/Ca ) against z − za is a line with slope −wso /Dso . Thus, provided Dso is independently estimated, this method enables the estimation of the settling velocity from measured concentration proﬁle. Its applicability is limited to low concentrations at which the settling velocity can be taken as constant. The time-variation of vertical proﬁles of concentration under nonequilibrium is site-speciﬁc and is not treated further due to the length of treatment. It suﬃces to say that interesting phenomena can occur depending on the nature of the wave ﬁeld and bottom sediment. For example, Li and Parchure47 used Eq. (27.32) to corroborate the observation that over the open-coast mudbanks of Kerala in southwestern India, turbidity decreases as wave action increases soon after the onset of the monsoonal season. This is so because ﬂuid mud is generated and as a result waves are signiﬁcantly damped. 27.10. Resuspension Modes Modes of resuspension interactively depend on the concentration proﬁle. The vertical structure of concentration is conveniently subdivided into four zones (Fig. 27.11). In the upper zone the suspension layer (DSL) is dilute, and is characterized by Newtonian ﬂow behavior. The lower zone is occupied by the benthic nepheloid layer (BNL), which contains ﬂuid mud. In the benthic suspension layer (BSL), the concentration is intermediate between DSL and BNL. The suspension in BSL is non-Newtonian but the concentration is not high enough for settling to be hindered. Finally, at the bottom a consolidating bed (CB) occurs. It possesses an eﬀective stress but is soft enough (i.e., not fully consolidated) for the sediment to be susceptible to resuspension when wave forcing is suﬃciently strong. The four zones are dynamically linked by particle settling, by coalescence or deposition of particles-in-ﬂuid parcels, and by entrainment of these parcels. In the

Fig. 27.11.

Sediment concentration zones and resuspension modes.

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absence of BNL and BSL, DSL is sustained by erosion of particles from CB. Settling particles deposit onto the bed, unless the near-bed ﬂuid stresses are high enough to hinder or prevent deposition. These transport processes are sometimes called classical erosion and deposition associated with CB.48 When BNL occurs but BSL is (practically) absent, sediment entrainment occurs due to mixing between ﬂuid mud and the dilute suspension. The settling parcels, if and when they reach BNL, will coalesce into BNL. Similarly, coalescence causes downward exchange of sediment between DSL and BSL. Exchange processes can also occur among BSL, BNL, and CB that change their thickness and concentration without participation by DSL. 27.11. Florida Setting Possible interactions between waves and sediment are indeﬁnitely large, and have not been explored suﬃciently either in the open sea or interior environments. However, useful pointers can be gleaned from limited studies in some of Florida’s shallow lakes, which are typically laden with black mud or muck. This material has a high percentage of amorphous plant debris. Its origins are terrigenous sources in the watershed, the lacustrine littoral zone, and the lake itself when it is large. A characteristic feature of the material is that it tends to be refractory, but at the same time highly reactive with respect to nutrients including nitrogen and phosphorus.5 Organic matter reduces the density of inorganic clayey material in such a way that the solids volume fraction ϕ of soft beds varies nonlinearly with the fraction of organic matter. From mass balance, ϕ=1−

ρm − ρ D ρw

(27.55)

where ρD is the dry density of the bed material (and equivalent to concentration C when sediment is suspension). Soft bed sediments were sampled at sites shown in Fig. 27.12. Sampling was carried out using grab-samplers and push-corers, with the cores varying in thickness between ∼ 0.1 and ∼ 1.5 m. The following approximate relationships for densities representing mean trends were obtained49 ρm = 912.2 exp(−0.123 OC) + 1046

(27.56)

ρD = 1448 exp(−0.0992 OC) + 92

(27.57)

where OC is percent organic content of sediment (obtained from loss of ignition). Equations (27.56) and (27.57) are applicable in the range of OC from 1% to 81%, and indicate a rapid decline in ρm and ρD as OC increases from 1% to about 25%. An artifact of the forms of these equations is that at freshwater density (ρw = 1,000 kg m−3 ) ρs varies nonmonotonically with OC, and therefore cannot be used to calculate ρD from Eq. (27.57) (when OC ≥ 25%). Thus, at OC = 1%, ρs = 2,549 kg m−3 , and at OC = 25%, ρs decreases to 1,704 kg m−3 . From then on ρs increases and reaches 1,999 kg m−3 at OC = 81%. Gowland et al.49 recommend that for OC ≥ 25%, ρD may be assumed constant at 1,704 kg m−3 as an approximation. In salt water (taking ρw = 1,030 kg m−3 ), ρs decreases from 2,489 kg m−3

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Fig. 27.12. Ref. 49).

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Some lakes and other sites in Florida laden with black mud or muck (adapted from

at OC = 1% to 1,246 kg m−3 at OC = 81%. Accordingly, in this case Eq. (27.57) can be used to calculate ρD over the entire range 1% ≤ OC ≤ 81%. 27.12. Wave Damping Using characteristic parameters for Newnans Lake (Table 27.6), calculated wave damping coeﬃcients are given in Table 27.7. Assumed causes of dissipation include laminar boundary layer, turbulent boundary layer, viscoelastic bed, and poroelastic bed (Table 27.2). For comparison purposes, percent reductions in wave energy ∆E = 100(a20 − a2 )/a20 are also given. The wave amplitude (a) is conveniently taken as the value at the (2.1-m deep) site of the measurement platform (Fig. 27.1(a)), and a0 is a reference wave amplitude at an upwind distance of 5.5 km north of the platform. The wave-generating wind speed was taken to be 8 m s−1 . Coulomb friction, viscous mud, and viscoelastic (Jeﬀreys) models provide competing mechanisms for signiﬁcant energy loss. It should be noted that the poroelastic bed formulation and the Kelvin–Voigt viscoelastic formulation in Table 27.2 are for inﬁnite mud depth so they are believed to over estimate energy loss. Wave damping coeﬃcients for viscous model and viscoelastic model are obtained from the solution of a semi-analytical model which is applicable from very thin to thick muds and is presented in Ref. 33. Low speciﬁc permeability (1010 m2 ) suggests that for application in Newnans, Lake, the poroelastic model may be no less qualitative than the viscoelastic representation. At the same time, given low bed elasticity (G = 300 Pa) we may infer that the viscoelastic bed oﬀers no noteworthy advantage over a viscous bed. In fact the top ∼ 20 cm active layer of muck was practically a ﬂuid (mean ρm = 1,040 kg m−3 ). Accordingly, the viscous mud model appears to be is reasonable, which also renders

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Characteristic parameters for Newnans Lake.

T (s)

a (m)

h (m)

d (m)

G (Pa)

ηw (Pa.s)

ηm (Pa.s)

ρw (kg m−3 )

ρm (kg m−3 )

Ksp (m2 )

δc

1.2

0.05

1.3

0.3

300

1 × 10−3

2.4

1,000

1,200

1 × 10−10

0.5

Table 27.7.

Wave attenuation coeﬃcients and percent reductions in energy.

Laminar Turbulent Percolation+ Coulomb boundary boundary viscous BL friction Viscoelastic loss loss (KV) Quantity layer layer∗

Viscous mud

Viscoelastic (Jeﬀreys)

ki (m−1 )

7 × 10−6

6 × 10−7

1 × 10−5

6 × 10−4

3 × 10−1

1 × 10−4

1 × 10−4

∆E(%)

7

1

11

99

100

66

66

∗ Based

on wave friction factor fw = 0.1 over uneven lake bottom.

Table 27.8. Characteristic dimensionless numbers for Newnans Lake. δw /h

δm /d

M aG

η /ηm

0.0005

0.09

0.14

0.002

analysis of wave damping easier because viscosity based models are simpler to use than poroelastic or viscoelastic ones. In the qualitative sense one can arrive at a similar conclusion as above regarding the bed by examining the magnitudes of characteristic dimensionless parameters (Table 27.8). Observe that δw /h = 0.0005 δm /d = 0.09 implies that dissipation is mainly in the mud, with practically little contribution from water. The Mach number M aG is about 0.14, which means that wave speed in water would be about seven times that in mud, a manifestation of the signiﬁcant role of mud as an energy dissipater. The very low value (0.002) of η /ηm , the ratio of loss due to mud elasticity to mud viscosity, corroborates the inference that the muck was more like a ﬂuid than a solid.

27.13. Mud Response Wave-induced acceleration of bottom mud, total pressure, and pore pressure were measured at a “deep” site in the lake (Fig. 27.1(a)) 0.2 m below the mud–water interface.12 The vertical amplitude of acceleration is plotted against the wind speed in Fig. 27.13. There is an evident trend of increasing amplitude with wind speed. Acceleration is a manifestation of heave, and as seen, 3.2 m s−1 may be taken as the threshold speed for the onset of heave. The eﬀective normal stress σv in mud is

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800

Vertical acceleration (m s-2)

0.010 Data

0.008 0.006 0.004 Critical wind speed for muck heave = 3.2 m s-1

0.002 0

Fig. 27.13.

1

2

3 4 Wind speed (m s-1)

5

6

Vertical acceleration in mud versus wind speed in Newnans Lake.

2.0 Pore water pressure (kPa)

Data

1.5 1.0 0.5 0 -0.5 -1.0 -1.5 4

Fig. 27.14.

Liquefied muck Critical wind speed for liquefaction 7.1 m s-1

5

6

7 8 Wind speed (m s-1)

9

10

Eﬀective stress in mud versus wind speed in Newnans Lake.

given by σv = σv − ∆u

(27.58)

where σv is the total normal stress (pressure) and ∆u is the pore pressure. Representative values of σv are plotted as a function of wind speed in Fig. 27.14. With increasing wind the resulting bottom shear stress can destabilize the bed particle matrix and result in a reduction in σv . When the matrix is completely broken σv vanishes and the bed is liqueﬁed. Liquefaction is seen to occur at wind speeds higher than about 7 m s−1 . Since the actual wind speed in the lake is often close to (or in excess of) this value, the top layer of mud is typically found in the liqueﬁed state. As expected, the mud begins to heave at a lower wind speed (3.2 m s−1 ). As long as heaving is gentle, the structural integrity of the bed is retained. At the same time, however, energy dissipation can cause signiﬁcant wave damping. The diﬀerence between the wave amplitude required for soft mud to heave, and higher amplitude necessary for mud entrainment can be used as the basis to design submerged, shore-parallel mud berms for protection of the shoreline against storm

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waves. Oﬀshore of Mobile Bay in Alabama, the crest elevation of such a berm above the seabed was so chosen as to enable incoming waves passing over the berm to heave the mud without entraining it. In this way a stable, wave-energy absorbing berm was constructed. It was 2.75 km long, 300 m wide, and 6 m high. The ambient mean water depth was 6.5 m. Wave measurements indicated that during a storm, incoming wave of 0.75 m amplitude experienced a 46% reduction in amplitude across the berm, which remained practically uneroded.50,51

27.14. Resuspension In Newnans Lake the wave period is only on the order of 1 s. Referring to Fig. 27.11, the consolidating bed participates in the resuspension process only along the shallow (<∼ 1 m depth) littoral periphery of the lake. In the central portion (>∼ 1 m) the sediment remains in suspension without participation by the bed. 27.14.1. Shallow zone In order to assess resuspension at a “shallow” (0.25 m deep) site in the lake (Fig. 27.1(a)), Eq. (27.32) must be solved for the time-dependent concentration proﬁle, with appropriately calibrated expressions for the erosion ﬂux coeﬃcients and the settling velocity. Pertinent measurements and analysis are described elsewhere.49 Parameters for simulation of concentration proﬁle are summarized in Table 27.9. The organic-rich bottom is conveniently treated as bed (as opposed to ﬂuid mud) in this analysis. For the numeric solution of Eq. (27.32), the water column was discretized into 30 sublayers and the time-step for simulation was taken as 60 s. The resulting increase in concentration under constant wave action is shown in Fig. 27.15. At 90 min, the initially uniform (measured) concentration of 0.04 kg m−3 is seen to have evolved into a signiﬁcantly bottom-weighted proﬁle approaching an equilibrium shape. The measured and simulated concentrations were both close to 0.4 kg m−3 , a high value indicating the presence of an easily erodible bed surface. An interesting feature of the simulated proﬁles, not captured by the measurements because Table 27.9.

Parameters for concentration proﬁle simulation.

Parameter Water depth, h (m) Wave amplitude, a (m) Wave period, T (s) Bed particle density, ρs (kg m−3 ) Bed dry density, ρD (kg m−3 ) Free settling velocity, wo (m s−1 ) Velocity coeﬃcients, af , bf , nf , mf Feo (kg N−1 s−1 ) Bed shear strength, τs (Pa)

Value 0.25 0.025 1 2,059 100 0.1 × 10−5 0.7, 7.9, 2.5, 1.4 1.82 × 10−3 0.07

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0.25

Elevation (m)

0.20

Data Initial 90 min

0 min 30 min 60 min 90 min

0.15 0.10 0.05 0

0

0.25

0.50

0.75

1.00

1.25

1.50

Concentration (kg m-3) Fig. 27.15.

Concentration proﬁle evolution at the shallow site in Newnans Lake.

they were not made suﬃciently close to the bed, is the development of a highconcentration layer within the ﬁrst 1–2 cm above bottom. Concentration in this layer exceeded 1 kg m−3 . Persistent of such a layer can have signiﬁcant eﬀect on benthic biota.

27.14.2. Deep zone At the deep site (Fig. 27.1(a)), wave-induced bed shear stresses were too small to erode the sediment. We will conveniently assume that neither the consolidating bed (CB) nor the benthic nepheloid layer (BNL) contributed to resuspension, and that sediment exchange mainly occurred between the benthic suspension layer (BSL) and the dilute suspension layer (DSL). As seen from a typical concentration proﬁle in Fig. 27.16, BSL does not have a well-deﬁned thickness. However, its mean height He can be taken as 0.80 m based on the equal area assumption, which idealizes the water column as composed of a distinct BSL beneath sediment-free water (i.e., DSL with zero suspended matter). The mean concentration in BSL is 0.07 kg m−3 . Over the several months of measurement of suspended sediment concentration proﬁle at this site, it was found that 0.80 m was a representative mean value within the range of about 0.2 to 1 m for He .4 Taking 100 kg m−3 as the dry density of BNL (Table 27.9), the equivalent height of eroded matter from BNL would be (0.07/100) × 0.80 = 0.00056 m = 0.56 mm. In other words, wind energy, mainly via wave generation, was expended in expanding a 0.56 mm thick BNL with a dry density of 100 kg m−3 to 0.80 m high BSL with a concentration of 0.07 kg m−3 . In order to provide a mechanistic explanation for this transformation, Vinzon and Mehta52 used the steady state form of the turbulent kinetic energy balance to obtain the equilibrium height He of BNL as a function of

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Wave-Induced Resuspension of Fine Sediment

Fig. 27.16.

803

Estimation of eﬀective BSL height at the deep site in Newnans Lake.

wave properties. The semi-theoretic equation was shown to be He = 0.65

(a3wb ks )3/2 w T 3 ρsρ−ρ gws φv w

1/4 (27.59)

where ks is the bed roughness and awb is the amplitude of wave-induced water parcel excursion close to the bed and related to the velocity amplitude uwb by awb =

uwb . σ

(27.60)

For estimating He using Eq. (27.59) we will choose the following representative values: a = 0.035 m, T = 1.4 s, and h = 1.6 m (height of water above the assumed BSL–DSL interface), ks = 0.1 m, ρs = 1,700 kg m−3 , ρw = 1,000 kg m−3 , ws = 1.15 × 10−4 m s−1 , and ϕv = 70/(1,700 × 1,000) = 0.041 × 10−3 . This yields He = 0.68 m, which is within the measured range (0.2 to 1 m). 27.15. Concluding Comments A critical feature in predictive modeling of wave damping by mud is the characterization of interaction between orbital wave motion in mud and the eﬀect this has on the wave itself. What should be the thickness of participating mud at the beginning of wave action? How do mud properties change as wave action continues? These are the types of questions that are only qualitatively understood at present, and that too based on very limited evidence from laboratory experiments and a few ﬁeld investigations. Fuller empirical evidence and concomitant theoretical

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analyses are fruitful subjects for further research. Eﬀorts are presently being made to simulate bed erosion as “continuous-phase” modeling, in which the modeling domain extends to depths at which the mud is entirely stationary. In this physicsbased approach, technical demands remain signiﬁcant because modeling changes in state from a bed to a suspension (and vice versa) requires signiﬁcant treatment of inter-particle forces, and damping of turbulence by suspended matter. Nevertheless, owing to its simplicity, the treatment presented here is likely to continue for acquiring estimates of wave damping and resuspension in typical engineering investigations.

References 1. M. Gloor, A. W¨ uest and M. M¨ unnich, Benthic boundary mixing and resuspension induced by internal seiches, Hydrobiol. 284, 59–68 (1994). 2. J. Bloesch, A review of methods used to measure sediment resuspension, Hydrobiol. 184, 13–18 (1994). 3. D. P. Hamilton and S. F. Mitchell, An empirical model for sediment resuspension in shallow lakes, Hydrobiol. 317, 209–220 (1996). 4. M. Jain, A. J. Mehta, E. J. Hayter and J. Di, Fine sediment resuspension and nutrient transport in Newnans Lake, Florida, Sediment and Ecohydraulics: INTERCOH 2005, eds. T. Kusuda, H. Yamanishi, J. Spearman and J. Z. Gailani (Elsevier, Amsterdam, 2007), pp. 295–311. 5. R. Kirby, Organic-rich ﬁne sediments in Florida part I: Sources and nature, Estuarine and Coastal Fine Sediment Dynamics, eds. J. P.-Y. Maa, L. P. Sanford and D. H. Schoellhamer (Elsevier, Amsterdam, 2007), pp. 147–166. 6. F. Jiang and A. J. Mehta, Some observations on ﬂuid mud response to water waves, Dynamics and Exchanges in Estuaries and the Coastal Zone, ed. D. Prandle (American Geophysical Union, Washington, DC, 1992), pp. 351–376. 7. L. H. Holthuijsen, Waves in Oceanic and Coastal Waters (Cambridge University Press, Cambridge, England, 2007). 8. M. Jain and A. J. Mehta, Lake Monroe WRV assessment: Comments on transfer of detrital material and sediment loads, Report UFL/COEL/MP-2006/001, Coastal and Oceanographic Engineering Program, University of Florida, Gainesville (2006). 9. I. R. Young and L. A. Verhagen, The growth of fetch limited waves of ﬁnite depth. Part I: Total energy and peak frequency, Coast. Eng. 29, 47–77 (1996). 10. W. Kranenburg, Wave damping by ﬂuid mud — An extension of the SWAN model, MSc thesis, Delft University of Technology, Delft, The Netherlands (2007). 11. J. M. Hamrick, Theoretical and computational aspects of sediment transport in the EFDC model, Technical Memorandum, Tetra Tech, Inc., Fairfax, VA (2000). 12. M. Jain, A. J. Mehta, E. J. Hayter and W. G. McDougal, A study of sediment and nutrient loading in Newnans Lake, Florida, Report UFL/COEL-2005/002, Coastal and Oceanographic Engineering Program, University of Florida, Gainesville (2005). 13. H. A. Barnes, J. F. Hutton and K. Walters, An Introduction to Rheology (Elsevier, Amsterdam, 1989). 14. P. Cousot and J. M. Piau, On the behavior of ﬁne mud suspensions, Rheol. Acta 33(3), 175–184 (1994). 15. M. Casson, A ﬂow equation for the pigment-oil suspensions of the printing ink type, Rheology of Disperse Systems, ed. C. C. Mills (Pergamon, Oxford, 1959), pp. 84–104.

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16. W. B. Dade and A. R. M. Nowell, Moving muds in the marine environment, Proc. Coastal Sediments’91, eds. N. C. Kraus, K. J. Ginerich and D. L. Kriebel (ASCE, New York, 1991), pp. 54–71. 17. D. C.-H. Cheng, Viscosity-concentration equations and ﬂow curves for suspensions, Chem. Ind. 17, 403–406 (1980). 18. A. W. Sisko, The ﬂow of lubricating greases, Ind. Eng. Chem. 50(1), 1789–1792 (1958). 19. A. J. Mehta, Mudshore dynamics and controls, Muddy Coasts of the World: Processes, Deposits and Function, eds. T. Healy, Y. Wang and J.-A. Healy (Elsevier, Amsterdam, 2002), pp. 19–60. 20. J. A. Liggett, Fluid Mechanics (McGraw-Hill, New York, 1994). 21. M. Isobe, T. N. Huynh and A. Watanabe, A study on mud mass transport under waves based on an empirical rheology model, Proc. 22nd Int. Conf. Coast. Eng., ASCE, Reston, VA (1992), pp. 3093–3106. 22. R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (Prentice-Hall, Englewood Cliﬀs, NJ, 1984). 23. P. L.-F. Liu, Damping of water waves over porous bed, J. Hydra. Div. ASCE 99(12), 2263–2271 (1973). 24. T. Yamamoto and S. Takahashi, Wave damping by soil motion, J. Waterway Port Coast. Ocean Eng. 111(1), 62–77 (1985). 25. H. MacPherson, The attenuation of water waves over a non-rigid bed, J. Fluid Mech. 97(4), 721–742 (1980). 26. M. Jain, Wave–mud interaction in shallow waters, PhD thesis, University of Florida, Gainesville (2007). 27. J. C. Winterwerp and W. G. M. van Kesteren, Introduction to the Physics of Cohesive Sediment in the Marine Environment (Elsevier, Amsterdam, 2004). 28. O. C. Zienkiewicz, C. T. Chang and P. Bettess, Drained, undrained, consolidating and dynamic behaviour assumptions in soils, Geotechnique 30(4), 385–395 (1980). 29. A. J. Mehta, D. J. A. Williams, P. R. Williams and J. Feng, Tracking dynamic changes in mud bed due to waves, J. Hydra. Eng. 121(5), 504–506 (1995). 30. K.-N. Hwang, Erodibility of ﬁne sediment in wave dominated environments, MS thesis, University of Florida, Gainesville (1989). 31. K. Liu and C. C. Mei, Eﬀect of wave-induced friction on a muddy seabed modeled as a Bingham-Plastic Fluid, J. Coast. Res. S15, 777–789 (1989). 32. T. van Kessel and C. Kranenburg, Gravity current of ﬂuid mud on sloping bed, J. Hydra. Eng. 122(12), 710–717 (1996). 33. M. Jain and A. J. Mehta, Role of basic rheological models in determination of wave attenuation over muddy seabeds, Contin. Shelf Res. 29(3), 642–651 (2009). 34. R. D. Evans, Empirical evidence of the importance of sediment resuspension in lakes, Hydrobiol. 284, 5–12 (1994). 35. M. A. Ross, A. J. Mehta and C. P. Lin, On the deﬁnition of ﬂuid mud, Proc. Hydra. Eng. Conf., ASCE, New York (1987), pp. 231–236. 36. M. H. Mian and E. K. Yanful, Tailings erosion and resuspension in two mine tailings ponds due to wind waves, Adv. Environ. Res. 7, 745–765 (2003). 37. Y. Li, Sediment-associated constituent release at the mud–water interface due to monochromatic waves, PhD thesis, University of Florida, Gainesville (1996). 38. A. M. Teeter, Clay-silt sediment modeling using multiple grain classes; part I: Settling and deposition, Coastal and Estuarine Fine Sediment Transport Processes, eds. W. H. McAnally and A. J. Mehta (Elsevier, Amsterdam, 2001), pp. 157–171. 39. E. Wolanski, T. Asaeda and J. Imberger, Mixing across a lutocline, Limnol. Oceanograp. 34(5), 931–938 (1989).

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40. E. Wolanski, R. Gibbs, P. Ridd, B. King, K. Y. Hwang and A. Mehta, Fate of dredge spoil, Cleveland Bay, Townsville, Proc. 10th Austalasian Conf. Coast. Ocean Eng., Auckland, New Zealand (1991), pp. 63–66. 41. J. Jiang, An examination of estuarine lutocline dynamics, PhD thesis, University of Florida, Gainesville, FL (1999). 42. F. G. Marv´ an, A two-dimensional numerical transport model for organic-rich cohesive sediments in estuarine waters, PhD thesis, Heriot-Watt University, Edinburgh, UK (2001). 43. N. K. Ganju, Trapping organic-rich ﬁne sediment in an estuary, MS thesis, University of Florida, Gainesville (2001). 44. W. H. Munk and E. A. Anderson, Notes on a theory of the thermocline, J. Marine Res. 7, 276–295 (1948). 45. M. A. Ross, Vertical structure of estuarine ﬁne sediment suspensions, PhD thesis, University of Florida, Gainesville (1988). 46. R. C. F. G. Costa, Flow-ﬁne sediment hysteresis in sediment-stratiﬁed coastal waters, MS thesis, University of Florida, Gainesville (1989). 47. Y. Li and T. M. Parchure, Mudbanks of the southwest coast of India VI: Suspended sediment proﬁles, J. Coast. Res. 14(4), 1363–1372 (1998). 48. J. C. Winterwerp, A. W. Bruens, N. Gratiot, C. Kranenburg, M. Mory and E. A. Toorman, Dynamics of concentrated benthic suspension layers, Fine Sediment Dynamics in the Marine Environment, eds. J. C. Winterwerp and C. Kranenburg (Elsevier, Amsterdam, 2002), pp. 41–53. 49. J. E. Gowland, A. J. Mehta, J. D. Stuck, C. V. John and T. M. Parchure, Organicrich ﬁne sediments in Florida part II: Resuspension in a lake, Estuarine and Coastal Fine Sediment Dynamics, eds. J. P.-Y. Maa, L. P. Sanford and D. H. Schoellhamer (Elsevier, Amsterdam, 2007), pp. 167–188. 50. A. J. Mehta and F. Jiang, Some observations on water wave attenuation over nearshore underwater mudbanks and mud berms. Report UFL/COEL/MP-93/01, Coastal and Oceanographic Engineering Program, University of Florida, Gainesville (1993). 51. T. N. McLellan, M. K. Pope and C. E. Burke, Beneﬁts of nearshore placement, Proc. Third Ann. Natl. Beach Preser. Tech. Conf., Florida Shore and Beach Preservation Association, Tallahassee (1990). 52. S. B. Vinzon and A. J. Mehta, Mechanism for formation of lutoclines by waves, J. Waterway Port Coast. Ocean Eng. 124(3), 147–149 (1998).

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Chapter 28

Suspended Sand and Bedload Transport on Beaches Nobuhisa Kobayashi Center for Applied Coastal Research University of Delaware, Newark, DE 19716, USA [email protected] Andres Payo Graduate School of Science and Technology University of Kumamoto, 2-39-1, Kurokami Kumamoto, 860-8555, Japan Bradley D. Johnson US Army Engineering Research and Development Center 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA Simple formulas for the cross-shore and longshore transport rates of suspended sand and bedload on beaches are proposed by synthesizing available data and formulas. A combined wave and current model based on the time-averaged continuity, momentum, and energy equations for water is improved and used to provide hydrodynamic input to the proposed sand transport model. The model is compared with spilling and plunging wave tests conducted in a large wave basin using ﬁne sand. The numeric model predicts the measured longshore suspended sand and total transport rates within a factor of about 2. The longshore bedload transport rate is predicted to be small. The predicted cross-shore sand transport rates are relatively small on the quasi-equilibrium beaches in these tests. The computed beach proﬁle change under 10-h wave action is less than about 5 cm. The proposed sediment transport model will need to be veriﬁed using additional data but no bedload data is available in surf zones and reliable suspended load data is scarce.

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28.1. Introduction The predictive capability of sediment transport on beaches is essential for coastal sediment management, beach nourishment and storm damage reduction, and dredging of navigation channels. A large number of studies have been conducted to obtain sediment transport data in laboratories and on natural beaches and to develop predictive sediment transport models. However, none of the existing transport models are reliable and robust unlike coastal wave and current models that have been improved steadily for the past 30 years. The major reason for this discrepancy is that no dynamic equation is available to describe the motion of a large number of sediment particles. Consequently, existing sediment transport models are essentially empiric and dependent on reliable sediment data. Unfortunately, sediment dynamics on beaches are highly complex and involve wide ranges of morphologic scales in time and space. Correspondingly, available sediment transport models have become more complex but less robust. In view of the empiric nature of any sediment transport model, it is desirable to develop a simple but robust model that synthesizes the existing formulas for sediment transport on beaches. Furthermore, the sediment transport model should be very eﬃcient computationally because the model will need to be calibrated and veriﬁed using extensive data sets. This chapter presents our recent eﬀorts to develop and calibrate a sand transport model that is suited for practical applications but contains the basic mechanics of sand suspension and bedload movement on beaches. The hydrodynamic input required for the sand transport model is limited to the variables of irregular waves and currents which can be predicted eﬃciently and fairly accurately using a combined wave and current model based on time-averaged continuity, momentum, and energy equations. More advanced but computationally-demanding wave and current models may not improve the accuracy of the sand transport model with errors of a factor of about 2. Moreover, practical coastal sediment problems require the prediction of sediment transport rates for a duration of days to years. The computational eﬃciency is hence essential for practical applications. This chapter is organized as follows. Section 28.2 presents the combined wave and current model which is presently limited to the case of alongshore uniformity and unidirectional irregular waves. Section 28.3 shows new formulas for the crossshore and longshore transport rates of suspended sand and bedload where the proposed formulas are presently limited to uniform sand. The beach proﬁle evolution is computed using the standard continuity equation of bottom sand. Section 28.4 compares the sediment formulas with the spiling and plunging wave tests conducted in the Large-scale Sediment Transport Facility at the US Army Engineer Research and Development Center. Section 28.5 summarizes the ﬁndings of this chapter and discusses future work.

28.2. Combined Wave and Current Model The combined wave and current model in the following is an extension of the time-averaged model developed by Kobayashi et al.1 on the basis of the Dutch

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Fig. 28.1.

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Incident irregular waves and wind on beach of alongshore uniformity.

models by Reniers and Battjes2 and Ruessink et al.3 Figure 28.1 shows obliquely incident irregular waves on a straight shoreline where the cross-shore coordinate x is positive onshore, and the longshore coordinate y is positive in the downwave direction. The depth-averaged cross-shore and longshore velocities are denoted by U and V , respectively. Incident waves are assumed to be unidirectional with θ is the incident angle relative to the shore normal and uniform in the longshore direction. The wave angle θ is assumed to be in the range of |θ| < 90◦ to ensure that the incident waves propagate landward. The wind speed and direction at the elevation of 10 m above the sea surface are denoted by W10 and θw , respectively. The governing equations for water under the assumption of alongshore uniformity are averaged over a number of wind waves to obtain the time-averaged equations. The beach is assumed to be impermeable where Kobayashi et al.4 included a permeable layer for the case of normally incident waves. The depth-integrated continuity equation of water requires that the cross-shore volume ﬂux is zero in the absence of wave overtopping where Kobayashi et al.5 analyzed irregular wave overtopping of permeable slopes. The requirement of zero cross-shore volume ﬂux is expressed as ¯U ¯+ h

gση2 cos θ + qr cos θ = 0 Cp

(28.1)

¯ is the mean water depth given by h ¯ = (¯ where h η − zb ), with η¯ the mean free surface ¯ is the mean cross-shore velocity, which elevation and zb the bottom elevation; U is negative and oﬀshore because cos θ > 0; g is the gravitational acceleration; ση , the standard deviation of the free surface elevation η; Cp , the linear wave phase ¯ corresponding to the spectral peak period Tp ; velocity in the mean water depth h and qr , the volume ﬂux of a roller on the front of a breaking wave. The second term on the left-hand side of Eq. (28.1) is the onshore volume ﬂux due to linear waves

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propagating in the direction of θ (e.g., Ref. 6) where the representative period of irregular waves is chosen as the peak period Tp speciﬁed at the seaward boundary x = 0. Snell’s law is used to obtain the wave direction θ sin θ/Cp = constant

(28.2)

¯ and Tp speciﬁed at where the constant value is obtained from the values of θ, h, x = 0. Reﬂected waves are neglected in this model. The cross-shore and longshore momentum equations are expressed as dSxx η ¯ d¯ = −ρg h − τbx + τwx dx dx dSxy = −τby + τwy dx

(28.3) (28.4)

where Sxx is the cross-shore radiation stress; ρ, water density; τbx , cross-shore bottom stress; τwx , cross-shore wind stress on the sea surface; Sxy , shear component of the radiation stress; τby , longshore bottom stress; and τwy , longshore wind stress on the sea surface. The wind shear stresses may not be negligible especially outside surf zones on natural beaches.7 Linear wave theory for progressive waves are used to estimate Sxx and Sxy 1 2 ; Sxy = (nE + Mr ) cos θ sin θ Sxx = (nE + Mr ) cos θ + E n − (28.5) 2 with n = Cg /Cp ;

E = ρgση2 ;

Mr = ρCp qr

(28.6)

¯ and Tp; E, speciﬁc wave where Cg is the linear wave group velocity based on h √ energy with the root-mean-square wave height deﬁned as Hrms = 8ση ; and Mr , momentum ﬂux of a roller propagating with the phase velocity Cp . The time-averaged bottom shear stresses are written as τbx =

1 ρfb U Ua ; 2

τby =

1 ρfb V Ua ; 2

Ua = (U 2 + V 2 )0.5

(28.7)

where fb is the bottom friction factor, and the overbar indicates time averaging. The bottom friction factor fb is of the order of 0.015 and should be calibrated using longshore current data because of the sensitivity of longshore currents to fb . The equivalency of the time and probabilistic averaging is assumed to express τbx and τby in terms of the mean and standard deviation of the depth-averaged velocities U and V expressed as U = σT FU ;

V = σT FV ;

Ua = σT Fa ;

Fa = (FU2 + FV2 )0.5

(28.8)

with FU = U∗ + r cos θ;

FV = V∗ + r sin θ;

U∗ =

¯ U ; σT

V∗ =

V¯ σT

(28.9)

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¯ and V¯ are the depth-averaged cross-shore and longshore currents; σT , the where U standard deviation of the oscillatory (assumed Gaussian) depth-averaged velocity UT with zero mean; and r, the Gaussian variable deﬁned as r = UT /σT whose probability density function is given by 2 1 r f (r) = √ exp − . (28.10) 2 2π Linear progressive wave theory is used locally to express UT in terms of the oscillatory free surface elevation (η − η¯) Cp UT = ¯ (η − η¯) h

(28.11)

which yields the standard deviation σT of the oscillatory velocity UT σT = Cp σ∗ ;

¯ σ∗ = ση /h.

(28.12)

¯ /σT and V∗ = V¯ /σT are of the order of unity or less. The It is noted that U∗ = U standard deviations of U and V are given by σU = σT cos θ;

σV = σT | sin θ|

(28.13)

where cos θ > 0 but sin θ can be negative. Substitution of Eq. (28.8) into Eq. (28.7) yields 1 ρfb σT2 Gbx ; 2

τby =

FU Fa f (r)dr;

Gby =

τbx = with

Gbx =

∞

−∞

1 ρfb σT2 Gby 2

∞

−∞

FV Fa f (r)dr

(28.14)

(28.15)

which must be integrated numerically. The wind shear stresses are expressed as 2 τsx = ρa CD W10 cos θw ;

2 τsy = ρa CD W10 sin θw

(28.16)

where ρa is the air density (ρa 1.225 kg/m3 ); CD , the drag coeﬃcient, W10 , the 10-m wind speed; and θw , the wind direction deﬁned in Fig. 28.1. The formula by Large and Pond8 is used to estimate CD where CD = 0.0012 for W10 < 11 m/s and CD = (0.00049 + 0.000065 W10) for W10 ≥ 11 m/s. It is noted that the measured values of CD during tropical cyclones by Powell et al.9 indicated no increase of CD with the increase of W10 much above 25 m/s. In short, available data is insuﬃcient to estimate CD for extreme wind conditions. The wave energy equation is written as dFx = −DB − Df ; dx

Fx = ECg cos θ

(28.17)

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N. Kobayashi, A. Payo and B. D. Johnson

where Fx is the cross-shore energy ﬂux based on linear progressive wave theory, and DB and Df are the energy dissipation rates due to wave breaking and bottom friction, respectively. The energy dissipation rate DB due to wave breaking in Eq. (28.17) is estimated using the formula by Battjes and Stive,10 which was modiﬁed by Kobayashi et al.11 to account for the local bottom slope and to extend the computation to the lower swash zone. The breaker ratio parameter γ in this formula is typically in the range of γ = 0.5–1.0,1 but should be calibrated to obtain a good agreement with the measured cross-shore variation of ση if such data is available. The energy dissipation rate Df due to bottom friction is expressed as Df =

1 ρfb Ua3 . 2

Substitution of Ua given in Eq. (28.8) into Eq. (28.18) yields ∞ 1 Fa3 f (r)dr Df = ρfb σT3 Gf ; Gf = 2 −∞

(28.18)

(28.19)

where Fa and f (r) are given in Eqs. (28.8)–(28.10). The energy equation for the roller represented by its volume ﬂux qr may be expressed as3 d 2 ρCp qr cos θ = DB − Dr ; Dr = ρgβr qr (28.20) dx where the roller dissipation rate Dr is assumed to equal the rate of work to maintain the roller on the wave-front slope βr of the order of 0.1. Use is made of the empiric formula for βr proposed by Kobayashi et al.11 who included the local bottom slope eﬀect. If the roller is neglected, qr = 0 and Eq. (28.20) yields Dr = DB . The roller eﬀect is included in the subsequent computation to improve the agreement for the longshore current.1 Equations (28.1)–(28.20) are the same as those used by Kobayashi et al.17 who neglected the wind shear stresses in Eqs. (28.3) and (28.4) and used linear shallow¯ 0.5 in Eq. (28.11). Substitution of Eqs. (28.12) water wave theory with Cp = (g h) and (28.13) into Eq. (28.1) yields ¯ Cp qr gh ¯ (28.21) U = − 2 σU σ∗ 1 + Cp gση2 which reduces to the equation used by Kobayashi et al.1 in shallow water. The landward-marching computation starting from x = 0 outside the surf zone is the same as before. Approximate analytic equations of Gbx , Gby , and Gf given by Eqs. (28.15) and (28.19) are obtained in the following to reduce the computation time and improve the numeric stability. The function Fa given in Eq. (28.8) with Eq. (28.9) is rewritten as 2 0.5 (28.22) Fa = (r − rm )2 + Fm with rm = −(U∗ cos θ + V∗ sin θ);

Fm = V∗ cos θ − U∗ sin θ.

(28.23)

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Equation (28.22) is approximated as Fa = (r − rm ) + |Fm |

for r ≥ 0

Fa = −(r − rm ) + |Fm | for r < 0.

(28.24)

Substituting Eq. (28.24) into Eqs. (28.15) and (28.19) and integrating the resulting equations analytically, we obtain approximate expressions for Gbx , Gby , and Gf 2 (U∗ − rm cos θ) + U∗ |Fm | Gbx = (28.25) π 2 (V∗ − rm sin θ) + V∗ |Fm | (28.26) Gby = π 2 2 2 2 2 2 Gf = 2 (28.27) + (1 + U∗ + V∗ )|Fm | + U + V∗2 + 2rm π π ∗ which depends on sin θ (cos θ > 0 assumed), rm and Fm where Eq. (28.23) yields U∗ = −(rm cos θ + Fm sin θ) and V∗ = (Fm cos θ − rm sin θ). For the case of normally incident waves with no wind, sin θ = 0 and V∗ = 0. Equations (28.25)–(28.27) yield Gbx = 1.6 U∗ , Gby = 0, and Gf = (1.6 + 2.4 U∗2 ). For this case, Eq. (28.4) requires τby = 0 and Eq. (28.14) yields Gby = 0. As a result, Eq. (28.26) is exact. For sin θ = 0 and V∗ = 0, Gbx and Gf given by Eqs. (28.15) and (28.19) can be integrated analytically as presented by Kobayashi et al.4 who approximated the analytic expressions of Gbx and Gf as Gbx = 1.64 U∗ and Gf = (1.6 + 2.6 U∗2 ). These approximate equations are very similar to the above equations obtained from Eqs. (28.25) and (28.27). For the case of | sin θ| 1 and |U∗ | |V∗ |, Eq. (28.26) can be approximated as Gby = V∗ (0.8 + |V∗ |). Using ﬁeld data and probabilistic analyses, Feddersen et al.12 obtained Gby = V∗ (1.162 + V∗2 )0.5 . The diﬀerence between these two approximate equations for Gby is less than 20% for |V∗ | < 1.4, which is typically satisﬁed. Finally, the approximate values of Gbx , Gby , and Gf given by Eqs. (28.25)– (28.27) are compared with the exact values of Gbx , Gby , and Gf obtained by the numeric integration of Eqs. (28.15) and (28.19). The percentage error Ep is deﬁned as Ep =

100 × |exact value − approximate value| . |exact value|

The values of EP for Gbx , Gby , and Gf are computed for the ranges of | sin θ| < 1, |rm | < 1, and |Fm | < 1. The maximum value of Ep with respect to sin θ for given rm and Fm is obtained and plotted in Fig. 28.2. The percentage error increases with the increase of |rm | and |Fm | because of the approximation of Fa given by Eq. (28.24), which is exact only when rm = 0 and Fm = 0. The maximum error in Fig. 28.2 is less than 35%. This error is probably less than the uncertainty of the bottom friction factor fb involved in τbx , τby , and Df in Eqs. (28.14) and (28.19). Consequently, Eqs. (28.25)–(28.27) are adopted here for computational eﬃciency and stability. It is noted that the longshore momentum equation (28.4) is solved numerically to obtain τby and Gby by use of Eq. (28.14). Equation (28.26) is solved

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Fm

1

10

0

20

20

Fm

10

30

30

20

20 10

20 10

10

10

10

20

10

30

20

30

20

10

20 10

10

10 20

0 20

10 10 20

30

30

−0.5

20

30

30

20

Fm

10

30

10

−1 −1

20 20

10

10

−1 1

10

10 10

20 10

−1 1 0

10

30

20

10

0 rm

0.5

1

Fig. 28.2. Contours of maximum percentage error with respect to sin θ as a function of rm and Fm for Gbx (top), Gby (middle), and Gf (bottom).

analytically to obtain the longshore current V¯ for the computed Gby . This implicit numeric procedure improves the numeric stability of the computation marching in x (similar to time).

28.3. Sediment Transport Model The combined wave and current model predicts the cross-shore variations of the hydrodynamic variables used in the following sediment transport model for given beach proﬁle, water level, and seaward wave conditions at x = 0. The bottom sediment is assumed to be uniform and characterized by d50 , the median diameter; wf , sediment fall velocity; and s, sediment speciﬁc gravity. First, the cross-shore variation of the degree of sediment movement is estimated using the critical Shields parameter ψc 13 which is taken as ψc = 0.05. The instantaneous bottom shear stress τb is assumed to be given by τb = 0.5 ρfb Ua2 with Ua given in Eq. (28.8). The sediment movement is assumed to occur when |τb | exceeds the critical shear stress, ρg(s − 1)d50 ψc . The probability Pb of sediment movement 2 can be shown to be the same as the probability of (r − rm )2 > Fb2 = (Rb2 − Fm ) −1 0.5 where Rb = [2g(s − 1)d50 ψc fb ] /σT and rm and Fm are deﬁned in Eq. (28.23). For the Gaussian variable r given by Eq. (28.10), Pb is given by Fb − rm Fb + rm 1 1 √ √ + erf c for Fb2 > 0 Pb = erf c (28.28) 2 2 2 2

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and Pb = 1 for Fb2 ≤ 0 where erf c is the complementary error function. The value of Pb computed from x = 0 located outside the surf zone increases landward and ﬂuctuates in the surf and swash zones, depending on the presence of a bar or a terrace that increases the local ﬂuid velocity. Second, the cross-shore variation of the degree of sediment suspension is estimated using the experimental ﬁnding of Kobayashi et al.11 who showed that the turbulent velocities measured in the vicinity of the bottom were related to the energy dissipation rate due to bottom friction. Representing the magnitude of the instantaneous turbulent velocity by (Df /ρ)1/3 with Df = 0.5ρfb Ua3 in light of Eq. (28.18), the probability Ps of sediment suspension is assumed to be the same as the probability of (Df /ρ)1/3 exceeding the sediment fall velocity wf . The prob2 ) with ability Ps is then equal to the probability of (r − rm )2 > Fs2 = (Rs2 − Fm 1/3 Rs = [(2/fb ) wf /σT ] and is given by 1 Fs − rm Fs + rm 1 √ √ (28.29) Ps = erf c + erf c for Fs2 > 0 2 2 2 2 and Ps = 1 for Fs2 ≤ 0. If Ps > Pb , use is made of Ps = Pb assuming that sediment suspension occurs only when sediment movement occurs. Fine sands on beaches tend to be suspended once their movement is initiated. Third, the suspended sediment volume Vs per unit horizontal bottom area is estimated by modifying the sediment suspension model by Kobayashi and Johnson14 Vs = Ps

eB Dr + ef Df (1 + Sb2 )0.5 ; ρg(s − 1)wf

Sb =

dzb dx

(28.30)

where Sb is the cross-shore bottom slope; and eB and ef are the suspension eﬃciencies for the energy dissipation rates Dr and Df due to wave-breaking and bottom friction in Eqs. (28.17) and (28.20), respectively. Use is made of eB = 0.002 and ef = 0.01 adopted by Kobayashi et al.1 but the value of eB is uncertain and should be calibrated if Vs is measured.15 The sediment suspension probability Ps is added in Eq. (28.30) to ensure Vs = 0 if Ps = 0. The term (1 + Sb2 )0.5 is the actual bottom area per unit horizontal bottom area and essentially unity except for very steep slopes. The cross-shore and longshore suspended sediment transport rates qsx and qsy are expressed as ¯ Vs ; qsx = aU

qsy = V¯ Vs

(28.31)

where a is the empiric suspended load parameter. The parameter a accounts for the onshore suspended sediment transport due to the positive correlation between the time-varying cross-shore velocity and suspended sediment concentration. The value of a increases to unity as the positive correlation decreases to zero. For the three small-scale equilibrium proﬁle tests conducted by Kobayashi et al.,11 a was of the order of 0.2. Use is made of a = 0.2 in the subsequent computation. The cross-shore suspended sediment transport rate qsx is negative (oﬀshore) because the ¯ is negative (oﬀshore). On the other hand, the longreturn (undertow) current U shore suspended sediment transport rate qsy in Eq. (28.31) neglects the correlation between the time-varying longshore velocity and suspended sediment concentration, which appears to be very small if the longshore current V¯ is suﬃciently large.

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Fourth, the formulas for the cross-shore and longshore bedload transport rates qbx and qby are devised somewhat intuitively because bedload in the surf zone has never been measured. The time-averaged rates qbx and qby are tentatively expressed as qbx = B(U 2 + V 2 )U ;

qby = B(U 2 + V 2 )V

(28.32)

where B is the empiric parameter. Equation (28.32) may be regarded as a quasisteady application of the formula of Meyer-Peter and Mueller (e.g., Ref. 16). Substitution of U and V given in Eq. (28.8) with Eqs. (28.9) and (28.10) into Eq. (28.32) yields qbx = BσT3 (b∗ + U∗ V∗2 + 2Fm sin θ)

(28.33)

qby = BσT3 [V∗ (1 + U∗2 + V∗2 ) − 2rm sin θ]

(28.34)

where b∗ = (3U∗ + U∗3 ), and Fm and rm are deﬁned in Eq. (28.23). Equations (28.33) and (28.34) yield qbx = b∗ BσT3 and qby = 0 for normally incident waves with sin θ = 0 and V∗ = 0. The expressions of B and b∗ are obtained by requiring that qbx = b∗ BσT3 reduces to the onshore bedload formula proposed by Kobayashi et al.17 for normally incident waves. This formula was shown to synthesize existing data and simple formulas. The proposed formulas are written as bPb σ 3 (1 + U∗ V∗2 + 2Fm sin θ)Gs g(s − 1) T

(28.35)

bPb σ3 [V∗ (1 + U∗2 + V∗2 ) − 2rm sin θ] g(s − 1) T

(28.36)

qbx = qby =

where b is the empiric bedload parameter, and Gs , the bottom slope function. The sediment movement probability Pb given in Eq. (28.28) accounts for the initiation of sediment movement. It is noted that b∗ = 1 in Eq. (28.35) to compensate for the limitations of Eq. (28.32) and the Gaussian distribution of the horizontal velocity used in Eqs. (28.9) and (28.10) as discussed by Kobayashi et al.17 They calibrated b = 0.002 using the 20 water tunnel tests of Ribberink and Al-Salem18 and the four large-scale wave ﬂume tests of Dohmen-Janssen and Hanes.19 However, these tests were conducted for nonbreaking waves and the assumed value of b = 0.002 is uncertain in surf and swash zones. The bottom slope function Gs was introduced by Kobayashi et al.17 to account for the eﬀect of the steep cross-shore slope Sb on the bedload transport rate and is expressed as Gs = tan φ/(tan φ + Sb )

for − tan φ < Sb < 0

Gs = (tan φ − 2Sb )/(tan φ − Sb ) for 0 < Sb < tan φ

(28.37) (28.38)

where φ is the angle of internal friction of the sediment and tan φ 0.63 for sand.20 For Sb = 0, Gs = 1. Equation (28.37) corresponds to the functional form of Gs

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used by Bagnold21 for steady stream ﬂow on a downward slope with Sb < 0 where the downward slope increases qbx . Equation (28.38) ensures that Gs approaches negative inﬁnity as the upward slope Sb approaches tan φ and that Eqs. (28.37) and (28.38) reduce to Gs (1 − Sb / tan φ) for |Sb | tan φ. Equation (28.35) with Gs given by Eqs. (28.37) and (28.38) implies that the bedload transport rate qbx is positive (onshore) for Sb < (tan φ)/2 and negative (oﬀshore) for Sb > (tan φ)/2. Use is made of |Gs | < Gm = 10 to avoid an inﬁnite value in the computation. The computed proﬁle change is not very sensitive to the assumed value of Gm because the beach proﬁle changes in such a way to reduce a very steep slope except in the region of scarping. The landward marching computation of the present time-averaged model ends ¯ is less than 1 cm. at the cross-shore location x = xm where the mean water depth h The probabilistic model by Kobayashi et al.22 could be used to predict the irregular wave runup distribution but no reliable data exists for suspended sand and bedload transport rates in the zone which is wet and dry intermittently. Consequently, the following simple procedure is adopted to deal with the zone with the bottom slope Sb > tan φ. The cross-shore total sediment transport rate qx = (qsx + qbx ) at x = xm is denoted by qxm . If qxm is negative (oﬀshore), qx is extrapolated linearly to estimate qx on the scarped face with Sb > tan φ qx = qxm (xe − x)/(xe − xm ) for xm < x < xe

(28.39)

where xe is the landward limit of the scarping zone with Sb > tan φ. The extrapolated qx is in the range of qxm ≤ qx ≤ 0 and the scarping zone is eroded due to the oﬀshore sediment transport. This simple procedure does not allow onshore sediment transport due to overwash. Finally, the cross-shore beach proﬁle change is computed using the continuity equation of bottom sediment for the case of alongshore uniformity (1 − np )

∂zb ∂qx + =0 ∂t ∂x

(28.40)

where np is the porosity of the bottom sediment which is assumed to be np = 0.4, and t, slow morphologic time for the change of the bottom elevation zb . Equation (28.40) is solved using an explicit Lax-Wendroﬀ numeric scheme (e.g., Ref. 23) to obtain the bottom elevation at the next time level. This computation procedure is repeated starting from the initial bottom proﬁle until the end of a proﬁle evolution test. The computation time is of the order of 10−3 of the test duration. 28.4. Comparison with Data The present numeric model is compared with the spilling and plunging wave tests conducted in the large-scale sediment transport facility at the US Army Engineer Research and Development Center.24 Pumps were used to circulate the wave-induced longshore current and establish the alongshore uniformity of hydrodynamics and morphology on the middle section of the quasi-equilibrium beach

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818 Table 28.1. wave tests.

Wave conditions at x = 0 for spilling and plunging

Breaker

d (cm)

Hrms (cm)

θ (deg)

Tp (s)

fb

γ

Spilling Plunging

76.8 77.3

18.2 18.9

10 10

1.5 3.0

0.02 0.02

1.0 0.7

Initial

t=10 h

40 zb (cm)

Spilling 0 −40 −80 40 zb (cm)

Plunging 0 −40 −80 0

4

8

12

16

x (m) Fig. 28.3. Quasi-equilibrium beach profile (solid line) and computed profile after 10-h wave action (dash line) for spilling and plunging wave tests.

comprised of uniform ﬁne sand with d50 = 0.15 mm, s = 2.65, and wf = 1.65 cm/s. Table 28.1 lists the wave conditions at the seaward boundary x = 0 and the values of the bottom friction factor fb in Eq. (28.7) and the breaker ratio parameter γ for DB in Eq. (28.17). These values of fb and γ are the same as those used in the previous comparison by Kobayashi et al.1 The still water depth d, the root√ mean-square wave height Hrms = 8ση and the incident unidirectional wave angle θ at x = 0 were practically the same for both tests. The major diﬀerence was the spectral peak period Tp of the incident waves and the resulting breaker patterns. The quasi-equilibrium beach proﬁles of the two tests are shown in Fig. 28.3. Comparisons are made of the measured and computed cross-shore variations of the mean and standard deviation of the free surface elevation η, cross-shore velocity U , and longshore velocity V as well as the suspended sediment volume Vs per unit horizontal bottom area. Vs is predicted using Eq. (28.30) with the sediment suspension probability Ps and the bottom slope eﬀect, which were not included previously. These comparisons are assessed in light of those presented by Agarwal et al.25 and Kobayashi et al.1 The degree of agreement is found to be practically the same. The modiﬁcations presented in this chapter change the computed results very little. However, the present computation is more eﬃcient and stable because of the

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Cross−shore sand transport rate (cm2/s)

Cross−shore sand transport rate (cm2/s)

qbx

qsx

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qx

0.02 0 −0.02

Spilling

0.05 0 −0.05 Plunging −0.1 0

4

8 x (m)

12

16

Fig. 28.4. Computed cross-shore sand transport rates qbx , qsx , and qx = (qbx + qsx ) for spilling and plunging wave tests.

use of the analytic expressions of Gbx , Gby , and Gf given by Eqs. (28.25)–(28.27). The roller eﬀect represented by the roller volume ﬂux qr improves the agreement for the longshore current V¯ as shown in the previous comparisons. The computed sediment transport rates with the roller eﬀect are presented in the following. Figure 28.4 shows the computed cross-shore variations of qbx , qsx , and qx for the spilling and plunging wave tests. The cross-shore bedload transport rate qbx is positive (onshore), whereas the cross-shore suspended sand transport rate qsx is negative (oﬀshore). The absolute values of qbx and qsx are larger in the breaker zone near x = 4 m and near the still water shoreline, especially for the plunging waves. The computed total sand transport rate qx = (qbx +qsx ) is positive (onshore) except in the zone near the shoreline where qx < 0. The absolute value of qx is less than about 0.05 cm2 /s but qx = 0 is required on the equilibrium beach. Consequently, the proﬁle evolution is computed using the measured quasi-equilibrium proﬁle as the initial proﬁle. The initial proﬁle is exposed to the wave conditions listed in Table 28.1 for 10 hours. The computed proﬁle is shown in Fig. 28.3. The computed change of the bottom elevation zb is less than about 5 cm. The subtle proﬁle change is diﬃcult to predict and measure accurately. It is noted that the ﬂuid velocity and suspended sand concentration were not measured synchronously in these tests, resulting in no measurement of qsx . Figure 28.5 shows the computed cross-shore variations of qby , qsy , and qy = (qby + qsy ). The longshore bedload transport rate qby is positive (downwave) and small in comparison with the longshore suspended sand transport rate qsy . The longshore total sand transport rate qy is dominated by qsy in these tests using the ﬁne sand with d50 = 0.15 mm and wf = 1.65 cm/s. The longshore sand transport rate qy is larger under the plunging waves by a factor of more than 2. The comparison of Figs. 28.4 and 28.5 indicates that the longshore sand transport qy exceeds the cross-shore transport rate qx by a factor of about 5.

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qsy

Longshore sand transport rate (cm2/s)

qby 0.1

qy

Spilling

0.05

Longshore sand transport rate 2 (cm /s)

0 0.3

Plunging

0.2 0.1 0 0

4

8 x (m)

12

16

Fig. 28.5. Computed longshore sand transport rates qby , qsy , and qy = (qby + qsy ) for spilling and plunging wave tests.

Measured

Computed

qsy(cm2/s)

0.1 Spilling 0.05

0 qsy (cm2/s)

0.3

Plunging

0.2 0.1 0 0

4

8 x (m)

12

16

Fig. 28.6. Measured and computed longshore suspended sand transport rates for spilling and plunging wave tests.

Figure 28.6 compares the measured and predicted longshore suspended sand transport rates. The measured qsy was obtained by integrating the product of the measured mean longshore velocity and mean sand concentration vertically as explained by Kobayashi et al.1 The numeric model overpredicts qy in the middle of the surf zone by a factor of about 2 perhaps because Eq. (28.30) does not account for the downward decay of turbulence generated by broken waves landward of the breaker zone.

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Spilling

0.05

0

qy (cm2/s)

0.3

Plunging

0.2 0.1 0 0

4

8 x (m)

12

16

Fig. 28.7. Measured and computed longshore total sand transport rates for spilling and plunging wave tests.

Figure 28.7 compares the measured and predicted longshore total sand transport rates. The measured qy was obtained from the volume of sand collected in each of 20 downdrift bottom traps. This time-averaged model predicts the large sand transport rate near the shoreline fairly well. The overall agreement is within a factor of about 2. The diﬀerence in the cross-shore distributions of qy under the spilling and plunging waves is predicted by the present time-averaged model.

28.5. Conclusions The combined wave and current model for obliquely incident waves by Kobayashi et al.1 is improved by including the ﬁnite-depth eﬀect on the relationship between the oscillatory horizontal velocity and free surface elevation as well as the wind stresses for future ﬁeld applications. The numeric integrations involved in the bottom shear stresses and energy dissipation rate are replaced by suﬃciently accurate analytic expressions. These modiﬁcations improve the computational eﬃciency and numeric stability of the wave and current model. The sediment transport model developed by Kobayashi et al.11,17 for normally incident waves is extended to include obliquely incident waves and longshore currents. The probabilities of sediment movement and suspension are proposed to account for the initiation of sediment movement and suspension under irregular waves. The cross-shore and longshore suspended sediment transport rates are expressed in terms of the suspended sediment volume per unit horizontal area and the cross-shore and longshore currents. The onshore transport due to the correlation between the cross-shore velocity and suspended sand concentration is taken into account. The cross-shore and longshore bedload transport rates are estimated using a formula similar to that by

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Meyer-Peter and Mueller that reduces to the bedload formula by Kobayashi et al.17 for normally incident waves. The new sediment transport model coupled with the improved wave and current model is compared with spilling and plunging wave tests conducted in a large wave basin using ﬁne sand. The coupled model predicts the measured longshore suspended sand and total transport rates within a factor of about 2. The longshore bedload transport rate is predicted to be small in comparison with the corresponding suspended sand transport rate. The cross-shore sand transport rates were not measured in these quasi-equilibrium beach tests. The coupled model predicts only slight beach proﬁle changes after 10-h wave action in these tests. However, subtle proﬁle changes are diﬃcult to predict and measure accurately. The coupled numeric model will need to be compared with ﬁeld data to demonstrate its utility for practical applications. The assumptions of longshore uniformity and uniform sediment are too restrictive on natural beaches. The cross-shore one-dimensional model may be extended to a horizontally two-dimensional model without much diﬃculty but available sediment transport data is scarce in surf and swash zones. For the prediction of long-term morphologic changes, a method for long-term and spatial averaging will need to be devised to compute morphologic changes for years.

Acknowledgments This study was supported by the MORPHOS Project of the US Army Corps of Engineers, Coastal and Hydraulics Laboratory. The second author was supported by the Spanish Postdoctoral Scholarship MEC/FULBRIGHT.

References 1. N. Kobayashi, A. Agarwal and B. D. Johnson, Longshore current and sediment transport on beaches, J. Waterw. Port Coast. Ocean Eng. 133(4), 296–304 (2007). 2. A. J. H. M. Reniers and J. A. Battjes, A laboratory study of longshore currents over barred and non-barred beaches, Coast. Eng. 30, 1–21 (1997). 3. B. G. Ruessink, J. R. Miles, F. Feddersen, R. T. Guza and S. Elgar, Modeling the alongshore current on barred beaches, J. Geophys. Res. 106(C10), 22451–22463 (2001). 4. N. Kobayashi, L. E. Meigs, T. Ota and J. A. Melby, Irregular breaking wave transmission over submerged porous breakwaters, J. Waterw. Port Coast. Ocean Eng. 133(2), 104–116 (2007). 5. N. Kobayashi and F. J. de los Santos, Irregular wave seepage and overtopping of permeable slopes, J. Waterw. Port Coast. Ocean Eng. 133(4), 245–254 (2007). 6. R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (World Scientiﬁc, Singapore, 1984). 7. S. Lentz, R. T. Guza, S. Elgar, F. Feddersen and T. H. C. Herbers, Momentum balances on the North Carolina inner shelf, J. Geophys. Res. 104(C8), 18205–18226 (1999).

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8. W. G. Large and S. Pond, Open ocean momentum ﬂux measurements in moderate to strong winds, J. Phys. Oceanogr. 11, 324–336 (1981). 9. M. D. Powell, P. J. Vickery and T. A. Reinhold, Reduced drag coeﬃcient for high wind speeds in tropical cyclones, Nature 422, 279–283 (2003). 10. J. A. Battjes and M. J. F. Stive, Calibration and veriﬁcation of a dissipation model for random breaking waves, J. Geophys. Res. 90(C5), 9159–9167 (1985). 11. N. Kobayashi, H. Zhao and Y. Tega, Suspended sand transport in surf zones, J. Geophys. Res. 110(C12009) (2005), doi: 10.1029/2004JC002853. 12. F. Feddersen, R. T. Guza, S. Elgar and T. H. C. Herbers, Velocity moments in alongshore bottom stress parameterization, J. Geophys. Res. 105(C4), 8673–8686 (2000). 13. O. S. Madsen and W. D. Grant, Quantitative description of sediment transport by waves, Coast. Eng. 1976, Proc. 15th Coast. Eng. Conf., American Society of Civil Engineering, Reston, VA (1976). 14. N. Kobayashi and B. D. Johnson, Sand suspension, storage, advection and settling in surf and swash zones, J. Geophys. Res. 106(C5), 9363–9376 (2001). 15. N. Kobayashi and Y. Tega, Sand suspension and transport on equilibrium beach, J. Waterw. Port Coast. Ocean Eng. 128(6), 234–248 (2002). 16. J. S. Ribberink, Bed-load transport for steady ﬂows and unsteady oscillatory ﬂows, Coast. Eng. 34, 59–82 (1998). 17. N. Kobayashi, A. Payo and L. D. Schmied, Cross-shore suspended sand and bedload transport on beaches, J. Geophys. Res. 113(C07001) (2008), doi: 10.1029/ 2007JC004203. 18. J. S. Ribberink and A. A. Al-Salem, Sediment transport in oscillatory boundary layers in cases of rippled beds and sheet ﬂow, J. Geophys. Res. 99(C6), 12707–12727 (1994). 19. C. M. Dohmen-Janssen and D. H. Hanes, Sheet ﬂow dynamics under monochromatic nonbreaking waves, J. Geophys. Res. 107(C10), 3149 (2002), doi: 10.1029/ 2001JC001045. 20. J. A. Bailard, An energetics total load sediment transport model for a plane sloping beach, J. Geophys. Res. 86(C11), 10938–10954 (1981). 21. R. A. Bagnold, An approach to the sediment transport problem from general physics, U.S. Geol. Surv., Prof. Paper 422-I (1966). 22. N. Kobayashi, F. J. de los Santos and P. G. Kearney, Time-averaged probabilistic model for irregular wave runup on permeable slopes, J. Waterw. Port Coast. Ocean Eng. 134(2), 88–96 (2008). 23. R. B. Nairn and H. N. Southgate, Deterministic proﬁle modelling of nearshore processes. Part 2. Sediment transport and beach proﬁle development, Coast. Eng. 19, 57–96 (1993). 24. P. Wang, B. A. Ebersole, E. R. Smith and B. D. Johnson, Temporal and spatial variations of surf-zone currents and suspended sediment concentration, Coast. Eng. 46, 175–211 (2002). 25. A. Agarwal, N. Kobayashi and B. D. Johnson, Longshore suspended sediment transport in surf and swash zones, Coast. Eng. 2006, Proc. 30th Coast. Eng. Conf., World Scientiﬁc, Singapore (2006).

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Chapter 29

Headland-Bay Beaches for Recreation and Shore Protection John R.-C. Hsu∗,† , Melissa M.-J. Yu, F.-C. Lee and Richard Silvester∗ Department of Marine Environment and Engineering National Sun Yat-sen University, Kaohsiung 80424, Taiwan, ROC ∗ School of Civil and Resource Engineering University of Western Australia Nedlands, WA 6009, Australia † [email protected] Headland-bay beaches, natural or man-made, have curved planform and distinct compartments. These beaches bound by or behind rocky outcrops or artificial structures are not only aesthetically beautiful, but also the most recognizable coastal feature from the sky on a clear day. Despite they are ubiquitous in many countries in the world, most of them are in dynamic conditions which require continuous sediment supply to maintain their stability, an almost impossible scenario nowadays. However, it has been recently accepted that a bay beach in static equilibrium can remain stable without requiring sediment supply in a persistent swell environment, from which an empiric relationship termed parabolic bay shape equation was developed in the late 1980s, initially aiming for shore protection. Upon the introduction of this empiric bay shape equation and the software developed later in the 2000s for its ready applications, several practical examples are given to demonstrate its engineering applications which include the verification of bay beach stability, mitigation of beach downdrift of a harbor breakwater or inlet jetty, and design of bayed beach for recreation.

29.1. Introduction Headland-bay beaches, natural or man-made, are one of the most recognizable coastal features on maps, hydrographic charts, aerial photographs, and satellite imageries (e.g., Google Earth). They have curved planform and distinct compartments bounded by or behind rocky outcrops and headlands, which can be observed on exposed or sheltered sedimentary coasts, from oceanic edge to river bank. Those created as a by-product upon the installation of an artiﬁcial structure are in the 825

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form of salient or tombolo which can be observed in the lee of harbor breakwaters, detached breakwaters and inclined or composite groins. Despite these beaches have occupied about 50% of the world coastline1 and are ubiquitous in many countries in the world, most of them are in dynamic or even unstable condition which require continuous sediment supply to maintain their stability in a persistent swell environment. Although headland-bay beaches have been called in a variety of names since the 1900s (zeta-curved bays,2 half-heart-shaped bays,3 logarithmic spiral beaches,4 crenulated-shaped bays,5,6 curved or hooked beaches,7 pocket beaches,8 and headland-bay beaches4,9–11 ; see Refs. 12 and 13), a common feature among them is the asymmetric shape, which is characterized by a curved shadow zone, a gently curved transition and or a relatively straight tangential portion at its downdrift end. In terms of stability, a bay beach may be either in static equilibrium, dynamic equilibrium, or unstable.12–14 Static equilibrium is reached when the predominant waves are breaking simultaneously around the whole bay periphery, hence littoral drift is almost nonexistent and additional external sediment is not required to maintain its long-term stability. On the other hand, for bay beaches in dynamic equilibrium, balance in sediment budget is required to maintain their stability; otherwise, shoreline could retreat as supply reduces. Should supply diminish, then a bay beach in dynamic equilibrium may recede toward the limit deﬁned by static equilibrium. Finally for bays in unstable condition, which is often produced by wave sheltering due to structural addition on a beach, a curved shoreline will result with accretion in the lee accompanied by erosion downdrift, hence the so-called natural reshaping.13 With a plethora of names and indentations associated with headland-bay beaches, coastal geologists, geographers, and coastal engineers have attempted to derive an empiric expression to ﬁt part or whole of the bay periphery, since the 1940s. Most notably these include logarithmic spiral model,4 hyperbolic-tangent shape model,11 and parabolic bay shape model.15 These models diﬀer in mathematic expressions, coordinate system and origin, controlling parameters, and wave direction related to wave diﬀraction point. The former two models consider only the ﬁtting of geometry to the shoreline rather than stability, without looking at wave direction and the positioning of a headland in relation to the beach. On the other hand, the latter (i.e., the parabolic bay shape) relates the shoreline change to the tip of a headland or wave diﬀraction point upcoast, which is also the origin of a quadratic expression in this model, hence relocation of this control point by manmade extension can be assessed. This parabolic bay shape equation15 has received recognition in the new Coastal Engineering Manual16 as well as the Spanish Coastal Modeling System (SMC; Refs. 6 and 17) for project evaluation and design of recreational beaches. With the immense pressures nowadays from population growth and economic activities, coastal strips have become a prime target for industrial and residential development, harbors, navigation jetties, and recreation. Not only a natural embayment requiring veriﬁcation of stability prior to any proposal for man-made utilization or improvement, but also a plane straight coast may even be converted into an embayment with an addition of artiﬁcial structures. In fact, artiﬁcial bayed beaches have been created for recreation and tourism in the Mediterranean countries, especially in Spain, France, Italy, and Israel since the 1960s

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(e.g., Refs. 18–20). By creating a bay beach in static equilibrium with adequate storm beach buﬀer, its stability would not be aﬀected by any reduction in sediment supply, except the onslaught of a ﬁerce storm wave which could erode the berm with material being transported oﬀshore to form a bar. However, beach recovery is often ensured if adequate storm buﬀer width is provided. This is nature’s way of self-defense for her beach. The main purpose of this chapter is to demonstrate several geomorphic engineering aspects of the headland-bay beaches based on the concept of the parabolic bay shape equation,15 using software MEPBAY21 and SMC6,17 that have recently developed for applying this empiric equation. Examples to be presented include veriﬁcation of beach stability on natural and artiﬁcial embayment, especially those at downdrift of navigation jetties, mitigation of beach erosion by headland control, and the design of a bay beach for recreation.

29.2. Parabolic Bay Shape Equation and Software Hsu and Evans15 have developed an empiric model for bay beach called “parabolic bay shape equation” from ﬁtting the planform of 27 mixed cases of 14 prototype and 13 model bays believed to be in static equilibrium. This empiric equation is in the nondimensional form of a second-order polynomial, Rn = C0 + C1 Rβ

β θn

+ C2

β θn

2 .

(29.1)

Equation (29.1) has two primary physical parameters, the reference wave obliquity angle β (measured in radians) and the control line length Rβ (see deﬁnition sketch in Fig. 29.1). The former is the wave obliquity measured at the updrift

Fig. 29.1. Deﬁnition sketch of parabolic bay shape equation, showing the primary and dependent parameters (modiﬁed from Refs. 12, 13, and 15).

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control (diﬀraction) point from the local incident wave crest to the control line, which links the diﬀraction point and a downdrift control point on the more straight part of the curved beach; or that between the downdrift tangent and the control line. It is worth noting that any point on or near the straight downdrift part of the curved beach could be conveniently chosen as a downdrift control point, without causing sensitive variation in the resulting value of Rn /Rβ . This can be determined visually from a map, vertical aerial photograph, or on a planning sketch, once an observer has grasped the physical insight of wave propagation toward a beach with a headland. In addition, Eq. (29.1) also contains two dependent parameters, which are radius Rn to any point on the curved shoreline angled θn from the same incident wave crest line radiating out from the point of wave diﬀraction. The three C coefﬁcients in this equation were generated by regression analysis to ﬁt the peripheries of the 27 prototype and model bays and vary with the reference wave angle β.12,13 In recent years, the values C coeﬃcients have been modiﬁed to include the straight downdrift part of the embayment,22–24 or can be further simpliﬁed to ease its manual calculation for Rn /Rβ . Manual application of Eq. (29.1) is straightforward. For example, (1) ﬁrst choosing a downdrift control point on a more straight part of the embayment or an appropriate location on a curved shoreline and linking it to the updrift control point (i.e., point of wave diﬀraction) to form the control line; (2) determining the reference wave angle β and the control line length Rβ ; (3) calculating a set of rays and polar-angles (Rn , θn ) on the embayed shoreline, starting from θn = β at the downdrift control point; and (4) ﬁnally sketching the resulting shoreline in static equilibrium. The user may then visually examine the stability of the bay beach by the nearness of the existing shoreline to that predicted in static equilibrium. Should they coincide with each other, or very close, then the existing beach may be classiﬁed as in static equilibrium or in quasi-static equilibrium. In this condition, the stability of this embayment would not be aﬀected by any reduction in sediment supply externally or internally, except the onslaught of a ﬁerce storm. If the existing shoreline is seaward of that predicted, then the beach is said to be in dynamic equilibrium, in which shoreline could remain unchanged from the balance in sediment budgets. However, shoreline in dynamic equilibrium may retreat as net supply decreases. Moreover, an unstable situation which implies the existing shoreline is landward of the predicted and natural beach reshaping would occur within the embayment due to wave sheltering by structures, in which accretion in the lee accompanying by erosion downdrift. However, manual applications of Eq. (29.1) are rather repetitive and laborious. To improve the eﬃciency of application, this empiric equation has been integrated into software called MEPBAY21 and SMC.6,17 First, MEPBAY was developed as educational software in the University of Vale at Itajai, Brazil, which can be downloaded from http://siaiacad05.univali.br/∼meppe. Wave height information is not required upon applying MEPBAY; hence, the application procedure is straightforward. On the other hand, the parabolic bay shape equation has been integrated into SMC (Coastal Modeling System), a numeric modeling package developed by the Ocean and Coastal Research Group (GIOC) in the University of Cantabria

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for the Spanish Ministry of Environment during the period of 1996–2003. This model includes wave-current–sediment calculations for short- and long-term beach and proﬁle changes, and incorporates a beach design module. Comparing with the MEPBAY, SMC is much sophisticated because calculations of wave-current ﬁeld are required while applying it for the design of a headland-bay beach in static equilibrium. SMC can be downloaded after registration from http://www.smc.unican.es. Examples of its application to beach design will be given in Sec. 5 later in this chapter. Klein et al.21 demonstrated three categories of bay beach stability using MEPBAY. Their examples are reproduced here to assist the readers in applying MEPBAY and post assessment of beach stability. The ﬁrst example is for a bay beach in static equilibrium at Taquaras–Taquarinhas in the central-north coast of Santa Catarina, Brazil because the existing shoreline planform is almost identical to the static bay shape given by the parabolic equation (Fig. 29.2). Second example depicts the dynamic equilibrium condition at the southern part of Balne´ ario Cambori´ u Beach, also in Santa Catarina, Brazil, where the existing shoreline planform is seaward of that predicted (Fig. 29.3) and its current stability has been maintained by sediment from updrift and/or riverain source. The third example in Fig. 29.4 indicates the eﬀect of man-made structure on Porto Beach in Imbituba, Santa Catarina, Brazil, where downdrift erosion has occurred accompanying by accretion in the lee of the structure. Nearshore current is responsible for the transport of sediment from downdrift to the lee, rather than the usual longshore current. In this case, the existing beach is landward of the static bay shape predicted by the parabolic model using MEPBAY. On the other hand, the SMC6,17 software was developed as part of Spanish Coastal plan, the State Coastal Oﬃce and the Coastal and Ocean Research Group (GIOC) of the University of Cantabria for a new Spanish Beach Nourishment and

Fig. 29.2. Example of a bay beach in static equilibrium at Taquaras–Taquarinhas in Santa Catarina, Brazil.21

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Fig. 29.3. Example of a bay beach in dynamic equilibrium at Balne´ ario Cambori´ u in Santa Catarina, Brazil.21

Fig. 29.4. Example of a bay beach in unstable condition or natural reshaping at Porto Beach in Imbituba, Santa Catarina, Brazil.21

Protection Manual (SBM) between 1995 and 2003. The coastal modeling system (SMC) allows coastal technicians to develop coastal engineering projects based on a work methodology, a database of littoral morphodynamic information and numeric tools. The many diﬀerent numeric models included in the SMC permit for a speciﬁc engineering project, the design and evaluation of coastal alternatives in diﬀerent scales for short-term (hours–days), medium-term (seasonal changes), and long-term (years–decades). The organization logo of the SMC in Spanish is depicted in Fig. 29.5.

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Organization logo of the Spanish SMC. Source: http://www.smc/unican.es.

The SMC is a user-friendly system. It is structured in ﬁve modules25 : (1) A preprocess module which generates all of the input data for the short-, middle-, and long-term numeric models. This module obtains (for any location along the Spanish coast including the islands) all the Spanish bathymetry (Baco), wave directional regimes (Odin), tidal information (Atlas), and the littoral ﬂooding risk. (2) The short-term module includes beach proﬁle elevation (Petra) and numeric evolution morphodynamic models (Mopla) for monochromatic and irregular input waves, in a process on a scale of hours to days. (3) The middle- and long-term modules allow the analysis of the middle-term processes (seasonal changes) and long-term response of the system on a scale of years. (4) The bathymetry renovation module permits easy updating of the actual bathymetry including diﬀerent elements (sand ﬁlls in equilibrium beaches: plan and proﬁle, coastal structures, etc.) to evaluate the different alternatives proposed using the numeric models. (5) The tutorial module (Tic) includes the theoretic background in a numeric system and provides some data process systems for time series (e.g., buoys and tidal gauges). This module supports the science-based documents and it is subdivided into four items: dynamics, coastal processes, coastal structures, and environmental impact. Nowadays, the SMC is a nationwide numeric package in Spain, with regular training course held every three months in Madrid, and several Spanish-speaking countries in the Latin America.

29.3. Test of Stability on Natural and Man-Made Bay Beaches Headland-bay beaches in natural condition diﬀer not only in shape and size, but also with diﬀerent degrees of sediment availability and exposures to the prevailing waves. They may appear in the form of single curved embayment with a straight portion

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downdrift of a headland, an asymmetric to symmetric form bounded between two headlands, or even as salient or tombolo in the lee of oﬀshore island. On bay beaches with man-made structures, the structural components, similar to the natural headlands or outcrops, are now in the form of single groin, straight or curved, inlet jetties or composite groins, or even detached breakwater or harbor breakwater. It is highly recommended that natural or man-made embayment be tested against its stability, prior to any modiﬁcation to the existing headland in order to mitigate or avoid possible negative eﬀect on the beach. 29.3.1. Stability of natural bay beaches The ﬁrst example for the stability of a natural embayment is displayed in Fig. 29.6. This Google Earth imagery shows the Bicheno headland and the beaches in its lee in the NE of Tasmania, Australia. A salient planform is visible in the lee of the Bicheno headland. The center of the main beach bounded between the tips of the headland marked as B and C is located at 148◦ 17 11 E and 41◦ 51 48 S. Upon applying MEPBAY to these wave diﬀraction points, respectively, it can be shown both the upper and lower parts of this main beach are in static equilibrium. The same result can be found for the bay beach in the lee of headland A. Physically, these bay beaches are not only aesthetically attractive, but also stable. Despite having shoals in the vicinity of the headlands, their inﬂuence on local wave

Fig. 29.6. Example of natural bay beaches in static equilibrium at Bechino, Tasmania, Australia. Courtesy of Google Earth imagery.

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propagation seems to be negligible. However, with a bay beach in static condition, suggestion to modify the position of any tip (e.g., serving as a new jetty for marina) would render the bay beach unstable, which is a scenario that should be avoided at all cost. In addition, the same beach planform can be maintained if the Bicheno headland is replaced by a straight detached breakwater spanning across tips A and B, together with single groin with tip at C. 29.3.2. Stability of man-made bay beaches Since there are many bay beaches have been created in the form of salient or tombolo in the lee of detached breakwaters with variable gaps between two consecutive units, it would be useful to see whether these beaches conform to the static bay shape, even if they were designed and constructed without using the parabolic equation (29.1). The same principle can be extended to bay beaches between T-head groins, where salient or tombolo planform can be found. Mamzar Beach Park in Dubai, United Arab Emirates (Fig. 29.7) was developed in the 1980s. The site consists of three detached breakwaters with length ranging from 220 to 260 m with clear gaps about 230 m, and two terminal groins, altogether over a stretch of 1.4 km long. Despite these tombolo beaches were constructed without the use of the parabolic bay shape equation, which was derived in 1989, the planform of each and every embayment can now be veriﬁed using MEPBAY and is found to be in static equilibrium. With the hindsight, it may be perceived that, once the tips of the headlands and shoreline limits have been determined, the stable bay shape in each compartment can be constructed with conﬁdence.

Fig. 29.7. Example of ﬁtting static bay shape to man-made tombolo beaches constructed in the 1980s at Mamzar Beach, Dubai, UAE. Courtesy of Google Earth imagery.

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29.4. Mitigation of Erosion Downdrift of Navigation Jetties Straight groins have been used as inlet jetties or training walls to maintain a navigable channel linking an inland harbor or lagoon with the ocean in many places. Beach downdrift of these structures may be already curved or could evolve into a bay periphery, initially in a dynamic condition. One basic principle associated with the jetty construction is to minimize negative impact on the downdrift beach, by positioning its tip appropriately with the right length. The ﬁrst example is for the Doran Beach on Bodega Bay, about 34 km SW of Santa Rosa on the Paciﬁc coast of California (Fig. 29.8). Without details of local bathymetry, wave conditions, and ﬁeld monitoring, it is not known whether this beach is in static or dynamic condition. However, a preliminary assessment using MEPBAY to the satellite imagery indicates that the Doran Beach was in dynamic condition, as the static bay shape predicted (dotted line marked as A D) is landward of the existing beach and its equilibrium could be maintained by suﬃcient sediment supply from updrift coast and/or riverain source in the back lagoon where the Bodega harbor is located.26 It is not known when this picture was taken, but with the two inlet jetties in place, the southern jetty can be applied to test the stability of Doran Beach at the time after the installation of these jetties. First, application of MEPBAY with the tip B renders the predicted static bay shape landward of the existing waterline. This is similar to that before the jetty construction, but this later static bay shape marked as B D is closer to the existing shoreline than the previous A D; hence, less potential erosion overall. It can be envisaged that beach may retreat if sediment supply reduces in the future. However, for a bay beach in dynamic condition, a

Fig. 29.8. Eﬀect of repositioning the tip of an inlet jetty on the stability of Doran Beach in Bodega Bay, California. Courtesy of Google Earth imagery.

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feasible remedy to mitigate or prevent potential erosion is by extending point B to point C, thus converting the stability from dynamic to quasi-static or static equilibrium, since the resulting bay periphery C D is in better agreement with the existing shoreline. By this way, which can be called geomorphic engineering,27 the beautiful bay beach can be saved without hard structures on it. Another example to illustrate the importance of a proper jetty length and positioning of its tip on downdrift beach is given in Fig. 29.9. This picture displays the 1,900-m long inlet jetty commencing from Zuniga Point and the entire bay beaches fringing the Coronado Bay, from North Island to Coronado, Glorietta and further southward to the famous Imperial Beach. Application of MEPBAY with the most southern tip of the Zuniga jetty as the updrift control point on the Google Earth imagery reveals that Coronado Bay foreshore can be classiﬁed as in dynamic condition because the existing shoreline is seaward of the predicted static bay shape marked as A D. However, the existing bay shape deviates from the predicted curve A D starting at Glorietta Beach, where local erosion has been reported in the recent years. Similar to Doran Beach presented in Fig. 29.8, local foreshore erosion may be mitigated if the present jetty tip A is extended to point B, thus making the predicted static bay shape B D closer to the existing beach planform. This geomorphic approach may be considered as a feasible option in the future, if required, for mitigating potential erosion on a downdrift beach.28

Fig. 29.9. Example of mitigating potential downdrift erosion by extending or repositioning the tip of Zuniga jetty in Coronado Bay, San Diego, California. Courtesy of Google Earth imagery.

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29.5. Design of Bay Beaches for Recreation and Shore Protection The parabolic bay shape equation has the potential to improve the stability of an existing bay beach, and therefore, can help design a static bay shape for shore protection and recreation. This will create a win–win situation for a local community. Ideally, a bay beach design requires nourishment to be protected by artiﬁcial headlands and incorporates with the ﬁll placement with suﬃcient storm beach buﬀer to reproduce a static equilibrium planform. The artiﬁcial headland may be a member of composite groins29 (T -, L-, and Y -shaped; ship anchor or ﬁsh-tailed) or detached breakwater, in addition to the fulﬁllment of cross-shore proﬁle equilibrium. 29.5.1. Application of MEPBAY With the aid of the MEPBAY, a bay beach in static equilibrium can be designed once the persistent incident wave direction is known or perceived. Initially, wave height and wave period are not required, neither numeric calculations of wave propagation and longshore currents at the preliminary design stage. This approach not only save the valuable time and budget, but also provide reasonably accurate result for a bay beach to be created. For example, Fig. 29.10 is an oblique view of a recreational bay beach produced by partial nourishment at Shirarahama Beach, Wakayama Prefecture in Japan. The T -shaped groin head is 170 m long with a 130 m trunk to help stabilize the initial ﬁll volume of 65,000 m3 imported from overseas. The entire project had taken more than 10 years to complete. During this time span a series of scientiﬁc tasks were undertaken, ranging from numeric simulations, ﬁeld monitoring programs, and laboratory experiments.30 With the hindsight, the parabolic bay shape equation and MEPBAY can be readily applied to verify the

Fig. 29.10. An oblique view of a recreational bay beach with T -head groin, submerged detached breakwater and nourishment at Shirarahama, Wakayama, Japan. Courtesy of Prof. T. Yamashita.

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Fig. 29.11. Fitting a static bay shape to Shirarahama Beach, Wakayama, Japan. Courtesy of Dr. T. Mishima.

resulting beach planform, which is shown to be very close to static equilibrium (Fig. 29.11). For preliminary study, this process can be completed in a matter of minutes on a scaled map or a vertical aerial photograph rather than in months or years. The discrepancy between the ﬁeld data and static bay shape behind the Gongenzaki point is due to the presence of a 40-m long submerged detached breakwater and the natural reefs nearby. Another example of designing single bay beach which is stabilized by a pair of straight groins or revetment walls can be found on South Island, SE of the tourist island of Sentosa, Singapore (Fig. 29.12). The pair of nonparallel groins protects an artiﬁcially nourished beach. This design was ﬁrst initialed in the 1990s, and now can be viewed on Google Earth. Even the picture reveals the project is still in the construction stage, the striking agreement with the static bay shape at high tide water level is evident. 29.5.2. Application of SMC Instead of the visual assessment applying the MEPBAY on a vertical aerial photograph or a scaled map, the Spanish SMC integrates the concept of static bay shape together with a series of numeric computations of waves and currents and sediment transport. Currently, the SMC, which contains speciﬁcally a module for bay beach in static equilibrium in plan and proﬁle equilibrium, is the only numeric package for beach design among the software tools currently available worldwide. This methodology diﬀers from the conventional approach of implying equilibrium in planform and beach proﬁle separately. With this innovative approach, sediment

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Fig. 29.12. Artiﬁcial bay beach under construction on South Island, Singapore showing predicted static bay shape at high tide level. Courtesy of Google Earth imagery.

leaking oﬀshore can be minimized as the headland tip position is linked to the natural equilibrium proﬁle. Moreover, Gonzalez and Medina6 proposed a method using local wave ﬂux to modify the downdrift control point to the original parabolic bay shape deﬁnition (see Eq. (29.1) and Fig. 29.1). Example of this design concept is illustrated in Fig. 29.13 for Poniente Beach and San Lorenzo Beach, in the lee of Gij´on Harbor, Spain, aiming to preserve these beaches following the expansion of harbor facilities. Another example applying the SMC for bay beach design is available at Dapeng Bay, Pindong County in southern Taiwan. One of the options put forward to the authority is shown in Fig. 29.14, which calls for a spur groin of 200-m long butting out from the tip of the southern inlet jetty to the Dapeng Bay lagoon, with a 300-m long detached breakwater about 800 m to the south of this jetty, as well as a short detached breakwater about 100 m further southward. To accomplish these three headland-bay beaches, the total volume of nourishment calculated by the SMC is about 660,000 m3 . A salient planform is expected in the lee of the 300-m long detached breakwater in the middle of this coastal sector. The SMC also outputs the new bathymetry from simultaneous equilibrium in planform and beach proﬁle, as illustrated in Fig. 29.15. Based on the results of this expected distribution of bottom contours, the SMC then performs a series of numeric calculations on waves (for swells and storms) and longshore currents (not shown in this chapter). With the unique operational system inherited with the SMC and onscreen modiﬁcation of bottom contours via the tracking of the mouse on the screen, graphic outputs are usually available in a matter of minutes for most cases of bay beach design, once the bathymetry and wave conditions are pre-processed

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Fig. 29.13. Design of static bay shapes for recreation at Poniente Beach and San Lorenzo Beach, Gijon, Spain. Courtesy of SMC software.

Fig. 29.14. Design of static bay shapes for recreation on an existing bathymetry at Dapeng Bay in southern Taiwan.

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Fig. 29.15. An SMC output of the modiﬁed bathymetry following the proposed headland layout of static bay shapes at Dapeng Bay in southern Taiwan.

and resided in the system. For engineering applications, a feasible option can then be made after examining a series of trial options.

29.6. Conclusions The shoreline along the boundary between the land and the sea may be either straight over a suﬃcient length or curved in plane separated by headlands, natural or man-made. The curved beaches are aesthetically beautiful and very stable, if static equilibrium is reached, compared to their straight counterparts. As more than 70% of the world’s sandy beaches have retreated31 due to natural and man-made factors, it would seem worthwhile to consider the more permanent geomorphic form of the bay beach in static equilibrium, for which a parabolic bay shape equation was developed.15 On shore protection, traditional hard structures and artiﬁcial nourishment without adequate structural protection may not be the best or long-term solution for an eroding beach, as the diminishing trend of sediment supply has prevailed or is imminent on almost all the sandy beaches worldwide. A practical and feasible approach is to emulate a bay beach in static equilibrium in a natural environment that has remained stable without the need of additional supply in a persistent swell condition. This concept has been labeled as Headland Control.12,13 For places receiving storm wave attack from time to time, a bay beach may survive if an adequate storm buﬀer is provided incorporating headland control methodology. If a bay beach is already in static equilibrium, it should be spared from any proposal for large-scale development because an unstable condition would eventuate with the ongoing erosion problem. However, should a bay beach be in dynamic equilibrium, it can be converted into static equilibrium by installing an artiﬁcial headland near the tip of the present headland.26

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Based on the static bay beach concept and with the aid of software MEPBAY21 and SMC6,17,25 demonstrated in this chapter, it should become obvious that the parabolic bay shape model is beneﬁcial for a wide range of practical applications in coastal management and shore protection, ranging from the veriﬁcation of bay beach stability in natural or man-made condition, for devising a plan of headland layout to stabilize the sandy beach bounded between them, for mitigating potential beach erosion downdrift of harbor breakwater and inlet jetty, and for the design of bay beaches for recreation.

Acknowledgments The authors wish to acknowledge Prof. R. Medina and Prof. M. Gonzalez, Coastal and Ocean Research Group (GIOC), University of Cantabria, Spain, for the provision of SMC implemented in this study. JRCH also thanks the support from Asian Paciﬁc Ocean Research Center, NSYSU, Taiwan.

References 1. A. Short and G. Masselink, Handbook of Beach and Shoreface Morphodynamics, ed. A. Short (John Wiley & Sons, New York, 1999), p. 230. 2. G. H. Halligan, Proc. Limnol. Soc., New South Wales 31, 619 (1904). 3. R. Silvester, Nature, Paper 4749, 467 (1960). 4. W. E. Yasso, J. Geology 73, 702 (1965). 5. R. Silvester and S. K. Ho, Proc. 13th Int. Conf. Coast. Eng., ASCE (1972), p. 1394. 6. M. Gonzalez and R. Medina, Coast. Eng. 43, 209 (2001). 7. C. C. Ree and P. D. Komar, J. Sediment. Petrol. 45, 866 (1975). 8. R. Silvester, Proc. Inst. Civil Eng., Part 2, 45, 677 (1980). 9. P. H. LeBlond, J. Sediment. Petrol. 49, 1093 (1979). 10. J. D. Phillips, Marine Geol. 65, 21 (1985). 11. L. J. Moreno and N. C. Kraus, Proc. Coast. Sediments’99 (1999), p. 860. 12. R. Silvester and J. R.-C. Hsu, Coastal Stabilization: Innovative Concepts (Prentice Hall, New Jersey, 1993). 13. R. Silvester and J. R.-C. Hsu, Coastal Stabilization (Reprint from Silvester and Hsu, 1993 by World Scientific, Singapore, 1997). 14. J. R.-C. Hsu, Encyclopedia of Coastal Science, ed. M. L. Schwartz (Springer, 2005), p. 415. 15. J. R.-C. Hsu and C. Evans, Proc. Inst. Civil Eng., Part 2, 87, 557 (1989). 16. USACE, Coastal Engineering Manual, US Army Corps of Engineers, Coastal Engineering Research Center, Washington DC (2002). 17. M. Gonzalez and R. Medina, Proc. 29th Int. Conf. Coast. Eng., ASCE (2002), p. 844. 18. L. Tourmen, Proc. 11th Int. Conf. Coast. Eng., ASCE (1968) p. 556. 19. J. M. Berenguer and J. Enriquez, Proc. 21st Int. Conf. Coast. Eng., ASCE (1988), p. 1411. 20. MOPU, Coastal Action, Director General of Port and Coast, Spain (1988), 143pp. 21. A. H. F. Klein, A. Vargas, A. L. A. Raabe and J. R.-C. Hsu, Comput. Geosci. 29, 1249 (2003).

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22. S. K. Tan and Y. M. Chiew, J. Waterw. Port Coast. Ocean Eng. 120, 145 (1944). 23. M. Serizawa, T. Uda, T. Kumada, T. San-Nami and K. Furuika, Ann. J. Coast. Eng. 47, 601 (2000) (in Japanese). 24. K. Sakai, A. Kobayashi, T. Kumada, T. Uda and M. Serizawa, Proc. Coast. Sediments’03 (2003), p. 1. 25. M. Gonzalez and R. Medina, Proc. 2005 Int. Coast. Plan. Forum, Kaohsiung, Taiwan (2005), p. 57. 26. J. R.-C. Hsu, L. Benedet, A. H. F. Klein, A. L. A. Raabe, C.-P. Tsai and T.-W. Hsu, J. Coast. Res. 24, 198 (2008). 27. D. R. Coates, Geomorphology and Engineering, ed. D. R. Coates (George Allen & Unwin, London, 1980), p. 3. 28. J. R.-C. Hsu, J.-C. Chu, S.-R. Liaw and C.-Y. Lee, Proc. 30th Int. Conf. Coast. Eng., ASCE (2006), p. 3762. 29. K. R. Bodge and S. Howard, Proc. 30th Int. Conf. Coast. Eng., ASCE (2006), p. 3756. 30. Y. Tsuchiya, Y. Kawata, T. Yamashita, T. Shibano, M. Kawasaki and H. Habara, Proc. 23rd Int. Conf. Coast. Eng., ASCE (1971), p. 3426. 31. E. C. F. Bird, Beach Management (John Wiley & Sons, Chichester, 1966).

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Chapter 30

Beach Nourishment Robert G. Dean Department of Civil and Coastal Engineering University of Florida, Gainesville, FL, USA [email protected] Julie D. Rosati U.S. Army Corps of Engineers Coastal and Hydraulics Laboratory, Mobile, AL, USA [email protected] Beach nourishment comprises the placement of sediment in the nearshore to advance the shoreline seaward and is usually placed in response to beach erosion which may be naturally or anthropogenically induced. Nourishment with compatible sediment has the advantage of maintaining the beach system in a nearnatural condition and can provide beneﬁts to biota and upland property, the latter by buﬀering storm erosion and damage. The processes associated with beach nourishment are becoming better understood such that project design can be accomplished with simple or more complex models. The more simple models, called one-line models, represent the changes in the entire active beach proﬁle by a single contour and consider the proﬁle to be displaced without change in form in response to a change in volume. The transport equation completes the governing equations. Pelnard–Consid`ere (PC) combined these two equations into a linearized second-order diﬀerential equation. Although the PC equation generally is not useful directly for design, it provides the basis for the derivation of a number of fundamental results. This chapter considers beach nourishment evolution to be represented as planform evolution and proﬁle adjustment.

30.1. Introduction Beach nourishment comprises the placement of large quantities of sediment in the nearshore to advance the shoreline seaward. The placement may extend the prenourishment beach seaward or occur as an oﬀshore deposit with the expectation that 843

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Fig. 30.1.

Beach and proﬁle (or berm) nourishment.

this deposit will both provide a degree of sheltering and gradually be transported landward by wave action (Fig. 30.1). The beneﬁts of beach nourishment can be fairly broad including improving recreational opportunities,1 providing protection against major storms,2 esthetics (beaches are more attractive than coastal armoring), and enhancement of ecologic functions, such as increasing habitat. The economic viability of beach nourishment depends on a number of factors including, but not limited to: (1) the value of the upland that is to be protected including both the structures and the income producing value of a wide recreational beach (e.g., Ref. 1), (2) the background erosion rate, (3) the availability of relatively low-cost quality sediment, and (4) the value of ecologic enhancements. Generally, the beaches most ideally suited for nourishment are those in which the value of the upland resources to be protected or enhanced are substantial and the background erosion is low such that with a moderate amount of sediment input, the system can be “tipped” back to neutrality. In evaluating an eroded area to determine the appropriateness of beach nourishment, the cause of the erosion should be identiﬁed to the degree possible. In some cases, the cause is anthropogenically induced such as interruption of the longshore sediment transport by a jettied inlet and in such cases, it is desirable to reinstate the transport. In other cases, the cause may be the result of constructing upland infrastructure too far seawarda and the associated diﬃculties of maintaining a beach and in still other cases, the erosion may be completely natural. The most appropriate response to beach erosion is always more likely to be identiﬁed with a thorough understanding of the cause of the erosion that has led to the need for nourishment. This chapter provides a review of the planform and proﬁle processes related to beach nourishment. The governing equations are introduced to the degree that they are known. The combination of the linearized longshore sediment transport equation and the continuity equation results in a second-order partial diﬀerential equation a In some cases, coastal construction has occurred during periods when the beach was at an abnormally seaward position. With the return of more normal conditions, the beach translated landward and the construction was threatened. Thus, nourishment was considered as an option. In other cases prior to good coastal management, construction to expand the existing hotel facilities encroached on the active beach.

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from which considerable basic information regarding nourishment interrelationships can be determined. Informative references on beach nourishment include: Delft Hydraulics Laboratory,3 Dean and Yoo,4 Stauble and Kraus,5 National Research Council,6 Houston,7 Davis et al.,8 Gravens et al.,9 Hanson et al.,10 and Dean.11

30.2. Morphologic Settings The need for beach nourishment may occur in a variety of morphologic settings and the analysis and design challenges diﬀer for each of these. Four diﬀerent morphologic settings are considered in this chapter: (1) a pocket beach in which there is little exchange with the adjacent beaches; (2) a long uninterrupted beach, probably the most common of the four; (3) a beach downdrift of a partial or complete littoral barrier; and (4) a barrier island subject to the combined processes of “rollover,” subsidence, and longshore sediment transport gradients. The unique characteristics of each of these settings are described brieﬂy below. 30.2.1. Pocket beach A pocket beach which is essentially isolated from the adjacent coastal system is the most simple of the four cases, whereas the other three cases involve both cross-shore and longshore considerations; if the entire pocket beach is nourished, this setting requires consideration only of cross-shore processes, namely proﬁle equilibration. If only a portion of the longshore dimension of the pocket beach is nourished, planform evolution will occur until the beach has advanced approximately uniformly over the longshore dimension of the pocket beach. Pocket beaches may be “leaky” such that sand will be transported around the headlands to the adjacent coastal system. Also, the planform orientations of pocket beaches often oscillate in response to seasonal or longer term changes in wave direction. In such cases, the tendency for leakage of sand to the adjacent coastal system is increased by nourishment which advances the proﬁle seaward. A speciﬁc example of designing artiﬁcial headlands to form pocket beaches has been described by Bodge and Olsen.12 30.2.2. Long straight beach We will see that nourishment on a long straight beach requires consideration of both longshore and cross-shore processes. However, if the nourishment is carried out with compatible sand (i.e., a similar grain size as the native), considerable simpliﬁcations result. In particular, the equilibrium proﬁle shape of the nourished project can be considered to be the same as the native beach and the planform evolution is only weakly dependent on wave direction. The weak dependence on wave direction is very valuable since in most locations the wave height has been quantiﬁed to a much better degree than wave direction. Also, the forcing can be represented by an “eﬀective” wave height.

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30.2.3. Nourishment downdrift of a littoral barrier This is a fairly common setting as a littoral barrier causes the downdrift beach to be erosionally stressed resulting in a need for nourishment. The additional complication relative to a long straight beach is the very strong dependence on wave direction and perhaps the complex interaction of structures and shorelines, primarily by downdrift sheltering and diﬀraction from the littoral barriers.

30.2.4. Nourishment of a barrier island subject to longshore sediment transport, overwash, and subsidence This setting is the most complex of the four and requires stochastic consideration of the detailed time series. In addition to cross-shore considerations including seaward transport of sand, landward overwash of sand must be taken into account as well as the subsidence of the island and resetting of conditions as the barrier island retreats. Methods for treating this situation are in their nascent stage and will thus be presented in general terms.

30.3. Considerations, Methods, and Findings Beach nourishment results in a system that is out of equilibrium due to the protuberance in the planform and a proﬁle that is generally oversteepened. This disequilibrium induces both alongshore and cross-shore sediment ﬂows that tend to restore equilibrium. Because of the complexity of the sediment transport processes attending beach nourishment, these processes are usually considered separately in terms of planform and cross-shore evolution. Usually the timescales of crossshore evolution are much shorter than the longshore processes associated with planform evolution and most designs consider the cross-shore processes to occur instantaneously. In the following, we consider the two components of sediment transport separately that are responsible for these two modes of evolution shown in Fig. 30.2.

30.3.1. Planform evolution This portion of our treatment considers the beach to have equilibrated in the cross-shore direction. Later, we will discuss an adjustment to account for the cross-shore equilibration during the early stages of planform evolution. As noted, beach nourishment represents a planform anomaly in the shoreline and nature responds by inducing longshore sediment transport processes that reduce the planform anomaly. This is best visualized by considering waves advancing directly toward shore. While prior to nourishment, the longshore sediment transport was zero, with the nourishment project, the sediment transports at the two ends of the project are away from the project such that the anomaly is diminished.

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Sand transport “losses” and beach proﬁles associated with a beach nourishment

30.3.1.1. Governing equations The two equations governing planform evolution are the conservation equation (also called the “continuity equation”) and the sediment transport equation. The conservation equation simply performs a “book-keeping” role to ensure that no sediment is lost or gained. This basic equation is written as ∂∀ 1 ∂Qx =− . ∂t (h∗ + B) ∂x

(30.1)

In which ∀ is deﬁned as the volume per unit beach length or “volume density,” x is the axis in the longshore direction with positive to the right as an observer faces seaward, t is time, h∗ and B are the closure depth (discussed later) and the berm

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height, respectively, and Qx is the total in-place longshore sediment transport rate. A number of longshore sediment transport equations are available. For purposes here, we choose the so-called “CERC Equation” expressed as 5/2 g/κ Hb sin 2(β − αb ) Qx = K 16(S − 1)(1 − p)

(30.2)

in which Hb is the eﬀective breaking wave height, β and αb are the azimuths of the outward normal to the shoreline and the azimuth of the direction from which the breaking wave arrives, respectively, S is the speciﬁc gravity of the sediment, p is the in-place porosity, g is gravity, κ is the ratio of breaking wave height to water depth (of order unity), and K is the so-called sediment transport coeﬃcient (of order unity) and can be considered to vary with sediment size as shown in Fig. 30.3. These equations are now suitable for calculating evolution of a beach nourishment planform and can be incorporated into a numeric model with the most direct and simple being an explicit one-dimensional model in which the sediment transport is applied at the grid lines during one-half time step while the shoreline positions are held constant and then in the next one-half time step, the transport is held constant and the shoreline positions are modiﬁed in accordance with the transport equation. This explicit formulation allows internal features and boundary conditions such as groins or bypassing at inlets to be handled with ease. However, this solution scheme is subject to the Courant condition which relates the maximum time step, ∆tmax , grid spacing, ∆x, and, in a broad sense, the energy level of the

Sediment Transport Coefficient, K

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Median Sediment Diameter (mm) Fig. 30.3. Relationship of sediment transport coeﬃcient with median grain diameter, D50 . From del Valle et al.,13 published with permission from the American Society of Civil Engineers.

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system. This Courant condition is ∆tmax ≤

∆x2 2G

(30.3)

where the term G is the “longshore diﬀusivity” given by G=K

5/2 g/κ Hb 8(S − 1)(1 − p)(h∗ + B)

(30.4)

which can be seen to be closely related to the sediment transport that would occur at a relative wave approach angle, β − αb = 45◦ . With the speed of modern computers, the Courant condition is usually not restrictive for one-line numeric modeling; in Florida (which has a rather mild wave climate), we normally use a longshore spacing of 150 m and a time step of one day. However, it is possible to develop implicit equations that allow much larger time steps (or smaller grid spacings), but implementing internal boundary conditions such as discussed above is more diﬃcult. 30.3.1.2. Closure depth Although Eqs. (30.1) and (30.2) presented above are suﬃcient to describe the longshore ﬂows (transport equation) and changes in volume (continuity equation), a third equation is required to establish the approximate oﬀshore limit to which the sand is distributed during beach nourishment. The depth associated with this oﬀshore limit is called the “closure depth,” h∗ , deﬁned as the depth limit at which the proﬁle ﬂuctuations are small on an annual basis.14 As an example, if compatible sand is placed on the beach, the closure depth establishes the approximate depth limit to which this sand is distributed seaward. Naturally, the greater the closure depth, the less the additional dry beach width upon proﬁle equilibration. An equation for the closure depth, h∗ was proposed by Hallermeier14 as follows: h∗ = 2.28He −

68.5He2 gTe2

(30.5)

in which He is deﬁned as the “eﬀective” signiﬁcant wave height which is equaled or exceeded six hours per year at the location of interest and Te is the associated wave period. It is noteworthy that this equation does not include any dependency on sediment grain size. We will see later that the nourishment sand size relative to the native sand size is a critical factor in the performance of the project in terms of the additional dry beach width.b For the case of compatible sands, in which it can be expected that the nourishment proﬁle will equilibrate to the native proﬁle, the additional width of the equilibrated dry beach, ∆y0 , can be shown to be related to the nourishment volume density (volume per beach length), ∀, and the closure b Most nourishment projects are judged by the additional dry beach width and the longevity of this width.

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Fig. 30.4. sands.

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Proﬁle translation, ∆y0 , associated with volume density addition, ∀, with compatible

depth, h∗ , and berm height, B, as (see Fig. 30.4) ∆y0 =

∀ (h∗ + B)

(30.6)

where all variables are in consistent units. An early design rule of thumb was that “the placement of one cubic yard of sediment as nourishment results in the advancement of the beach by one foot.” Inspection of Eq. (30.6) will show that this rule of thumb implies that the sum (h∗ + B) = 27 ft. This is a reasonable value for much of the east coast of the USA; however, the values on the Paciﬁc Coast and Gulf of Mexico would be greater and less than this value, respectively. 30.3.1.3. Background erosion Beach nourishment projects are usually constructed in areas where background erosion is present. Generally, the best representation of the background erosion is to assume that it will continue at the same rate as before construction of the project and can be superposed on the shoreline change due to the evolution of the project. Thus, denoting the background shoreline change as ∆yB and the shoreline change due to project evolution as ∆yE , the total shoreline change, ∆yT is simply: ∆yT = ∆yE + ∆yB , where ∆yB is usually negative. 30.3.1.4. The Pelnard–Consid`ere methodology The one-line model approach described earlier allows the treatment of complex initial planforms, internal conditions, and time-varying wave climates through numeric modeling. However, application of the one-line numeric model does not assist in understanding the interrelationships among the variables such as wave height, alongshore project length and longevity. The Pelnard–Consid`ere (hereafter PC) methodology15 comprises the combination of the linearized transport equation and the equation of continuity into a single linear second-order partial diﬀerential equation which allows the investigation of many nourishment interrelationships.

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The PC equation will not be derived here; however, the basis is approximating the term sin 2(β − αb ) by 2(β − αb ) and further approximating β by −(∂y/∂x), resulting in the PC equation below which is recognized as the linear second-order heat conduction equation,c ∂y ∂2y =G 2 ∂t ∂x

(30.7)

where G is deﬁned in Eq. (30.4). The solution of this equation requires two boundary conditions and the speciﬁcation of an initial condition. The PC form of the planform evolution equation has signiﬁcant utilities including the availability of many known solutions. One of the most useful is the case of an initially uniform nourishment over a shoreline length, . The solution can be presented in nondimensional form with the nondimensional time, t , longshore length, x , and shoreline displacement, y expressed as t = 16Gt/2 , y = y/Y , and x = x/, respectively. The solution for this case is presented in Fig. 30.5 and the solution is given by Eq. (30.8). 1 1 1 (30.8) y (x , t ) = erf √ (2x + 1) − erf √ (2x − 1) 2 t t where the term erf is the so-called “error function” deﬁned as z 2 2 e−u du erf(z) = √ π 0

(30.9)

in which z is the argument of the error function and u is a dummy variable. It is possible to integrate Eq. (30.8) to determine the volume remaining in the project area

Fig. 30.5. Nondimensional shoreline evolution for initial rectangular platform. Note that the initial and subsequent planforms are symmetric about x = 0 and the solutions for x < 0 are not shown. cA

detailed derivation of the PC equation may be found in Dean.11

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Fig. 30.6.

Proportion of material remaining in region placed M (t) versus the parameter

√

Gt/.

at any future time (assuming that an appropriate value of G is known). The result of this integration is shown in Fig. 30.6 where the proportional volume remaining in the project area is denoted by M (t). The volume remaining is a very important quantity to the design engineer as a key indicator of the project performance. The PC equation and its associated solutions provide a number of valuable insights to beach nourishment. Several are illustrated in the following. Available space does not permit a detailed discussion for which the reader is referred to Dean.11 Based on Fig. 30.6, a simple rule of thumb can be developed for the volumetric “half-life” of a project with an √ initial rectangular planform. We see from Fig. 30.6 that the half-life occurs when Gt50% / ≈ 0.46. Thus, considering reasonable values for the variables (h∗ + B = 27 ft (or 8.2 m), K = 0.77, κ = 0.78, S = 2.65, and p = 0.35), we ﬁnd t50% = K

2 Hb2.5

(30.10)

where K = 8.8 year ft2.5 /mile2 in the English system where the breaking wave height is in feet, the project length is in miles, and the time is in years, or K = 0.17 year m2.5 /km2 where the breaking wave height is in meters, the project length is in km, and the time is in years. For example in the English system, for a project with a length of one mile acted upon by an eﬀective wave height of one foot, onehalf the volume would have spread outside the placement area in a period of 8.8 years. Similarly, in the metric system, for a project of 1 km length acted upon by an eﬀective wave height of 1 m, one-half the volume would have spread outside the placement area in a period of 0.17 years.d Inspection of the PC equation [Eq. (30.7)] demonstrates that it is insensitive to wave direction. Thus, somewhat counter intuitively, the project evolves independently of wave direction. It can also be shown that for projects constructed with d Note

that these results do not consider background erosion.

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compatible sediments, the project centroid will remain ﬁxed.e If the nourishment is conducted with sand that is ﬁner or coarser than the native, the centroid of the planform anomaly moves downdrift or updrift, respectively. Many projects are renourished when project evolution establishes some minimum criterion. In the absence of background erosion, it can be shown for this case that the renourishment intervals increase with renourishment number. However, the renourishment interval is adversely aﬀected by increasing background erosion rates such that with large background erosion rates, the resulting renourishment intervals decrease with increasing renourishment number. 30.3.2. Cross-shore considerations Two aspects of cross-shore processes are relevant to design: (1) the time required for the constructed proﬁle to approximate the equilibrium proﬁle and (2) the additional dry beach width provided by the nourishment as a function of time. Although the former is relevant, in many cases, the time is short (say two years or so) relative to that of planform evolution and it is often considered in design that the proﬁle equilibration occurs instantaneously. For compatible nourishment sediments, the additional dry beach width is related to the nourishment volume density through Eq. (30.6); however, if the nourishment sediments are ﬁner or coarser, it is necessary to account for this diﬀerence via another procedure. The approach recommended here to account for nourishment sediment with a diﬀerent size than the native is through the equilibrium beach proﬁle relationship represented by the form h(y) = Ay 2/3

(30.11)

in which h(y) is the water depth at a seaward distance, y, and A is the so-called “proﬁle scale parameter” which depends on the median sediment size, D or the sediment fall velocity, w, as presented in Fig. 30.7 and Table 30.1. The proﬁle represented by Eq. (30.11) is monotonic and concave upward and the proﬁle steepness increases with sediment size. For noncompatible sediments it is useful for design purposes to consider both the native and nourishment sediments to be represented by Eq. (30.11). It is thus possible to show that there are three types of nourished proﬁles: (1) intersecting, (2) nonintersecting, and (3) submerged, as shown in Fig. 30.8. Figure 30.9 illustrates the eﬀects of nourishment with the same volume density and sediments that are coarser, the same and ﬁner than the native. In panel (a) of Fig. 30.9 the nourishment sediments are coarser than the native and the additional dry beach width is 92.4 m. With compatible sediments [panel (b)], the additional beach width is 45.3 m, approximately half that of panel (a). Panels (c) and (d) present the results of further decreases in sediment size such that the additional dry beach width in panel (c) is 15.3 m and zero in panel (d). The relationships between the initial additional dry beach width, ∆y0 , horizontal distance to the oﬀshore limit e The PC equation is based on a linearized sediment transport equation. If the full transport equation is considered, for compatible sediments, there is a very small movement of the nourishment centroid in the updrift direction!

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Fig. 30.7. Variation of beach proﬁle scale parameter, A, with sediment size, D, and fall velocity, w. From Dean,16 published with permission from the American Society of Civil Engineers.

Table 30.1.

Summary of recommended A values (units of A parameter are m1/3 ).

D (mm)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.063 0.100 0.125 0.145 0.161 0.173 0.185 0.194 0.202 0.210

0.0672 0.103 0.127 0.1466 0.1622 0.1742 0.1859 0.1948 0.2028 0.2108

0.0714 0.106 0.129 0.1482 0.1634 0.1754 0.1868 0.1956 0.2036 0.2116

0.0756 0.109 0.131 0.1498 0.1646 0.1766 0.1877 0.1964 0.2044 0.2124

0.0798 0.112 0.133 0.1514 0.1658 0.1778 0.1886 0.1972 0.2052 0.2132

0.084 0.115 0.135 0.153 0.167 0.179 0.1895 0.198 0.206 0.2140

0.0872 0.117 0.137 0.1546 0.1682 0.1802 0.1904 0.1988 0.2068 0.2148

0.0904 0.119 0.139 0.1562 0.1694 0.1814 0.1913 0.1996 0.2076 0.2156

0.0936 0.121 0.141 0.1578 0.1706 0.1826 0.1922 0.2004 0.2084 0.2164

0.0968 0.123 0.143 0.1594 0.1718 0.1838 0.1931 0.2012 0.2092 0.2172

Notes: 1 The A values above, some to four places, are not intended to suggest that they are known to that accuracy, but rather are presented for consistency and sensitivity tests of the eﬀects of variation in grain size. 2 As an example of use of the values in the table, the A value for a median sand size of 0.24 mm is A = 0.112 m1/3 . To convert A values to feet units, multiply by 1.5.

of the initial equilibrated proﬁle on the native proﬁle, W∗ (= (h∗ /AN )2/3 ),f the berm height, B, and the “ﬁll” (i.e., nourishment) and “native” sediment scale parameters, AF and AN , respectively, can be expressed generically as ∆y0 =f W∗

f It

∀ AF h∗ , , BW∗ AN B

(30.12)

is essential that in the equations to follow, the value of W∗ be based on AN rather than the AF value.

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Fig. 30.8. Three generic types of nourished proﬁles. Adapted from Dean,17 published with permission from the Journal of Coastal Research.

where the subscripts “F” and “N” refer to “ﬁll” and “native” sand, respectively. It is noted that these equations must be solved iteratively for ∆y0 . For intersecting proﬁles, the volume added is indicated as ∀1 and the equation is 5/3 ∆y0 3 h∗ ∆y0 1 ∀1 = + (30.13) 3/2 2/3 BW∗ W∗ 5 B W∗ AN 1 − AF and the volume for nonintersecting proﬁles as ∀2 , with the relationship   3/2 5/3 3/2   3 h∗ ∆y0 ∀2 ∆y0 AN AN + = + − .  BW∗ W∗ 5 B  W∗ AF AF

(30.14)

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Fig. 30.9. Eﬀect of nourishment material scale parameter, AF on width of resulting dry beach. Four examples of decreasing AF all with the same volume density are shown. From Dean,17 published with permission from the Journal of Coastal Research.

If the nourishment sediment is coarser than the native, for the smaller volumes, intersecting proﬁles can result and with increasing volumes, the proﬁles will become nonintersecting. The critical volume separating intersecting/nonintersecting proﬁles is indicated by the subscript “c1 ” 3/2 ∀ AN 3 h∗ (30.15) 1− = 1+ BW∗ c1 5B AF and applies only for (AF /AN ) > 1. For larger nondimensional volume densities than given by Eq. (30.15), the proﬁles are nonintersecting. Also of interest, the critical

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Fig. 30.10. Additional dry beach width versus volume of sand added per unit beach length. h∗ = 6 m and B = 2 m.

volume of sand that will just yield a ﬁnite shoreline displacement for nonintersecting proﬁles (AF /AN < 1), is indicated by the subscript “c2 ” as

∀ BW∗

= c2

3 h∗ 5B

AN AF

3/2

AN −1 . AF

(30.16)

Figure 30.10 presents the additional dry beach width as a result of nourishment with sediments of three diﬀerent sizes. The upper curve is for the case of nourishment with sediments coarser than the native and shows that for the smaller volume densities, the proﬁles are intersecting. At a critical volume density of approximately 570 m3 /m, the proﬁles become nonintersecting. For compatible sediments, the additional beach width is approximately linearly related to the volume density. For sediments ﬁner than the native, a threshold volume is required to achieve a nonzero beach width. It is noted that the results presented here are for nourishment with a single sediment size and therefore are not fully representative of actual conditions. Thus, these results should be considered as qualitative guidance rather than fully appropriate for design. More complete applications of equilibrium beach proﬁle methodology considering nourishment with a range of sediment sizes documents that even for nourishment with a median sediment ﬁner than the native, a ﬁnite additional beach width will always occur even for small volume densities of beach nourishment. Figure 30.11 presents results in the same form as Fig. 30.10 and includes additional results for sediments characterized by a range of sediment sizes. The quantity σ represents the so-called sorting of the sediments which is the standard deviation of the sediments with the size represented on a log (phi) scale (see Ref. 18).

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Fig. 30.11. Variation of shoreline displacement with volume density and storing characteristics, h∗ = 6 m and B = 2 m.

30.4. Two Examples of Nourishment Projects on Open Sandy Coastlines Examples will be presented of two beach nourishment projects on open coasts. These include the Delray Beach, FL project which has been nourished and monitored for more than 30 years and the Perdido Key, FL project. 30.4.1. Delray Beach, FL beach nourishment project The Delray Beach nourishment project is located on the lower east coast of Florida with no nearby obstructions and was ﬁrst nourished in 1973 and by the year 2000 had been renourished three times as shown in Fig. 30.12. The abcissa represents the volume of sand remaining within the project area and the two lines (for background erosion rates of 0 and 2 ft per year, respectively) are predictions based on methodology developed for the State of Florida in which the only free variable is the berm height as the other variables including eﬀective wave height and closure depth are pre-speciﬁed and are not available for “tuning” to better match the measurements. It is seen as discussed earlier, that even though the volume retained within the project area is increasing with renourishment number, the renourishment interval is increasing. Further informative description of this project is provided by Beachler.19 30.4.2. Perdido Key FL beach nourishment project The Perdido Key beach nourishment project is located in western Florida and was constructed in 1990. Both subaerial and proﬁle (berm) nourishment placements

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Fig. 30.12. Measured and predicted nourishment volumes at Delray Beach, Florida. Project length = 2.7 miles, Hb = 1.2 ft, background erosion = 0, and 2 ft/year, h∗ + B = 21 ft.

Fig. 30.13. Perdido Key nourishment area with eastern boundary adjacent to Pensacola Pass showing proﬁle locations.20

consisted of compatible sand. The eastern end of the project is located adjacent to Pensacola Pass which has been deepened for navigational purposes. Figure 30.13 presents the project location and the beach and proﬁle nourishment locations. Figure 30.14 presents the measured and predicted proportions of volume remaining within the eastern and western halves of this nourishment project and the total for this project. As for the case of predictions of the Delray Beach nourishment project, the predictions here are based on a procedure developed for the State of Florida in which all parameters are ﬁxed except quantiﬁcation of the local

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Fig. 30.14. Comparison of blind-folded predictions of proportional volume changes with monitoring data for Perdido Key Beach Nourishment Project. Background erosion = 0.11

berm height. The eﬀects of storms in reducing volume within the project area are evident. Also, the eﬀects of greater losses from the eastern half of the project which is adjacent to Pensacola Pass compared to the western half are noted. However, the diﬀerences in measured performances between the eastern and western project halves are less than predicted.

30.5. Barrier Islands on Soft Sediments Barrier islands overlying poorly consolidated substrates experience increased rates of relative sea level rise because of the decrease in elevation due to subsidence. These types of barrier island systems can exist near river deltas, as deltaic sediments are reworked by coastal processes, and on the open coast as islands have migrated landward over bay sediment or a peat substrate. Potentially, one-third of the more than 3,600 km of continental US barrier islands consists of a sandy island morphology that overlies poorly consolidated sediment.21,22 The weight of a barrier island compresses (or consolidates) the underlying substrate as a function of the loading and characteristics of the sediment, with the rate of consolidation initially rapid then decreasing with time.23 As barrier islands migrate and are transformed by waves, currents, and winds, deposition of sediment into the bay and longshore transport of sediment creates new loading on a previously poorly consolidated substrate. The result is a barrier island system that becomes more vulnerable to storm surge and erosion through time by loss in elevation through subsidence and overwash. Beach nourishment can mitigate volumetric losses to these islands, although the weight of the beach ﬁll will increase consolidation. Present design guidance for beach nourishment over poorly consolidated substrates is identical to that for

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nonconsolidating sediments, with the subsidence partially factored in through consideration of the historic erosion rate and relative sea level rise. The time-dependent consolidation due to loading by the placement of beach nourishment, future maintenance that renourishes the beach, and the time-dependent nature of consolidation are not considered in present design practice. Lowering of a barrier island by subsidence is compounded as barrier sand is transported into the bay by overwash during storms. The existing barrier elevation is reduced, making future overwash more likely, and the overwash deposit (“washover”) begins to load the previously unconsolidated substrate. The newly loaded sediment base then begins the primary consolidation process. Over long periods of time, these barrier islands are eroded and distorted by successive storms, potentially migrating into the bay. They ultimately may become subaqueous, such as Ship Island Shoal in Louisiana.24 To illustrate the signiﬁcance of a poorly consolidated substrate on barrier island migration and beach nourishment, two sets of examples are presented. First, an example of sediment core data from three barrier islands in Virginia is presented to demonstrate the magnitude and engineering signiﬁcance of the consolidation and loading process on human timescales. Next, a two-dimensional barrier island evolution model is applied with and without a consolidating substrate, and with a large-scale beach nourishment project, to illustrate how the loading due to an external beach ﬁll can alter the island elevation, and subsequently the migration and overwash processes. 30.5.1. Data from Virginia In a study of Virginia barrier islands, Gayes25 surveyed the barrier and beach proﬁle, and took four cross-sectional proﬁles and associated sediment cores across three barrier island systems overlying compressible substrates: Assawoman, Metompkin, and Wallops Islands, Virginia, USA. One example of these data is presented in Fig. 30.15. These data show the compaction of clay and silt beneath the overlying sandy barrier island, which occurred as the islands migrated landward from 3.8 to 4.8 m/year. If consolidation of the underlying substrate had not occurred, the sand– clay/silt interface would lie at approximately the mean high water (MHW) line. Void ratios (volume of voids divided by volume of solids) of the bay sediment are greater than those of the clay and silt underlying the sandy barrier island, supporting the interpretation of consolidation due to the loading of the island. Based on the measurements and island migration rates, these barrier island systems have experienced consolidation between 0.1 and 3.5 m over 35 to 40 years. 30.5.2. Two-dimensional model of migration, consolidation, and overwash A two-dimensional barrier island evolution model that incorporates migration, consolidation of the underlying substrate, and overwash (MCO model) of the island over time periods extending from years to decades was applied to demonstrate how consolidation

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Fig. 30.15.

Assawoman Island, Virginia, cross-section #1 (modiﬁed from Ref. 25).

may factor into barrier island evolution. The MCO model calculates erosion, overwash due to wave runup, overwash due to inundation of the island, subsequent migration, and time-dependent consolidation of the substrate due to new loading. The theory and operation of the model are described by Rosati et al.26 The model was validated with data from the Virginia barrier islands described previously.27 Model applications compared the migration, dune lowering, and consolidation of a barrier island with sediment restoration using a one-time, large infusion of sediment versus a more traditional approach of smaller volumes of sediment placed incrementally through time. Both applications totaled the same volume of introduced sediment over the 100-year period; however, the entire volume of the largescale restoration was placed at the start of the simulation, whereas the incremental volume was set at 10% of the total volume, placed every 10 years, for 100 years. The total simulation length was 200 years with no additional nourishment placed during the last 100 years. The large nourishment has the advantage of providing more protection, but it also causes more consolidation, thereby reducing the barrier dune elevation. The simulations were conducted with a cross-section similar to barrier islands in Louisiana: a barrier island with dune height = 1.5 m and width = 1.5 km, average storm surge (including astronomic tide) equal to the zeroth-moment deepwater wave height = 1.5 m, and average peak wave period equal to 5.6 s. The total volume of sediment placed on the barrier was 2,500 m3 /m. For the incremental placement, 10% of this volume (250 m3 /m) was placed on the barrier island every 10 years for 100 years. Typical nourishment volume densities in the USA are of the order 250 m3 /m [Ref. 11, p. 23]; thus, this example is representative of a periodic US beach ﬁll project. The simulations were run for 200 years, as shown in Fig. 30.16.

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L=Large Fill I=Incremental Fill

Consolidation Dune Elevation, m

5

863

C=Consolidation D=Dune Elevation M=Migration

450 400 350

L,D

300

4 I,M

250

3

200 150

2

Migration, m

6

FA

I,D

100 1

L,C I,C L,M

0

0

50

100

150

50 0 200

Elapsed Time, years Fig. 30.16. Application of MCO for incremental (250 m3 /m every 10 years) and large-scale (2,500 m3 /m at t = 0) beach nourishment.

As expected, the large-scale restoration simulation had the largest consolidation value, 0.76 m, as compared to 0.4 m for the incremental nourishment. However, the large-scale restoration experienced no migration, as compared to 420 m for the incremental nourishment. The large-scale restoration also had the higher ﬁnal barrier elevation after the 200-year simulation, 4 m, as compared to the incrementally placed nourishment, which had a ﬁnal barrier elevation equal to 1 m. Simulations were conducted for half these quantities (1,250 m3 /m for the large-scale restoration, and 125 m3 /m for each incremental nourishment). These simulations had similar responses relative to each other, except the diﬀerences were not as pronounced (for the large-scale and incremental nourishments, respectively: ﬁnal consolidation: 0.6 and 0.2 m; ﬁnal migration 14 and 660 m; ﬁnal dune elevation: 2.5 and 0.8 m). Thus, for the conditions evaluated in this example, a large-scale restoration with beach nourishment decreases barrier migration and results in higher barrier dune elevations than a more traditional, incrementally placed beach ﬁll project. For actual design, the eﬀects of planform evolution would also require consideration.

30.6. Summary and Conclusions Beach nourishment provides an approach to beach stabilization that can enhance recreation, storm protection, and habitats for nesting sea turtles and other biota that reside in the beach and dune environment. The mechanics of beach nourishment along sandy shorelines can be represented in terms of the planform evolution and cross-shore processes. These two issues are usually considered separately in design with the recognition that proﬁle equilibration normally occurs with a shorter timescale than the planform evolution and calculations of planform evolution are generally carried out as if the proﬁle adjustment is instantaneous.

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A valuable pedagogic understanding of the evolution of beach nourishment planforms is obtained by an examination of the PC equation and the associated solutions. The PC equation is a combination of the linearized sediment transport equation and the conservation of sediment equation. For projects constructed with compatible sediments, it is shown that the evolution of a nourishment project placed on a long unobstructed beach is nearly independent of wave direction and that the longshore position of the centroid in the planform anomaly remains ﬁxed. Nourishment with sediments ﬁner and coarser than the native results in downdrift and updrift migration of the planform centroid, respectively. Because beach proﬁle steepness increases with sand size, the most signiﬁcant proﬁle design issue is the additional dry beach width after proﬁle equilibration. Procedures for applying equilibrium beach proﬁle methodology are presented which demonstrate the signiﬁcant role of sand size in design. Monitoring results from two constructed beach nourishment projects are presented illustrating evolution in diﬀerent settings. The processes associated with nourishment of barrier islands founded on soft sediments are examined. These systems are predominant in river delta systems and can provide sheltering of productive marshlands. The feature of relatively rapid subsidence diﬀerentiates this setting from the more traditional beach nourishment project. It is demonstrated that nourishment with large quantities of sediment rather than periodic nourishment with small individual quantities can contribute substantially to project performance. Future beach nourishment design methodology will be improved in general through detailed monitoring of constructed projects. Speciﬁc design needs include: (1) improving quantiﬁcation of the variation of the sediment transport coeﬃcient with sediment size and other parameters; (2) quantifying timescales of proﬁle equilibration; and (3) determining the role of the range of sediment sizes in project performance. Placing large quantities of sediment in the nearshore environment results in a large physical perturbation which induces strong cross-shore and longshore sediment transport signals and oﬀers the opportunity through monitoring to better understand natural and altered nearshore systems.

References 1. J. R. Houston, The economic value of beaches — A 2002 update, Shore and Beach 70(1), 9–12 (2002). 2. R. G. Dean, Realistic economic beneﬁts from beach nourishment, Twenty-first Int. Conf. Coast. Eng., ASCE, Chapter 116, Malaga, Spain, June 1988, pp. 1558–1572. 3. Delft Hydraulics Laboratory, Manual on artiﬁcial beach nourishment, Rijkswaterstaat, The Delft Hydraulics Laboratory Centre for Civil Engineering Research, Codes and Speciﬁcations, Report No. 130 (1987), 195 pp. 4. R. G. Dean and C.-H. Yoo, Beach nourishment performance predictions, ASCE J. Waterw. Port Coast. Ocean Eng. 118(6), 567–586 (1992). 5. D. K. Stauble and N. C. Kraus (eds.), Special volume on beach nourishment engineering and management considerations, Proc. Coast. Zone Man., American Society of Civil Engineers (1993).

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6. National Research Council, Beach nourishment and protection, Committee on Beach Nourishment and Protection, Marine Board (National Academy of Science, 1995), 334 pp. 7. J. R. Houston, Beach ﬁll design, Advances in Coastal and Ocean Engineering, ed. P. L.-F. Liu (World Scientiﬁc Publishing Company, 1996), Vol. 2, pp. 199–230. 8. R. A. Davis, P. Wang and B. Silverman, Comparison of the performance of three adjacent and diﬀerently nourished beaches on the Gulf Peninsula of Florida, J. Coast. Res. 16, 396–407 (2000). 9. M. B. Gravens, B. A. Ebersole, T. L. Walton and R. A. Wise, Beach ﬁll design, In Coastal Engineering Manual, Part V, Coastal Project Planning and Design Chapter V-4, Engineer Manual 1110-2-1100, ed. D. L. Ward (U.S. Army Corps of Engineers, Washington, DC, 2002). 10. H. Hanson, A. Brampton, M. Capobianoco, H. H. Dette, L. Hamm, C. Laustrup, A. Lechuga and R. Sapnhoﬀ, Beach nourishment projects, practices, and objectives — A European overview, Coast. Eng. 47, 81–111 (2002). 11. R. G. Dean, Beach Nourishment Theory and Practice (World Scientiﬁc, 2002), 397 pp. 12. K. R. Bodge and E. J. Olsen, Aragonite beach ﬁll at Fisher Island, FL, Shore and Beach 60(1), 3–8 (1992). 13. R. del Valle, R. Medina and M. A. Losada, Dependence of coeﬃcient K on grain size, ASCE J. Waterw. Port Coast. Ocean Eng. 119(5), 568–574 (1993). 14. R. J. Hallermeier, Uses for a calculated limit depth to beach erosion, Proc. 16th Int. Conf. Coast. Eng., ASCE, Hamburg (1978), pp. 1493–1512. 15. R. Pelnard-Consid`ere, Essai de Th´eorie de l‘Evolution des Formes de Rivage en Plages de Sable et de Galets, 4th Journees de l’Hydraulique, Les Energies de la Mer, Question III, Rapport No. 1 (1956). 16. R. G. Dean, Coastal sediment processes: Toward engineering solutions, Proc. Coast. Sediment., ASCE (1987), pp. 1–24. 17. R. G. Dean, Equilibrium beach proﬁles: Characteristics and applications, J. Coast. Res. 7(1), 53–84 (1991). 18. W. G. Krumbein and W. R. James, A lognormal size distribution model for estimating stability of beach ﬁll material, Technical Memorandum No. 16, U.S. Army Coastal Engineering Research Center (1965). 19. K. E. Beachler, The positive impacts to neighboring beaches from the Delray beach nourishment program, Proc. 6th Natl. Conf. Beach Preserv. Techn., The State of the Art of Beach Nourishment 6, 223–238 (1993). 20. P. A. Work, L.-H Lin and R. G. Dean, Perdido Key beach nourishment project: Gulf Islands National Seashore. First post-nourishment survey — Conducted 22–26 September 1990, Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL (1991). 21. A. Morang, L. Gorman, D. King and E. Meisburger, Coastal classiﬁcation and morphology, Coastal Engineering Manual, Chapter 2, Part IV, Engineer Manual 1110-21100 (U.S. Army Corps of Engineers, Washington, DC, 2003). 22. O. Pilkey, A Celebration of the World’s Barrier Islands (Columbia University Press, 2003), 309 pp. 23. T. H. Wu, Soil Mechanics (Allyn and Bacon, Inc., Boston, MA, 1966), Chap. 5, pp. 91–117 . 24. S. Penland, R. Boyd and J. R. Sutter, Transgressive depositional systems of Mississippi delta plain: A model for barrier shoreline and shelf sand development, J. Sed. Petrol. 58, 932–949 (1988).

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25. P. T. Gayes, Primary consolidation and subsidence in transgressive barrier island systems, MSc thesis, Department of Geosciences, Pennsylvania State University (1983). 26. J. D. Rosati, G. W. Stone, R. G. Dean and N. C. Kraus, Restoration of barrier islands overlying poorly-consolidated sediments, South-Central Louisiana, Gulf Coast Association of Geological Societies Transactions 56, 727–740 (2006). 27. J. D. Rosati, R. G. Dean, N. C. Kraus and G. W. Stone, Morphologic evolution of subsiding barrier island systems, Proc. Int. Conf. Coast. Eng. (2006), pp. 3963–3975.

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Chapter 31

Engineering of Tidal Inlets and Morphologic Consequences Nicholas C. Kraus US Army Engineer Research and Development Center Coastal and Hydraulics Laboratory, 3909 Halls Ferry Road Vicksburg, MS 39180, USA [email protected] Tidal inlets are part of the coastal sediment-sharing system, and an inlet will modify the nearshore and estuary morphology, as well as the up-drift and downdrift beaches. Morphologic response to an inlet varies over several time and spatial scales. This chapter discusses the inlet morphology and its related functional design considerations that must balance navigation and shore-protection requirements. The ﬁrst half of this chapter reviews the selected material on the morphology of inlets and introduces empiric predictive expressions found useful for engineering. The second half of the chapter concerns aspects of engineering of tidal inlets.

31.1. Introduction A tidal inlet is a short, narrow waterway connecting a bay, estuary, or similar body of water with a larger water body such as a sea or ocean upon which the astronomic tide acts. A tidal inlet is distinguished from other possible embayments and coastal inlets in that the inlet channel is primarily maintained by the tidal current.1 Water ﬂow through a coastal inlet can be caused by the tide, wind,2 long-period seiching,3 and by river discharge. For water bodies connected to an ocean or large sea by an inlet, the astronomic tide is typically the major forcing for water movement and scouring of the inlet channel. In areas where the wind can be strong and tidal range small, such as the coast of Texas, USA, wind-generated ﬂow through an inlet can often dominate the tidal signal, as can seasonal variations in water level at inlets that experience moderate tide range.4 A tidal inlet can be in a natural state, or its channel can be dredged as needed for navigation, for improving ﬂushing, or for keeping the inlet open. Navigable inlets are usually stabilized by one or, more commonly, two jetties. 867

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Tidal inlets serve an essential ecologic function in exchanging water, nutrients, and sediment between the lagoon and ocean, as well as being a conduit for water-borne biomass exchange. Inlets are often managed as part of the transportation system for commercial, military, and recreational vessel traﬃc. Maintenance dredging of inlet entrance channels and construction of jetties act to preserve navigable depth and protect the channel and vessels transiting it from sediment shoaling and waves. These activities also stabilize the cross-sectional area, location, and orientation of the channel. Channel dredging, jetties, and breakwaters disrupt natural sediment transport pathways among the inlet, adjacent beaches, and estuary. The resulting morphologic responses can compromise the integrity of the beaches and estuary and, ultimately, endanger the inlet itself. For example, the jetties at many of the larger stabilized coastal inlets in the USA were constructed around the turn of the 20th century, with federal jetties in the Great Lakes being the oldest in dating to the 1840s. When these early jetties were constructed, knowledge of coastal processes was limited. Main concerns or challenges were to furnish a reliable navigation channel and perform construction in the marine environment.5 Many of the earlier inlet stabilization projects were built on the shifting sediments of tidal ﬂats and estuaries, far from infrastructure and development. The coast of the USA was relatively unpopulated, so consideration of the beaches adjacent to the inlets was minimal. With increased utilization of the coast for residences, businesses, recreation, and nature preserves, and in recognition of the great environmental signiﬁcance of beaches and estuaries, the relation between tidal inlets and their surroundings came forward in the latter half of the 20th century. Present-day engineering practice recognizes that inlets must be managed within a sediment-(sand-)sharing system. A sediment budget serves as the foundation for coastal engineering actions at inlets.6–10 For example, “inlet management plans” including sediment budgets are required by the states of Florida and North Carolina. The intent of this chapter is to serve as a resource about concepts and information on inlet engineering and morphologic responses, with focus on navigable inlets (see also Ref. 11). The ﬁrst half of the chapter reviews selected material on understanding of the morphology of inlets and introduces some empiric predictive expressions that are useful for engineering. The second half of the chapter concerns engineering of tidal inlets. Material includes engineering considerations for designing new inlets and modifying and maintaining existing tidal inlets.

31.2. Overview of Tidal Inlet Processes This section reviews selected features of FitzGerald13 can be consulted for diﬀerent many other references to the literature. Gerritsen14 is still relevant in establishing pathways at tidal inlets.

tidal inlet morphology. Hayes12 and overviews than presented here and for The seminal article by Bruun and terminology and elucidating sediment

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31.2.1. Inlet terminology An inlet evolves to a characteristic planform morphology that is controlled by the geometry of the estuary, net and gross longshore sediment transport rates, relative strength of wave action to tidal ﬂow, geologic controls such as presence of nonerodible bottom and sediment type, presence and conﬁguration of jetties, engineering activities in the channel, estuary, and adjacent beaches, channel width and depth, storm magnitude and frequency, and other factors. Although it would appear from the preceding that inlets exhibit a diverse range of morphologic characteristics, there are many commonalities, as well as notable distinctions. The morphology of a typical medium-sized inlet on the northeast coast of the USA was captured in the photograph shown in Fig. 31.1, taken during calm wave conditions at the federally maintained Shinnecock Inlet, located on the south shore of Long Island, New York, and connecting Shinnecock Bay with the Atlantic Ocean. The range of the predominantly semidiurnal tide is about 1 m. Average signiﬁcant wave height is about 0.8 m, and sediment size is medium to coarse on this glacially inﬂuenced coast. Although it may appear that the ﬂood shoal in Shinnecock Bay is much larger than the ebb shoal, bay bottoms are typically shallow and ﬂat (say, from 2 to 4 m deep), whereas the nearshore bottom under the ebb shoal slopes into much deeper water. Thus, the thickness and volume of the ebb shoal can be much greater than that of the ﬂood shoal. The main channel of Shinnecock Inlet cuts

Fig. 31.1. Shinnecock Inlet, New York, April 22, 1997. The picture was taken during a time of small waves, revealing the inlet morphology.

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870

Fig. 31.2.

Deﬁnition sketch for tidal inlet morphology (modiﬁed from Ref. 15).

through or displaces the ebb shoal along the axis of the inlet. The island on the west side of the ﬂood shoal likely consists of material placed during early dredging of the navigation channel. At Shinnecock Inlet, longshore transport is directed strongly to the west, making the ebb shoal asymmetric. The main morphologic features of a natural (nonengineered) tidal inlet are sketched in Fig. 31.2. Of these, the inlet channel, ebb shoal, and ﬂood shoal are typically of interest to navigation and to the integrity of the adjacent beaches. The ebb shoal serves as a pathway for sediment bypassing naturally around the inlet, and it is sometimes mined as a sand source for beach ﬁll. At many inlets, the channel must be maintained at greater depth through the ebb shoal (sometimes called the outer bar or entrance bar) than in the inlet and bay because of breaking waves on the shoal, which cause greater vertical excursion of vessels. The ﬂood shoal often encroaches on the inlet channel or is located where a waterway passes, and channels around or through ﬂood shoals must be dredged. The planform of the ebb shoal and ﬂood shoal can vary greatly according to the relative action of waves (littoral transport) and tidal range. Coastal geologists refer to the ebb- and ﬂood-tidal shoals as deltas in analogy to the shape of river deltas. Here, the terminology shoal is employed to emphasize the typical littoral, as opposed to riverine, provenance of shoals at tidal inlets, which is also in accord with navigation usage. Galvin16 classiﬁed inlet planform conﬁgurations according to relative strength of longshore sediment transport rates (typically as a volume per year), in which QR and QL , represent right- and left-directed transport, respectively, as viewed from

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the inlet and facing the ocean (Fig. 31.3). The conﬁgurations were termed as overlapping oﬀset, up-drift oﬀset, down-drift oﬀset, and negligible oﬀset. Representative locations of the main channel and the ebb (E) and ﬂood (F) shoals are indicated. An overlapping oﬀset inlet forms by spit growth from the direction of strongly dominant longshore sediment transport (QR in top drawing of Fig. 31.3). The inlet channel becomes longer, unless the spit growth is contained in some way. Fire Island Inlet, New York (Fig. 31.4; Ref. 17) is a clear example of an overlapping oﬀset inlet. Eventually, such a channel becomes so long that friction weakens the tidal ﬂow, and the spit grows toward shore to close the inlet. The inlet channel may not be ﬁxed by a jetty; in the case of Fire Island Inlet, the predominant transport was so strong

Fig. 31.3. Planforms of inlets (modiﬁed and expanded from Ref. 16). Lengths of arrows denote relative strength of longshore sediment transport rate.

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Great South Bay Captree Island Oak Is.

Cedar Is.

Gilgo Is.

Dike Cedar Beach Fire Island Inlet

Fire Islands

Sexton Island

Oak Beach

Lighthous

Democrat Point Gilgo Beach

Fire Island Jetty 2 km

Atlantic Ocean Fig. 31.4. Fire Island Inlet, Long Island, New York. The inlet migrated 8 km east since 1827, and a jetty placed on Democrat Point in 1939–1941 was impounded and bypassed within 10 years. Shorelines oriented approximately east-west.

that sediment buried the relatively short jetty and continued spit development down drift. The channel may cut through the ebb shoal, which tends to orient toward the down-drift shoreline in a submerged extension of spit elongation. If there is an unequal but adequate (available) up-drift source of sediment, the inlet may be oﬀset with only minor spit growth and not tend to exhibit strong spit growth. Bypassing will occur through a relatively well-connected ebb-tidal shoal. Shinnecock Inlet, New York, and Ocean City Inlet, Maryland, are examples of updrift oﬀset inlets. If there is an inadequate or limited source of sediment from the predominant direction, then a down-drift oﬀset might occur such that the downdrift side protrudes, in part because of attachment by or feeding from the ebb shoal and its bypassing bars, and by local wave and transport reversal through wave refraction.18 Thus, inlets can have a seaward down-drift oﬀset even without jetties. This is the mesotidal morphology described by Hayes.19 Finally, for Fig. 31.3, if the longshore transport rate is balanced or weak, a natural inlet will tend to be wide and consist of multiple migrating channels and shoals. Hayes19 and Davis and Hayes20 characterized inlet planform morphology according to the relative strength of tide and waves, as depicted in Fig. 31.5, with tidal range serving as a surrogate for tidal prism (volume of water entering or leaving an inlet in the corresponding half tidal cycle) or tidal current (which moves the sediment). Tide-dominated inlets tend to have larger ebb shoals that include channel margin bars similar to dual jetties. Bypassing at tide-dominated inlets can be through tidal bypassing, in which the sediment enters the channel on one side at ﬂood tide and a portion eventually returns to the opposite side at ebb tide (Fig. 31.6). Large volumes of sand can also be added to the barrier segment on either side of the inlet by major reorientation of the outer channel that isolates a portion of the ebb shoal from strong tidal ﬂow.21 Sand bodies can move onshore over the shallower portion of the ebb shoal exposed to breaking waves. During storms, portions of the channel– margin bars or other features of the ebb shoal may break oﬀ and migrate onshore.

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6 tion d ma te for ) a r w e n i i (lo m arr fb ted Do ) it o – high na i m i l e m ( . e d (tid Do prox Ti rgy e – Ap Ene ated) d Tid e Mix domin y Energ Mixed inated) m o d (wave

Mean Tidal Range,m

5 4 3 2

ted Wave Domina

1 0 0

1

2

Mean Wave Height, m Fig. 31.5.

Tide Wave

Classiﬁcation of tidal inlet morphology (after Ref. 20).

Transitional; Mixed Shoal Main transport pathway Secondary pathway

Fig. 31.6. Representative planform morphology and mode of bypassing of natural inlets, depending on wave or tide dominance.

Wave-dominated inlets tend to be ringed by a semi-circular ebb shoal (Fig. 31.7). Bypassing of sediment around wave-dominated inlets mainly occurs through bar bypassing; sand moves around the shoal with the longshore current that is generated by waves breaking on it. On wave-dominated coasts, the ﬂood shoal tends to be large, because a large amount of littoral sand is brought to the inlet, which can be swept inside by the tidal current, as well as bypassed through tidal bypassing.

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Fig. 31.7. High-resolution survey of ebb-tidal shoal at Ocean City Inlet, Maryland, 2004, exhibiting a clear concentric shoal characteristic of a wave-dominated inlet. Longshore transport is from north to south.

Mixed-energy inlets share the features of each of the tide-dominated and wavedominated idealized end states (Fig. 31.6; Ref. 22). Anthony and Orford23 investigated mixed-energy inlets on sediment-deﬁcient coasts. For wave-dominated and mixed-dominance inlets, the semi-circular ebb-tidal shoal can be subdivided as depicted in Fig. 31.8 into the ebb shoal proper, bypassing bars to each side of the ebb shoal proper, and attachment bars connecting the bypassing bars to the shore.24 The ebb shoal was subdivided because the location and size of the ebb shoal proper is related primarily to the tidal jet. In contrast, the bypassing bars and attachment bars are controlled by sediment transport produced primarily by breaking waves. For Shinnecock Inlet (Fig. 31.1), the strong predominant direction of longshore transport to the west pushes the up-drift attachment bar toward the beach that is fully impounded directly adjacent to the east jetty. The down-drift attachment bar is located about 1 km to the west of the inlet.

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Updrift Attachment Bar

Ma rg in Ma Sp Ch rg it an in ne Sp l it

Updrift Bypassing Bar

Fl o

od

Eb b

Ebb Shoal Proper

Downdrift Bypassing Bar

Downdrift Attachment Bar

Flood Shoal Fig. 31.8. Inlet entrance morphology, East Pass, FL, 1990, facing the Gulf of Mexico (adapted from Ref. 25).

31.2.2. Quantification of natural inlet bypassing For maintaining or reestablishing the sediment bypassing rate, knowledge of the mode of sediment bypassing is helpful in assessing options and procedures for bypassing. As discussed above, Bruun and Gerritsen14 and Bruun26 identiﬁed three mechanisms for natural sediment bypassing at tidal inlets: (1) wave induced sand transport along the periphery of the ebb delta (bar bypassing); (2) transport of sand in channels by tidal currents, and (tidal bypassing); and (3) by the migration of tidal channels and sand bars. To predict bypassing type, they deﬁned a ratio r as: r=

P M

(31.1)

where P is the tidal prism at spring tide and M is the total transport rate arriving at the inlet in one year (so that the numerator and denominator of Eq. (31.1) have units of volume). The parameter r expresses the relative strength of tidal ﬂow that

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Table 31.1. Inlet bypassing and classiﬁcation of channel area cross-sectional stability (modiﬁed from Bruun and Gerritsen14 and supplemental information). r-value

Channel stability

r < 20

Unstable. Inlet may be closed by deposition of sediment during a storm. Not typically a navigable channel. Highly variable channel in location and area, with multiple channels possible. Dredging and jetties typically required to maintain navigable depths. Clear main channel and well-developed ebb shoal. Reasonable stable channel.

r = 20–50

r = 50–150 r > 150

Dominant bypassing mode Bar bypassing

Bar bypassing; may have several bars

Bar bypassing and tidal bypassing Episodic bypassing, tidal bypassing

acts to sweep the inlet sediment as opposed to the volume of sediment brought to the inlet entrance by longshore transport during a year. Their observations led to the classiﬁcation of inlets according to mechanism of sand bypassing and stability of the channel (Table 31.1). Implications for navigation over the ebb shoal or entrance bar based on their classiﬁcation are also noted in Table 31.1. Another mechanism for natural sand bypassing at inlets is episodic bypassing, by which a portion of the ebb-tidal shoal or down-drift bypassing bar detaches from the main body and migrates to the down-drift shore.27–29 Episodic bypassing initiates after the ebb shoal, typically of a transitional or wave-dominated inlet, grows large and is disturbed by a storm. A large river ﬂood can also discharge excess sediment that is not in equilibrium with the prevailing typical tidal-river current and waves, causing at least a portion of the new material to gradually migrate to the down-drift shore under wave action. Kraus and Lin30 attribute an increased rate of migration and tendency for closure of the San Bernard River Mouth, Texas, to a large volume of sediment discharged from the Brazos River, located 5.6 km up drift (to the north), during an exceptional ﬂood in early 1992. Hands and Shepsis31 document periodic bifurcation and detachment of a large spit entering Willapa Bay, Washington, in association with large El Ni˜ no that occurs about every seven years. The detached portion of the spit supplied material to the ebb-shoal complex at this wide, natural inlet. FitzGerald et al.21 posit nine conceptual mechanisms for natural sand bypassing at tidal inlets, including those with jetties. 31.2.3. Empiric geomorphic relations Despite the hydrodynamic and morphodynamic complexity of tidal inlets, many empiric relations are available for estimating bulk characteristics, typically pertaining to inlet equilibrium. Table 31.2 summarizes selected empiric relations for predicting inlet morphology. The table indicates that the tidal prism is a dominant factor controlling (tidal) inlet morphology. It is this long-term average that is referred to as “equilibrium” in Table 31.2. Vincent and Corson49 give empiric relationships for other geometric features of natural inlets. Carr and Kraus25 investigate

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Example empiric and theoretic equilibrium relationships for tidal inlet morphology.

Author

Morphologic feature

Relation

LeConte,32

Channel cross-sectional area, AC (note: LeConte, Riedel and Gourlay, and Hume and Herdendorf consider the longshore transport rate magnitude)

AC = C1 P

Escoﬃer37

Inlet cross-sectional area stability

Closure curve

Floyd38

Channel depth and minimum depth over the ebb shoal

Jarrett39

Channel cross-sectional area, with and without jetties

AC = C2 P n

Bruun and Gerritsen14

Inlet stability, sand bypassing type

P/M

Walton and Adams,40 Marino and Mehta,41 Hicks and Hume42

Ebb shoal volume, VE (note: relationships diﬀer according to wave climate)

VE = C3 P m

Shigemura43

Throat width, We (“e” denoting equilibrium)

We = C7 P

Gibeaut and Davis44

Ebb shoal area, AE

AE = C4 P k

Kraus45

Channel cross-sectional area relation including longshore sediment transport rate in C2 , see Eq. (31.8).

AC = C2 P n

Carr de Betts,46 Carr de Betts and Mehta47

Flood shoal area, AF , and volume, VF

AF = C5 P p VF = C6 P q

Buonaiuto and Kraus48

Limiting depth hCr over crest of ebb shoal; limiting slopes around ebb shoal

HCr ∼ P j

O’Brien,33,34 Riedel and Gourlay,35 Hume and Herdendorf36

—

P : tidal prism; A: area; V : volume; subscripts C, E, and F : channel, ebb shoal, and ﬂood shoal, respectively; C: empiric or derived coeﬃcient; j, m, n, p, q: empiric or derived power; W : width of inlet throat; M : gross longshore transport in a year.

empiric relations for tidal inlet morphologic asymmetries with implications for maintenance of navigation channels and sediment bypassing to the adjacent beaches. 31.2.4. Maximum tidal current necessary for inlet channel stability A tidal inlet having a stable channel cross-sectional area on a sandy coast tends to have a mean-maximum velocity through it of approximately 1 m/s.37,50,51 This is the current velocity necessary to maintain cross-sectional area: i.e., sweep material from the channel to maintain depth. The mean-maximum velocity is the average of a regularly occurring maximum velocity, as would occur on spring tides. If the discharge is solely related to the tidal prism and there is a sinusoidal tide with one (the ﬁrst) harmonic component (Keulegan and Hall52 gave a simple correction for

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including the third harmonic), the maximum discharge Dm and tidal prism P are related as: T /2 2π Dm sin P = t dt (31.2) T 0 where T is the tidal period, and t is time. The integration gives: Dm =

π P. T

(31.3)

Tidal prism can also be calculated as the product of the eﬀective bay surface area served by the subject inlet and the tidal range. It can also be obtained from a computation of water discharge, as through a numeric model. By deﬁnition of a discharge, the mean-maximum velocity Vmm is: Vmm =

Dm AC

(31.4)

where AC is the minimum inlet channel cross-sectional area below mean sea level (msl). Although reﬁnements have been made in empiric predictive equations relating AC and P (see next section), it is convenient to consider the linear relation from O’Brien34 : AC = CP

(31.5)

where AC is in m2 , P is in m3 , and C = 6.6 × 10−5 m−1 . Then34 : Vmm =

π . CT

(31.6)

For a pure semi-diurnal inlet, T = 12 h, 25 min = 44,712 s. Equation (31.6) yields Vmm = 1.06 m/s, in agreement with empirical observations. For a tide that is primarily diurnal, the tidal period is 89,424 s, giving Vmm = 0.53 m/s. The implication is that an inlet in a diurnal tidal setting may require a smaller mean-maximum tidal velocity to maintain channel cross-sectional area stability as compared to inlets in a semi-diurnal setting, which are more common because of the prevalent semidiurnal tide. Jarrett39 compiled annual average maximum ebb velocities for 70 inlets on the Atlantic coast, 38 inlets on the Gulf coast, and 28 inlets on the Paciﬁc coast of the USA. The compilation gives mean-maximum velocities of 1.17, 0.75, and 1.06 m/s, respectively, for the inlet groups. Most inlets on the Gulf coast experience a diurnal tide and a reduced longshore transport rate (see next section) as compared to the Atlantic and Paciﬁc coasts. The trend thus agrees qualitatively with the simple calculation given above. 31.2.5. Tidal prism-inlet channel area relations It is conceptually reasonable that the equilibrium area of a tidal inlet is determined by a balance between the transporting capacity of the inlet current and the littoral

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or longshore transport. In a short discussion paper, LeConte32 found a quantitative relationship between the inlet cross-sectional area and tidal prism based on his observation of a small number of harbor and inlet entrances on the Paciﬁc coast of the USA. He gave the equations, AC = 1.1 × 10−4P for unprotected entrances, and AC = 1.4×10−4P for inner harbor (protected) entrances (metric units). The empiric coeﬃcients have units m−1 . In forming these and similar equations to follow, it is assumed that the channel cross-sectional area is at or near its equilibrium value. The observations of LeConte32 are remarkable in that not only did he deduce a direct (linear) relation between channel cross-sectional area and tidal prism, but also that the empiric coeﬃcient is larger if less sediment is driven by waves to the inlet entrance. Unprotected harbors are exposed to wave action and longshore transport of sediment, whereas inner harbor entrances would be protected or sheltered to some degree from wave action. Therefore, for the same value of tidal prism, protected inlets can have larger channel cross-sectional area. The work of LeConte32 was followed by that of O’Brien33,34 and Johnson,53 among others. The O’Brien34 relation for nine inlets on the Atlantic coast, 18 on the Paciﬁc coast, and one inlet on the Gulf coast of the USA, is AC = 6.6 × 10−5 P (metric units). Jarrett39 comprehensively analyzed the relation between spring or diurnal tidal prism and inlet channel cross-sectional area. He compiled 162 data points for 108 inlets, with 59 inlets located on the Atlantic coast, 25 on the Paciﬁc coast, and 24 on the Gulf coast of the USA. Jarrett’s objective was to determine if tidal inlets on all three coasts of the USA follow the same tidal prism–inlet area relationship, and to investigate the change in that relationship for stabilized and nonstabilized (natural) inlets. The results are summarized in Table 31.3, referring to the equation: AC = CP n .

(31.7)

Among other observations, Jarrett39 noted that the smaller waves on the Gulf of Mexico coast relative to those on the Paciﬁc coast and most of the Atlantic Ocean coast would produce smaller rates of longshore sediment transport. Kraus45 derived a form of Eq. (31.7) by consideration of a balance of sand transport by the channelclearing inlet current and channel inﬁlling by longshore sand transport, resulting in Table 31.3. Inlet-area and spring or diurnal tidal prism regression values found by Jarrett39 for AC = CP n (area units of m2 , prism units of m3 ) tidal inlets on US coasts. All inlets Location All inlets Atlantic coast Gulf coast Paciﬁc coast

No jetty; single jetty

Two jetties

C

n

C

n

C

n

1.576 × 10−4 3.039 × 10−5 9.311 × 10−4 2.833 × 10−4

0.95 1.05 0.84 0.91

3.797 × 10−5 2.261 × 10−5 6.992 × 10−4 8.950 × 10−6

1.03 1.07 0.86 1.10

7.490 × 10−4 1.584 × 10−4 Insuﬀ. data 1.015 × 10−3

0.86 0.95 Insuﬀ. data 0.85

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n = 0.9, and:

C=

4/3

απ3 N 2 We Qg T 3

0.3 (31.8)

in which α is the transport-related coeﬃcient on order of 0.1–1, N is the Mannings coeﬃcient, We is the width of the inlet at equilibrium, and Qg is the gross longshore transport rate arriving at the inlet. The prediction for C indicates that this quantity increases for wider inlets and for smaller longshore transport rates (sheltered coasts or coasts with small transport rates in general, such as on the west coast of Florida). Also, the inverse dependence on T suggests that inlets located on coasts having a predominantly semi-diurnal tide should be more stable than those experiencing a diurnal tide, all other conditions being equal. This prediction is contrary to that given in the preceding section. The data points for all inlets tabulated by Jarrett39 are plotted in Fig. 31.9. The equation at the top of the ﬁgure is that given by Jarrett, and the one on the bottom of the ﬁgure was computed by the author. There is a small diﬀerence, mainly because several points appear to have been omitted by Jarrett in his correlation calculation. The solid lines are the predictive equations which overlap, and the dashed lines give 95% conﬁdence limits. Byrne et al.,54 Riedel and Gourlay,35 and Hume and Herdendorf36 studied the inlet channel stability on sheltered (protected) coasts and demonstrated that larger values of the empiric coeﬃcient C (in accord with Ref. 32) and smaller values of n in Eq. (31.7) apply to coasts with limited littoral transport. The aforementioned three studies also indicate that the mean-maximum velocity required to maintain stability of the inlet channel is less (reaching approximately one-third less) than the typical 1 m/s required to maintain a channel on an exposed coast.

10

6

US, All Inlets A = 1.576 * 10-4* P0.95 5

MSL Area, m

2

10

10

10

4

3

A = 1.528 * 10

-4

0.954

*P

R2=0.925

10

2 5

10

6

10

10

7

8

10

9

10

10

10

3

Tidal Prism, m

Fig. 31.9.

Data on inlet channel area and spring tidal prisms (after Ref. 39).

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MSL Area, m

2

10

10

881

7

NZ Rivers Inlets, NZ

6

NZ Harbors

5

NZ Bays

10

4

US, All Inlets

10

US & NE Coast of NZ

3

Northeast Coast of North Island, New Zealand Rivers, New Zealand

10

Inlets, New Zealand

2

Bays, New Zealand Harbors, New Zealand

10

1 5

10

10

6

10

7

10

8

Tidal Prism, m

9

10

10

10

10

11

3

Fig. 31.10. Trends from Jarrett39 for US inlets and New Zealand inlet categories as compiled by Hume and Herdendorf.36

Trend lines from the data of Jarrett39 and of Hume and Herdendorf36 are plotted in Fig. 31.10. For the same value of tidal prism, channel cross-sectional areas for New Zealand entrances in sheltered areas (bays, protected inlets) tend to plot higher than those for unprotected or unsheltered entrances. Entrances on the (unsheltered) northeast coast of New Zealand plot on top of the all-inlet trend line for US inlets.

31.2.6. Tidal prism versus ebb shoal volume relationships Ebb shoals form under a balance of sediment transport produced by the ebb ﬂow of the inlet and by the longshore current created by waves and wind. A portion of the sediment transported toward the inlet by waves is forced oﬀshore by the ebb current, where it accumulates. If the inlet closes, the maintaining tidal force is lost, and the material in the ebb shoal is transported onshore by the waves, a process called ebb-shoal collapse or abandonment, whereby a portion of the ebb shoal welds to shore. If jetties are built at the location of an existing inlet, or if the existing jetties are extended oﬀshore, the ebb jet will be conﬁned and sustain greater velocities to a greater distance oﬀshore. Some areas to the sides of the inlet that had been within the inﬂuence of the ebb current prior to jetty construction or extension will no longer be exposed to the current. As a consequence of conﬁnement of the ebb jet, portions of the original ebb shoal located in areas no longer covered by the ebb current will migrate onshore under wave action. Also, the portion of the shoal remaining in front of the now-stronger ebb jet will be translated further seaward.55 Buijsman et al.56 document the century-long collapse of the northern

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lobe of the large ebb shoal at Grays Harbor, Washington, after long jetties were built in the late 1890s to the early 1900s. The collapsing shoal increased beach width to the north of the inlet by more than a kilometer. The ebb shoal in front of the ebb jet migrated seaward to the edge of the continental shelf as a result of the jetty construction. Collapse of an ebb shoal can give a false impression as to the net direction of longshore transport. For example, at Grays Harbor the regional net direction of transport is to the north,57 yet the wide beach to the north created by the collapsed ebb shoal suggests (incorrectly) that the net transport is to the south. Once an ebb shoal develops, if it is not translated too far oﬀshore by the ebb jet; it provides an eﬃcient pathway for sand to bypass around the inlet during times of larger waves that can break on the shoal. Therefore, the volume of an ebb shoal will approach an equilibrium value, after which the sediment transported to the ebb shoal by the wave-and wind-induced longshore current will be bypassed to the down-drift beach or transported to the channel and then to the ﬂood shoal. It was proposed several times, but to the author’s knowledge never executed, to artiﬁcially build an ebb shoal at the location of a newly cut inlet so as to more quickly reestablish natural sand bypassing. Walton and Adams40 examined the volumes of ebb shoals as a function of tidal prism by analyzing 44 inlets on the Atlantic, Gulf, and Paciﬁc coasts of the USA that were judged to be near or at equilibrium. They estimated the volume from nautical charts and other surveys by assuming that the ebb shoal formed on top of parallel-depth contours that can be estimated from those far from the inlet, a procedure developed by Dean and Walton.58 They further classiﬁed the inlets according to slope of the continental shelf (location) and wave climate. The volume of the ebb shoal for inlets on the mildly exposed coasts was larger than those on highly exposed coasts for the same tidal prism. Walton and Adams40 hypothesized that larger waves would tend to drive the sediment toward the shore. Others have conﬁrmed the essential ﬁnding of Walton and Adams40 that the volume of the ebb-tidal shoal is related to the tidal prism. Marino and Mehta59 examined 18 inlets on the east coast of Florida. They found that tidal prism was a leading parameter controlling ebb-shoal volume, with the ratio of inlet width to depth being a secondary factor. Hicks and Hume42 conducted a similar analysis for 17 inlets in New Zealand and conﬁrmed the overall results of Walton and Adams. Hicks and Hume42 also considered the angle between the ebb jet and the shoreline in their correlations. For the 44 inlets, Walton and Adams40 found: VE = CE P 1.23

(31.9)

where VE is the ebb-shoal volume in m3 , CE = 2.121 × 10−2, and P is in m3 . The data and correlation equation are given in Fig. 31.11, together with the 95% conﬁdence limits. The best-ﬁt equation diﬀers slightly from the original equation of Walton and Adams40 because of discrepancies between data they tabulated and locations of points in their ﬁgures.

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10

10 Volume of Sand in Outer Bar / Shoal, m

3

Tidal Prism-Outer Bar Storage Relationship for All Coasts 10

10

10

9

VE = 2.121 * 10-2 P1.1673

8

7

Exposure

10

6

High Moderate Mild

10

5 5

10

6

10

7

8

10

10

9

10

10

10

3

Tidal Prism, m

Fig. 31.11.

Volume of ebb-tidal shoal as function of tidal prism, all coasts (after Ref. 40).

31.2.7. Tidal prism versus flood-shoal volume and area relationships It is diﬃcult to unambiguously identify the volume and area of a ﬂood shoal. The absence of a hydrodynamic force to balance the ﬂood ﬂow that is comparable to waves at an ebb shoal allows the sediment to be transported deep within a bay, particularly during spring tide and storms, creating a thin layer over a wide area that is diﬃcult to distinguish from the natural bay bottom. As a practical matter, channels are often dredged through and around ﬂood shoals, with the sediment sometimes placed as islands near the shoals, causing potential double counting. If the bay perimeter is located near the inlet, wetlands may form in the margins of the ﬂood shoal, making it diﬃcult to distinguish the ﬂood shoal from features such as washover fans and wetlands. Despite these challenges, Carr de Betts46 and Carr de Betts and Mehta60 analyzed 67 inlets in Florida and obtained correlations between ﬂood tidal shoal volume and tidal prism, and between the ﬂood-tidal shoal area and tidal prism. This work distinguished the ﬂood shoal as comprised of a near-ﬁeld deposit and a far-ﬁeld deposit, which together give the total volume. The near-ﬁeld deposit is the visible portion of a ﬂood shoal that may be, for example, bat-wing shaped. It was hypothesized that the near-ﬁeld deposit is an equilibrium form that can reach nonﬁlling, nonscouring equilibrium depth with typical ﬂood tide current. Additional sediment arriving at the ﬂood shoal will spread out around and pass it as a thin layer, forming the far-ﬁeld deposit. Quantitative relations were obtained between ﬂood shoal area and spring tidal prism, for which it is noted that correlation coeﬃcients were low (R2 values in the range of 0.21–0.39) for widely scattered data that exhibited a broad trend. The

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results pertaining to Florida conditions are typiﬁed by mild waves, and small tidal prisms, and hard limestone bottom at many Atlantic coast inlets. In the following, the ﬁrst subscript, F denotes ﬂood shoal, and the second subscripts N , F , and T denote near ﬁeld, far ﬁeld, and total, respectively. 31.2.7.1. Flood shoal volume versus spring prism VF N = 4.056 × 103 P 0.314 VF F = 1.5337 × 104 P 0.314 VF T = 2.0389 × 10 P 4

0.296

(31.10) .

In these equations, volume and prism are expressed in units of m3 . It is possible for these equations to give a total volume less than the sum or the near ﬁeld and far ﬁeld, demonstrating variability in the data available to the study. 31.2.7.2. Flood shoal area versus spring prism AF N = 1.4532 × 104 P 0.254 AF F = 3.4122 × 104 P 0.244 AF T = 4.7585 × 10 P 4

0.249

(31.11) .

In these equations, area is expressed in m2 and prism in m3 . 31.2.8. Depth over ebb shoal The depth over the ebb shoal is of interest for navigation channel design and as a basic bulk parameter characterizing ebb shoals. Floyd38 examined the maximum depth of the channel hC at the entrances to tidal rivers and the maximum depth of the ebb shoal hE (which is referred to as the saddle of the entrance bar). The terminology saddle denotes the lowest point on the ebb shoal or entrance bar. Floyd38 compiled data from several rivers each in Australia, the USA, and New Zealand, and two river entrances from other countries, and found the following simple relation, with depth referenced to mean tide level (approximately the same as msl): hE = 0.5 hC .

(31.12)

The equation was valid irrespective of whether the river ﬂow was or was not trained by structures. Floyd38 concluded that it is not possible to increase the depth over the ebb shoal with jetties, and that the greatest depth across a bar can only be obtained by increasing the depth in the channel, as perhaps by dredging. Buonaiuto and Kraus48 analyzed the bathymetry of 18 inlets around the coast of the USA to determine a predictive expression for the minimum depth over crest hCr of the ebb shoal. It was reasoned that, because both incident waves and the tidal prism are expected to be controlling independent variables for ebb shoal development, a parameter combining both average annual signiﬁcant wave height HS and

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885

12 Best Fit Grays H.

95% Confidence Limit

10

hc, m (MLLW)

Columbia R. 8 Willapa Bay St. Marys Ent. 6

Lake Worth Inlet Charleston Harbor

Moriches Inlet

4 New Pass 1995 Shinnecock Inlet

2 New Pass 1997

0 0

50

100

(H sP)

150

200

250

1/4

,m

Fig. 31.12. Minimum depth over crest of an ebb shoal and product of average annual signiﬁcant wave height and tidal prism as (HS P )1/4 .

tidal prism would provide the best predictive capability. The parameter (HS P )1/4 , which has units of meters, was devised to represent the combination. Correlation with the data determined the following predictive relations (metric units): hCr = 0.27 + 3.6HS hCr = 0.0063P 0.35 hCr = −0.066 + 0.046(HS P )

(31.13) 1/4

.

For these equations, depth is measured with respect to mean lower low water, because waves would inﬂuence the bottom most at this lower tide level. Linear regression coeﬃcients R2 for the above equations were 0.81, 0.83, and 0.87, respectively. Figure 31.12 is the correlation plot corresponding to the last of the three predictive equations. 31.3. Engineering Inlets Under Conflicting Requirements Morphologic response to an inlet typically has a long timescale and great spatial extent. Sediment pathways are shared with the adjacent beaches and estuaries in a complex hydrodynamic environment. Therefore, consequences of the presence of inlets, both natural and engineered, can be subtle and far-ranging. It is also diﬃcult to transfer experience and monitoring results among inlets because of diﬀerent balances of kinds and strengths of the acting processes and conditions, of which tidal range, wave height and direction, wind, river ﬂow if any, bay or estuary surface

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area, sediment type and surrounding sedimentary structure, types of structures, conﬁguration of the estuary, and other factors play a role. Within this environment, modern inlet engineering attempts to balance two conﬂicting requirements: (i) Maintain inlet stability (minimal dredging) while assuring sediment bypassing continues to the adjacent beaches. (ii) Provide inlet navigability (strong tidal current to scour the channel) while assuring navigation reliability and safety (avoiding an excessively strong inlet current). The ﬁrst conﬂict concerns the design objective of protecting the channel from excessive sediment shoaling so that navigable depth is maintained for the longest possible time. Traditionally, this was done by building long jetties to block inﬁltration by sediment moving alongshore. Jetties also promote a stable location and orientation of the channel, and partially shelter vessels from waves in the surf zone. However, long jetties will interrupt longshore transport and deprive the down-drift beach and, perhaps, the estuary of sediment. In the USA, older long jetties have been deteriorating, and it is a signiﬁcant question as to whether they should be rehabilitated. The second conﬂict concerns the design requirement of promoting a tidal current of adequate strength to contribute to maintenance of channel depth by scour. On the other hand, safe navigation for smaller vessels requires moderate ebb current so that steep waves that pose a navigation hazard are not created by the wave–current interaction. Numerous considerations enter in these conﬂicting requirements, and some are discussed below. 31.3.1. Engineering situation Engineering situations can be classiﬁed into three categories as a new inlet or relocated inlet, modiﬁcation of an existing inlet, and maintenance of an existing inlet. Only a few aspects of each are discussed to illustrate some of the issues. Dean61 analyzes many of the interactions between jettied inlets and beaches. 31.3.1.1. New or relocated inlet In design of a new or a relocated inlet, one has the opportunity to address the two conﬂicts with maximum ﬂexibility. For a new inlet, one must consider: (i) Navigation channel reliability and maintenance. (ii) Formation of new ebb and ﬂood shoals (removes sediment from the littoral system). (iii) Long timescale for establishing natural bypassing. (iv) Channel stability (dredging requirements). (v) Navigation reliability (tidal current, entrance dimensions, etc.). (vi) Timescale and extent of response of adjacent beaches. (vii) Response of bay or estuary to storm surge; change in bay ﬂushing, etc. (viii) Optimized construction and dredging maintenance costs.

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In construction of a new inlet, the ebb and ﬂood shoals can be assumed to be formed by sand transported from the adjacent beaches. Although this will be a long-term process, the volume removed from the beaches should be considered in the sediment budget. For a relocated inlet, abandonment of the old ebb shoal can be an advantage in nourishing the down-drift beach and factored into the sediment budget. Kana and Mason62 and Kana and McKee63 discuss the beneﬁt for shore preservation of moving Captain Sams Inlet, South Carolina, a pioneering and successful eﬀort. VilaConcejo et al.64 discuss morphologic change for relocation of an inlet in Portugal and also review the literature. Kraus et al.17 perform a regional sediment management study and evaluated tradeoﬀs of relocating Fire Island Inlet, New York (Fig. 31.4). Cialone et al.65 discuss response of the ﬂood shoal at Barnegat Inlet, New Jersey, to construction of a south jetty almost parallel to the existing north jetty and dredging of a shoal that had formed in the entrance of the previous arrowhead conﬁguration jetties. As a response to this and other engineering actions, the historic growth of the ﬂood shoal halted. 31.3.1.2. Response of adjacent beaches If an inlet experiences a dominant direction of longshore sediment transport, the typical response of the adjacent beaches is up-drift accretion and down-drift erosion. If the inlet is in a nodal zone of longshore transport such that the long-term net rate varies around zero, shoreline response as moderate accretion on both sides can result from jetty construction. However, as opposed to the situation where jetties interrupt appreciable longshore sediment transport (large net transport rate), an equilibrium shoreline conﬁguration at nodal points may be reached within relatively few years, as found by Komar et al.66 for Paciﬁc northwest coast inlets and Williams et al.67 for Packery Channel, on the Texas coast. Porous jetties can cause erosion of the up-drift beach by allowing sediment to leak through to the inlet channel, increasing dredging maintenance.61 This loss may be deleterious to the down-drift beach if sand entering the channel is lost from bypassing around the ebb shoal. Sand tightening of porous jetties near to shore can provide an immediate beach-growth enhancement.68 Dean69 and Dalrymple70 examined shoreline response near inlets through analytic investigations with the one-line (shoreline) model. The approach of Dalrymple70 allowed sand to pass over, around, or through the jetties. However, both these works are limited in not considering cross-shore transport, such as occurring at the down-drift jetty on an isolated beach, as discussed below. Jetty construction at an existing inlet may conﬁne the ebb-tidal current and push the ebb shoal oﬀshore from its original location (Sec. 31.2.6). Flanks of the ebb shoal not located in the ebb jet may migrate onshore and give the appearance of accretion by longshore transport on the down-drift side of the inlet, until the abandoned portions of the ebb shoal serving as a source of sediment are fully depleted. The down-drift beaches then begin to erode. Bruun71,72 distinguishes the near-ﬁeld adjustment and far-ﬁeld adjustment of the down-drift shoreline at inlets. The near ﬁeld is the shoreline reach between the down-drift (and possibly up-drift) jetty and the attachment bar (Fig. 31.13) and at

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Fig. 31.13. Inferred sediment pathways and chronically eroding, sediment-isolated down-drift beach (area “3”), Shinnecock inlet (from Ref. 73).

many inlets near-ﬁeld recession of the shoreline is chronic and requires special measures of shore protection.73 This erosion may thin barrier islands to the point that breaching adjacent to the inlet becomes a concern. The far-ﬁeld shoreline response can extend many kilometers beyond the inlet. The existence and extent of the shoreline adjustment depend in great part on (1) length of jetties, (2) placement frequency and location of material dredged from the channel or bypassed mechanically, (3) balance of net and gross longshore sediment transport rates, and (4) elapsed time after jetty construction, among many factors. Shoreline-change numeric models can give an estimate of adjustment of the shoreline to be expected. Such modeling must include the anticipated conﬁguration of the ebb-tidal shoal in the wave transformation. 31.3.1.3. Modification of an existing inlet Typical modiﬁcations of inlets are the construction, tightening, and lengthening of jetties, and deepening and widening of the navigation channel. For such modiﬁcations, one must consider, for example: (i) (ii) (iii) (iv) (v)

Constriction or focusing of the ebb-tidal jet. Increase or decrease in the ebb current velocity. Translation of the ebb shoal further oﬀshore. Interruption of sediment bypassing. Recovery rates and interruption of sediment pathways if ebb or ﬂood shoals are mined. (vi) If there is a major rehabilitation, the ﬂanks of the ebb shoal might collapse and migrate on shore (Grays Harbor, Washington; Charleston, South Carolina).

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Walther and Douglas74 document recovery of the ebb shoal at Boca Raton Inlet, Florida, which was mined as a source for beach nourishment. Buttolph et al.75 observationally and numerically investigated oﬀshore migration of the ebb shoal at Ocean City Inlet, Maryland, that resulted from raising the outer portion of the south jetty. This study was aided by high-resolution bathymetry surveys (Fig. 31.7). 31.3.1.4. Maintenance of an existing inlet navigation channel Maintenance of navigation channels includes the infrastructure such as jetties and breakwaters. Mobilization for dredging is a great expense, calling for the longest possible dredging cycle. Considerations include: (i) Inlet locational stability. (ii) Channel cross-section stability. (iii) Annual dredging maintenance volume and material placement location (reestablishing sediment bypassing). (iv) Interruption of sediment bypassing (erosion of down- or up-drift beaches). (v) Possible change in tidal current speed and inﬂuence on navigation. (vi) Translation and growth of the ebb shoal. (vii) Growth or decline of the ﬂood shoal. (viii) Sedimentation in bay channels. (ix) Potential for breaching down drift of the inlet (avoided through bypassing and, possibly, placement of structures along the sediment-isolated beach). (x) Integrity of the jetties and breakwaters, including factors such as elevation, permeability, scour, ﬂanking, and stone dislodgement during storms.

31.4. Inlet Reservoir Model of Shoal Growth and Sediment Bypassing Maintenance of navigation channels through tidal inlets must consider means of bypassing of sediment from the inlet to the adjacent beaches (where the sediment originated). Hydraulic and mechanical bypassing preserves the pathways of sediment in the littoral zone within the context of the natural sediment-sharing system of inlets and beaches. Seabergh and Kraus76 discuss sediment bypassing strategies and review the literature. Stive et al.77 introduced an “aggregate” model of long-term and wide-scale change in estuaries and inlets, based in part on empiric predictions of morphologic features for the Dutch coast. Here, the Inlet Reservoir Model24 is discussed as one tool for assisting in understanding of the time-dependent inlet sediment budget in support of engineering and management activities. Applications include prediction of ebb and ﬂood shoal growth for new and relocated inlets,17,78 for estimating recovery of an ebb or ﬂood shoal to be mined for beach ﬁll,79 for verifying bypassing actions and consequences of shoal mining,80 and examining complex and seasonal sediment pathways.81

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31.4.1. Morphology concepts for the inlet reservoir model The reservoir model is based on the conservation of sand volume, ability of the engineer to identify morphologic features and sediment pathways, existence of an equilibrium volume of morphologic features, and a “reservoir assumption” that is described below. Bypassing bars grow in the direction of predominant transport, similar to growth of a spit. At inlets with left- and right-directed longshore transport or with a small tidal prism, two large bypassing bars can emerge from the ebb shoal, creating a nearly concentric halo about the inlet entrance. As the bypassing bar merges with the shore, an attachment bar is created, thereby transporting sand to the beach (Fig. 31.8). At this point in evolution of the ebb-shoal complex, substantial bypassing of sand can occur. In the context of the Inlet Reservoir Model, the ebb-shoal complex is deﬁned as consisting of the ebb-shoal proper, one or two ebb-shoal bypassing bars (depending on the balance between left- and right-directed longshore transport), and one or two attachment bars. These features are schematically shown in Fig. 31.8. The model distinguishes between the ebb-tidal shoal proper, typically located in the conﬁne of the ebb-tidal jet, and the ebb-shoal bypassing bars that grow toward the shore from the ebb shoal, principally by the longshore transport of sediment by wave action. Previous authors (e.g., Ref. 40) combined the ebb shoal proper and the bar(s) protruding from it into one feature referred to as the ebb shoal. For the Inlet Reservoir Model, the shoal and bypassing bars are distinguished because of the different formation processes. When an inlet forms, a shoal ﬁrst becomes apparent within the conﬁnes of the inlet ebb jet.24 Bypassing bars form later by sediment transported oﬀ the shoal through the action of breaking waves and wave-induced longshore current (tidal and wind-induced currents can also play a role). In simpliﬁed applications of the model, however, the ebb shoal and bypassing bar can be treated as a unit.

31.4.2. Mathematic representation Morphologic features such as shoals and channels can be described mathematically by analogy to a series of reservoirs or beakers, as depicted in Fig. 31.14. Sand arrives to the ebb shoal at a rate Qin , equivalent to the right-directed transport QR . Also, the volume VE in the ebb shoal tends to increase while possibly bypassing some amount of sand to create a down-drift bypassing bar. The volume of sand in the shoal (reservoir) can increase until it reaches an equilibrium volume VEe (the subscript e denotes equilibrium) according to the transporting conditions. Sand leaks to the bypass bar from inception of the shoal and, after equilibrium is achieved (the reservoir is full), all sand brought to the ebb shoal is bypassed in the direction of transport at the particular time. Similarly, the bypassing bar volume VB grows as it is supplied with sediment by the littoral drift and the ebb shoal, with some of its material leaking (bypassing) to the down-drift attachment bar. After the bypassing bar reaches equilibrium volume VBe , all sand supplied to it is contributed to the volume of the attachment bar VA . The attachment bar transfers sand to the adjacent beaches. After it reaches

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Fig. 31.14.

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equilibrium volume VAe , all sand supplied to it by the bar is bypassed to the beach. The model thus requires values of the input and output rates of transport from each morphologic feature and their respective equilibrium volumes. Complicated sediment pathways, multiple connections, and time-varying transport can be represented.79–81 Here, a simple analytic version of the model (closed-form solution) is reviewed.24 In applications, the governing equations are solved numerically and can treat complex, time-dependent conditions. The continuity equation governing change of the volume VE of the ebb-tidal shoal is: dVE = Qin − (QE )out dt

(31.14)

where t is the time and (QE )out is the rate of sand leaving (going out of) the ebb shoal. The input transport rate is assumed known; for example, it could be the right-directed transport QR or the left-directed transport QL . The remaining unknown is the output transport rate. The reservoir model assumption is that the output rate is proportional to the input rate times the volume of sand in the beaker divided by the equilibrium volume. Therefore, rate of sand leaving or bypassing the ebb shoal (QE )out , is speciﬁed as: (QE )out =

VE Qin VEe

(31.15)

in which Qin is taken to be constant here, although this is not necessary in a numeric model. For the present situation, Eqs. (31.14) and (31.15) give: dVE VE = Qin 1 − . dt VEe

(31.16)

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With initial condition VE (0) = 0, the solution of Eq. (31.16) is exponential growth: VE = VEe 1 − e−αt (31.17) in which: α=

Qin . VEe

(31.18)

The quantity 1/α is a characteristic timescale for growth of the ebb shoal. For example, if Qin = 1 × 105 m3 /year and VEe = 2 × 106 m3 , which are representative values for a small inlet on a moderate-wave coast, then 1/α = 20 years. The shoal would be predicted to reach 50% and 95% of its equilibrium volume after 14 and 60 years, respectively, under the constant imposed transport rate. The parameter α is essentially the inverse of the r-parameter introduced by Bruun and Gerritsen14 [Eq. (31.1)]. This simple situation for constant input longshore transport magnitude and direction gives the volume of the bypassing bar as: VB = VBe 1 − e−βt ,

β=

Qin , VBe

t = t −

Qin , VAE

t = t −

VE Qin

(31.19)

and the volume of the attachment bar, as: VA = VAE 1 − e−γt ,

γ=

VB . Qin

(31.20)

The coeﬃcients 1/β and 1/γ function similarly to 1/α in representing timescales for the bypassing bar and attachment bar, respectively. The quantities t and t in Eqs. (31.19) and (31.20) are lag times that account for a delay in development of the respective features. After formation of an inlet, a certain time is required for the bypassing bar to receive a signiﬁcant amount of sand from the shoal and a longer time for the attachment bar or beach to receive sand as it moves around the inlet from the up-drift side (delays). The following are obtained for the bypassing rate of the bar (QB )out , which is equal to the input of the attachment (QA )in , and the bypassing rate of the attachment (QA )out , which is the input to the beach, (Qbeach )in : VE VB Qin = (QA )in VEe VBe

(31.21)

VE VB VA Qin = (Qbeach )in . VEe VBe VAe

(31.22)

(QB )out = (QA )out =

The rate (QA )out describes the amount of sand reaching the down-drift beach as a function of time and is a central quantity entering beach nourishment and shoreprotection design near inlets. Figure 31.15 illustrates the sediment pathways developed for Sebastian Inlet, Florida, accounting for seasonality in wave direction and longshore sediment

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Fig. 31.15. Sediment pathways conceptualized for Sebastian Inlet, FL (E: ebb shoal; B: bypass bar; A: attachment bar; C: channel; T: sand trap; F: ﬂood shoal; Y: bay; SS : south ﬁllet; IN : north ﬁllet; IS : south ﬁllet; and O: oﬀshore loss).

transport.81 The inlet opens to the Atlantic Ocean on a north–south trending coast. Predictions shown in Fig. 31.16 agree with most measurements of ﬂood and ebb shoal volumes at Sebastian Inlet, but underestimate volumes determined for the late 1980s. The calculations and measurements match for the past decade with respect to the overall consequence of sand bypassing. The simulation shown in Fig. 31.16 includes sediment volume removed from the sand trap to represent sand-bypassing projects conducted between 1972 and 1999. Zarillo and Brehin82 give an update on the status of Sebastian Inlet and its morphologic evolution.

31.5. Elements of Tidal Inlet Hydrodynamics and Modeling This chapter concerns morphologic change around and engineering of tidal inlets. The hydrodynamics at a tidal inlet, the water movement that transports sediment and determines inlet morphologic forms, could not be discussed due to space limitations. Also, advanced numeric modeling of inlets was not covered. Here, some elements of these subjects are presented for completeness. The hydrodynamics of inlets is fascinating, for which much mathematic elegance has been devoted. Keulegan83 developed the basic one-dimensional equation of motion that is used today. Seabergh84 investigated some predictions of the

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Fig. 31.16. Inlet Reservoir Model simulation of sediment volumes at Sebastian Inlet, Florida, 1950–2050. Solid symbols indicate shoal volumes estimated from topographic data and analysis of aerial images. Solid arrows indicate sand bypass events.

Keulegan approach for inlet channel stability. The reader is directed to Chap. 13 of Dean and Dalrymple85 and to Seabergh86 for thorough reviews of simple tidal inlet hydraulics that can be of great aid in understanding and engineering design. Seelig and Sorensen87 demonstrate the utility of such an approach. First-order analysis of inlet stability rests on the important “Escoﬃer stability curve”37 that depends on a calculation of the current through an inlet. Seabergh and Kraus88 discuss properties of the Escoﬃer stability curve and provide a desk-top PC program to calculate it. The program is based on the analytic solution for an inlet current given by DiLorenzo,89 which represents overtides — higher harmonics of the dominant tide component generated through tidal wave shoaling (see Ref. 90). The Escoﬃer stability curve also requires a predictive relation for the minimum channel area for a given tidal prism, as discussed in Sec. 31.2.5. Sediment pathways and morphology change around an idealized dual-jetty inlet similar to Shinnecock Inlet were investigated by Militello and Kraus91 with a sophisticated numeric model. This work demonstrates signiﬁcant diﬀerences in sediment transport depending on the wave climate as either typical or a storm condition. Recently, Fortunato and Oliveira92 report an interesting two-dimensional numeric model application investigating inlet stability and minimization of channel dredging. Consideration of nonlinear processes associated with tidal ﬂats is included. Such works are among many demonstrating the engineering utility of numeric models of inlet hydrodynamics and morphology change.

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31.6. Concluding Discussion In considering options for design or maintenance of a navigable tidal inlet, two contradictory requirements of inlets must be balanced or reconciled in the engineering design. These are (1) maintaining inlet stability while assuring natural functioning of sediment bypassing around the inlet, and (2) providing inlet navigability while promoting safe navigation. Requirement 1 implies inlet morphology retains equilibrium plan form and depths over which sediment moves in an eﬃcient way, whereas Requirement 2 implies that inlets must be dredged to a necessary depth in support of navigation. A channel deeper than the natural channel depth will intercept more of the sediment moving toward it, which must subsequently be dredged. Sediment may be jetted farther oﬀshore by the constraining jetties, depriving the beaches of that material or delaying its arrival. If the inlet has a shallow channel, sediment can cross or bypass easily, but the depth may not be adequate for navigation. Williams et al.67 describe the design and functioning of a new inlet, Packery Channel on the Texas coast, for which monitoring during its ﬁrst three years has indicated no need to dredge and no signiﬁcant negative response of the adjacent beaches. The inlet was designed with awareness of many of the considerations described in this chapter,93 in particular that the jetties not intercept all sand moving alongshore and that the hydraulic eﬃciency (entrance width to depth ratio) be less than 100.39 The main issues in sediment management are interruption of the littoral drift by the inlet jetties; creation, growth, and mining of ebb- and ﬂood-tidal shoals; and resultant changes in position of the shoreline, which may advance seaward on the up-drift side and recede on the down-drift side. Optimal placement of beach-quality material on the adjacent shores that is removed from the channel during newwork dredging (original dredging) and maintenance dredging operations, as well as mechanical bypassing of littoral material that is blocked by the up-drift jetty, are also central elements of a sediment-management plan. Dean61 has discussed such processes and associated policies. Inlet design and sediment management considerations are therefore linked through interruption of the littoral drift, dredging of an inlet, increase in tidal current through the dredged channel, and the water and sediment circulation around the inlet. Larger tidal inlets can evolve over hundreds of years, indicating that regional responses must be appreciated or anticipated in engineering design. Kraus et al.17 took a regional sediment management approach in hypothetic relocation of Fire Island Inlet, New York. Among several aspects, ebb-shoal collapse as a form of beach nourishment was found to yield a large beneﬁt for a chronically eroding down-drift beach. Regional applications will typically involve multiple tidal inlets to the same or connecting bays. Changes in one inlet can cause a response in the others. Batten et al.94 made a morphologic study documenting the decrease in size of Pass Cavallo, the natural tidal inlet to Matagorda Bay, Texas. This inlet is estimated to be about 2,600 years old and has been at the same location for the past 200 years. Opening of the deep-draft Matagorda Ship Channel to the bay in the early 1960s “captured the tidal prism”, causing a loss of prism through Pass Cavallo and reduction in size

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of the ebb-tidal shoal. Seabergh95 investigated the stability of two inlets serving the same bay system in Guatemala, taking an engineering approach with Escoﬃer stability diagrams. The science and engineering of tidal inlets are a challenge, and it is hoped that this chapter will be of some small assistance to those interested in this complex coastal environment. Acknowledgments This work was performed as an activity of the Inlet Morphology and Channels work unit of the Coastal Inlets Research Program administered by Headquarters, US Army Corps of Engineers. Permission was granted to publish this information. I am indebted to Julie Dean Rosati, William Seabergh, and Dr. Gary Zarillo for thoughtful and helpful reviews. References 1. D. M. FitzGerald, Tidal inlets, Encyclopedia of Coastal Science, ed. M. L. Schwartz (Springer, 2005), pp. 958–964. 2. W. A. Price, Reduction of maintenance by proper orientation of ship channels through tidal inlets, Proc. 2nd Coast. Eng. Conf., Council on Wave Research (1952), pp. 243–255. 3. R. M. Sorensen and W. Seelig, Hydraulics of Great Lakes inlet — Harbors systems, Proc. 15th Coast. Eng. Conf., ASCE (1976), pp. 1646–1665. 4. N. C. Kraus, Coastal inlets of Texas, USA, Proc. Coast. Sediments’07, ASCE (2007), pp. 1475–1488. 5. The Engineer School, The Fundamentals of Seacoast Jetties (U.S. Army, Fort Humphrey, VA, 1932), 43 pp. 6. K. R. Bodge, Inlet impacts and families of solutions for inlet sediment budgets, Proc. Coast. Sediment’99, ASCE (1999), pp. 703–718. 7. K. R. Bodge and J. D. Rosati, Sediment management at inlets and harbors, Coastal Engineering Manual, Part 5, Chap. 6, Engineer Manual 1110-2-1100 (U.S. Army Corps of Engineers, Washington, DC, 2002). 8. J. D. Rosati and N. C. Kraus, Advances in coastal sediment budget methodology — With emphasis on inlets, Shore Beach 67(2&3), 56–65 (1999). 9. J. D. Rosati and N. C. Kraus, Sediment budget analysis system (SBAS), ERDC/CHL CHETN-XIV-3, U.S. Army Engineer Research and Development Center, Vicksburg, MS (2001). 10. J. D. Rosati, Concepts in sediment budgets, J. Coast. Res. 21(2), 307–322 (2005). 11. N. C. Kraus, Coastal inlet functional design: Anticipating morphologic response, Proc. Coast. Dynamics’05, ASCE, CD-ROM (2006), 14 pp. 12. M. O. Hayes, Geomorphology and sedimentation patterns of tidal inlets: A review, Proc. Coast. Sediments’91, ASCE (1991), pp. 1343–1355. 13. D. M. FitzGerald, Geomorphic variability and morphologic and sedimentologic controls at tidal inlets, J. Coast. Res. SI23, 47–71 (1996). 14. P. Bruun and F. Gerritsen, Natural bypassing of sand at coastal inlets, J. Waterw. Harbor. Div. 85, 401–412 (1959).

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15. R. A. Davis and D. M. FitzGerald, Beaches and Coasts (Blackwell Science, 2004), p. 419. 16. C. J. Galvin, Jr., Wave climate and coastal processes, Water Environments and Human Needs, ed. T. Ippen (Parsons Lab. for Water Res. and Hydrodynamics, MIT, 1971), pp. 28–78. 17. N. C. Kraus, G. A. Zarillo and J. F. Tavolaro, Hypothetical relocation of Fire Island Inlet, New York, Proc. Coast. Sediments’03, World Sci., CD-ROM (2003), 14 pp. 18. M. O. Hayes, V. Goldsmith and C. H. Hobbs, III, Oﬀset coastal inlets, Proc. 12th Coast. Eng. Conf., ASCE (1970), pp. 1187–1200. 19. M. O. Hayes, Barrier island morphology as a function of tidal and wave regime, Barrier Islands, ed. S. P. Leatherman (Academic Press, 1979), pp. 1–28. 20. R. A. Davis, Jr. and M. O. Hayes, What is a wave dominated coast? Mar. Geol. 60, 313–329 (1984). 21. D. M. FitzGerald, N. C. Kraus and E. B. Hands, Natural mechanisms of sediment bypassing at tidal inlets, ERDC/CHL CHETN-IV-30, U.S. Army Engineer Research and Development Center, Vicksburg, MS (2001). 22. D. M. FitzGerald, Sediment bypassing at mixed energy tidal inlets, Proc. 18th Coast. Eng. Conf., ASCE (1982), pp. 1094–1118. 23. E. J. Anthony and J. D. Orford, Between wave- and tide-dominated coasts: The middle ground revisited, J. Coast. Res. SI36, 8–15 (2002). 24. N. C. Kraus, Reservoir model of ebb-tidal shoal evolution and sand bypassing, J. Waterw. Port Coast. Ocean Eng. 126(3), 305–313 (2000). 25. E. E. Carr and N. C. Kraus, Morphologic asymmetries at entrances to tidal inlets, ERDC/CHL CHETN-IV-33, U.S. Army Eng. Res. and Develop. Center, Vicksburg, MS (2001), http://chl.wes.army.mil/library/publications/chetn. 26. P. Bruun, Stability of Tidal Inlets (North Holland Pub. Co., Amsterdam, 1960), 123 pp. 27. T. W. Kana, M. L. Williams and D. Stevens, Managing shoreline changes in the presence of nearshore shoal migration and attachment, Proc. Coast. Zone’85, ASCE (1985), pp. 1277–1294. 28. D. M. FitzGerald, Shoreline erosional-depositional processes associated with tidal inlets, Hydrodynamics and Sediment Dynamics of Tidal Inlets, eds. D. G. Aubrey and L. Weishar (Springer, 1988), pp. 186–225. 29. D. J. Gaudiano and T. W. Kana, Shoal bypassing in South Carolina tidal inlets: Geomorphic variables and empirical predictions for nine mesoscale inlets, J. Coast. Res. 17(2), 280–291 (2000). 30. N. C. Kraus and L. Lin, Coastal processes study of the San Bernard River Mouth, Texas: Stability and maintenance of the mouth, TR ERDC/CHL-02-10, U.S. Army Engineer Research and Develop Center, Coastal and Hydraulics Lab., Vicksburg, MS (2002). 31. E. B. Hands and V. Shepsis, Cyclic channel movement at the entrance to Willapa Bay, Washington, USA, Proc. Coast. Sediments’99, ASCE (1999), pp. 1522–1536. 32. L. J. LeConte, Discussion on river and harbor outlets, notes on the improvement of river and harbor outlets in the United States, Paper No. 1009 by D. A. Watts, Transactions ASCE 55, 306–308 (1905). 33. M. P. O’Brien, Estuary and tidal prisms related to entrance areas, Civil Eng. 1(8), 738–739 (1931). 34. M. P. O’Brien, Equilibrium ﬂow areas of inlets on sandy coasts, J. Waterw. Port. Harbor. Div. 95(WW1), 43–52 (1969). 35. H. P. Riedel and M. R. Gourlay, Inlets/estuaries discharging into sheltered waters, Proc. 17th Coast. Eng. Conf., ASCE (1980), pp. 2550–2562.

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36. T. M. Hume and C. E. Herdendorf, Morphologic and hydrologic characteristics of tidal inlets on a headland dominated, low littoral drift coast, Northeastern New Zealand, Proc. Skagen Symp. J. Coast. Res. SI9, 527–563 (1990). 37. F. F. Escoﬃer, The stability of tidal inlets, Shore Beach 8(4), 114–115 (1940). 38. C. D. Floyd, River mouth training in New South Wales, Australia, Proc. 11th Coast. Eng. Conf., ASCE (1968), pp. 1267–1281. 39. J. T. Jarrett, Tidal prism-inlet area relationships, GITI Report 3, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS (1976). 40. T. L. Walton and W. D. Adams, Capacity of inlet outer bars to store sand, Proc. 15th Coast. Eng. Conf., ASCE (1976), pp. 1919–1937. 41. J. N. Marino and A. J. Mehta, Sediment trapping at Florida’s east coast inlets, Hydrodynamics and Sediment Dynamics of Tidal Inlets, eds. D. G. Aubrey and L. Weishar (Springer-Verlag, 1988), pp. 284–296. 42. D. M. Hicks and T. M. Hume, Morphology and size of ebb tidal deltas at natural inlets on open-sea and pocket-bay coasts, North Island, New Zealand, J. Coast. Res. 12(1), 47–63 (1996). 43. T. Shigemura, Tidal prism-throat width relationships of the bays of Japan, Shore Beach 49(3), 34–39 (1981). 44. J. C. Gibeaut and R. A. Davis, Jr, Statistical geomorphic classiﬁcation of ebb-tidal deltas along the west-central Florida coast, J. Coast. Res. SI18, 165–184 (1993). 45. N. C. Kraus, Inlet cross-sectional area calculated by process-based model, Proc. 26th Coast. Eng. Conf., ASCE (1998), pp. 3265–3278. 46. E. E. Carr de Betts, An examination of ﬂood tidal deltas at Florida’s tidal inlets, MS thesis, Coastal and Oceanographic Eng. Dept., University of Florida, Gainesville, FL (1999). 47. E. E. Carr de Betts and A. J. Mehta, An assessment of inlet ﬂood deltas in Florida, Proc. Coast. Dynamics’01, ASCE (2001), pp. 252–262. 48. F. S. Buonaiuto and N. C. Kraus, Limiting slopes and depths at ebb-tidal shoals, Coast. Eng. 48, 51–65 (2003). 49. C. L. Vincent and W. D. Corson, Geometry of tidal inlets: Empirical equations, J. Waterw. Port Coast. Ocean Div. 107(1), 1–9 (1981). 50. P. Bruun, Tidal Inlets and Littoral Drift (Univ. Book Co., Oslo, Norway, 1968). 51. P. Bruun, Tidal inlets on alluvial shores, Port Eng., Vol. 2, Chapter 9, ed. P. Bruun (Gulf Pub., 1990), pp. 810–929. 52. G. H. Keulegan and J. V. Hall, A formula for the calculation of tidal discharge through an inlet, U.S. Army Corps of Engineers, Beach Erosion Board Bulletin 4, 15–29 (1950). 53. J. W. Johnson, Characteristics and behavior of Paciﬁc coast tidal inlets, J. Waterw. Harbor. Coast. Eng. Div. 99(WW3), 325–339 (1973). 54. R. J. Byrne, R. A. Gammisch and G. R. Thomas, Tidal prism-inlet area relations for small tidal inlets, Proc. 17th Coast. Eng. Conf., ASCE (1980), pp. 2517–2533. 55. J. Pope, Ebb delta and shoreline response to inlet stabilization, examples from the southeast Atlantic coast, Proc. Coast. Zone’91, ASCE (1991), pp. 643–654. 56. M. C. Buijsman, G. M. Kaminsky and G. Gelfenbaum, Shoreline change associated with jetty construction, deterioration, and rehabilitation at Grays Harbor, Washington, Shore Beach 71(1), 15–22 (2003). 57. M. R. Byrnes, J. L. Baker and N. C. Kraus, Coastal sediment budget for Grays Harbor, Washington, Proc. Coastal Sediments’03, World Sci., CD-ROM (2003), 14 pp. 58. R. G. Dean and T. L. Walton, Sediment transport processes in the vicinity of inlets with special reference to sand trapping, Estuarine Research II, ed. L. E. Cronin (Academic Press, 1973), pp. 129–149.

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59. J. N. Marino and A. J. Mehta, Inlet ebb tide shoals related to coastal parameters, Proc. Coast. Sediments 87, ASCE (1987), pp. 1608–1622. 60. E. E. Carr de Betts and A. J. Mehta, An assessment of inlet ﬂood deltas in Florida, Proc. Coastal Dynamics ’01, ASCE (2001), pp. 252–262. 61. R. G. Dean, Sediment interactions at modiﬁed coastal inlets: Processes and policies, Hydrodynamics and Sediment Dynamics of Tidal Inlets, eds. D. G. Aubrey and L. Weishar (Springer, 1988), pp. 412–439. 62. T. W. Kana and J. E. Mason, Evolution of an ebb-tidal delta after an inlet relocation, Hydrodynamics and Sediment Dynamics of Tidal Inlets, eds. D. G. Aubrey and L. Weishar, Lecture Notes on Coastal and Estuarine Studies 29 (Springer, 1988), pp. 382–411. 63. T. W. Kana and P. A. McKee, Relocation of Captain Sams inlet — 20 years later, Proc. Coast. Sediments’03, World Sci., CD-ROM (2003), 12 pp. 64. A. Vila-Concejo, O. Ferreira, B. D. Morris, A. Matias and J. M. A. Dias, Lessons from inlet relocation: Examples from southern Portugal, Coast. Eng. 51, 967–990 (2004). 65. M. A. Cialone, W. C. Seabergh and K. D. Watson, Flood shoal response to inlet modiﬁcations at Barnegat Inlet, New Jersey, Proc. Coast. Sediments’99, ASCE (2003), pp. 1434–1449. 66. P. D. Komar, J. R. Lizarraga-Arciniega and T. A. Terich, Oregon coast shoreline changes due to jetties, J. Waterw. Harbor. Coast. Eng. Div. 102(WW1), 13–30 (1976). 67. D. D. Williams, N. C. Kraus and C. M. Anderson, Morphologic response to a new inlet, Packery Channel, Corpus Christi, Texas, Proc. Coast. Sediments’07, ASCE (2007), pp. 1529–1542. 68. C. G. Creed, E. J. Olsen and K. R. Bodge, Performance of an interim sand-tightening measure at an inlet jetty, Proc. 7th Natl. Conf. Beach Preserv. Tech., FSBPA (1994), pp. 374–388. 69. R. G. Dean, A framework for sediment management practices at jettied inlets, Proc. Coast. Sediment’03, World Sci., CD-ROM (2003), 13 pp. 70. R. A. Dalrymple, An extended one-line model for jettied inlets, Proc. Coast. Sediments’03, World Sci., CD-ROM (2003), 12 pp. 71. P. Bruun, The development of downdrift erosion, J. Coast. Res. 11(4), 1242–1257 (1995). 72. P. Bruun, Bypassing at littoral drift barriers, Encyclopedia of Coastal Science, ed. M. L. Schwartz (Springer, 2005), pp. 210–215. 73. H. Hanson and N. C. Kraus, Chronic beach erosion adjacent to inlets and remediation by composite (T-head) groins, ERDC/CHL CHETN IV-36, U.S. Army Engineer Research and Development Center, Vicksburg, MS (2001). 74. M. P. Walther and B. D. Douglas, Ebb shoal borrow area recovery, J. Coast. Res. SI18, 221–223 (1993). 75. A. M. Buttolph, W. G. Grosskopf, G. P. Bass and N. C. Kraus, Natural sand bypassing and response of ebb shoal to jetty rehabilitation, Proc. 30th Coast. Eng. Conf., Ocean City Inlet, Maryland, USA, World Sci. (2007), pp. 3344–3356. 76. W. C. Seabergh and N. C. Kraus, Progress in management of sediment bypassing at coastal inlets: Natural bypassing, weir jetties, jetty spurs, and engineering aids in design, Coast. Eng. J. 45(4), 533–563 (2003). 77. M. J. F. Stive, M. Capobianco, Z. B. Wang, P. Ruol and M. C. Buijsman, Morphodynamics of a tidal lagoon and the adjacent coast, Physics of Estuaries and Coastal Seas, eds. J. Dronkers and M. Scheﬀers (Balkema, 1998), pp. 397–407.

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78. K. M. Erickson, N. C. Kraus and E. E. Carr, Circulation change and ebb shoal development following relocation of Mason Inlet, North Carolina, Proc. Coastal Sediments’03, World Sci., CD-ROM (2003), 13 pp. 79. A. Militello and N. C. Kraus, Shinnecock Inlet, New York, site investigation, Report 4, evaluation of ﬂood and ebb shoal sediment source alternatives for the west of Shinnecock interim project, New York, TR CHL TR-98-32, Coastal Inlets Research Program, U.S. Army Research and Develop. Center, Coastal and Hydraulics Laboratory, Vicksburg, MS (2001). 80. M. A. Dabees and N. C. Kraus, General methodology for inlet reservoir model analysis of sand management near tidal inlets, Proc. Coast. Dynamics 05, ASCE, CD-ROM (2006), 14 pp. 81. G. A. Zarillo, N. C. Kraus and R. K. Hoeke, Mophologic analysis of Sebastian Inlet, Florida: Enhancements to the tidal inlet reservoir model, Proc. Coast. Sediments’03, World Sci., CD-ROM (2003), 14 pp. 82. G. A. Zarillo and F. G. A. Brehin, Hydrodynamic and morphologic modeling at Sebastian Inlet, FL, Proc. Coast. Sediments’07, ASCE (2007), pp. 1297–1310. 83. G. H. Keulegan, Tidal ﬂows in entrances: Water level ﬂuctuations of basins in communication with seas, Committee on Tidal Hydraulics TB-14, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS (1967). 84. W. C. Seabergh, Long-term coastal inlet channel area stability, Proc. Coast. Sediments’03, World Sci., CD-ROM (2003), 8 pp. 85. R. G. Dean and R. A. Dalrymple, Coastal Processes with Engineering Applications (Cambridge University Press, 2002), 475 pp. 86. W. C. Seabergh, Hydrodynamics of tidal inlets, Coastal Engineering Manual, Part 2, Chap 6, Engineer Manual 1110-2-1100, U.S. Army Corps of Engineers, Washington, DC (2002). 87. W. N. Seelig and R. M. Sorensen, Numerical model investigation of selected tidal inletbay system characteristics, Proc. 16th Coastal Eng. Conf., ASCE (1978), pp. 1302– 1319. 88. W. C. Seabergh and N. C. Kraus, PC program for coastal inlet stability analysis using Escoﬃer method, CETN-II-11, U.S. Army Engineer Waterways Experiment Station, Coastal and Hydraulics Laboratory, Vicksburg, MS (1997). 89. J. L. DiLorenzo, The overtide and ﬁltering response of small inlet/bay systems, Hydrodynamics and Sediment Dynamics of Tidal Inlets, eds. D. G. Aubrey and L. Weishar (Springer, 1988), pp. 24–53. 90. C. T. Friedrichs and D. G. Aubrey, Non-linear tidal distortion in shallow well-mixed estuaries: A synthesis, Est. Coast. Shelf Sci. 27(5), 521–545 (1988). 91. A. Militello and N. C. Kraus, Numerical simulation of sediment pathways at an idealized inlet and ebb shoal, Proc. Coast. Sediments’03, World Sci., CD-ROM (2003), 14 pp. ´ 92. A. B. Fortunato and A. Oliveira, Case study: Promoting the stability of the Obidos Lagoon inlet, J. Hydraul. Eng. 133(7), 816–824 (2007). 93. N. C. Kraus and D. J. Heilman, Packery Channel feasibility study: Inlet functional design and sand management, Report 1 of a two-part series. TR TAMU-CC-CBI-9606, Conrad Blucher Institute for Surveying and Sci., Texas A&M U.-Corpus Christi, Corpus Christi, TX (1997), 106 pp. 94. B. K. Batten, N. C. Kraus and L. Lin, Long-term inlet stability of a multiple inlet system, Proc. Coast. Sediments’07, ASCE, Pass Cavallo, Texas (2007), pp. 1515–1528. 95. W. C. Seabergh, Approaches to understanding multiple-inlet stability, Proc. Coast. Sediments’07, ASCE (2007), pp. 1391–1404.

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Chapter 32

Water and Nutrients Flow in the Enclosed Bays Yukio Koibuchi∗ and Masahiko Isobe Graduate School of Frontier Sciences The University of Tokyo, Environmental Building #662, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8563, Japan ∗ [email protected] Enclosed bays are one of the most productive areas in the sea, and they have long provided us with uncountable beneﬁts. Since enclosed bays are the boundary domain between land and open sea, they are also signiﬁcant for global material cycles. However, eutrophication has been occurring in many enclosed bays around the world. To maintain sustainable utilization of such bays’ ecosystems, advanced interdisciplinary research is needed. This chapter describes the mechanisms of water quality variation under typical currents and introduces an ecosystem model as a tool for the integrated management of enclosed bays.

32.1. Introduction Enclosed bays have always been natural environment next to urban areas, and provided us with uncountable beneﬁts. Due to their particular geographic feature, the enclosed bays are stable when compared with open seas, and cannot be easily inﬂuenced by ocean weather conditions. Because of this, many harbors are located in enclosed bays, and hence many big cities in the world, even today, are located around enclosed bays.1 Enclosed bays also provide ﬁsh and shellﬁsh resources, and navigation and recreation potentials. In addition, the suitability of coastal zone for various human activities has led to reclamation of shallow water areas on mildly sloping bottom.2 However, compared with the long history of our relationship with enclosed bays, our understanding of these areas is not so deep enough. This may be because they have an extremely wide variety of environments depending on location and time since they have so complicated geographic features bordering inland and outer ocean and hence are aﬀected by both of them. Recently, deterioration of water quality has been occurring in many enclosed bays in the world. The damages caused by the events of red tides, harmful algal bloom, and decrease in dissolved oxygen in the bottom water are occurring 901

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frequently. They include degradation of aquatic ecosystems and loss of aquatic resources, such as seagrass beds, and ﬁsh and shellﬁsh. This is caused by excess inﬂow of nutrients resulting in increase of their concentrations called eutrophication. However, the events are inﬂuenced by weather conditions, ocean current, accelerated population growth, household and industrial wastes, nutrients coming from agricultural lands, sewage treatment plants, acid rain, and others.3 Furthermore, interactions of these factors are complicated and further deteriorate water quality. Therefore, it is diﬃcult to distinguish between natural and anthropogenic eﬀects. This raises diﬃculty in recovering the water quality and ecosystem, and managing resources in enclosed bays. This chapter describes how the water quality and ecosystems of enclosed bays are changed by the characteristics of current condition. Section 32.2 explains the eutrophication phenomena. The relationship between the water motion and ecosystem in enclosed bays are described in Sec. 32.3. Section 32.4 shows the outline of ecosystem modeling as a tool for integrated management. Concluding remarks and problems are given in Sec. 32.5.

32.2. Eutrophication of Enclosed Bays To understand the bay’s ecosystem and water quality comprehensively, we must have enough understanding of biologic, chemical, and physical processes. Nutrients enter the bay from river, atmosphere, industry and wastewater treatment plants, and transferred by the eﬀect of advection and diﬀusion in the bay. Nutrients are also transformed to organic matter by phytoplankton uptake or mineralized by adsorption. Even if it settles down on the seabed, it will be recycled as suspended matter by wave-induced ﬂow or remineralized by bacteria. This chapter deals with red tide, blue tide, and hypoxia, which are the main results of eutrophication. These have serious inﬂuence on the ecosystem of enclosed bays and material recycling in parallel to the water quality change. 32.2.1. Ecosystem of enclosed bays The ﬁrst trophic level in the food chains of the pelagic ecosystem of a bay is phytoplankton. While major plants on land are various grasses and trees, the phytoplankton has equivalent function to these in the sea, being organisms capable of photosynthesis as well as benthic algae.4 Although phytoplankton is so small that it can only be seen by a microscope, it is the origin of ecosystem in a bay. Through feeding, remaining organisms obtain through feeding material and energy including trace elements such as nitrogen and phosphorus that are indispensable elements to make an organic body. There are many species of phytoplankton. They contribute 95% of marine primary production.5 Planktonic algae can be classiﬁed in terms of its form, features of photosynthesis, and multiplication. The principal taxa of planktonic producers in most of the world’s bays are diatoms, dinoﬂagellates, cyanobacteria, and other bacteria.

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Diatoms can multiply very rapidly and dominate in nutrient-rich marine waters. Their distinguishing feature is a hard mineral shell made by polymerized silicic acid. They ﬂoat in the water column or attach to surface of sediment. Diatoms are major contributors to production, especially in spring blooms. Dinoﬂagellates have a single cell and are smaller than diatoms. They drift with the water by using ﬂagella to move. Diatoms do not have swimming capability and they passively follow the ﬂow. Therefore, dinoﬂagellates with swimming capability apparently seem to have an advantage. However, since consuming energy by swimming and a bodily organs become complicated, dinoﬂagellates are inferior in multiplication capability compared with diatoms. So, it cannot be said that dinoﬂagellate has advantageous for survival compared with diatom. The amount of phytoplankton is maintained under a struggle for survival in environmental change.6,7 All higher order biota in the ecosystem can use stably the organic matter which phytoplankton produces. Since nutrients, such as nitrogen and phosphorus, are especially abundantly supplied from a river to bays, much phytoplankton can live and it enables other various aquatic biota to become abundant through preying on this. For this reason, enclosed bays serve as remarkable area where the commercial ﬁsh and shell ﬁsh production is high, although the area is small compared with an open ocean. 32.2.2. Eutrophication and red tide Nutrients are essential substances and support an abundance of primary production and a range of species of bays. However, as excess nutrient input fosters phytoplankton blooms, transparency will decrease and various eutrophication phenomena will also be generated.8 The term “eutrophic” means well nourished; thus, “eutrophication” refers to the natural or artiﬁcial addition of nutrients to water bodies and to the eﬀects of added nutrients.9 The reduction of transparency diminishes seaweed stock. Moreover, some limited phytoplankton species which were not seen conventionally increase extensively. Then, ﬁsh stocks change from commercial species to other species.3 As a result, these bring oxygen deﬁcits in bottom waters of many areas and ultimately the death of much aquatic life. It will lead to various serious damages to ﬁshing, commerce, and recreation. This adverse eﬀect chain is often found in shallow coastal bays having low freshwater inﬂow and restricted tidal ranges. However, even larger estuaries have fallen victim to hypoxia and anoxia, resulting in a series of undesirable events. In many cases, the beginning of a eutrophication phenomenon is the coloring phenomenon of the sea water which is produced by the excess increase and accumulation of phytoplankton. This phenomenon is called red tide.10 The color of the sea changes, but red tide is not necessarily red. Sometimes it is dark reddish-brown to brown. Therefore, red tide is deﬁned not by sea colors but by concentration of chlorophyll a. These deﬁnitions diﬀer among bays, for example, red tide is deﬁned for chlorophyll a concentration higher than 50 µg/l in Tokyo Bay. Chlorophyll a is the photosynthetic pigment of phytoplankton and there are many kinds of other chlorophyll, for example, chlorophyll b and chlorophyll c.

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Chlorophyll a is used as an index of the amount of phytoplankton since all phytoplankton hold it. Usually, even if phytoplankton increases considerably to the level of red tide, other organisms do not suﬀer harm by itself. However, there are some species which have a bad inﬂuence on the human body or on marine organisms, and are called “harmful algae.” An excess increase of these species is called as harmful algal bloom (HAB).11 When it is condensed by shellﬁsh, and it causes shellﬁsh poison; furthermore, damage extends to farmed ﬁsh, or to laver culture because of shortage of nutrients. Even if direct damage does not occur, the balance of an ecosystem will collapse due to phytoplankton increasing by red tide, and various related phenomena will occur. 32.2.3. Control factor of red tide occurrence Although prediction of standing stock biomass is generally diﬃcult due to the variation of biologic phenomenon, it is possible to predict the variations of phytoplankton to some extent from factors such as water temperature,12 light,13,14 nutrients,15 and stratiﬁcation.16,17 When the photosynthesis process by phytoplankton is simpliﬁed, it is shown as follows: 106CO2 + 106H2 O + 16NH3 + H3 PO4 → (CH2 O)106 (NH3 )16 (H3 PO4 ) + 106O2.

(32.1)

This formula represents a process which produces organic matters and oxygen from nutrients (nitrogen and phosphorus), carbon dioxide, water by using the energy of light. Figure 32.1 shows (a) the variations of wind speed and its direction, (b) water temperature at the surface, middle, and bottom, (c) daily solar radiation and precipitation, (d) chlorophyll a at the surface, (e) nitrogen at the surface, and (f) phosphorus at the surface.18 The increase in chlorophyll a occurred only at the time when salinity is low and nitrogen is suﬃcient. Thus, the increase in the phytoplankton in this area was generated when nutrients were supplied from river. However, nitrogen was exhausted with the increase in phytoplankton and became undetected after the phytoplankton increase. Phosphorus was not exhausted by phytoplankton blooms. Moreover, phosphorus increases after the middle of July when phytoplankton did not increase. Therefore, it is hard to think that phosphorus is the limiting factor of phytoplankton. Nitrogen is a limiting factor for a phytoplankton increase in this area. Under a suﬃcient nutrient condition, phytoplanktons produce their biomass with a nearly constant P:N:C ratio of 1:16:106 (by molecular ratio), where P denotes phosphorus; N, nitrogen; and C, carbon. This ratio is called the Redﬁeld ratio.19 For example, when there are 20 mol of nitrogen and 1 mol of phosphorus in sea water, 4 mol of nitrogen will remain after phytoplanktons assimilate all phosphorus. Then, they cannot increase further. This kind of nutrient is called limiting nutrient and it changes with location and time.

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Water Solar Radiation Wind Temperature (MJm-2day-1) (m/s) (°C) Chlorophyll a ( µ g/l)

(d)

Nitrogen ( µ mol/l)

(c)

0 50 100 150

Precipitation (mm/day)

(b)

Phosphate M) ) (( µ mol/l

(a)

8 0 -8 30 20 10 0 Surface 32 Middle 30 Bottom 28 26 24 22 40 30 20 10 0 30 20 NH4 4−N NO2 2−N NO33−N 10 0 2 PO -P 4 1.5 1 0.5 0 7/1 7/6 7/11 7/16 7/21 7/26 7/31 8/5 8/10 8/15 8/20 8/25 8/30

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Fig. 32.1. (a) Wind vector, (b) solar radiation and precipitation, (c) water temperature, (d) chlorophyll a, (e) nitrogen, and (f) phosphate from July to August 2001.

Figure 32.2 shows the bathymetry of the Tokyo Bay, Japan and observation sites. Figure 32.3 shows dissolved nutrients at the two stations and variations of river discharge.20 Station 1 is located near the river mouth in the west side of the Tokyo Bay. Station 2 is located at the head of the bay about 15 km from the river mouth. From these ﬁgures, the timing of the high nitrate and phosphate concentration in the surface layer coincides with increases in the river discharge. Phosphate concentration showed less temporal ﬂuctuations than nitrate and the decrease in phosphate corresponds with the increase of chlorophyll a. This is more similar at Stn. 2 than Stn. 1. Since Stn. 1 is closer to the river mouth, it has relatively small variations in the consumption by phytoplankton compared with Stn. 2. N–P ratio is estimated from these ﬁgures, and it has exceeded 16 in general, except for the seabed during summer. Hence, a limiting factor of phytoplankton is phosphorus in this area. For the nutrient limitation, we should consider not only for the N–P ratio but also the other nutrients and their absolute concentration. The limiting factor derived from the Redﬁeld ratio suggests a possibility of relative shortage, and absolute values suggest an overall nutrient restriction.

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Arakawa Kyuu Edogawa River River Edogawa River Sumidagawa River

(km) 60

Makuhari

Stn.1 50

40

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Tamagawa River Turumigawa River

Stn.2

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Ichihara

Stn.3

30 20m

N

Kanazawa 20

Yokosuka 10 40m

Kanaya

0 0

10

20

30

40 (km)

Fig. 32.2. Bathymetry of the Tokyo Bay, Japan, and its adjacent coastal area showing the location of the measurement in the bay.

For example, in case of Tokyo Bay, it is judged as the phosphorus limitations from the Redﬁeld ratio. However, phosphorus concentration is high enough in this area, and it is rare for nutrients to restrict the increase of phytoplankton. Figure 32.4 shows the relation between the chlorophyll a and solar radiation at the head of Tokyo Bay from April to October in 1999. Chlorophyll a increases clearly coincided with the increase in solar radiation. In this observation site, since nutrients suﬃciently exist, they do not become the limiting factor of phytoplankton increase. As a result, variations of phytoplankton can be mostly explained by solar radiation. 32.2.4. Oxygen dynamics in bays In an oligotrophic system which is characterized by low nutrient loading, phytoplankton productivity is nutrient limited. Supplied nutrients are rapidly consumed by phytoplankton’s uptake. Ambient nutrient concentration remains low and phytoplankton biomass is stable and controlled by grazing. As a result, the ecosystem

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Fig. 32.3. Isopleths of (a) nitrate (NO3 -N), (b) ammonia (NH4 -N), (c) phosphate (PO4 -P), (d) silica (SiO2 -Si) at Stn. 1 (left), and Stn. 2 (right). (e) River discharge of Kyu-Edogawa River (m3 /s) and (f) chlorophyll a at Stn. 2.

Solar radiation (MJ/day)

30 25 20 15 10 5 0

Stn.1 Stn.2

80 60 40

Oct\28

Oct\18

Oct\8

Sep/28

Sep/8

Sep/18

Aug/29

Aug/19

Aug/9

Jul/30

Jul/20

Jul/10

June/30

June/20

June/10

May/31

May/21

May/11

May/1

Apr/21

Apr/11

20 0 Apr/1

Chlorophyll a ( µ g/l)

120 100

Fig. 32.4. Relation between the solar radiation (top) and chlorophyll a (bottom) at the head of Tokyo Bay in 1999.

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of the bay will be stabilized, and, various ﬁsh and shellﬁsh will be stabilized and live abundantly. In contrast, once nutrient load is increased beyond the level absorbable by phytoplankton, then the aquatic environment becomes an eutrophic system, phytoplankton growth rate becomes nutrient saturated, and changes in nutrient supply can only cause relatively slow changes in phytoplankton biomass. As a result, phytoplankton biomass varies mainly with solar radiation, temperature, and the thickness of the mixed layer.21 Phytoplankton production exceeds the grazing capacity of zooplankton and large ﬂuctuations in phytoplankton biomass may occur. Phytoplankton grazers, such as copepods, whose life cycle is in the order of days to weeks, cannot respond to these quick ﬂuctuations in phytoplankton biomass, and a portion of the organic matter remains ungrazed.22–24 Subsequently, it sinks from the euphotic zone into subpycnocline waters where it is rapidly metabolized.25 Most parts of a bay does not have enough depth for degradation of phytoplankton during falling on the bottom. Subsequent sedimentation of the bloom also constitutes a major input to the benthic ecology. Even if only 1% of the organic matter produced by photosynthesis in the euphotic zone is deposited to sediment in an ocean, it reaches to 10–50% and 90–99% of the organic matter deposited on the seabed is decomposed by the microorganisms on the seabed.26 As a result, instability of plankton population has important ecologic consequences. Such accumulations of ungrazed organic material often lead to microbial-dominated food webs, which is characterized by a greater decomposition and oxygen consumption. Oxygen-depleted water occurs under stratiﬁed conditions. Higher temperature or lower salinity blanket of water overlies the heavier, lower temperature, or higher salinity water in the bay. This overlying water prevents reaeration of oxygendepleted water in the lower layer by suppressing vertical exchange. So, oxygendepleted water is induced by both a physical process and a biochemical process. The eﬀect of oxygen-depleted water to aquatic ecology had been well documented. Diaz and Rosenberg27 reported that a high mortality of benthos may occur under 1.1–2.0 mg/l dissolved oxygen. Once oxygen-depleted water occurs, ﬁsh tends to shunt, to avoid it. In contrast, shellﬁsh like oysters and clams cannot escape and will die because they have small transportation capability. As a result, a nonbenthos area will be formed in the large portion of inner-bay during the summer in Tokyo Bay. Bivalves such as oysters and clams have the important role to decrease the suspended solid. So, once they decrease, transparency will fall and seaweed will become extinct. Furthermore, when phytoplankton is decomposed on the seabed, nutrients will be emitted to the sea water again and a further increase of phytoplankton will be caused. Thus, oxygen-depleted water causes remarkable damage to the ecosystem inside the bay. Figure 32.5 shows observation results of oxygen-depleted water in Tokyo Bay during the summer in 2003. The oxygen-depleted water (oxygen concentration is less than 3 ml/l) is widely spread at the center part of inner bay. Oxygen-depleted water is transported by wind-driven currents, and changes the distribution considerably.28 This oxygen-depleted water is in the inner bay about six months from early summer to early autumn. Figure 32.5 also shows bivalves distribution in 2003 and bivalves

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Fig. 32.5. Dissolved oxygen distribution at the bottom for August 2003 (left) and benthic bivalve distribution on the seabed in November 2003 (right).

do not exist at the center part of the bay. This area is almost the same as the distributions of oxygen-depleted water.29

32.3. Influence of Circulation Processes on Water Quality and Ecosystem Water quality is inﬂuenced by a ﬂow directly and indirectly. For example, since phytoplankton drifts underwater, it is inﬂuenced strongly by the ﬂow of the bay. Motion of bay waters also exerts strong inﬂuences on the distribution of dissolved oxygen in bays. These kinds of inﬂuences are direct and obvious. In contrast, even if the ﬂow is very small, it still has some residual current, ﬂowing in one direction. In this case, nutrient loads are transported to distant locations and, probably, the water exchange rate of the bay also increases. Then a permissible nutrient loading which the bay can receive will also alter through the change of water exchange rate. In this section, we explain the role of physical processes and the relationship between water motion and nutrient cycles. 32.3.1. Dominant physical processes in bays Figure 32.6 shows the results of current measurement at the center of Tokyo Bay (Stn. 2 in Fig. 32.2). The top panel shows the water level, the second one shows wind, the next two panels show the instantaneous currents at surface and bottom layers, respectively, and the last two panels show the calculated residual current in surface and bottom layers, respectively. The residual current shown in this ﬁgure is obtained by low-pass ﬁltering from the instantaneous current data measured by the

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(a) Water level (m)

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(b) Wind (m/s)

(c) Instantaneous current surface

10m/s

20cm/s

(d) Instantaneous current bottom (e) Residual current surface (f) Residual current bottom

Oct1

Oct2

Oct3

Oct4

Oct5

Oct6

Oct7 (2005)

Fig. 32.6. Results of measurement of (a) water level, (b) wind, (c) instantaneous currents at surface, (d) bottom layers, (e) residual currents at surface, and (f) bottom layers, respectively, illustrating the dependency of instantaneous currents on tide and residual currents on wind stress at Stn. 1 in 2005.

acoustic Doppler current proﬁler (ADCP). The ADCP measures the velocity proﬁle by using Doppler shift of ultrasonic waves, and can measure the ﬂow without contacting and disturbing velocity ﬁeld.30 Its maintenance is easy. Therefore, ADCP has commonly been used to measure the ﬂow ﬁeld in recent years. From the top ﬁgure, we can see the oscillations of water level, with a period of approximately half a day. Instantaneous currents are also oscillating with the semi-diurnal or diurnal periods at the surface and bottom. This oscillation pattern of current is called the tidal current and is the most dominant component of a ﬂow in bays. The water of a bay is moving continuously by the tidal rhythms. So we need to measure currents continuously to extract the residual component. For a unidirectional ﬂow like a river ﬂow, we can estimate the discharge by using instantaneous measurement results. However, bay water moves back and forth during ebb and ﬂood tides. So we cannot determine the average movements of the bay water only by the instantaneous current. Residual currents which are extracted by ﬁltering out the tidal component from instantaneous currents are often used for discussion of a ﬂow pattern, because it is more adequate when we discuss the net transportation of substances.31 From these ﬁgures, the dependency of temporal residual currents on the wind is also clear. Since the sea surface has not so many obstacles compared with that of the land and the roughness is also smaller than the land, the sea wind is often

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twice as strong as that on land. Surface sea water is dragged by such strong winds. Then the surface water drags further the lower layer, and the ﬂow is generated in the down-wind direction. Occurrence of wind-driven currents is very incidental compared with that of tidal currents. Especially, the strong wind which blows at typhoons or hurricanes mixes bay water and has a strong impact to the water quality and ecosystems of bays. Although a ﬂow in the down-wind direction is easy to understand, density structure and rotation of the earth make it more complicated. To simplify our discussion, let us consider the rectangle bay which is very small and has a uniform depth. It is assumed that Coriolis force and stratiﬁcation eﬀect can be neglected. When the wind blows over this bay, it generates a shear stress at the sea surface. The surface water moves in response to the wind-shear stress; the water surface of the leeward side becomes higher and the windward side becomes lower (see Fig. 32.7). If the water pressures at the seabed are compared at this time, the water pressure of the leeward will become high because of the eﬀect of wind stress. As a result, a ﬂow which returns to the upwind direction occurs at the seabed. Consequently, a vertical circulation, for which the direction of the ﬂow are opposite at the surface and bottom, occurs. In this mechanism, the water near the seabed as well as the surface responds to a wind in comparatively short time, and moves to the upwind direction. When the bay water is stratiﬁed and has a pycnocline (a layer with a large density gradient), the transfer of motions is prevented. Therefore, a comparatively strong circulation is formed only in the upper layer. Moreover, after the wind stopped, the pycnocline which was inclined by the wind-induced ﬂow may vibrate, and a strong ﬂow may occur. Consequently, even when constant wind blows on a certain bay, the resulting water motions depends on the strength of stratiﬁcation and its depth.

Fig. 32.7.

Wind-induced current profile and a horizontal distribution of water level.

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One might consider that the surface water would be forced to move in the same direction as the wind. However, the water normally moves in diﬀerent direction similar to the wind when Coriolis force has signiﬁcant eﬀect. If a wind blows continuously beyond half a day in bays, the inﬂuence of the Coriolis force will become signiﬁcant. It is one of the apparent forces and produced by the earth’s rotation. It acts rightward and perpendicular to movement in the northern hemisphere, and leftward in the southern hemisphere. Therefore, if the surface water is dragged by the wind, the lower layer water will change the direction to the righthand side in the northern hemisphere, so that a spiral may be drawn in the vertical direction as shown in Fig. 32.8. This ﬂow is called Ekman transport and its vertical distribution of the ﬂow is called Ekman spiral. Consequently, sea water is transported in the direction diﬀerent from a wind direction by Ekman transport which is conﬁned within top 10–100 m of the water column. The direction is 90◦ to the right of the wind in the northern hemisphere and 90◦ to the left in the southern hemisphere. As shown in Fig. 32.8, the velocity decreases exponentially in the vertical direction. Since the velocity decays to 1/e times at z = hE , where hE is called an Ekman layer thickness and is expressed by hE = 2KZ /f , in which KZ denotes the eddy viscosity and f denotes the Coriolis parameter. Since the ﬂow direction changes with the wind direction through Ekman transport, as shown in Fig. 32.9, upwelling occurs for wind parallel to the coastline and with the coastline on the left-hand side. Conversely, if water is driven toward the coast by Ekman transport, downwelling occurs at the coast. Once upwelling occurs, bottom water which has high concentrations of nutrients and deleted with oxygen appears to the surface layer and it has large inﬂuence on the water quality and ecosystem of the bay.

Fig. 32.8. Ekman current profile for a zonal (eastward) wind in the northern hemisphere. Note the surface velocity makes the 45◦ angle with the wind vector and the turning of the current to the right of the stress vector with depth.

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Fig. 32.9. Upwelling and downwelling situations in the bay: (left) coastal downwelling induced by a wind parallel to the coast with the coast to its right (in the northern hemisphere) and (right) coastal upwelling.

32.3.2. Wind-induced flows and its influence on phytoplankton variation As mentioned before, wind-driven currents are also one of the important currents for water quality and ecosystem in bays. Figure 32.10 shows the variations of (a) wind, (b) daily solar radiation, chlorophyll a [Stn. 1 (c), Stn. 2 (d), and Stn. 3 (e)], water temperature [Stn. 1 (f), Stn. 2 (g), and Stn. 3 (h)], and salinity [Stn. 1 (i), Stn. 2 (j), and Stn. 3 (k)].32 Station 1 is located at the inner most part of east side of the bay, Stn. 2 at the center part of east side of the bay, and Stn. 3 at the west part of the inner part of Tokyo Bay (see Fig. 32.2). Chlorophyll a content varied rapidly between 0 and 100 µg/l. Chlorophyll a increased during the ﬁve periods marked by A–E in Fig. 32.10(c)– 32.10(e). The increase of chlorophyll a was observed almost simultaneously at the three stations, except at Stn. 3 during A and E. So, the general variation of phytoplankton is almost the same inside the bay. The maximum chlorophyll a occurred in period B at the three stations. For these ﬁve periods, when the north wind began to blow, the blooms stopped rapidly. This is because a strong north wind caused outward transport from the inner bay and upwelling at the east side of the bay, resulting in advection and dispersion of phytoplankton in the water column. For example, when a north wind blew, the surface water temperature at Stn. 1 fell rapidly as shown in Fig. 32.10(f) (on April 20, 25, 30, May 15, June 9, and 19). In contrast, the bottom water temperature at Stn. 2 rose sharply as shown in Fig. 32.10(g) (on April 25, 30, May 15, June 9, and 19). On the other hand, the south wind caused the contrary phenomena, and the bottom layer temperature rose sharply at Stn. 1 (on May 5, 20, 28, June 3, 11, and 24). In contrast, the surface temperature fell at Stn. 2. These responses to the wind seemed to have occurred because the two stations are located on the east and west sides of the inner bay: while downwelling occurs at one station, upwelling occurs at the other. These responses are particularly clear at Stn. 2, since it is located at the innermost part

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Fig. 32.10. Time series of (a) wind speed, (b) daily solar radiation, chlorophyll a [Stn. 1 (c), Stn. 2 (d), and Stn. 3 (e)], temperature [Stn. 1 (f), Stn. 2 (g), and Stn. 3 (h)] and salinity [Stn. 1 (i), Stn. 2 (j), Stn. 3 (k)], from April to June 1999. The wind speed is smoothed by a 6-h running mean.

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of the bay. Variations of chlorophyll a at Stn. 3 resembled that of Stn. 2, as these stations are located on the same side. Thus, wind-driven currents are one of the dominant factors aﬀecting the change in phytoplanktons in bays. Phytoplanktons decreased temporarily by upwelling. However, they also transport nutrients from the bottom water to the surface at the same time. Therefore, if conditions are satisﬁed, red tides, blooming of phytoplankton will occur again.17 32.3.3. Estuarine circulation and nutrient cycles Estuarine circulation is also one of the important ﬂow structures in bays. Figure 32.11 shows a schematic diagram of estuarine circulation. River water ﬂows toward the mouth of the bay after being emitted from the river mouth, and it spreads across the surface like a veil since the river water has small density compared with bay’s saline water. To cancel the density diﬀerence between the river water and the saline water, much sea water is drawn in river water. It continues until river water becomes the same density as the surrounding sea water and, as a result, a vertical circulation is formed in bays. This circulation ﬂux is several to 10 times greater than the river ﬂux.33 Thus, estuarine circulation induces outward currents on a surface, and inward currents near the bottom. The current is small compared with the tidal currents as explained previously. Since estuarine circulations are in the ﬁxed direction, its material transport eﬀect is very large over a long time scale in spite of its small current speed. Estuarine circulation also plays an important role in nutrients cycles in stratiﬁed bays. The organic matters deposit on the seabed after phytoplankton blooms or the river runs oﬀ. It is decomposed by bacteria in the seabed. These nutrients are supplied from the seabed under anoxic condition in summer. Seasonal variations in nutrient concentrations in bottom water are consistent with these phenomena, indicating that high concentrations of phosphate and ammonium in bottom water (Fig. 32.3). ←

Ocean

River Mouth Low Saline Water

Halocline Entrainment High Saline Water

Fig. 32.11.

Cross-section view of two-layer estuarine circulation in a bay.

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These nutrients that are supplied from the seabed during summer are transported to the head of the bay by the estuarine circulation. Furthermore, they are again uptaken into the upper water, and are used for the increase in the phytoplanktons in the euphotic zone. Estuarine circulation plays a key role in the nutrient cycles of stratiﬁed estuaries by increasing the retention, recycling the nutrients, and stabilizing the ecosystem.8 32.3.4. Blue tide When oxygen-depleted water is formed at the seabed, a high concentration of nutrients and hydrogen sulﬁde will be accumulated in oxygen-depleted water by bacterial and chemical processes.34 Once upwelling occurs by wind-induced ﬂow, a hydrogen sulﬁde and oxygen react on the sea surface causing a sea surface discolored milky blue.35 This phenomenon is called blue tide and it is often seen in late autumn in Tokyo Bay after strong southward wind.36 Since blue tide is induced by the seabed water which is hypoxic or anoxic, shellﬁsh in the shallow water region may be faced with a shortage of oxygen, and this may cause serious damage for the ecosystem. Moreover, since blue tide can happen in a broad area, it causes serious damage for ﬁshes. As seen so far, red tide, oxygen-depleted water, and blue tide are related mutually. Once red tide occurs under eutrophic condition, oxygen-depleted water may occur at the seabed, and nutrients and sulﬁde are supplied from sediment. Moreover, when the nutrient-rich water is transported by an upwelling, nutrients from the seabed are supplied to a euphotic layer, and the occurrences of mass mortality by oxygen shortage during blue tide result in further increase of nutrients. This process accelerates red tide, and ﬁnally induces a negative spiral of ecosystems of the bay. Therefore, it is necessary to understand each phenomenon and the relationships among these phenomena to restore the water quality and ecosystem.

32.4. Numeric Simulations of Enclosed Bays The phenomena in bays are not only physical phenomena but also biologic and chemical ones, and they are related to each other. So, the ecosystem and water quality of bays are highly complicated. To deal with these complex systems quantitatively, numeric modeling has been developed since 1960s along with developments of computers. Modeling is also important for applications such as understanding and prediction of bays’ water quality and ecosystems. A water quality model in bays is comprised of a three-dimensional circulation model and an ecosystem model that describes pelagic and benthic aspect of nutrients cycling. A physical model solves essentially Navier–Stokes equation with the forcing (the wind stress, Coriolis force, and buoyancy force) under adequate approximations. In order to include the density eﬀects, conservation equations for temperature and salinity are also solved. In order to obtain a realistic prediction for vertical stratiﬁcation, turbulent closure model is also employed.38 As a result, the gravitational, wind-driven, and topographically induced ﬂows can be reproduced within

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physical numeric models. Although many three-dimensional hydrodynamic models are developed in this decade. POM (princeton ocean model),39 CH3D (curvilinear hydrodynamics in three dimensions),40,41 and ROMS (Regional Ocean Modeling System)42,43 are widely used. Wind-driven tides and estuarine circulation are inﬂuenced by geometry and bathymetry, whereas natural bays are rarely regular in shape. Therefore, a computational grid is required to ﬁt to a natural geometry and bathymetry more accurately. For this reason, the governing equations are not diﬀerent among models, but grid systems were altered along with the progress of models. The simplest horizontal computational grid is the rectangular grid with ﬁxed spacing. This is extended to a nested grid system in which the ﬁner grids are used in regions to yield detailed information. Recently, curvilinear coordinate systems are used widely. These systems allow greater ﬂexibility than rectangular grid systems. Figure 32.12 is an example of horizontal curvilinear coordinate systems. The Chesapeake Bay has a typical complex geometry as well as other bays, and thus a horizontal curvilinear coordinate system is advantageous.43

Fig. 32.12. (a) Bathymetry of the Chesapeake Bay and its adjacent coastal area. Depths are in meters. (b) A horizontal curvilinear coordinate system designed for resolving the complex coastlines and the deep channel in the bay (from Ref. 37).

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Fig. 32.13. An example of vertical grid system: (a) a Cartesian or z-coordinate vertical grid and (b) sigma-coordinate vertical grid system.

For vertical coordinate systems (shown in Fig. 32.13), a Cartesian grid (zcoordinate vertical grid) and a sigma-coordinate grid are widely used. The Cartesian grid is equivalent to the rectangular horizontal grid. However, it is sometimes more accurate than the sigma-coordinate, featuring the presence of steep bottom topography. The sigma-coordinate system is convenient in that it can introduce an essentially “ﬂatting out” the variable bottom at z = −h(x, y). The sigma-coordinate system has long been used, in both meteorology and oceanography.44,45 Various kinds of models are also proposed for an ecosystem model46–50 ; it also solves conservation equations for relevant components with appropriate source and sink terms. If C is the concentration of some component in a model, the time rate of variation of C is given by: dC = Fin − Fout + R dt

(32.2)

where Fin and Fout are the ﬂuxes into and out of the target control volume which is calculated by using the physical model results, and R is the net increase or decrease due to internal production and removal of biogeochemical eﬀect. These models provide a quantitative description of the inﬂuences of physical circulation on biologic and chemical processes in bays. Figure 32.14 shows schematic interactions of a lower trophic ecosystem model which is used in Tokyo Bay.51 This model has 13 state variables: phytoplankton, zooplankton, nutrients (nitrogen, phosphorus, and silica), labile detritus, and refractory detritus for each nutrients and dissolved oxygen as well as sedimentation process of particulate organic material. For example, certain labile compounds which are rapidly degraded, such as sugars and amino acids in the particulate organic matter deposited on the sediment surface are decomposed readily; others such as cellulose are more refractory, or resistant to decomposition. Figure 32.15 shows the calculation results of an annual budget of nitrogen and phosphorus in Tokyo Bay. The

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Detritus N,P,Si NO3-N

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Fig. 32.14. Idealized nutrients cycling in Tokyo Bay’s ecosystem in the model of Koibuchi et al.51 Cycling between the 13 state variables: phytoplanktons, zooplanktons, nutrients (nitrogen, phosphorus, and silicate), labile detritus, and refractory detritus for each nutrients and dissolved oxygen as well as sedimentation processes of particulate organic material.

Fig. 32.15. Summary of fluxes and process rates calculated in Tokyo Bay, January 1999 to January 2000. Units are given in ton per year for each element.

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annual budget is useful in the understanding of nutrient cycles. Nitrogen is supplied to considerable degree from rivers, since atmospheric nitrogen input is signiﬁcant around urban area. Phytoplanktons uptake the nitrogen and sink to bottom waters, where they are decomposed by heterotrophic processes which consume oxygen. At the head of the bay (between lines 1 and 2 in Fig. 32.15), about 40% of nitrogen sunk as a detritus and 20% of it is lost by denitriﬁcation into the atmosphere. Ammonia released from sediment reaches 20%. About 60% of nitrogen load ﬂows out from the bay. In contrast, the atmospheric phosphorus input to the bay is negligible compared to the contribution from other sources. Phosphate released from the sediment is of the same amount as that discharged from the rivers, and is transported to the head of the bay by estuarine circulation. As a result, phosphate in the inner bay remains high. In contrast, nitrogen is mainly supplied from the river mouths and transported quickly out of the bay. In conclusion, the nitrogen and phosphorus showed important diﬀerences in the mechanisms of cycling in the bay. Nutrients regeneration and release from the sediment is an important source for phytoplankton growth and equal to the contributions from rivers. Especially, phosphorus is largely retained within the system through recycling between sediment and water. These results denote the diﬃculty of improvement of eutrophication in bays only by the construction of sewage treatment plants.

32.5. Conclusion This chapter dealt with the variation mechanisms of water quality and an ecosystem under typical currents in enclosed bays. Since enclosed bays are the boundary domain of land and open seas, these are complicated phenomena. In order to understand the recycling of nutrients for integrated management of bays ecosystem, advanced interdisciplinary research is needed. Over the last half of the century, constructions of a sewage treatment plant has been performed to reduce nutrients load for the maintenance of water quality of bays. However, there is a limitation in nutrient removal in a sewage treatment plant by the present technology. In order to stop negative spiral of eutrophication phenomena, an onsite countermeasure also needs to be carried out. Enclosed bays forming the edge of our world in many ways are next to our life and are not as so vast as open seas. So, if our understanding of phenomena of bays can be deepened, we will have a potential to control or redesign them and therefore move toward a new, more stable, and attractive environment.

References 1. H. J. Walker, The Coastal Zone. In the Earth as Transformed by Human Action. Local and Regional Changes in the Biosphere over the Past 300 Years, eds. B. L. Turner I, W. C. Clark and R. W. Kates (Cambridge University Press, Cambridge, 1990). 2. I. Valiela, Marine Ecological Processes (Springer, NewYork, 1995). 3. G. Ærtebjerg, J. H. Andersen and O. S. Hansen, Nutrients and Eutrophication in Danish Marine Waters (Ministry of the Environment, National Environmental Research Institute, 2003).

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4. T. R. Parsons, M. Takahashi and B. Hargrave, Biological Oceanographic Processes, (Butterworth-Heinemamm Limited, 1984). 5. E. S. Nielsen, Marine Photosynthesis with Special Emphasis on the Ecological Aspects, Elsevier Oceanography Series (Elsevier, Amsterdam, 1975). 6. G. T. Evans, Limnol. Oceanogr. 33, 1027 (1988). 7. H. Nomura, Oceanogr. Jap. 7(3), 159 (1998). 8. T. C. Malone, L. H. Crocker, S. E. Pike and B. W. Wender, Mar. Ecol. Prog. Ser. 48, 235 (1988). 9. B. B. Jøgensen and K. Richardson, Eutrophication in Coastal Marine Ecosystem (A.G.U., 1996). 10. J. L. Maclean, Marine Pollution Bulletin 20(7), 304 (1989). 11. G. M. Hallegraeﬀ, Phycologia 32(2), 79 (1993). 12. R. W. Eppley, Fish. Bull. 70, 1063 (1972). 13. K. S. Baker and R. Frouin, Limnol. Oceanogr. 32, 1370 (1987). 14. J. H. Steele, Limnol. Oceanogr. 7, 137 (1962). 15. S. E. Jørgensen, Handbook of Environmental Data and Ecological Parameters (Elsevier, 1979). 16. H. U. Sverdrup, J. Cons. Explor. Mer. 18, 287 (1953). 17. Y. Koibuchi, M. Gomyo, J. Sasaki and M. Isobe, Proc. Coast. Eng. 47, 1071 (2000). 18. Y. Koibuchi, Coast. Eng. VII, 101 (2005). 19. A. C. Redﬁeld, Deep Sea Res. 2(Suppl.), 115 (1955). 20. Y. Koibuchi, J. Sasaki and M. Isobe, Proc. Coast. Eng. 47, 1066 (2000). 21. K. Furuya, K. Takahashi and H. Iizumi, J. Oceanogr. 49, 459 (1993). 22. R. Gaudy, Mar. Biol. 25, 125–141 (1974). 23. H. Liu, M. J. Dagg and S. Strom, J. Plankton Res. 27, 647–662 (2005). 24. A. Tsuda and H. Sugisaki, Mar. Biol. 120, 203 (1994). 25. M. Dagg, Deep Sea Res. 40, 1431–1445 (1993). 26. A. Waite, P. K. Bienfang and P. J. Harrison, Mar. Biol. 114, 131 (1992). 27. R. J. Diaz and R. Rosenberg, Oceanography and Marine Biology: An Annual Review, 33, 245 (1995). 28. T. Kuramoto and K. Nakata, Bull. Coast. Oceanogr. 28, 140 (1991). 29. M. Ishii and Y. Shoji, Bull. Chiba Prefect. Fish. Res. Cent. 4, 35 (2005). 30. G. F. Appell, P. D. Bass and M. A. Metcalf, J. Oceanic Eng. OE-16, 390 (1991). 31. J. Candela, R. C. Beardsley and R. Limeburner, J. Geophys. Res. 97, 767 (1992). 32. Y. Koibuchi and M. Isobe, Coast. Eng. J. 49(4), 461 (2007). 33. S. Unoki, J. Oceanogr. 7(5), 283 (1998). 34. E. Suess, Nature 288(20), 260 (1980). 35. Y. Koibuchi and M. Isobe, Ann. J. Coast. Eng. 52, 896 (2005). 36. M. Matsuyama, K. Touma and A. Ohwaki, Bull. Coast. Oceanogr. 28(1), 63 (1990). 37. M. Li, L. Zhong and W. C. Boicourt, J. Geophys. Res. 110, C12004 (2005), doi: 10.1029/2004JC002585. 38. G. L. Mellor and T. Yamada, Rev. Geophys. 20(4), 851 (1982). 39. A. F. Blumberg and G. L. Mellow, A coastal ocean numerical model, Proc. Int. Symp. Mathematical Modeling of Estuarine Physics, eds. J. Sunderman and K. P. Holz (Springer-Verlag, 1980). 40. B. H. Johnson, K. W. Kim, R. E. Heath, N. N. Hseish and H. L. Butler, J. Hydraul. Eng. 119, 2 (1993). 41. H. V. Wang and B. J. Johnson, Water Qual. Ecosyst. Model. 1, 51 (2000). 42. P. MacCready, R. D. Hetland and W. R. Geyer, Cont. Shelf Res. 22, 1591 (2002). 43. M. Li, L. Zhong and W. C. Boicourt, J. Geophys. Res. 110, C12004 (2005). 44. N. A. Phillips, J. Meteorology 14(2), 184 (1957).

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45. N. G. Freeman, A. M. Hale and M. B. Danard, J. Geophys. Res. 77(6), 1050 (1972). 46. J. N. Kremer and S. W. Nixon, A Coastal Marine Ecosystem: Simulation and Analysis, Ecological Studies (Springer-Verlag, Heidelberg, 1978). 47. M. J. R. Fasham, H. W. Ducklow and S. M. Mckelvie, J. Mar. Res. 48, 591 (1990). 48. F. Chai, R. C. Dugdale, T. H. Peng, F. P. Wilkerson and R. T. Barber, Deep-Sea Res. II49, 2713–2745 (2002). 49. M. J. Kishi, M. Kashiwai, D. M. Ware, B. A. Megrey, D. L. Eslinger, F. E. Werner, M. Noguchi-Aita, T. Azumaya, M. Fujii, S. Hashimoto, D. Huang, H. Iizumi, Y. Ishida, S. Kang, G. A. Kantakov, H.-C. Kim, K. Komatsu, V. V. Navrotsky, S. L. Smith, K. Tadokoro, A. Tsuda, O. Yamamura, Y. Yamanaka, K. Yokouchi, N. Yoshie, J. Zhang, Y. I. Zenko and V. I. Zvalinsky, Ecol. Model. 202, 12 (2007). 50. B. deYoung, M. Heath, F. E. Werner, F. Chai, B. A. Megrey and P. Monfray, Science 304, 1463–1466 (2004). 51. Y. Koibuchi, J. Sasaki and M. Isobe, Proc. Coast. Eng. 48, 1076 (2001).

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Chapter 33

Socioeconomic and Environmental Risk in Coastal and Ocean Engineering Miguel A. Losada∗, Asunci´on Baquerizo†, Miguel Ortega-S´ anchez‡ § and Juan M. Santiago Grupo de Din´ amica de Flujos Ambientales CEAMA - University of Granada, Avda. del Mediterr´ aneo s/n 18006 Granada, Spain ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Elena S´ anchez-Badorrey Departamento de Mec´ anica de las Estructuras e Ingenier´ıa Hidr´ aulica University of Granada, Granada, Spain [email protected] It is broadly admitted socioeconomic progress is exhausting the energy, water, and coastal zone resources. To overcome this trend, a more serious and rigorous slogan must drive the progress today: (1) socioeconomic and environmental progress must be concomitant, (2) use of basic resources must be minimized, and (3) operationality and safety of the human works must be maximized. These new demands ask for a new coastal and ocean engineering philosophy, regarding the socioeconomic and environmental impact of the human intervention during its useful life. However, coastal and ocean engineering must deal with the environmental events and their random nature. Thus, the response to the problem has to include the associated uncertainty, among others, to the occurrence of the atmospheric and maritime agents and to the response of the systems. In this chapter, a summary of some of the new design principles and tools that can help to match the society demands are presented. Based on risk analysis and decision theory, the problem of an integrated coastal and harbor management is formulated. The new approach is applied to evaluate the probability distribution of the coastline in V years in a stretch of coast in the south of Spain. Next, the † Corresponding

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evaluation of the risk during the ship passage in the harbor entrance channel is considered. Again, based on risk analysis and decision theory, the problem is formulated and solved to evaluate the stoppage probability along the entrance channel to the harbor of Motril of a bulkcarrier during a storm.

33.1. Introduction Throughout history man’s relationship with the sea has been both intense and fraught with diﬃculty. In this sense the sea had been and still is a source of immense richness and great potential for mankind. It oﬀers maritime routes that from the beginning of time have fomented commerce and communication between civilizations. On the less positive side, it has brought illnesses and epidemics, and opened the door for marauders and invasions. In this way the coast has played an important role in the history, culture, and economy of countries with coastal boundaries. Until the middle of the 20th century this relation was symbiotic. The impacts produced in the majority of cases were either reversible or absorbed by the sea. In recent decades, anarchic and unsustainable use of the sea and its resources has becoming the rule rather than the exception, something that is causing the progressive deterioration of the nearshore system. The most evident consequences of the increase in activities and uses in such a limited space are: (1) intensive land use, (2) increase of loads and pressure on the coastal ecologic trace, (3) massive transfers to the sea, and (4) decrease of the biodiversity in both mediums. The socioeconomic progress of the Western civilization began its growth after the Second World War. At that time the slogan was “development must not stick at nothing.” In the early 70s of the last century, the ﬁrst alarms arrived with the ﬁrst oil crisis. The response was a new doctrine: development in view of the oil (energy) price. Fifteen years later arrive the next alarm, the environment which halted the progress: we are killing the planet earth. The introduction of environmental studies to forecast the environmental damage and, just in case, to develop a new area in exchange for a damaged area, proportionated a solution to proceed forward. However, data obtained during the ﬁrst years of this century conﬁrms that the actual socioeconomic progress is exhausting the energy, water, and coastal zone resources. A more serious and rigorous slogan must drive the progress today: (1) socioeconomic and environmental progress must be concomitant, (2) use of basic resources must be minimized, and (3) operationality and safety of the human works must be maximized. These new demands ask for a new coastal and ocean engineering philosophy based on new design principles, considering the socioeconomic and environmental impact of the human intervention during its useful life. However, coastal and ocean engineering must deal with the environmental events, and their random nature. Thus, the response to the problem has to include the associated uncertainty, among others, to the occurrence of the atmospheric and maritime agents. One method of eﬀectively dealing with this situation is through Integral Coastal Zone and Harbor Management (ICZ&HM), based on a rational decision-making process in which economic, social, and environmental factors are considered.

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In this chapter, a summary of some of the new design principles and tools that can help to match the society demands are presented. It is only the start, but it will be a long way before the maritime works are optimized and their uncertainty bounded. The chapter is organized as follows. First, the problem of an integrated coastal and harbor management is formulated within a new framework. Next, the morphologic evolution of a stretch of coast is considered,1 and the main tools developed for the risk analysis are given. The evaluation of the risk during the ship passage in the harbor entrance channel is analyzed. Again, the probabilistic analysis of the problem is solved. Two case studies are presented. Firstly, the probable evolution of the coastline in V years for a stretch of coast in the south of Spain is analyzed. Secondly, the stoppage probability along the entrance channel to the harbor of Motril of a bulkcarrier during a storm is evaluated. The chapter concludes with a discussion and some recommendations.

33.2. Formulation One of the objectives of the integral management of the coastal zone is to establish the dynamic interactions between the economy and the environment within the coastal system. While environmental conditions make a certain coastal zone attractive, the economic agents interact with the physical environment and modify it, altering the natural evolution process. This interaction may have signiﬁcant environmental and economic consequences that need to be evaluated to exploit the coastal zone in a rational and sustainable way. This situation is eﬀectively dealt with through ICZ&HM, based on a rational decision-making process in which economic, social, and environmental factors are considered. Integral management has gone through various stages. Its initial phase focused on controlling territorial uses. The following stage was more concerned with the complementary analysis of diﬀerent national jurisdictions but it is now more oriented to combine all elements aiming at generating a speciﬁcally integrated set of research documents.2 A case in point is the system established by the OCDE, based on the set of Driving Forces, Pressures, States, Impacts, and Responses (DPSIR). However, these elements have negative aspects such as: (1) they are static tools and (2) they lack well-deﬁned temporal and spatial scales and which interrelated, so it seems impossible to deduce scales of imbalances. Here, we present a tool for the management of the coastal and harbor zone. The model deals with a territorial subset of the coastal area and covers a medium- to long-term time interval (several decades). The model can be formulated mathematically and solved in a logic, sequential way in time. The model begins with an evaluation of the socioeconomic and environmental quality of the territory, the availability of resources (water, energy, land, landscape, etc.) and the speciﬁcation of the objectives to be pursued in the short, medium, and long terms, on the basis of the values of certain representative indicators. This information, which characterizes the initial scenario, is used to design management strategies.

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Next, it is hypothesized that the morphologic processes in coastal and harbor areas are cumulative in the sense that the response to a certain sea state is the initial condition, in the mathematic sense, for the next sea state. It is possible to imitate the processes during a certain period of time by using as a forcing mechanism the local wave climate simulated as a series of consecutive sea states. The model can be launched with the ﬁrst sea state and the initial condition. The cumulative forcing of the series of states during V years is then accounted for by subsequently applying the evolution model with the prediction made with the previous state as an initial condition. The resulting ﬁnal situation after V years is considered as the ﬁnal outcome of the experiment. Finally, it is possible to apply probabilistic techniques to analyze the sample space obtained with the outcomes of several repetitive numeric experiments. For every new situation, the fulﬁllment of management targets are assessed by means of legal requirements, environmental restrictions, shortage of resources, and so on. The failure events are characterized in terms of their frequency of occurrence, magnitude, and consequences. Therefore, it is possible to quantify the associated risks, information that helps to redeﬁne the management strategies in the decisionmaking process. The medium- and long-term management can therefore be formulated as an optimization problem with the constrictions that come from the socioeconomic and environmental requirements derived from the limited availability of natural resources (land, water, and energy).

33.3. Morphologic Risk in V Years Natural phenomena are not exempted from uncertainty. In particular, those processes in which climatic agents are involved inherit their stochastic character at diﬀerent temporal and spatial scales. This is the case for coastal morphodynamics where changes occurring at short, medium, or large spatial scales are clearly associated to the corresponding time scales of the forcing mechanisms.19 In the past few decades, a big eﬀort has been made to model the physical and geomorphologic processes governing the medium- and long-term natural changes of the coastal zone and its alteration by human activities. Most of the tools and concepts (e.g., HUMOR UE Project) deal with simpliﬁed geometries and stationary input conditions that can be considered as “state models” in the sense that they can only analyze the morphodynamic response to a certain climatic state deﬁned by a constant energetic level. Inherently, their application relies in the hypothesis that if the climatic state lasts for enough time, the morphology will arrive to a stable morphodynamic equilibrium that will remain until a new climatic state moves oﬀ it to another morphodynamic state. Although these models are capable of describing the main morphologic processes in the coastal zone, coastal morphology does not only respond to average climatic conditions but also to extreme events (impulses) and, therefore, in a long-term basis, the littoral morphology is the result of the sequential action of a random number of storm and nonstorm events.

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In this framework, deterministic approaches cannot cope with the inherent randomness of the processes and any predictive model for the evolution of the littoral zone over years or decades has to be based upon statistic tools capable of dealing with the uncertainty of the prognosis. To overcome this limitation, a procedure for the assessment of intrinsic uncertainty to be applied for the long-term prediction of the evolution of a certain morphologic feature driven by climatologic agents, was proposed by Payo et al.3 The methodology is based on the random simulation of climatic states and the sequential modeling of the morphologic state response and is suitable to be applied, with forecasting purposes, to morphodynamic problems evolving in time and space. The overall procedure is summarized in Fig. 33.1. 33.3.1. Monte Carlo simulation of the maritime climate For the simulation of the maritime climate it is required to know the joint probability distribution functions (pdf) of the random forcing variables involved in the process. The pdf can be estimated from a climatic database that should contain discrete information of the evolution of the variables during a long enough period of time. In order to guarantee the statistic signiﬁcance of the pdf, the records of the database should comprise at least a period of time of the same duration than the length of the climatic forcing to be reproduced. For those processes that are mainly driven by the wave action under storm conditions, the proposed methodology assumes that each storm is an independent realization or run of the same underlying stochastic process. In such a way that each year is considered as a random sequence of runs and pauses. The choice of the variables depends on the physical process that is being studied. For the simulation of each sequence of runs, the following sea state random variables are selected: • • • •

Number of storms per year, ns . Interarrival time between storms, Sc . Duration of each storm, Ds . Peak of the storm, deﬁned as the maximum signiﬁcant wave height in the storm, Hp . • Mean zero up-crossing period at the peak of the storm, Tp . • Meteorologic tide during the storm, ηs . • Astronomic tide, ηa . In addition to the choice of the level that deﬁnes the occurrence of a storm, another threshold is chosen as the minimum value of the signiﬁcant wave height for which the morphologic changes are valuable. A representative value in between those upper and lower thresholds is taken to simulate the morphologic evolution under mild climatic forcing conditions. A Monte Carlo simulation of the wave climate during the period of time that is object of study, V , measured in years (see Fig. 33.2), will provide a sample containing N possible outcomes of the forcing during V years. Each of them can be discretized as a series of sea states characterized by the values of the signiﬁcant wave height, peak period, mean water level, etc.

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Diagram of the long-term shoreline evolution model. Fig. 33.1.

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Sketch of the simulation of wind-waves-related random variables. Fig. 33.2.

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33.3.1.1. Morphologic response to V years of climatic forcing The simulation of the changes induced by the cumulative climatic action on the morphologic unit during the V years is then performed by sequentially solving the model launched with the initial morphologic situation for the ﬁrst climatic state and for the following states with the model output of the previous state. Repeating this step for every simulated wave climate, a sample with N possible morphologic situations after V years of climatic forcing is obtained. To illustrate the procedure, the long-term evolution of the shoreline will be addressed with the aid of the solution of the one-line model with time-dependent boundary conditions proposed by Payo et al.4 A brief summary of the boundary value problem is presented next. 33.3.1.2. One-line quasi-analytic solution Larson et al.5 give a comprehensive survey of analytic solutions of the one-line model available in the literature. Most of them address the simpliﬁed geometries and wave climatic conditions. Therefore, they are not always able to reproduce realistic problems. One of such limitations comes from the adoption of a constant diﬀusion coeﬃcient which presumes the alongshore uniformity of the sediment transport rates. Moreover, the sediment balance that leads to the one-line governing equation, only accounts for longitudinal transport induced by wave breaking. To include the eﬀect of other sources of sediment (e.g., artiﬁcial nourishment or river discharge) a nonhomogeneous equation has to be solved. Analytic solutions usually require permanent boundary conditions, limiting the analysis to time-independent conditions. In addition, only for a few analytic expressions of the initial beach planform is possible to ﬁnd an explicit solution of the problem. To overcome these limitations, Payo et al.4 proposed a quasi-analytic solution for small breaking angles, expressed as an expansion of the orthogonal set of functions of the Sturm–Liouville problem that arises from the resolution of the homogeneous equation by separation of variables. In a coordinate system (x, y) with the x-axis following the orientation of the shore and the y-axis pointing oﬀshore, the sediment balance in a control volume leads to the equation governing the temporal evolution of the shoreline ys (x, t): ∂ ∂ys (x, t) = ∂t ∂x

∂y (x, t) ε(x) S + w(x, t) ∂x

(33.1)

where ε is the diﬀfusion coeﬃcient and w(x, t) includes the alongshore variation of wave breaking height and breaking angle, θb , and a source/sink term, q(x, t)/D, where D is the depth of closure, w(x, t) =

∂ q(x, t) − (ε(x)θb (x)). D ∂x

(33.2)

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The boundary value problem (BVP) consists in the resolution of Eq. (33.1) in the domain x ∈ [0, l] and t ≥ 0 with the initial condition yS (x, 0) = f (x),

x ∈ [0, l]

(33.3)

and the boundary conditions ∂yS (0, t) = g(t), ∂x

t≥0

∂y (l, t) = h(t), c3 yS (l, t) + c4 S ∂x

t≥0

c1 yS (0, t) + c2

(33.4)

g(t) and h(t) describe the behavior of the shoreline at the extremes of the beach, depending on the physical restrictions imposed. If a groin is partially blocking the sediment transport, it is assumed that the volume of sand bypassing the obstacle is proportional to the distance from the tip of the groin to the depth of closure isobath. A detailed analysis of the boundary conditions is given by Payo et al.4 33.3.1.3. Statistic analysis of the sample space Let ys (x, t = 0) be the initial position of the shoreline and ys (x, t) be its orientation at any time t. Because of the stochastic nature of the climatic agents, ys (x, t) can be considered as a stochastic process that, over periods of years to decades, is controlled by the most severe energy ﬂux regime. Once a sequence of sea states representative of the climatic conditions during a certain period of time say, V years, is available, it is possible to estimate the shoreline orientation after V years of wave forcing, y V (x), by sequentially solving the BVP starting with ys (x, t = 0), and using the response to a sea state as the initial condition for the next one. Repeating the experiment N times, the sample space of N equally likely outcomes is obtained. This sample can be analyzed by applying the empiric orthogonal functions (EOFs) technique as presented next. 33.3.1.4. Principal component analysis Let us call x = (x1 , x2 , . . . , xM )T the column vector containing the M alongshore positions at which the shoreline is simulated. Each outcome of the experiment can be expressed as another vector yi = (yi1 , yi2 , . . . , yiM )T for i = 1, . . . , N , where yij = yiV (xj ), j = 1, . . . , M . The M × N matrix Y = (y1 , . . . , y N ), contains, therefore, a sample of the M -random vector that deﬁnes the shoreline position after V years:   V 1 V y1 (x ) y2V (x1 ) · · · yiV (x1 ) · · · yN (x1 )  y V (x2 ) y V (x2 ) · · · y V (x1 ) · · · y V (x2 )  2   1 i N   .. .. .. .. .. ..     . . . . . .  (33.5) Y =  y V (xj ) y V (xj ) · · · y V (xj ) · · · y V (xj )  .   1 2 i N   .. .. .. .. .. ..     . . . . . . M M 1 M V (x ) y1V (x ) y2V (x ) · · · yiV (x ) · · · yN

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The principal component analysis technique provides a set of M -dimensional vectors {e1 , . . . , ek , . . .}, k ≤ M , called EOFs that are linear combinations of the original variables. The ﬁrst principal component accounts for as much as the variance of the original variables as possible. The second one explains as much of the remaining variance and so on. Therefore, retaining the ﬁrst p components, it is possible to account for most of the variation of the original variables in such a way that Y = (y1 , . . . , y N ) can be estimated by a linear combination of these vectors, yi = µ + zi1 e1 + · · · + zip ep

i = 1, . . . , N

(33.6)

where µ = (µ1 , µ2 , . . . , µM )T is a column vector with the mean values of the shoreline position at the selected M alongshore locations, µi = E[y(xi )]

(33.7)

and the coeﬃcients aﬀecting the eigenfunctions, zik , are called the scores, that are also obtained in the analysis. Equation (33.6) suggests the approximate shoreline position after V years of climatic forcing as a linear combination of p random variables Z1 , . . . , Zp : y( x) = µ + Z1 e1 + · · · + Zp ep in such a way that the matrix Z containing the scores   z11 z12 · · · z1p  z21 z22 · · · z2p     Z= .. .. ..   ..  . . . .  zN 1

z22

(33.8)

(33.9)

· · · zN p

provides a sample of the random vector (Z1 , . . . , Zp ) which allow the estimation of the joint distribution functions of the component variables. 33.3.1.5. Prediction of the derived variables: Assessment of uncertainty Once the joint distribution f (Z1 , Z2 , . . . , Zp ) of the components of the random vector is known, the assessment of the uncertainty of any of the events deﬁned through the shoreline position can be performed, e.g., the volume of sand lost, and the maximum advance/retreat of the shore. Let L(y( x)) be a function deﬁned by means of the shoreline position. According to Eq. (33.8), L depends on the random variables Z1 , Z2 , . . . , Zp . If A is a Borel set of Rp , the probability of any event of the sample space associated with the random variable L can be calculated as follows: f (Z1 , Z2 , . . . , Zp )dZ1 dZ2 · · · dZp . (33.10) Pr(L ∈ A) = A

In order to show the potential use of the methodology, the distribution function of two random variables deﬁned in terms of the ﬁnal shoreline position is calculated next for p = 2.

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33.3.1.6. Distribution function of the random variable increment of dry beach after V years The random variable that measures the increment of area of the dry beach is deﬁned as follows: l l y V (x)dx − f (x)dx A= 0

=

l

0

l

µ dx + Z1

0

e1 dx + · · · + Zp

0

l

l

ep dx −

0

f (x)dx.

(33.11)

0

By retaining only the ﬁrst two eigenfunctions, l y( x)dx = Aµ − Ao + Z1 E1 + Z2 E2 A=

(33.12)

0

l where the initial area of the beach, Ao = 0 f (x)dx, the area of the mean beach l l dx and Ei = 0 ei dx i = 1, 2 are known values. A is a random position, Aµ = 0 µ variable that depends on Z1 and Z2 and therefore its distribution function can be obtained as follows: Pr(A ≤ a) = Pr(Z1 E1 + Z2 E2 ≤ a − Aµ + Ao )

∞

= −∞

dZ1

a−Aµ +Ao E2

E

−Z1 E1

2

−∞

f (Z1 , Z2 )dZ2 .

(33.13)

33.3.1.7. Distribution function of the random vector describing the shoreline position From Eq. (33.6) it is possible to estimate the joint distribution function of the random vector y( x) Pr(y( x) ≤ α ) = f (Z1 , Z2 )dZ1 dZ2 (33.14) Sα e

where

M Sαe = (Z1 , Z2 ) | e11 Z1 + Z2 e12 ≤ α1 − µ1 ; · · · ; eM 1 Z1 + Z2 e2 ≤ αM − µM . (33.15)

For the particular case in which αi = α, i = 1, . . . , M the probability of the variable Ymin , minimum advance/retreat of the shoreline can be evaluated: x)} ≤ α) F (α) = Pr(Ymin ≤ α) = Pr(min{y( = 1 − Pr(min {y( x)} > α) = f (Z1 , Z2 )dZ1 dZ2

(33.16) (33.17)

Sα

with

M Sα = (Z1 , Z2 ) | e11 Z1 + Z2 e12 > α − µ1 ; · · · ; eM 1 Z1 + Z2 e2 > α − µM .

(33.18)

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33.4. Risk Analysis During the Ship Passage in Entrance Channels One of the objectives in the management of entrance channels is to guarantee safety and operationally levels that minimize the risk of accidents as well as to optimize the use and exploitation of the infrastructure. In recent years the evaluation of both levels has become necessary for the following reasons: (1) the constant increase of the size of the ﬂeet; (2) the increase of maritime traﬃc and the competitiveness between harbors; (3) the high cost of construction and maintenance of the infrastructures; and (4) the increasing social sensitivity toward the environmental impact caused by accidents. Evidence of this interest can be seen in the recent revisions in safety policy carried out by PIANC and USACE.6 At the present time, there is a general lack of tools capable of accurately calculating the operational levels of entrance channels from a probabilistic point of view which is a necessary information to calculate the risk of failure (probability of occurrence times consequences) during a ship passage, a year or, the useful life of the navigation channel. In this section, a model for the calculation of the probability of failure during a ship passage is presented. Here, the “passage” is deﬁned as any access or departure of a ship from harbor areas in a speciﬁc set of environmental (sea, wind, and currents), morphologic and maneuvering conditions (e.g., ship velocity and path). In actual practice, real-time management (i.e., the refusing or acceptance of the passage base on possible violation of the navigation safety requirements) is carried out solely on the basis of the experience of pilots and ship captains. The model is based on ROM 0.07 and calculate the “stoppage probability” under a given set of environmental, morphologic, and maneuvering conditions that, in general, will have a random behavior. For that purpose: (1) the possible stoppage mechanisms of the channel and (2) a mathematic expression, called here “veriﬁcation equation,” that describes the relation between the stoppage mechanisms and the environmental, morphologic, and maneuvering random variables, are needed. The procedure is summarized in Fig. 33.3. Moreover, following the procedure of the previous section plus ship arrival– departure simulator, the proposed model can be used to evaluate the passage probability during a year and calculate the harbor operational cost due to the navigation channel, and the goodness of the channel design.

33.4.1. Temporal and spatial state scales in one-way entrance channels during the passage In analogy to the “sea state” concept the “passage state scales” are introduced. The “state scales” of a passage are space and timescales in which the behavior of the signiﬁcant random variables can be assumed, respectively, homogeneous and stationary from a statistic point of view. The time-state scale is represented by tst and the length-state scale by lst .

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Fig. 33.3. Scheme of the probabilistic model and methodology for the estimation of safety depth requirements.

33.4.1.1. Temporal-passage state scales In the case of entrance channels, the deﬁnition of the time-state scale depends on the local variability of the environmental, morphologic, and maneuvering variables. Assuming that the: (1) meteorologic conditions are deﬁned by a sea state with duration Tstsea , (2) passage occurs during a water column state with duration Tstwc , (3) bed morphology is deﬁned by a stationary morphologic state within a duration Tstmor, (4) maneuvering conditions and ship response are stationary during a period of time Tstship called “maneuvering state”,

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then the duration of the time-state scale tst can be deﬁned as: tst = min Tstsea , Tstwc , Tstmor , Tstship , TC .

(33.19)

TC denotes the total duration of the passage. Typically, Tstship ≤ TC ≤ Tstsea ≤ Tstwc ≤ Tstmor . Thus, the time-state scale used to be determined by the maneuvering conditions, and not by the environmental or morphologic ones. Consequently, ship tst ∼ = Tst . 33.4.1.2. Space-passage state scale If, similarly to the time-state scale, we assume that at each location of the channel it is possible to ﬁnd a homogeneous: (1) (2) (3) (4)

sea state within a distance Lsea st , water column state within a distance Lwc st , morphologic state within a length Lmor , st maneuvering state within a distance Lship st ,

then the space-state scale lst can be deﬁned as: ship wc mor lst = min Lsea st , Lst , Lst , Lst , LC .

(33.20)

≤ Lsea ≤ Lwc LC denotes the total length of the channel. Generally, Lship st st st ≤ mor Lst ≤ LC . Thus, the space-state scale is determined by the maneuvering conditions and ship response to the climate agents. Consequently, it can be deﬁned as ship ship ship ship . lst ∼ = Lst . Note that Lst = f Tst , V Then, in order to calculate the stoppage probability along an entrance channel, the passage is spatially divided into “subsets” and temporally into “navigation states.” 33.4.1.3. Entrance channel subsets A channel “subset” is deﬁned as the area in which the statistic homogeneity of the signiﬁcant random variables can be assumed. In practice, this condition is applied in its weak form.8 According to the state scales deﬁnition, the typical scale in which the homogeneity of the signiﬁcant random variables can be assumed is the local state scale lst . Besides the homogeneity condition, the other two statistic requirements for the subset deﬁnition are that the set of subsets should be complete, exhaustive and mutually exclusive, and the statistic independence of the subsets. 33.4.1.4. Entrance channel navigation states A “navigation state” is deﬁned as the time interval in which, for a given subset, the signiﬁcant random variables can be assumed statistically stationary. The duration of each navigation state is represented by tns and can be calculated as: tns = min(tst , τ )

(33.21)

where τ is the average permanence time of the ship in the subset, and tst is the local time-state scale.

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Taking into account that τ is deﬁned as the ratio between the subset length ship , we found that, in and the average velocity of the ship in the subset, τ = lsp /V ship ship general, τ ∼ = tst because lst ∼ = Lst and tst ∼ = Tst . Hence, usually, the navigation state duration is determined by the maneuvering conditions. Thus, each subset can be divided into a set of navigation states. Besides the stationary condition, a complete, exhaustive, and mutually exclusive set of navigation states concurrent with the passage through the subset can be deﬁned as the union of possible: (1) water column states, (2) sea states, (3) morphologic states, and (4) maneuvering states during the passage. Arranging this information, for a passage under known climatic, morphologic, and maneuvering conditions, a complete set of navigation states in a subset can be deﬁned by means of: (1) A water column state characterized by the average local mean water column level and typical deviation. (2) A sea state characterized by the state descriptors (i.e., mean wind velocity and direction, characteristic wave height, and period and direction) and probability functions of the signiﬁcant climatic agents. (3) A set of morphologic states characterized by state descriptors (i.e., mean bed slope, and mean height of bed morphology), and their spatial probability distributions. (4) A set of maneuvering states characterized by state descriptors and/or probability distributions of the maneuvering conditions (i.e., mean velocity, direction, trim, path, and human factor) and of the signiﬁcant ship movements. 33.4.1.5. Predominant stoppage mechanisms The predominant stoppage mechanisms for one-way entrance channels are: insuﬃcient “under-keel clearance,” related with the touching bottom failure mode, and insuﬃcient “way-mark clearance,” related with the failure caused by partial or total exceedence of the channel way marks. Following the ROM 0.0 methodology,7 in each subset and navigation state, the safety requirements of each of the stoppage mechanisms are deﬁned as a safety margin relation (or veriﬁcation equation) between the relevant random variables of the system. The terms of the veriﬁcation equations are averaged in a characteristic timescale of the subset. The “under-keel clearance” is deﬁned as the minimum distance between the keel of the ship and the channel bottom. In terms of a safety margin relation, the veriﬁcation equation reads, 0 SK = hwc − dship − SK

(33.22)

where hwc represents the sum of all the environmental and morphologic parameters and variables that may induce a change in the water column height; dship is the sum of all the parameters and variables that may contribute to the “actual” ship 0 draught and, SK represents an extra safety margin term in which we include the model uncertainty contribution to the under-keel clearance.9 The “way-mark clearance” is deﬁned as the minimum distance between the hull of the ship and the borders (way-marks) of the entrance channel. As in the previous

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case, the veriﬁcation equation of the way-mark clearance is deﬁned in terms of a safety margin equation as: 0 SW = wc − ws − SW

(33.23)

where wc represents the minimum distance between the entrance channel borders and the center of mass of the ship hull; ws is the sum of all the random variables and parameters that may contribute to the variation of the minimum distance between 0 represents an extra security the ship and the “way-marks” of the channel, and Sw margin due to model uncertainty. The model considers two “natural” veriﬁcation approaches for the stoppage domain calculation: (1) A “clearance state approach” in which the terms of the veriﬁcation equations are averaged in a characteristic timescale of the subset, tSC . (2) An “instantaneous clearance approach” in which the average timescale is equal to a characteristic timescale of the short-time random variables. The instantaneous clearance timescale is represented by tIC . The instantaneous clearance approach is only strictly necessary when the safety requirements are not fulﬁlled in the state clearance approach, or when the levels of use and exploitation of the channel have to be optimized. 33.4.1.6. Stoppage probability during the passage Once the passage is spatially divided into subsets and temporarily into navigation states, the overall stoppage probability P passage (i.e., stoppage probability for the entire passage) can be calculated by making use of the statistic independence of the subsets as: P passage = 1 −

M

(1 − P m )

(33.24)

m=1

where P m is the total stoppage probability in the subset CSm , and M is equal to the total number of subsets. The probability P m can be expressed as: M m m P m = PK + PW − PK∩W

(33.25)

m m is the stoppage probability related to the under-keel stoppage mode; PW where PK m is the stoppage probability related to the way-mark stoppage mode; and PK∩W is the joint probability of both stoppage mechanisms. Any of the above probabilities (henceforth Pim ) can be calculated as:

Pim =

Nm P Sim ≤ 0N Snm · P N Snm

(33.26)

n=1

where P [Sim ≤ 0|N Snm ] represents the stoppage probability of the stoppage mechanism i in the subset CSm conditioned to the occurrence of the navigation state N Snm ; P [N Snm ] is the probability of the navigation state during the passage through

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Nm m the subset CSm . Note that, n=1 P [NS n ] = 1 according to the completeness requirement of the navigation state deﬁnition. Finally, Nm is the total number of possible navigation states during the passage through the subset. In the state clearance approach, the probability P [Sim ≤ 0|NS m n ] is calculated as the probability associated to the stoppage domain of the state clearance approxim mation of the veriﬁcation equation Sim (i.e., Sim ∼ ) during the environmental, = SIC,i morphologic, and maneuvering conditions deﬁned by the navigation state NS nm . Assuming the statistic independence of both approaches (based on the condition tIC tSC ), it can be expressed as: m m m m m P Sim ≤ 0NS m n ≈ P SIC,i ≤ 0 SSC,i ≤ 0, NS n · P SSC,i ≤ 0 NS n .

(33.27)

The probabilities deﬁned by Eqs. (33.26) and (33.27), can be calculated by integrating the joint probability function of the signiﬁcant random variables using Monte Carlo or FORM numeric methods (ﬁrst order reliability methods). A description of these methods can be found in Refs. 11 and 12. 33.4.1.7. Safety criteria for one-way entrance channels The safety criteria for passages along one-way entrance channels can be speciﬁed by one of the following conditions: P passage ≤ Pmax P

m

≤ Pmax

(33.28) with m = 1, . . . , M

(33.29)

where Pmax represents the maximum stoppage probability admissible for the stoppage mechanism under consideration. The choice of the Pmax value depends: (1) on the management point of view adopted (e.g., that of harbor pilots, ship captains, and harbor authorities); and as recommended by Rom 0.0,7 (2) on the economic, social, and environmental impact of the possible occurrence of an accident (i.e., on the operational nature of the channel.13 33.4.1.8. Risk analysis for the design Following the same procedure as described for the morphologic risk of V years of climate forcing, the ship passage model can be applied for the optimization of the channel design and management. Once the environmental climate is simulated, the arrival–departure of the boats during useful life is further simulated based on the exploitation forecasting (number of refusing and acceptance of the passage) on the basis of possible violation of the navigation safety requirements are computed. This is the sample needed to calculate the frequency of stoppage of the design entrance channel and duration of the waiting. Moreover, the consequences of the stoppage can be evaluated and the risk computed. Repeating the experiment M times, the probability function of the variables and the risk value can be obtained. The ﬁnal decision should be taken considering the total cost (construction and maintenance

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and eventually reparation) of the entrance channel versus probability. The use of Bayesian or assimilation techniques data during the exploitation of the entrance channel will help to facilitate the adjustment of parameters of the probabilistic models and the reduction of the uncertainty.

33.5. Case Study 1: La Barrosa Beach As a management application of the proposed methodology, in this section, the erosion problem of a stretch of the coast is addressed. The case study is Playa de la Barrosa, a ﬁne sand (D50 = 0.35 mm), 6-km-long beach in C´ adiz (Spain) facing the Atlantic Ocean with an average alignment N335. The beach is limited at its northernmost extreme by Torres Bermejas Cliﬀ, and it ends at the change of alignment occurring at Punta del Frailecillo. At this location, the erosion problem is more severe and it has aﬀected the toe of the ﬁrst dune ridge (Fig. 33.4). One of the proposed alternatives to prevent the erosion is to build at Punta del Frailecillo, a 150-m-long groin from the toe of the dune, perpendicular to the beach mean alignment. The methodology proposed is applied for the analysis of the eﬀect after 50 years of climatic forcing, of the construction of the groin. For the application of the quasi-analytic one-line model, Torres Bermejas is modeled as a 120-m-long groin barrier from the toe of the dune. The groins are

Fig. 33.4.

Location map and aerial view of La Barrosa beach.

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Definition sketch of the reference frame and initial position.

partially blocking the alongshore sediment transport that is assumed to take place over the entire beach proﬁle up to the depth of closure (Fig. 33.5). 33.5.1. Climatic database The joint distribution functions of the random variables involved in the process were obtained from climatic data of the Gulf of C´ adiz, at a hindcasting point called WANA1055045 (latitude, 36 25; longitude, −6.25.), provided by Puertos del Estado (http://www.puertos.es), WANA Project. This point is located at approximately 450 m depth and covers a period of time from 1970 to 1995. Every 3 h, the following data among other, are provided at each location: • Time of the prediction. • Signiﬁcant wave height estimated from the zero order moment of the spectrum, Hm0 . • Peak period, Tp . • Mean wave direction, θ. This wave prediction model includes refraction, waves growth due to the wind, dissipation due to bottom friction and wave breaking, and nonlinear interactions. The grid used in the region of interest is a ﬁnite diﬀerences grid with a spatial resolution of 0.25◦ . The input data of WAM model14 are wind ﬁelds at 10-m-height coming from HIRLAM model run at the Spanish National Meteorological Institute every 6 h with a spatial resolution of 0.5◦ . 33.5.1.1. Wave climate The dominant climatologic actions occur during the pass of low pressure systems traveling from west to east generating a gradual temporal of the atmospheric pressure, wind, and wind waves. The most probable storm track is the SW and the net annual ﬂux of wave energy along the single cell is almost aligned to the normal to the shoreline. Moreover, some low pressure systems tracks along higher latitudes. In these cases, usually only swell waves arrive at the coast. To analyze the extreme sea-state regime, a POT (peaks over threshold) analysis was performed with the sea state curves, deﬁning a storm event as that formed by the consecutive sequences of sea states in which the signiﬁcant wave height exceeds the threshold value of Hm0 = 2 m. The mean number of storms per year was calulated

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Fig. 33.6.

Joint probability density function of Hm0 and Tp .

and gave a value around λ = 9. For every storm event, the duration of the storm, the maximum value of Hm0 achieved and the corresponding values of the mean zero upcrossing, Tz , and the mean direction, θ were calculated. Figure 33.6 shows that over the threshold value, the most probable sea state can be swell with Tz = 12–14 s or sea with Tz = 8–10 s. 33.5.2. Application results Wave climatic conditions during a period of time of V = 50 years are obtained with a Monte Carlo simulation technique, by using the distribution functions of the random variables involved in the proccess. Once these variables were simulated, the climatic conditions during the V years were divided into a series of sea states of 3 h. For every sea state, the numeric code REFDIF15 was used to propagate a monochromatic wave train with the values of the signiﬁcant wave height, the mean zero upcrossing period, and the mean direction of propagation of the state. The result of wave propagation allows to obtain the position of the breaker line and the wave angle at breaking along the shore, therefore providing the functions ε(x), θb (x) that are needed to solve the quasi-analytic oneline model. This approach overcomes the problem of over-prediction of the diﬀusion coeﬃcient in the classic formulation as pointed out by Falqu´es.16 The procedure is used for the simulation of the cumulative response of the shoreline to the sequence of storms after V = 50 years. The experiment is repeated

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Fig. 33.7. Initial shoreline position (dashed line) and some final shoreline positions after 50 years of climate forcing.

Fig. 33.8.

Fig. 33.9.

First four eigenfunctions.

Explained variance as a function of the number of components retained.

N = 200 times. Figure 33.7 shows some of the outcomes of the experiment that were used for the analysis. Figure 33.8 shows the ﬁrst four eigenfunctions and Fig. 33.9 represents the variance explained by retaining the terms up to each of them. It can be appreciated that with the ﬁrst two ones it is possible to account for more than 99% of the total variability.

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Fig. 33.10.

Plot of pairs of values of the scores of the first three eigenfunctions.

Fig. 33.11.

Joint density function of the variables Z1 and Z2 .

Figure 33.10 plots the pairs of scores of every two eigenfunctions for the ﬁrst three ones. The variables Z1 and Z2 are strongly correlated and therefore, are not independent. It is possible, however, to infer their joint density functions from the samples of values of the scores as shown in Fig. 33.11. The distribution function of the random variable that measures the increment of area of the dry beach, is shown in Fig. 33.11. Figure 33.12 shows the distribution function of Ymin where the integral in Eq. (33.17) is estimated with a Monte Carlo technique.

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Fig. 33.12. in V years.

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Probability distribution function of the random variable increment of dry beach

33.6. Case Study 2: Motril Navigation Channel The Harbor of Motril is located on the SE coast of Spain (Mediterranean sea) and consists of a main dock protected from wave action by the mound breakwater (850 m length) on the west side of the harbor. To improve the harbor operativity, the existing dock was enlarged by lengthening the sloping breakwater 580 m and dredging the dock area to 11.5 m depth. The typology of the new breakwater is vertical impermeable. The principal entrance channel is NE oriented. The total length of the channel is 3,000 m. Only the last 600 m of the channel are dredged. The nominal channel width and depth in this area are 100 and 11.5 m, respectively. The harbor location and NE entrance path described above are plotted in the upper part of Fig. 33.13. Below, a picture of the present harbor conﬁguration is shown. In the image, the vertical typology of the enlarged breakwater can be distinguished.18

33.6.1. Passage description Most of the maritime traﬃc (around 56%) of the Motril harbor are handled by Bulkcarriers of 10.000 TPM, length (140 m), draught (8.5 m), and beam (20 m). In this case study the passage of a Bulkcarrier during the occurrence of an SW storm event in low water level (LWL) conditions is considered. The starting point of the passage has been located 2,800 m far from the enlarged breakwater’s head. The vessel tries to follow the NE path plotted in Fig. 33.14. The mean passage duration is Tc = 17.5 min. The passage sea state IS characterized, in deepwater, by a narrow band Jonswap spectrum with signiﬁcant wave height Hs = 3.8 m, peak period Tp = 9 s, and shape factor γ = 3.3. The mean sea state duration is Tstsea = 1.8 h. As TC Tstsea , the statistic descriptors of the sea state can be considered stationary during the passage. The mean wave direction is SW; no directional dispersion is considered. The agitation conditions along the passage have been calculated by means of a modiﬁed mild slope linear wave propagation model. Wave reﬂection from bathymetry (up to second order in bed slope) and coastal or harbor structures is

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Fig. 33.13.

Motril harbor: location, current configuration, and NE entrance channel.

Fig. 33.14. Stoppage probability during the passage for the “ideal” reflective breakwater case (KR = 1): state clearance, instantaneous clearance approaches, and real time solution. passage passage passage = 50.3%; PIC = 42.4%; and PRT = 42.7%. PSC

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included. The old mound breakwater has been characterized by a reﬂection coefﬁcient KR = 0.2. The coastline has been supposed fully dissipative. To study the inﬂuence of the enlarged vertical breakwater reﬂectivity on the stoppage probability along the entrance channel, we have assigned two “ideal” reﬂectivities to the vertical breakwater: (1) total reﬂection with reﬂection coeﬃcient KR = 1 and (2) full dissipation with KR = 0. Mean wind and current velocities are considered negligible during the passage. Their possible eﬀects on the maneuvering conditions and ship response have been included as model uncertainties. The nominal value of the local depth along the channel was obtained from bathymetric data in LWL. The relative error in the depth measurements is 5%. The morphologic state during the storm event is considered stationary (i.e., Tstmor TC ). The presence of small-scale bedforms and wave-induced morphology has been included in the under-keel veriﬁcation equation through the Gaussian ηmor term. 33.6.2. Modeling of ship passage Accurate modeling of maneuvering conditions in real environments is quite diﬃcult. Proof of that is the continuous eﬀort on dynamic ship response and vessel control modeling. Although some commercial models give a deterministic answer to the problem, the stochastic description of the maneuvering conditions (ship velocity, path, etc.), including the inﬂuence of the human factor and propulsion systems is not, to the authors’ knowledge, completely solved. Here, a crude description of the maneuvering conditions based on the empiric models described in the Spanish recommendations for navigation ways10 and deterministic maneuvering models are adopted. This choice does not have inﬂuence in the overall methodology. The mean vessel velocities during the passage are calculated with a numeric fast-time simulation program for ship maneuvering. Nominal values of trim and squat are also calculated by applying10 empiric models. With respect to the passage duration, all the above-mentioned random variables have medium-time variability scales. Their statistic behavior has been supposed Gaussian. The possible deviation from the mean path has been considered also a medium-time scale and Gaussian distributed variable with zero mean and standard deviation B/2. The short-time scale ship response is calculated with a spectral, linear model including damping, restoring, and forcing terms. The inﬂuence of the human factor on the maneuvering conditions is included through: (1) the ship velocity uncertainty, (2) the vessel position uncertainty, and (3) the maneuvering under-keel clearance v correction term, Mmax . In general, this term can be expressed as a function of the local under-keel clearance, SK : 0 , 0 if SK < SK v (33.30) = Mmax 0 f (SK ) if SK ≥ SK . As a ﬁrst choice, the maximum stoppage probability is set equal to Pmax = 10%. From Fig. 33.14, it is concluded that the passage under consideration cannot be considered safe for any of the management criteria. Consequently, after checking it with the harbor pilot, the access should be denied.

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33.6.3. Influence of breakwater reflectivity on the passage In the last years, the classic rubble-mound breakwater used traditionally in the Mediterranean countries is being changed to vertical breakwater with apparently lower cost and less cost environmental impact. However, wave reﬂection and agitation conditions along entrance channels may be enhanced by vertical breakwaters, as well as formation of bars around the structure.17,18 This is the case of Motril harbor. The wave climate in two control points of the mean NE path is shown. For the vertical breakwater, wave agitation conditions are signiﬁcantly increased (up to 35% in P2 and 48% in P1), in Fig. 33.15. Furthermore, the ratio of the local and incident wave heights is plotted. The increase of the agitation conditions due to the almost normal incident of wind waves on the breakwater is maximum in this case (locally, h/hin ∼ = 2). The interference (egg box pattern) between the incident and reﬂected wave trains is signiﬁcant particularly in the last 1,500 m of the entrance channel (i.e., about 15 wavelengths). As a consequence, the stoppage probability along the passage for the reﬂective breakwater case is signiﬁcantly increased. The overall stoppage probability in the reﬂective case is P passage = 42.4%. For the mound breakwater it reduces to P passage = 25.6%. The increase in the stoppage probability has the following consequences: (1) Decrease of the usage and operativity levels of the entrance channel. (2) Increase of the depth requirements to obtain the same level of operativity. Following the procedure shown in Fig. 33.3, the depth requirements for reﬂective and mound breakwaters have been calculated. In Fig. 33.16, the mean and 95% upper conﬁdence bands of the current bathymetry and the estimated depth requirements for the current passage are compared. The results show that, for the passage considered, the volume of dredging for the reﬂective breakwater case may be up to 42% higher than for the mound breakwater. Both aspects, and their impact on the economical and environmental cost of entrance channels construction and dredging maintenance, should be taken into account in the entrance channel and breakwater design. The application of present methodology can help to ﬁnd the optimum design.

33.7. Discussion The validity and ﬁeld of application of the overall methodology and each of the steps are questioned in this section: (1) Data processing. The analysis is based on the observation that a signiﬁcant change in coastal morphology and ocean engineering is only valuable when the energy of the sea state exceeds a certain level. This hypothesis allows the analysis of wave climate to be performed from a POT regime. The value of the threshold has to be based on expert judgment, and several values were tested.

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Fig. 33.15. Agitation conditions along the Motril entrance route. (a) Deepwater and entrance channel wave climate for the mound and reflective breakwater cases. (b) Ratio between local and incident wave height along the mean entrance path (detail). Sea-state model as monochromatic wave with Tp = 9 s, Hs = 3.8 m.

(2) Data uncertainty. The simulation of the climatic forcing starts with the joint distribution functions of the random variables involved in the process which are themselves based on the surrogated data. To this respect, the validity of the model results depends on the length and quality of the climatic database. As previously mentioned, for the climatic temporal variability at the diﬀerent timescales to be assimilated into the models, the time series should comprise a period of time at least equal to the period to be simulated. Fortunately, hindcast techniques provide long time series, over 40 years, of climatic statistic descriptors (i.e., HIPOCAS UE project).

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Fig. 33.16. Safety depth requirements along Motril entrance channel. Mean values and 95% upper confidence bands of current entrance channel bathymetry and safety depth requirements (for the passage considered). Reflective breakwater case versus mound breakwater case.

(3) Sample representativity. Additionally, the size of the sample simulation has to be large enough to guarantee the establishment of the results, particularly at the tails of the distribution functions. In this context, it would mean that the joint probability density function of the random vector which components are the scores of the eigenfunctions, are required to be accurate. To this end, it would be necessary to repeat the experiment until establishment is reached. (4) The proposed methodology consists of four sequential steps that include: data processing, simulation of the forcing and the response of the system, EOF analysis, and assessment of the uncertainty of associated events after a given period of time. The procedure has a general character in the sense that it can be applied to many physical processes evolving in time and space. It is also not dependent on the particular tools employed in each step, provided that all the information necessary for the following steps can be obtained from the previous ones. (5) The synthetic database of the forcing mechanism (wave climate) can also be obtained from any simulation process, like autoregressive models. Moreover, the data processing and simulation can be replaced by any technique capable of providing a sample of the diﬀerent responses of the system. In the case of shoreline related problems, a sample with the position of the shoreline measured from video images would also be feasible for a shorter-term analysis.19 In relation to the one-line and ship movement model, they are just two of the

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available methods for the analysis of shoreline and ship response to a climatic forcing, but many other tools can be used. (6) Finally, the EOF technique can be replaced by any of the methods available in the literature for the analysis of structure in data sets.

References 1. A. Baquerizo and M. A. Losada, Human interaction with large scale coastal morphological evolution. An assessment of the uncertainty, Coast. Eng. 55, 569–580 (2008), doi: 10.1016/j.coastaleng.2007.10.004. 2. B. Cicin-Sain and R. W. Knetch, Integrated Coastal and Ocean Management: Concepts and Practices (Island Press, Washington, D.C., 1989). 3. A. Payo, A. Baquerizo and M. A. Losada, One-line model with time dependent boundary conditions, ed. J. M. Smith, Proc. 28th Int. Conf. Coast. Eng. ICCE, ASCE, Vol. 3, World Scientiﬁc (2002), pp. 3046–3052. 4. A. Payo, A. Baquerizo and M. A. Losada, Uncertainty assessment of long term shoreline prediction, ed. J. M. Smith, Proc. 29th Int. Conf. Coast. Eng. ICCE, ASCE, Vol. 16, World Scientiﬁc (2005), pp. 2087–2096. 5. M. Larson, H. Hanson and N. C. Kraus, Analytical solutions of the one-line model of shoreline change near coastal structures, J. Waterw. Port Coast. Ocean Eng. 123(4), 180–191 (1997). 6. W. O. Gray, J. Waters, A. Blume and C. Landsburg, Channel design and vessel manoeuvrability — Next steps, International Workshop on Channel Design and Vessel Manuverability (2001). 7. ROM 0.0, General procedure and requirements in the design of harbor and maritime structures, Part I (Ministerio de Fomento, Spain, 2001). 8. D. E. Newland, An Introduction to Random Vibrations and Spectral Analysis, 1st edn. (Longman Group Limited, London, 1975). 9. S. Haver and T. Moan, On some uncertainties related to the short term stochastic modelling of ocean waves, App. Ocean Res. 5(2), 93–108 (1983). 10. ROM 3.1.99, Proyecto de configuraci´ on mar´ıtima de los puertos; Canales de acceso y areas de flotaci´ ´ on (Ministerio de Fomento, Spain, 1999) (in Spanish). 11. O. Ditlevsen and H. O. Madsen, Structural Reliability Methods (Wiley, Chichester, NY, 1996). 12. E. Castillo, M. A. Losada, R. M´ınguez, C. Castillo and A. Baquerizo, An optimal engineering design method that combines safety factors and failure probabilities: Application to rubble mound breakwaters, J. Waterw. Port Coast. Ocean Eng. 130(2), 77–88 (2004). 13. M. A. Losada and M. I. Benedicto, Target deseign levels for maritime structures, J. Waterw. Port Coast. Ocean Eng. 131(4), 171–180 (2005). 14. H. G¨ unther, S. Hasselman and P. A. E. M. Janssen, Wamodel Cycle 4, Technical Report No. 4, Deutsches KlimaRechenZentrum, Germany (1991). 15. J. T. Kirby and R. A. Dalrymple, An approximate model for nonlinear dispersion in monochromatic wave propagation models, Coast. Eng. 9, 545–561 (1986). 16. A. Falqu´es, On the diﬀusivity in coastline dynamics, Geophys. Res. Lett. 30(21), 211910.1029/2003GL017760 (2003). 17. A. Baquerizo and M. A. Losada, Longitudinal current induced by oblique waves along coastal structures, Coast. Eng. 35, 211–230 (1998).

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18. E. S´ anchez-Badorrey, M. A. Losada and J. Rodero, Sediment transport patterns in front of reﬂective structures under wind wave-dominated conditions, Coast. Eng. 55, 685–700 (2008), doi: 10.1016/j.coastaleng.2007.10.004. 19. M. Ortega-S´ anchez, M. A. Losada and A. Baquerizo, On the development of largescale cuspate features on a semi-reﬂective beach: Carchuna beach, Southern Spain, Marine Geol. 198, 209–223 (2003).

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Chapter 34

Utilization of the Coastal Area Hwung-Hweng Hwung Tainan Hydraulics Laboratory National Cheng Kung University, Tainan 70101, Taiwan [email protected] The history of human utilization of the coastal area can be traced to hundreds years ago, when they consumed the natural resources through fishing and pasturing. In the previous century, human beings built harbors and dikes to prevent storms and waves. When it came to the past decades, large-scaled reclamation lands were developed for the purposes of relaxation, aquaculture, and industry. Consequently, a large part of ecologic environment in the coastal area was seriously sabotaged. Therefore, this chapter attempts to categorize and explain different types of utilization around the world, to thoroughly review the impacts caused by local constructions and other factors, and to scheme applicable programs for response and reparation. Last, the monitoring data of land subsidence and ecologic transformation in the newly developed industrial area in western Taiwan serves as good lessons to the ocean engineers that they should work with other specialists of environment, ecology landscape, and management to solve the related problems. Only by doing this, the abundant resources in the coastal area could be preserved and utilized in a sustainable way.

34.1. Introduction The coastal area is the entity of land and water approximate to both terrain and sea. According to the US Commission on Marine Science, Engineering, and Resources (1969), the deﬁnition of the coastal area is “the part of the land aﬀected by its proximity to the sea, and that part of the sea aﬀected by its proximity to the land as the extent to which man’s land-based activities have a measurable inﬂuence on water chemistry and marine ecology.” Many researches on marine biologic resource have demonstrated that abundant ecologic resources, including planktons, algae, and nutritional salt, are contained in the coastal area of natural wetland, river mouth area, and lagoon. These resources not only serve as the food to aquatic animals and zoobenthos, but also provide functions like water puriﬁcation, preventing wave and sand drift and are equipped with landscape value. This area is a very sensitive and crucial part in the process of ecologic evolution of the earth. 953

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Earlier utilization of the coastal area was out of human instincts and was based on its natural condition. A natural gulf with perfect shield from violent storms in the outer sea often served as a port for ships looking for protection. Fishery was also restricted to the coastal area. Van de Ven1 mentioned in his Man-Made Lowlands: History of Water Management and Land Reclamation in the Netherlands that in the centuries after 800, the Dutch became the decisive factor in the formation and deformation of the land. The peat areas behind the coast were reclaimed by artiﬁcial drainage, and in the newly reclaimed peat areas, both arable farming and animal husbandry were practiced. That is the earliest recorded utilization of coastal area by humankind with engineering. Under their special circumstances, the Dutch have invested enormous eﬀorts in the development of coastal area and never stopped their ﬁghting with the nature until this century. Other cases of human utilization of the coastal area can be found in the History and Heritage of Coastal Engineering, edited by Kraus,2 which documented the history of coastal engineering and development of 15 industrialized countries. Before the mid-20th century, the utilization of the coastal area was restricted to its natural conditions. However, after the Second World War, coastal engineering doubled its speed of advancement and evolved into multiple applications. Wiegel et al.3 reported that it was not until the late 1960s that the environmental awareness began to develop in the USA and nurtured the constitution of The Federal Coastal Zone Management Act in 1972, which served as a legal basis of administration on the utilization and development of the coastal area. After the Second World War, in the mid-20th century, the world economy accelerated, the number of population boosted, and the needs of lands were consequently desperate. Therefore, human beings began to expand their activities and construction to the shores. The over-exploitation ﬁnally led to a vital damage to the coastal ecosystem. As a consequence, in the 1970s, urbanized countries worldwide launch a series of regulations to control the development, utilization, and management of the coastal area. Based on the principle of sustainable access to the coastal area, this chapter intends to organize the technological engineering and the experiences that ever been adopted by countries worldwide in the utilization of the coastal area to work as a reference for the future entree into the coastal area.

34.2. The Categories of Coastal Utilization Because of diﬀerent coastal conditions, needs of lands, requirements for economic development, and the viewpoints toward ecologic protection, the utilization of coastal area, the land, and the sea, are diﬀerent from country to country. Combining the articles published in international journals in related ﬁelds, utilization of coastal area could be sorted into the following categories: 34.2.1. Special protective zone of coastal area Many advanced countries with environmental consciousness, such as the USA, Canada, Britain, the EU, Japan, Australia, and Taiwan, have set up laws for the

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development, utilization, and management of the coastal area, including protection of their speciﬁc ecologic environment. In these nations, special zones were proclaimed for the sustainable development of the region. 34.2.2. Shipping and delivering Ninety percent of the international cargos were delivered by ships. The coastal areas undoubtedly became the major ports of global navigation channels, and the lands approximate to the sea turned out to be competitive operational spaces for cargo transference and storage. 34.2.3. Harbor construction Countries build harbors for merchant ships, ﬁshery boats, touring yachts, industrial vessels, and military ﬂeet. According to their functions, harbors could be classiﬁed as commercial harbors, ﬁshery harbors, yacht harbors, industrial harbors, and military harbors. Construction and utilization of harbors are the most signiﬁcant deploitations on the shores. 34.2.4. Aquaculture and fishery Due to a huge demand on the consumption of seafood, humans develop technology from coastal ﬁshery to inland breeding to increase the production. Japan and Taiwan are the two major countries on the lead. However, inland breeding has caused a lot of problems that gradually drove the business to the outer sea. Currently, Japan, Taiwan, Norway, Spain, and the USA have endeavored to push forward the new technology of the reproduction of coastal resources. 34.2.5. Reclamation area Large spaces of reclamation area were created in the estuary from the sediments transport of the river to the shore. These spaces were later reclaimed by the governments for the utilization of agriculture, ﬁshery, and pasturage. Recently, Japan and Taiwan governments even planned to construct the reclamation area for industrial uses. 34.2.6. Power plant and industrial park Both nuclear power stations and ﬁre power plant need huge amount of cooling water. Therefore, most of the power plants were built in the shore for an easy access to the sea water. In some countries, for a better management of industrial pollution and more convenient import and export of raw materials and products, industrial parks were established in the neighborhood of coastal area. 34.2.7. Waste water discharge Many industrialized countries discharged the processed waste water into deep sea through ocean outfalls so that they could reduce the pollution from the industrial

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and urban drainage within their borderline. As a result, many sewage treatment plants and associated factories were built along the coast for convenient operation and management. 34.2.8. Leisure and recreation Construction in the coastal area for leisure purposes includes beach facilities and tourist hotels. In these places, popular recreational water activities, such as surﬁng, underwater touring, and diving, have covered the whole coastal area from land to sea. 34.2.9. Exploration of ocean energy and resources Due to a global shortage of natural resources, many countries have turned their attention to the development of ocean energy, including the construction of tide and current power generator, wave power generator, ocean thermal energy, and conservation. The USA and Japan had developed technology for the utilization of deep ocean water two or three decades ago. Recently, Norway, Taiwan, and Korea also dedicated to the development and utilization of this kind of resource.

34.3. The Environmental Impacts The development and utilization of coastal area, because of the engineering operation, often caused impacts in the region in certain dimensions. Sometimes, the inﬂuence appeared immediately; sometimes, it took long time to be observed. Among those impacts, the change of natural environment is inevitable and often covers a wide scope of land and sea, aﬀecting ecologic, hydrologic, and geologic system and the landscape of the region. In addition, issues derived from coastal development and utilization, including inland drainage, sewage disposal, traﬃc ﬂow, and air pollution control, are crucial problems needed to be solved one by one. Moreover, some positive and negative social and economic eﬀects will also emerge at the same time during the process. Because the cause and eﬀect of coastal development and utilization cannot be evaluated in a short period, thorough investigation, exhaustive study, and long-term monitoring of the operation are necessary to every case related to the development and utilization of the coastal area. It is necessary to provide a systematic explanation on what kinds of environmental impact will be triggered by the development and utilization of the coastal area, under construction or in operation. 34.3.1. Ecologic transformation of the land Utilization of the coastal area sometimes results in a huge dimensional land arrangement or constructions, which often caused tremendous damages on the native fauna and ﬂora. It was not unusual that the equilibrium of the local ecosystem was sabotaged because of misusage of the constructive materials and the invasion

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of foreign species. This is a cross-ﬁeld issue and needs consultation from the professional specialists before the construction really takes place. 34.3.2. Regional drainage discharge and flood Huge dimensional utilization of the coastal area deﬁnitely will bring considerable inﬂuences to the rivers and drainage system in the neighborhood. Sometimes, the detour or replanning of new drainage system became necessary. Without thorough consideration and comprehensive investigation, huge dimensional coastal construction will damage the normal drainage discharge outside the developmental area, even cause ﬂoods when there are storms; this will result in countless environmental disputes. This is an important issue that should not be undervalued or ignored. 34.3.3. Discharge of waste water and environmental loading Development of any kind of coastal area will deﬁnitely lead to signiﬁcant increase of domestic sewage or industrial waste water, no matter the area is sketched for agricultural, ﬁshery, pasturage, leisure, residential, or industrial usage. Without wholedimensional planning of all kinds of waste water discharge, the environment might end up with irreparable disasters. In addition, the loading of environmental pollution should also be taken into serious investigation and analysis to retain vigilant control of the pollution emission in the developmental area. 34.3.4. The influence of shore protection structures Even in front of the invasion of storm tides, lives of human and livestock and properties were not endangered under natural circumstances. However, if the coastal area is developed without proper protective measures, widespread damages will be inﬂicted whenever storm comes. Constructions on the shore not only change its own structure, but also alter the landscape and sabotage the conﬁguration balance of the shallow water area, leading to its local transformation. As a consequence, considering the importance of shore protection, most of the countries have set up rules in the development of the coastal area to preserve its function of sheltering the inland. Recently, the ocean engineers have also learned to consider the landscape and ecosystem when exploring their constructive plans of shore protection. 34.3.5. Erosion and accumulation of coastal topography The construction of harbor, breakwater, dike, submerge breakwater, or other protection works in the coastal area often lead to unbalanced sedimentation in the district; that is, erosion on one side of the underwater structure and accumulation on the other side. Such a kind of alteration might change the ﬂow ﬁelds, produce vortex, and results in the scour of the foundation that will endanger the structure. On the other hand, due to the shortage of resources on the land, many countries use sea sands as materials for public construction and for ﬁlling the foreshores to

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create lands. Over usage of the sea sands not only changes the slope of the seaﬂoor, but also accelerates the scour of the neighboring coastal area. 34.3.6. Transformation and influence of coastal ecosystem No matter what development is conducted in the inland or shallow sea area, it will definitely cause great impact on the regional ecosystem. The drainage from households and factories will deteriorate the neighboring water and ravage the environment. If the development happens in the shallow sea, the water will be seriously disturbed and the environmental conﬁguration will be changed, which will lead to enormous transformation of the ecosystem. Plankton, phytoplankton, groundorganisms, and nutritional salt will be fatally damaged. It often takes a very long time for the devastated area to recover to its original faces. Some particular species might stop growing because of the environmental change and might just disappear forever. Therefore, a thorough investigation before the development plan is indispensable. 34.3.7. Reduction in fishery catch and the economic loss Ecologic transformation resulted from the development of coastal area inevitably will lead to a reduction in the catch of ﬁsh. The deterioration of the ﬁshery area sometimes will drive the ﬁshermen into public protests. Among all the developmental plans, the construction of power plants and ocean outfall system bring the most obvious changes to the area. Therefore, an assessment of the economic loss and compensation of the ﬁsheries should be taken into serious consideration. 34.3.8. Land subsidence of the developed area Most of the developmental coastal area need huge amount of backﬁlls. Although the geology of the region could be improved, the nonuniform land subsidence is inevitable. Because it is very diﬃcult to predict the proportion and the districts of the subsidence, it caused a lot of trouble to the public-sector engineering, especially in the design of common ducts, which decide the eﬃciency of the urban drainage system. The construction of Kansai Airport in Japan and Yu-Lin Oﬀshore Industrial Park in Taiwan served as the best examples of land subsidence in diﬀerent areas.

34.4. Strategy Because of a widespread awareness of the importance of environmental protection, intellectuals have achieved consensus on the sustainable growth. Hence, the coastal area was developed with a cautious attitude. Advanced countries, like the USA, Canada, Britain, France, Germany, Spain, Portugal, Italy, Denmark, Japan, South Africa, Austria, and Taiwan, not only made laws to control the utilization of the coastal area, but also established oﬃcial divisions to be in charge of the assessment,

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evaluation, administration, and followup of the developmental program of the coastal area. As a matter of fact, any developmental project will inevitably bring detrimental inﬂuence to the environment during the process of construction. Therefore, the utilization of the coastal area needs to be supervised with diﬀerent methods in different periods (i.e., before construction, under construction, and after construction and during the operation). On the other hand, the dimension of management and control could be further separated into institutional, legal, and technological ﬁelds. Following are illustrations of the issues and management during the process of development and utilization of the coastal area. 34.4.1. The environmental impact assessment on coastal development Most countries handled environmental impacts caused by the development of the coastal area with great cautions. Although the jurisdictions might diﬀer from country to country, the spirits of evaluation and management are the same. Here are the standards of operation of some advanced countries that could serve as reference to the assessment of the situations at the coastal area after development (Fig. 34.1). First, an environmental impact statement, including related data collected for evaluation, should be addressed before the development. After the evaluation, in the second stage, a public explanation meeting should be conducted with local residents. Finally, a monitoring program focused on the operation afterward needs to be approved. 34.4.2. The collection and investigation of environmental background information Development and utilization of the coastal area will cause certain impacts on the environment. The larger the dimension of the developmental plan, the greater the impact it will induce. These impacts could change the physical, chemical, and ecologic elements of the coastal area, transforming its landscape and deteriorate its cultural and socioeconomic functions. Hence, scrupulous environmental evaluations must be performed according to the standards of assessment laid by individual country. That is why prior collection and investigation of environment-related information is indispensable whenever there is a developmental plan to be executed. Environmental conditions — including those in the categories of oceanography, geology, hydrology, water quality, ecology, living environments, humanity and culture, and social economy — vary from country to country; so is the environmental law. Therefore, the time interval and the quantity of data assembled for evaluation also need to be adjusted according to speciﬁc circumstances. Nevertheless, environmental background information covering four seasons of a whole year is the basic requirement for the job so that a more comprehensive overview of environmental features could be well presented. Moreover, investigation in ocean and coastal area could contain the following items: (i) wave climate, (ii) current measurement, (iii) ﬂoating drogue, (iv) estuary

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The developer submits draft proposal. Researcher institutes and or consultant companies collect data, conduct initial investigation, and planning. Phase I: The developer submits environmental impact statement to Environmental Protection Administration for review.

Approve those projects that have minor influence on environment.

Disapprove those projects that are adverse to the maintenance of environmental quality.

Phase II: To hold the kiko meeting for the environmental impact evaluation. At least one year or four seasons complete investigation, research, analysis, and detailed planning. Provide environmental impact assessment report. To hold a public explanation meeting to listen to the local residents’ opinions. Review environmental impact assessment report – approval, disapproval, and conditional approval. Monitor o to and a d track t ac the t e environmental e v o e ta impact pact assessment assess e t of o the t e constructing co st uct g project. p oject. Conduct an environmental monitoring and an environmental impact assessment after completion and operation of a development activity.

Fig. 34.1. The procedures of environmental impact assessment on coastal development and utilization.

survey, (v) sediment transport, (vi) nearshore current, (vii) coastal ﬂying sand, (viii) STC proﬁle, (ix) coastal water quality, (x) ground water and water quality, (xi) regional water discharge, (xii) geologic investigation, (xiii) ocean ecology investigation, (xiv) zooplankton and phytoplankton, (xv) ecology of benthos, (xvi) heavy metal in benthos, (xvii) ﬁshing economy, (xviii) air quality, (xix) noise, (xx) traﬃc condition, and (xxi) database and GIS establishment. Complete environmental background information not only serves as the basis of impact evaluation and preliminary engineering design, but also functions as the reference to the recovering work in case the ecosystem is sabotaged. 34.4.3. The response to the environment impact during construction Engineering operation on lands or in the coastal area inevitably will disturb the local environment. Nowadays, people who are dedicated to environmental protection keep closed eyes on the issues that will cause any kind of damage to their homeland. Sometimes their caring will lead to serious protest, which might impede or even terminate

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the progress of construction. Therefore, scrupulous evaluation and planning before any developmental program is extremely important. Constructive routes should be consciously arranged not only to evade the residential and sensitive inland area, but also to detour from the ﬁshery area and dense sea ways. Therefore, the scope of damage could be reduced to its minimum and many unnecessary disputes could then be avoided. Locations that are specially sensitive to external forces need to be protected with particular attention so that immediate inﬂuence or sabotage would be reduced. The explosions of environment-related issues mentioned above sometimes evolve into storms of public demonstration. When it happens, evidences and professional discriminations are important to clarify the responsibilities. In taking preventive measure, government divisions of environmental protection had made up laws and eﬀective monitoring programs in the regulation of the frequency and location of the control and management of the construction. According to the past experiences, when disputes arouse on the responsibilities of environmental pollution, whether it occurred during the construction or produced by long-term existed sources, the cases often be ﬁled to the courtroom. Therefore, when programming constructive projects in sensitive areas, especially in the estuary, real-time monitoring should be specially emphasized besides the regular engineering process corresponding to national regulations. 34.4.4. Response to the deficiency of major construction project Defects always exist in traditional utilization of the coastal area or small-scaled developmental project. Although they might lead to some negative inﬂuence on the afterward operation, the eﬀects are temporary, mild, and easy to be amended or improved. However, major construction works in the coastal area, especially those land-making projects like Kansai Airport in Japan and Chang-Hua Industrial Park and Yun-lin Oﬀshore Industrial Park in Taiwan are developed through earthwork or land reclamation that might cause tremendous and immediate alteration in the landscape. Some impacts continue its progress and become long-term, complicated issues for correction. Such impacts include the transformation of the local ecosystem, resulting from sand-dredging, land reclamation, or replanting of foreign vegetations. Certain unique and precious species might thus been sabotaged and unable to be recovered. This is a signiﬁcant issue that requires extraordinary attention from the engineers of ocean and coastal area. On the other hand, land subsidence is a common engineering problem in the newly reclaimed area because of its loose density and inadequate bearing capacity. Therefore, soil improvement in such area should be conducted with mindful concentration. Take the “Formosa Sixth Naphtha Cracking Project” in Yun-lin Oﬀshore Industrial Park as an example (Fig. 34.2). Figure 34.2(a) demonstrated the reclamation coastal area before construction, whereas Fig. 34.2(b) revealed the industrial area eight years after the developmental work. In that area, 2,134 ha were reclaimed by 63,290,000 m3 sand, and the altitude was made up to +4.5 m from diﬀerent levels of depth. Hence, the geology must be amended to fortify its fundamental weight capacity and to prevent the liquefaction of the soil.

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SPT-N (after DC)

Fig. 34.2. “Formosa Sixth Naphtha Cracking Project” in Yun-lin Oﬀshore Industrial Park. (a) Before construction. (b) Construction completed (No. 6 Naphtha Cracking Project).

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SPT-N value (N value of standard penetration test) by dynamic consolidation method.

From the report of Formosa Plastics Group,4 there were two major important methods applied to the reinforcement of geologic structure during construction, which were “dynamic consolidation method” and “preloading method.” Test results after the practice of above two methods, SPT-N value (N value of standard penetration test) in Fig. 34.3 and CPT-Qc value (Qc value of cone penetration test) in Fig. 34.4, demonstrated remarkable improvement in the weight capacity of reclaimed land. Even the anti-liquefaction ability of the land was signiﬁcantly advanced evaluated by the standards of Seed et al.5 On September 21, 1999, a 7.3 magnitude (Richter scale) earthquake happened in central Taiwan; the site of Formosa Sixth

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CPT-Qc value (Qc value of cone penetration test) by preloaded soil.

Naphtha Cracking Project was only 50 km from the epicenter. However, not a single building in the area was damaged and no soil-liquefaction happened. The lesson told us how important the geologic reinforcement is when the coastal area is under construction. Ocean engineers need to put special emphasis on this work. 34.4.5. Response to the land subsidence Sand used for the development and construction in the coastal area are transported from inland or dredged from seabed. Under the pressure of ﬁlled soil, the land will continue to subside. Other factors like constructive activities, heavy facilities in the household or factories, and excessively-loaded vehicles also contribute a lot to the nonstop land subsidence. Reviewing the cases of Kansai Airport in Japan and Formosa Sixth Naphtha Cracking Project,6,7 reasons of land subsidence can be generalized as the following types: (i) Subsidence caused by environmental forces, including the natural consolidation phenomenon in the alluvium or similar situation appeared due to the overdrawing of underground water for the aquaculture in the near sea. (ii) Subsidence caused by construction. Loading of backﬁlls and newly erected ediﬁces are the major reasons that cause subsidence of the newly developed lands. (iii) Unexpected land subsidence, including those caused by earthquakes and plate movements. Figure 34.5 demonstrates the subsidence records of two stations in Miaoli of “Formosa Sixth Naphtha Cracking Project.” It is clear to identify that the subsidence of lands reach 22 and 27 cm from 1997 to 2005, and the process is still continuing. Land subsidence in the coastal area brings countless detrimental impacts, including (i) sabotaging the ﬂood control mechanism in the area; (ii) breaking the

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pipelines, roads, and drainage system; (iii) increasing the amount of reclamation; and (iv) producing leaks on the dike, bank, and breakwater, spoiling their functions of preventing storm waves from the outer sea, and then block the operation of the port. Therefore, when launching the utilization and development program in the coastal area, engineers need to be especially careful about the possible land subsidence caused by the construction work and need to have backup plans when such kinds of situation happened so that the damage to the public facilities and structures could be reduced to the minus. 34.4.6. Scouring around the head of the breakwater Take the construction of Yun-Lin Oﬀshore Industrial Park in Taiwan as an example. Since the completion of Miaoli industrial harbor in October 2000, the water depth at the head of breakwater have deepen to 46 m from original design of 26 m due to scouring. The stability and safety of breakwater are thus threatened. Figures 34.6 and 34.7 show the bathymetry around the head of breakwater before and after the construction. It is clear to see that the serious scouring around the head of breakwater is apparent after construction. To prevent the scouring hole from expanding and aﬀecting the safety of breakwater, Tainan Hydraulics Laboratory (THL) was entrusted to conduct a detailed ﬁeld investigation and a series of physical experiments. Based on these results, satisfactory solutions are proposed by Hwung et al.8 The analysis of ﬁeld data identiﬁes that the strong vortex induced by interaction of tidal current is the major mechanism causing this scouring. This is further conﬁrmed by numeric simulation and hydraulic model test. Moreover, to alleviate this scouring problem, it is suggested to use large sand bag ﬁlled with sand and place on the bottom of scouring area and near the head of breakwater. It turned out to be

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The contours of bathymetry before construction in Miaoli.

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The scouring area in Miaoli area gradually enlarges dated March 2002.

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an eﬀective and economic way to prevent the sea bottom from scouring. This case indicates that the construction of coastal structures will inevitably change the ﬂow ﬁeld and further generate unexpected impacts on the environment. Therefore, the success of coastal development activities hinge on the sound investigation plans. 34.4.7. Change of flow field in nearshore area As mentioned previously, the scouring hole is found near the head of breakwater in Yu-Lin oﬀshore industrial park due to the construction of Miaoli port. In addition to the immediate remediation for such phenomenon and the investigation of generating mechanism, THL also conducted a series of velocity measurements to understand and quantify how the ﬂow ﬁeld changes. According to the long-term ﬁeld data recorded by automatic record current meter Model 108 MKIII/308 distributed in Miaoli port and Sin-Sing area between 1993 and 2005, it is found that the average amplitude of current velocity decreases signiﬁcantly after the construction of west breakwater of Miaoli port. Moreover, the scattering of current amplitude is seen to be smaller and is shown in Fig. 34.8(a). On the contrary, the average current velocity tends to increase gradually since October 1999 and can be seen in Fig. 34.8(b).

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Moreover, the direction of current directs between 25◦ and 90◦ no matter if it is under high tide or low tide, as shown in Fig. 34.8(c). This is a dramatic diﬀerence before and after the construction of west breakwater. Based on the long-term data analysis, it is clearly shown that the coastal development activity will change the ﬂow ﬁeld and even cause the change of ocean characteristics. This change will further aﬀect the ocean environment and ecology. More attention should be paid on this issue. 34.4.8. Primary production change of ocean environments Large coastal development activities will inevitably inﬂuence the ocean environments and in turn bring crucial impacts on the ocean ecology and local primary production. Such ecologic variation needs long-term monitoring data to be identiﬁed. Figures 34.9 and 34.10 are some results of Tainan Hydraulics Laboratory,9 which revealed the monitoring data of zooplankton and noctiluca during September 1993 and December 2001 in Tai Shi area. It was clear that the amount of zooplankton and noctiluca remained abundant during the ﬁrst three years of construction. However, their numbers slumped after 1996 (Fig. 34.9). Similarly, quantity of both zooplankton and noctiluca decreased since the construction of Sin-Sing area. Until now, 3000

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there is no single indication of recovery. Such outcome provided vivid evidence of how great inﬂuence the developmental activities will bring to the ocean ecosystem. Investigation of species like crabs, shrimps, ﬁsh eggs, and larvae in Fig. 34.10 also shows similar situations caused by constructions. 34.5. Conclusion In the 20th century, every country is desperate for a rapid economic development. As a consequence, more and more high-tech products were produced while fewer and fewer resources were left. Energy and water resources become needed; even food has turned out to be scarce. When the resources on the lands have been utilized to their limits, people turn to look for development in the coastal area. Expansion of aquaculture industries, construction of harbors, nuclear and steam power plants, development of industrial park, recreational activities, and the technology of ocean outfalls are examples of the utilization of the coastal area. However, no matter what kinds of human activities will inevitably cause impacts to nature. Sometimes, the damage is forever and can never be retreated. In the new era, ocean and coastal engineers need to have visionary foresight and cross-ﬁelded thoughts to maintain sustainable utilization of the coastal resources.

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In achieving this goal, they must: (a) based on their professional trainings, lower the degree of compromise that might be caused to the coastal area when construction is in progress; (b) cooperate with specialists of environment, ecology, and landscape so that the utilization of the coastal area could merge with its natural surroundings; (c) keep in mind the idea of sustainable management when conducting construction programs in the costal area instead of the traditional engineering methods; and (d) ﬁle the development project with ample budget to uphold the monitoring program in the area after construction so that the transformation of each stage will be clearly documented as future reference. In the immense universe, we have only one planet to live; that is our hometown for nurturing the future generations. We, ocean and coastal engineers, must dedicate ourselves to maintaining the ecosystem of the coastal area, extenuating the impacts of construction to the natural environment and sticking to the vision of sustainable management of the coastal area. Nevertheless, one proverb we must always keep in mind: there is no such thing as free lunch. References 1. G. P. Van de Ven, Man-Made Lowlands: History of Water Management and Land Reclamation in the Netherlands, 3rd revised edn. (Uitgeverij Matrijs, 1996). 2. N. C. Kraus, History and Heritage of Coastal Engineering, Prepared under the auspices of the Coastal Engineering Research Council of the American Society of Civil Engineers (1996). 3. R. L. Wiegel and T. Saville, Jr., History of Coastal Engineering in the USA, Prepared under the auspices of the Coastal Engineering Research Council of the American Society of Civil Engineers (1996). 4. Formosa Plastics Group, The Collections of Construction Engineering on Formosa Sixth Naphtha Cracking Project: Geological Engineering Chapter, edited by the Department of Construction. Headquarter of Management, Formosa Plastics Group (2000) (in Chinese). 5. H. B. Seed, K. Tokimatsu, L. F. Harder and R. M. Chung, The influence of SPT procedures in soil liquefaction resistance evaluation, J. Geotech. Eng., ASCE, 111(12), 1425–1445 (1985). 6. H. H. Hwung and S. C. Hsiao, The utilization and impact of reclamation in Taiwan Western Coast, Asian and Pacific Coasts 2005, 4–8 September 2005, Jeju, Korea (2005), pp. 24–38. 7. H. Endo and K. Oikawa, Construction of a Large-Scale Man-Made Island for the Kansai Int. Airport. Soft Seabed Deposit, Kansai International Geotechnical Forum, Int. Exchange Committee of Japanese Society of Soil Mechanics and Foundation Engineering, Kansai, Japan (1990), pp. 50–70. 8. H. H. Hwung, K. S. Hwang and J. Y. Liou, The scouring around the head of west breakwater at Miaoli Industrial harbor, Proc. 5th Taiwan–Japan Joint Seminar on Natural Hazards Mitigation (2002), pp. 229–245. 9. Tainan Hydraulics Laboratory, Field investigation and numerical analysis for the integrated reclamation of industrial district on out-bank of Yun-Lin, Bulletin No. 330, National Cheng Kung University, Tainan, Taiwan (2004) (in Chinese).

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Chapter 35

Ocean Wave Climates: Trends and Variations Due to Earth’s Changing Climate Paul D. Komar College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, Oregon 97331, USA [email protected] Jonathan C. Allan Coastal Field Oﬃce Oregon Department of Geology and Mineral Industries 313 SW 2nd, Newport, Oregon 97365, USA [email protected] Peter Ruggiero Department of Geosciences Oregon State University Corvallis, Oregon 97331, USA [email protected] Storm intensities and the heights of ocean waves are being aﬀected by Earth’s changing climate, including the extreme waves generated by both hurricanes and extratropical storms, attributed to global warming. The increases in wave heights have been documented by buoy data collected in the North Atlantic and North Paciﬁc, with the rates of increase being important to engineering design and in the development of coastal-hazard zones. Reviews are presented about the climate controls and investigations that have documented the wave-height increases. The evidence is variously presented as progressive increases in the annual averages of the measured signiﬁcant wave heights, and decadal shifts in histograms for the full ranges of measured waves. A review is also provided of the statistical techniques available that account for these climate-induced changes in the extreme-value projections used in engineering and management applications.

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35.1. Introduction The heights, periods, and energies of storm-generated waves are the dominant environmental factors that aﬀect coasts, with their most extreme measured occurrences having resulted in the erosion of beaches and backshore properties, and damaged infrastructure such as harbor jetties and breakwaters. Important is the development of a wave climate, a documentation of the wave conditions experienced during past decades, with projections of the most extreme waves that could occur in the future, potentially posing a threat to the coast. The establishment of wave climates has been of fundamental importance to coastal engineers and scientists, the former requiring them in the design of ocean and coastal structures so they can withstand the forces of the most severe storm waves and continue to function through their desired lifetimes. The requirements of coastal scientists are similar, for example, in the establishment of coastal-hazard zones (setback lines) that will maintain homes and other developments safe from extreme storms. This places demands on the development of the wave climate, ideally that the wave data span several decades of buoy measurements or stormhindcast assessments, so that reasonably conﬁdent projections can be made of the most extreme waves expected during a century or longer (e.g., the 100-year event, that having a 1% probability of occurrence during any year). These become the design parameters in applications; their underestimation could lead to the failure of a structure, whereas an overestimation would result in construction at an unnecessarily inﬂated cost. Similarly, an underestimation of the most extreme waves and associated beach processes could result in recommended setback distances that are insuﬃcient, leading to the eventual loss of homes and other developments. The establishment of wave climates has become more of a challenge due to Earth’s changing climate, recognized mainly in the context of global warming that according to the assessments of nearly all climatologists is the result of the release of greenhouse gases by humans into the atmosphere. Most of the focus on potential coastal impacts from global warming has been the expected accelerated rates of sea-level rise as projected by reports of the International Panel on Climate Change. However, changes in the global climate can also be expected to produce an intensiﬁcation of storms, resulting in higher-generated waves and progressive shifts in wave climates toward more extreme conditions. This had already been documented in the northeast Atlantic1,2 and Paciﬁc3–5 for waves generated by extratropical storms, and for Atlantic hurricanes with increasing wave heights having been measured by buoys along the US East Coast.6,7 This necessitates that wave climates account for multi-decadal, progressive increases in storm intensities and their generated waves, including shifts in the extreme design conditions needed in engineering and coastalmanagement applications. This chapter is an expansion of that by Goda,8 which provided a basic summary of wave climates and their analysis techniques, the objective here primarily being to review the eﬀects of Earth’s changing climate and how it can be dealt with in projections of future extreme wave conditions. This chapter begins with a summary of the climate related factors that aﬀect storm intensities and wave generation, including long-term factors such as global warming and accompanying increases in

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ocean water temperatures, but also the existence of interannual events such as a major El Ni˜ no and climate cycles such as the Paciﬁc Decadal Oscillation (PDO). The presentation then turns to the available sources of wave data, and a review of the studies that have documented progressive increases in wave heights and their climate controls. Of particular interest is how these climate-controlled changes have aﬀected the coastal-wave climates, and how assessments of extreme wave conditions required in applications can account for those changes.

35.2. Earth’s Changing Climate There has been growing concern regarding Earth’s changing climate and its environmental consequences, by the public at large and by climatologists whose research has been directed toward the causes and consequences of those changes. The products of that research are of direct interest in this chapter in so far as the changing climate has aﬀected the intensities of storms, accounting for trends of increasing heights and periods of waves they have generated. Here, we provide a summary of the climate controls, followed in subsequent sections by the resulting trends of measured waves. The climate change of principal concern is the progressive global warming that has increased the numbers and intensities of hurricanes, and likely also has intensiﬁed the extratropical storms at higher latitudes. Other climate controls include variations in water temperatures and atmospheric-pressure diﬀerences that determine the strengths of the storms, and episodes of interannual climate extremes such as the development of major El Ni˜ nos in the Paciﬁc that intensify the storms and alter their tracks, temporarily aﬀecting the wave conditions along the western shores of North America. Each of these climate controls needs to be recognized for its potential eﬀects on wave climates, and how they might change in future decades. The progress of global warming is well established through worldwide measurements of atmospheric temperatures, which on average have increased over the decades, though some debate remains concerning whether this increase is due to human activities that have resulted in the release of greenhouse gases into the atmosphere, or represents some long-term natural climate cycle. Of interest to the intensities of storms and heights of the waves they generate are whether ocean-water temperatures have increased in parallel with the atmosphere. This connection has a direct consequence to the potential increases in the intensities of tropical cyclones (hurricanes in the Atlantic and eastern Paciﬁc, typhoons in the northwest Paciﬁc, and cyclones in the southwest Paciﬁc). It is well recognized that elevated water temperatures in the tropics serve as the fuel for the heat engine that is responsible for the generation and intensity of hurricanes.9 As a consequence, global warming can be expected to result in progressively greater numbers of tropical cyclones and their average intensities. It is well established that during the recent decades sea-surface temperatures over most tropical oceans have increased. In the North Atlantic the most consistent increase has taken place since about 1970, of interest in that as will be reviewed below buoy measurements of hurricane-generated waves have documented progressive increases spanning that period. Mann and Emanuel10 undertook statistical

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analyses of the North Atlantic water temperatures, ﬁnding that there is a close correspondence with the decadal-smoothed numbers of hurricanes, both having experienced dramatic increases since 1970. Trenberth and Shea11 analyzed water temperatures in the tropical Atlantic between June and October 2005, the year with record numbers of high-category hurricanes, including Katrina and Rita in the Gulf of Mexico, and found that the average temperature was 0.92◦C above the average recorded between 1901 and 1970, the 70-year period that predated the recent surge in worldwide temperatures. Of particular signiﬁcance is the study by Emanuel,12 who measured the potential destructiveness of hurricanes in the North Atlantic, quantiﬁed as the total power dissipation (TPD) of each storm that represents the integration of the magnitude of the cube of its surface wind speeds over the surface area aﬀected by the storm and spanning its lifetime. His graph of the annual variations in the total TPD from 1930 through 2002 for the annual numbers of tropical storms and hurricanes closely followed the same pattern as a curve for the sea-surface temperatures measured during September (the peak of the hurricane season). Some debate remains among climatologists as to the quality of the measurements documenting the wind speeds and atmospheric pressures within hurricanes, and how our ability to collect that data has improved over the decades.13 Of particular importance has been the use of satellites beginning in the 1970s, and there appears to be consensus among climatologists that the intensities of hurricanes have increased since the advent of that technology to monitor the storms. However, uncertainty persists prior to 1970 due to the poorer quality of the data and because the technology has evolved, which might systematically have aﬀected the stormintensity assessments. The sophisticated global-climate numerical models developed by climatologists yield a clear indication of the expected increases in hurricane intensities due to global warming, and also indicate that the intensities of extratropical storms at high latitudes in the Northern Hemisphere will increase. However, complicating those progressive trends are modes of climate variability that aﬀect both the atmosphere and ocean, a variability that can range from years to decades. The greatest source of interannual variability in the weather and climate is related to the occurrences of strong El Ni˜ nos as part of the full range of climate conditions from El Ni˜ nos to La Ni˜ nas. The primary attention has been directed toward major El Ni˜ no events such as those that occurred during 1982–1983 and 1997–1998, which resulted in more intense winter storms in the northeast Paciﬁc, whose tracks crossed the coast of California rather than having followed their normal paths at more northerly latitudes.14 The result was extreme wave conditions and property erosion along the shores of southcentral California. Enhanced erosion also occurred at the latitudes of northern California, Oregon, and Washington, due in large part to a shift in the wave directions compared with their normal wave climates, the waves arriving more from the south–southwest than normal, resulting in the northward displacement of sand on beaches and “hot spot” erosion north of headlands and tidal inlets.15–18 This illustrates that wave directions and their variations can also be an important component of wave climates. At the same time it has been shown that the occurrence of a major El Ni˜ no in the Paciﬁc results in decreased hurricane activity in the Atlantic, reducing coastal impacts along the US East and Gulf Coasts19,20 ; this

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decrease is caused by the formation of high-altitude winds that shear the tops oﬀ hurricanes before they can fully develop. An El Ni˜ no occurs at irregular intervals of approximately 2–7 years, although the average is once every 3–4 years, and generally each event persists for about a year. Other years represent “normal” conditions, or range into the development of a La Ni˜ na. This range is characterized by a number of changes in the environmental conditions, in the oceans and atmosphere, one of the most important being that the eastern equatorial Paciﬁc sea-surface temperatures (SST) are anomalously warm during an El Ni˜ no, whereas during a La Ni˜ na the water is unusually cold. This change is a response to diﬀerences in atmospheric pressures in an east–west direction along the equator through the tropics, and its eﬀect on the Trade Winds, an atmospheric variation that is described by the Southern Oscillation; this combination of ocean and atmosphere components is commonly referred to as the El Ni˜ no–Southern Oscillation or ENSO climate event. The range from El Ni˜ nos to La Ni˜ nas can be characterized by environmentally based indices, the simplest being the SST in the eastern equatorial Paciﬁc, but more advanced indices incorporate additional environmental responses. Figure 35.1 is a graph of the annual variations in the Multi-variate ENSO Index (MEI), which includes a number of responding environmental factors21 ; the more positive the value of MEI the stronger the intensity of the El Ni˜ no, with the major 1982–1983 and 1997–1998 El Ni˜ nos seen to be distinct peaks in the graph, whereas a negative value denotes the occurrence of a La Ni˜ na. It is evident in this diagram that there has been multi-decadal cycles of alternating dominance by El Ni˜ nos versus La Ni˜ nas, with more frequent occurrences and greater intensities of El Ni˜ nos from about 1976 to 1998. In contrast, the period from 1946 to 1976 was dominated by La Ni˜ nas. This

Fig. 35.1. Variations in the MEI through the 20th century, a measure of the occurrence of El Ni˜ nos versus La Ni˜ nas.

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cycle corresponds to the Paciﬁc Decadal Oscillation (PDO), deﬁned on the basis of a multi-decadal pattern of monthly SST anomalies in the North Paciﬁc poleward of 20◦ N, that is at higher latitudes than the environmental factors used to deﬁne the MEI and occurrences of El Ni˜ nos. The PDO cycle was formulated by Mantua et al.22 as a climate control on the abundance of salmon at those higher latitudes, but from the cycles of El Ni˜ nos seen in Fig. 35.1 it is apparent that the PDO also describes the variations in that climate event, and therefore the changes in tracks of storms and the waves they generate in the northeast Paciﬁc. Of interest in the context of wave climates is that the “warm” PDO phase dominated by frequent and strong El Ni˜ nos, 1976 to about 1998, corresponds approximately to the time frame of buoy measurements of waves that have served as the primary basis in the development of wave climates along the shores of the USA. From the recent apparent shift to the “cold phase” of the PDO in 1999, with the likelihood of there now being more frequent La Ni˜ nas, fewer strong El Ni˜ nos, we can expect that there will be responses in the US West Coast wave climates that could persist for the next 25 years: for example, reduced extreme wave events along the coast of California compared with the past 25 years. On the other hand, this shift in the PDO could become a factor in more frequent and intense hurricanes in the Atlantic. Another important index describing the climate of the Paciﬁc, relevant to storm intensities and wave generation, is the North Paciﬁc Index (NPI), the diﬀerence between the atmospheric pressures within the Aleutian Low and Hawaiian High. It can be expected that the greater the diﬀerence in these pressures, the stronger the Westerlies blowing at mid-latitude to high latitude across the Paciﬁc and the stronger the intensities of extratropical storms that follow tracks generally from west to east, crossing the western shores of the USA and Canada. Thus, one could anticipate that the NPI would provide some degree of prediction of the heights of waves occurring along those coasts, and as will be reviewed below this has been demonstrated to be the case. However, there is a reasonably strong correlation between the MEI and the NPI, even though their assessments incorporate environmental conditions at diﬀerent latitudes. This correlation conﬁrms the observation that the occurrence of a major El Ni˜ no has climate consequences at all latitudes, not just near the Equator, and extends beyond the area of the Paciﬁc Ocean, across the North American continent and into the Atlantic. One signiﬁcance of this correlation is that in analyses of the climate controls on storm-generated waves, there may be dependencies between measured wave heights and multiple indices, requiring an interpretation as to which has been the primary control. There is a comparable set of climate indices for the North Atlantic that potentially exert controls on the intensities of storms, both extratropical storms and hurricanes, although those relationships are not as ﬁrmly established as in the Paciﬁc and there is some debate as to their analyses and interpretations. Comparable to the NPI for the North Paciﬁc, the atmospheric pressure diﬀerence between the Iceland Low and Azores High deﬁnes the North Atlantic Index (NAI), with research having shown that an increase in this index produces higher intensity storms and the heights of the waves they generate.23 The Atlantic Multi-decadal Oscillation (AMO) is somewhat analogous to the PDO in the Paciﬁc, based in practical terms

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on the SST variability at high latitudes in the North Atlantic.24,25 Its variations are attributed to changes in water currents as part of the global ocean circulation, which carries warm surface water via the Gulf Stream into the northern latitudes of the Atlantic where it cools and sinks, returning as cold water at depth southward across the equator. The North Atlantic deepwater formation is diﬃcult to measure and to track over time, so assessments of the AMO are made in terms of the SST variability since it is related to the overturning circulation. In its warm phase, the decades when the temperature is elevated, some researchers have associated the AMO with increased hurricane activity,26 but others have disputed this correlation and even the analysis procedures used to deﬁne the AMO. Research by climatologists of extratropical storms in the North Paciﬁc has documented a progressive increase in their intensities over the decades, possibly as far back as the 1950s.27 It has been suggested that this increase is the result of global warming and higher temperatures of the ocean’s surface water, but it also has been suggested that the increase in storm intensities is aﬀected by particulate pollution in the atmosphere, originating in southeast Asia. At the same time it is clear that there is a signiﬁcant interannual to decadal variability in storm intensities and wave climates in the northeast Paciﬁc, associated with the range from major El Ni˜ nos to La Ni˜ nas that aﬀects the intensities of storms, the tracks they follow, and the latitudes at which they cross the western shore of North America. There are similar trends of increasing intensities and variability of extratropical storms in the North Atlantic, related to the climate indices reviewed above, but of particular interest in the Atlantic is the evidence that the numbers and intensities of hurricanes have increased since the 1970s, posing a major threat to the eastern shores of North and Central America. As reviewed below, these changes in Earth’s climate and its storms are reﬂected in the wave climates for the North Atlantic and Paciﬁc.

35.3. Wave Data Sources The data required for the development of a wave climate are preferably derived from programs that for decades have collected hourly or daily measurements of wave spectra, signiﬁcant wave heights based on the energy content of the spectra, and a representation of the dominant wave periods such as the peak-energy period within the wave spectra. Such programs rely mainly on the deployment of moored buoys, or recording devices attached to a ﬁxed structure such as an oﬀshore oil platform. Lacking such direct measurements, or to supplement that data if the records span too few years to establish a satisfactory wave climate, wave-hindcast techniques have been employed based on wave assessments derived from storm fetch areas, wind speeds, and the durations of the storms. Beyond these standard sources of wave data, information on long-term changes in the wave climate may be obtained from visual observations from ships, although the quality of that data is questionable. Measurements of waves along the coasts of the USA are derived primarily from buoys deployed by NOAA’s National Data Buoy Center (NDBC), and their hourly measurements are commonly used to deﬁne wave climates. Furthermore, the 25- to 35-year records available from many NDBC buoys now have suﬃcient lengths to

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permit the analyses of trends of increasing or decreasing wave heights, the presence of climate-controlled cycles, or annual variations due to events such as a major El Ni˜ no. Prior to August 1979 the data were collected at 3-h intervals, while since that time the measurements have been hourly. Otherwise, the collection methods and analysis procedures have for the most part remained the same over the decades. In 1984 NDBC changed their analysis procedures to improve the inclusion of longperiod motions in the wave spectra. The details of that change and how it aﬀected the spectra and derived signiﬁcant wave heights has been analyzed by Earle et al.28 The change is important only where there is a strong mix of swell derived from distant storms and locally-generated sea having a wide spectrum of lower wave periods, this causing the ﬁxed-hull accelerometers commonly used by NDBC to introduce spurious energy into the combined spectra. This problem was largely accounted for by analysis procedures prior to 1984, but it became apparent that further improvements were needed. Earle et al.28 produced wave spectra analyzed by both the pre- and post-1984 procedures to illustrate the potential diﬀerences, and their examples indicate that post-1984 the reported signiﬁcant wave heights could in the most extreme cases be increased by roughly up to 10%, the actual increase depending on the relative proportions of sea and swell being measured. This diﬀerence can be expected to be most important in analyses of speciﬁc stormwave events having broad spectra, but would likely have a small eﬀect on waves dominated by swell and on monthly to yearly averages needed in wave climates. In our investigations of climate-controlled trends of increasing wave heights along the US West and East Coasts, reviewed below, we did not detect any upward “steps” in the data trends in 1984 that might have resulted from a change in the analysis procedures. Few direct measurements of waves by buoys have been collected in the southern hemisphere, spanning suﬃcient lengths of time for the development of wave climates. For this reason wave hindcasts based on meteorologic records have been the primary source of data for the establishment of wave climates. This is illustrated by the hindcasts undertaken by Gorman et al.29 for the coasts of New Zealand. Their analyses employed the wave-generation model WAM (WAve Model) based on data for the daily winds across the expanse of the South Paciﬁc, the latitudes from 10◦ S near the Equator southward to the coast of Antarctica, and 100◦ E to 220◦E in longitude, New Zealand being at the center of that area. The hindcasts were made at 3-h intervals for the 20-year period from 1979 through 1998, with graphical results presented for eight sites around the New Zealand Coast. The distributions of hindcast signiﬁcant wave heights showed good agreement with data derived from satellites, although the hindcasts tended to underestimate the occurrences of the largest wave events; it is uncertain whether this diﬀerence represents an error in the satellite measurements or the hindcast results. Gorman et al.29 did not identify a statistically signiﬁcant trend of changing wave heights, which might have reﬂected a long-term climate control. The longest records of ocean wave heights are those that have been collected by visual estimates from ships, for some areas extending back into the 18th century, but generally only for major shipping routes, with global coverage being available only since about 1950. While this data is sometimes incorporated into wave climates,

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its quality remains open to question. Nevertheless, these data have proved to be valuable in studies of climate controls on variations in wave heights spanning centuries, extending into the past well beyond the analyses based on buoy measurements or wave hindcasts. An example of such analyses is that of Gulev and Grignorieva,30 of wave heights in the North Atlantic and North Paciﬁc from the 1890s to 2002, relating their long-term variations to the climate indices reviewed in the previous section.

35.4. Trends of Increasing Ocean Wave Heights Wave climates are most easily deﬁned on coasts that are dominated by one type of storm system; examples are the North Paciﬁc shores of Canada and the USA, and those of Europe, where the waves are generated by extratropical storms. Other coasts in eﬀect have two wave climates, the Atlantic shore of the USA being a prime example, with one climate consisting of waves from extratropical storms (Northeasters) and the second being waves generated by tropical cyclones (tropical storms and hurricanes). These two types of storm systems are fundamentally diﬀerent in their modes of formation, with largely separate seasons of dominance, and can be expected to have distinctive climate controls on their intensities and generated waves. They will therefore be considered separately in this section and throughout the remainder of this chapter whenever dealing with coasts that are impacted by waves having these diﬀerent origins. 35.4.1. Extratropical storm waves Extratropical storms are formed at relatively high latitudes by cold air masses moving down from subpolar regions and colliding with warmer air masses, the strongest storms developing during the winter. Considerable attention has been given to the occurrence in the recent decades of increasing wave heights in the North Atlantic generated by these storms. The earliest observations suggesting such an increase came from visual estimates and wind measurements used in wave-hindcast analyses, but the study by Carter and Draper1 in 1988 provided the ﬁrst positive documentation based on wave records obtained from a Shipborne Wave Recorder ﬁtted in the Seven Stones Light Vessel located oﬀshore from Land’s End, the southwestern tip of England. A graph of the annual averages of the measured signiﬁcant wave heights from 1962 through 1986 is shown in Fig. 35.2, having been updated by Bacon and Carter.2 The linear regression yields a slope of 0.022 m/year for the average rate of increase, having a standard error of ± 0.005; the Wilcoxon statistic ranking test was used to conﬁrm the existence of a progressive increase in wave heights. Further conﬁrmation for this increase in the North Atlantic came from the study by Bouws et al.31 in 1996, based on analyses of a collection of more than 20,000 hand-drawn wave charts produced between 1960 and 1988 by the Koninklijk Nederlands Meteorologisch Instituut in the Netherlands. Of signiﬁcance to the consistency of the results, the techniques used to prepare the charts remained essentially

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Fig. 35.2. The increase in annually averaged signiﬁcant wave heights measured by a wave recorder oﬀshore from southwestern England.2

the same throughout those 28 years, and were even undertaken by the same individuals. The charts were prepared by a combination of meteorologic analyses and direct observations of winds and waves. The resulting charts cover the entire North Atlantic, but the published analyses focused on two areas, west of Ireland and northeast of Newfoundland. For the annual averages the regression for the area west of Ireland yielded a rate of increase of 0.027 m/year, in close agreement with the measurements by the Seven Stones Light Vessel. The rate of increase in the western Atlantic near Newfoundland was similar, 0.023 m/year. However, analyses of wavebuoy data further to the south along the East Coast of the USA did not ﬁnd an increase for the winter wave heights, those generated by extratropical storms.7 A review of the storms and changing wave conditions in the North Atlantic was undertaken in the late 1990s by the European project WASA (Waves and Storms in the North Atlantic), to examine the hypothesis of worsening storms and wave climates during the 20th century.32 Their overall conclusion was that while the wave heights have increased in the recent decades as measured by the Seven Stones Light Vessel, the storms and waves earlier in the century had undergone signiﬁcant cycles on timescales of decades. It was further concluded that the presentday storm intensities and wave heights are comparable to those at the beginning of the 20th century. Their conclusions are generally supported by the study of Gulev and Grignorieva30 involving analyses of the visual assessments of wave heights from ships collected during the past two centuries. In the northeast Atlantic the annual averages of the signiﬁcant wave heights showed a distinct increase after about 1955, again in agreement with the Seven Stones Light Vessel data, but earlier in the century there were clear cycles that correlated with variations in the North Atlantic Oscillation (NAO), with the wave heights at their maximum early in the century being some 20 cm higher than at present.

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Attempts have been made to account for the measured wave-height increases in the North Atlantic since the 1960s through comparisons with potential climate controls. Bacon and Carter23 correlated the Seven Stones Light Ship data with the NAI (the atmospheric pressure gradient between the Azores High and Iceland Low), expected to be important to winds in the North Paciﬁc. The measured pressure gradients were found to have a high degree of interannual variability during the ﬁrst half of the 20th century, but beginning in about 1960 there was a distinct progressive increase with a strong correlation with the annual and monthly averages of measured wave heights. Similar increases in wave heights have been found in the northeast Paciﬁc, ﬁrst documented by Allan and Komar3–5 in 2000 with data from six NDBC buoys along the US West Coast from south-central California northward to the coast of Washington (and one buoy in the Gulf of Alaska). Figure 35.3 graphs the decadal increases in annual averages of the winter wave heights measured by the series of buoys, “winter” being taken as the months of October through March, the dominant season of signiﬁcant storms. The highest rate of increase is seen to have occurred on the Washington Coast, slightly less oﬀshore from Oregon, with northern to central California being a zone of transition having still lower rates of wave-height increases, until oﬀ the coast of south-central California there has not been a statistically signiﬁcant change in the wave heights. Figure 35.4 and Table 35.1 provide more detailed assessments of the wave-height increases oﬀ the Washington Coast by NDBC buoy #46005, the series progressively representing more extreme assessments of the measured signiﬁcant wave heights. It is evident in Fig. 35.4 that the more extreme the assessment the greater the rate of increase, with the averages of the winter waves having increased at a rate 0.032 m/year, while the annual averages of the ﬁve largest measured signiﬁcant wave heights had increased at a rate of 0.095 m/year. This progression is more evident in Table 35.1 where additional regressions are included, together with results from tests of the statistical signiﬁcance of the linear regressions. Included is an analysis of the annual averages of the signiﬁcant wave heights for the entire year, its regression yielding a rate of increase of 0.018 m/year, nearly identical to that found in the North Atlantic by the Seven Stones Light Vessel (Fig. 35.2). Also included in Table 35.1 is a column listing the ratios for the series of rates of waveheight increases to this 0.018 m/year value representing the entire year. These ratios are seen to progressively increase from 1.8 for the averages of the winter waves to 6.0 for the regression of the single maximum measured signiﬁcant wave height each year. It is seen in Table 35.1 that for the series of linear regressions, only those up through the analysis of the annual averages of the ﬁve largest signiﬁcant wave heights are statistically signiﬁcant at the 95% level. The increasing scatter of the data for still more extreme assessments is evident in Fig. 35.4, for the maximum annual measured signiﬁcant wave heights, accounting for the loss of statistical signiﬁcance based on linear regressions. However, as will be reviewed in Sec. 35.6, this lack of statistical signiﬁcance most likely is due to the nonsymmetric distribution of the most extreme wave heights, as analyses employing advanced analysis models are statistically signiﬁcant even for these annual maximum wave heights.

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Fig. 35.3. Annual averages of the “winter” signiﬁcant wave heights measured by buoys along the US Paciﬁc Coast.5

Allan and Komar5 analyzed the climate controls that might account for the increasing wave heights in the northeast Paciﬁc, but were unable to ﬁnd a correlation with the climate indices. This included the NPI, the diﬀerence in atmospheric pressures between the Aleutian Low and Hawaiian High. However, the NPI was

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Fig. 35.4. Regressions for wave heights oﬀ the coast of Washington; the more extreme the assessment, the greater the rate of increase.5

found to explain a signiﬁcant portion of the annual variability in the data above and below the regression lines, basically having accounted for part of the data scatter seen in Figs. 35.3 and 35.4. The MEI equally accounted for this variability, expected given the correlation between the NPI and MEI discussed earlier. The conclusion was that a major El Ni˜ no, as measured by MEI, tends to result in higher wave conditions all along the US West Coast, with the annual average plotting above the regression lines representing the net decadal increases. At the same time, it is recognized that a strong El Ni˜ no aﬀects the meteorology of storms even at high latitudes in the Paciﬁc, and this accounts for the correlation with the NPI.

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P. D. Komar, J. C. Allan and P. Ruggiero Table 35.1. Linear regressions for the trends of increasing wave heights measured oﬀ the Washington Coast (NDBC buoy #46005).5

Wave heights Annual average Winter average Five largest Three largest Annual maximum

Rate (m/year)

Ratio of rate to annual average

Statistic signiﬁcance*

0.018 0.032 0.095 0.103 0.108

1.0 1.8 5.3 5.7 6.0

SS SS SS NSS NSS

*SS: Statistically signiﬁcant at the 95% level; NSS: Not statistically signiﬁcant.

Although the focus of this review has been on the changing wave heights over the decades, Allan and Komar5 also documented similar increases in the wave periods measured by the US West Coast NDBC buoys, expected from the intensiﬁcation of the storms. The signiﬁcance of this is that the range of nearshore processes on beaches, such as wave runup elevations, depend on both wave heights and periods, so increased runup elevations since the 1970s would have resulted from increases in both parameters, important to coastal erosion and ﬂooding. 35.4.2. Waves generated by tropical cyclones As discussed earlier the “fuel source” for tropical cyclones is the high water temperatures found in near-Equatorial oceans, so their initial development is limited to the summer to early fall months at low latitudes, although once formed their tracks can take them to higher latitudes (e.g., hurricanes in the Atlantic have made landfalls on the coast of Nova Scotia and Newfoundland). As noted above, there is general agreement among climatologists that hurricane numbers and intensities have increased at least since the 1970s. Recent investigations have documented that the heights of waves generated by hurricanes in the North Atlantic have increased, again since the 1970s when the NDBC buoy data became available. Komar and Allan6,7 analyzed the measurements from three buoys in deepwater along the US Atlantic Coast, and one buoy in the central Gulf of Mexico; the Atlantic buoys are located oﬀshore from Cape May, New Jersey, Cape Hatteras, North Carolina, and Charleston, South Carolina. Separate analyses were undertaken for the winter season dominated by extratropical storms and the summer season of hurricanes. The hurricane-generated waves were analyzed for the months of July through September, dominated by tropical cyclones, while the waves of extratropical storms were based on records from November through March; transitional months such as October were not included, when both types of storms could be expected to have been important in wave generation. There was not a statistically signiﬁcant change over the decades in the heights of waves generated by extratropical storms, but statistically signiﬁcant increases were found for the hurricane-generated waves. The increases measured by the three Atlantic buoys for the summer hurricane seasons are documented in Fig. 35.5, graphs

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Fig. 35.5. Annual averages for signiﬁcant wave heights greater than 3 m generated by hurricanes, measured by buoys along the central US Atlantic Coast.7

of the annual averages that included only occurrences when the signiﬁcant wave heights were greater than 3 m, it having been shown that those higher waves during the summer can be attributed in almost all cases to speciﬁc hurricanes, whereas the lower waves represent calm intervals between storms. It is seen that there has been a latitude dependence, with the highest rate of increase having occurred in the south: 0.059 m/year (1.8 m in 30 years) for the Charleston buoy; 0.024 m/year for the Hatteras buoy; and 0.017 m/year for Cape May. It will be shown in the following section that the most extreme waves measured by these buoys during the summer hurricane seasons have increased from on the order of 7.5-m signiﬁcant wave heights early in the buoy records to greater than 10 m during the 1996–2005 decade. This increase in hurricane-generated wave heights could depend on several factors, including changes from year to year in the numbers and intensities of storms, their tracks that determined whether they traveled northward through the Atlantic where their generated waves could be recorded by these buoys, and how distant the hurricanes were from the buoys, whether they passed far oﬀshore within the central Atlantic or approached the US shore and possibly made landfall. Analyses by Komar and Allan7 indicated that all of these factors have been important to the observed wave-height increases, but the increased hurricane intensities found by Emanuel12 in his analyses of measured wind speeds within the storms provided the best explanation for the progressive increases seen in Fig. 35.5, since the numbers and tracks of the storms showed considerable variability from year to year.

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There has been some documentation of measured waves generated by typhoons in the northwest Paciﬁc, indicating a similar increase to that found in the North Atlantic. The coast of Japan is heavily instrumented with wave-measurement systems, with the earliest data having been collected in the 1970s. However, due to the irregular outline of that coast, a number of stations are partially sheltered from the sea and this unfortunately is true for several that have the longest records. Okada et al.33 found in these data sets that when they limited their analysis to the autumn, the season when typhoon-generated waves reach their maximum on the Paciﬁc Coast of Japan, the average rate of wave-height increase was about 0.01 m/year, an orderof-magnitude agreement with that found by Komar and Allan6,7 for the Atlantic hurricanes. Okada et al.33 suggested that the increase on the Paciﬁc Coast of Japan had in part been caused by increased typhoon activity, but with some contribution by extratropical storms that cross Japan during the autumn and winter. Their ﬁndings have been supported by Sasaki et al.34 based on wave-hindcast analyses, their focus having been on recent increases in summertime extreme wave heights, which they attribute in part to typhoon intensiﬁcation during El Ni˜ nos.

35.5. Wave-Height Histograms and Extreme Values While trends in annual averages of measured wave heights, as graphed in Figs. 35.3– 35.5, document their progressive decadal increases, another meaningful depiction of the changing wave climate is provided by histograms for the full range of measured signiﬁcant wave heights. Histograms in particular permit one to focus on increases in the highest measured waves, and to relate them to the results from extreme value analyses. The traditional depiction of a wave-height histogram is to graph the frequencies (percentages) of occurrence for individual signiﬁcant wave heights within the range of measurements, this approach having the advantage of normalizing the histogram to the total number of observations, with the summation being 100%. There are disadvantages, however, as the most extreme but rarely measured wave heights typically have frequencies less than 0.1%, the signiﬁcance of which is obscure, and when graphed using a linear scale these extreme waves disappear on the diagram, unfortunate since they are most important to the wave climate in terms of applications. In our investigations of climate controls on wave climates we have adopted an alternative procedure by graphing the actual numbers of observations, and employing a log scale that has the eﬀect of emphasizing the rare but extreme wave heights.35 In this manner we can directly identify the occurrence of even the single most extreme measured signiﬁcant wave height measured over the years, which graphs as a 100 = 1 occurrence. An example is shown in Fig. 35.6 based on the hourly signiﬁcant wave heights measured by the NDBC buoy seaward of Cape Hatteras, North Carolina; the compiled data are for the “winter” months of November through the following March, the season when the waves have been generated by extratropical storms (Northeasters). In contrast to the decadal increase in the heights of summer waves

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Fig. 35.6. Histograms for the range of signiﬁcant wave heights generated by extratropical storms in the Atlantic, a histogram for the percentages of occurrence and a graph for the actual numbers of height measurements.35

generated by hurricanes (Fig. 35.5), the winter waves have not experienced a statistically signiﬁcant change, so we can compile a histogram for its full record of measurements to obtain an assessment of the winter-wave climate.6,7,35 Figure 35.6 includes two depictions of the histograms for this wave record, one being the conventional graph as percentages of observations using a linear scale, the second being our alternate approach of graphing the actual numbers of observations of measured signiﬁcant wave heights with a log scale. Comparing the two it is apparent that the highest measured waves do not show up on the conventional frequency histogram, their frequencies being less than the thickness of the lines used to prepare the graph. However, those extreme waves are readily apparent when graphed as the actual numbers of observations on a log scale, even though they represent only 1 or 2 occurrences among the thousands of signiﬁcant wave heights measured over the years. It is seen in this example that the highest measured signiﬁcant wave height during the 30 years of buoy operation was 11 m, with heights down to about 8.5 m representing 10 or fewer observations. Nothing from that range of wave heights appears on the conventional frequency graph, even though they are of greatest relevance in applications. Figure 35.6 also illustrates that it is convenient to compare the histogram with the results derived from an extreme value assessment, in this example ranging from the 25- to 100-year projected extremes based on the ﬁt to a Weibul distribution of the annual maximum signiﬁcant wave heights.35 Having been graphed with a log scale we can directly examine the actual numbers of measured occurrences of those extreme value projections during the time frame of data collection. The 25-year projected extreme is 11 m, and it is seen that two measured signiﬁcant wave-height

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occurrences in 30 years corresponded to that projection. The 100-year projected extreme is 11.5 m, that having a 1% probability of occurrence. The probabilities of these extremes bear no direct relationship to the percentage frequencies of the conventional histogram included in Fig. 35.6, so a comparison is not as meaningful as permitted by this alternative approach to graphing numbers of occurrences. With a decadal increase in the summer wave heights measured by the US East Coast buoys (Fig. 35.5), the waves generated by hurricanes, the documentation of the changing wave climate involves a comparison between histograms for the ranges of measured signiﬁcant wave heights from early in the buoy’s record to the recent decade. This is seen in Fig. 35.7, again for the Cape Hatteras buoy,6,7,35 with the number of years of data included in the 1976–1991 histogram having been lengthened until the number of observations is nearly the same as in the 1996–2005 decade. Both histograms have maximums centered on about 1.5 m, the modes of the wave-height histograms, representing the calm periods that dominate the summer wave records between hurricanes. To the right of that mode, for values greater than about 2 m, there has been a shift in the numbers of higher waves during the 1996– 2005 decade compared with the earliest buoy measurements, with a substantial increase in the most extreme signiﬁcant wave heights that had been measured. This increase in numbers of wave-height occurrences between the two decades in the medium-height range of 2–5 m, where the curves are reasonably regular, mainly represent measured signiﬁcant wave heights derived from distant hurricanes, with the heights having decreased by wave dispersion as they traveled to the recording buoy. Irregularities in the histograms become noticeably greater for wave heights in excess of 5–6 m; these more extreme waves have been generated primarily by

Fig. 35.7. Histograms for the hurricane waves measured by the Cape Hatteras buoy, comparing the numbers of occurrences from early in the record versus the recent decade, 1996–2005.35

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hurricanes that passed in close proximity to the recording buoy, the numbers of occurrences generally involving fewer than 10 for a speciﬁc wave height, often only 1 or 2, with the irregularities in the histograms resulting from the sensitivity on a log scale to these low numbers of observations. It is seen in Fig. 35.7 for the Cape Hatteras buoy that in the 1976–1991 histogram the highest signiﬁcant wave height generated by hurricanes during the summer had been 7.8 m, with an increase to 10.2 m in the 1996–2005 histogram. Although this is consistent with the general increase in hurricane-generated waves measured by this buoy, these highest values are in the range of the statistical 25- to 50-year extreme value projections, with the 100-year extreme expected to be slightly higher. With the buoy record of measurements having been only 30 years, measured signiﬁcant wave heights of 8–10 m could potentially have occurred at any time during those decades, when the track of a fairly intense hurricane came near the recording buoy, although the probability of that occurring has gradually increased with the average intensities of the Atlantic hurricanes. There is the potential for misinterpretations in comparing a pair of histograms as in Fig. 35.7, in that there appears to be a marked diﬀerence in areas beneath the two curves. The common interpretation (usually mistaken) is that this is due to a greater number of measurements having been included in the 1996–2005 histogram compared with that for 1976–1991. However, this diﬀerence persists even when the data are graphed as percentages, normalized to the numbers of observations, with each graph then having an identical area representing 100%. Furthermore, in some of our analyses (as seen below in Fig. 35.8) the histogram for the earlier decade had greater numbers of observations than the more recent decade, but the disparity in

Fig. 35.8. Histograms documenting changes in the ranges of measured signiﬁcant wave heights oﬀ the coast of Washington.

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apparent areas beneath the curves remains much the same as seen here in Fig. 35.7. This apparent diﬀerence in areas is produced by the use of a log scale that in eﬀect warps the true areas beneath the curves, having accentuated the relatively rare measurements of the most extreme waves so the area beneath that portion of the curve is expanded.35 This eﬀect is analogous to the commonly used Mercator projection of the spherical Earth onto a ﬂat map, with the resulting expansion of areas located at high latitudes; for example, Greenland’s area on a Mercator map is much too large compared with a land mass close to the Equator. According to that analogy, the counterpart to the artiﬁcial expansion of Greenland is the expansion in Fig. 35.7 of the areas represented by rare but highest measured wave heights. With our investigations having focused on regions where the heights of waves have progressively increased due to Earth’s changing climate, such comparisons of pairs of histograms invariably result in the shift of the more recent histogram to greater numbers of high waves, as seen in Fig. 35.7, with a larger apparent area beneath that curve. Another example of a pair of histograms comparing their shifts in response to the changing climate is presented in Fig. 35.8, based again on data from NDBC buoy #46005 oﬀ the Washington Coast in the US Paciﬁc Northwest, used in the analyses of the decadal increases in Fig. 35.4 and Table 35.1. While the shift to higher extreme waves is again apparent in this pair of histograms, here the modes of greatest numbers of occurrences, centered on 3- to 4-m signiﬁcant wave heights, have also shifted to higher values and there is even a suggestion that the lowest waves, those during the summer, have experienced an increase. From the comparison of the overall shapes of the two histograms in Fig. 35.8 for the Washington data, it is apparent that the skewness has increased with time; this is also seen in Fig. 35.7 for the East Coast hurricane-generated waves. This developing skewness accounts for the systematic pattern in decadal trends seen in Fig. 35.4 and Table 35.1 for the Washington data, where the more extreme the wave-height assessment the higher the rate of increase according to the regressions. Taken together the analyses provide guidance as to the magnitudes of increase of extreme value projections, even though the histograms in Fig. 35.8 individually represent only about a decade of data, two short to derive meaningful extreme-value projections beyond the 25-year event.

35.6. Statistical Models of Trends in Extreme Wave Heights Formal statistical analysis procedures are being developed for applications to timevarying changes in data populations, with many of those applications directed toward the environmental consequences of global warming (e.g., temperatures, rainfall, and river discharges). Such investigations represent a signiﬁcant advance over classic extreme value theory, and have begun to be applied to the seasonal to decadal variations in ocean-wave climates. Statistical procedures are now available that involve a direct extension of the conventional extreme value analyses for a stationary population, an extension that does not assume the wave climate has

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remained the same over the decades, thereby being able to account for progressive changes in the extreme value assessments.36–38 Figure 35.9 shows the results of such an application to the Washington buoy (NDBC #46005), with the data analyzed being the ﬁve highest signiﬁcant waves measured each year (not their averages as analyzed in Fig. 35.4 to determine the trends in annual averages). This analysis represents an application of the r largestorder statistical model to extreme value analyses,36,39 with time being modeled as a covariate via the location parameter of the generalized extreme value (GEV) distribution. The three lines included in Fig. 35.9 are for the 25-, 50-, and 100-year projections, each increasing at a rate of approximately 0.09 m/year as determined by the nonstationary extreme value analysis. The r largest-order statistical model is employed due to the relatively short time series of annual maxima for this buoy. A nonstationary application of the annual maximum method (AMM), based on the single highest measured signiﬁcant wave height each year, yields much the same results as in Fig. 35.9, with the rate of increase having been slightly greater, 0.1 m/year. This result for the increase in the extreme value assessment is in good agreement with the linear regression for the highest measured signiﬁcant wave heights (Fig. 35.4 and Table 35.1), but by having modeled the extreme values this rate of increase is statistically signiﬁcant whereas that simple least-squares regression in Table 35.1 was not, most likely due to the fact that extreme values are not normally distributed so that a least squares analysis is not a reliable method for identifying the trends.36

Fig. 35.9. Statistical analyses of the increasing extreme value projections with a rate of increase of approximately 0.09 m/year, based on the ﬁve highest signiﬁcant wave heights measured each year by the Washington buoy.

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Fig. 35.10. Statistical analyses including a GEV model that yielded a rate of increase of 0.0.085 m/year for the 25-year projection, and a model that accounts for the monthly variations.38

One of the most advanced statistical models to deﬁne the varying wave climate is illustrated by the analyses of M´endez et al.38 Their model is based on a timedependent version of the Peak Over Threshold model, and permits a consideration of the annual cycle in signiﬁcant wave heights, long-term trends, and the inclusion of dependencies on climate indices. Their application was again to the NDBC #46005 buoy oﬀ the coast of Washington, with their analysis results shown in Fig. 35.10, ﬁtted to the data through 2005 and with its projection up to the year 2020 as a prediction of future wave climates. The data included in the analysis are those that exceeded the 99.5% quantile in the distribution of signiﬁcant wave heights, this threshold signiﬁcant wave height being 8.1 m. Each of the curves included in Fig. 35.10 represents the evolution of the 25-year return period of the significant wave height. The simplest model (GEV1) is comparable to that presented in Fig. 35.9 in having determined only a long-term trend; the result yielded a rate of increase of 0.085 m/year, a lower rate because M´endez et al.38 assumed an exponential rate of increase in extreme waves rather than a linear rate of increase as applied in Fig. 35.9, and due to diﬀerences in the levels of high-wave conditions being analyzed. The most-advanced analyses by M´endez et al.38 included modeling the annual cycles (Fig. 35.10), with the results up through 2005 having also included a dependence on the climate indices for those years, but without that dependence for future projections since those indices are unknowable. Although this advanced model provides the best overall assessment of the variations in the wave climate, the net rate of increase is reduced to 0.035 m/year when including the seasonal variability, which essentially is the value of the rate of increase in the annual averages of the winter measured signiﬁcant wave heights found in the regression analyses of Allan and Komar,5 graphed in Fig. 35.4. While M´endez et al.38 correctly concluded that their model provides a rigorous analysis of the long-term variability of Washington’s extreme wave climate, including the controls as predicted by the climate indices, its potential use depends on the nature of the application. For example, if the application is directed toward the extraction of wave energy in a “wave farm” (which is under consideration for the Washington and neighboring coasts), the advanced model in Fig. 35.10 would have particular relevance in accounting for the winter variability in power generation. On the other hand, if the concern is the stability of a

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structure to survive Washington’s extreme wave environment, whether the structure is the wave-energy extraction unit to be employed in that wave farm, or the construction of a harbor breakwater, then the higher magnitudes of the extremes and greater rates of projected increases provided by Figs. 35.4 and 35.9 become more relevant. This would also be the case for the development of coastal-hazard zones, similarly dependent on extreme values of the waves and related nearshore processes such as the wave runup on beaches.40,41

35.7. Summary and Discussion The development of wave climates is of fundamental importance to coastal engineers and scientists, having applications that range from the design of ocean structures to the development of coastal-hazard zones that not only account for the present-day measured ranges in wave heights and periods, but also project the most extreme occurrences expected in the next 25–100 years. The main requirement has been a wave record at least on the order of 35 years to project meaningful extremes, data derived from buoys or wave-hindcast analyses. Otherwise, the analyses used in developing wave climates were well established and had become reasonably routine. That status changed with the recognition of Earth’s variable and evolving climate. It is now realized that the development of a wave climate based on the assumption of a static environment could considerably miss the mark in projecting the future conditions, potentially leading to the failure of the designed structure or episodes of coastal erosion that exceed the recommended hazard zone. Our awareness of the consequences of Earth’s changing climate has centered mainly on the eﬀects of global warming and the expected accelerated rates of sealevel rise as projected in the reports of the International Panel on Climate Change. The importance of the potential consequences of the elevated water levels cannot be overstated, with its implications to the enhanced erosion of coasts being well recognized and accounted for in the establishment of hazard zones. However, the change in the global climate has also produced an intensiﬁcation of storms, resulting in higher generated waves and progressive shifts of the coastal wave climates toward more extreme conditions. As reviewed in this chapter, dramatic increases in stormgenerated waves have been documented by wave measurements in the North Atlantic and Paciﬁc, waves generated by both extratropical storms at high latitudes and tropical cyclones (hurricanes). This necessitates that the wave climates account for such progressive increases in storm intensities and their generated waves, including shifts in the extreme design conditions used in engineering and coastal-management applications. In this chapter we reviewed the types of wave-height analyses that need to be included in documentations of wave climates, analyses of the decadal trends in annual averages as seen in Figs. 35.2–35.5, and comparisons of shifts in the histograms of ranges of measured signiﬁcant wave heights toward higher values, (Figs. 35.7 and 35.8). In addition, analyses of the changes in wave periods also need to be included, since from the intensiﬁcation of storms the wave heights and periods have been found to increase in parallel, with both wave parameters combining to

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aﬀect the magnitudes of the ranges of nearshore processes, including the wave runup elevations on beaches, an important component in hazard-zone assessments. There can also be climate-controlled variations on the storm locations and therefore on the directions of wave arrival at the coastal site of interest, this for example being important along the US Paciﬁc Coast during El Ni˜ nos, having been a primary factor in localized “hot spot” beach and property erosion. All of these possibilities need to be considered in the development of wave climates, due to the multiple environmental consequences of Earth’s changing climate. A problem remains in that in some instances, while we can document with conﬁdence that the ocean wave heights have been progressively increasing, we are not always certain as to the underlying climate controls. A full understanding of this can be a challenge, as natural climate cycles may also be present, extending over decades or longer, an example being the range from El Ni˜ nos to La Ni˜ nas and how their relative frequency of occurrences is aﬀected by the Paciﬁc Decadal Oscillation (PDO), the dominance of one over the other being on the order of a 25-year cycle. That period in the PDO cycle is eﬀectively the span of most records available from wave buoys, such that in analyses of the wave climates along the Paciﬁc Coast of the USA we recognized that the wave data we work with had been dominated by major El Ni˜ nos that occurred during the past 25 years.5 With the likelihood of there having been a recent shift in the PDO so there might now be more frequent La Ni˜ nas, it can be expected that this change will aﬀect the wave climates, but we are uncertain as to how and to what extent. At the same time, although the heights of buoy-measured waves have progressively increased along the US West Coast, we are not 100% certain as to its cause. Therefore, uncertainties exist as to projections of wave climates into the future. In our assessments of coastal-hazard zones along the US Paciﬁc Northwest, our response to this uncertainty has been to base the environmental projections on the expected level of development of the site, the more extensive the proposed development the greater the expanse of the recommended hazard zone. The development of wave climates is no longer a routine undertaking, as the ocean engineer or coastal scientist needs to account for progressive increases in wave heights, periods, and changing wave directions, as well as the projected increase in mean sea levels. With the remaining uncertainty in the climate controls, yet with an intense level of research underway by climatologists, it can be expected that our approaches to evaluating wave climates will continue to evolve.

References 1. 2. 3. 4. 5. 6. 7. 8.

D. J. T. Carter and L. Draper, Nature 332, 494 (1988). S. Bacon and D. J. T. Carter, Int. J. Climatol. 11, 545–558 (1991). J. C. Allan and P. D. Komar, EOS, Trans. Am. Geophys. Union 81, 561–567 (2000). J. C. Allan and P. D. Komar, WAVES2001 Conf., ASCE (2001). J. C. Allan and P. D. Komar, J. Coast. Res. 22, 511–529 (2006). P. D. Komar and J. C. Allan, EOS, Trans. Am. Geophys. Union 86, 301 (2007). P. D. Komar and J. C. Allan, J. Coast. Res. 24 (2008). Y. Goda, Handbook of Coastal and Ocean Engineering (Gulf Publishing Co., 1990).

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9. K. A. Emanuel, Devine Wind: The History and Science of Hurricanes (Cambridge University Press, 2005). 10. M. E. Mann and K. A. Emanuel, EOS, Trans. Am. Geophys. Union 87, 233 (2006). 11. K. E. Trenberth and D. J. Shea, Geophys. Res. Lett. 33 (2006), doi: 10.1029/ 2006GL026894. 12. K. A. Emanuel, Nature 436, 686–688 (2005). 13. C. W. Landsea, B. A. Harper, K. Hoarau and J. A. Knaﬀ, Science 313, 452–454 (2006). 14. R. J. Seymour, R. R. Strange, D. R. Cayan and R. A. Nathan, Proc. 19th Int. Conf. Coast. Eng., ASCE (1984), pp. 577–592. 15. P. D. Komar, Shore Beach 54, 3–12 (1986). 16. D. L. Revell, P. D. Komar and A. H. Sallenger, J. Coast. Res. 18, 792–801 (2002). 17. J. C. Allan, P. D. Komar and G. R. Priest, Shoreline mapping and change analysis: Technical considerations and management implications, J. Coast. Res. SI38, 83–105 (2003). 18. P. Ruggiero, G. M. Kaminsky, G. Gelfenbaum and B. Voigt, J. Coast Res. 21, 553–578 (2005). 19. J. J. O’Brien, T. S. Richards and A. C. Davis, Bull. Am. Meteorol. Soc. 77, 773–774 (1996). 20. S. B. Goldenberg and L. J. Shapiro, J. Climate 9, 1169–1187 (1997). 21. K. Wolter and M. S. Timlin, 17th Climate Diagnostics Workshop, CIMMS and School of Meteorology, Univ. of Oklahoma (1993), pp. 52–57. 22. N. J. Mantua, S. R. Hare, Y. Zhang, J. M. Wallace and R. C. Francis, Bull. Am. Meteorol. Soc. 78, 1069–1079 (1997). 23. S. Bacon and D. J. T. Carter, Int. J. Climatol. 13, 423–436 (1993). 24. T. L. Delworth and M. E. Mann, Climate Dynam. 16, 661–676 (2000). 25. M. E. Mann and J. Park, J. Geophys. Res. 99, 25819–25833 (1994). 26. G. D. Bell and M. Chelliah, J. Climate 19, 590–612 (2006). 27. N. E. Graham and H. F. Diaz, Bull. Am. Meteorol. Soc. 82, 1869–1893 (2001). 28. M. D. Earle, K. E. Steele and Y. H. Hsu, Proc. Oceans ’84 (1984), pp. 725–730. 29. R. M. Gorman, K. R. Bryan and A. K. Laing, NZ J. Marine Freshwater Res. 37, 589–611 (2003). 30. S. K. Gulev and V. Grignorieva, Geophys. Res. Lett. 31, L24302 (2004). 31. E. Bouws, D. Jannick and G. J. Kowen, Bull. Am. Meteorol. Soc. 77, 2275–2277 (1996). 32. WASA, Bull. Am. Meteorol. Soc. 79, 741–760 (1998). 33. K. Okada, Y. Suzuki, S. Utsunomiya and Y. Watanabe, Proc. 21st Int. Conf. Coast. Eng., ASCE (1998), pp. 972–984. 34. W. Sasaki, S. I. Iwaswaki, T. Matsuura and S. Izuka, Geophys. Res. Lett. 32, L15607 (2005). 35. P. D. Komar and J. C. Allan, Shore Beach 75, 1–5 (2007). 36. X. Zhang, F. W. Zwiers and L. Guilong, J. Climate 17, 1945–1952 (2004). 37. C. Stefanakos and G. A. Athanassoulis, Environmetrics 17, 25–46 (2006). 38. F. J. M´endez, M. Men´endez, A. Luce˜ no and I. J. Losada, J. Geophys. Res. 111, C07024 (2006), doi: 10.1029/2005JC003344. 39. C. Guedes Soares and M. G. Scotto, Coast. Eng. 51, 387–394 (2004). 40. P. D. Komar, J. C. Allan and P. Ruggiero, Coast. Disasters ’08 Conf., ASCE (2008). 41. P. Ruggiero, Coastal Disasters ’08 Conf., ASCE (2008).

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Chapter 36

Sea Level Rise: Major Implications to Coastal Engineering and Coastal Management Lesley Ewing California Coastal Commission∗ 45 Fremont Street, SF, CA 94105 USA Viterbi School of Engineering∗ University of Southern California, LA, CA 90089 USA [email protected] Now rolls the deep where grew the tree. Alfred Lord Tennyson Water level variability has been one of the most signiﬁcant aspects of coastal factors for coastal engineering and coastal management. Current factors inﬂuencing local water level include tides, wind waves, tsunamis, atmospheric forcing, seiching, storm surge, and local uplift and subsidence. Changes in global sea level have inﬂuenced coastal conditions over geologic time. Changes in eustatic sea level have been relatively small for the past several thousand years, but there is evidence from global records, trends in sea level and atmospheric-ocean models that sea level is now rising more quickly that has been experienced in the recent past, and this trend will continue or increase during the next 100 years. Coastal engineering and coastal management eﬀorts will need to adjust to these changing sea level trends. Certain situations may be best addressed by individual responses, such as armoring, beach nourishment, or retreat. In most cases, the use of several complementary approaches and the application of the major principles of sustainability may be more appropriate.

36.1. Introduction Water level variability is one of the most signiﬁcant aspects of the coast that distinguishes coastal land from most other lands. Local water level variability inﬂuences ∗ This report has not been reviewed by the California Coastal Commission or Viterbi School of Engineering.

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where someone will place their beach towel for a day of sun and surf; it eﬀects where diﬀerent species can exist comfortably; where it is possible to launch a boat or where it seems safe to erect buildings, roads, and infrastructure. Just as water level variability is one of the basic features that diﬀerentiates coastal land from other land, it is also one of the features that diﬀerentiates coastal engineering and coastal zone management from their inland counterparts. Water level variability is also one of the components of the coast that is most diﬃcult to forecast. The multiple, often independent forces and system dynamics that drive water level at a speciﬁc location contribute to this diﬃculty. The observed water level at a speciﬁc location is the cumulative result of global and local factors including oceanic conditions, atmospheric forcing, storm surge, waves, tides, land subsidence or uplift, wind, and river ﬂows. Recent examination and discussion of global warming have focused attention on the long-term and large-scale variability of water levels encompassed by the term global or eustatic sea level change. Although global sea level rise is occasionally viewed as a new phenomenon, even a cursory examination of historic and geologic sea level trends shows that global sea level has only rarely been stable or static. In fact, the most typical characteristic of global sea level over time has been that it does change, rising and falling in response to varying climatic and tectonic conditions. The current interest in global warming has directed new attention to the ﬂuctuations in global sea level, the ensuing changes in local sea level and the likely engineering and management responses. Global sea level is but one of the factors that contribute to observed local sea level and that inﬂuence decisions on coastal engineering and coastal zone management. To understand the role of global sea level change in coastal engineering and management, it is important to understand the overall dynamics of water level along the coast from the global to the local scale, how localized water level variability has traditionally been incorporated into coastal engineering and management, and the implications of future variability for these traditional and innovative engineering and management approaches.

36.2. Geologic and Historic Changes in Eustatic Sea Level Global climate is inﬂuenced by changes to both the ocean and the atmosphere. During the current Quaternary period, large-scale variability in sea level has been controlled by glacial and climate change. During periods of cooling and glacial advance, many millions of cubic meters of water are stored on land-based glaciers and polar ice. Remaining ocean water is highly saline, cold, and dense. During periods of warming and glacial retreat, melt water and calving from land-based glaciers expand the volume of ocean water and increasing atmospheric temperatures transfer heat into the ocean. Sea level rises during interglacial periods due to the increased water volume. Sea level during the Quaternary has been strongly inﬂuenced by glacial advance and retreat; however, over much of the geologic record, the prime control on

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global eustatic sea level has been the overall rate of global plate movements — speciﬁcally changes in the rates of mid-oceanic spreading.1 Spreading rates of the mid-oceanic ridges range from 2 to 6 cm/year, causing a global rise in sea level up to 1 cm/1,000 year.2 During periods of rapid spreading, large portions of the ocean basin are composed of young, hot basaltic crust. This young crust is both thicker than the older crust and rises higher on the underlying asthenosphere, reducing the volume of the ocean basin. Shallower, smaller ocean basins displace ocean water into the continental land masses. Rapid seaﬂoor spreading causes eustatic high stands of sea level and during periods of slow spreading rates, the eustatic sea level stands were low. One of the highest identiﬁed sea level stands occurred in the Cretaceous period (130–65 million years BP), during fragmentation of Gondwanaland and opening of the Indian Ocean and Mediterranean Sea, when much of interior North America, Europe, and Asia was covered by shallow continental seas. Reconstructions of tectonics and paleo-climate can only show large trends. Temporal trends in sea level are often plotted as a continuum, but actual sea level variability most likely occurred as fairly discrete and sudden changes.3 This variability has been evident in reconstructions of sea level rise following the most recent interglacial period, the Late Wisconsin Glaciation. This glaciation was followed by a rapid rise in sea level, increasing by approximately 120 m from the glacial low level.4 From 18,000 B.P. to about 6,000 B.P. the rate of sea level rise was about 9.0 mm/year. Over the past 6,000 years, sea level rise slowed to a rate of approximately 0.5 mm/year and to a rate of 0.1 to 0.2 mm/year during the past 3,000 years.5 Recently, Yu et al.6 reconstructed the Holocene sea level history for the Swedish Baltic coast and identiﬁed a rapid rise in sea level ca. 7,600 B.P. that ranged from 3.0 to 7.2 m above present sea level, potentially linked to decay of the Labrador sector of the Laurentide Ice Sheet. The Yu research and other studies provide indications that sea level change has been episodic over many diﬀerent time scales. Over the past 100–150 years, tide gages around the world have provided measurements of local changes in sea level. Since 1992, these measurements have been supplemented by altimeter measurement of the open ocean water level taken by the TOPEX/Poseidon and Jason satellites. As shown in Fig. 36.1 (IPCC 2007, Fig. 5.13), sea level has shown a rise of about 1.7 ± 0.5 mm/year when averaged over the 20th century and 1.8 ± 0.5 mm/year when averaged over the more recent time period from 1961 to 2003. Much of the sea level change examination by the IPCC Fourth Assessment Report (2007) attempted to bridge the gap between the geologic record of sea level change and the more recent tide gage and satellite data. Archeologic sources, well data and information from ﬁsh pond elevations from the Mediterranean area, have established that sea level was at a near still stand from about 3,000 to 2,000 year B.P. until the 19th century. The current rise in sea level seems to coincide with the industrial revolution, with a noticeable rise in sea level occurred in the time period between 1850 and 1950. Thus, the observed increase in sea level of 1.7–1.8 mm/year seems to be only a recent global condition and represents an acceleration of 1.3 ± 0.3 mm/year/year for the past 150 years.7

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Fig. 36.1. Annual averages of the global mean sea level (mm). The thin black line with standard deviations is the reconstructed sea level since 1870 (from Ref. 7); the shaded gray curve shows coastal tide gage measurements since 1950 (from Ref. 8) and the solid black line is based on satellite altimetry (from Ref. 9). The reconstructed sea level ﬁeld and the tide gage measurements are deviations from their averages for 1961 to 1990; the satellite altimetry is the deviation from the average of the reconstructed sea level for the period 1993 to 2001. Error bars show 90% conﬁdence. (From Ref. 10; Fig. 5.13).

Short-term localized sea level conditions can deviate greatly from average conditions and from global conditions. Localized observed sea level is dependent on climatic events such as the occurrence of El Ni˜ nos, frequency of storm surge or atmospheric forcing, pulses of river water and short-term thermal warming or cooling. Satellite altimetry data for the 1990s (a time period that included the inﬂuences of 1997/1998 El Ni˜ no) showed a 3 mm/year rise in sea level in the open ocean for the decade of recorded observations and the global coastline had a sea level rise rate of 4 mm/year9 for global coastal waters. Some are tempted to see these short-term rates as an indication that the anthropogenic increases in atmospheric temperature linked to greenhouse gas emissions have led to acceleration in sea level rise; however, there are many limitations in the extrapolation of short-term sea level rates for long-term trends. 36.2.1. Future trends in sea level Future trends in water level will depend on a number of factors — all of which have their own uncertainties. Despite the signiﬁcance of individual tidal components to the daily changes to the coast and to the function of coastal systems, most existing projections of future sea level, such as the IPCC reports, address mean

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sea level as the only component. In most situations, changes in mean sea level will provide a good ﬁrst-order approximation for changes in the other tidal constituents. Furthermore, the range provided in most projections can cover the variation in the individual constituents. Over the past century, global temperature has increased about 0.6◦ C and mean sea level has risen by 15–18 cm globally.11 State-of-the-art climate models predict that global temperature may rise by 2–9◦ F (1–5◦ C) over the next 100 years.12 There is a large uncertainty in these estimates of future temperature change and even greater uncertainty about the sea level response to this warming; however, there is strong certainty that a future rise in global temperature will be accompanied by a rise in sea level. The uncertainty is only for the rate and amount of rise that could occur by any speciﬁc time in the future. Future changes in global sea level will depend, among other factors, upon: • • • •

future global temperature; lag time between atmospheric changes and oceanic reactions; thermal expansion of ocean water; eﬀects of atmospheric temperature changes on Antarctica, Greenland, and other glaciers.

The two main factors contributing to global sea level (eustacy) remain thermal expansion (steric eustacy) and the addition of water to the oceans from glacial melting. Both of these factors can cause spatial variability in observed global water levels. Satellite and tide gage information have found spatial variability in the recent changes in sea level and large-scale atmospheric–oceanic models have been in general agreement with these observations. However, observations and model data sets available at the time of the IPCC Third Assessment Report13 were not able to fully match observed water level changes with thermal expansion or glacial melting. The IPCC Fourth Assessment Report found a better matching of the various data sets and observations and concluded that thermal expansion and melting of land-based glaciers each accounted for about half of the sea level rise observed from 1993 to 2003. These agreements provide conﬁdence in modeling of future trends in sea level attributable to changes in atmospheric conditions and global warming. The thermal capacity of the oceans is about 1,000 times that of the atmosphere, with the transfer of heat between the atmosphere and the ocean providing a buﬀering mechanism to minimize extreme ranges of atmospheric temperature.13 Levitus et al.14 found that the net heat uptake by the oceans since 1955 has been approximately 20 times more than the atmosphere. Most of the heat has been stored in the upper ocean layers. Estimated thermal energy of that layer has increased over the time period from 1955 to 2003, by 10.9 ± 3.1 × 1022 J or 0.14 ± 0.04 Wm−2 . Such an increase in energy corresponds to an increase in average temperature of 0.1◦ C and rise in sea level due to thermal expansion of approximately 0.42±0.12 mm/year. These trends have not been spatially uniform; approximately half the warming has occurred in the Atlantic Ocean. Some shallow equatorial areas and high-latitude areas have experienced cooling over the same period from 1955 to 2003.15 Thermal

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expansion for the past decade (1993–2003) alone has led to an average rise in sea level of 1.6 ± 0.5 mm/year. The future rate and volume of glacial change is the largest uncertainty for estimating future sea level. Prior to satellite imaging, it had been diﬃcult to get detailed data on the extent and mass of glaciers. The IPCC Fourth Assessment Report provides estimates that glaciers (glaciers and ice caps, Greenland Ice Sheet and the Antarctic Ice Sheet) contributed approximately 1.2 mm/year to sea level rise from 1993 to 2003. The observed thermal expansion and glacial melt for 1961 to 2003 could not be combined to account for most of the observed rise in sea level, leaving +0.7 ± 0.7 mm/year as an unaccounted diﬀerence between observed rise in sea level and contributions. Human activities have contributed to changes in the available water and water sequestration. Table 36.1 shows the general distribution of water. The ocean contains 96.5% of the total water reserves, and fresh water is only 2.5% of the water reserves. Of the available fresh water, approximately 69% is locked up in glaciers; approximately 30% is available as groundwater or soil moisture, leaving only 1% of the fresh water available as surface water or in the atmosphere. Gornitz16 has estimated that deforestation, wetland losses, and fossil fuel combustion release the sea rise equivalent of 0.4–0.9 mm/year, and sequestration in reservoirs and inﬁltration below reservoirs has provided the equivalent of a sea level drop of 1.3–1.8 mm/year. This overall reduction in potential sea level by 0.9 ± 0.5 mm/year has been a significant inﬂuence on current sea level, especially when compared with the general rise in sea level of 1.7–1.8 mm/year. Accelerated melting of the glaciers and ice sheets could lead to substantial increases in sea level. Melting of the ice sheets on Greenland could result in up to 7 m of additional sea level rise and melting of the glaciers and ice sheets in Antarctica could result in an additional 20 m of sea level rise.17 Research and observations conﬁrm that polar ice caps and glaciers are melting at rates far higher than previously; however, there is little long-term information on glacial change to put the current conditions into context. Few researchers have been able to provide any time periods for these occurrences and accelerated loss of polar ice caps and glaciers has not been included in any of the current IPCC predictions for future sea level. Figure 36.2 and Table 36.2 provide the most recent IPCC scenarios for future sea level rise.18

Table 36.1.

Global water reserves (103 km3 ).16

Glaciers, snow, and ice sheets Rivers Lakes Swamps Groundwater and soil Oceans Atmosphere

24,364 2 176 11 23,400 1,338,000 13

360 km3 of additional ocean water is equivalent to 1 mm of sea level rise.

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Fig. 36.2. Time series of global mean sea level (deviation from the 1980 to 1999 mean) in the past and as projected for the future. For the period before 1870, global measurements of sea level were not available. The estimates of the past show the uncertainty in the estimated long-term rate of sea level change; the instrumental record is the reconstructed global mean sea level from tide gages and for the period from 1991 to present from satellite altimetry, with gray shading to denote the range of variations from a smooth curve. The projections of the future sea level are model predictions for the IPCC scenario A1B, relative to the 1980 to 1999 mean and it has been calculated independent of observations. Beyond 2100, the projections are increasingly dependent on future greenhouse gas emissions scenarios. Over many centuries or millennia, sea level could rise by several meters (IPCC, FAQ 5.1, Fig. 1).

Table 36.2. Projected globally averaged surface warming and sea level rise 2090 to 2099 relative to 1980–1999 (Ref. 13, Table SPM-3). Temperature change (◦ C) Case Constant Year 2000*** B1 scenario A1T scenario B2 scenario A1B scenario A2 scenario A1F1 scenario

Sea level rise (m)

Best estimate*

Likely range*

Model-based range**

0.6 1.8 2.4 2.4 2.8 3.4 4.0

0.3–0.9 1.1–2.9 1.4–3.8 1.4–3.8 1.7–4.4 2.0–5.4 2.4–6.4

NA 0.18–0.38 0.20–0.45 0.20–0.43 0.21–0.48 0.23–0.51 0.26–0.59

*These estimates are assessed from a hierarchy of models that encompass a simple climate model, several Earth Models of Intermediate Complexity (EMICs) and a large number of Atmosphere Ocean Global Circulation Models (AOGCMs). **Excludes future rapid dynamic changes in ice ﬂow. ***Year 2000 constant composition is derived from AOGCMs only.

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The Emission Scenarios of the IPCC Special Report on Emission Scenarios (SRES) A1: The A1 scenarios describe a future world of very rapid economic growth, global population that peaks mid-century and declines thereafter, and a rapid introduction of new and more eﬃcient technologies. Major underlying themes are convergence among regions, capacity building and increased cultural and social intersections, with a substantial reduction in regional diﬀerences in per capita income. The three A1 scenarios are distinguished by their technological emphasis: (A1F1) non-fossil energy sources; (A1T) a balance across all sources; A1B where balance is deﬁned a not relying too heavily on one particular energy source and similar improvement rates apply to all energy sources and end use technologies. A2: The A2 scenarios describe a very heterogeneous world. The underlying theme is self-reliance and preservation of local identities. Fertility patterns across the region converge very slowly, resulting in continuously increasing population. Economic development is primarily regionally oriented and per-capita economic growth and technological change is more fragmented than other scenarios. B1: The B1 scenarios describe a convergent world with the same global population as in the A1 scenarios which peaks in mid-century and declines thereafter, but with rapid change in economic structure toward a service and information economy, with reductions in material intensity and the introduction of clean and resource eﬃcient technologies. The emphasis is on global solutions to economic, social and environmental sustainability, including improved equity but without additional climate incentives. B2: The B2 scenarios describe a world in which the emphasis is on local solutions to economic, social and environmental sustainability. It is a world with continuously increasing global population, at a rate lower than A2, intermediate levels of economic development, and less rapid and more diverse technological change than in the B1 or A1 scenarios. While the scenario is also oriented towards environmental protection and social equity, it focuses on local and regional levels. All scenarios should be considered equally sound. The SRES scenarios do not include additional climate incentives which means that no scenarios are included that explicitly assume implementation of the United Nations Framework Convention on Climate Change or the emissions targets of the Kyoto Protocol.

36.3. Coastal Water Level Dynamics Most coastal engineering and management decisions need to consider the anticipated rise in future global sea level. This global rise in sea level will not be an isolated component of a design or management decision; it must be considered in association with other changes in water level. Coastal residents, visitors, and observers are aware of many of the changing water levels that occur throughout the day, from season to season, over a period of years, or at irregular intervals. The main factors inﬂuencing these changes are astronomic tides; El Ni˜ no/Southern Oscillation events (ENSO); atmospheric pressure variability; wind and storm surge; waves; tsunamis; river discharge; land subsidence and land uplift, in addition to the historic eustatic sea level change. These factors vary in both spatial and temporal signiﬁcance. Table 36.3 provides an overview of these factors, many of which are discussed below.

36.3.1. Tides Tides and tidal currents are caused by the gravitational forces of the moon and the sun, the centrifugal force from the rotation of the earth and interaction or

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Factors leading to changes in local sea level.

Factor

Time scale

Tides

Semi-diurnal M2 (12.42 h) Diurnal K1 (23.93) Monthly Mf (13.7 days) Epoch (18.61 years) Hours to months A few days Weeks to months Seconds Minutes to hours Seconds to minutes

Atmospheric forcing Storm surge River ﬂows Wind waves Long-period tsunamis Land uplift/subsidence

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Spatial scale

Water level change

Global, with local ampliﬁcations

0.2–10+ m

Regional/basin wide Regional/basin wide Local Local Regional/basin wide Local to regional

−0.7 to 1.3 m Up to 8 m Up to 1 m Variable Up to 10 m Variable

interference of land masses with these deepwater tides. People recognized a connection between the moon, the sun, and the tides several thousand years ago. One of the ﬁrst documented descriptions of tides was presented in the 17 books of Geography of Strabo (64 BC to 21 AD) in which he cites subsequently lost writings by Posidonius of Rhodes: “Now he (Posidonius) asserts that the motion of the sea corresponds with the revolution of the heavenly bodies and experiences a diurnal, monthly and annual change, in strict accordance with the motion of the moon.”19

These early writings detailed the observed connections between the daily, monthly, and annual tidal variations; however, it was sixteen centuries, and many tidal observations and tidal theories later, that Newton expressed the gravitational forces between astronomic bodies with the equation, F = (Gm1 m2 /r2 ). The detailed examination and expansion of tidal harmonics continues to this day. Tidal ﬂuctuations are often the main changes to water level that are considered in coastal engineering and design. Tidal ﬂuctuations have been included routinely in wetland restoration projects, where the habitat ranges of wetland vegetation are determined in large part by the amount of time the plants will be submerged. Vegetation zonations tend to follow the tidal zonations, where the intertidal zone, the subtidal zone, and the supra tidal zone provide habitat for diﬀerent plant palettes. Beach construction and shoreline access is often planned for low tides to maximize available dry beach area. The Port of Oakland, in Oakland, California, installed two new cranes that were shipped across the Paciﬁc Ocean, through the Golden Gate and under the Bay Bridge. Delivery was timed for low tide to provide adequate clearance for the cranes to ﬁt safely under the bridge. Navigation charts reference all water elevations to a low tide datum in an attempt to prevent ships from running aground during low tide periods. High tide too can be important for coastal engineering and coastal management. Flick20 found that during the 1982/1983 El Ni˜ no storms, most of the storm events that caused the greatest damage were those storms that occurred during high tide. Wood21 looked through the 341-year record of coastal ﬂooding throughout North America and found more than 100 incidents when ﬂooding occurred due to storms occurring at the same time as the perigean spring tides. Likewise, some of the worst ﬂooding from Hurricane Katrina occurred with the combination of high storm surge

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and high tide.22 Tides are signiﬁcant to many aspects of coastal engineering, design, and management. Changes to eustatic still water levels can be expected to alter tides in several ways. As inundation areas move inland with a rise in sea level, the area of land exposed to tidal inﬂuence will shift inland; however, there will not be a consistent shift of the tide range or of the tidal constituents. Flick et al.23 analyzed data from long-term stations in the US tide gage network and found that most open ocean areas have experienced a change in tidal range and the rate of change of the tidal constituents does not parallel the rate of change of the mean. For example, the mean and diurnal tide ranges at San Francisco increased at a rate of 60 mm/century and mean high tide and mean higher high tide constituents increased 19% faster than mean sea level. Changes in the tidal range and tidal constituents will alter the coastal elevations exposed to tidal inﬂuence. An increase in the high tide components will expose more areas to periodic inundation. Increases in tidal range will increase tidal currents and the intertidal zone. The dynamics of tidal variability with sea level have not been well deﬁned, but, future tidal variability can be expected to occur with global climate change.

36.3.2. Atmospheric pressures and storm surge Atmospheric pressures and storm surge are meteorologic forcings that can inﬂuence daily or seasonal water levels. At the longer-term, regional scale, El Ni˜ no events can be accompanied by sustained low pressure systems and a rise in sea level that can persist for several weeks. During the 1997/1998 El Ni˜ no system, Nerem and Mitchum24 observed a 20-mm rise and then drop of the global mean sea level attributable to the El Ni˜ no/Southern Oscillation event. During Hurricane Katrina, atmospheric pressure dropped by 66 mb in 48 hours to 902 mb25 and this low pressure system likely resulted in a short-term rise in sea level of almost 0.9 m, independent of surge.26 Storm surge also can cause a large, short-term increase in water level. The compilation by the ASCE Hurricane Katrina External Review Committee found that peak water levels from storm surge and the low-pressure system reached 6 m above still water level in south Plaquemines Parish and reached 3.7 m above still water level at the entrances to the canals along the New Orleans Lakeshore. Table 36.4 shows the Saﬃr–Simpson hurricane scale and associated storm surge.

Table 36.4.

Saﬃr–Simpson hurricane scale.

Category

Wind speed km/h (mph)

1 2 3 4 5

119–153 (74–95) 154–177 (96–110) 178–209 (111–130) 210–249 (131–155) ≥250 (≥156)

Storm surge m (ft) 1.2–1.5 1.8–2.4 2.7–3.7 4.0–5.5 ≥5.5

(4–5) (6–8) (9–12) (13–18) (≥18)

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The areas of future storm surge will depend on changes in sea level insofar as sea level will change inundation areas and water depth. However, since surge is primarily inﬂuenced by wind conditions, sea level rise would be expected to cause only minor changes to the water level variability that is associated with atmospheric pressure or storm surge. However, there are some indications that global warming could change both the frequency and intensity of storms. These changes would have signiﬁcant inﬂuence on the resulting water levels and such changes would greatly alter coastal engineering and coastal management options in areas subject to large water level variability due to meteorologic forcing from atmospheric pressure or surge. 36.3.3. River flows Rivers are well recognized as sources of coastal sediment. During times of peak ﬂows, large plumes of fresh water from rivers pulse into the ocean, ﬂoating on top of the salty, denser ocean water. These pulses of fresh water alter the surface elevation of the ocean in the area adjacent to the river mouth. Coastal riverine ﬂooding is of enormous concern to coastal engineers and managers. Flood conditions can cause rivers to migrate from existing channels and carve new pathways to the ocean, can alter wetland, estuarine and lagoonal habitats, water quality and sedimentation. River ﬂooding can both spread fertile sediments onto the ﬂoodplain as occurred for years along the Nile River, but river ﬂooding can also be extremely damaging to structures and infrastructure along its course. The oceanic changes to sea level as a result of ﬂood pulses of fresh water are not well studied and are of less concern to coastal management interests that is inland ﬂooding. Satellite altimetry is beginning to examine these river pulses in detail. As more is learned about these phenomena, their signiﬁcance to coastal management may become more apparent. 36.3.4. Wind waves Waves are often the most recognized aspects of the coast and the source of most coastal damage. They are the major force for the changes and dynamics of the coast, moving sediment, setting up nearshore currents and carrying vast amounts of energy through the ocean basins. Most waves along the coast are wind waves, developed by the transfer of energy from the atmosphere into the water column as winds blow across the vast expanse of the open ocean. Complex water surfaces develop as wave trains leave the initial generation location, being inﬂuenced by other atmospheric conditions and interacting with other wave trains, and locally-generated waves. The initial energy and the duration and area of the initial forcing all inﬂuence the resulting wave heights. The nearshore bathymetry and local conditions inﬂuence the transformation of deepwater waves into nearshore, shallow water waves. Wave heights along the Paciﬁc coast are inﬂuenced greatly by bottom depths and for most locations along the coast, the heights of nearshore waves are “depth limited.” Wave energy is proportional to the square of the wave height. For a 1-m high wave in the surf zone, the energy per unit area would be approximately 1,250 Nm/m2 ; if the wave period is 6 s, the energy at the coast would be about 4,000

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watts per meter of shoreline.27 The energy at the coast from a 2-m wave would be approximately 5,000 N-m/m2 , transferring four times the energy of a 1-m wave. In addition, Kanoglu and Synolakis28 have shown that small increases in water elevation, such as those from sea level rise, may have substantial, nonlinear eﬀects on runup and impacts on vertical walls. Small changes in water level can cause signiﬁcant changes in wave energy and the potential for shoreline damage from wave forces. Any rise in sea level, the 0.2–0.6 m rise over the 21st century modeled by the IPCC scenarios would cause signiﬁcant changes in nearshore wave energy. And, all IPCC scenarios exclude future dynamic changes in glaciers and polar ice caps, so the modeled changes in sea level would be low estimates if glacial and polar melting accelerates. Wave energy is sensitive to water depth and large rises in sea level will greatly inﬂuence coastal conditions; a sea level rise of 6 or 7 m would transform the coast in far greater ways than a rise of 0.2–0.6 m. There is some evidence that global warming and climate change will contribute to changes in the intensity and frequency of storms. Graham and Diaz29 have found an intensiﬁcation of storm activity in the northeast Paciﬁc that may be related to or indications of changing climatic conditions. A rise in global temperature may increase not only sea level and the transformation of wave energy, but could change the characteristics of available wave energy. 36.3.5. Tsunamis/seismic waves Seismic waves are the infrequent long waves that can be generated by certain large earthquakes,30 submarine landslides, eruptions of underwater volcanoes, explosions, meteor strikes, or other large oceanic disturbances. General relationships between magnitude, energy, and wave runup have been developed,31 but occasionally the initial conditions that set up a tsunami are modiﬁed by local conditions and contributory elements. Tide gages in Japan recorded amplitudes of about 25 cm from the 17 July 1998 tsunami, yet at Papua New Guinea, runup exceeded 10 m, causing more than 2,100 fatalities. Synolakis et al.32 identiﬁed a large co-seismic submarine slump that, when modeled with the initial event, accounted for the extreme conditions experiences in New Guinea. There have been some recent or recorded extreme events such as the 2004 Indian Ocean tsunami, the 1964 Great Alaskan tsunami, and others, that have provided devastating examples of extreme events. Since extreme events are rare, many coastal locations augment observed tsunami experience with modeling and exploration of the geologic record to develop information on extreme tsunami wave heights. Modeling by Synolakis et al.33 of the California coast based on potential extreme events from potential teletsunamis, nearshore earthquake events or submarine landslides indicate extreme inundation heights up to 13 m. Such extreme events must ﬁrst be considered for emergency preparedness and evacuation planning. Education has been a fundamental aspect of tsunami preparedness. There is debate now about the extent to which tsunamis can or should be incorporated into coastal engineering and coastal management. The 2004 Indian Ocean tsunami

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clearly showed the life safety beneﬁts of vertical evacuation and the importance of considering potential tsunami forces for the design of critical facilities.34 NOAA’s National Tsunami Hazard Mitigation Program has initiated development of design guidelines for vertical evacuation structures. And, for the development of a new community in a tsunami-prone area of Humboldt Bay, CA,35 tsunami evacuation routes are being included in the circulation patterns for the community and potential tsunami inundation levels are being used to site habitable structures and critical facilities. Tsunami inundation and runup will be altered by rising sea level. For most situations, there is little detailed guidance for how to include tsunamis eﬀects into coastal engineering and management. Engineering and management eﬀorts for tsunamis clearly require a long-term focus, and this long-term perspective should consider whether the combined inﬂuence of tsunamis and sea level rise will be important. 36.3.6. Seiches Seiches are oscillating standing waves that develop in harbors, lakes, or partially enclosed water bodies. The main characteristic of a seiche is that the wave continuing pendulum fashion after the initial force has stopped. The initial forcing can come from winds blowing across the surface of the water body, waves, tides, and earthquakes. Seiches often set up in lakes or enclosed harbor basins where the oscillations depend on the natural periods of the harbor and the energy from the forcing event.36 Most lakes develop seiches in response to wind, tides, or seismic disturbances. Lake Erie can experience a 4.5-m change in water level due to wind-induced seiching.37 Often the vertical displacement has a small amplitude and the main evidence of seiching is the occurrence of the large horizontal displacements that can be quite damaging to moored ships. The lowest natural frequency for harbor seiching is proportional to the square root of water depth. Seiching is sensitive to changes in water depth and will be sensitive to changes in local sea level, with harbor responses and resonance being an area of signiﬁcant concern. Seismic-induced seiching is one of the more intriguing seiching conditions. The 1964 Alaska Earthquake caused oscillations in water bodies throughout North America and the world. Many locations in the Gulf of Mexico area recorded seismic seiching, but seiching was also recorded as far away as Australia. McGarr and Vorhis38 analyzed data on all seiches known to have been caused by the 1964 Alaskan Earthquake. Maximum recorded amplitudes were fairly small, all under 0.5 m, but in enclosed reservoirs, oscillations lasted more than 2 h. McGarr and Vorhis concluded that the distribution of seiches was more dependent upon geologic and seismic factors than hydrodynamic ones. More recently, Barberopoulou et al.39 examined long-period eﬀects from the 2002 Denali Earthquake on water bodies throughout the Puget Sound, where vertical oscillations reached 0.8 m near Seattle. As with harbor seiching, the most damaging aspect of most seismic seiching comes from horizontal forces; nevertheless, the vertical oscillations are short-term, potentially signiﬁcant changes in water level.

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36.3.7. Uplift and subsidence The present day coast is a tectonic collage of various terrains formed by millennia of plate movements, subduction, uplift, glacial rebound, and sediment buildup. Continental uplift and subsidence is ongoing for most coastlines of the world, occurring gradually though glacial rebound or more rapidly through seismic uplift or subsidence. Parts of the Alaskan coast have such rapid rates of uplift that localized sea level is dropping. Human actions such as ﬂuid extractions can also cause land subsidence. The shoreline near Long Beach, CA experienced over 3 m of localized subsidence due to oil extraction, before ﬂuid reinjection was initiated to slow this eﬀect. Castle et al. found subsidence rates in the Los Angeles Basin averaged over 10 mm/year from the mid-1940s to the mid-1970s due to groundwater extraction, oil extraction, and natural sediment compaction.40 Such changes in land elevation, natural or human induced, must be considered in any examination of localized changes in sea level. 36.4. Coastal Engineering, Coastal Zone Management, and Sea Level Rise 36.4.1. Potential consequences of sea level rise The most obvious consequence of a large rise in sea level will be changes in areas that are submerged. Lands that now are only wet at high tide could be wet most of the day. Structures that are built above the water, like docks and piers, will be closer to the water, or eventually submerged. A second consequence will be an increase in wave energy since wave energy is a function of the square of wave height and wave height is strongly inﬂuenced by water depth. The consequences of a signiﬁcant rise in sea level are far reaching. The amount of sea level rise modeled by any of the IPCC scenarios would cause enormous changes to the coast over the next 100 years. These changes can be expected to be larger and more rapid if there is any increase or acceleration in future melting of glaciers and polar ice caps (an aspect of sea level change that was excluded from the IPCC sea level trends). Along the Paciﬁc coast, an analogy for sea level rise is thought to be El Ni˜ no, where a signiﬁcant rise in sea level might be an extreme El Ni˜ no or an El Ni˜ no on steroids. One of the factors that contributed to the amount of damage caused by the 1982/1983 El Ni˜ no was that several storms coincided with high tide events and the elevated water levels (from tides and low pressure system combined) brought waves further inland than would have occurred otherwise. Table 36.5 shows a summary of the major consequences of a rise in sea level. 36.4.1.1. Ports, harbors, and marine facilities Much of the infrastructure of a port or harbor will be aﬀected by a change in sea level. So too will marine terminals and oﬀshore structures. All of the horizontal elements, such as the decking of wharves and piers, will be exposed more frequently to uplift forces larger than those occurring now. Compared to current conditions, ships will ride higher at the dock and cargo-handling facilities will have less access

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Major consequences of sea level rise.

Ports and harbors • • • • •

Wharves can experience more uplift from waves — increased storm damage Ships will be higher than the docks — cargo facilities will be less eﬃcient Breakwaters and jetties will have less freeboard — increased storm damage Channels will be deeper — decreased need for dredging Tidal prism will increase — greater scour of all foundations

Seawalls and other engineered structures • More regular exposure to wave energy — increased storm damage • Exposure to deeper water — greater foundation scour Groundwater • Increased salt water intrusion to coastal aquifers Wetlands • • • •

More areas will be inundated Saltwater/freshwater interface and zone of brackish water will migrate inland Tidal prism will increase — potentially greater scour and removal of sediment Gains or losses in area will depend on: Ability of wetland to migrate inland Ability of wetland to migrate up, with greater trapping of sediment Overall change in tidal range

Beaches • More area will be inundated • Rule of thumb is that 1 of rise will cause about 50 to 100 beach loss Coastal bluﬀs • More regular exposure to wave energy — increased bluﬀ retreat • Exposure to deeper water — greater scour and undercutting

to the holds of ships. Loading and unloading may have to be scheduled for low tide periods to allow greatest access into the ship, or else mooring and cargo handling facilities will need to be elevated. If breakwaters or jetties protect the harbor, these structures will become less eﬃcient as water levels increase. The breakwaters and jetties will need to be enlarged and heightened to keep up with the rise in sea level, or the harbor will have to accept a higher level of overtopping and storm surge, and a higher probability of storm damage. The increase in water level could also increase the tidal prism of the harbor, resulting in increased scour at the foundations of any structures in the harbor. So, it may also be necessary to reinforce the base of the breakwater or jetty to insure stability. Changes in water depth can also alter harbor seiching/harbor resonance characteristics. Harbors with resonance problems may develop diﬀerent or new seiching problems; harbors without recorded seiching problems may develop them. Beneﬁts that could occur from a rise in sea level would be the opportunity for harbors to accommodate deeper draught ships and a decrease in dredging to maintain necessary channel depths.

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36.4.1.2. Seawalls and other engineered shoreline protection Seawalls will experience many of the same eﬀects as oﬀshore structures. The foundations would be exposed to greater scour and the main structures would be exposed to greater and more frequent wave forces. Overtopping and runup will be greater and more frequent, with possible nonlinear ampliﬁcation of the impact force.41 As with breakwaters and jetties, these structures will need to be reinforced to withstand these greater forces, or a lower level of protection will have to be accepted for the backshore property. 36.4.1.3. Wetlands Coastal wetlands will be greatly modiﬁed by changes in sea level; however, the consequences will vary with the diﬀerent wetland areas. Overall, there will be greater areas of inundation. The change in the intertidal area will depend on local topography, the future change in tidal range, and the ability of the wetland to migrate both up and inland. Historically, many wetlands have accommodated the rise in sea level by increasing the base elevation. Sediment collects in the roots and vegetative mass of the wetland and provides a substrate for new growth. If the rate of sediment entrapment equals the rate of sea level rise, the wetland will remain fairly constant. If the rate of sedimentation exceeds the rate of sea level rise, the wetland will convert to a wet meadow or other system with more supratidal vegetation. If the sediment rate is less than the rise in sea level, the wetland will transition to intertidal and subtidal habitat. Wetland changes also will be aﬀected by inland development. Historically, wetland areas migrated both upward and landward as they were inundated. If the inland area has a slope and soil composition that can support a wetland and is not already developed, then inland migration may be possible. If there is a steep bluﬀ or some type of ﬁxed development such as a highway or bulkhead inland of a wetland, inland migration will not be possible and the wetland area will diminish over time. Another physical change to wetlands in response to a rise in sea level is an increase in the tidal currents, with the potential for increased scour. For estuarine systems there will be a shift in the location of the saltwater–freshwater interface, and an inland movement of the zone of brackish water. Wetlands have survived for millennia by moving up and inland. If existing development or other barriers restrict inland migration, as is often the case, wetland areas will narrow and eventually be converted to subtidal and submerged habitat. Various climate change models have attempted to examine changes in run-oﬀ and sedimentation from global warming. Two global circulation models predict both increases in the number of storm events entering the US from the Paciﬁc and an increase in run-oﬀ state-wide. Along coasts such as California, where rivers are a major source of coastal sediment supplies, the eﬀects from increased runoﬀ and sediment loadings will greatly modify coastal conditions. There is considerable uncertainty about the eﬀects of climate change to hydrologic cycles, particularly on the regional and watershed level and the changes to coastal sediment supplies may change greatly from current levels. For example, some models

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indicate that, as less precipitation is stored as snow, run-oﬀ in California could increase by about 25% over the next 25–35 years and by 100% over the next 100 years.42 This could increase vertical stratiﬁcation in the water column and aﬀect circulation in estuaries, lagoons, and nearshore coastal areas. Increased runoﬀ could also increase transport of nutrients, suspended sediments, and bedload to the coast.

36.4.1.4. Beaches and coastal bluﬀs Open coastal landforms like beaches and bluﬀs will be exposed to greater and more frequent wave attack. There will more potential for erosion and shoreline retreat. For gently sloping beaches, the general rule of thumb (Bruun rule) is that 50–100 m of beach width will be lost from use for every meter of sea level rise. This accounts only for inundation and does not address the increased erosion and backshore changes when these areas are exposed to more frequent wave attack. If additional coastal sediment supplies do not increase to match the rates of inundation, or if beaches are not able to roll over or migrate inland, large expansions of dry beach area could be lost. Coastal bluﬀs too can be greatly changed. As coastal bluﬀs lose the buﬀering beneﬁts of coastal beaches, bluﬀs will be exposed more and more to wave attack. The increased water levels would cause increased wave energy at the base of coastal bluﬀs and an increase in the frequency of wave attack. Collins et al.43 have identiﬁed empiric threshold wave levels that can trigger bluﬀ failure. As sea level and wave energy increase, there will be an increase in the number of days when bluﬀ failure is possible. While precipitation induced failures are predicted to occur at a constant level, the wave-induced failures are predicted to occur in an increasing frequent basis. For Paciﬁca, CA, one of the study sites, the failure triggering events could reach over 165% of the current number of failures per year.

36.4.2. Responses to sea level rise Responses to a rise in sea level will vary. The actions that can be taken in response to sea level rise include hard engineering, such as seawalls, revetments, breakwaters, levees, and other structures built to protect inland areas, soft engineering, such as beach nourishment, dune nourishment or vegetated buﬀers, accommodation/ adaptation, and retreat. There may be appropriate times for each response to be considered. However, some responses, such as soft engineering or retreat, are often best applied on a large or multi-lot scale, while hard engineering often can be applied to individual properties, regardless of the actions taken on surrounding properties. Due to the uncertainty associated with sea level change and the fact that this is but one of the factors that will aﬀect the coast in the future, most of the experts who are developing sea level response scenarios are recommending that solutions be as ﬂexible as possible. A second recommendation is for “no regret” solutions that

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will be beneﬁcial for many reasons and not just to accommodate a single prediction of future sea level rise.44 36.4.2.1. Hard engineering responses Hard engineering has been the default response to many coastal hazards. Hard structures provide a barrier between ocean and the property that is being threatened. As sea level rises, it can be expected that many property owners and communities will consider hard engineering as a response to future increases in sea level. Existing structures would be fortiﬁed to withstand new wave conditions. New seawalls, bulkheads, revetments, breakwaters, and levees could be considered for those areas of the coast that are not now armored. This is consistent with the general response to erosion caused by each El Ni˜ no, where property owners have reinforced the armoring that already existed and introduced armoring in areas that had not yet been protected. When local subsidence decreased the eﬀectiveness of the Redondo, CA Breakwater, a new layer of armor rock was placed in it to restore it to its former level of protection and eﬃciency. Thus, the hard engineering responses have been used historically to address changes in sea level or to potential damage from storms. In most situations, armoring can be designed to protect against future storm conditions. For large increases in sea level, the direct cost of this engineering will be enormous, possibly causing property owners to question the reasonableness of the continued investment in armoring (Table 36.6).

Table 36.6.

Responses to sea level rise.

Hard engineering • Heighten existing structures or add new layers of armor • Build more seawalls/bulkheads/revetments/breakwaters/groins/jetties Soft engineering • Nourish beaches with sand (restore historic sources or ﬁnd new sources) • Perched beaches Accommodation/adaptation • Elevate structures • Switch to salt tolerant crops Retreat Planning and regulatory responses to sea level rise • • • • • • • • • •

Setbacks for new development Engineering design of new structures on the coast Wetland buﬀers Notify property owners of risks with building on the coast Prohibit future shoreline protection for new development Monitor for subsidence from ﬂuid withdrawal projects Studies of shoreline change Encourage beach nourishment and opportunitistic nourishment Coastal land acquisition Public education programs

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36.4.2.2. Soft engineering Soft engineering is usually thought to be beach or dune nourishment or the creation of perched beaches. Over the short-term, beach nourishment will continue to be a very eﬀective way to protect the backshore and at the same time provide access and a recreational beach. A single beach nourishment eﬀort is not a permanent response; it must be maintained on a regular basis to remain eﬀective. This can allow an opportunity for periodically reviewing the eﬀectiveness of this response. Beach nourishment can provide some level of protection while keeping open other options. For locations with a high demand for recreational beach area, but where there is not an adequate volume of sand to supply a nourished beach, a perched beach can be an eﬀective option. Natural beaches depend upon sand from a zone of active transport that extends oﬀshore to water depths of about 10 m below mean lower low water. A perched beach reduces the oﬀshore dimension of the beach and can provide a cost-eﬀective way to maintain the dry sand component of the beach without also building up the vast oﬀshore portion. Long-term eﬀectiveness of a perched beach would require maintenance of the sill at the toe of the beach, as well as routine additions of sand to maintain the dry beach elevation above water level. Both hard engineering and soft engineering options are ways to work against sea level rise. The hard engineering options will strengthen existing bluﬀs or structures so that they will be able to withstand the increased energy of a higher sea. The soft engineering responses will provide a buﬀer between the waves and the backshore and thus reduce the energy and damage that would occur at the backshore.

36.4.2.3. Accommodation/adaptation Another approach to varying sea level is to accommodate or adapt to the changes. Coastal systems have used adaptation for millennia as a response to climate change. Adaptive capacity refers to the ability of a system, individual, organization, or community to adjust to changing conditions, take advantage of opportunities to cope with the consequences, with the result that damages from the climate change can be moderated. If structures are at risk, they can be raised so they will not be inundated. If agricultural lands are at risk, there may be options to switch to salt tolerant vegetation. Islands or spits could be elevated to keep pace with sea level rise. Some of the more accepted adaptation techniques include the use of elevation, setbacks and buﬀers to provide physical separation between development and either hazard or a resource that needs to be protected from development impacts. Rolling easements have been suggested as a way to site development in a hazardous area and then move the development back from the hazard when the separation decreases. Approaches for adaptation or accommodation often require more “handson” involvement than most hard engineering approaches. Accommodative/adaptive responses are rarely one-time actions; to be most eﬀective they require continued ﬂexibility and modiﬁcation to meet the ongoing conditions.

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36.4.2.4. Retreat Historically, one of the natural responses to rising sea level has been retreat. Wetlands have demonstrated this as an eﬀective response to sea level rise. Retreat has also been used in some situations to protect development or movable property. During the 1982/1983 El Ni˜ no storms, a mobile home subdivision in Paciﬁca, CA lost the seaward most row of lots. The response was to pull the facilities landward and reduce the number of available sites. During the storms of 1997/1998, a different part of Paciﬁca, CA was attacked by waves and 10 homes were eventually removed from the bluﬀ top. In southern Big Sur, portions of Paciﬁc Coast Highway have been threatened by bluﬀ retreat. Small amounts of revetment have been built to protect the most threatened portions, but the overall plan is to relocate several miles of the road to a safer, more inland location. Such examples of retreat may be more common in the future, as the threat from coastal erosion becomes more pervasive and the long-term costs of in-place protection increase. 36.4.3. Planning and regulatory responses for sea level rise “Society can neither control, nor at present precisely determine, the sensitivity of the earth’s climate system to rising GHG concentrations. As a result, society must consider the implications of a range of climate sensitivities when evaluating the risks of climate change and devising policies to manage the one factor we can control: our own GHG emissions.”45

Coastal zone management and land use planning are important tools for addressing sea level rise. New development can be located far enough away from erosion hazards or bluﬀ retreat, through either elevation or setbacks, to assure site stability for foreseeable future conditions. Individual property owners can undertake many of these actions voluntarily, but in many situations, safe setback and elevation design conditions are required through the regulatory process. Regulatory eﬀorts also can establish wetland buﬀers to allow future inland or upland migration of wetlands. Deed restrictions concerning hazardous conditions can be used to notify current and future property owners that the conditions at a site may worsen with time. Along coastal areas where hard engineering can reduce the quality or quantity of public beach access and recreation, planning agencies can use the idea of “assumption of risk” to put property owners on notice that they will need to address the future consequences of sea level rise without actions such as seawalls or revetments that would diminish the public beach. Few states have passed special regulations to address sea level rise. Instead, most states have coastal programs that address sea level rise as an ongoing coastal condition and they modify or adapt current regulatory mechanisms to cover the eﬀects of sea level rise. Texas has a rolling easement program that relocates the public land boundary to the current line of vegetation. States like South and North Carolina and Massachusetts have prohibited the construction of any hard shoreline armoring. This limits the responses that can be used to address sea level rise to soft engineering, accommodation, and retreat. Maine has regulations that prohibit rebuilding structures that have been damaged by storms if the new structure could

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reasonably be expected to be damaged within the next 100 years. This regulation covers damage from sea level rise or other hazards. The US Army Corps of Engineers is now incorporating adaptive planning for new projects, requiring consideration of a range of sea level conditions in the feasibility analysis of all coastal and estuarine projects. The planning guidance requires, “A sensitivity analysis should be conducted to determine what eﬀect (if any) changes in sea level would have on plan evaluation and selection. The analysis should be based, as a minimum, on the extrapolation of the local, historical record of relative sea level rise as the low level and Curve 3 from the NRC report [a rise of 1.5 m by 2100] as the high level.”46

Sustainability and integrated coastal management are part of a new trend in coastal management and they oﬀer opportunities to address many future coastal concerns. The European Union has adopted eight principles of best practice for integrated coastal zone management:47 (1) (2) (3) (4) (5) (6) (7) (8)

A broad “holistic” perspective (thematic and geographic) A long-term perspective Adaptive management during a gradual process Reﬂect local speciﬁcity Work with natural processes Participatory planning Support and involvement of all relevant administrative bodies Use of a combination of instruments.

During times of rapidly changing sea level, the coastal zone is the area that experiences the most dynamic changes from both rising water levels and adjustments from the inland areas. There is great uncertainty in the potential changes to water level from global warming and this uncertainty will be combined with ongoing uncertainties concerning water level changes from tides, waves, seiching, river ﬂows, atmospheric forcing, uplift, and subsidence. Engineering or planning responses that are based upon any single ﬁxed water level will have great diﬃculty maintaining eﬀectiveness during a period of rising sea level. The principles of sustainability and reliance upon multiple adaptive responses to the array of changing coastal conditions can provide many useful approaches for dealing with future changes in sea level. 36.5. Conclusions Global sea level has been rising since the end of the last ice age about 18,000 years ago. The rate of rise has varied and for short periods of time, such as the Little Ice Age, sea level has even dropped. However, based on current climate models and projections of greenhouse gas emissions, it is expected that future sea level will rise at a greater rate than it has over the past hundred years. The continued rise in sea level will increase inundation of low coastal areas. Nearshore wave heights and wave energy will increase, increasing the potential for storm damage, beach erosion, and bluﬀ retreat. Ports and harbors will have reduced cargo transfer capability as ships ride higher along the dock. Wetlands may be

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inundated if they are not able to migrate either upward or landward. Almost all coastal systems will be aﬀected; even groundwater aquifers will be a greater risk from saltwater intrusion. Sea level rise is not a new phenomenon; it has been a major component of coastal change throughout time. While there are a number of possible responses to future changes in sea level, in many situations, response ﬂexibility will be important. Reliance on a single response for a certain water level may be useful while that water level occurs. But, as either the water level continues to rise or the coastal land subsides, the ﬁxed response will become less eﬃcient and be less appropriate to the new conditions. Along most coastal areas, it is likely that a combination of hard engineering, soft engineering, accommodation/adaptation, and retreat responses will be considered to address sea level rise. There are situations where each response may be appropriate and well suited. In all coastal projects, it is important to recognize and accept that there will be changes in sea level and in other coastal processes over time. Careful review, siting and permitting of new projects on the coast can increase the likelihood that these projects will be able to adapt and change to accommodate future coastal hazards. Public education and eﬀorts to alert property owners to the dynamic nature of the coast will be important. The recent interest in coastal sustainability oﬀers planning opportunities for addressing the consequences of sea level rise. Sustainability principles espouse the need for integrated planning and the incorporation of a holistic approach to the coast, with strong consideration of local conditions and the need for adaptive planning to adjust to changing conditions. Acknowledgments The author would like to thank Meg Caldwell and Costas Synolakis for conversations past and future about the engineering and management concerns relating to a world of changing sea level. Thanks to Eli Davidian and Susan Hansch for their input throughout, Mark Johnsson for his geologic input and generously sharing material from his library, and Dr. Kim for starting this project. Portions of this paper have been developed from earlier work: (1) L. Ewing, J. Michaels, and R. McCarthy (1990) Draft Report: Planning for an Accelerated Sea Level Rise along the California Coast, California Coastal Commission, San Francisco, CA and (2) California Coastal Commission Staﬀ (2001) Overview of Sea Level Rise and Some Implications for Coastal California. References 1. J. Kennett, Marine Geology (Prentice-Hall, Englewood Cliﬀs, 1982). 2. W. C. Pittman, III, The eﬀect of eustatic sea level changes in stratigraphic sequences at Atlantic margins, Am. Assoc. Petrol. Geol. Memoir, 29 pp., in J. Kennett, op. cit. 3. J. Kennett, op. cit.

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4. R. G. Fairbanks, A 17,000-year glacio-eustatic sea level record: Inﬂuence of glacial melting rates on the younger dryas event and deep-ocean circulation, Nature 342(6250), 637–642 (1989). 5. J. A. Church, J. M. Gregory, P. Huybrechts, M. Kuhn, K. Lambeck, M. T. Nhuan, D. Qin and P. L. Woodworth, Changes in sea level, IPCC, Third Assessment Report (Cambridge University Press, Cambridge, 2001), http://hdl.handle.net/10013/epic. 15081. 6. S.-Y. Yu, B. E. Berglund, P. Sandgren and K. Lambeck, Evidence for rapid sea-level rise 7600 yr ago, Geo. Soc. Am. 35, 891–894 (2007), doi: 10.1130/G23859A.1; 3 ﬁg. 7. J. A. Church and N. J. White, A 20th century acceleration in global sea-level rise, Geophys. Res. Lett. 33, L01602 (2006), doi: 10.1029/2005GL024826, in IPCC (2007). 8. S. J. Holgate and P. L. Woodworth, Evidence for enhanced coastal sea level rise during the 1990s, Geophys. Res. Lett. 31, L07305 (2004), doi: 10.1029/2004GL019626. 9. E. W. Leuliette, R. S. Nerem and G. T. Mitchum, Calibration of TOPEX/Poseidon and Jason altimeter data to construct a continuous record of mean sea level change, Mar. Geodesy 27(1–2), 79–94 (2004). 10. IPCC observations: Climate change and sea level rise, Fourth Assessment Report (2007); J. A. Church and N. J. White, op. cit.; S. J. Holgate and P. L. Woodworth, Evidence for enhanced coastal sea level rise during the 1990s, Geophys. Res. Lett. 31, L07305 (2004), doi: 10.1029/2004GL019626; E. W. Leuliette, R. S. Nerem and G. T. Mitchum, Calibration of TOPEX/Poseidon and Jason altimeter data to construct a continuous record of mean sea level change, Mar. Geodesy 27(1–2), 79–94 (2004). 11. IPCC, Third Assessment Report (2001). 12. C. B. Field, Confronting Climate Change in California: Ecological Impacts on the Golden State (Union of Concerned Scientists Publication, Cambridge, MA, 1999). 13. IPCC, Observations: Oceanic Climate Change and Sea Level (2007). 14. S. Levitus, J. I. Antonov and T. P. Boyer, Warming of the World Ocean, 1955–2003, Geophys. Res. Lett. 32, L02604 (2005), doi: 10.1029/2004GL021592, in IPCC (2007). 15. S. Levitus, op. cit. 16. V. Gornitz, Impoundment, groundwater mining, and other hydrologic transformations: Impacts on global sea level rise, Sea Level Rise: History and Consequences, Vol. 75, eds. B. C. Douglas, M. S. Kearney and S. P. Leatherman (Academic Press, San Diego, International Geophysics Series, 2001). 17. J. T. Overpeck, B. L. Otto-Bliesner, G. H. Miller, D. R. Muhs, R. B. Alley and J. T. Keihl, Paleoclimatic evidence for future ice-sheet instability and rapid sea-level rise, Science 311, 1747–1750 (2006); J. M. Gregory, P. Huybrechts and S. C. B. Raper, Climatology threatened loss of Greenland ice-sheet, Nature 428, 616 (2004), doi: 10.1038/428616a. 18. IPCC Observations: Climate change and sea level rise, Fourth Assessment Report (2007). 19. D. E. Cartwright, Tides: A Scientific History (Cambridge University Press, Cambridge, 2000). 20. R. E. Flick, Comparison of California tides, storm surges, and mean sea level during the El Ni˜ no winters of 1982–83 and 1997–98, Shore Beach, J. ASBPA 66(3), 7–11 (1998). 21. F. J. Wood, The strategic role of perigean spring tides in nautical history and North American coastal ﬂooding, 1635–1976, U.S. Department of Commerce, NOAA (1978), Coastal Processes with Engineering Implications, eds. R. G. Dean and R. A. Dalrymple (Cambridge University Press, 2002).

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22. American Society of Civil Engineers Hurricane Katrina External Review Panel, The New Orleans Hurricane Protection System: What Went Wrong and Why (ASCE Publications, 2007). 23. R. E. Flick, J. Murray and L. C. Ewing, Trends in United States tidal datum statistics and tide range, J. Waterw. Port Coast. Ocean Eng., ASCE, 129(4), 155–164 (2003). 24. R. S. Nerem and G. T. Mitchum, Observation of sea level change from satellite altimetry, Sea Level Rise: History and Consequences, Vol. 75, eds. B. C. Douglas, M. S. Kearney and S. P. Leatherman (Academic Press, San Diego, International Geophysics Series, 2001). 25. NOAA National Weather Service, Service Assessment, Hurricane Katrina August 23–31, 2005, June 2006, http://www.weather.gov/os/assessments/pdfs/Katrina.pdf. 26. American Society of Civil Engineers Hurricane Katrina External Review Panel, op. cit. 27. R. G. Dean and R. A. Dalrymple, Coastal Processes with Engineering Applications (Cambridge University Press, 2002). 28. U. Kanoglu and C. E. Synolakis, Long wave runup on piecewise linear topographies, J. Fluid Mech. 374, 1–28 (1998). 29. N. E. Graham and H. F. Diaz, Evidence for intensiﬁcation of North Paciﬁc winter cyclones since 1948, Bull. Am. Meteorol. Soc. 82, 11869–11892 (2001). 30. R. L. Wiegel, Oceanographic Engineering (Prentice-Hall, Englewood Cliﬀs, NJ, 1964). 31. K. Iida, Magnitude, energy, and generation of tsunamis, and catalogue of earthquakes associated with tsunamis, Proc. Tsunami Mtgs. Assoc, with the 10th Pac. Sci. Cong., Monograph No. 24, IUGG, 1963 (cited in R. L. Wiegel, op. cit.). 32. C. E. Synolakis, J.-P. Bardet, J. C. Borrero, H. L. Davies, E. A. Okal, E. A. Silver, S. Sweet and D. R. Tappin, The slump origin of the 1998 Papua New Guinea tsunami, Proc. Royal Soc. (London), Ser. A 458, 763–789 (2002). 33. C. E. Synolakis, J. Borrero and R. Eisner, Developing inundation maps for Southern California, Proc. Solutions to Coastal Disasters, ASCE (2002). 34. C. E. Synolakis and L. Kong, Runup measurements of the December 2004 Indian ocean tsunami, EERI Spectra 22(S3), S67–S91 (2006). 35. Humboldt County Community Development Services, Samoa Town Master Plan, Recirculation Draft 2: Tsunami Vulnerability Evaluation (2007), http://co.humboldt. ca.us/planning/samoa/docs/. 36. R. L. Wiegel, Oceanographic Engineering (Prentice-Hall, Englewood Cliﬀs, NJ, 1964). 37. S. J. Bolsenga and C. E. Herdendorf (eds.), Lake Erie and Lake St. Clair Handbook (Wayne State University Press, Detroit, 1993). 38. A. McGarr and R. C. Vorhis, Seismic Seiches from the March 1964 Alaska Earthquake, U.S. Geol. Surv. Profess. Pap. 544E, E1–E43 (1968). 39. A. Barberopoulou, A. Qamar, T. L. Pratt and W. P. Steele, Long-period eﬀects of the Denali Earthquake on water bodies in the puget lowland: Observations and modeling, Bull. Seism. Soc. Am. 96(2), 519–534 (2006). 40. R. O. Castle, M. R. Elliott, J. P. Church and S. H. Wood, The evolution of the Southern California uplift, 1955 through 1976, U.S. Geol. Prof. Paper #1362 (1984). 41. U. Kanoglu and C. E. Synolakis, op. cit. 42. D. Scavia et al., Climate change impacts on US coastal and marine ecosystems, Estuaries and Coasts 25(2), 149–164 (2002). 43. B. D. Collins, R. Kayen and N. Sitar, Process-based empirical prediction of landslides in weakly Lithiﬁed coastal cliﬀs, San Francisco, California, USA, Landslides and Climate Change — Proceedings of the International Conference on Landslides and Climate Change, eds. R. McInnes, J. Jakeways, H. Fairbank and E. Mathie, Isle of Wight, United Kingdom, May 2007, Taylor & Francis, London, pp. 175–184.

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44. D. M. Mageean, A. Constable and M. D. Van Arsdol, Jr., Impacts of rising sea level on coastal populations in California and Maine, ISA Research Committee #24, Montreal, Canada (1998). 45. D. Cayan, A. L. Luers, M. Hanemann, G. Franco and B. Croes, Scenarios of climate change in California: An overview, CEC-500-2005-186-SF (2006). 46. US Army Corps of Engineers, ER 1105-2-100 Planning Guidance Notebook, Appendix E, Section IV.E-24.k, April 2000. 47. Commission of the European Communities, Communication from the Commission to the Council and the European Parliament on Integrated Coastal Zone Management: A Strategy for Europe; Annex 1 (COM(2000) 547 final, 2000; http://ec.europa.eu/ environment/iczm/comm2000.htm).

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Chapter 37

Sea Level Rise and Coastal Erosion Marcel J. F. Stive∗ , Roshanka Ranasinghe∗,†,§,¶ and Peter J. Cowell‡ ∗

Department of Hydraulic Engineering Delft University of Technology P. O. Box 2600 GA, Delft, The Netherlands [email protected] † Department of Water Engineering UNESCO-IHE, Delft, The Netherlands ‡ School of Geosciences, University of Sydney NSW 2006, Australia § Department of Environment and Climate Change Sydney, NSW 2001, Australia ¶ [email protected]

IPCC projections indicate that the rate of sea level rise (SLR) during the 21st century may be about an order of magnitude greater than the 20th century rate of 1–2 mm/year. This accelerated SLR will in turn result in much faster coastline retreat with particularly severe impacts on low-lying areas. The socioeconomic impact of such accelerated coastline retreat could be massive due to the rapid growth of coastal communities and infrastructure over the past ﬁve or six decades. The method most commonly used to estimate coastline retreat due to SLR is the simple two-dimensional mass conservation principle known as the Bruun rule. However, in view of the high level of predictive accuracy that is clearly needed to facilitate informed planning decisions for the future, can we continue to depend on the Bruun rule? This chapter discusses the evidence for and against the Bruun rule and suggests alternative methods that may be more suitable for the 21st century.

37.1. Introduction The IPCC1 projections for 21st century sea level rise (SLR) range from 0.18 to 0.79 m by 2090–2099 relative to 1980–1999, including an allowance of 0.2 m for uncertainty associated with ice sheet ﬂow. Very recent research also suggests that the measured SLR over the last decade is under predicted by the IPCC models,2 and that a maximum SLR of 1.4 m by 2100 (relative to 1990 levels) is not unlikely.3 1023

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Compared to the 1–2 mm/year rate of SLR that was experienced in the last century, these 21st century projections constitute an order-of-magnitude increase in the rate of SLR. It has been long known that, in the absence of other compensatory mechanisms, any rise in the mean sea level will generally result in the retreat of unprotected coastlines.4 Fortunately, the still-stand (slow) conditions of SLR during the last century has resulted in slow and mostly manageable coastline retreat (recession). However, the potential order-of-magnitude increase in the rate of SLR in the 21st century is likely to result in much faster coastline recession. The socioeconomic impact of such accelerated coastal recession could be massive due to the unprecedented growth of coastal communities over the past 50 years or so which has led to billions of dollars worth of developments and infrastructure within the coastal zone. To ensure the safety of growing coastal communities and to avoid massive economic losses in the future, it is now imperative that any predictions of coastal recession due to SLR be highly accurate. The IPCC has successfully raised awareness on political and societal levels, which in many nations has resulted in including SLR scenarios in new designs of shore protection works, both hard (structures) and soft (nourishment) or combinations thereof.5 In this context, it is crucial that the impact of SLR in low-lying coastal areas be quantiﬁed accurately. The inundation of such low-lying areas will result in signiﬁcant coastline retreat, the magnitude of which is governed by the local coastal slope. As coastal slopes in such areas may be as mild as 1 in 1,000, the resulting coastline retreat could be three orders of magnitude greater than the rate of SLR. Quantifying the impact of SLR on dune and barrier coasts is less straightforward. This is because the response of dune and barrier coasts to SLR is a complex morphodynamic issue. The most commonly used method to quantify the recession due to SLR at such coastal locations is the simple two-dimensional mass conservation principle known as the Bruun rule, which predicts a landward and upward displacement of the cross-shore proﬁle in response to a rise in the mean sea level. For many of the world’s coastlines, where the nearshore beach slope is about 0.01–0.02, the Bruun rule predicts a coastline retreat between 50 × SLR and 100 × SLR, which are proportionalities that are commonly used as a rule of thumb. Although, coastal scientists and engineers have been routinely using the Bruun rule for almost ﬁve decades, mainly due its simplicity and the lack of any other easyto-use alternative methods, it has been receiving some heavy criticism in the recent past.6,7 While the many attempts to verify the Bruun rule against ﬁeld and laboratory data over the last four decades (e.g., Refs. 8–12) have qualitatively conﬁrmed the basic concept of the Bruun rule, not many have resulted in convincing quantitative comparisons between measurements and Bruun rule predictions.13 Zhang et al.14 who undertook a large-scale study of a 220-km stretch of the US East coast is the only study which resulted in reasonably good comparisons between measured shoreline recession and Bruun rule predictions. However, does even a good comparison between measured and predicted values under present still-stand SLR conditions (1–2 mm/year SLR) mean that the Bruun rule is conclusively validated? Based on the observations made in the USA, the Netherlands, the Mediterranean, and Australia, Stive15 showed that the net natural

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shoreline change due to cross-shore processes is ∼1 m/year, whereas the net natural shoreline change due to longshore processes is ∼0.1–1 m/year. However, the Bruun rule predicts a recession rate between 0.1 and 0.2 m/year for the current long-term SLR rate of about 2 mm/year (using the rule-of-thumb approach). Thus, the Bruun eﬀect (i.e., the coastal recession due to eustatic SLR alone) is, at best, an order of magnitude less than the observed net natural shoreline changes. This means that the Bruun eﬀect are likely to be overridden by other coastal processes under the present still-stand SLR conditions. Therefore, any good comparisons between shoreline recessions measured in the last century and the corresponding Bruun rule predictions are likely to be fortuitous and cannot be considered as conclusive veriﬁcation of the Bruun rule. However, it should be noted that the quantitative accuracy of the Bruun rule has been validated for contemporary systems that have undergone a rapid relative SLR due to subsidence.9,16 Holocene coastal evolution modeling using the Bruun concept under considerable rates of SLR and fall also lends convincing support to the concept.17,18 It is also noteworthy that large- and small-scale laboratory tests of dune erosion under high storm surge levels (e.g., Refs. 19 and 20) suggest a Bruun-type response at the much shorter timescales of storm duration (hours-day). Although the available evidence suggests that the Bruun rule is most likely conceptually correct, arguably it addresses only one potentially important eﬀect of a range of eﬀects. If no other sediment sources or sinks are present or if no other sediment transport gradients in cross-shore and longshore directions prevail, the Bruun eﬀect is the only operational eﬀect. However, this idealized situation is the exception rather than the rule, and there are other eﬀects of SLR on the coastal sediment budget, which are generally much larger, or at least of the same order of magnitude as the Bruun eﬀect. In the following section we present the available evidence pointing to the presence of mechanisms other than the Bruun eﬀect, which may play an equal or more crucial role in governing coastline retreat/advance.

37.2. What Is the Evidence? A general point of view, triggered by Bird,21 is that since 70% of the world’s sandy beaches are in a state of erosion, global SLR has to be the most probable cause (cf. Ref. 22). However, there are numerous coastal systems that have been accretive in the Holocene, even though sea level was rising. A few examples are the Australian coast,23 deltaic coasts (Mississippi, Ebro, Po, Yangtze, and many other deltas at earlier stages of the Holocene) and composite coasts such as the US Northwest Washington coast and the Dutch coast.17 This implies that there must be a number of other processes that can override the Bruun eﬀect, which is generally erosive, to such an extent that the resultant response is coastline advance. In contrast, many other coasts experience larger erosion than is explained by the Bruun eﬀect. This implies that there must be coastal processes other than the Bruun eﬀect that contribute to coastline retreat/advance. The important question then is, what are these other processes and are they likely to be aﬀected by the accelerated SLR that is likely to occur in the 21st century?

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The other processes that may govern coastline retreat/advance may be collectively referred to as “sediment availability,” which is implicitly included in earlier ﬁrst approximation kinematic models of long-term (millennia) coastal change.24 Swift25 extended Curray’s ideas into a general framework for long-term coastal change entailing transgression (landward retreat) and regression (seaward advance) of the shoreline due to SLR and fall, respectively, with corresponding tendencies toward retrogradation and progradation due to net sediment losses or gains from alongshore gradients in sediment transport. Cowell et al.26 show how Swift’s concepts can be quantiﬁed and related back to the Bruun rule, when upper shoreface sediment balance is considered. Cowell et al. assume that, to a ﬁrst approximation, the upper shoreface is form invariant relative to mean sea level over time periods for which proﬁle closure occurs (1 year).27 The upper shoreface is represented by an arbitrary, but usually concave-up, proﬁle h(x) to a depth h∗ (a morphologically active depth) and a length L∗ , in which x is the distance from the shore.28 Assuming that the cross-shore proﬁle shape remains constant over time, sediment-volume conservation for proﬁle kinematics requires (for a Cartesian coordinate system with seaward and upward directions positive) that ∂h ∂h + cp =0 ∂t ∂x

(37.1)

where cp is the horizontal proﬁle displacement, or via h = MSL-zb , where MSL is mean sea level and zb is the bottom level: ∂zb ∂zb ∂M SL + cp = ∂t ∂x ∂t

(37.2)

where cp is the horizontal translation rate of the shoreline position. The sedimenttransport balance equation for a ﬁxed spatial control volume is ∂zb ∂qx ∂qy + + +s=0 ∂t ∂x ∂y

(37.3)

where qx and qy are the cross-shore and alongshore sediment transports, and s is a local source or sink. These equations may be combined to yield cp = −

∂M SL ∂t

∂h ∂x

−1

−

∂qx ∂qy − ∂h ∂y

∂h ∂x

−1

−s

∂h ∂x

−1 (37.4)

or, after cross-shore integration over L∗ , cp h∗ =

∂M SL ∂Qy L∗ − (qx,sea − qx,dune ) − −s ∂t ∂y

(37.5)

in which Qy is the alongshore transport integrated over L∗ . In the absence of littoral transport gradients and other sources or sinks (including sand exchanges with the lower shoreface and backbarrier), the above

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transgression

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7200

E A T R T R E

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750s

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source m 3 m -1a -1

20

retrogradation

m 3 m -1a -1

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N C E A V A D progradation and regression

10

Fig. 37.1. Evolution of the central Netherlands coast (Hoek van Holland to Den Helder) as a time trajectory in sediment-supply/accommodation phase space (abscissa and ordinate, respectively, scaled in cubic meters per year per meter of shoreline). Numbers along the trajectory indicate time (years BP); suﬃxes n and s denote north and south of Haarlem, respectively (after Ref. 26).

reduces to the standard Bruun rule4,13 : ∂M SL cp = ∂t

L∗ . h∗

(37.6)

Equation (37.5) is similar to the Dean and Maurmeyer’s29 generalized version of the Bruun rule, an analytic precursor of the coastal-tract concept. The source and sink terms in Eq. (37.5) allow the qualitative Curray–Swift model of coastal evolution to be quantiﬁed as a time trajectory in sediment source/sink phase space: e.g., evolution of the well-documented central Netherlands coast between Hoek van Holland and Den Helder in Fig. 37.1. The trajectory is derived by applying Eq. (37.5) and based on (a) estimates derived from radiometric data by Beets et al.,30 for the period 5,000–0 years BP and (b) the results of reconstruction simulations for 7,200–5,000 BP. The line separating advance and retreat of the coast is ﬁtted for the trajectory in the top-right quadrant, with its mirror image assumed for the bottom-left quadrant in the absence of other data. The trajectory bifurcates after 2,000 BP because diﬀerences develop in rates of shoreline change averaged alongshore north and south of Haarlem. The shape of the advance/retreat–threshold curve demonstrates that coastal evolution is governed mainly by (a) sediment supply (+/−) under near-still-stand sea-level conditions (such as those predominating in the late Holocene), and (b) change in accommodation space when sea-level changes rapidly (such as during global glaciation and deglaciation).

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What do we learn from this evidence? In periods of near-still-stand sea-level conditions the Bruun eﬀect is operational, but is commonly overridden by the sediment availability terms in Eq. (37.5). Therefore, an understanding of the relative magnitudes of these sediment availability mechanisms and the Bruun eﬀect appears to be crucial in attempting to predict future coastline retreat/advance accurately. Sediment availability mainly consists of three potential contributions: (a) cross-shore contributions, (b) alongshore contributions, and (c) other sources/sinks such as inlets and deltas. General quantiﬁcations of these three contributions are presented in the following three sections (Secs. 37.3–37.5).

37.3. Cross-Shore Processes A general estimate of the contribution of cross-shore processes to coastline retreat/advance can be obtained by considering geologic reconstructions and associated sediment balances. Geologic reconstructions of the Australian23 and the Dutch coast26 have strengthened the hypothesis31 that middle shoreface wave-induced sediment transport is generally onshore on concave-shaped shorefaces. This is associated with wave asymmetry and wave-boundary-layer-induced net ﬂow.32 While this is a contribution that results in shoreline advance, there are also two contributions that result in shoreline retreat, which should be generally considered. One contribution is due to aeolian loss (i.e., wind-driven onshore transport of sediment that is lost from the active upper-shoreface proﬁle) and the other is a virtual loss due to the Bruun eﬀect. The Bruun rule can quantify this virtual loss, theoretically. This latter loss due to the Bruun eﬀect amounts to 50–100 times the SLR rate (in m3 /m/unit time). Under recent near-still-stand SLR conditions of typically 20 cm/century (2 mm/year), the Bruun eﬀect leads to a shoreline retreat of 0.1–0.2 m/year, and an associated virtual loss of 1–2 m3 /m/year (assuming an active proﬁle slope h∗ /L∗ of 1/50–1/100 and an active proﬁle height of 10 m). The loss due to aeolian transport is often an order-of-magnitude larger.26 Therefore, on average, long-term cumulative losses due to SLR and aeolian loss in cross-shore direction are 5–10 m3 /m/year. However, Australian and Dutch observations indicate net cross-shore gains of the order of 0.1–10 m3 /m/year leading to coastline advance. Hence, onshore asymmetry- and boundary-layer-induced onshore transport on the middle shoreface should amount to 5–20 m3 /m/year. In the absence of longshore sediment transport gradients one might therefore observe net chronic (i.e., long-term) and extreme event-driven ephemeral shoreline changes as indicated in Table 37.1. Chronic changes are due to long-term processes such as the Bruun eﬀect and aeolian losses, while ephemeral changes are those that are associated with extreme storm events, which cause dune erosion. The ephemeral changes will be restored in the long term, if no upper shoreface losses due to alongshore transport gradients occur.33 The values given in Table 37.1 are validated for the Dutch and Australian coasts, and are expected to be generally applicable for moderate (lower bound values) to high (higher bound values) energy coasts.

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Table 37.1. Typical cross-shore losses (negative) and gains (positive) and associated shoreline changes (negative = retreat).15 Ephemeral processes∗ Chronic processes Eﬀective proﬁle height Rate of shoreline change due to chronic processes Rate of shoreline change due to ephemeral processes (extreme events or adverse years)∗ Rate of shoreline change due to Bruun eﬀect only ∗ The

O (−102 ) m3 /m/year or/extreme event O (+101 ) m3 /m/year 10 m O (+1) m/year O (−10) m/year or/extreme event O (−0.1) m/year

net ephemeral loss will be zero, as these losses will be recovered in time.

What can we conclude from these cross-shore process quantiﬁcations? Under the present SLR conditions the Bruun eﬀect is at least an order-of-magnitude smaller than contributions from other processes and is therefore negligible. Obviously, if the rate of SLR increases ﬁve- or ten-fold, then the losses due to the Bruun eﬀect will be of the same order-of-magnitude as the gains due to chronic accretionary processes (e.g., wave asymmetry and boundary layer ﬂow), which may even reverse the net shoreline change from advance to retreat.

37.4. Longshore Processes In this section we consider shoreline changes due to gradients in longshore sediment transport. Here, we distinguish low- and high-energy coasts in terms of wave energy, and assume wave-induced surf zone longshore ﬂow to be the driving agent. This is a reasonable assumption along coasts that are not inﬂuenced or interrupted by coastal inlets and associated tidal basins or major engineering structures. Tables 37.2 (low-energy coasts) and 37.3 (high-energy coasts) summarize typical longshore sediment transport rates (integrated cross-shore over the surf zone) associated with chronic (ambient transport gradients) and ephemeral (storms) processes, natural and human-induced length scale variations and associated longshore transport gradients, and resultant (net) shoreline changes. The length scale of natural variations, such as coastline curvature, is usually an order-of-magnitude larger than human-induced length scale variations (e.g., harbor moles and shore protection structures). Tables 37.1–37.4 indicate that cross-shore eﬀects dominate coastline change along low-energy coasts, whereas along high-energy coasts cross-shore and longshore eﬀects are comparable and are of equal importance where coastline change is concerned. In the case of human-induced changes, the eﬀects of cross-shore and longshore processes are comparable on low-energy coasts, while on high-energy coasts longshore eﬀects are dominant. This may explain why cross-shore impacting structures, such as oﬀshore breakwaters and perched beaches, perform better on lowenergy coasts than on high-energy coasts. It is also clear that both on low- and high-energy coasts the Bruun eﬀect is of similar or lower magnitude when compared to other eﬀects. However, if the rate of SLR increased by ﬁve- or ten-fold,

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Table 37.2. Typical longshore sediment transport rates (integrated across the surf zone) along low-energy coasts (e.g., the Mediterranean coast) and associated shoreline changes (negative: retreat).15 Ephemeral processes Chronic processes Length scale of natural variations Length scale of human-induced variations Transport gradients due to natural variations Transport gradients due to human-induced variations Eﬀective proﬁle height Rate of shoreline change due to natural processes Rate of shoreline change due to human-induced processes Rate of shoreline change due to Bruun eﬀect only

O (105 ) m3 /year or/extreme event O (104 ) m3 /year 10 km (long-term scale) 1–10 km (medium-term scale) 1 m3 /m/year 1–10 m3 /m/year 10 m O (+ or −0.1) m/year O (+ or −0.1–1) m/year O (−0.1) m/year

Table 37.3. Typical longshore sediment transport rates (integrated across the surf zone) along high-energy coasts (e.g., Holland coast, eastern US coast) and associated shoreline changes (negative: retreat).15 Ephemeral processes Chronic processes Length scale of natural variations Length scale of human-induced variations Transport gradients due to natural variations Transport gradients due to human-induced variations Eﬀective proﬁle height Rate of shoreline change due to natural processes Rate of shoreline change due to human-induced processes Rate of shoreline change due to Bruun eﬀect only

O (106 ) m3 /year or/extreme event O (105 ) m3 /year 10–100 km (long-term scale) 1–10 km (medium-term scale) 1–10 m3 /m/year 10–100 m3 /m/year 10 m O (+ or −0.1–1) m/year O (+ or −1–10) m/year O (−0.1) m/year

Table 37.4. Typical longshore sediment transport rates (integrated across the surf zone) along a high-energy barrier coast, e.g., the Frisian Wadden coast or the eastern US coast and associated shoreline changes (negative: retreat).15 Ephemeral processes Chronic processes Length scale of natural variations Length scale of human-induced variations Transport gradients due to natural variations Transport gradients due to human-induced variations Eﬀective proﬁle height Rate of shoreline change due to natural processes Rate of shoreline change due to human-induced processes Rate of shoreline change due to Bruun eﬀect only

O (106 ) m3 /year or/extreme event O (0.5*106 ) m3 /year 10 km (long-term scale) 1–10 km (medium-term scale) 50 m3 /m/year 50–500 m3 /m/year 10 m O (−5) m/year O (−5–50) m/year O (−0.1) m/year

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increased erosion or decreased advance due to an enhanced Bruun eﬀect will be noticeable. 37.5. Backbarrier Sources and Sinks The eﬀect of backbarrier sources and sinks are represented by the last term on the right-hand side in Eq. (37.5). This term could play a signiﬁcant role when the backbarrier consists of a river, estuary, or tidal lagoon. In case of a river or an estuary there may be a natural supply of sediment to the coastal system that can compensate for all cross-shore and longshore losses mentioned above in Secs. 37.3 and 37.4. In this case, a delta will form and evolve with time. The delta evolutionary characteristics will depend on the relative role of waves, tides, and river ﬂow. In the Holocene, many deltas have been outbuilding as a result of abundant sediment supply due to erosion of the catchment basin. Over the past ﬁve decades many deltas have started to disintegrate due to human intervention in the form of damregulated river discharge that decimates downstream sediment supply. In case of a tidal lagoon or an estuary with low freshwater inputs, the backbarrier tidal basin area may act as a source or a sink for the coastal sediment budget. Classic examples of sink behavior are the Frisian Wadden basins along the Dutch and West German North Sea coast. Dronkers34 analyzed the net sediment transport behavior of these basins and showed that these basins are generally ﬂood dominant, i.e., there is a tendency to accumulate sediment within the basin as SLR, restoring dynamic equilibrium geometry. Stive and Wang35 further analyzed this response and showed that, in this case, the Bruun rule can be extended as follows to express the impact of SLR on inlet-inﬂuenced coasts: cp =

∂M SL L∗ ∂M SL Ab + ∂t h∗ ∂t h∗ Lac

(37.7)

where Ab is the tidal basin area and Lac is the length of the adjacent coast impacted. In the above equation the ﬁrst term on the right-hand side expresses the Bruun eﬀect and the second term expresses the basin accommodation eﬀect. The Bruun eﬀect is exceeded by the basin eﬀect when: Ab > L∗ Lac

(37.8)

Typical orders of magnitude for L∗ and Lac are 1 and 10 km, respectively, meaning that the direct impact of basin areas larger than O (10 km2 ) on coastline retreat overrides the Bruun eﬀect. Friedrichs and Aubrey36 presented a similar analysis for a large number of schematized tidal basins in eastern USA. They showed that depending on basin hypsometry, tidal basins could be either ﬂood or ebb dominant. This implies that SLR may lead to both importing and exporting basins. Figure 4 of Zhang et al.14 indicates that long stretches of coastline, which they denote as inlet-inﬂuenced, experience stronger recession rates than the noninlet-inﬂuenced stretches of coastline where the Bruun rule was used. This gives rise to the hypothesis that backbarrier basins along that coastline are ﬂood dominant. However, when the basins are

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ebb dominant, SLR may cause an export of sediment. This will decrease or even compensate for the Bruun eﬀect to such an extent that shoreline advance may occur. If SLR forces positive accommodation in ﬂood-dominant basins, surf zone generated sediment transport will be diverted into the tidal basin by ﬂood currents. These sediments will be trapped in the ﬂood delta. Typical barrier length scales are O (10 km). Therefore, typical values for shoreline changes in the vicinity of backbarrier sinks can be estimated as shown in Table 37.4. Table 37.4 indicates that along inlet interrupted coastlines the tidal basins linked to the inlets have a considerable inﬂuence on coastline change. Interestingly, the magnitude of tidal basin inﬂuenced coastline change (as indicated above) is signiﬁcantly larger than the coastline changes estimated by Zhang et al.14 along their noninlet-inﬂuenced coastline (cf. Fig. 37.3 of Ref. 14). In essence, therefore, while the underlying concepts of the Bruun rule have been veriﬁed, its quantitative accuracy remains unveriﬁed; largely due to the presence of other coastal processes that easily override the Bruun eﬀect under present still-stand SLR conditions. Furthermore, the numerous restrictive assumptions associated with the Bruun rule preclude its application in most natural environments, and especially in the vicinity of inlets linked to tidal basins larger than 10 km2 . However, what is needed now, in view of the massive socioeconomic impacts along the world’s highly developed coastal margins that are likely to result from accelerated SLR, is a robust and widely applicable method that can be conﬁdently applied to obtain accurate predictions of coastline retreat due to SLR. From the above discussion, it is apparent that the Bruun rule is of limited use in this context. Therefore, alternative methods to address this critical issue must now be considered. In this regard, a promising alternative philosophy is the “coastal tract” concept presented by Cowell et al.26,37

37.6. An Innovative Approach: The Coastal Tract The coastal tract approach introduces the concept of a meta-morphology, deﬁned as the morphologic composite comprising the lower shoreface, upper shoreface, and backbarrier (where present). It is the ﬁrst-order system within a cascade hierarchy that provides a framework for aggregation of processes in modeling the evolution of coastal morphology over decades to millennia (low-order coastal change). This type of coastal change involves parts of the coast normally ignored in predictions required for management of coastal morphology: i.e., shoreline evolution linked to behavior of the continental shelf and coastal plain. The coastal tract approach adopts a temporally and spatially cascading framework, where appropriate boundary conditions and internal dynamics are deﬁned to separate low-order from higher-order coastal behavior for site-speciﬁc applications. This procedure involves preparation of a datamodel by templating site data into a structure that complies with scale-speciﬁc properties of any given predictive models. Each level of the coastal-tract cascade is a self-contained system that shares sediments with other levels. This sediment sharing constrains morphologic responses of the system on given temporal and spatial scales. The internal dynamics of these responses involve morphologic coupling of the upper shoreface to the backbarrier

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and to the lower shoreface. The coupling mechanisms govern systematic lateral displacements of the shoreface, and therefore determine trends in shoreline advance and retreat. These changes manifest as the most fundamental modes of coastal evolution upon which higher-order (shorter-term, i.e., subdecadal scale) changes are superimposed. Prediction of shoreline change adopts diﬀerent approaches, depending on the space and timescale over which predictions are required. For short-term (subdecadal) coastal change (event and synoptic-scale changes occurring over hours through seasons to years), the focus is generally on the local sediment dynamics. These aﬀect the shoreline plan form and the cross-shore proﬁle (e.g., shoreline and proﬁle models) in response to ﬂuctuations in environmental conditions (i.e., the wave climate, littoral sediment budgets, sea level, and the eﬀects of anthropogenic activities). Theoretical and empirical approaches to these subdecadal timescales generally focus on changes to the upper shoreface (deﬁned loosely as the active zone; cf. Ref. 38), which correlate with shoreline movements. These changes are moderated by littoral sediment budgets and by sediment “production” via shoreline erosion cutting into onshore sand reserves (e.g., eroding dunes or cliﬀs), or through artiﬁcial nourishment of beaches. The practical imperative for long-term prediction (decades or longer) requires an expanded scope as included in the coastal tract concept that includes the lower shoreface and the interaction between the shoreface and backshore environments (Fig. 37.2). The upper shoreface has cross-shore length scales that are typically two to three orders of magnitude less than for the lower shoreface (depicted in Fig. 37.2). This scale diﬀerence means that changes on the lower shoreface are associated with disproportionately larger changes on the upper shoreface, due to mass continuity for sediment exchanges between the two zones.39,40 The upper shoreface is subject to a similar interaction with the backshore, which comprises a morphologically active zone located between the upper shoreface (ocean beach) and

Fig. 37.2. Physical morphology encompassed by the coastal tract (after Ref. 37, see text for explanation).

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the mainland. This zone may variously include dunes, wash over surfaces, ﬂood-tide deltas, lagoon basins, tidal ﬂats (Fig. 37.2A), mainland beaches (Fig. 37.2B), and ﬂuvial deltas (Fig. 37.2C). Each of these may be present or absent, depending on local conditions, especially the regional substrate slope.17 The sediment exchanges depicted by the arrows in Fig. 37.2 occur in principle during any average year and on all longer timescales. These exchanges are summarized schematically in Fig. 37.3 that diﬀerentiates sediment ﬂuxes into sand and mud fractions. For coastal change on any scale, antecedent morphology, sea-level change, ﬂuvial discharges, and littoral sediment budgets can be regarded as boundary conditions for the coastal area of interest. For subdecadal prediction of horizontal movements in the upper shoreface, sand exchanges with the lower shoreface (Fig. 37.3B) are usually ignored because these ﬂuxes are so small that resulting morphologic change is negligible: i.e., the annual closure-depth concept.27,41 The ﬂuxes of ﬁne sediments (Fig. 37.3C and 37.3D) are not directly relevant to the upper-shoreface sediment budget because mud deposition there is negligible. For long-term predictions, like on the scale of climate change, however, none of the internal sediment exchanges depicted in Fig. 37.3 can be ignored. This is because systematic residual ﬂuxes, that are small on the subdecadal timescale, eventually cumulate through time enough to produce nonnegligible (i.e., measurable) morphologic changes. Moreover, the changes in morphology of the backbarrier, lower shoreface, and upper shoreface cause these three zones to interact dynamically: i.e., the sediment exchanges themselves become inﬂuenced by the morphologic changes.

Fig. 37.3. Schematic representation of mechanisms steering the location of the upper shoreface (after Ref. 37).

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37.7. Conclusions The characteristics of coastline response to SLR suggested by the Bruun rule seem conceptually correct. However, under present near-still-stand SLR conditions, the Bruun eﬀect can easily be subordinate to a host of other processes. Therefore, any good comparisons between shoreline recessions measured in the last century under near-still-stand SLR and corresponding Bruun rule predictions are likely to be fortuitous and cannot be considered as conclusive veriﬁcation of the Bruun rule. Furthermore, due to the many restrictive assumptions associated with the Bruun rule, it is not applicable in most natural environments, as exempliﬁed by the exclusion by Zhang et al.14 of almost 70% of the study area from their analysis. The applicability of the Bruun rule is particularly compromised in the vicinity of tidal basins associated with inlets as the Bruun eﬀect can be overridden when the basin area is larger than 10 km2 .15 Clearly, the Bruun rule is not suitable to obtain exact and site-speciﬁc predictions of coastal recession due to SLR, particularly in view of the accelerated SLR projected for the 21st century. At best, any predictions obtained via the Bruun rule should be considered only as broadly indicative, order-of-magnitude estimates that are not suitable for direct use in making planning decisions. The only robust solution to the problem lies in comprehensive bottom-up (small-scale, process-based) and top-down (large-scale, behavior-based) numerical modeling that adopts the coastal tract philosophy as a conceptual template. Once comprehensively validated by ﬁeld data, such numerical models can be strategically applied to determine quantitative forcing-response relationships of complex, nonlinear coastal processes. These relationships can then be aggregated and/or parameterized and embedded into a robust and easy-to-use numerical model, which accounts for at least the primary physical processes governing coastline response to SLR. Such a process-based approach would constitute a signiﬁcant step forward from the Bruun rule and is likely to provide more scientiﬁcally robust and reliable predictions of coastline retreat due to future SLR. In closure we note that here we have only discussed the impact of a change of the rate of SLR on coastline evolution and not the impacts associated with other climate change driven impacts such changes in regional hurricane or typhoon climates, changes in dominant wave direction, storm frequency and intensity,42 and ENSO, NAOs, and SOI oscillations (cf. the chapter by Komar et al. and Ref. 43), which may have local and/or regional impacts as important as or even exceeding the impact due to SLR alone.

Acknowledgments M.J.F.S. was supported by the project “Sustainable Development of North Sea and Coast” (DC-05.20) of the Delft Cluster research project dealing with sustainable use and development of low-lying deltaic areas in general and the translation of specialist knowledge to end users in particular.

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References 1. IPCC, Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Summary for Policymakers (2007), 18 pp. 2. S. Rahmstorf, A. Cazanave, J. Church, J. Hansen, R. Keeling, D. Parker and R. Somerville, Science 316, 709 (2007). 3. S. Rahmstorf, Science 315, 368 (2007). 4. P. Bruun, Sea-level rise as a cause of shore erosion, J. Waterw. Harbor. Div., ASCE, 88, 117–130 (1962). 5. L. Hamm, M. Capobianco, H. H. Dette, A. Lechuga, R. Spanhoﬀ and M. J. F. Stive, A summary of European experience with shore nourishment, Coast. Eng. 47(2), 237–264 (2002). 6. O. H. Pilkey and J. A. G. Cooper, Society and sea level rise, Science 303, 1781–1782 (2004). 7. J. A. G. Cooper and O. H. Pilkey, Sea level rise and shoreline retreat: Time to abandon the Bruun rule, Global Planet. Change 43, 157–171 (2004). 8. P. A. Rosen, A regional test of the Bruun rule on shoreline erosion, Marine Geol. 26, M7–M16 (1978). 9. E. B. Hands, The great lakes as a test model for proﬁle responses to sea level changes, Handbook of Coastal Processes and Erosion Boca Raton, ed. P. D. Komar (CRC Press, 1983), pp. 176–189. 10. C. H. Everts, Sea level rise eﬀects on shoreline position, J. Waterw. Port Coast. Ocean Eng. 111(6), 985–999 (1985). 11. O. H. Pilkey and T. W. Davis, An analysis of coastal recession models: North Carolina coast, Sea-level Fluctuation and Coastal Evolution, SEPM, eds. D. Nummedal, O. H. Pilkey and J. D. Howard (Society for Sedimentary Geology) Special Publication No. 41 (Tulsa, Oklahoma, 1987), pp. 59–68. 12. R. G. Dean, Beach response to sea level change, The Sea, Vol. 9 (Wiley, 1990), pp. 869–887. 13. SCOR Working Group 89, J. Coast. Res. 7(3), 895 (1991). 14. K. Zhang, B. C. Douglas and S. P. Leatherman, Climat. Change 64, 41 (2004). 15. M. J. F. Stive, Climat. Change Editorial Comment 64, 27 (2004). 16. N. Mimura and H. Nobuoka, Veriﬁcation of Bruun rule for the estimate of shoreline retreat caused by sea-level rise, Coastal Dynamics 95, eds. W. R. Dally and R. B. Zeidler (American Society of Civil Engineers, New York, 1995), pp. 607–616. 17. P. J. Cowell, P. S. Roy and R. A. Jones, Simulation of LSCB using a morphological behaviour model, Marine Geol. 126, 45–61 (1995). 18. J. E. A. Storms, Event-based stratigraphic simulation of wave-dominated shallowmarine environments, Marine Geol. 199(3–4), 83–100 (2003). 19. H. J. Steetzel, Cross-shore transport during storm surges, PhD. thesis, Delft University of Technology (1993). 20. M. R. A. van Gent, E. M. Coeveld, D. J. Walstra, J. van de Graaﬀ, H. J. Steetzel and M. Boers, Dune erosion tests to study the inﬂuence of wave periods on dune erosion, Proc. Int. Conf. Coast. Eng., ASCE, San Diego (2006). 21. E. C. F. Bird, Coastline Changes (Wiley & Sons, New York, 1985), p. 219. 22. S. P. Leatherman, K. Zhang and B. C. Douglas, Sea level rise shown to drive coastal erosion, EOS Trans. 81(6), 55–57 (2000). 23. A. D. Short, A survey of Australian beaches, Keynote Lecture, Coastal Sediments 2003, Clearwater, Florida (2003). 24. J. R. Curray, Transgressions and regressions, Papers in Marine Geology, ed. R. C. Miller (McMillan, New York, 1964), pp. 175–203.

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25. D. J. P. Swift, Continental shelf sedimentation, Marine Sediment Transport and Environmental Management, eds. D. J. Stanley and D. J. P. Swift (Wiley, New York, 1976), pp. 311–350. 26. P. J. Cowell, M. J. F. Stive, A. W. Niedoroda, D. J. P. Swift, H. J. de Vriend, M. C. Buijsman, R. J. Nicholls, P. S. Roy, G. M. Kaminsky, J. Cleveringa, C. W. Reed and P. L. De Boer, The Coastal-Tract (Part 2): Applications of aggregated modelling to lower-order coastal change, J. Coast. Res. 19(4), 828–848 (2003). 27. R. J. Nicholls, W. A. Birkemeier and G.-H. Lee, Evaluation of depth of closure using data from Duck, NC, USA, Marine Geol. 148, 179–201 (1998). 28. R. G. Dean, Equilibrium beach proﬁles: Characteristics and applications, J. Coast. Res. 7, 53–84 (1991). 29. R. G. Dean and E. M. Maurmeyer, Models of beach proﬁle response, CRC Handbook of Coastal Processes and Erosion, eds. P. Komar and J. Moore (CRC Press, Boca Raton, 1983), pp. 151–165. 30. D. J. Beets, L. van der Valk and M. J. F. Stive, Holocene evolution of the coast of Holland, Marine Geol. 103, 423–443 (1992). 31. P. J. Cowell, M. J. F. Stive, P. S. Roy, G. M. Kaminsky, M. C. Buijsman, B. G. Thom and L. D. Wright, Shoreface sand supply to beaches, Proc. 27th Int. Conf. Coast. Eng., ASCE (2001), pp. 2495–2508. 32. A. J. Bowen, Simple models of nearshore sedimentation, beach proﬁles and longshore bars, The Coastline of Canada, ed. S. B. McCann, Geological Survey of Canada Paper, 80-10 (1980), pp. 1–11. 33. J. H. List and A. S. Farris, Large-scale shoreline response to storms and fair weather, Proc. Coast. Sediments ’99, ASCE, Reston, VA (1999), pp. 1324–1338. 34. J. Dronkers, Morphodynamics of the Dutch delta, Physics of Estuaries and Coastal Seas, eds. J. Dronkers and M. Scheﬀers (Balkema, Rotterdam, 1998), pp. 297–304. 35. M. J. F. Stive and Z. B. Wang, Morphodynamic modeling of tidal basins and coastal inlets, Advances in Coastal Modeling, ed. C. Lakhan (Elsevier, 2003), Chapter 13, pp. 367–392. 36. C. T. Friedrichs and D. G. Aubrey, Non-linear tidal distortion in shallow well-mixed estuaries: A synthesis, Est. Coast. Shelf Sci. 27, 521–545 (1988). 37. P. J. Cowell, M. J. F. Stive, A. W. Niedoroda, H. J. De Vriend, D. J. P. Swift, G. M. Kaminsky and M. Capobianco, The Coastal-Tract (Part 1): A conceptual approach to aggregated modelling of low-order coastal change, J. Coast. Res. 19(4), 812–827 (2003). 38. M. J. F. Stive and H. J. De Vriend, Modelling shoreface proﬁle evolution, Marine Geol. 126, 235–248 (1995). 39. P. S. Roy, P. J. Cowell, M. A. Ferland and B. G. Thom, Wave dominated coasts, Coastal Evolution: Late Quaternary Shoreline Morphodynamics, eds. R. W. G. Carter and C. D. Woodroﬀe (Cambridge University Press, Cambridge, 1994), pp. 121–186. 40. P. J. Cowell, D. J. Hanslow and J. F. Meleo, The shoreface, Handbook of Beach and Shoreface Morphodynamics, ed. A. D. Short (Wiley, Chichester, 1999), pp. 37–71. 41. R. J. Hallermeier, A proﬁle zonation for seasonal sand beaches from wave climate, Coast. Eng. 4, 253–277 (1981). 42. R. Ranasinghe, D. Lord, D. Hanslow and K. McInnes, Climate change impacts on NSW coasts and estuaries, Proc. Coasts and Ports ‘07, Melbourne, VIC, Australia, CD ROM published by Engineers Australia (2007). 43. R. Ranasinghe, R. McLoughlin, A. D. Short and G. Symonds, The southern oscillation index, wave climate, and beach rotation, Marine Geol. 204, 273–287 (2004).

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Chapter 38

Coastal Flooding: Analysis and Assessment of Risk Panagiotis Prinos∗ and Panagiota Galiatsatou† Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki 54124, Greece ∗ [email protected] † [email protected] This chapter deals with ﬂood risk analysis and assessment. The conceptual model source pathway receptor consequence for ﬂood risk analysis is presented and its components are analyzed. The methodology to extract the predicted probability of coastal ﬂooding from risk sources and pathways, as well as the expected damages from risk receptors are introduced and examined. Reliability analysis of a coastal system is also brieﬂy discussed. Quantitative methods to deﬁne acceptable ﬂooding probabilities on the level of the protected area are presented. Tools such as cost–beneﬁt analysis, utility models, and the life quality index are introduced to deﬁne the “tolerable” risk of ﬂooding.

38.1. Introduction A ﬂood is a temporary covering by water of land normally not covered by water. This deﬁnition includes ﬂoods from rivers, mountain torrents, Mediterranean ephemeral water courses, and ﬂoods from the sea in coastal areas and may exclude ﬂoods from the sewerage systems. Worldwide, over 65% of the large cities with more than 2.5 million inhabitants are located in coastal areas, deltas, and estuaries and almost 40% of the population of our planet is living within a 100-km-wide coastal strip. In the past, ﬂooding disasters have struck human society all over the world and in the future they may be expected to do so again. Flooding still remains one of the most widely distributed natural hazards in the whole world. Flood risk is the combination of the probability of a ﬂood event and of the potential adverse consequences to human health, the environment, and economic activity associated with the ﬂood event. For coastal areas, the hazards that contribute to the risk of ﬂooding are related to the water surface elevation process. 1039

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The surface elevation can be described by the water level elevation and a number of parameters for individual waves, e.g., the wave height, the wave period, the wavelength, and the wave steepness. Risk can be deﬁned in a number of diﬀerent ways.1 Some deﬁnitions of risk are the following: (1) risk is the probability of an unwanted event; (2) risk is the consequence of an unwanted event; (3) risk is the function of the probability and the eﬀect; and (4) risk is deﬁned as risk = probability × consequence. The ﬁrst two deﬁnitions are not well applicable because the risks with small probabilities and very large consequences and those with great probabilities and small consequences cannot be described properly. The third deﬁnition is more general and helps assigning a weight to the consequence of an unwanted event, which depends on the seriousness of the consequence. This is especially important for small probabilities and great consequences and can include matters such as risk aversion in the assessment of risk. The deﬁnition of risk as product of probabilities and consequences (fourth deﬁnition) is simpler and oﬀers a good basis for comparisons of risks. Following the fourth deﬁnition, risk is a combination of the chance of a particular event, with the impact that the event would cause if it occurred. Risk has two components: the chance or probability of an event occurring and the impact or consequence associated with the event. The consequence can be desirable or undesirable; however, risk is typically concerned with the likelihood of an undesirable consequence. The probability component of the risk consists of1 : (a) the probability that the unwanted event occurs, (b) the probability that the unwanted event leads to a possible eﬀect, and (c) the probability that this eﬀect leads to the considered consequence. For example, the probability of ﬂooding at a deﬁned area consists of the probability that a dike slips, the probability that water ﬂows into the polder, when the water level is high and the probability of damage and casualties due to this eﬀect. The total probability component of risk amounts to: P (E1 ) · P (E2 | E1 ) · P (G | E1 ∩ E2 ), where E1 is the unwanted event (e.g., collapse of dike), E2 is the eﬀect (e.g., inundation of polder), and G is the consequence (e.g., number of dead people by drowning). The probability P (E1 ) · P (E2 | E1 ) is usually the probability of failure Pf . The consequence of ﬂooding is an impact such as economic, social, or environmental damage/improvement. It can be expressed quantitatively (e.g., monetary value), by category (e.g., high, medium, and low), or descriptively. Typical descriptions of consequence are: (a) economic damage, (b) number of people/properties aﬀected, (c) occurrence of speciﬁed event, and (d) degree of harm to an individual (injury, stress, etc.). The consequence of an unwanted event can be both deterministic and statistic. If a statistic eﬀect is involved, a probability density function can be deﬁned for the risk, for which the expected value can be determined. Failure to recognize the full spatial extent of the consequences will bias the decision-making process and could lead to suboptimal decisions. Equally, the temporal scale of the consequences must be explicitly considered. For example, the impact of ﬂooding should not only be considered with reference to the physical time for the ﬂoods to recede, but also the time taken to reestablish community businesses

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as well as stress- and health-related issues that may persist in ﬂooded communities.2 Only with proper consideration of the variety and temporal and spatial scale of the consequences can appropriate decisions be made. To consider the signiﬁcance of risk, reference must be made not only to the numeric value of the product of probability and consequence, but also to how it will be perceived by society or the individual. Also, in some cases it is important to understand the nature of risk, distinguishing between rare, catastrophic events and more frequent less severe events. For example, risk methods adopted to support the targeting and management of ﬂood warning represent risk in terms of probability and consequence, but low probability/high consequence events are treated very diﬀerently to high probability/low consequence events. Flood risk management deals with the analysis and governance of the ﬂood hazards, the ﬂood vulnerability (the resilience of a particular group of risk receptors) and the resulting ﬂood risk. It considers all natural and societal processes related with the ﬂood issues of a coastal cell. Based on such a risk analysis, the tolerability of risk has to be assessed and risk reduction options to be deﬁned. Decision-making, implementation, and control of actions are also parts of ﬂood risk management. Flood risk management includes the analysis and assessment of risks and the formulation and implementation of risk reduction options. Risk analysis includes hazard as well as vulnerability determination and estimates the integral risk. Risk assessment deals with the evaluation and weighing of costs and beneﬁts to derive a tolerable level of risk, taking risk perception into account. Risk reduction deals with planned and realized measures and instruments supposed to decrease risks before (pre-ﬂood), during the event and after an event (post-ﬂood). Figure 38.1 shows the three tasks included in ﬂood risk management. Risk analysis is divided into hazard, vulnerability, and risk determination and constitutes the main subject of interest of the present chapter. It deals with extreme

Flood risk management

Risk analysis

Risk assessment

Risk reduction

Hazard determination

Vulnerability determination

Risk determination

Risk perception

Risk weighing

Pre-Flood risk reduction

Flood event risk reduction

Post-flood risk reduction

Probability and characteristics (e.g. water depth, flow velocity, duration) of flood events

Potential social, economic, and ecological damage depending on value and susceptibility to a certain type of hazard

Probability of certain social, economic, and ecological damage to a certain hazard

Overall view of risk held by a person or group depending on cultural and personal values, experiences, and feelings

Agreement on tolerability of risk, weighing benefits, and costs depending on individual or collective perception and interest

Physical measures, regulatory, financial, and communicative instruments to reduce the risk by prevention and/or preparedness

Physical measures, regulatory instruments, and communicative activities to reduce the risk of an ongoing event

Physical measures, regulatory, financial, and communicative instruments to deal with existing flood damages

Fig. 38.1.

Scheme of tasks and components of ﬂood risk management.3

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coastal events, their characteristics, and their probability of occurrence and the potential damage (social, ecologic, and economic) that they may cause if they occur, depending on the susceptibility of the society to a certain type of hazard. Risk analysis encompasses investigations on the impact of climate and societal change. Risk determination is the main outcome of this component of the ﬂood risk management process. In comparison to the “objective” analysis, risk assessment includes individual and collective perception and weighing of risk and will also be further analyzed in the present chapter. There are psychologic–cognitive and sociologic–cultural approaches of explicating risk perception, the overall view of risk depending on cultural and personal values, experiences, and feelings. Risk assessment also covers the evaluation and weighing of risk. Evaluation is dedicated to determine eﬀectiveness, eﬃciency, robustness, ﬂexibility, and other criteria, whereas weighing means to balance probable costs of risk and risk reduction measures with expected beneﬁts by using ﬂood-prone areas.3 Finally, risk reduction includes pre-ﬂood, ﬂood event, and post-ﬂood risk reduction, namely physical measures, regulatory, ﬁnancial, and communicative instruments to reduce the risk by prevention and/or preparedness, by coping with the risk of an ongoing event and by dealing with existing ﬂood damages, respectively. As far as the overall principle of risk reduction is concerned, ﬂood risk management is based on a shift from resistance against ﬂoods toward a resilience of the entire ﬂood risk system. The system-based risk analysis provides the prerequisites for a holistic ex ante analysis of the outcomes of risk reduction.3 Since ﬂood risk management has been indicated as a societal task, the decision making and development process of responsible actors come into the foreground. Decision-making is fundamentally about making choices. The societal decision process concerning the protection against ﬂooding disaster shows an increasing demand on the performance of the ﬂood defence system over time. Society is a continuous process of decision making on the protection against ﬂooding. Developments in science and technology have led to a demand for a quantitative method for judging ﬂooding risk. The decision process in a risk-based approach relies upon a variety of decision criteria and must be able to distinguish the merits and demerits of an action over another (i.e., it must be able to compare the risk and performance of one option with another). The conventional engineering distinction between failure and nonfailure has for many years been translated to the design of ﬂood defences via the concept of a “design load,” almost always expressed in terms of a return period. In the current design practice three approaches exist2 : (a) a deterministic, where design is on the whole based on the concept of design loads, be they wave heights or water levels for coastal areas and where precise, single values are used for all variables; (b) a deterministic including sensitivity testing, where deterministic outcomes are tested by systematically varying input values; and (c) a probabilistic, where the variability of input values is taken into account to provide a probabilistic result. In cases (a) and (b), the design equations or constraints are written in terms of safety factors. Safety factors have the advantage of being easily interpreted in terms of their physical or engineering meaning, but do not give clear information on the reliability of the

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structure. However, the deterministic methods are aimed at preventing the worst, without taking the likelihood of the various circumstances into account. The main advantage of the probabilistic-based design [approach (c)] is that the reliability of the structure can be evaluated if the statistics are well deﬁned. However, it has the shortcoming of being very sensitive to tail assumptions, and in some cases, as, for example, rubble-mound stability, runup, overtopping, and geotechnical stability, parametric dependencies and statistics are diﬃcult to deﬁne. Applications of approach (c) in coastal engineering have been published among others by Bakker and Vrijling,4 Burcharth et al.,5 and Voortman et al.6 In the last few years all methods have been improved by applying optimization techniques.7 The reliability design method with partial safety factors is superior to the conventional methods, in the sense that it takes into account the uncertainties involved in the load and resistance parameters. Nevertheless, it cannot predict the amount of damage when it occurs, just like the conventional methods.8 The beneﬁt of a risk-based approach, compared to the other approaches to design or decision making, is that it deals with outcomes. A risk-based approach enables informed choices to be made based on comparison of the expected outcomes and costs of alternative courses of action. This is distinct from, for example, a standards-based approach that focuses on the severity of the load that a particular ﬂood defence is expected to withstand. Using a risk-based approach, risk assessment often involves more complete representation of the ﬂood/erosion system. This more holistic approach will be able to account for: (a) complex physical mechanisms, spatial, and temporal variations in natural hazards (e.g., natural variability in wind, wave, rainfall, and water level conditions); (b) descriptions based on sparse/incomplete data; and (c) multiple stakeholders with diﬀering, often conﬂicting, values and objectives.2 The decision process in a risk-based approach is split into three levels.1 Level A incorporates decision making on the appropriate probability of ﬂooding on the level of protected areas. On level A, the cost of protection on the level of the full system and the consequences of ﬂooding in the protected area are considered. Level B consists of the decision making on the failure probability per individual structure in the protection system for a given value of the probability of ﬂooding. The minimization of the construction costs of the full system is utilized for the allocation of a given probability of ﬂooding to individual sections of a system. Finally, level C includes decision making on the geometry of an individual ﬂood defence structure for a given probability of failure. Considering the cost of elements of the cross-section and a set of failure modes, the construction costs of an individual section of the protection system are minimized. Risk-based design is performed only on level A. On levels B and C, reliability-based design is applied. Section 2 deals with ﬂood risk analysis. The conceptual model source–pathway– receptor–consequence for ﬂood risk analysis is presented and its components are analyzed further. The methodology to extract the predicted probability of coastal ﬂooding from risk sources and pathways, as well as the expected damages from risk receptors are introduced and examined. Reliability analysis of a coastal system is also brieﬂy discussed in the subsection dealing with the pathways of risk. Section 3 concentrates on risk assessment. Quantitative methods to deﬁne acceptable ﬂooding

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probabilities on the level of the protected area are presented. Tools such as cost beneﬁt analysis, utility models, and the life quality index are introduced to deﬁne the “tolerable” risk of ﬂooding. Section 4 presents the main ideas and conclusions of the chapter, recommending an approach to risk management that embraces all sources, pathways, and receptors of risk and considers combinations of structural and nonstructural solutions. 38.2. Flood Risk Analysis 38.2.1. The conceptual model for flood risk analysis For a risk to arise there must be a hazard that consists of a “source” or initiator event (i.e., high wave height), a “receptor” (e.g., cliﬀ top or ﬂood plain properties), and a pathway between the source and the receptor (i.e., ﬂood routes including defences, overland ﬂow, or landslide). A hazard does not automatically lead to harmful outcome, but identiﬁcation of a hazard does mean that there is a possibility of harm occurring. High wave conditions and high sea levels (tide and surge) oﬀshore (including processes of wave generation and the interaction of waves with each other), and transformed to nearshore (loosely deﬁned as the zone in which the seabed inﬂuences wave propagation and includes shallow water eﬀects such as shoaling, depth refraction, interaction with currents, and depth-induced wave breaking) are typically considered as the “source” of coastal ﬂooding. Shoreline responses (including response of beaches and defences to waves, wave structure interaction, overtopping, overﬂowing, and breaching), and ﬂood inundation and propagation are considered as the pathways of coastal ﬂooding. The receptors of coastal ﬂooding are considered as property, people, and the environment. In the context of coastal ﬂooding these terms are identiﬁed in Fig. 38.2. In Fig. 38.2, Pfc is the predicted overall probability of failure for ﬂood defences and E(D) is the expected economic and noneconomic damages in the ﬂood prone areas. The predicted ﬂood risk probability is deﬁned as: Rfc = Pfc · E(D). Risk Sources Storm surge Waves Heavy rainfall

Risk Pathways Loads and resistances Defence failures Inundation

Predicted flooding c probability Pf

Risk Receptors People and property Ecological impact Risk perception

Expected damages E(D)

Predicted flood risk R fc = Pfc . E(D)

Fig. 38.2.

Methodology of calculating ﬂood risk.9

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Risk Sources

Hydrological/ hydraulic boundary conditions, including uncertainties and focusing on extremes and joint probabilities

Modeling of loading and resistance of each defence structure

Pfc

Fig. 38.3.

The link between risk sources and the predicted probability of failure.

38.2.2. Risk sources Risk sources describe the events that cause risk, such as storm surges, waves, heavy rainfall, or combinations of them. Description of risk sources will help to deliver the hydrologic and hydraulic boundary conditions which will be needed to describe the loading of ﬂood defence structures or will already be the key input for the probability of ﬂooding. Figure 38.3 shows the link between the predicted probability of ﬂooding and the risk sources, as presented within the framework of the European Program FLOODsite (Integrated Flood Risk Analysis and Management Methodologies). Flood risk sources are associated to the occurrence of extreme events. Some recent ﬂoods can be classiﬁed as extreme events with a recurrence interval of once in a hundred years or less. Extreme value methods are powerful statistic methods for drawing inference about the extremes of a process, using only data on relatively extreme values of the process. Extreme value methods are usually utilized for the purpose of extrapolation to levels more extreme than those which have been observed. The statistic methodology is motivated by a well-established mathematic theory [extreme value theory (EVT)], which relies on the assumption that the limiting models suggested by the asymptotic theory continue to hold at ﬁnite but extreme levels. Nevertheless, a crucial assumption in ﬁtting distribution functions to data is that the data are independent and identically distributed (iid). In situations where the probability of extreme levels of an environmental process need to be estimated, statistic methods based on univariate EVT are widely adopted: e.g., Davison and Smith.10 These models can be used for several predictions of engineering interest, such as, for example, extreme values during a certain time frame.11 Coles12 concentrates on statistic inference for extremes and uses extreme value (EV) methodologies to analyze diﬀerent data sets and to produce statistic inference for extremes. The papers of Smith,13 Tawn,14 Coles and Tawn,15 and Walshaw and Anderson16 are representative of the analysis of oceanographic, wind, and other sort of data using EV methods. Block maxima and exceedances over high thresholds (peaks over threshold, POT approaches) are used, according to data availability, to extract design values for diﬀerent structures. When block maxima are utilized, the generalized extreme value (GEV) distribution is ﬁtted to a sample of annual maximum values of the variable under consideration over a period of time. Because of limited annual-maxima data, quantile estimates corresponding to large return periods tend to be highly uncertain. An extension of the classical extreme value analysis leads to the incorporation of a

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larger amount of data in the inference, which is expected to reduce uncertainty. This approach, referred to as the (POT), is widely used in the ﬁeld of hydrology.17 The choice of the threshold should be based on physical and meteorologic information. For POT methods, independence requires a suitable combination of both threshold and minimum separation time between events. There have been many proposals for the distribution functions of extreme wave heights. The current consensus is to choose a function best ﬁtting to the extreme wave data among several candidates, which include the Fisher–Tippett type I (FT-I), Fisher–Tippett type II (FT-II), and Weibull distributions. A diﬃculty in selecting a distribution function best ﬁtting to a data set of storm wave heights is the lack of information on the population distribution of extreme events. Goda8 (Fig. 38.4) exhibits the diﬀerence between the distribution functions of the partial duration and annual maximum series data in terms of return period for a given wave height. The 50-year wave height is set at 8.0 m. Three distributions FT-II and two Weibulls are shown with their converted annual maximum distributions. Because of the deﬁnition of annual maxima, its return period must be greater than 1.0 year. Except for the zone of short return period, the distribution for the annual maximum series exhibits little diﬀerence from that for the partial duration series. When an extreme value model (GEV or POT) is ﬁtted to the data of waves, storm surges or any other variable of interest, the parameters of the applied model are estimated using diﬀerent methods. Three of them are the maximum likelihood estimation (MLE), the L-moments (LM), and the Bayesian estimation procedure.

Fig. 38.4. Comparison of the distribution functions of partial duration and annual maximum series data for the 50-year wave height.8

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Wave height estimates using ML, Bayesian, and LM estimation procedures.18

Galiatsatou18 (Fig. 38.5) estimates median and 95% conﬁdence intervals of return level estimates for wave heights, when extreme value parameters are estimated with (a) maximum likelihood (ML), (b) Bayesian with noninformative prior distributions, and (c) L-moments (LM) estimation procedures. The choice of the water level distribution is another subject which has received considerable interest. The observed water level is the sum of a deterministic astronomic tidal component and a stochastic meteorologically induced component, the surge component. Surges are generated by a combination of air pressure and the wind ﬁeld over the surrounding region. The observed water level is modeled either after an analysis into its components or as a single variable. Galiatsatou and Prinos19 show estimates of the median and 95% conﬁdence intervals of surge (raw and log-transformed data) return levels, using the Bayesian approach and the MLE procedure to estimate the parameters of the extreme value models ﬁtted to two stations oﬀ the Dutch coast. Figure 38.6 shows return level plots for the station Eld (Eierlandse Gat) at the northern part of the Dutch coast in the North Sea. Figure 38.6(a) presents estimates of the surge level for raw data, when noninformative prior distributions for the parameters in the Bayesian framework are utilized and Fig. 38.6(b) gives corresponding estimates when information from a neighboring station is used. Spatial dependence of a single process (e.g., storm surge) has been recently included in the estimation of return levels, using extreme value methodologies. Galiatsatou and Prinos20 use a “spatial linkage” assumption to extract results of surge (return) levels for a process of daily storm surges, taking account of the degree of spatial dependence among nine sites of the Dutch coast in the North Sea. Return levels and their standard errors are extracted under the assumption of ξ-linkage and compared to the ones when no parameter linkage is considered. Variance and

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(a)

(b)

Fig. 38.6. Surge levels against return period for the Eld station based on ML-estimates (dashed line) and the posterior Bayesian distributions (solid line) for the raw data, using (a) noninformative prior distributions and (b) information from the neighboring station Son.19

eﬃciency gains in the model parameters are estimated and discussed further in the paper. In Voortman,21 two diﬀerent possibilities of modeling the distribution of water levels are examined: (a) by direct statistic analysis of the observed water levels and (b) by combining the probability distribution of wind speed with a relevant physical model. Realizing that physical models are often imperfect, in option (b) an estimate of the model uncertainty is necessary. Where the source consists of one or more variables (e.g., coastal ﬂooding caused by extreme wave heights and water levels), it is necessary to consider their joint probability. Joint exceedance probability refers to the chance of two or more partially related variables occurring simultaneously. Joint exceedance combinations of wave heights and sea levels with a given chance of occurrence are deﬁned in terms of sea conditions in which a given wave height is exceeded at the same time as a given water level (or its surge component) being exceeded. There are diﬀerent levels of complexity for joint probability methods but all require some assessment of the dependence between variables. A relatively simple method uses the marginal distributions of the variables involved and an assessment of their dependence. The joint return period is expressed in terms of combinations of the marginal return periods. The main disadvantage of the method is that the joint return period of the variables (e.g., wave height and water level) is not the same as the return period of the response variable. A more sophisticated approach involves ﬁtting a probability distribution to the joint distribution of variables and extrapolating the joint density function. The beneﬁt of this approach is that the return period of the response function can be easily determined. The main disadvantage is that this approach needs a large amount of concurrent data of the variables involved. For the second approach Multivariate EVT is used to extract joint exceedance probabilities and joint return periods. Multivariate Extreme Value Theory (MVE) is used to describe the joint distribution of two or more variables, and appropriate methodology has only been

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developed in recent years. Coles and Tawn,22,23 Joe et al.,24 Zachary et al.,25 and Schlather and Tawn26 give the main aspects of MVE with applications to oceanographic and other sort of datasets. Fang and Hogben,27 Haver,28 Athanassoulis et al.,29 Morton and Bowers,30 De Haan and De Ronde,31 Repko et al.,32 and Ferreira and Soares33 describe the joint probability distribution function of longterm hydraulic conditions. Especially when the main interest is in the design of ﬂood defence structures, the extreme conditions are important, which implies that the dependence between hydraulic conditions needs to be accounted for. The joint probability analysis of extreme waves and water levels thus is signiﬁcant in order to estimate more accurately the extreme environmental loading on a coastal structure. Because wind setup (storm surge) and wave conditions depend on the same driving force, a strong dependence between them is observed under extreme conditions. Galiatsatou18 compares diﬀerent pairs of bivariate observations of extreme waves and surges with reference to joint exceedance probabilities, in order to ﬁnd the most severe sea state caused by the two variables. A bivariate logistic model and a sequential estimation procedure are used for this case to extract joint exceedance probabilities. The parameters of the margins of the bivariate distribution are deﬁned by maximum likelihood (ML), Bayesian with ﬂat prior distributions, and L-moments (LM) estimation procedures. Figure 38.7 presents the eﬀect of diﬀerent marginal parameters to the joint return levels of surges and wave heights for return periods of 1, 10, 50, and 100 years, for the most severe sea state. Voortman6 describes a model for the hydraulic boundary conditions, where the JPDF (joint probability density function) of hydraulic conditions nearshore is written as a function of properties of the wind ﬁeld, geometry of the North Sea basin, astronomic tide, and nearshore bathymetry. A description of the boundary conditions on the basis of parametric models found on physical concepts is favored in the thesis over pure statistic methods. EVT is progressively utilized to compute limit state equations, which correspond to diﬀerent failure modes of coastal and oﬀshore structures. The input parameters of the limit state function are the stochastic load and strength parameters

Fig. 38.7. Bivariate return levels for surges and wave heights with three methods of estimating marginal parameters (ML, Bayesian, LM) (a) Bayesian and ML estimators and (b) ML and LM estimators.18

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corresponding with the failure mode, like wave heights and wave periods. It should be noted that oceanographic data like wave heights and periods experience dependence, which leads to the use of joint probability analysis to describe in a better way the wave climate and make probabilistic design and risk-based optimization possible. Galiatsatou and Prinos34 implement four bivariate models (bivariate lognormal, bivariate log-normal with correction for wave skewness, a parametric distribution based on transformations to normality, and bivariate logistic distribution) to the data of signiﬁcant wave height and wave period from the ﬁeld station Eld oﬀ the Dutch coast and compare them mainly on their ability to correctly describe the data and on their behavior when extrapolation is of interest. In practice such multivariate problems are often simpliﬁed to a univariate setting, focusing on the relevant variable for the structure under consideration (structure variable approach). Brunn and Tawn35 compare the method which uses the joint distribution of wave heights and water levels and the structure variable approach in the general framework of estimating the risk of coastal ﬂooding in ∆1 ∆2 ∆3 northwest Europe. The problem is structured as: W −→ X(O) −→ X(N) −→ Z, where W is the wind ﬁeld over the northwest European continental shelf; X(O) and X(N) are the sea state variables oﬀshore and nearshore, respectively; Z is the impact variable on the ﬂood defence system; and ∆1 , ∆2 , and ∆3 is a series of complex hydrodynamic numeric models. The vector X(O) = (X1 , X2 ) is deﬁned as the vector of signiﬁcant wave height and water level. The structure variable Z is analyzed with the nested numeric model ∆3 (∆2 ) approximated by: Z = 0.3X1 + X2 − υ, where υ is deﬁned as the height of the dike and ﬂooding is assumed if Z > 0. In case of using the joint distribution of waves and water levels, the dependence model of Ledford and Tawn36 characterizes the asymptotic structure of the joint survivor function, which enables the identiﬁcation of a whole class of models, for which the limiting dependence of the componentwise maxima is independence. Figure 38.8 shows (a) estimates of the design parameter υp for the failure region 0.3X1 + X2 − υp estimated using the joint probabilities method and the structure variable approach and (b) the failure regions {(x1 , x2 ) : 0.3x1 + x2 > v} with v = 2.5, 3.5, 4.5, and 5.5: the regions are shown after transformation to logged unit Fr´echet marginal scales [log10 (Y1 ), log10 (Y2 )]. In both (a) and (b), (X1 , X2 ) are the Dutch oﬀshore data and the thresholds used correspond to the 80% empiric quantiles of the variables involved. The analysis of oﬀshore variables (wave heights, water levels, and wave periods) and the extraction of high enough return levels for them, seems to be a suitable compromise in the analysis of extreme events. Contrarily to the associated nearshore variables, oﬀshore data are regionally homogeneous (e.g., on the 50-km scale) as they correspond to observations at a depth at which complex local interactions between the processes do not occur.35 One statistic analysis is required, from which estimates at diﬀerent sites can be obtained by suitably changing the complex hydrodynamic and hydraulic numeric models which propagate wave heights and water levels near the shore. The assessment of the wave and water level conditions near the coast is essential for the estimation of ﬂood characteristics. Using the output of the analysis as input to nearshore wave and water level models, the wave climate next to the coastline is produced.

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Fig. 38.8. (a) Estimated design parameter vp for the failure region 0.3X1 + X2 − vp : estimates for the JPM (—) and SVM (. . . . . . .) and (b) failure regions {(x1 , x2 ) : 0.3x1 + x2 > v} with υ = 2.5, 3.5, 4.5, and 5.5.35

For nearshore water level predictions (tide and surge), the ﬁrst-generation models are 2DH models providing results of tide and surge components across a given area. They solve the nonlinear shallow water equations and use inputs of wind ﬁelds and atmospheric pressure over the modeled area. More advanced models include the eﬀects of breaking waves, causing setup of water levels in nearshore areas. MIKE21 HD/NHD is a representative model from this category. The secondgeneration models are 3D models that include the eﬀects of temperature and salinity, in addition to the characteristics of ﬁrst-generation models. POLCOMS, TELEMAC 2D, and FINEL 3D are second-generation nearshore water level models. For nearshore wave prediction there are phase-resolving and phase-averaging models. The ﬁrst-generation models of the former category (e.g., “mild slope” models) are 2DH models which provide instantaneous surface elevations over a given area. They include a linear representation of refraction, mild shoaling, and an approximate representation of diﬀraction. First-generation models of the latter category are 2DH wave-tracing models that provide results at a point or in an area and they have a linear representation of refraction and shoaling (e.g., COSMOS). The second-generation models (Boussinesq models) are 2DH models which include nonlinear representation of diﬀraction, refraction, and mild shoaling. The secondgeneration models are 2DH models which provide averaged results of tide and surge components across a given area. Examples are the MIKE21 NSW and the STWAVE models. The third-generation models are like the second-generation ones, but include an explicit representation of the nonlinear transfer of energy resulting from the primary wave–wave interaction frequencies (e.g., SWAN). Response probability refers to the occurrence of a particular response (such as overtopping or failure), which in turn depends on the joint occurrence of water levels

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and waves. From a very large number of combinations of wave heights and water levels with the same joint return period, only one will be the worst case for each response. The probability of occurrence of the response function (e.g., overtopping or force) calculated from the worst-case combination of wave heights and water levels will be higher than the joint exceedance probability. In practice, a small margin of safety is added to the joint exceedance probability predictions to try to oﬀset this discrepancy with the return period of the response.37 A more detailed analysis of the pathways of risk and estimation of the predicted probability of ﬂooding is given in the following sections. A ﬁrst approach to determine the probability of extreme events and their implication for ﬂooding would be to analyze the joint probability of strong winds and high water levels, or large waves and high water levels. However, since ﬂooding often is a result of the combined eﬀect of the mean water level and the wave properties through the runup, it is also worthwhile to analyze the runup level, which is the sum of the mean water level and the runup height. Runup height is a critical quantity to estimate when assessing the probability of ﬂooding since it deﬁnes the highest elevation to which the waves might reach. The runup height is normally referenced to the still water level and it includes the wave setup.38 The earliest formula for wave runup height R was developed by Hunt39 : R tan β = Ho Ho /Lo

(38.1)

where β is the beach slope, L the wavelength, and o denotes deepwater conditions. The runup height calculated for every value in the calculated wave time series can be added to the applicable still-water level given by the water level time series. These generated data may constitute a base for determining the risk of ﬂooding including overtopping of defence structures at various alongshore locations in the study area. Hanson and Larson38 used the equation of Hunt to determine runup heights in the southern Baltic Sea for the period 1982–2004, from which the runup levels were derived by taking into account the water levels. Empiric distribution functions were then ﬁtted to the data to extrapolate the total water level (runup level) to high return periods. Figure 38.9 illustrates the empiric distribution function for the annual maximum runup level plotted with Gringorten’s formula together with a ﬁtted Gumbel distribution. 38.2.3. Risk pathways Risk pathways describe the way how the risk travels from the source to the receptors. This part of a risk analysis includes loading and failure modes of ﬂood defences, morphologic changes, breaching initiation and breach growth, reliability analysis, and ﬂood inundation (Fig. 38.10). All these processes help to understand the performance of the entire ﬂood defence system and its components and therefore contribute signiﬁcantly to obtain an estimate of the overall probability of ﬂooding. In this section, the emphasis is on the evaluation of the physical consequences of the coastal storm, (i.e., erosion and ﬂooding). At ﬁrst the probability of ﬂooding of

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Fig. 38.9. Annual maximum runup level plotted against the reduced value from the Gumbel distribution using the Gringorten plotting position formula together with a linear ﬁt.38

Risk Pathways Joint probability distributions from flood risk sources Morphological changes

Loading and failure modes

Breaching initiation and growth

Flood inundation

Reliability analysis: Pf

E(D) Performance of entire defence system and its components, including breach growth as a key issue to provide initial conditions for the assessment of flood wave propagation and inundation

Fig. 38.10.

Pfc

The link between risk pathways and the predicted probability of failure.9

the area is estimated, taking into account the existing morphology. Although this is a static approach, it is possible to introduce some kind of dynamics by including the existing knowledge in local coastal response (e.g., observed variability of beach proﬁles protecting the hinterland). Then, the beach response to the selected storm is

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Storm-induced proﬁle changes: eroded volume ∆V (m3 /m) and beach retreat ∆X (m).

estimated and the inundation of the area after considering the beach modiﬁcation, (e.g., beach lowering and ﬂattening and coastal retreat) is evaluated.40 The main variables used to characterize storm-induced proﬁle changes: eroded volume ∆V (m3 /m) in the inner part of the beach and, beach retreat, ∆X (m) are shown in Fig. 38.11. Beach evolution models describe cross-shore and longshore transport and consequently, the change of coastal morphology. One-line models (e.g., Ref. 41), and coastal area models (e.g., Ref. 42) tend to be based on representations of physical processes and typically include forcing by waves and/or currents, a response by sediments to this energy input, and a morphologic updating module.43 COSMOS, DELFT 2D/3D, LITCROSS SBEACH are some cross-shore beach evolution models, while GENESIS is a well-known longshore model of the same category.44 Walkden and Hall45 have recently developed a long-term model of the erosion and proﬁle development of soft rock shores, called SCAPE. This models the development of the shore platform, beach, talus, and cliﬀ at a series of representative cross-shore proﬁles, each of which is represented by a column of elements. Cowell et al.46 simulated large-scale coastal change using a morphologic behavior model. The method involved extensive parameterization of processes into simplistic behavior-oriented models, avoiding the need for process-based determination, but reliant on observation over greater periods of time. Behavior-oriented models have been successfully applied along coastlines where data spanning many decades were available. There are also hybrid methods being applied which combine, for example, deterministic process based models with parameterized behavior.43 As part of a full systems approach to ﬂood forecasting, it will be important to recognize the inﬂuence of defences. Beach morphology is often a crucial component of a sea defence. The assessment of sea defence assets seldom takes account of the

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morphologic variability of the bed levels and slopes in front of the defence. Changes in bed morphology can have a signiﬁcant inﬂuence on the wave conditions reaching the defence and thus, enhance ﬂooding risks. Numeric models that are used to predict the evolution of the shape of the shoreline thus play a key role in assessing the safety of coastal developments. Wave-driven overtopping of beaches or seawalls is a highly variable process, sensitive to defence structure shape and composition, to water levels and wave conditions. AMAZON-CC, NEWMOTICS, and SKYLLA are some examples of this category of models. Empiric formulae for the calculation of overtopping are also known. Apart from defence overtopping, the understanding of the likelihood of a breach is critical if reliable forecasts are to be made regarding the ﬂood inundation depths and hence risk to life. Predicting breach growth and maximum size is problematic and at present beyond the capabilities of most numeric tools. However, breach events represent the most signiﬁcant of ﬂood scenarios and are of considerable importance in determining ﬂood risk, with two issues of primary importance to the forecaster: (a) breach probability and, (b) breach size and invert level.44 HR BREACH and NWS BREACH belong to the category of shoreline responsebreaching models. Some advanced third-generation ﬂood inundation models (e.g., FINEL 2D–3D nested) include the breaching process. Flood inundation models, which combined with digital terrain models, describe the processes taking place in the ﬂood plain, have shown signiﬁcant progress in the latest years.44 The basis for ﬂood inundation models is a digital terrain map (DTM) of the ﬂood plain area. DTMs are often produced based on laser scan data and from joining survey and laser scan data. Empiric methods are often described as pure mapping. No physical lows are involved in the simulations performed. They are rather simple methods compared to the others, with low cost, but they provide poor estimates of ﬂood risk in large low lying or extensive areas where ﬂows through a breach may be critical in determining the ﬂood extent. First-generation models are essentially 1D models used with a 2DH grid. These models calculate water level in each ﬂood cell at given output steps and therefore enable the duration of ﬂood to be estimated. In cases where the ﬂood plain is extensive, such models can give poor results, because they do not consider the propagation of ﬂoodwater within each cell. Infoworks RS, Infoworks CS, and ISIS are typical examples of this category of models. Second-generation models are 1D/2DH hybrid models and fully 2D. The models of this category use the St Venant equations to model channel ﬂow; however, a 2D continuity equation is used to approximate ﬂow over the ﬂood plain area. HYDROF and LISFLOOD-FP are second-generation models of method 2D− , which means that they are 2D excluding the law of conservation of momentum for the ﬂoodplain ﬂow. MIKE21 and TELEMAC 2D are 2D models, which solve the 2D shallow wave equations. TELEMAC 3D is an available ﬁnite element 3D model included in the category of second-generation models. Third-generation models simulate breaching in 3D with the ﬂood inundation in 2D. They provide better simulations of the ﬂood inundation as the ﬂow velocities at the boundary are accurately simulated. FINEL 2D is a 2DH ﬁnite element model of medium cost and runtime and high accuracy. Third-generation models such as CFX, FLUENT,

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and PHOENIX are usually applied for local predictions of 3D velocity ﬁelds in ﬂoodplains. In a lot of cases, ﬂooding occurs when there is failure of a defence. This can arise from a functional failure (the conditions exceed those for which the defence is designed) or a structural failure (some element or component does not perform as intended under the design conditions). Structural failures are more dangerous because they are unexpected. Excessive overtopping leading to erosion of the back of the crest of the defence, or damage to the toe layers, and toe failure where erosion of the foreshore of the defence occurs to such an extent that the structure is undermined and collapses.43 In structural design, the failure probability is important to evaluate the reliability of the structure. 38.2.3.1. Reliability analysis of coastal structures Reliability is often expressed as a probability of proper functioning. It is the probability that the limit state, just before failure occurs, is not exceeded. Using limit states, the general form of the reliability functions can be deﬁned as Z = R − S, where R is the strength or more generally the resistance to failure and S is the load or more generally that which leads to failure. The reliability is the probability P (Z > 0) and is therefore the complement of the probability of failure: P (Z > 0) = 1−Pf . The strength is seldom known exactly and can best be modeled as a random variable. The probability distribution of the strength is often determined on grounds of test results. However, it is not wise to base the type of the probability distribution type solely on statistic material. It is important to analyze which components comprise the strength. For example, if the strength is deﬁned as a sum or a product of a number of variables, a normal or a log-normal distribution, seem to be the reasonable choices. If the strength is deﬁned by the minimum of a large number of variables, selecting one of the asymptotic extreme distributions is justiﬁed.1 The reliability of an element depends on the margin between the resistance to failure and the loads. In the Netherlands, a level-classiﬁcation of the calculation methods is proposed, which distinguishes three levels: (a) the level III methods calculate the probability of failure, by considering the probability density functions of all strength and load variables; (b) level II entails linearizing the reliability function in a carefully selected point; and (c) at level I an element is considered suﬃciently reliable if a certain margin is present between representative values of strength and load and no failure probabilities are calculated. To choose from the diﬀerent calculation methods for determining the probability of failure, it is desirable to compare these methods as regards a number of points, such as the inﬂuence of the number of variables, the desired accuracy of the failure probability calculation, the shape of the failure space, the statistic dependence of the base variables, and the statistic dependence of the reliability functions. In case of system analysis, it is important to divide the system into subsystems. Mostly, subsystems involved are a set of a number of physical components or processes. Furthermore, a subsystem can be distinguished for every function of the system. For such a division the functional requirements of the system are also the requirements for the functioning of the subsystems. The physical components

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or processes can constitute elements of the diﬀerent functional subsystems, thus relating these to each other. The FMEA (failure modes and eﬀects analysis) is a qualitative analysis, which entails systematically writing down the failure modes of the components and of the subsystems and the consequences for the functioning of the system. The FMEA is usually carried out in the design phase of a system or process, with the objective of identifying the components and subsystems that require improvements as regards reliability. An FMEA becomes a FMECA by adding the so-called criticality matrix. In this matrix the diﬀerent failure modes and consequences are related to each other and the consequences are classiﬁed according to gravity. Furthermore, the frequencies of the diﬀerent failure modes are estimated.1 For a system that consists of statistically independent elements, the probability distribution of the lifespan can be determined with the expected values of the lifespan of the elements, by means of the calculation rules for probabilities of compound events. If the elements are statistically dependent, the calculations have to be carried out with the conditional probability distributions of the lifespans of the elements. 38.2.4. Risk receptors and their vulnerability Risk receptors describe who is receiving the risk, such as people living in ﬂood prone areas, properties, environmental assets, representing the three dimensions of sustainability.3 This category also deals with ecologic impacts and risk perception. Vulnerability refers to the resilience of a particular group of risk receptors, e.g., people, property, and the environment, and their ability to respond to hazardous conditions. For example, elderly people may be less able to evacuate in the event of a rapid ﬂood than young people.2 It can be expressed in terms of functional relationships between expected damages regarding all elements at risk and the susceptibility and exposure characteristics of the aﬀected system, referring to the whole range of possible ﬂood hazards.47 Figure 38.12 shows a methodology to estimate the expected damages of ﬂooding which are related to the receptors of ﬂood risk and their vulnerability. Guidelines and methods for socioeconomic ﬂood damage evaluation, socioeconomic modeling, and risk perception are included in the procedure of expected damage quantiﬁcation. All these factors will help to describe direct and indirect losses and to create a framework to integrate these consequences. The methodology for the estimation of the expected damages of ﬂooding, described in Fig. 38.12, is part of the FLOODsite (Integrated Flood Risk Analysis and Management Methodologies) approach to analyze ﬂooding risk. It should be noted that a distinction is made between direct and indirect damage due to ﬂooding. Direct damage is deﬁned as damage caused by contact with the ﬂood water. This includes, for example, damage to buildings, economic goods and dykes, loss of standing crops and livestock in agriculture.48 Indirect damage is a consequence of direct damage. This category includes damage, which occurs as a further consequence of the ﬂood and the disruptions of economic and social activities, and can aﬀect areas quite larger than those actually inundated. The ﬁrst step in a quantitative analysis of ﬂooding consequences should be the quantiﬁcation

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Vulnerability (Risk Receptors) Flood wave propagation and inundation Guidelines and methods for socioeconomic flood damage evaluation

Socioeconomic evaluation + modeling

Risk perception, community behavior, and resilience

Direct/ indirect and tangible/ intangible losses, including degradation andresilience

E(D)

Fig. 38.12.

The link between risk receptors and the expected damage of coastal ﬂooding.

of the direct damage. A further categorization of ﬂood damage refers to the fact whether the damages can be speciﬁed easily in monetary terms (tangible damage) or whether the damage eﬀects are better quantiﬁed by nonmonetary measures (intangible damage), like number of lives lost or square meters of ecosystems aﬀected by pollution.48 Regarding loss of life (direct intangible damage), some empiric models were developed by Jonkman et al.49 and Penning-Rowsell et al.50 for the context of their nations. Indirect tangible damage is really diﬃcult to model and thus, it is rarely considered in practical vulnerability analyses. Intangible and indirect damages have been rarely considered to date, due to methodologic diﬃculties. The damage due to ﬂooding is inﬂuenced by two factors: the size of the ﬂooded area and the water level in the area. These two factors can be extracted using one of the ﬂood inundation models described in the section dealing with pathways (see Fig. 38.10). From an economic methodologic point of view, direct tangible ﬂood damages are relatively easy to measure and to model ex ante because the damage can be related to market prices. In every ﬂood damage evaluation study, at ﬁrst, an appropriate approach based on the scale, the study objective, the availability of resources, and the availability of pre-existing data, has to be found. Then, an appropriate damage category is determined (e.g., buildings and inventories). It is common place to use the most important damage categories to reduce the eﬀort of the study. The major tasks of ﬂood damage evaluation, the estimation and calculation of economic values of potentially damaged tangible goods, and the gathering of sound information are then carried out. The intensity of diﬀerent ﬂood types by means of various inundation and ﬂood characteristic indicators (e.g., depth and area of inundation, time of occurrence, and velocity), the gathering of land use data for the area at risk and information on the value of assets at risk belong to this stage of damage evaluation. Finally, all information gathered previously is brought together and ﬂood damages

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to be expected from a certain ﬂood event are calculated. Depending on the choice to work with relative or absolute damage functions the percentage of property value approach needs to be distinguished from the absolute of direct damage estimate approach, which is more precise but much more resource and data intensive.48 The damage potential of a speciﬁc area represents the maximum possible amount of damage which may occur if the area becomes inundated. To estimate the consequences of coastal ﬂooding, vulnerability aspects must be considered to calculate the proportion of damage potential which will ﬁnally materialize (i.e., to determine expected damages). In many aspects a vulnerability factor is derived for the most important vulnerability indicators having a substantial impact on the degree of damage produced during a ﬂood event. In some vulnerability analyses, such a factor is derived from expert knowledge and empiric data on ﬂood damages and then expressed on a scale between 0 (no loss at all) and 1 (total loss) to quantify the expected damage reduction for certain categories of elements at risk.47 The most important vulnerability indicator in current ﬂood damage analysis is the exposure indicator “inundation depth.” The distribution of ﬂooding damage can generally be described by a probability density function with two or more peaks. The conditional distribution of the ﬂooding damage can be found by deﬁning damage factors as a function of ﬂooding properties. The damage factor21 is deﬁned as: c=

D Dmax

(38.2)

where D is the ﬂooding damage and Dmax is the maximum possible ﬂooding damage (damage potential). The maximum possible ﬂooding damage is generally taken equal to the value of the full inventory of the area. To ﬁnd the distribution of the consequences of ﬂooding, it is necessary to ﬁnd the distribution of the damage factor for every type of ﬂooding damage. If the damage factor is explicitly written as a function of water level, ﬂooding speed (speed of water level increase), current velocity, wind speed, and wind direction, the distribution of the damage factor can be found from the joint probability distribution of the ﬂooding properties by application of probabilistic methods. Delft hydraulics provides a function for ﬂooding damage to dwellings which is a function of the water level only (Fig. 38.13). Combination of the damage function with the distribution of the water level inside the area directly leads to the distribution of the damage factor. The casualty factor is estimated to be in agreement with the general damage factor: ccas =

Nd Np

(38.3)

where Nd is the total number of casualties and Np is the population size in the aﬀected area. The methodology described above can be used to establish the probability distribution of the casualty factor. An alternative methodology for the risk of loss of life is the loss of life expectancy at birth caused by ﬂooding. This measure of loss of life is used in the model of Nathwani et al.51 The problem of quantifying the eﬀects of ﬂooding risk on the life expectancy at birth is limited to ﬁnding

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Fig. 38.13.

Damage function for dwellings.21

the expected value of the added probability of death as a consequence of ﬂooding. The added probability of death can be found from the casualty factor ccas and the ﬂooding probability Pﬂood by: Pd = ccas|ﬂood · Pﬂood .

(38.4)

Economic consequences consist also partly of direct economic damage and partly of indirect economic damage. Damage factors, introduced above, can be used to quantify this kind of direct damage. Indirect economic damage is deﬁned as the damage caused by the primary damage due to relations that exist between the ﬂooded area and the rest of the economy. The estimates of economic consequences of ﬂooding depend on the choice of the scope of the study and on the deﬁnition of cost. It may be clear that quantiﬁcation of the macro-economic eﬀects of ﬂooding requires extensive modeling. Within the European Research Program FLOODsite (Integrated Flood Risk Analysis and Management Methodologies) methods of ﬂood damage evaluation in four European countries: England, the Netherlands, the Czech Republic, and Germany, which feature very diﬀerent histories of ﬂood protection policy and different institutional settings, were reviewed. It can be noted that the major diﬀerences relate to the damage categories considered, the degree of detail, the scale of analysis, the application of basic evaluation principles (e.g., replacement cost versus depreciated cost), and the application or nonapplication of results in beneﬁt–cost and risk analyses.52 In England and Wales, there is a long history of developing and applying methods of ﬂood damage evaluation. Diﬀerent methods for ﬂood damage evaluation are used at diﬀerent spatial scales (national, regional, intermediate, and local level). In almost all levels standard damage data developed by the Flood Hazard Research Center (FHRC, Middlesex University) are used for ﬂood damage evaluation.

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The newest and updated version is the multi-colored manual (MCM).53 The MCM provides methods for the quantiﬁcation of many damage categories in monetary terms, not only for direct tangible damages, but also for indirect and some intangible losses, particularly regarding the damages to residential and nonresidential properties. Regarding residential properties, at ﬁrst a classiﬁcation is carried out after type, age, and social status. After that a typical inventory is compiled for each of these altogether 100 types. The depreciated value of the complete building structure inclusive its inventories are then determined according to replacement costs and market prices. On the basis of the assessment of loss adjusters, the susceptibility of these assets to inundation depth is then assessed and ﬁnally absolute depthdamage functions are derived. Depth-damage functions show the total damage of the valuable property (e.g., buildings, cars, and roads) or its relatively damaged share as a function of inundation depth. Moreover, long and short durations are distinguished. Figure 38.14 shows the damage functions for diﬀerent types of houses. This standard depth-damage data is used for damage evaluation on all spatial levels. At all spatial levels there are also examples for the use of the Social Flood Vulnerability Index (SFVI) as a measure for the coping capacity of the ﬂood aﬀected population. The index was developed by FHRC (Flood Hazard Research Center) including indicators for vulnerable groups and persons like elderly people, lone parents as well as persons with preexisting health and ﬁnancial deprivation problems. FHRC also developed a FORTRAN-based software for the calculation of damages called ESTDAM, which is not GIS based. In the Netherlands, especially inundation depth is needed for the damage evaluation. In case of residential buildings, impacts of velocity and waves are also considered. Regarding casualties, three diﬀerent inundation characteristics are taken into account: velocity, rise rate, and inundation depth. The amount of tangible damage for each grid cell is calculated by the formula: S=

n

ai ni Si

(38.5)

i=1

Fig. 38.14.

Absolute damage functions for diﬀerent residential house types (sector mean).53

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where ni is the number of units of category i, Si is the maximum damage per unit in category i, and ai is the damage factor category i. The three essential parts of the damage evaluation are: the gathering of land use information, the estimation of maximum damage amounts per unit of each category, and the derivation of (depth-) damage functions. Altogether 11 damage functions are derived from a study by Vrouwenvelder.52,54 The damage functions are mostly depth-damage functions. In the Czech Republic, the methods of ﬂood damage evaluation focus also on the estimation of direct, tangible damages. For buildings, the formula for the estimation of damages to an individual building ﬂoor underwater is: DAMAGE = H · C · %p · A

(38.6)

where H is the height of each ﬂoor of an individual building, C is the price of a cubic meter of the building, A is the ground ﬂoor area of the building, and %p is the percentage of damage according to damage function. The cost of health problems due to ﬂooding is the only intangible loss category, which is tried to be included in monetary terms. Vulnerability analyses in the area St. Peter-Ording in Germany are performed using a methodology which estimates the socioeconomic damage potential on the basis of oﬃcial statistics and indicators. The calculation of the possible damages is mainly based on the so-called depth-damage functions. Thereby it is assumed that standardized functions represent a dependency between the key parameter ﬂood level at the individual objects and the expected damage. The functions were based on data of past ﬂood events and describe either the monetary damage or the percentage of the total value of the object. Some depth-damage functions used in the MERK project are presented in FLOODsite.52 In addition, there are categories (e.g., gross value added, forest land), whose damages are not deﬁned by the ﬂood depths. Here, speciﬁc other parameters, such as ﬂood duration, determine the damage expectation. 38.3. Flood Risk Assessment Tolerability refers to willingness to live with a risk to secure certain beneﬁts and in the conﬁdence that it is being properly controlled. To tolerate a risk means that we do not regard it as negligible, or something we might ignore, but rather as something we need to keep under review and reduce still further if and as we can.2 In the past, three criteria for risk acceptance were developed: (a) the personal acceptance of risks, (b) the social acceptance, and (c) the economic criterion. The personally accepted risk level is deﬁned as the frequency of suﬀering a certain degree of injury as a result of an event, accepted by an individual. The social risk concerns the risks for the total population. Society looks at the total consequence of an event, including the number of casualties, material and economic damage, and the loss of immaterial [matters]. Generally, the consideration of social consequences in the case of safety problems is limited to the number of casualties as a result of an event. More often, the social consequence is considered the total material damage. This deﬁnition is more suitable for an economic optimization of the risk level to be

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accepted. It is recognized that the limited consideration of the consequences in the form of the number of casualties or the economic damage is not suﬃcient to describe the social perception of the loss. If the economic optimal risk is greater than the personally or socially accepted risk, the economic optimum will not be applied as a testing criterion.1 Quantifying acceptable, individual and group risk has been the subject of much debate in the recent years and a number of techniques exist to elicit indicative acceptable standards for ﬂood and coastal defence. The approach that is being widely promulgated in the UK is that the general form or framework for acceptability criteria should be represented as a three-tier system with: (a) an upper bound on individual or societal risk levels, beyond which risks are deemed unacceptable; (b) a lower bound on individual or societal risk levels, below which risks are deemed not to warrant concern; and (c) an intermediate region between the previous two, where further individual and societal risk reduction are required to achieve a level deemed “as low as reasonably practicable,” the so-called ALARP principle.2 To achieve a wide consensus on the acceptable ﬂood risk, it is indispensable that the various methods, rules, and tools to be developed in the advanced ALARP framework are robust and transparent.55 The description of the acceptability of the risk can be a function of the consequence. This function oﬀers a maximum allowable probability of exceedance for all possible consequences. If the damages are expressed in monetary terms, the target ﬂooding probability Pft may be formulated as a cost-optimization problem. Most of the diﬃculties arise when trying to evaluate the so-called intangible losses, such as human injury, loss of life, and environmental and cultural losses caused by ﬂooding. In CUR,1 a simple formulation of the personally accepted risk, in case that decease is the most important consequence of the failure of a system or of an accident is given: Pﬁ ≤

βi · 10−4 Pd|ﬁ

(38.7)

where Pﬁ is the probability of an accident as a result of activity i, Pd|ﬁ is the probability of decease after an accident, and βi is the policy factor that is determined on the extent to which participation in the action is voluntary and/or beneﬁcial to the individual, varying from 10 to 0.01. In CUR,1 a formulation of the risk acceptance at a local level is given: Pf ij = 1 − FNdij (n) ≤

Ci n2

(38.8)

for all n, where Ci =

βi · 100 √ k · Na

in which Na is the number of independent locations where the activity takes place, and k is the risk aversion index. The measures for acceptable risk of loss of life by Vrijling et al.56 are given as direct limitations on the probability distribution of the yearly number of casualties. The distribution of the yearly number of ﬂooding

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casualties can be found from the ﬂooding probability and the damage factor for casualties. Quantitative methods to deﬁne acceptable ﬂooding probabilities on the level of the protected area have been obtained from literature. Three commonly used methods are21 : cost–beneﬁt analysis, utility analysis, and life quality analysis. In cost–beneﬁt analysis, a decision is deemed acceptable if the beneﬁts exceed the cost of protection. Costs and beneﬁts are expressed completely in monetary terms. The cost beneﬁt model can be expressed as: Bref (Pﬂood , T ) = −I(Pﬂood ) +

T t=0

T

Pﬂood (cd cb b(t) + cd d(t)) b(t) − t (1 + r) (1 + r)t t=0

(38.9)

where Pﬂood denotes the ﬂooding probability, I is the function of the cost of protection, T is the reference period chosen in the design stage of the protection system, b(t) are the beneﬁts at time t, d(t) is the maximum ﬂooding damage at time t, cd is a factor deﬁning the part of the indirect economic damage caused by the direct economic damage, cb is a factor describing the macroeconomic eﬀects of ﬂooding, and r is the market interest rate, including inﬂation. Rules have to be established that indicate whether a given value of the ﬂooding probability is acceptable in view of resulting value of cost–beneﬁt model and/or constraint on the ﬂooding probability itself. Figure 38.15 shows an example based on the work of van Dantzig21 of the optimization of the ﬂooding probability (acceptable ﬂooding probability), combining the cost–beneﬁt model with an acceptance rule which considers the sum of costs and beneﬁts over the reference period to be nonnegative.

Fig. 38.15. Analysis of acceptable ﬂooding probability by requiring a nonnegative value of the sum of costs and beneﬁts.21

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Utility models form a mathematically consistent basis for decision making on ﬂood protection. To be useful for decision-making purposes the utility model should be deﬁned in such a way that the model reﬂects the preferences of the society. In utility analysis, nonmonetary measurable consequences are expressed in a subjective measure of the usefulness of the decision (utility). The same is done for the monetary consequences, so that ultimately all costs and consequences are measured in the same variable. The expected value of utility over the reference period T is given as: Uref (Pﬂood , T ) = uexp;0 (Pﬂood ) +

T

uexp(t)

t=1

uexp (Pﬂood , T ) = (1 − Pﬂood ) · u + Pﬂood · u

b(t) v(1 + r)t

(38.10a)

(1 − cd cb )b(t) − cd d(t) v(1 + r)t

(38.10b)

where u is an utility function of the exponential type, r is the risk aversion, v is a normalization factor, and all other symbols are deﬁned as previously. Figure 38.16 shows the total expected utility over the reference period as a function of ﬂooding probability. Maximization of the expected utility over T gives an optimum ﬂooding probability. The life quality method, proposed by Nathwani et al.51 appears closely related to utility methods. Decisions concerning risk are judged using a social indicator called the life quality index that combines the eﬀects of the decision on the life expectancy

Fig. 38.16.

Total expected utility over the reference period as a function of ﬂooding probability. 21

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Fig. 38.17.

Application of life quality index to a case study after van Dantzig.21

at birth and the gross domestic product per capita. The summed life quality over the reference period is deﬁned as: L(Pﬂood , ccas , T ) =

T

1(t, Pﬂood , ccas )

(38.11a)

t=0

1(Pﬂood , ccas , T ) = g(t)w (e(Pﬂood , ccas ))1−w

(38.11b)

where ccas is the damage factor for casualties, g is the gross domestic product per capita, e is the life expectancy at birth, and w is the relative lifetime spent working. Figure 38.17 shows the application of life quality index to a case study after van Dantzig. 38.4. Conclusions Risk is a combination of the chance of a particular event, with the impact that the event would cause if it occurred. Risk has two components: the chance or probability of an event occurring and the impact or consequence associated with the event. For a risk to arise there must be a hazard that consists of a “source” or initiator event (i.e., high wave height and storm surge), a “receptor” (e.g., people and property), and a pathway between the source and the receptor (i.e., ﬂood routes including defences). Flood risk management includes the analysis and assessment of risks and the formulation and implementation of risk reduction options. A measure of the ﬂood

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risk level which is appropriate for the decision making under consideration can be formulated by calculating the diﬀerence Rfc − Rft of the predicted and the acceptable risk of coastal ﬂooding, the remaining risk. The EVT is used to estimate return levels and joint probabilities of the sources of ﬂooding risk. It is important to ﬁnd good statistic models that give accurate parameter predictions and corresponding uncertainty estimates, and is therefore of interest to ﬁnd both marginal probability distributions as well as joint probability distributions for combined parameters. Runup height is a critical quantity to estimate when assessing the probability of ﬂooding since it deﬁnes the highest elevation to which the waves might reach. Risk pathways include loading and failure modes of ﬂood defences, morphologic changes, breaching initiation and breach growth, reliability analysis, and ﬂood inundation. All these processes help to understand the performance of the entire ﬂood defence system and its components, and therefore contribute signiﬁcantly to obtain an estimate of the overall probability of ﬂooding. At the beginning, the probability of ﬂooding of the area is estimated, taking into account the existing morphology. Then, the beach response to the selected storm is estimated and the inundation of the area after considering the beach modiﬁcation (e.g., beach lowering and ﬂattening and coastal retreat) is evaluated. Beach evolution, shoreline response, and ﬂood inundation models are utilized for this purpose. Knowledge gaps in the science of coastal morphology lead to uncertainty in the solutions derived by engineering methods. This is because the methods are based on an incomplete understanding of the physical processes that govern the morphodynamics of dunes and beaches. Filling these knowledge gaps is expected to reduce the uncertainty in our understanding and improve our predictions of morphology and ﬂood risk. In risk analysis, a good analysis of the functioning of the system and the subsystem is of great importance. The reliability of a system is the extent to which a system meets the requirements and is determined by the reliability of the elements and by the relations between those elements. The reliability of an element depends on the margin between the resistance to failure and the loads. Reliability-based design is deﬁned as a design approach where the probability of failure is used as a measure of the performance of the structure. A maximum failure probability is deﬁned and the structure should meet the requirement. Vulnerability refers to the resilience of a particular group of risk receptors, e.g., people, property, and the environment and their ability to respond to hazardous conditions. The damage due to ﬂooding is inﬂuenced by two factors: the size of the ﬂooded area and the water level in the area. Damage factors are used to estimate the conditional distribution of the ﬂooding damage as a function of ﬂooding properties. The most important vulnerability indicator because of coastal ﬂooding in current ﬂood damage analysis is the exposure indicator “inundation depth.” Thus, the calculation of the possible damages in most European countries is mainly based on the so-called depth-damage functions. These functions describe either the monetary damage or the percentage of the total value of the object related to inundation depth.

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Three criteria were developed to deﬁne acceptable risk: the personal acceptance of risks, the social acceptance, and the economic criterion. Maximum allowable probabilities of exceedance for all possible consequences and all risk categories are given in the literature. Three commonly used quantitative methods to deﬁne acceptable ﬂooding probabilities on the level of the protected area are cost–beneﬁt analysis, utility analysis, and life quality analysis. A general purpose of the ﬂood risk management framework is providing a basis for rational decision making. Recording diﬀerent variants with associated risks, costs, and beneﬁts, in a matrix or decision tree serves as an aid for making decisions. Once the predicted ﬂood risk (Rfc ) and the acceptable ﬂood risk (Rft ) are obtained, a measure of the ﬂood risk level that is appropriate for the decision making under consideration can be formulated as a function of cost and further intangible losses. For instance, a risk scale G = Rfc − Rft /Rft = 0 shows an optimum risk level. Negative G-values mean overdesign, while positive G-values mean underdesign. If the ﬂood risk management framework is used in the design of processes and objects, the steps are often repeated several times with adjusted system speciﬁcations to obtain an optimal design. An economic optimization calculates the costs of the process or object and the risk with every iteration step. The design is optimal when the costs are minimal. A risk-based approach oﬀers signiﬁcant advantages compared to the current practice. A ﬂood-risk approach allows the choice of the safety levels to be further rationalized if the consequences of ﬂooding and the costs of protection are made explicit. Risk-based approaches exist also in other ﬁelds where safety levels have to be deﬁned, so that a risk-based approach to ﬂooding safety opens the probability of comparison of risk levels. The lack of adequate data to specify parameters probabilistically and the diﬃculty in communicating uncertain results to the public and other professionals are the main drawbacks of a risk-based approach. However, the skepticism as to the ignorance of the techniques can be combated by routine application of risk techniques. In a risk analysis framework uncertainty is understood and attempts are made to handle it transparently.

Acknowledgment This work was supported by the European Community’s Sixth Framework Programme through the grant to the budget of the Integrated Project FLOODsite, Contract GOCE-CT-2004-505420.

Disclaimer This chapter reﬂects the authors’ views and not those of the European Community. Neither the European Community nor any member of the FLOODsite Consortium is liable for any use of the information in this chapter.

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42. T. J. Chesher, H. M. Wallace, I. C. Meadowcroft and H. N. Southgate, PISCES-A Morphodynamic Coastal Area Model, First Annual Report, HR Wallingford Report SR337 (1993). 43. FLOODsite/Integrated Flood Risk Analysis and Management Methodologies — Predicting Morphological Changes in Rivers, Estuaries and Coasts, Report T05-07-02 (2007). 44. DEFRA/Environment Agency Flood and Coastal Defence R&D Programme — Best Practice in Coastal Flooding, R&D Technical Report FD2206/TR1, Workshop Version issued 24/09/03, HR Wallingford Report TR 132 (2003). 45. M. J. A. Walkden and J. W. Hall, A predictive mesoscale model of the erosion and proﬁle development of soft rock shores, Coast. Eng. 52, 535–563 (2005). 46. P. J. Cowell, P. S. Rou and R. A. Jones, Simulation of large-scale coastal change using a morphological behaviour model, Marine Geol. 126, 45–61 (1995). 47. F. Messner and V. Meyer, Flood damage, vulnerability and risk perception — Challenges for ﬂood damage research, UFZ discussion papers, Department of Economics, Germany (2005). 48. F. Messner, Evaluating ﬂood vulnerability — Scope of approaches and challenges to research, Flood risk management research — From extreme events to citizens involvement, Proc. Eur. Symp. Flood Risk Manag. Res. (EFRM 2007), Dresden (DE), IOER (2007), pp. 75–82. 49. S. N. Jonkman, P. van Gelder and J. K. Vrijling, Loss of life models for sea and river ﬂoods, Flood Defence 2002, eds. B. Wu et al. (Science Press, New York Ltd., 2001), http://www.waterbouw.tudelft.nl/public/gelder/paper120b-v10210.pdf (2002). 50. E. C. Penning-Rowsell, P. Floyd, D. Ramsbottom and S. Surendran, Estimating injury and loss of life in ﬂoods: A deterministic framework, Nat. Haz. 36, 43–64 (2005). 51. J. S. Nathwani, N. C. Lind and M. D. Pandey, Aﬀordable safety by choice, the life quality method, Inst. Risk Res. University of Waterloo, Canada (1997). 52. FLOODsite/Integrated Flood Risk Analysis and Management Methodologies — National Flood Damage Evaluation Methods — A Review of Applied Methods in England, the Netherlands, the Czech Republic and Germany (2005). 53. E. C. Penning-Rowsell, C. Johnson, S. Tunstall, S. Tapsell, J. Morris, J. Chatterton, A. Coker and C. Green, The beneﬁts of ﬂood and coastal defence: Techniques and data for 2003, Flood Hazard Research Centre, Middlesex University (2003). 54. A. C. W. M. Vrouwenvelder, Project uncertainty-analysis, Delft, TNO 97-CON-R0935 (1997). 55. A. Kortenhaus and H. Oumeraci, Probabilistic risk analysis as a basic tool for integrated coastal zone management, 6th Int. Conf. Hydrosci. Eng. (ICHE), Brisbane, Australia (2004). 56. J. K. Vrijling, W. van Hengel and R. J. Houben, Acceptable risk as a basis for design, Reliab. Eng. Sys. Safety 59, 141–150 (1998).

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Chapter 39

Physical Modeling of Tsunami Waves Michael J. Briggs Coastal and Hydraulics Laboratory US Army Engineer Research and Development Center 3909 Halls Ferry Rd., Vicksburg, MS 39180-6199, USA [email protected] Harry Yeh∗ and Daniel T. Cox† School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331-3212, USA ∗ [email protected] † [email protected] This chapter summarizes the physical modeling of tsunami waves. The ﬁrst section contains an overview of historic studies. The next section describes equipment and procedures used in laboratory studies for generating, measuring, and analyzing tsunami waves. The third section summarizes some early National Science Foundation-funded laboratory studies of runup on vertical walls, plane beaches, and circular islands that were conducted at the Coastal and Hydraulics Laboratory. The next section describes the latest Network for Earthquake Engineering Simulation-funded, the state-of-the-art tsunami wave-making facility at Oregon State University, and several recent studies. Finally, the last section is a summary and conclusion of laboratory modeling of tsunami waves.

39.1. Introduction Tsunamis are long waves relative to the ocean (on the order of 20 times ocean depth) as their wave periods are much longer than typical gravity waves. In the deep ocean, one barely notices the tsunami as it passes underneath at the speed of a jet plane. As the tsunami approaches shallower water, it begins to “feel” (i.e., react to) the bottom as the entire water column is moving. It undergoes refraction, diﬀraction, reﬂection, and breaking. The tsunami increases in height and steepness with complicated currents and multiple wave trains. Edge waves may even develop depending on beach slope and bathymetry, coastline irregularity, and incident wave 1073

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direction. Reﬂections from adjacent shorelines may aﬀect the number of tsunami waves and their amplitudes. The wave moves ashore as a fast-rising tide or as a bore (if broken) ﬂooding everything in its path and producing massive destruction due to the movement of large objects, such as trees, cars, and trains. Historically, people have worried about far-ﬁeld tsunamis caused by the tectonic events or earthquakes on the ocean ﬂoor. They are known to “export death and destruction.”1 The sudden movement and displacement of the Earth’s plates deforms the seaﬂoor and produces tsunami waves on the ocean surface that travel great distances to distant coastlines. Although volcanic eruptions and asteroid impacts can trigger tsunamis, these are far less likely. Recent research indicates that many tsunamis can be generated from co-seismic landslides, triggered by the violent shaking of the Earth’s crust.2 Concern has recently focused not only on the landslide-generated tsunami, but also on potential near-ﬁeld or local tectonic sources. Both these tsunami scenarios have the characteristic of far less travel time (and reduced warning time for evacuation) from the generation area of the tsunami to the coastal inundation site. The Indian Ocean Tsunami of December 26, 2004 was a “wakeup call” about the dangers of tsunamis.3 In the past 400 years, it is estimated that more than a million people have died by tsunamis, with billions of dollars in damages. Prior to the Indian Ocean Tsunami, Synolakis4 reported that 12 major tsunamis had occurred around the Paciﬁc Rim in the past 10 years, causing more than 3,000 deaths and $1B in damages. Wave runup is the most devastating hazard associated with tsunamis, yet it is the least understood. The Intergovernmental Oceanographic Commission of UNESCO (IOC) Tsunami Glossary deﬁnes runup as the diﬀerence between the elevation of maximum tsunami penetration (inundation line) and the sea level at the time of the tsunami.5 It further describes runup as the elevation reached by seawater measured relative to a reference datum. Vertical elevation can be as high as 30 m and usually is on the order of 10 m.6 A good understanding of tsunami wave runup is essential in calculating forces on coastal structures and predicting inundation zones for coastal ﬂooding, especially as this impacts civil defense evacuation plans and routes.7 Physical models are one of the best tools for understanding the complex physics governing tsunami wave transformation and runup. Numeric models are constantly being improved, but require good laboratory or ﬁeld data to validate the latest enhancements. Field data are becoming more available for a variety of scenarios, but collecting and verifying data quality is diﬃcult.8 Laboratory experiments have the advantage of controlled input parameters and conditions. Especially with largerscale models (closer to prototype), scale and boundary eﬀects become less of a problem. Also, laboratory models have the advantage of enabling people to see and hear the processes as they occur. In 1992, the National Science Foundation (NSF) funded a three-year research program to better understand three-dimensional (3D) tsunami wave runup9 based in part on recommendations from the International Workshop on Long Wave Runup.10 Goals of this project were to (a) advance the theoretic and numeric modeling techniques for simulation of 3D free surface ﬂows, (b) generate small- and large-scale experimental databases for veriﬁcation and modiﬁcation of numeric models, and

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(c) improve the predictive models used for tsunami hazard mitigation by including improved reproduction of the important physical parameters in tsunami runup models. This joint research study included participants from Cornell University, Harvard University, University of Washington, University of Southern California, and the Coastal and Hydraulics Laboratory (CHL) of the US Army Engineer Research and Development Center (ERDC). This tsunami research was the ﬁrst to be systematically designed to include multiple institutions with multi-disciplinary expertise. Due in part to the large number of destructive tsunamis in the 1990s, the NSF funded the Network for Earthquake Engineering Simulation (NEES) for earthquake and tsunami research in the academic community in 2003. Oregon State University (OSU) is one of the engineering schools in the USA selected for tsunami research, especially for laboratory modeling. This chapter summarizes physical modeling of the hydrodynamics of tsunamis. The ﬁrst section contains an overview of historic laboratory studies, with particular emphasis on early pioneering research. The next section describes the physical model design, equipment, and procedures for generating, measuring, and analyzing tsunami waves. The third section summarizes the NSF-funded, large-scale laboratory studies conducted at CHL to better understand the runup on vertical walls, plane beaches, and circular islands. The fourth section describes the latest stateof-the-art tsunami wave-making facilities at Oregon State University that are part of NEES. Finally, the last section summarizes physical modeling and what we can expect in future studies. It is, of course, not possible in this short chapter to describe all the results and provide comprehensive data lists. Interested readers may contact any of the authors for additional information and data.

39.2. Overview of Historic Studies The ﬁrst systematic laboratory experiments relevant to the study of tsunamis were introduced by Russell11 after he accidentally observed the generation of a solitary wave in August 1834 while investigating ship waves. His historic remark is as follows: I believe I shall best introduce this phenomenon by describing the circumstances of my own ﬁrst acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped — not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violate agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-deﬁned heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original ﬁgure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase

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of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my ﬁrst chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears; when I have since found to be an important element in almost every case of ﬂuid resistance, and ascertained to be the type of that great moving elevation of the sea, which, with the regularity of a planet, ascends our rivers and rolls along our shore.

This discovery of solitary waves motivated Russell to build a water tank for generating waves of translation by the addition of a volume (7.29 P) by dropping weights at one end of the water tank. The breadth of the tank was 30.5 cm and the still water depth was 13 cm. The original sketch of his experiments is shown in Fig. 39.1. Russell conﬁrmed that the propagation speed c could be predicted accurately by c = g(h + a), where h is the quiescent water depth and a is the height of the wave crest above the water level. While he measured the characteristics of a solitary wave for its permanent and stable form, Russell also performed the experiments by the removal of a volume by lifting weights as shown in Fig. 39.2. His experiments generated a radiating oscillatory wave train, and the propagation speed was slower than the equivalent wave of translation (solitary wave). In 1972, more than 100 years later, Hammack12 performed his experiments at Caltech generating waves by volume displacement, just like the experiments by Russell. Hammack’s experiments were speciﬁcally designed for tsunamis, generating

Fig. 39.1.

Scott Russell ﬂume experiment for generating solitary waves.

Fig. 39.2.

Russell solitary wave time series.

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waves by displacing vertically a small end portion (61 cm long) of the tank ﬂoor. The tank was 38.6 cm wide and 31.7 m long. Unlike Russell’s experiments, the tsunami generation was precisely controlled by a hydraulic servo-system, and the displacement of water at the ﬂoor was a closer mimic of the real seismic seaﬂoor displacement. Hammack’s experiments were the ﬁrst to use the KdV-soliton theory in a real ﬂuid environment.13 The theory predicts that an initial water displacement will evolve in the farﬁeld to generate a ﬁnite number of solitary waves — the larger one leads the smaller ones — followed by oscillatory waves. Hammack and Segur14 and Tadepalli and Synolakis15 later demonstrated by their scaling argument that such theoretic prediction may not be realizable in the real world. They noted that their ﬂume was not physically long enough for the solitons to accurately sort into separate waves. The Earth does not have suﬃcient size for a real tsunami, typically generated by co-seismic faulting, to evolve into the separation of solitons. The real tsunami is often too linear and too nondispersive. Tsunami generation in a 3D basin was ﬁrst performed by Takahashi and Hatori.16 Tsunamis were generated in a 25 m × 40 m × 0.6 m deep water tank by rapid dilation of an elliptic-shaped, rubber membrane driven by compressed air. Wave heights and propagation speeds were measured in their experiments. Besides co-seismic fault displacement, tsunamis can also be triggered by submarine and subaerial landslides. Perhaps, the ﬁrst systematic and careful experiments were those performed by Wiegel.17 In his experiments, rigid bodies of several shapes, sizes, and weights were used to slide down the submerged sloping surface. Wiegel found that the generated wave has what is now referred to as an N -wave proﬁle; the leading wave has an elevation followed by a depression wave of amplitude greater than the leading elevation.18 Note that his focus on tsunamis propagating oﬀshore was evidently motivated by the 1946 Aleutian tsunami. Wiegel’s keen foresight to examine a potential submarine landslide associated with this event is surprising — nearly 50 years later, several scientists discussed a conjecture of a submarine landslide scenario for the 1946 Aleutian event (e.g., Refs. 18 and 19). Similar experiments in an even smaller laboratory apparatus were conducted 50 years later by Watts.20 Raichlen and Synolakis21 conducted other experiments in a similar manner — sliding a rigid body on an inclined surface. They used a relatively wide tank so that marginal 3D runup motions along the beach could be examined, although the alongshore propagation distance was too short to examine their evolution. Fritz et al.22 studied subaerial landslides by sliding granular material instead of a rigid body. Detailed velocity ﬁelds were measured with the particle imagery velocimetry (PIV) technique. Runup of tsunami onto a sloping beach is often modeled by imposing an incident solitary-wave form for convenience, taking advantage of a solitary wave being a permanent and stable wave form. Although earlier attempts had been made to study this problem (e.g., Refs. 23 and 24), Synolakis25 was the ﬁrst to demonstrate careful validation of the theory with his experiments. The key to generating a “clean” solitary wave is the precise control of wave-paddle motion using “nonlinear” wavemaker theory; such an algorithm was given by Goring and Raichlen.26 This algorithm was later generalized for the creation of arbitrary long waves by Synolakis.27

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Another incident-wave model often used for studying tsunami runup is a bore. Miller28 may be the ﬁrst to systematically examine the runup of a bore that was generated by a ﬂat vertical piston. This type of bore generation may generate unwanted simple depression waves when the piston stops. To circumvent this contamination, Yeh et al.29 generated a single bore by lifting a vertical gate that initially separated the quiescent water on the beach from the deeper water behind the gate. The laboratory observations of the maximum runup locations were always signiﬁcantly lower than the predictions made by the nonlinear shallow water theory. Aiming at evaluating tsunami forces on onshore structures, Cross30 measured the forces caused by the incident surge impact on a vertical wall. He generated a single bore using the dam-break method. Arnason31 (also see Ref. 32) investigated boreimpinging forces on vertical columns as well as the detailed ﬂow ﬁelds surrounding the column with the combination of laser Doppler velocimetry (LDV) and PIV. There are many benchmark experiments for harbor resonance. Following the pioneering work by Ippen and Raichlen,33 Ippen and Goda34 and Lee35 performed careful laboratory experiments on harbor resonance induced by waves that can be similar to the tsunami problem. In general, historic experiments were performed primarily to explore characteristics and behaviors of tsunamis. On the other hand, a majority of recent experiments aims at providing adequate benchmark data sets for validation of numeric models. For example, benchmarking exercises for numeric models were conducted at the 1995 Friday Harbor Workshop36 and the 2004 Catalina Island Workshop.37 Objectives of laboratory experiments have evolved together with advances in measuring instruments.

39.3. Physical Model Design, Equipment, and Procedures 39.3.1. Model design In general, the smaller the scale factor (i.e., closest to prototype), the better as it minimizes the scale eﬀects due to dominant forces not being in proper proportion. A typical scale factor is Lr = Prototype length/model length = 100, or 100 m in the prototype is equivalent to 1 m in the model. Of course, a value smaller than this, like 50 or 25 is even better, but the model costs are usually much higher and diﬃcult to justify except for sediment transport studies. Larger scales can be used, but other concerns due to measurement error and capillary eﬀects can adversely aﬀect the model. Laboratory eﬀects are a concern due to artiﬁcial boundaries in a physical model like walls. Models are usually undistorted (i.e., the vertical and horizontal scales are the same). An example of a distorted scale model is the large physical model (0.4 hectares) of the ports of Los Angeles and Long Beach at the CHL where the vertical scale is 100 and the horizontal scale is 400. The purpose of this model is to study small amplitude, long wave energy and harbor resonance, so a distorted scale model works for these longer wave periods (with relatively small vertical velocities and accelerations) and could work for tsunami waves. A distorted model can reduce the

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inconsistency (i.e., viscous scale eﬀect) caused by the Reynolds number discrepancy, but introduces complex interpretation of data due to the violation of geometric similitude. The scale factor aﬀects the design and cost of the experiment including the selection of ﬂume or basin, model size, wavemaker, and instrumentation. To achieve a dynamically similar ﬂow condition in a scaled model, the Froude number F , the Reynolds number R, and the Euler number E should match those of the prototype. When the ﬂow is transient, the Strouhal S number should be matched as well. These parameters are deﬁned as: V F =√ ; G

R=

ρV ; µ

E=

2pf ; ρV 2

S=

Vt

(39.1)

where V is the characteristic velocity, g is gravitational acceleration, is a representative geometric distance or length, ρ is ﬂuid density, µ is dynamic viscosity, pf is the characteristic ﬂuid pressure, and t is the characteristic time. For a scaled model in the tsunami tank, it is impractical to use a diﬀerent ﬂuid because of its basin size. All the dynamic similitude parameters can usually be matched except the Reynolds number. For experiments under Froude law, wave period and velocity scale as the square root of the length scale factor Lr . Although R cannot normally be matched in scaled experiments with that of the prototype, it is customary to assume that, if the model R is large enough to generate fully developed turbulent ﬂows, the model results are relatively insensitive to R. The prototype R is typically on the order of 107 , using ﬂow depth as the characteristic length . In the case of tsunami basin experiments of 50 scale, the R would be about 3 × 104 . Hence, it is anticipated that scale eﬀects associated with R should not be signiﬁcant. 39.3.1.1. Flume or basin Flumes are two-dimensional (2D) and basins are 3D. Thus, basins have more parameters (relative to the 3D) that can be varied relative to ﬂumes. The dimensions of the ﬂume or basin must be adequate for the desired model scale. Scales are often smaller in ﬂumes since the walls may be higher and the wavemaker has a longer stroke. Flumes can have wall eﬀects due to the relative closeness of the ﬂume sides, however. Waves can only be unidirectional in ﬂumes. Flumes often have viewing transparent (plexiglass) windows for observing the wave patterns and transformation. Basins are much wider and often larger than ﬂumes, meaning higher construction costs and water usage occur, but the model can be larger and 3D. Waves can be multi-directional and more realistic because the wave transformation mechanisms of wave refraction and diﬀraction can be simulated. Both ﬂumes and basins should have some form of passive wave absorption along the boundaries to minimize unwanted wave reﬂections from walls (not present in the prototype) and reﬂective coastal structures (i.e., vertical walls that have less space to dissipate in models than in the prototype). Although this absorption is usually less critical for tsunamis than for wind waves, it is still useful to minimize the wasted time between runs that can be lost waiting for the water level to still out. In basins it is possible to have open boundaries on one or more sides that help minimize

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unwanted reﬂections. It is common to use rock, metal shavings, synthetic “horse hair,” or wave absorber frames around the basin perimeter to reduce reﬂections from model structures and basin walls. Longer waves typically require wider wave absorbers to be eﬀective. 39.3.1.2. Model size and layout Most physical models in tsunami studies consist of scaled oﬀshore bathymetry and a shoreline to simulate a beach, island, or coastal structure such as a breakwater or seawall. The models are usually ﬁxed bed since sediment transport issues are not considered. The ﬁxed bed model is typically constructed using templates and sand with a thin mortar cap or veneer. If a movable bed is used, then scaling issues become a concern since it is diﬃcult to properly scale both the sediment and the model with the same scale factor. Sediment grains can be minimized only so much before changing from noncohesive (i.e., sand) to cohesive behavior (i.e., mud). Model size is governed by the scale factor, dimensions of the ﬂume or basin, and wave and current-generating capability of the laboratory equipment. The orientation of the model is usually ﬁxed in a ﬂume, since it is 2D and consists of a cross-section of the bathymetry and coastal structure. The main concern is the distance between the wavemaker, shoreline, and other structures. The tsunami wave needs suﬃcient distance to develop and transform. In a 3D basin, orientation is important. One usually works with the scale factor, wavemaker size and capabilities, and basin walls to optimize the “ﬁt.” Again, the distance between the wavemaker and model is an important consideration and reﬂections need to be minimized.

39.3.2. Wavemaking equipment 39.3.2.1. Far-field tsunamis Waves from far-ﬁeld tsunamis are generated in the laboratory with a programmable wavemaker. Flumes have unidirectional wavemakers with one paddle. The 3D basins may have multi-directional wavemakers with a number of independent paddles that can realistically simulate wave directional spreading and the eﬀect of source length and orientation (Fig. 39.3). The wavemaker can be hydraulic or electric, but should have vertical paddle(s) that extend the entire water depth and work in piston mode to accurately simulate the long-wave characteristics of tsunami waves. In ﬂumes the wavemaker is usually ﬁxed to a wall at one end. Wavemakers in basins may be either portable or ﬁxed to a wall. The wavemaker length is determined by the width of the ﬂume or basin. If the wavemaker does not extend the entire width of the basin or the basin is open on the sides, the wavemaker should be suﬃciently long to minimize boundary (i.e., reﬂections) and end (i.e., diﬀracted wave energy) eﬀects. The wavemaker stroke and maximum velocity are important because they determine the height and wavelength of the wave that can be simulated (scale factor again). Of course, larger waves can be simulated as the stroke increases. Wavemakers are computer-controlled to (a) perform digital to analog (D/A) conversion for paddle(s) at run time, (b) monitor paddle displacement and feedback,

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Multi-directional wavemaker with multiple wave boards at CHL. Fig. 39.3.

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(c) update control signals, and (d) collect and evaluate wave heights from active wave absorption systems. Active wave absorption (although important in wind wave studies) is not required for tsunami studies as usually only one wave is created and the experiment is completed by the time the reﬂected wave returns to the wavemaker. 39.3.2.1.1. Solitary wave simulation Tsunami waves are usually simulated in a physical model by solitary waves for convenience because of their permanent form wave characteristics.25,38,39 Although the solitary wave is a single wave, it consists of a complex spectrum of frequencies that allows for elegant analysis and reliable generation in the laboratory. Further, it propagates over constant depth without appreciable changes, so it allows for consistent referencing of its oﬀshore wave height. Synolakis25 and others have used the height-to-depth ratio H = H/d to describe solitary waves. The surface proﬁle η(x, t) = η(x, t)/d for a wave centered at x = X1 and time t = 0 is deﬁned as η(x, 0) = H sech2 γ(x − X1 )

(39.2)

√ where γ = 0.75H . A measure of the wavelength L can be deﬁned in terms of H and water depth d as L=

√ 2d arccosh 20 γ

(39.3)

so that it is equal to the distance between two points in the symmetric surface proﬁle (i.e., one near the front and one near the tail) where the height is only 5% of the height at the crest H. The N -wave is a special type of tsunami wave that has been shown to represent nearshore or local tsunamis better than the solitary wave.40 At generation most tsunamis are either leading-elevation or leading-depression N -waves. When close to shore, they retain their leading-elevation or leading-depression shape. 39.3.2.1.2. Generation procedures Because solitary waves are generated with a single positive stroke, it is usually best to ramp the wave board(s) back from their normal neutral position to be able to utilize the full stroke of the wavemaker. Another useful concept is to imbed the solitary wave control signal in a longer control signal, which includes a long ramp time and wait time before and after the main solitary wave to allow the water motion in the ﬂume or basin to “still” or cease to a minimal level. Finally, the control signal is converted to an analog signal with a digital-to-analog (D/A) rate of typically 20 Hz to drive the wavemaker. A faster D/A rate can be used, but it does not necessarily ensure a more accurate tsunami wave simulation in a largescale ﬂume or basin, and does require more computer storage. This D/A rate is often “hardwired” by the wavemaker and associated software.

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Changes in runup magnitude and in the shape of the runup tongue can be investigated in the 3D basin by varying the number of paddles used in each experiment and the eccentricity of the source. The number of paddles is referred to as the source length because this is the physical parameter most closely represented. The source width is modeled by the wave height, since smaller solitary waves have larger wavelengths for a given depth. When generating other long waves, the source width is represented by varying the stroke. Eccentricity is a parameter used to relate the cross-shore and longshore distances between the tsunami source and the coastal location. 39.3.2.2. Landslide generation A diﬀerent kind of wavemaker is required for simulating landslide-generated waves. Static and dynamically modeled landslide masses are used to create the wave by sliding down a ramp into the water by gravity, much as landslide masses do naturally. The ramp and associated control and monitoring hardware are usually specially designed and fabricated for the particular scale of the experiments (Fig. 39.4). Important parameters for the wave generation are (a) the initial location (i.e., submergence) of the sliding mass, (b) the shape and size of the landslide mass, (c) the initial acceleration (i.e., the slope of the beach surface and the weight of the mass), and (d) frictional properties of the landslide mass and sliding surfaces. It is also desirable to guide the falling mass using the prescribed tracking and minimize the friction of the sliding mass. For subaerial landslides, the speed of the mass entering into the water should be controlled with a mechanical or pneumatic device so that more realistic penetration speed can be realized.

Fig. 39.4.

Landslide generator used at OSU tsunami basin.

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39.3.3. Instrumentation 39.3.3.1. Wave gages Capacitance wave gages are typically used to measure surface wave elevations since they are very reliable. They are more robust and less temperature sensitive than resistance gages. The number of gages should be suﬃcient to adequately determine incident and transformed wave conditions without having to repeat runs. A grid layout can be used to optimize the placement of the gages. Incident gages (i.e., measure incident wave properties) are usually located parallel and normal to the wavemaker at a distance that may vary for each wave case since there is a certain amount of evolution in the solitary wave form as it travels. Prior to each run, the incident wave gages are often moved seaward from the toe of the island to a distance equivalent to half-a-wavelength (i.e., L/2) of the wave to be generated. This procedure ensures that the tsunami wave is always measured at the same relative stage of evolution. Other gages can be located in a cross-shore array to measure wave transformation normal to the wavemaker. These wave gages should be calibrated as necessary to ensure an acceptable level of accuracy. These multiple gage calibrations are necessary to rule out any drift in the gages due to temperature and humidity (especially in some laboratory environments) and error from ﬂoating debris in the water. The length of the measurement rods on the gages is usually determined by the water depth and wave height. The wave rods can be manually or automatically raised three or more heights or steps in the water column to obtain a reading. At CHL automatic Jordan controllers and stepper motors are used to raise and lower the rods through 22 steps: 11 up and 11 down. A linear least-squares averaging technique is used to obtain calibration coeﬃcients. In the event that the calibration coeﬃcients do not meet a minimum threshold for a particular gage(s), the calibration is selectively repeated for them. At CHL, data are sampled for a minimum of 10 samples per wavelength. Typically, a sampling rate of 10 Hz is adequate, especially for longer-period tsunami waves; because even at model scale many more than 10 points are collected for each wave. Higher sampling rates (i.e., 20–60 Hz) may give more deﬁnition of the recorded wave proﬁle, but not necessarily more accurate wave parameters. The main disadvantage is the collection and storage of more data, although this is not much of a problem with modern computer technology. It is a good practice to conduct a short “calibration” phase prior to the “production” phase to make sure the waves were created correctly and are being simulated accurately. It is also a good practice to repeat the individual wave cases, if possible, to ensure that the recorded values are correct and give some conﬁdence in the repeatability of measured values from a statistic standpoint. Analysis for wave data typically consists of plots of the measured surface elevation and calculations of wave height and period. Tsunamis are usually characterized by only a few wave forms, so it is relatively easy to measure the main or largest wave in the proﬁle. These calculations are usually done in the time domain by estimating the average wave height in the record.

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39.3.3.2. Velocity measurements Water velocity can be measured using an LDV or acoustic Doppler velocimeter (ADV). Both can be used seaward of the shoreline, but the LDV can be used to measure velocities in the runup “tongue” as it propagates shoreward when a glasswalled ﬂume is used. 39.3.3.2.1. Laser velocimeter A two-component LDV system measures two orthogonal components of ﬂuid velocity, usually in the plane of the ﬂow. It detects the Doppler shift in the scattered light, and this shift is converted to velocity using only the wavelength of the beam and the geometry of the optics. In a ﬂume application, the LDV system is mounted outside the ﬂume with laser beams focused at a point inside the ﬂume. The water is seeded with small particles of titanium dioxide, which scatter the laser beam as they move through the small sampling volume located at the intersection point of the laser beams. This seeding ensures a strong signal-to-noise ratio (SNR). A backward scatter LDV is required if the glass wall is only on one side of the ﬂume. LDVs are accurate and usually have their own specialized data collection and analysis software. The Doppler frequency range needs to be set to correspond with the velocity range. For example, the frequency range for a large-scale experiment with an Lr = 50 in the CHL ﬂumes was 33–333 kHz, corresponding to an eﬀective velocity range of 0–1.5 m/s, with an accuracy of ±1.5 cm/s. LDV velocity time series are typically collected at a sampling rate of 40 Hz. Sampling start and end times should be suﬃcient to capture the entire wave and any reﬂections from the beach that may be of interest. 39.3.3.2.2. Acoustic velocimeter One or more ADVs can be used to measure tsunami wave velocity components seaward of the shoreline. They are capable of 2D and 3D velocity point measurements. They operate on the Doppler principle just as the LDV system does, but utilize acoustic waves instead of light waves. An acoustic pulse is transmitted by the transducer, reﬂected from particles in the water, and sensed by the receivers. The speed of the water is determined by calculating the shift in the signal frequency. Seeding material is added to the water before and during data collection to ensure a strong SNR. The middle of the sampling volume is typically 5 cm from the receivers. Typical velocity resolution is 0.1 mm/s with an accuracy of 1% of measured velocity. They are usually mounted from above in a ﬂume and on tripods in basins. The main concern is to minimize the disturbance to the sampling volume due to the mounting mechanism. 39.3.3.3. Runup measurements 39.3.3.3.1. Manual methods The measurement of runup is labor-intensive if done manually. It is best to set up a “painted” grid on the bottom. In the ﬂume, the maximum runup distance can

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be calculated by locating the maximum excursion of the runup tongue by visual observation or photographic/video records. The water tends to “wet” the bottom “dry” beach so that it is relatively easy to accurately mark the maximum extent reached by the water even after the wave has returned and the experiment is over. The same procedure is used in the basin experiments, except that there are usually many more grid locations to measure. This is where the experiment becomes laborintensive as it is nearly impossible for one person to mark the maximum excursion on multiple grid lines before the “wetness” of the bottom is lost. These diagonal distances along the slope of the beach from the still water level are converted to maximum vertical runup using the bathymetric survey data and the geometry of the model beach. An alternative manual method is to use a rod and transit to measure maximum vertical runup directly at each grid line. This method is more accurate since the calculation of vertical distance is not based on target slopes and bathymetry. Finally, Raichlen and Synolakis21 used a video camera to capture the entire runup time history. 39.3.3.3.2. Runup gages Raichlen and Hammack, Jr.41 conducted some pioneering work in the development of runup gages. Synolakis42 used an array of surface-piercing, capacitance gages mounted in an aluminum frame. Both the distance between gages and the frame could be moved. Yeh et al.29 used a digital runup gage embedded in a model beach to study runup velocities of a bore propagating up the beach. His 35-cm-long gage consisted of eight rods spaced 5 cm apart with tips that projected no more than 1 mm above the beach surface. Briggs et al.43 constructed a runup gage with some new features to measure runup on a circular island. Rather than a continuous wire or rod placed along the bottom, this runup gage consisted of a series of 32 discrete surface-piercing vertical rods that were turned on or oﬀ by water contact (Fig. 39.5). The advantage of this design is that runup can be measured in the laboratory for uneven bottom conditions such as rubble mound breakwaters. Gage resolution was limited by the 1-cm minimum spacing between rods. Software was used to convert the wetted rod number to the appropriate vertical runup or rundown. The gage was positioned along a transect or grid line so that the still water level was approximately midway among the rods to enable measurement of runup and rundown. These automated

Fig. 39.5.

Runup gage for automated measurement of runup.44

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runup gages showed great promise as a tool for reducing the manual measurement of tsunami runup, a considerable time and cost savings.

39.3.4. Support equipment 39.3.4.1. Computer support One or more computers are usually located in a control room to provide computer support for the experiments. These computers provide (a) calibration, command, and feedback for the wavemaker; (b) calibration and synoptic data collection for the instrumentation; (c) control signal generation; and (d) data analysis. All datacollection computers should be time-synchronized and able to communicate with each other (i.e., ﬁberoptic network). 39.3.4.2. Water level controller Flumes and (especially) basins have a tendency to leak. A water level controller is important to maintain the water level within a certain tolerance. A point gage and automatic water level ﬂoat with solenoid valve are conﬁgured to turn on a small water supply to keep up with any water losses during the experiments. A tolerance on the water level of ±0.6 mm is suﬃcient to ensure accurate wave measurements. The controller mechanism should be located away from the measurement area to prevent any “ripples” from the ﬂow of water during or immediately preceding a run. Often, they are turned oﬀ during a run to avoid contamination from induced electric currents.

39.4. Physical Model Studies at CHL The CHL has been conducting tsunami research since the 1960s, with more than 27 publications documenting physical, numeric, and analytic modeling of far-ﬁeld tsunami hazards in the Paciﬁc Ocean. Two early physical modeling studies involved the work of Oswalt and Boyd45 and Senter.46 Oswalt and Boyd45 conducted a laboratory study of Hilo Harbor, HI, to evaluate steady ﬂow stability for a tsunami barrier. Senter46 studied the eﬀect of tsunami waves on a proposed Crescent City, CA, harbor design. In the 1970s and the 1980s, most of the tsunami research involved pioneering numeric models for tsunami ﬂood predictions in the Paciﬁc Ocean. The work of Houston et al.47 and Houston48 is still referenced today as a deterministic method for predicting ﬂooding due to tsunami inundation,1 although many numeric models provide both deterministic and risk-based predictions. In the early 1990s, a joint tsunami study was funded by the NSF to identify important physical parameters involved in 3D tsunami runup. Over the course of this study, four physical models of a vertical wall (ﬂume), plane beach (ﬂume and basin), and a circular island (basin) were investigated.7 The following sections summarize the important issues for these experiments.

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39.4.1. Vertical walls One of the ﬁrst experiments was a ﬂume study of tsunami runup on a vertical wall to study the eﬀect of complex bathymetry on this highly nonlinear phenomenon. This study was used as “Benchmark Problem 3” in the International Workshop on Long Wave Models36 to validate numeric models. Kanoglu and Synolakis49 actively participated and validated their analytic method using this data. Bathymetric contours from Revere Beach, MA, were molded in the ﬂume. This “model of opportunity” was selected to take advantage of a realistic complex bathymetry for a much reduced construction cost. The compound-slope, ﬁxed-bed bathymetry consisted of three diﬀerent slopes (rise:run = 1 : 53, 1:150, and 1:13) and a ﬂat section in the deep end (Fig. 39.6). The vertical wall was located at the landward end of the 1:13 slope. The water depth in the ﬂat section of the ﬂume measured 21.8 cm. Ten capacitance wave gages were used to measure surface wave elevations along the centerline of the ﬂume. The ﬁrst four gages were located in the constant depth region to measure incident wave conditions. The ﬁrst three gages remained ﬁxed while gage #4 was moved to measure incident wave conditions according to the procedure described earlier. Three normalized target wave heights H = H/d = 0.05, 0.30, and 0.70 were simulated for Cases A, B, and C, respectively. Note that the oﬀshore water depth d = 21.8 cm. Figure 39.7 shows the paddle trajectories for the three cases. When the waves reached the vertical wall, a plume of water would shoot upward (Fig. 39.8). The highest point of this excursion was visually noted through the glass walls of the ﬂume and manually recorded after each run. Wave breaking occurred for Cases B and C only. For Case B the wave broke at or near the wall. For Case C the wave broke between gages 7 and 8 (i.e., in front of the toe between the 1:13 and 1:150 slopes) before re-forming and shoaling to the vertical wall. The largest runup at each depth was recorded for Case B that experienced wave breaking only at or near the wall. Table 39.1 lists the maximum vertical runup recorded on the vertical wall for each case. Included are target and measured normalized wave heights H , runup heights R, and normalized runup R = R/d. 39.4.2. Plane beaches Tsunami wave evolution, uniformity, runup, and wave kinematics over a plane beach were studied in the next series of experiments. Both ﬁxed-bed, ﬂume, and basin

Fig. 39.6.

Test setup for measurement of runup on a vertical wall.36

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Fig. 39.7.

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Wavemaker control signals for three Cases A, B, and C.36

models were constructed and investigated to provide comparisons between small (ﬂume) and large-scale (basin) results. The ﬂume also provided data on velocities in the runup plume. Additional details can be found in Briggs et al.50,51 39.4.2.1. Flume setup A schematic of the 42.4-m-long, glass-walled ﬂume is shown in Fig. 39.9. This same ﬂume was used in the previous experiments, but with the vertical wall removed and diﬀerent bottom contours installed. The ﬂat area in front of the toe of the 1 vertical on 30 horizontal beach was located 21 m from the wavemaker. Water depth in the constant depth region was 32 cm. Tsunami waves were simulated as solitary waves using a vertical hydraulic wavemaker in piston mode. The 10 wave conditions ranged from nondimensional wave heights H = 0.01 to 0.50. Ten capacitance wave gages were used to measure surface wave elevations along the length of the ﬂume. The ﬁrst gage was located 15 m from the wavemaker and was moved prior to each run to measure incident wave conditions. Gages 2–10 formed a cross-shore transect in the center of the ﬂume with a spacing of 1 m, except for gage 2, which was positioned 60 cm from the toe of the slope. Gage 10 was located in 3.3-cm water depth.

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Fig. 39.8.

Plume of water shooting up against vertical wall at end of ﬂume. Table 39.1.

Maximum vertical runup on a vertical wall. Normalized height

Runup

Case

Target

Measured

R (cm)

R

A B C

0.05 0.30 0.70

0.039 0.264 0.696

2.74 45.72 27.43

0.13 2.10 1.26

A two-component LDV was used to measure two orthogonal components (i.e., horizontal and vertical) of ﬂuid velocity in the plane of the ﬂow. The LDV was mounted outside the ﬂume with four laser beams focused at a point approximately 9 cm from the inside face of the glass ﬂume wall. Time series of velocity were collected at a 40-Hz sampling rate, starting about 40 s before generation of the tsunami wave and continuing for 102 s. 39.4.2.2. Basin setup A complementary large-scale experiment was conducted in a 30-m-wide by 25-mlong wave basin (Fig. 39.10). It included a ﬂat section and a 1:30 sloping beach with plane parallel contours. The oﬀshore water depth in the undisturbed, constant

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Fig. 39.9.

Test setup in ﬂume for study of runup on a plane beach.43

Fig. 39.10.

Test setup in basin for study of runup on a plane beach.43

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depth region of the model was 32 cm, which is the same as the ﬂume experiments described earlier. The toe of the slope was located 12.4 m in front of the wavemaker. The CHL’s directional spectral wave generator (DSWG) was used to generate solitary waves. The electronically controlled DSWG was 27.4 m long and consisted of 60 paddles, 46 cm wide and 76 cm high. Briggs and Hampton52 provide additional details of the DSWG. The rear of the basin (behind the DSWG) was lined with wave-absorbing material backed by a concrete wall. The basin sides were lined with absorber material and the right side (looking in the positive x-direction) was open to the adjacent basin. The global coordinate system was aligned with the X-axis (X) perpendicular and the Y -axis (Y ) parallel to the DSWG. The origin was located at the mid-point of the DSWG, in line with the front surface of all paddles at their rest positions. A 12-m by 20-m grid was established between X = 12–24 m and Y = −10 to 10 m. Spacing between grid points in both x and y directions was 1 m, except for 50 cm spacing in the x-direction for grid points between X = 20 and 24 m. Also, grid points with 1-m spacing were laid out along the centerline (Y = 0) between the face of the DSWG and the toe of the slope. The smaller spacing was to allow more accurate measurements of runup on the beach. Thirty capacitance wave gages were used to measure surface wave elevations within this grid. The ﬁrst three gages were located at X = 3, 6, and 9 m along the centerline in the constant depth region to measure incident wave conditions. The same procedure of moving one of them prior to each run was followed to ensure they always recorded the wave at the same relative stage of evolution. The other two gages remained in their ﬁxed positions. Twenty-seven gages were positioned in three cross-shore transects in an 8-m by 6-m measurement area between the toe of the slope and the still water level (SWL) to measure wave evolution. Spacing between transects in the longshore direction was 3 m, with the center transect over the centerline and the other two on either side of center. Spacing between the nine gages in each transect was 1 m. Eight target (nondimensional) wave heights from H = 0.01 to 0.20 were simulated. Measured wave heights were approximately 70–85% of these target values. The decrease in measured wave height from the target was due to losses in the mechanical generation of the solitary waves because of gaps between the ﬂoor and the wavemaker. The reduction in measured wave height is not critical to the study as even in the most carefully controlled experiments, the wave height is usually less than target values. What is important is that the solitary wave proﬁle is not adversely aﬀected. All waves were nonbreaking until ﬁnal stages of transformation near the shoreline (where gentle spilling occurred), except for H = 0.20 waves, which broke nearshore. Changes in runup magnitude and in the shape of the runup tongue were investigated for selected cases in the basin by varying the number of paddles used in each experiment and the eccentricity of the source. The maximum vertical runup along the sloping beach was measured at each gridline above the SWL. These horizontal measurements parallel to the slope of the beach were converted to maximum vertical runup using the bathymetric survey data or a surveyor’s rod and transit.

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39.4.2.3. Results 39.4.2.3.1. Comparison of results Normalized maximum vertical runup R was compared between the ﬂume and the basin results as a function of normalized wave height H . Synolakis25 had postulated a runup law given by R = 2.831H 1.25

cot β

(39.4)

where β is the beach slope. His data represented nonbreaking waves between 0.005 ≤ H ≤ 0.04, whereas the ﬂume and basin data included this range and breaking waves from 0.04 ≤ H ≤ 0.4. Breaking occurred near the shoreline in both the ﬂume and the basin for a measured H > 0.04. The runup law slightly overpredicted wave runup, since it is based on previous laboratory results, which did not fully cover the range of these experiments. Agreement between the ﬂume and basin tests was not as good as expected, especially for the nonbreaking cases. Possible explanations included the fact that the source lengths and cross-shore distances between the wavemaker and the toe of the slope were diﬀerent between ﬂume and basin, and the ﬂume was more reﬂective than the basin. 39.4.2.3.2. Tsunami kinematics From the ﬂume results, normalized maximum vertical runup R was plotted versus a 2 /2gd using the peak runup velocity normalized energy-based parameter V = vpeak vpeak . The vpeak was measured parallel to the 1:30 bottom slope and slightly seaward of the intersection of the slope and the SWL for each of the tsunami wave cases. The LDV beam crossing point was positioned 1.1 cm above the bed, as close to the bed as the sampling volume could be located. Earlier measurements of horizontal runup velocity at the SWL, mid-depth, and 5 cm above the SWL showed that the variation in horizontal velocity between the crest region and mid-depth was not great, with a peak velocity at 5 cm above the SWL being only 15% greater than the peak velocity at mid-depth. A linear least-square ﬁt to the initial three values at H = 0.01, 0.04, and 0.36 gave an equation of the form R = 0.0013 + 0.7755V

(39.5)

with a correlation coeﬃcient r 2 = 0.99. Later, all 10 values of H were included and a linear relationship was still produced. A nonlinear relationship could also have been produced, but the high r2 indicated that the simplest relationship would work just as well. Of course, this equation should predict zero runup for zero velocity, but the small intercept is due to the curve ﬁtting. Friction losses amounting up to 20% of the total energy during runup were suggested as the mechanism accounting for the diﬀerences in the measured and predicted runup.

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39.4.2.3.3. 3D eﬀects Several 3D eﬀects on runup were investigated in the basin including source length, eccentricity, and amplitude evolution. The source length was varied from 1/4 to 4 DSWG modules, all centered on the DSWG centerline. Runup heights measured in the longshore direction along the beach showed very good uniformity for diﬀerent source lengths. The eﬀect of eccentricity of the source on runup was studied by varying the oﬀset of the source from the measurement points on the beach. The source length was varied from 1/4 to 1 module and oﬀset to the right of the center of the beach. Runup values were largest directly opposite the center of the source and decreased linearly with longshore distance due to diﬀraction. Runup showed a strong linear trend with source length, increasing as the source length increased. The ﬁnal results illustrated the evolution of maximum amplitude with crossshore distance in the basin. Test results showed that dimensionless wave height ηmax = ηmax /d increases as source length S increases and water depth h decreases, in agreement with earlier ﬁndings of Synolakis39 relative to Green’s law. This may be the ﬁrst instance where it has been proven that this evolution law is valid for 3D waves.43

39.4.3. Circular islands The Flores Island, Indonesia tsunami struck Babi Island in 1992 destroying two ﬁshing villages and killing 263 of its 1,093 population.53,54 The island has a conical shape, with a 2-km diameter and a summit elevation of 351 m. The island’s surrounding waters are deep with a steep sea-bottom slope. These villages were located in sheltered areas (coral reefs) on the lee (south) side of the island from the origin of the tsunami. So, why were the villages completely destroyed? This last set of experiments43,44,55,56 was conducted to better understand the complex physics involved in why the tsunami wave produced such unexpected destruction. These data were used as “Benchmark Problem 2” in the International Workshop on Long Wave Models.36 39.4.3.1. Setup The model island was constructed in the center of the same basin as previously used for the plane beach experiments (Fig. 39.11). The island had the shape of a truncated, right circular cone with diameters of 7.2 m at the toe and 2.2 m at the crest. The vertical height of the island was approximately 62.5 cm, with a 1 vertical on 4 horizontal beach slope. The water depth was set at 32 cm in the basin. The surface of the island and basin were constructed with smooth concrete. The basin was lined with wave absorbers to minimize unwanted reﬂections. Twenty-seven capacitance wave gages were used to measure surface wave elevations (Fig. 39.12). The ﬁrst four gages were located parallel to the wavemaker to measure incident wave conditions. The same procedure for locating the incident gages as described for previous experiments was followed. A measurement grid of

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Fig. 39.11.

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Circular island in DSWG basin with trapped edge wave wrapping around lee side.54

Fig. 39.12.

Test setup for circular island experiment.36

six concentric circles covered the island to a distance 2.5 m beyond the toe. Measurement points were located at the intersection of these concentric circles and the 90-degree radial lines. The spacing between grid points was a function of the water depth. 39.4.3.2. Wave conditions The full length of the DSWG was used to generate three solitary wave cases. Target normalized wave heights of H = 0.05, 0.10, and 0.20 were simulated for Cases A, B, and C, respectively. Measured wave heights were 90% of the target values due to losses in the generation of the waves. All waves were nonbreaking until ﬁnal stages

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of transformation near the shoreline (where gentle spilling occurred) except for the Case C wave, which broke nearshore. 39.4.3.3. Results Maximum vertical runup R was measured at 20 locations around the perimeter of the island. Sixteen locations were evenly spaced every 22.5 degrees around the perimeter. Four radial transects with uneven spacing were located on the backside of the island to improve the resolution. Changes in runup shape and magnitude were investigated by varying the water depth, wave height, source length (number of modules), and eccentricity of the source. Figure 39.13 is a polar plot of maximum vertical runup around the island for Case C. Waves approach the island from the bottom or 270 degrees. The island crest, waterline, and toe are shown for reference. Two runs are overlain, demonstrating excellent repeatability. When the tsunami wave hit the front side of the island, the wave was split into two components. The ﬁrst component wave propagated along one side of the island, whereas the second propagated along the other side. Refraction and diﬀraction caused the waves to bend around the island as edge waves with their maximum wave amplitude along the shoreline when the crest was perpendicular to the shoreline. This wave trapping only occurs when the tsunami wavelength is comparable to the

Fig. 39.13.

Polar plot of maximum vertical runup around the island for Case C.51

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island diameter. Shorter waves are only refracted and dissipated by breaking on the beach. Longer waves pass around the island without trapping. When these two waves collided on the lee side of the island (i.e., the shadow zone), substantial magniﬁcation occurred near the shore, resulting in signiﬁcant ﬂow runup onto the beach. The runup was slightly larger on the front of the island, but the ﬂow velocities were larger on the lee side due to this focusing eﬀect. The physical model and numeric simulations veriﬁed these mechanisms. This phenomenon is very interesting since most people would feel “safe” on the backside of an island, which is not normally exposed to wind waves.

39.5. NEES Tsunami Basin at Oregon State University (OSU) 39.5.1. Background In 2000, the NSF established the George E. Brown Jr. Network for Earthquake Engineering Simulation (NEES) program to (a) improve understanding of the eﬀects of earthquakes and tsunamis on civil infrastructure, (b) reduce the loss of life in the event of these natural disasters, and (c) mitigate damage. The NEES is a new collaboratory consisting of 15 experimental facilities, numeric software, and data repository. The aim of NEES is that of a “shared-use” collaboratory, meaning that NSF invests in the capital development and operating costs of relative few sites, and these sites are then accessible to researchers nationwide on a competitive basis. Of the 15 equipment sites, only one tsunami site was constructed, located at OSU’s O.H. Hinsdale Wave Research Laboratory. The tsunami wave basin (TWB) was designed following an NSF-sponsored Workshop for Tsunami Research Facilities held in Baltimore in 1998. In the workshop report, the authors indicated that “numeric models [for tsunami inundation] generally must be conﬁrmed (or refuted) using the results of 2D and 3D laboratory experiments where the characteristics of the generated wave can be accurately prescribed.” This conﬁrmation and validation includes the benchmarking type of experiments described earlier. The authors also noted “one important area of research is that which will lead to the design of buildings and building construction techniques to reduce the potential for tsunami damage to structures.” The participants identiﬁed these critical research areas: • • • • •

breaking wave processes, wave forces on structures, tsunami runup and overland ﬂow, sediment deposit and scouring, debris ﬂow and debris impact.

The report recommended both 2D and 3D facilities: a 2D ﬂume to minimize scale eﬀects and a 3D basin to allow a greater range of topics to be studied. To increase the use of the facility and funding levels, the report recommended a multi-purpose facility to allow coastal engineering studies, explore interdisciplinary research with earthquake engineering, and encourage international collaboration.

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Short-, medium-, and long-term goals for tsunami research.

Term Short

Medium Long

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Description Runup for breaking and nonbreaking long waves Resonant excitation of harbors and embayments Bathymetric focusing and defocusing of nonlinear waves Propagation of aerial and submarine landslides Transient sediment transport for long waves Impact forces on structures owing to objects Tsunamis eﬀects on individual and groups of buildings Collaboration with NTHMP and NOAA Veriﬁcation and validation of numeric models for tsunami inundation Collaboration with geotechnical community to study landslide tsunamis Development of comprehensive, interactive scenario simulations that integrate the physical aspects — generation, propagation, runup, and structure interactions — with societal issues, such as transmission of warnings to the public, evacuation, environmental impacts, rescue tactics, short- and long-term recovery strategies

Given limited funding, an addendum to the report indicated that “it is more important to the research needs of numerical model validation that an adequate three-dimensional facility be developed.” In 2003, the National Research Council released a report “Preventing Earthquake Disasters: The Grand Challenge in Earthquake Engineering.” Table 39.2 lists the short-, medium-, and long-term goals for tsunami research. 39.5.2. OSU facilities 39.5.2.1. Tsunami wave basin (TWB) The TWB was completed in 2003, designed as a one-of-a-kind, shared-use facility to understand the fundamental nature of tsunami inundation (Fig. 39.14). The TWB is 48.8-m long by 26.5-m wide by 2.1-m deep with a maximum water depth of 1.5 m. The wavemaker is of piston type with 29 segmented paddles and 30 beltdriven actuators. The period range is 0.5–10 s with a maximum stroke of 2.1 m and velocity of 2.0 m/s. The wavemaker is capable of generating normally incident, unidirectional, and multi-directional regular and irregular waves as well as solitary or tsunami-like waves. Supporting infrastructure includes (a) a 7.5-ton bridge crane, (b) instrumentation carriage spanning the 26.5 m width, (c) unistrut installed in the basin ﬂoor and side walls to secure models, and (d) two 4.2-m access ramps for large truck and heavy equipment access. A recent instrumentation upgrade provides a comprehensive suite of sensors to measure free surface, velocity, pressure, stress, turbidity, and depth. Data can be made available in near-real time via the Web, and six cameras are Web-enabled for remote observations. 39.5.2.2. Large wave flume The large wave ﬂume (LWF) measures 104-m long by 3.7-m wide by 4.6-m deep (Fig. 39.15) and is the largest of its kind in North America. The wavemaker is of

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Fig. 39.14.

TWB at OSU.

Fig. 39.15.

LWF at OSU.

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hinged type with programmable hydraulic actuator capable of generating regular, irregular, and user-deﬁned waves. The wavemaker is operated with periods ranging from 0.5 to 10 s and with a maximum wave height of 1.6 m at 4 s. In 2007, the facility received a $1.1M MRI award from NSF to install a large-stroke, piston-type wavemaker to dramatically increase its performance for hurricane waves, coastal storms, and tsunamis. The updated system components will consist of a new hydraulic actuator assembly, piston-type waveboard assembly, electronic control and software, and hydraulic distribution subsystems. To achieve the desired performance (4 m stroke at a speed of 4 m/s) with minimal foot print, a custom longstroke, hydraulically-controlled actuator assembly will be engineered for this application. The actuator assembly will consist of two, single-ended actuators mounted to oppose each other and a static support actuator assembly, bundled in a compact form factor. Two servo-valve assemblies, one controlling 250 gpm and a second controlling 400 gpm will be used for the regular (hurricane) and tsunami wave generation, respectively. 39.5.3. Projects Since the start of the NEES Tsunami Research Program, the facility has conducted ﬁve major research projects as of publication of this handbook: (1) Runup and rundown generated by 3D sliding masses by Profs. P. Liu, T.-R. Wu, F. Raichlen, C. Synolakis, and J. Borrero (Cornell University, Caltech, and University of Southern California), 2005. (2) Tsunami-structure interactions by Prof. P. Liu (Cornell University). (3) 3D tsunami evolution with landslides by Prof. H. Fritz (Georgia Institute of Technology, Savannah). (4) Performance-based tsunami engineering by Prof. R. Riggs (University of Hawaii). (5) Multi-scale tsunami model by Profs. P. Lynett and P. Liu (Texas A&M University and Cornell University). Most of these projects are new and work is ongoing. In most cases, the measurements are still being analyzed. The results will be archived and documented for each project for future tsunami wave studies. Two example summaries of landslide studies are presented to illustrate the level of research in the NEES projects and to highlight the diﬀerences between landslide and distant tsunamis. 39.5.3.1. Runup and rundown generated by 3D sliding masses Liu et al.57 conducted landslide experiments in the NEES LWF using a freely sliding wedge with two orientations and a hemispherical solid to simulate the landslide mass. Their initial positions ranged from totally aerial to fully submerged, with variable slide masses. The slide mass was instrumented to provide position and velocity time series. The time histories of water surface and runup were measured at a number of locations. Figure 39.16 shows the wedge shape sliding down the 1:2 slope into the TWB.

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Fig. 39.16. p. 1100).

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Model landslide mass sliding (right to left) into NEES TWB (from Project (1) on

The runup and rundown generated by the sliding mass was used to validate a new numeric model that is based on the large-eddy simulation (LES) approach. The Smagorinsky subgrid scale-model was used to provide turbulence dissipation and the volume of ﬂuid (VOF) method to track the free surface and shoreline movements. A numeric algorithm for describing the motion of the sliding mass was also implemented. Very good agreement was shown between the numeric predictions and laboratory measurements for the time histories of runup and generated waves from the wedge. Details of the complex 3D ﬂow patterns and free surface and shoreline deformations were illustrated by the numeric models. The maximum runup heights were discussed as a function of the initial slide elevation and speciﬁc weight. Finally, eﬀects of the TWB on the maximum runup was discussed. 39.5.3.2. 3D tsunami evolution with landslides The coupling between landslide motion and 3D tsunami waves is of critical importance given the local, strongly directional source mechanism and the characteristic transcritical landslide versus tsunami velocity Froude numbers. Nearly a half-century after the Lituya Bay, Alaska mega-event, Fritz58 has successfully replicated a fully 3D scale model of a tsunami created by a deformable landslide. Using a unique landslide-generated tsunami simulator that was installed at the NEES

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TWB at OSU (see Fig. 39.4), the researchers simulated the impact of landslides that occur both above and below the water’s surface. Fritz’s team constructed the landslide tsunami generator as “an open box” that is mounted on a steel slide and ﬁlled with up to 1,350 kg of gravel. The box accelerates down the slide by means of four pneumatic pistons. The granular mass is accelerated inside the box and released while the sled is slowed down pneumatically. The box is 2.1 m by 1.2 m by 0.3 m with subdivisions to adjust initial slide length and thickness, and is placed on a slide that can vary in length. The box itself is able to travel approximately 2 m before the gravel is released down the 2H:1V slope at initial velocities up to 5 m/s. The researchers have tested two types of landslides: those that fall into the water and those that occur beneath the surface. Sensors were placed on the simulator to measure the velocity of the gravel. Using cameras placed above and within the water, the researchers measured the shape, length, and thickness of the gravel masses while they were in motion. Wave gages were placed to measure the size and shape of the waves that were generated, including the lateral onshore runup. The recorded wave proﬁles were extremely directional, unsteady, nonlinear, and located mostly in the intermediate water depth wave regime. Among the principal diﬀerences between a tectonic-generated tsunami and a landslide-generated tsunami is that the latter has a strong directional component that can be devastating to the immediate area. Because it has a shorter wavelength, however, it dissipates quickly over a short distance. Landslide tsunamis exhibit a more dispersive and strongly directional propagation than tectonic tsunamis. Planar PIV was applied to the tsunami surface and revealed the fully 3D tsunami generation. Currently, more than 60 successful runs have been completed and the main tsunamigenic parameters identiﬁed that will serve as key benchmarks for numeric models. 39.6. Summary and Conclusions This chapter gave a brief overview of physical modeling of tsunami waves. Background information on historic laboratory studies was presented. Equipment and procedures used in physical models for generating, measuring, and analyzing tsunami waves were described. Physical model studies conducted at CHL in the early 1990s to better understand runup on vertical walls, plane beaches, and circular islands were also described. The NEES state-of-the-art tsunami facility at OSU was described in the last section. Since its inception, the NEES has hosted ﬁve major research projects. In general, the data and conclusions from these studies are still being processed and will be available to the tsunami research community in future years. Although numeric models have come a long way, the future of physical models for tsunami wave studies seems assured as one will always want to see the many complicated processes that occur as the waves interact with coastlines and structures. Acknowledgments The authors wish to acknowledge Headquarters, US Army Corps of Engineers, and the Oregon State University, and the National Science Foundation (0527520)

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for authorizing publication of this chapter. Special thanks to Profs. Fred Raichlen (Caltech Emeritus), Costas Synolakis (University of Southern California), Phil Liu (Cornell University), and Hermann Fritz (Georgia Tech, Savannah), Dr. Steven Hughes and Alex Sanchez (CHL), for assistance in the review of this chapter. References 1. C. E. Synolakis and E. N. Bernard, Tsunami science before and beyond boxing day 2004, Proc. Roy. Soc. Lond. A 364(1845), 2231–2265 (2006). 2. C. E. Synolakis, A. C. Yalciner, J. C. Borrero and G. Plafker, Modeling of the November 3, 1994 Skagway, Alaska tsunami, Solutions to Coastal Disasters, eds. L. Wallendorf and L. Ewing, ASCE (2002), pp. 915–927. 3. P. L.-F. Liu, P. Lynett, H. Fernando, B. E. Jaﬀe, H. Fritz, B. Higman, R. Morton, J. Goﬀ and C. Synolakis, Observations by the international tsunami survey team in Sri Lanka, Science 308, 1595 (2005). 4. C. E. Synolakis, Tsunami and seiche, Earthquake Engineering Handbook, eds. W.-F. Chen and C. Scawthorn (CRC Press, 2003), pp. 9-1–9-90. 5. International Tsunami Information Center (ITIC), IOC Tsunami Glossary, 3. Surveys and Measurements (2007). 6. J. C. Borrero, Field data and satellite imagery of tsunami eﬀects in Banda Aceh, Science 308(5728), 1596 (2005). 7. M. J. Briggs, J. Borrero and C. Synolakis. Tsunami disaster mitigation research in the United States, Int. Symp. Tsunami Disas. Mitigat. Fut., Kobe, Japan, January 2005, pp. 17–18. 8. C. E. Synolakis and E. A. Okal, 1992–2002: Perspective on a decade of post-tsunami surveys, Tsunamis: Case Studies and Recent Developments, ed. K. Satake, Adv. Natur. Technol. Hazards 23, 1–30 (2005). 9. M. J. Briggs, Joint Tsunami Runup Study, The CERCular, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, CERC-92-3 (1992), pp. 7–8. 10. P. L.-F. Liu, C. E. Synolakis and H. H. Yeh, Report on the international workshop on long-wave runup, J. Fluid Mech. 229, 675–688 (1991). 11. J. S. Russell, Report on Waves, Report of the 14th Meeting of the British Association for the Advancement of Science, held in York in September 1844, 311–391 plus 11 plates (1844). 12. J. L. Hammack, A note on tsunamis: Their generation and propagation in an ocean of uniform depth, J. Fluid Mech. 60, 769–799 (1973). 13. M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia (1981). 14. J. L. Hammack and H. Segur, The Korteweg–de Vries equation and water waves. Part 2. Comparison with experiments, J. Fluid Mech. 65, 289–314 (1974). 15. S. Tadepalli and C. E. Synolakis, Model for the leading waves of tsunamis, Phys. Rev. Lett. 77(10), 2141 (1996). 16. R. Takahashi and T. Hatori, A model experiment on the tsunami generation from a bottom deformation area of elliptic shape, Bull. Earthq. Res. Inst. 40, 873–883 (1962). 17. R. L. Wiegel, Laboratory studies of gravity waves generated by the movement of a submerged body, Trans. Am. Geophys. Union 36, 759–774 (1955). 18. J. M. Johnson and K. Satake, Estimation of seismic moment and slip distribution of the April 1, 1946, Aleutian tsunami earthquake, J. Geophys. Res. 102, 11765–11774 (1997).

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19. E. A. Okal, G. Plafker, C. S. Synolakis and J. C. Borrero, Near-ﬁeld survey of the 1946 Aleutial tsunami on Unimak and Sanak Islands, Bull. Seismolog. Soc. Am. 93(3), 1226–1234 (2003). 20. P. Watts, Water waves generated by underwater landslides, PhD. thesis, California Inst. of Tech., Pasadena, CA (1997). 21. F. Raichlen and C. E. Synolakis, Waves and runup generated by a three dimensional sliding mass, Submarine Mass Movements and Their Consequences, eds. J. Locat and J. Mienert (Kluwer Academic Publishers, 2003), pp. 113–119. 22. H. M. Fritz, W. H. Hager and H.-E. Minor, Landslide generated impulse waves, Exp. Fluids 35, 505–532 (2003). 23. R. L. Street and F. E. Camﬁeld, Observations and experiments of solitary wave deformation, Proc. 10th Conf. Coast. Eng., Tokyo, Japan (1966), pp. 284–301. 24. F. E. Camﬁeld and R. L. Street, Shoaling of solitary waves on small slopes, J. Waterw. Harbor. Div. Proc. 95(WW1), 1–22 (1969). 25. C. E. Synolakis, Runup of solitary waves, J. Fluid Mech. 185, 523–545 (1987). 26. D. Goring and F. Raichlen, The generation of long waves in the laboratory, Proc. 7th Coast. Eng. Conf., ASCE (1980), pp. 763–784. 27. C. E. Synolakis, Determining hydrodynamic force on accelerating plate in ﬂuid with free surface, J. Eng. Mech. 115(11), 2480–2492 (1989). 28. R. L. Miller, Experimental determination of run-up of undular and fully developed bores, J. Geophys. Res. 73, 4497–4510 (1968). 29. H. Yeh, A. Ghazali and I. Marton, Experimental study of bore runup, J. Fluid Mech. 206, 563–578 (1989). 30. R. H. Cross, Tsunami surge forces, J. Water. Harbor. Div. 93(4), 201–231 (1967). 31. H. Arnason, Interactions between an incident bore and a free-standing coastal structure, PhD. dissertation, University of Washington (2005). 32. H. Arnason, H. Yeh and C. Petroﬀ, Interactions between an incident bore and a vertical column (2007), http://oregonstate.edu/∼yehh/tsunamiforces/. 33. A. T. Ippen and F. Raichlen, Wave induced oscillations in harbors: The problem of coupling of highly reﬂective basins, Report No. 69, Hydrodynamics Laboratory, Massachusetts Institute of Technology (1962). 34. A. T. Ippen and Y. Goda, Wave induced oscillations in harbors: The solution for a rectangular harbor connected to the open sea, Report No. 59, Hydrodynamics Lab., MIT, July 1963. 35. J. J. Lee, Wave-induced oscillations in harbors of arbitrary geometry, J. Fluid Mech. 45, 375–394 (1971). 36. H. Yeh, P. L.-F. Liu and C. E. Synolakis, Long wave runup models, Second International Workshop on Long Wave Runup Models, Friday Harbor, San Juan Island, WA (World Scientiﬁc Publishing, 1996), pp. 1–403. 37. L.-F. P. Liu, H. Yeh and C. E. Synolakis, Advanced Numerical Models for Simulating Tsunami Waves and Runup, Advances in Coastal and Ocean Engineering, Vol. 10 (World Scientiﬁc Publishing Co., Singapore, 2008), 344 pp. 38. C. E. Synolakis, Are solitary waves the limiting waves in the long wave runup? 21st Conf. Coast. Eng., Costa del Sol, Malaga, Spain (1988), pp. 219–233. 39. C. E. Synolakis, Tsunami runup on steep slopes: How good linear theory really is, Nat. Haz. 4, 221–234 (1991). 40. S. Tadepalli and C. E. Synolakis, The runup of N-waves on sloping beaches, Proc. Roy. J. Soc. Lond. 246, 1–14 (1994). 41. F. Raichlen and J. L. Hammack, Jr, Run-up due to breaking and non-breaking waves, Proc. 14th Int. Conf. Coast. Eng., Copenhagen, Denmark, June 1974, pp. 1937–1955.

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42. C. E. Synolakis, The runup of long waves, PhD. thesis, California Institute of Technology, Pasadena, CA (1986). 43. M. J. Briggs, C. E. Synolakis, G. S. Harkins and S. A. Hughes, Large scale threedimensional laboratory measurements of tsunami inundation, tsunami: Progress, Prediction, Disaster Prevention and Warning, Recent Developments in Tsunami Research, Advances in Natural and Technological Hazards Research, eds. Y. Tsuchiya and N. Shuto (Kluwer Academic Publishers, 1995), pp. 129–149. 44. M. J. Briggs, C. E. Synolakis and G. S. Harkins, Tsunami runup on a conical island, Int. Symp. Waves — Physical and Numerical Modeling, IAHR, Vancouver, BC, Canada (1994), pp. 446–455. 45. N. R. Oswalt and M. B. Boyd, Steady-ﬂow stability tests of navigation opening structures, Hilo Harbor tsunami barrier, Hilo, Hawaii, U.S. Army Engineer Waterways Experiment Station, Technical Report No. 2-742, Vicksburg, MS (1966). 46. P. K. Senter, Design of proposed Crescent City Harbor, California, tsunami model: Hydraulic model investigation, U.S. Army Engineer Waterways Experiment Station, Technical Report H-71-2, Vicksburg, MS (1971), pp. 1–33. 47. J. R. Houston, R. D. Carver and D. G. Markle, Tsunami-wave elevation frequency of occurrence for the Hawaiian islands, WES Technical Report No. H-77-16, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, August 1977. 48. J. R. Houston, Tsunami elevation predictions for American Samoa, WES Technical Report No. HL-80-16, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, September 1980. 49. U. Kanoglu and C. E. Synolakis, Long wave runup on piecewise linear topographies, J. Fluid Mech. 374, 1–28 (1998). 50. M. J. Briggs, C. E. Synolakis and S. A. Hughes, Laboratory measurements of tsunami run-up, IUGG/IOC Int. Tsunami Symp., Wakayama, Japan (1993), pp. 585–598. 51. M. J. Briggs, C. E. Synolakis, G. S. Harkins and D. R. Green, Laboratory experiments of tsunami runup on a circular island, PAGEOPH 144(3/4), 569–593 (1995). 52. M. J. Briggs and M. L. Hampton, Directional Spectral Wave Generator Basin Response to Monochromatic Waves, WES Technical Report CERC-87-6, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS (1987), pp. 1–90. 53. H. Yeh, F. Imamura, C. Synolakis, Y. Tsuji, P. Liu and S. Shi, The Flores Island tsunamis, EOS Trans. 74(33), 369–373 (1993). 54. H. Yeh, P. L.-F. Liu, M. J. Briggs and C. E. Synolakis, Propagation and ampliﬁcation of tsunamis at coastal boundaries, Nature 372(24), 353–355 (1994). 55. P. L.-F. Liu, Y.-S. Cho, M. J. Briggs and C. E. Synolakis, Runup of solitary waves on a circular island, J. Fluid Mech. 302, 259–285 (1995). 56. K. Fujima, M. J. Briggs and D. Yuliadi, Runup of tsunamis with transient wave proﬁles incident on a conical island, Coast. Eng. J. 42(2), 175–195 (2000). 57. P. L.-F. Liu, T.-R. Wu, F. Raichlen, C. E. Synolakis and J. Borrero, Runup and rundown generated by three-dimensional sliding masses, J. Fluid Mech. 536, 107–144 (2005). 58. H. M. Fritz, Physical modeling of landslide generated tsunami, Caribbean Tsunami Hazard, eds. A. Mercado-Irizarry and P. L.-F. Liu (World Scientiﬁc, Singapore, 2006), pp. 308–324.

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Chapter 40

Laboratory Simulation of Waves Etienne P. D. Mansard∗ and Michael D. Miles∗ Canadian Hydraulics Centre, National Research Council Canada Ottawa, ON, K1A 0R6, Canada During the past two decades, a large number of multidirectional wave facilities had been built around the world. In parallel, wave generation and analysis techniques have also advanced so that it is now possible to ensure realistic wave conditions that mimic the nature in laboratory basins and ﬂumes. With these capabilities, testing of coastal and oﬀshore structures can be carried out with greater accuracy ensuring thereby optimal designs for structures in terms of cost and safety. This chapter provides a brief review of the wave generation and analysis techniques that are commonly used for simulating uni- and multidirectional waves and also shares the experience gained at the Canadian Hydraulics Centre of the National Research Council Canada, in this ﬁeld.

40.1. Introduction Physical modeling is still the best means of design optimization of coastal and oﬀshore structures, in spite of the fact that it was speculated in the 1980s that the numerical modeling will take over, making physical models redundant. The numerical models have indeed improved over the years that it is now possible to use them to solve a large range of practical problems. However, with the trend of locating marine structures in deeper waters and in more severe environment the numerical models do not seem to cope with some of the environmental factors and/or their associated complexities in a reliable fashion. Hence, it is expected that the physical modeling techniques will continue to be relevant. Furthermore, the use of a combination of both numerical and physical modeling techniques in large projects is becoming more common. In such situations, numerical models are used to study the simpler components of the problem while physical models are employed for solving the complex components. There are also some situations where physical models are used to validate the numerical modeling results on some test cases, before applying the numerical tools to ﬁnd solutions for the entire project. ∗ Retired

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Waves being the most important input to any physical model study, correct simulation of waves in laboratory basins is crucial. This chapter brieﬂy reviews the wave generation techniques commonly used in major laboratories and shares the experience of the Canadian Hydraulics Centre (CHC) of the National Research Council Canada (NRC) in this ﬁeld.

40.2. Main Steps in Laboratory Simulation of Waves The ﬂowchart (Fig. 40.1) lists the main steps involved in laboratory simulation of waves. They include: • characterization or deﬁnition of the sea state, • synthesis of wave train, • preparation of wave machine driving signal,

Fig. 40.1.

Example of a ﬂowchart of laboratory simulation of waves.

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• wave generation and data sampling, • wave analysis. A description of the various steps involved in simulating unidirectional waves will be discussed ﬁrst while the aspects of multidirectional wave generation will be presented subsequently.

40.3. Simulation of Unidirectional Waves 40.3.1. Characterization of the sea state The very ﬁrst step in any model study of coastal and oﬀshore structures is the choice of the design wave conditions to be used for testing purposes. These design wave conditions are often wave conditions that may have a return period of 50, 100, or 1,000 years. The longer the return period, the higher would be the design wave height and costlier (but safer) would be the structure. It is common to design coastal structures to withstand 1:100 year wave conditions. Extreme value analysis is used to estimate the wave heights for large return periods from available recorded or hindcast data pertaining to that site. A review of these techniques can be found in publications, such as Goda et al.1 and Sarpkaya and Isaacson.2 Generally, the sea state to be used in a model study is characterized by its signiﬁcant wave height, peak period, and a parametric spectrum. The well-known parametric spectra are: • Bretschneider spectrum or ITTC (International Towing Tank Conference) or ISSC (International Ship Structure Congress) spectrum. • Pierson–Moskowitz spectrum that represents fully developed seas. • JONSWAP spectrum that represents fetch limited conditions. • Scott spectrum that provides a good ﬁt to observations made in Persian Gulf, North Atlantic, and West Coast of India. • Ochi and Hubble spectrum that represents coexistence of sea and swell. • TMA spectrum, an extension of the JONSWAP spectrum for shallow water situations. The Pierson–Moskowitz spectrum is a special case of the Bretscheneider spectrum while the JONSWAP spectrum is a modiﬁed version of the Pierson– Moskowitz spectrum. 40.3.2. Synthesis of wave train 40.3.2.1. Random phase method The technique that is most commonly used to synthesize a wave train from a given variance spectral density (commonly called also as spectrum) is the random phase spectrum method. It consists of pairing the amplitude spectrum derived from the

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given variance spectral density with a phase spectrum created by a random number generator and then obtaining a time series of desired length by inverse Fourier transform. Since random phases are used in this procedure, time domain characteristics such as wave grouping, cannot be controlled by this synthesis technique. However, the spectral characteristics are well reproduced, as it uses the amplitude spectrum from the target spectral density. Another method that uses the random phases, but has no complete control over the spectral characteristics, is called the random complex spectrum method. Its synthesis procedure is as follows. A Gaussian distributed white noise complex spectrum, with a standard deviation of 1, is ﬁrst generated and then ﬁltered using the amplitude spectrum derived from the target spectral density. Subsequent inverse Fourier transform results in a time series of desired record length. Unlike the previous method, the waves produced by this technique will not match exactly the desired spectral density. Obviously, this method does not also exercise control on the time domain characteristics. However, both these synthesis methods based on random phases have their own proponents. Funke and Mansard3 describe the rationale associated with each of these methods. One of the requirements during the synthesis of a wave train from a spectral density is the choice of the length of wave train to be created. Since the Gaussian distribution is used to describe the probability distribution of prototype water surface elevations and the Rayleigh distribution is used to describe the distribution of wave heights, the longer the wave train, the better is the ﬁt with the above functions. However, long wave trains can increase the aggregate testing time and thereby the cost of a physical model study. Hence, a suitable compromise is required. The length of the wave train is also characterized by the number of waves contained in that train and choices may range between 200 and 1,000 waves in a wave train. Often smaller lengths of wave trains (or smaller number of waves) are chosen to conduct sensitivity studies, such as penetration of wave energy in a harbor basin. Longer records are used where it is critical to ensure an appropriate distribution of wave heights. For instance, in breakwater stability studies, simulation of appropriate values of H1/10 and Hmax . are important. In general, the longer is the wave record, the larger would be the wave heights, such as H1/10 and Hmax . Studies using relatively short records (i.e., 200 waves) in shallow or breaking wave situations could potentially result in wrong interpretation of the test results (see Ref. 4). In CHC, wave records containing at least 1,000 waves are used in most studies. 40.3.2.2. Other methods of synthesis Besides the random phase method described above in Sec. 40.3.2.1, several other approaches could be used to establish the target wave train shown in Fig. 40.1. They include: • use of prototype data, • synthesis of a grouped wave train, • synthesis of episodic or freak waves.

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40.3.2.2.1. Use of prototype data As can be seen in the ﬂowchart (Fig. 40.1), simulation of prototype wave spectrum or prototype wave train is also an option in the simulation procedure. The reproduction of prototype wave data may be the best way of ensuring realistic waves in laboratory basins. However, this is often not practical because of lack of suﬃcient wave records for the site under consideration. Furthermore, their record lengths are often relatively short (i.e., 20-min-long record every 3 h). However, in situations where there are suﬃcient records measured in the proximity of a site that is being considered for development, prototype wave trains have been used. It is also common to endeavor to ﬁnd appropriate prototype wave trains when it is required to recreate a speciﬁc situation, such as damage to a marine structure. 40.3.2.2.2. Synthesis of a grouped wave train In the late seventies, NRC investigated the importance of wave grouping on the stability of ﬁxed and ﬂoating structures. For this purpose, Funke and Mansard5 developed a technique based on the concept of Smoothed Instantaneous Wave Energy History to create wave trains that can satisfy both the spectral characteristics of the desired spectrum as well as the desired time domain characteristics in terms of wave grouping. This technique has been extensively used to understand the physical processes associated with the interaction between wave groups and test structures. Johnson et al.6 successfully demonstrated that wave grouping is an important parameter to be reckoned with in studies of breakwater stability. Mansard and Pratte7 illustrated the importance of wave groups and the long waves generated by these groups on the slow drift oscillations of moored ships. Hence, this technique is indeed an appropriate tool for understanding physical processes associated with wave–structure interactions. 40.3.2.2.3. Synthesis of episodic or freak waves Funke and Mansard8 developed a technique that can generate a plunging type of breaking wave at any predetermined location in a ﬂume using the sweep frequency technique. This technique is used to subject test structures to severe breaking waves to study the interaction of extreme waves with structures. For instance, it was used to validate the capsizability of a communication buoy which otherwise was considered stable under normal wave conditions used in testing. NRC has also extended this technique to generate 3D episodic breaking waves in a multidirectional wave basin which have circular wave fronts and are focused both by frequency and wave direction.

40.3.3. Generation of wave machine control signal Before the synthesis and analysis of multidirectional seas, shown in the top right portion of Fig. 40.1 are discussed, the steps shown in the bottom portion of the ﬂowchart dealing with the generation and analysis of waves are discussed below.

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40.3.3.1. Capability of wave generator One of the ﬁrst tasks that is undertaken before simulating a sea state in any laboratory basin or ﬂume is to determine the capability of the wave machine to generate the desired sea state. This is often the critical step in determining the model scale factor that is needed to produce the design wave condition. Figure 40.2 shows the performance curves for one of the NRC wave generators, in a 2D ﬂume. The signiﬁcant wave heights that can be generated in this facility for peak periods ranging from 0.5 to 5.0 s are shown here for two diﬀerent operational water depths. The left envelope of these two curves represents the breaking limit deﬁned as a wave steepness 60% larger than that of a Pierson–Moskowitz spectrum for a fully developed sea. This limit is based on the maximum steepness observed in a sample of several hundred measured North Atlantic wave records. Waves with signiﬁcant wave heights beyond this threshold would be subject to breaking and therefore not achievable. The right-hand side portion of these curves indicates the inﬂuence of the wave machine stroke limit on the wave heights that can be generated. As long period waves require large strokes, the wave height becomes smaller as the wave period increases. These limits are based on the well-known Biesel theory that relates the displacement of the paddle required to generate a wave height of a given period, under a given water depth. 40.3.3.2. Creation of wave machine control signal Correct generation of a wave train in a laboratory facility depends very much on the accuracy of the techniques used to convert the target wave train into

Fig. 40.2.

Estimation of the capability of the wave generator.

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an appropriate wave generator control signal. This conversion requires static and dynamic calibration of the wave machine and its associated components such as servo controller and digital-to-analog ﬁlter. The hydrodynamic transfer function that computes the displacement required to produce a desired wave height is based on the well-known Biesel relationship. However, the dynamic response of the wave machine components can vary to a large extent from one facility to another and needs to be compensated for, to ensure a faithful reproduction of the desired wave train inside the laboratory basin. NRC uses a procedure called dynamic calibration to estimate the compensations required for correcting the dynamic response of the wave machine. This procedure is described below. The static calibration of the wave machine is ﬁrst established by driving the wave machine with known and slowly varying inputs of the voltage and then measuring the response in terms of paddle displacement. Then, a time series composed of wavelets of diﬀerent frequencies is fed into the system and the corresponding displacement of the paddle is monitored and analyzed. Figure 40.3 presents an example of the drive signal used for dynamic calibration, while Fig. 40.4 illustrates an expanded version of a portion of this time series (i.e., contained between 180 and 210 s). The cross-correlation of the input wave signal and the measured displacements of the paddle provides a complex transfer function that can be used to generate a wave board control signal that can ensure the desired waves. The phase and the amplitude response of this transfer function are shown in Fig. 40.5. The decreasing amplitude response and increasing phase diﬀerence, generally found in such systems when the frequency increases, can be seen clearly.

Fig. 40.3.

Wave machine drive signal for dynamic calibration.

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Fig. 40.4.

Fig. 40.5.

Portion of the drive signal presented in Fig. 40.3.

Complex transfer function obtained by dynamic calibration.

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The real-time control and data acquisition then ensure the appropriate control of the machine and the sampling of data by sensors such as wave gages.

40.3.4. Wave analysis 40.3.4.1. Conventional analysis of waves Figure 40.6 shows a typical wave analysis output of NRC from a test program for one of the wave gages used in the study. It lists some of the relevant spectral and time domain parameters derived from spectral and zero-crossing analysis. A list of notations of these parameters and their deﬁnitions can be found in IAHR/PIANC.9 The time series of the wave record, the spectrum of the measured time series, and the probability distribution of the measured wave heights are also presented. In the same ﬁgure, the target spectrum and the theoretical Rayleigh distribution are also overlaid for comparison purposes. It should be pointed out that although the target spectrum was a JONSWAP spectrum, the measured spectrum displays clearly nonlinear subharmonic and superharmonic components on the left and right side of the primary spectrum, respectively. This is to be expected given the shallow water conditions of this sea state (see values of the target sea state). A small reduction in the value of Hm0 from the

Fig. 40.6.

Sample analysis output of 2D waves.

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target value can be attributed to loss of energy during propagation along the basin and/or to some breaking. In studies where the inﬂuence of group-induced long waves is important, the long energy period energy (i.e., subharmonics) found in this ﬁgure will be analyzed in more detail. Note that in shallow water situations where substantive sub- and superharmonics are expected to be present, the second-order wave generation is recommended to be used to ensure correct reproduction of these harmonics. The techniques used for this purpose is beyond the scope of this paper. Readers are requested to refer to Barthel et al.10 and Sand and Mansard11 for these techniques. 40.3.4.2. Reflection analysis Reﬂection analysis is an important component of the analysis package for laboratory simulation of waves. It is used principally to design wave absorbers that can eﬀectively dissipate the incoming wave energy to minimize the boundary eﬀects (see Ref. 12), and also to estimate the reﬂection characteristics of coastal and oﬀshore structures. NRC developed several versions of techniques that can separate the incident and reﬂection characteristics, both in time and frequency domains (see Refs. 13–16). All these techniques are based on the least-squares method that requires information from three probes. It should also be pointed out that while the above techniques address 2D wave situations, Isaacson et al.17 had developed techniques for oblique waves. (Research is now underway in several laboratories to develop accurate techniques that can separate incident and reﬂected wave components in a multidirectional wave situation.) Waves reﬂected by test structures can propagate back to the paddle and then get re-reﬂected, corrupting the quality of data that prevails in the experimental setup. To overcome this diﬃculty active absorption techniques have been developed, and a description of these techniques is given in the following section.

40.4. Active Wave Absorption An increasing number of laboratory wave ﬂumes and basins are now equipped with wave machines that can function both as wave generators and absorbers. Such systems are called active wave absorbers to distinguish them from the passive wave absorbers that are also used in basins. Active absorbers are used mainly to improve wave quality by preventing waves reﬂected by the structure being tested from being re-reﬂected back toward the structure by the wave machine paddle. Thus, active absorption allows the desired incident wave ﬁeld to be maintained at the test structure, while preventing spurious wave energy from building up in the section of the ﬂume between the structure and the wave generator. Active absorption can also be used to greatly reduce the stilling time between tests in a ﬂume by removing lowfrequency waves, which would otherwise persist for some time. Similarly, it can also prevent spurious resonant oscillations in a basin or ﬂume during long duration tests.

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There are two main types of active wave absorption systems. The ﬁrst method uses an array of wave elevation probes located a few meters in front of the wave machine. Real-time reﬂection analysis of the wave probe signals is used to separate the incoming and outgoing wave trains and the incoming wave component at the position of the wave machine is then computed by linear dispersion theory. The corresponding paddle motion to absorb the incoming wave is then computed by using the linear hydrodynamic transfer function with compensation for the amplitude and phase response of the wave machine’s servo control system. The main advantage of this method is that it can be easily installed on existing wave machines without designing a new servo control system. The propagation delay between the wave probe array and the paddle gives the wave machine time to respond, so existing analog servo controllers with relatively large phase lags can still be used. No corrections are required for evanescent waves because the wave probes are located some distance from the paddle. One disadvantage of this method is that it relies on linear dispersion theory so nonlinear shallow water waves will introduce phase errors that reduce the absorption capability. The second method uses a wave elevation probe mounted on the face of the paddle and moving with it. This is the method most commonly used on segmented wave machines with one wave probe on each segment. The expected wave elevation at the paddle is subtracted from the total measured wave elevation and the residual wave elevation is assumed to be due to the incoming wave to be absorbed. The commanded paddle motion required to absorb the incoming wave is then computed by linear wave theory. The digital servo control system for this method must be very carefully designed and tuned so that there is very little phase lag over the full frequency range because the paddle motion must respond immediately to the measured incoming wave signal. The control system must also compensate for the amplitude and phase of the evanescent waves since the wave probe is mounted on the paddle. This is especially important at higher frequencies. Careful tuning may also be required to ensure a stable control system since phase lag must be kept to a minimum. Depending on the wave frequency and the distance between the wave machine and the reﬂecting structure being tested, the paddle motion required for absorption may be in phase with the motion required for wave generation. Consequently, the paddle motion required for simultaneous wave generation and absorption is larger than that required for wave generation alone although no additional power is needed. When designing new wave machines, the paddle stroke should typically be 30% larger than that required for wave generation alone if active absorption is to be used. The wave machine controller must also ensure that the paddle motions do not exceed the mechanical stroke limits and the percentage of active absorption control may have to be reduced when generating large waves to stay within the limits of the machine. The two main active wave absorption methods that have been used on segmented wave machines are known as quasi-3D and fully-3D. In the quasi-3D method, the direction of the incident waves approaching the wave machine is not measured but is set a priori based on the conﬁguration of the basin and the structure being tested. The resulting paddle motions will have the correct phase when absorbing

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an oblique wave but the amplitude may be in error. However, since the amplitude varies as cos(θ), the quasi-3D method still works very well in most cases since the amplitude error is small when the incident waves are within 20 degrees of the estimated direction. In fully-3D systems, the angle of the incident waves is measured directly so that the correct amplitude can be used. This can either be done by using a wave probe array in the basin or by using signals from wave probes mounted on several adjacent segments to determine the incident wave angle. Sch¨ aﬀer and Klopman18 provide a good review of the various active absorption techniques that have been used for segmented wave machines. The Canadian Hydraulics Centre of the NRC developed a 2D active wave absorption system for wave ﬂumes in 1999 that uses an array of three wave probes to measure the incoming and outgoing waves. This system provides a convenient and inexpensive way to add active absorption to an existing wave machine since the original servo controller can be used. CHC has recently also developed a new digital active wave absorption control system for segmented wave machines that uses a wave elevation probe mounted on each paddle. This system uses two drive signals per segment, which deﬁne the paddle motion for wave generation and the expected wave elevation at the probe including the evanescent component. The controller subtracts the expected wave elevation from the measured wave elevation to obtain the elevation of the incoming wave to be absorbed. Both of these methods have been tested experimentally using a special wave ﬂume equipped with two wave generators as shown in Fig. 40.7. In wave absorption tests, the waves are generated by wave machine A and absorbed by wave machine B. Standard reﬂection analysis of data from a wave probe array is then used to measure the reﬂection coeﬃcient of the active wave absorber. In other tests, wave machine B performs simultaneous generation and absorption while wave machine A remains stationary to provide a reﬂecting boundary. Some regular wave test results for the two types of active absorbers are shown in Fig. 40.8. The three-probe array method has good performance with an average

Fig. 40.7.

Flume conﬁguration for active wave absorber tests.

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Fig. 40.8.

Fig. 40.9.

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Active wave absorption test results for regular waves.

Irregular wave test results for a JONSWAP spectrum with Tp = 2.0 s.

reﬂection coeﬃcient of about 10% over the main range of wave periods from 1.0 to 3.0 s. However, the reﬂection coeﬃcient increases at longer wave periods due to phase errors associated with nonlinear shallow water waves. The new digital controller has very good performance over the full range of wave periods with an average reﬂection coeﬃcient of 5%. Some irregular wave test results for the new digital controller with a JONSWAP spectrum with a peak period of 2 s are shown in Fig. 40.9. It can be seen that the system also has excellent performance in irregular waves with an average reﬂection coeﬃcient of 4%. The results of a test of simultaneous generation and absorption are shown in Fig. 40.10. In this case, waves were generated at a resonant frequency of the ﬂume. It can be seen that the wave height steadily increases when active absorption is turned oﬀ but a stable standing wave is quickly established and maintained when active absorption is on.

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Fig. 40.10.

Resonant standing wave test results for T = 2.947 s.

Fig. 40.11.

Validation of the active wave absorption technique.

Figure 40.11 presents an example of the inﬂuence of active wave absorption in terms of stilling a ﬂume. The two curves show the elevations measured by a probe when the active absorption was turned oﬀ and also when it was turned on. During these tests, the wave generation was stopped at t = 100 s. Data was sampled for

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another 260 s. The two measured wave trains are almost identical for this ﬁrst 100 s, which means that the AWA has accurately identiﬁed that there are virtually no incoming waves to absorb during that period. The periods from 100 to 160 s consists mainly of the ﬁrst reﬂection of waves by the generator A (see Fig. 40.7). During this period, the AWA on wave height is approximately half as large as the AWA oﬀ, indicating the good absorption when AWA is on. Finally, a comparison of the results for the period 160–360 s shows that the AWA system has very eﬀectively reduced the residual wave energy in the ﬂume.

40.5. Simulation of Multidirectional Waves The importance of testing under multidirectional seas has been described in many publications such as Funke and Mansard,19 Mansard,20 Hoklie et al.,21 Franco et al.,22 and Stansberg et al.,23 and many multidirectional wave facilities have been built around the world during the past 15–20 years. In parallel, numerous techniques have been developed to simulate multidirectional waves for laboratory model investigations. The next sections will provide a brief review of the techniques used for the generation and analysis of multidirectional seas. They will also describe a numerical model that is used at NRC for designing experiments in an existing multidirectional wave basin or for designing new facilities. The ﬂowchart given earlier in Fig. 40.1 also provided a sketch of the diﬀerent steps involved in the generation of multidirectional seas. As it can be seen from that ﬁgure, the synthesis part of the target wave train is diﬀerent for generation of unidirectional and multidirectional seas, while the generation component through the wave machine is similar. Wave analysis will of course be diﬀerent for the two cases.

40.5.1. Generation of multidirectional (or 3D) waves The multidirectional spectral density of a sea state is given by: S(f, θ) = S(f ) · D(f, θ)

(40.1)

where S(f ) is a spectral density and D(f, θ) is the directional spreading function satisfying the relationship:

2π

D(f, θ) dθ = 1.

(40.2)

0

Figure 40.12 shows an example of a multidirectional spectrum. The main step involved in the generation of multidirectional seas is the choice of a directional spreading function, which describes the mean direction and the angular distribution of energy. The most commonly used spreading function is of the following form where Γ is the gamma function, θ0 is the mean wave direction, and s is the spreading index.

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Fig. 40.12.

An example of a directional spectrum.

This function can either be the same for all frequencies or the parameters θ0 and s may vary with frequency: π Γ(s + 1) cos2s (θ − θ0 ) for |θ − θ0 | < . D(f, θ) = √ 1 2 πΓ(s + 2 )

(40.3)

Figure 40.13 shows an example of a directional spreading function, which is nonuniform over the diﬀerent frequency ranges. Several techniques can be used to synthesize a time series from the directional spectral density given in Eq. (40.1). Among them, the most commonly used techniques include single summation and double summation models. Details of these two models can be found in publications such as Miles and Funke24 and Miles.25 Among these two techniques, many laboratories prefer the use of the single summation model. In this model, each frequency component can only travel in one direction, thus ensuring a spatially homogeneous ﬁeld in the basin for typical record lengths used in laboratory testing. On the other hand, in the double summation model, multiple wave directions exist at each discrete frequency resulting in a nonhomogeneous wave ﬁeld in the test section. Jeﬀerys26 and Miles and Funke24 discuss the relative merits and demerits of these models. In the ﬂowchart presented (Fig. 40.1), it is shown that the desired target multidirectional wave train could also be synthesized by combining a unidirectional wave record η(t) and a target spreading function. Alternatively, a time series of the water surface elevation η(t) and its associated orthogonal velocities u(t) and v(t) can also

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Nonuniform spreading function.

be used directly in the wave synthesis, since the values of u(t) and v(t) describe adequately the directional characteristics of the waves. The various synthesis techniques available at NRC for multidirectional wave generation can be summarized as follows: • Single summation method with equally spaced wave angles for speciﬁed wave spectrum and spreading function. • Single summation method with random wave angles for speciﬁed wave spectrum and spreading function. • Single summation method with random wave angles for speciﬁed wave train and spreading function. • Double summation method with equally spaced wave angles for speciﬁed wave spectrum and spreading function. • Discrete FFT wave synthesis for speciﬁed time series of wave elevation η(t) and horizontal velocities u(t) and v(t). The above methods perform not only the synthesis of water surface elevation, but also compute the required paddle motions based on the snake principle method of Sand and Mynett.27 These paddle motions are then compensated for dynamic and static transfer functions as in the case of unidirectional waves. The single summation method with random wave angles is most commonly used since it provides the most accurate frequency spectrum but equally spaced angle methods provide a more accurate spreading function at the expense of some spectral distortion. Test durations corresponding to 1-h full-scale are adequate when the spreading function does not vary with frequency. However, test durations of 5 h or more full-scale are

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required to accurately reproduce the directional spectrum in cases such as hurricane seas where the spreading function has large variations with frequency. For example, see Ref. 28. 40.5.2. Analysis of multidirectional waves 40.5.2.1. Analysis techniques A comprehensive review of the multidirectional wave analysis was carried out under the auspices of the Maritime Hydraulics Section of the International Association of Hydraulic Engineering and Research by Benoit et al.29 The diﬀerent methods of analysis can be classiﬁed according to the following categories: • • • • • •

Fourier decomposition method; Fitting of parametric models; Maximum likelihood methods (MLM); Maximum entropy methods (MEM); Bayesian directional method; and Deterministic analysis methods.

Among these methods, the MLM and MEM are those that are commonly used. The instrumentation that best corresponds to these methods is either a wave probe array or an η−u−v sensor. The experience gained at NRC, which used these two popular methods, is described below. Both the MLM and the MEM are based on cross-spectral analysis of the various sensor signals. The MLM assumes that D(f, θ) can be expressed as a linear combination of the cross-spectra, whereas the MEM estimates D(f, θ) by maximizing an entropy function subject to cross-spectra constraints. However, the MLM, which in fact is easier to implement, tends to provide an estimate of the spreading function wider than the target function. The NRC technique developed by Nwogu et al.,30 and based on MEM technique using η−u−v data, was found by Benoit et al.29 to be superior to other methods. This technique was later adapted to work with data from an array of wave gages, partly because of the fact that the current meters could be more expensive than wave gages and also susceptible to errors caused by contamination of the current-turbulent ﬂuctuations produced at the same frequencies as the wave-induced kinematics. The wave probe array consisted of ﬁve probes arranged in a trapezoidal fashion, and the water surface elevation data from the probe array were used to resolve the directional characteristics of the sea state. More recently, this method has been modiﬁed to use the wave slopes derived from the water surface elevation measured by these gages in the MEM analysis rather than using directly the water surface elevations. This modiﬁcation is triggered by the fact that the previous method was eﬃcient only over a limited frequency range where the energy contained in the spectrum was substantial. Computational eﬀort

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Fig. 40.14.

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Deﬁnition sketch for the wave probe array.

in terms of convergence of solution was also relatively high. It was also sensitive to some extent, to the spacing between gages. The new method is analogous to the η−u−v method since it uses the water surface elevation η, and the orthogonal surface slopes dη/dx and dη/dy. Figure 40.14 shows a deﬁnition sketch of the ﬁve-probe array.31 Gages A, B, C, and D are located on the circumference of a circle having a radius R and gage E is located at the center of the circle. The water surface elevation is derived from gage E, while the orthogonal slopes dη/dx and dη/dy are derived from the wave elevation diﬀerences between gages A and C and B and D, respectively. Numerical simulations were undertaken to test this technique (called 5-η MEM at NRC) on synthesized multidirectional waves with a JONSWAP spectrum and a cos2s spreading function. The traditional well-proven technique, called η−u−v MEM that uses η−u−v data was used for comparison purposes. Figure 40.15(a) shows the results of this comparison. The mean direction, the standard deviation of the spreading function, and the spreading function at the peak frequency estimated by these two techniques (5-η MEM and η−u−v MEM) are compared with the corresponding target (or imposed) function. The two analysis techniques ensure a good agreement with the target spreading function. In fact, the 5-η MEM result even comes closer to the target. Similarly, the values of standard deviation of the spreading function and the mean values of the direction resolved by these two techniques are nearly indistinguishable and also agree well with the target values. Following the good performance of 5-η MEM, additional investigations were undertaken to investigate the accuracy of the analysis if only four gages were used instead of ﬁve. In this case, the water surface elevations from gages A, B, C, and D will be averaged rather than using the information from gage E.

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Fig. 40.15. arrays.

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Numerical validation of wave slope method using data from four- and ﬁve-probe

Figure 40.15(b) shows comparisons obtained by a four-probe array, similar to the ones presented previously in Fig. 40.15(a). These ﬁgures show clearly that a four-probe array is quite adequate for resolving the directional characteristics of the sea state. The four-probe array will be less expensive than a ﬁve-probe array and in fact can lend itself for better structural support through the use of a supporting rod in the middle, instead of probe ﬁve. The preferred NRC method is therefore the four-probe array slope method called 4-η MEM. The optimal probe spacing required (i.e., value of R) as a function of the peak period wavelength (Lp ) to oﬀer reliable results was also investigated through numerical simulations.

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The results of these investigations suggested that a value of R/Lp ≈ 0.02 would be adequate to yield reliable results. More details on these investigations can be found in Cornett et al.31 40.5.2.2. Laboratory validation Extensive investigations were undertaken to validate this technique through basin tests even under severe situations such as breaking and nonlinear waves. Figure 40.16 shows a sample result from these investigations carried out under two diﬀerent water depths. Unlike in Fig. 40.15, where the values of standard deviation and mean direction were close to the expected values even in high- to low-frequency parts of the spectrum (where the relative energy content is lower than at the peak of the spectrum), the measured data shows that the accuracy of the results is poor in those regions. This is to be expected since the analysis accuracy depends highly on the relative magnitude of noise-to-signal ratio. Generally, in the high- and lowfrequency parts of the spectrum, one expects this ratio to be high, and therefore

(a) Fig. 40.16.

(b)

Directional wave analysis results for a set of wave basin data.31

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lesser accuracy of the analysis results. Furthermore, in this case, as it can be clearly seen from the variance spectral density measured at 10-m water depths, waves were highly nonlinear, whereas the entire analysis technique is based on linear theory. Given below are the conclusions that were drawn from basin studies conducted at NRC by Cornett et al.31 : The surface slopes MEM method (i.e., 4-η MEM) performs about as well as the η−u−v MEM over a wide range of wave conditions and water depths. Tests conducted with bimodal seas, nonlinear shallow water waves, and even breaking waves, indicate that both methods perform reasonably well, even under these challenging conditions. The surface slopes MEM oﬀers a number of operational advantages, such as calibration compared with alternative methods, and is a viable alternative for accurate and reliable directional wave analysis in the laboratory. However, the optimal array radius being dependent on the dominant wavelength could be perceived as a disadvantage of the surface slopes method.

Readers are requested to refer to Cornett et al.31 for more detailed information on the validation tests.

40.6. Numerical Modeling of Multidirectional Wave Basins Numerical models have been developed for evaluating the design options for new multidirectional wave basins but they can also be very useful tools for designing experiments in existing wave basins. They provide an eﬀective way to select the best wave generation method and basin conﬁguration to optimize the quality of the generated waves in a designated region of the basin for a particular type of model test. One such numerical model named WAGEN was originally developed by Isaacson and Qu32 for the case of oblique unidirectional regular waves generated by one or more segmented wave machines in a basin of constant depth. It is a linear diﬀraction model, which uses a large number of point wave sources to represent the action of the wave machine segments and the other fully reﬂecting or partially reﬂecting boundaries of the basin. Hiraishi et al.33 provides some results on the experimental validation of this model in a multidirectional wave basin. The WAGEN model was subsequently extended by CHC to cover the case of multidirectional irregular waves and wave machines with active wave absorption. It uses an iterative technique to compute the primary wave ﬁeld and the secondary wave ﬁelds produced by partial reﬂection of the primary waves from the passive wave absorbers in the basin as well as the re-reﬂection of any incident waves from the segmented wave generators. In the case of regular waves, WAGEN computes the wave height and the horizontal u- and v-wave velocity components over a speciﬁed x–y grid in the basin for a target wave train deﬁned by period, wave height, and wave propagation angle. In the case of multidirectional irregular waves, the target wave ﬁeld is typically synthesized by approximately 2,000 wave components with individual frequencies, amplitudes, directions, and phases. The amplitudes and phases are computed by the random phase method for a speciﬁed wave spectrum and the wave directions are selected at random based on the cumulative distribution deﬁned by a speciﬁed

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directional spreading function. The velocity potential Φ(x, y) is ﬁrst computed for each of these wave components and the ﬁnal velocity potential for the total wave ﬁeld is obtained by linear superposition. WAGEN then computes the signiﬁcant wave height Hs , the mean wave direction θm , and the directional spreading width σθ as functions of x and y on the basin grid. Time series of wave elevation η(t) and velocities u(t) and v(t) can also be computed at speciﬁed points in the basin so that directional wave spectra can be computed by the MEM. Most multidirectional wave basins have a segmented wave machine on one side and passive wave absorbers on the other three sides. The passive absorbers are essential to prevent reﬂected waves from propagating back to the model test site but they also cause substantial variations in wave height due to wave diﬀraction. There are also wave height variations due to reﬂection from the passive absorbers. WAGEN can be used to compute the wave height variation over the basin so that the location and orientation of the structure being tested can be chosen to minimize the eﬀects of diﬀraction and reﬂection. In some cases, partial length guide walls are used to reduce diﬀraction and WAGEN is also very useful for selecting the best length and position for such devices. For example, diﬀraction eﬀects can be reduced by using intentional reﬂection oﬀ a short guide wall at one end of a segmented wave generator (i.e., extending from X = 0 to 5 m in this case) for the case of unidirectional regular waves propagating at an angle of 30 degrees relative to the x-axis of the basin with a target wave height of 0.2 m (see Fig. 40.17). The wave generator is located on the left side of the basin. It can be seen that the corner reﬂection method has signiﬁcantly reduced the size of the diﬀraction zone on the lower right side of the basin. The increased wave height near the origin due to the corner reﬂection technique is also evident.

Fig. 40.17.

Contour plots of regular wave height with and without corner reﬂection.

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Fig. 40.18.

Normalized signiﬁcant wave height for irregular multidirectional waves.

A contour plot of the normalized signiﬁcant wave height computed by WAGEN for the case of a multidirectional JONSWAP spectrum with a spreading width of σθ = 30 degrees and a mean direction of 0 degrees is shown in Fig. 40.18. The segmented wave generator is located on the left side of this basin and passive absorbers with a reﬂection coeﬃcient of 0.1 are installed on the other three sides. These results show a useful working area about 5 m by 5 m at the center of the basin where the wave height is quite homogeneous but there are large variations in other parts of the basin due primarily to diﬀraction. Multidirectional wave basins designed for testing moving ship models usually have segmented wave machines installed on two adjacent sides to reduce the size of the diﬀraction zones. The normalized signiﬁcant wave height computed by WAGEN for a basin with segmented wave machines installed on the bottom and left sides are shown in Fig. 40.19. These waves have a JONSWAP spectrum with a spreading width of 30 degrees and a mean wave direction of 90 degrees relative to the x-axis. It can be seen that the use of two wave generators has greatly increased the size of the useful testing area where the wave height is homogeneous. These results also show how the wave height uniformity can be further improved when active wave absorption is used to prevent waves generated by the bottom wave machine from being reﬂected oﬀ the other wave machine on the left side. The corresponding mean wave direction computed by WAGEN for this case is shown in Fig. 40.20, which also demonstrates the improvement in wave quality due to active wave absorption.

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Laboratory Simulation of Waves

Fig. 40.19.

Normalized signiﬁcant wave height for a basin with two wave generators.

Fig. 40.20.

Mean wave direction for a basin with two wave generators.

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40.7. Conclusions Wave simulation technology has made great strides in the past two decades that it is now possible to ensure realistic ocean waves inside laboratory environments for testing purposes. Sophisticated analysis techniques ensure that the desired characteristics of the waves are faithfully reproduced. Advanced tools are also available for properly designing experiments in laboratory basins and for ensuring improved wave quality by preventing waves reﬂected by structures being re-reﬂected back to the structure by the wave generator. Further research is required to develop techniques that can analyze nonlinear multidirectional waves, separate the incident and reﬂected components from coexisting multidirectional waves and also to perform true 3D active wave absorption.

References 1. Y. Goda, P. Hawkes, E. P. D. Mansard, J. M. Martin, M. Mathiesen, E. Peltier, E. F. Thompson and G. Ph. VanVledder, Intercomparison of extremal wave analysis methods using numerically simulated data, Proc. Waves ’93 Conf., New Orleans, USA (1993). 2. T. Sarpkaya and M. Isaacson, Mechanics of Wave Forces on Oﬀshore Structures (Van Norstrand Reinhold Company Inc., New York, 1981). 3. E. R. R. Funke and E. P. D. Mansard, A rationale for the use of the deterministic approach to laboratory wave simulation, Proc. Sem. Wave Generation and Analysis in Lab. Basins, 22nd IAHR Conf., Lausanne, Switzerland (1987). 4. J. S. Readshaw, W. F. Baird and E. P. D. Mansard, Shallow water wave generation: An engineering perspective, Proc. Sem. Wave Anal. Generation, 22nd IAHR Conf., Lausanne, Switzerland (1987), pp. 397–410. 5. E. R. R. Funke and E. P. D. Mansard, On the synthesis of realistic sea states in a laboratory ﬂume, National Research Council Hydraulics Laboratory Technical Report LTR-HY-066 (1979). 6. R. R. Johnson, E. P. D. Mansard and J. Ploeg, Eﬀects of wave grouping on breakwater stability, Proc. 16th Int. Conf. Coast. Eng., Hamburg, Germany (1978), pp. 2228–2243. 7. E. P. D. Mansard and B. D. Pratte, Moored ship response in irregular waves, Proc. 18th Int. Conf. Coast. Eng., Capetown, South Africa (1982), pp. 2621–2640. 8. E. R. R. Funke and E. P. D. Mansard, SPLSH: A program for the synthesis of episodic waves, National Research Council Hydraulics Laboratory Technical Report LTR-HY065 (1979). 9. IAHR/PIANC, List of sea state parameters, PIANC Supplement to Bulletin No. 52, General Secretariat of PIANC, Rue de la loi 155, 1040 Brussels, Belgium (1986). 10. V. Barthel, E. P. D. Mansard, S. E. Sand and F. C. Vis, Group bounded long waves in physical models, Ocean Eng. 10(4), 261–294 (1983). 11. S. E. Sand and E. P. D. Mansard, Description and reproduction of higher harmonic waves, National Research Council Hydraulics Laboratory Technical Report TR-HY012 (1986). 12. W. W. Jamieson and E. P. D. Mansard, An eﬃcient upright wave absorber, Proc. ASCE Specialty Conf. Coast. Hydrodyn., University of Delaware, USA (1987).

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13. E. P. D. Mansard and E. R. Funke, The measurement of incident and reﬂected spectra using a least squares method, Proc. 17th Int. Conf. Coast. Eng., Sydney, Australia (1980), pp. 154–172. 14. E. P. D. Mansard, S. E. Sand and E. R. R. Funke, Reﬂection analysis of non-linear regular waves, National Research Council Hydraulics Laboratory Technical Report TR-HY-0ll (1985). 15. E. P. D. Mansard and E. R. R. Funke, On the reﬂection analysis of irregular waves, National Research Council Hydraulics Laboratory Technical Report TR-HY017 (1987). 16. E. P. D. Mansard, On the estimation of the incident wave train from reﬂection analysis, Proc. 9th Cong. Asian-Paciﬁc Div. Int. Assoc. Hydraul. Res., Singapore, 24–26 August 1994. 17. M. Isaacson, D. Papps and E. P. D. Mansard, Oblique reﬂection characteristics of rubble mound structures, Proc. 12th Canadian Hydrotech. Conf., Ottawa, Canada 1, 445–454 (1995). 18. H. Sch¨ aﬀer and G. Klopman, Review of multidirectional active wave absorption methods, IAHR Sem. Multidirectional Waves, 27th IAHR Cong., San Francisco (1997), pp. 159–182. 19. E. R. R. Funke and E. P. D. Mansard, On the testing of models in multidirectional seas, Presented at the 23rd Int. Conf. Coast. Eng., Venice, Italy (1992). 20. E. P. D. Mansard, Directional wave generation and its application, Proc. Symp. Real Sea ’98, Taejon, Korea (1998). 21. M. Hoklie, C. T. Stansberg and P. Werenskiold, Model tests with a single point mooring system in short-crested seas, Paper 4644, 15th Oﬀshore Technology Conference, Texas, USA (1983). 22. C. Franco, J. W. van der Meer and L. Franco, Multidirectional wave loads on vertical breakwaters, Proc. 25th Int. Conf. Coast. Eng., Orlando, Florida, USA (1996). 23. C. T. Stansberg, J. R. Krokstad and F. G. Nielsen, Model testing of the slow drift motion of a moored semi submersible in multidirectional waves, Proc. IAHR Sem. Multidirectional Waves Their Interaction with Structures, 27th IAHR Cong., ed. E. Mansard, San Francisco, USA (1997). 24. M. D. Miles and E. R. Funke, A comparison of methods for synthesis of directional seas, ASME J. Oﬀshore Mech. Eng. 111(1), 43–48 (1989). 25. M. D. Miles, A note on directional random wave synthesis by the single summation method, Proc. 23rd IAHR Cong., Vol. C, Ottawa, Ont., August 1989, pp. 243–250. 26. E. R. Jeﬀerys, Directional seas should be ergodic, Appl. Ocean Res. 9(4), 186–191 (1987). 27. S. E. Sand and A. E. Mynett, Directional wave generation and analysis, Proc. IAHR Sem. Wave Anal. Generation Lab. Basins, 22nd IAHR Cong., Lausanne, Switzerland (1987). 28. A. Cornett and M. Miles, Simulation of hurricane seas in a multidirectional wave basin, OMAE J. 113, 219–227 (1991). 29. M. Benoit, P. Frigaard and H. Sch¨ aﬀer, Analysing multidirectional wave spectra: A tentative classiﬁcation of available methods, Proc. IAHR Seminar on Multidirectional Waves and Their Interaction with Structures, 27th IAHR Congress, ed. E. Mansard, San Francisco, USA (1997). 30. O. Nwogu, E. P. D. Mansard, M. D. Miles and M. St. de Q. Isaacson, Estimation of directional waves spectra by the maximum entropy method, Proc. Sem. Wave Anal. Generation, 22nd IAHR Conf., Lausanne, Switzerland (1987), pp. 363–376.

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31. A. Cornett, M. Miles and D. Pelletier, Measurement and analysis of multidirectional waves using free surface slopes, Proc. 5th Int. Symp. Wave Measure. Anal., Madrid, Spain (2005). 32. M. Isaacson and S. Q. Qu, Predicted wave ﬁeld in a laboratory wave basin, Can. J. Civ. Eng. 17(6), 1005–1014 (1990). 33. T. Hiraishi, E. P. D. Mansard, M. D. Miles, E. R. Funke and I. Isaacson, Validation of a numerical diﬀraction model for multidirectional wave generation. Part 2: Experimental veriﬁcation of the model results, Presented at ISOPE-92 Conf., San Francisco, 14–19 June 1992.

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Chapter 41

Perspective on Coastal Engineering Practice and Education J. William Kamphuis Department of Civil Engineering Queen’s University, Kingston, ON, Canada K7L 3N6 [email protected] This perspective traces recent developments in coastal and civil engineering practice and in coastal engineering education. It notes that engineering has changed substantially and that engineers are not educated for the contemporary tasks they face. It states that changes must be made urgently, but they must be made within the conﬁnes dictated by a global market. It then oﬀers a number of alternatives to bring about closer cooperation between practice and education in order to provide better engineering education that is more relevant to the needs of engineering practice.

41.1. Introduction This is a perspective, a point of view. This essay is based on personal experience and it is essentially an opinion piece.a This chapter reviews the past development of coastalb engineering practice and education. It is seen that the two have been on divergent tracks, and observations of the past will be used to determine what can be done to bring about improvement in the relationship between the two in the near future. In the manner of wave hindcasting, this chapter determines some central tendencies in coastal engineering practice and education, based on observations of the past. This hindcast can be greatly improved if we would all contribute to the quality a The opinions are mainly based on North American experience, but the author’s extensive international experience, both in universities and in practice, indicates that other jurisdictions have similar problems and that both academia and engineering practice look up to and emulate American “success stories.” b The terms coastal engineering, civil engineering, and engineering are all used in this chapter, denoting if a point refers speciﬁcally to coastal engineering or is more general and applies to civil engineering or engineering.

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of the database. Hence this is a work in progress, which should really involve the thoughts and opinions of us all. This chapter extends Kamphuis,1 which presents a broad background to practice, education, and research in coastal engineering and management, tracing their origins and sociologic settings from historic times. The American Society of Civil Engineers (ASCE) is making very valuable contributions to address the mismatch between civil engineering practice and education. In fact, the ASCE views itself as being a leader in bringing about change in both practice and education of civil engineers. Since it is the civil engineering organization with the largest (global) exposure this is a natural step and one to be carefully followed. The ASCE has published a number of valuable documents by various authors from the civil engineering community. In addition, it is deﬁning a required body of knowledge for civil engineers.2,3 In a recent document titled “The Vision for Civil Engineering in 2025”,4 they deﬁne the required knowledge, skills, and attributes. This is a very helpful publication and will be quoted later. In that vein, Patricia Galloway, a recent past president of ASCE has just completed a book titled: “The 21st Century Engineer — A Proposal for Engineering Education Reform.”5 This very worthwhile and hard-hitting book makes many of the points made in this chapter. It has, however, a diﬀerent but complementary focus. Galloway writes as CEO of a large international engineering company and emphasizes diﬀerent subjects from this chapter that is written from the point of view of an academic/practicing engineer.

41.2. Impact of the Global Market Engineering projects and their designs must ﬁt within the existing socioeconomic system. A very important aspect of this is the global market and a short discussion of its impact will be given in this section. Global market forces aﬀect all aspects of engineering practice and education, which in turn involve research and teaching in the engineering schools and universities. Global market force calculations simplistically break large complex systems into connected series of commodities, which are essentially represented by simple $cost units.c The market operates by maximizing economic $ beneﬁt and minimizing $ cost through simplistic calculations such as: • $ proﬁt = ($ beneﬁts) − ($ cost units) • $ BCR = ($ beneﬁts)/ ($ cost units). use “$” as a shorthand symbol to denote money (in any denomination). $ cost calculations normally do not consider other cost deﬁnitions such as lifecycle cost. The focus is on $ return to the organizations and their shareholders. Therefore, we can describe economic cost simply as “$ cost.” All of us know that considering only $ costs and $ beneﬁts is dangerous, simplistic, and myopic. It generates and has generated much disaster, dislocation, and grief. But this is the common denominator of all transactions and, therefore, it is the stage on which engineering projects and education must play.

c We

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Engineering services are also computed into project cost via $ cost units. They are essentially commodities that are bought and sold. The market is global because many of the actors and many projects are international; impacts are transnational, and all purchases, also engineering services, are made globally. Social values are also global. Impact of global warming, consequences of emissions, and fallout of wars over water or unlimited access to fossil fuel have no national borders. Important ethical values such as safety, health, quality of life, and quality of the environment are normally introduced via regulations, and in contemporary jurisdictions via the public through stakeholder representation and regulation hearings. Such ethical/social aspects are also reduced to simple $ cost units in the project cost calculations. Universities are also conglomerates of $ economic units. They are normally funded through government grants, student’s tuition fees, and income from endowments, fund drives, and research. Universities work very hard to maximize the incoming $ funding on all these fronts and minimize their $ costs. They must. No one pays universities anymore simply to think and to provide quality education. The lofty ideals — a community of scholars with primary purposes of education of the young and the betterment of society — have long disappeared, except in mission statements. Unfortunately, the largest $ cost items for universities are . . . creating a reﬂective, thoughtful environment, and education of the students — the raisons d’ˆetre of universities. University income is maximized by accepting and graduating the maximum number of students. The $ costs to the universities of teaching students are minimized by increasing class sizes, student/teacher ratios, and student/space ratios. These calculations are forcing the universities away from being places of learning and quiet reﬂection. They have become factories that produce products (graduates and research papers) for a global market on a delicately balanced budget. The student/space ratio, in particular, severely restricts engineering education, which needs rather extensive laboratory space. Thus, coastal laboratories with their requirements for large spaces that often stand empty do not show up well on the student/space scale. Biological, geological, and other ﬁeld stations are also perceived as a liability. All this is purported not to aﬀect the quality of education. Even if this were so, the drive to maximize $ research income at minimum $ cost does aﬀect the education of engineering students. Research has really become just one type of machine within the university factory. Its primary purpose is to provide a product (research papers in this case) at minimum cost. The cost is kept low by paying graduate students and post-doctoral employees as little as possible and by experimenting as much as possible on computers, rather than in the ﬁeld or in the physical laboratory. The students and post-docs are considered ﬁrst as cost units and only secondly as young people who came to the university for higher education. The implications of this are described in greater detail below. Clearly, the above statements are generalizations. They are made to sketch the background against which decisions are made. Fortunately, there are many exceptions.

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41.3. Coastal Engineering Practice 41.3.1. Civil to coastal engineering Civil engineering began as a counterpoint to military engineering. It consisted of all engineering pertaining to civil projects. As more knowledge became available, this very broad discipline of civil engineering specialized into subdisciplines. Specialties as mining engineering and mechanical engineering were ﬁrst to separate themselves, to be followed later by electrical, chemical, metallurgical engineering, etc. The engineering tasks that were left over are now called civil engineering, which is essentially just another engineering specialty. In the late 19th and early 20th centuries the civil engineer was a generalist, a problem-solver, and a builder. The tasks consisted of design, layout, and construction of infrastructure — railways, roads, pipes, open channels, buildings, bridges, etc. Basic indispensable skills were technical drawing and surveying. Since civil engineering was still very broad, further specialization took place over time, particularly because research developed a deeper understanding of the principles involved. As more knowledge became available, the 19th and early 20th century civil engineer generalist evolved into the late 20th and 21st century technical expert engineer. By 1950, civil engineers were still capable of designing and building diﬀerent types of infrastructure, but that ended when they began to classify themselves as bridge engineers, structural engineers, hydraulic engineers, geotechnical engineers, etc. Specialties, such as coastal engineering, developed within these broad areas and coastal engineering is now splitting further into subspecialties, such as coastal structures, ﬂuid ﬂow, and sediment transport. As knowledge deepens, the ﬁelds of expertise narrows. As a result, “experts” from one such specialist area barely know the basics of the neighboring areas. By the mid-1970s, the ability to work in more than one speciﬁc narrow specialty area was essentially lost. The generalist engineer slowly disappeared and in time the civil engineering profession became a collection of experts, often functioning in (almost) noncommunicating technical silos. This development is encouraged by how research is carried on at the engineering educational institutions, as will be seen later. Few civil engineers are now capable of a broad perspective. Working familiarity with anything outside one’s narrow specialty is limited. And thus design of a bridge involves at least structural experts, hydraulics engineers, geotechnical engineers, transportation experts, and construction experts, who must all work together to produce a workable, safe, and economically viable bridge. But with individual engineers understanding only part of the project and really preferring to stay in their own niche area, an engineering or project manager is needed to manage the bridge design and construction — someone who can manage a team of engineering experts as well as the broader implications, such as ﬁnance, environment, and communication. Engineering (the application of ingenuity and knowledge to solve a problem) has evolved into employment of a number of commodities (experts in narrow ﬁelds). A project manager (essentially another engineering commodity employed by the owner) hires bits of technical expertise and puts it together. Coastal engineering,

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as it is practiced today, is similarly subject to these trends of specialization and “commodiﬁcation.”

41.3.2. Hypothetical coastal engineering example A hypothetical example will now illustrate the changing environment for coastal engineering. It concerns the historical development of a small tidal inlet. For centuries, small tidal inlets were simply left to follow their natural course. It was common knowledge that the inlet would be periodically blocked with sand and that it would become unblocked in due time or perhaps it would shift by some kilometers in either direction. The sparse population of ﬁshers in the area would simply adapt to the changes of the inlet. They had to adapt, because they did not have the equipment or the knowledge to do anything about the inlet and its migration. Then in the 19th and 20th centuries, many such inlets developed into centers of economic importance. They oﬀered ideal habitat for ﬁsh and shellﬁsh to spawn and grow. They oﬀered shelter for the ﬁshing boats that were becoming so large that they could no longer be simply dragged up the beach by human or animal strength. As a result, a ﬁshing and seafood industry developed in the bay behind the inlet. The inlet bay became viewed also as safe anchorage or mooring for larger vessels and this was the beginning of a small port, which in turn brought in other industry. Because of the port’s importance and because the military needed berthing and mooring space for its vessels, the port also became a naval station and all these new activities caused a large expansion of the local population. Roads and railroads were designed and built to support the industry and the port. Since the local economy and its infrastructure were now completely dependent on the tidal inlet’s location and the exchange of water through the inlet, it was decided to “stabilize the inlet.” The intention was to ﬁx the location of the inlet and maintain a minimum depth of water through the inlet, so ships of certain sizes could pass in and out at all times. All this became possible because there was now equipment powerful enough to accomplish this. Exactly how to stabilize an inlet was still not entirely clear, but there was always someone who either had done something like this before, or who had some training, was able to visualize what needed to be done and could put all the diﬀerent ideas together into a plan. This protocoastal engineer had very little training in the speciﬁc physical processes involved, but hed did have the ingenuitye to put the project together. Such early designs involved the construction of one or two structures built perpendicular to the shore on one or both sides of the inlet’s opening to the sea. The designs were based on experience at ﬁrst and later on some nebulous semiempirical formulae. The structures were usually built of rocks or wooden cribs ﬁlled

d Always a “he” at that time. e The word “engineer” is closely

related to the French word ing´ enieur, which essentially means “ingenious worker” or “worker with ingenuity.” Ingenuity is deﬁned as skill in contriving (as in ideas), skill in planning, inventing, etc., and cleverness.

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with rock. There were many problems with such designs and failures were common. No one had told the inlet not to move and therefore there were unexpected and unplanned breakthroughs and blockages, but eventually (by the 1960s) coastal engineers became quite proﬁcient at designing such inlets. They knew about stability of rock in and through the breaker zone. They were becoming familiar with the hydrodynamics (ﬂow) through inlets and the morphodynamics (the movement of sand and the change of shape) in and around such inlets. What began as a small, natural tidal inlet is now a small port city behind a “stable” tidal inlet. But there are many “new” problems, such as: • The original ingenious design had not correctly taken into account the movement of sediment along the shore (littoral drift), resulting in the collection of sand into a beautiful beach (accretion) on one side (the updrift side of the inlet), but because the sand can no longer move past the inlet, or is moved oﬀshore by the inlet structures, there is erosion of the shore downdrift of the inlet. • Much of the habitat for ﬁsh and shellﬁsh, which was the original reason for the development, has been destroyed through overﬁshing and degradation of the water quality from the industry and the ships.f 41.3.3. The tasks of a coastal engineer Once again, the small city behind the inlet wants to tender a contract to “stabilize the inlet.” This time it is more complex because a bridge was built across the inlet 30 years ago for the coastal highway. The bridge abutments are in danger of being eroded by the inlet, which has so far only been stabilized at the entrance, while the inlet channel is left free to move. Stabilization now means that the whole inlet channel will need to be armored to prevent it from migrating and destroying the bridge. A coastal engineer is hired to do the design and the problem can be readily solved by armoring the whole length of the channel and using greater reinforcements near the bridge. Problem solved? This is the 21st century. To obtain the necessary permits for the channel stabilization, many acts and regulations must be met, public meetings must be held, and stakeholders from various interests have been granted seats at the table. All their concerns need to be addressed and in fact, all these interested bystanders must eventually buy into the design. The actual engineering design with its calculations and ingenious synthesis of concepts is now only a minor part of the total task. Although the task of communicating about the design with the other players is at least as large as the design task, engineers are not trained very well in this aspect. Young people normally embark upon an engineering education because they are good at mathematics and are interested in building (designing) things. If they are lucky their education will ﬁll this expectation and focus on design (synthesis), which in turn requires extensive mathematics. If they are less fortunate, their training will have consisted of transmission of commonly used coastal engineering formulae and their education will have been more focused on analysis, which is what most f For example, an interesting history about the world-renowned New York oyster industry may be found in Kurlanski.6

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professors know best because that is what they do in their research. At the same time, students who were quite erudite communicators in elementary and secondary school (show and tell, science projects, compulsory language training, etc.) become less proﬁcient at both written and verbal communication during their engineering studies because it is apparently not required in their education and in a subsequent engineering career. As a result, most practicing engineers, because of their lack of training in communication and because they were educated to view their real work as design and application of mathematical expressions, are not interested in the communication part of the task. Yet the communication about a design must be done by the engineers. Any other type of “communication expert” would be severely handicapped to perform this communication task. They are not trained in the engineering and design concepts, which are not exactly easy to understand without adequate training. Reviewing the tasks of the contemporary coastal engineer of our example in more detail, she must ﬁrst be able to do the functional design of the inlet stabilization. This involves the application of knowledge about, for example: • Hydrodynamics: Tides, tidal exchange, wind generated waves, wave climate, long waves such as tsunamis and storm surge. • Sediment transport and morphology: Accretion and erosion along the sea shore, scour and deposition in the channel. • Design and analysis of inlet structures, scour protection around bridge foundations, channel revetments, etc. • Risk analysis about ﬂooding, storm surge, tsunamis, closing of the inlet, damage to ﬁsheries. • Environment: Impact, water quality, habitat. Because the engineer does not have detailed knowledge of all the above design aspects, she needs to communicate with experts in all the above areas where she has insuﬃcient expertise: climatologists, wave modelers, geologists, biologists, contractors, etc. This involves both multidisciplinary and interdisciplinaryg communication with the other scientists and engineers. Further, as part of contemporary functional design she must have an understanding and awareness of: • The social impact and limitations to the design, such as posed by land use, relevant laws and regulations. • Economic impacts: project costs, availability of people, construction materials and equipment, disruption of the existing economy, etc. But functional design is only the first part of the task of the engineer. The second task is to persuade the public and the various stakeholder groups to buy into the design and cooperate in bringing the design to fruition. That could involve g Multidisciplinary is deﬁned as interacting with related (science and engineering) disciplines, using each other’s expertise and understanding, while interdisciplinary is deﬁned as actually working together with related disciplines to solve a problem.

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among others: • Environmental groups who do not want further deterioration of the environment and who have concerns about “hard” channel protection. • Environmental activists whose only thesis is to leave nature alone at all cost. • Commercial ﬁshers, who are concerned, for example, that higher current velocities in the newly designed channel will endanger their boats and will prevent them from setting their nets. • Recreational ﬁshers/hunters, who do not want further deterioration of the wetlands high up in the inlet bay and who want to preserve the existing habitat for the ﬁsh and shellﬁsh and for the birds that feed on them (and for themselves as ﬁshers/hunters). • Taxpayer groups representing the residents in the erosion area downdrift of the stabilized inlet, who want to include upgrading of their eroded properties in the new design. • Parks authority and tourism operators who want to maintain their beautiful and lucrative beach areas updrift of the structures. • Surfers who want to maintain the unique surﬁng conditions. This part of the engineer’s communication task is transdisciplinaryh — the communication with planners, regulators, lawyers, and members of the public, whose cultures are very diﬀerent from engineering and who are at best nontechnical and at worst technophobic. Therefore, she must be able to express herself in common language and to listen to and understand diﬀerent points of view. 41.3.4. Design and decision making Figure 41.1 shows the decision making process of the past. The coastal engineer was hired produce a design. He incorporated whatever coastal science was needed, formulated, and designed the project, submitted the design to the decision makers and then, once the decision was made to proceed with the design, the engineer would supervise the implementation as agent for the decision makers, who were normally the project owners. Figure 41.2 shows the historical design process. The engineer would apply knowledge and data to make a preliminary design, which was normally modeled. A ﬁnal design would be produced as a result of the model studies. Then the design was implemented and possibly modiﬁed after implementation. The historical design process (Fig. 41.2) must be modiﬁed to include the steps needed to gain project approval. Figure 41.3 shows this modiﬁcation. It is clear that the approvals process is not simply attached to the earlier design process. It is an h Transdisciplinary communication for engineers is deﬁned as the interaction with disciplines who are not trained in sciences or engineering. From another vantage point: multidisciplinary and interdisciplinary (for engineers and scientists) means solving well-deﬁned problems; transdisciplinary involves solving ill-deﬁned problems, involving both the physical/environmental system and the socioeconomic system. Stabilizing an inlet is itself a well-deﬁned problem. Integration of the design into the socio-political context of regulators, politicians, and stakeholders surrounding the project makes it an ill-deﬁned problem.7

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Coastal Issue

Decision Makers Coastal Engineers

Project Formulation

Coastal Scientists

Project Design Implementation

Fig. 41.1.

Historical decision making process.

Desk Study Knowledge (Theory and Experience) and Prototype Data

Preliminary Design

Modeling

Fig. 41.2.

Design

Implementation

PostImplementation Design

Historical design process.

Desk Study Knowledge (Theory and Experience) and Prototype Data

Preliminary Design

Modeling

Design

Implementation

PostImplementation Design

Approval

Fig. 41.3.

Contemporary modiﬁcation to the design process.

integral and central part of the process, as shown by the arrows. Even information exchange at the time of the preliminary design is crucial to success of the project, as will be seen later. A sample of the decision making processes used at the present time is shown in Fig. 41.4. The heavy boxes show the two new players who have central roles in the

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Coastal Issue

Decision Makers Government

Coastal Manager

Law Coastal Engineers Project Formulation Alternatives

Coastal Scientists

Physics Chemistry

Theoretical and Empirical Relationships

Biology Geology Others

Modeling (uncertainties) Judgment Solution Governments

Approvals

Socioeconomic Input from Stakeholders

Nongov’t Orgs Interest Groups

Implementation

Citizens

Monitoring

Fig. 41.4.

Contemporary decision making process.

process: the stakeholders and the coastal manager. The stakeholders have already been discussed. The coastal manager is a central ﬁgure between the decision makers, the laws and regulations, and the design process. The coastal manager is ideally someone who communicates well with all the players and who can translate the information to the other parties so everyone understands each other. This position should be ﬁlled by someone with a background in science and engineering, since much of the translation involves the communication of socioeconomic facts to the technically trained engineers and translating technical details of the design to the socioeconomic partners, such as stakeholders, regulators, and lawyers. Yet, in practice, most coastal managers have little formal

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Stakeholders Different backgrounds, biases, risk perceptions and aversions, poor understanding of uncertainties

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Public involvement, Meetings, Press, etc

Pre-Engineering Knowledge: Theory, Experience, Some Data

Concepts Design Principles, Layouts

Resilience: Definitions, Requirements, Opportunities

PES Design: Theory, Experience, Data

System Design: Resilient PES, Resilient Interface Resilient SES

Communication

Decision Makers

Resilient System Fig. 41.5.

Contemporary design process.

technical training. Therefore, unless the engineers themselves are proﬁcient in translating their design into common terminology, while at the same time being able to understand the boundary conditions imposed by the socioeconomic framework, the project is headed for problems. The contemporary design process has become very complex. Figure 41.5 shows the complexity involved. It is clear that stakeholder and public involvement are integral parts of the contemporary design process, but it is very important that the stakeholders and public are part of the design process from the very beginning. They cannot be simply used as sounding boards because they must take ownership of the project and the later they appear on the scene, the more likely it is that they oppose the process and the project. It is essential to involve the public and the stakeholders even at the preliminary design stage, so that they feel like partners. This necessitates “pre-engineering.” No design has taken place at that time. But at the same time the engineers must understand the project well enough to be able to present the project outlines and central thrusts to the stakeholders. Then, with the subsequent input from the decision makers and stakeholders, the engineers work on a conceptual design. Recently, it has been realized that failures do occur for various reasons such as poor infrastructure design, rampant real estate development in dangerous areas, land subsidence, sea level rise, and global warming. Sometimes the failures can be disastrous and they often involve the coastal zone and coastal designs. Recent examples are the Indian Ocean Tsunami in 2004, the ﬂooding of New Orleans as a result of Hurricane Katrina in 2005, and the frequent ﬂooding of Bangladesh by monsoon runoﬀ and impact from cyclones.

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Even the inlet design in our example is subject to increased probability of failure because of: • • • • •

rampant real estate development close to the shore; land subsidence due to pumping of groundwater and hydrocarbons; a general decrease in alongshore sediment transport; increased exposure to storm surge as a result of global warming; habitat change in the ﬁshing areas.

Failure means that the system must be resilient. Resilience may be broadly deﬁned as: how does a system recover after failure? But the exact deﬁnition of resilience as it applies to a project depends on the socioeconomic context of the project, which very much involves the reactions and ﬂexibility of the local governments and populations. The next important design step after pre-engineering is, therefore, to deﬁne resilience as it pertains to the particular project and to explore the requirements as well as the opportunities to introduce resilience into the project. A detailed discussion of resilience is beyond the scope of this chapter. 41.3.5. System design In general, the project consists of two components. These are: — The physical/environmental system (PES). Design no longer involves only the physical aspects. Physical structures must be designed and built with due consideration of the environment in which it is placed. — The socioeconomic system (SES) within which the project must operate. The conceptual combination of the PES and SES for coastal engineering projects is shown in Fig. 41.6. The PES must be supported by the SES. In the simpler designs SES inﬂuence is small and limited essentially to the permitting process. In more complex designs, for example, the reconstruction of the New Orleans ﬂood protection, SES involvement is very large. The public and stakeholders must be involved in the discussions on the deﬁnition, requirements, and opportunities for resilience and they must be involved in the resilience design process itself. The system design sequence (the ﬁve middle boxes in Fig. 41.5) requires great communication skills (also listening skills) on the part of the engineers.

Fig. 41.6.

System representation.

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The engineers must also take on the diﬃcult task of simultaneously being leaders (in the design process) and servants of the public (in meeting regulations and designing a resilient system). It is quite clear that most engineers are not capable of doing this and, in fact, engineers are generally not interested in the communication/leader/servant tasks. This must change! Engineers must begin to view these expanded tasks as challenges and opportunities, instead of as threats and onerous burdens.

41.4. Engineering Schools/Universities The above section has shown that communication and leadership have become very large elements of the design process. But experience indicates that engineers’ facility with tasks not directly related to design calculations is insuﬃcient. To improve the training of engineers we must look at the universities and engineering schools. In this section we examine the engineering school/university systems to see what is feasible and how we can bring about improvement. An engineering professor’s task within the university is a combination of teaching, research, and administration. These three elements compete within the university and vie for each professor’s time and interest. University administration is not discussed in detail here, but engineers in high university administrative positions should be able to eﬀect important changes to the engineering curriculum, if they understand what changes are needed. But university administrations and also the engineers within administrations have not made many changes. Changes that would improve the quality and usefulness of the engineering graduates are quite distant concerns as the universities struggle to stay ﬁnancially viable within a competitive global market. The other two (very closely related) aspects of a professor’s task — research and teaching — are discussed in more detail below.

41.4.1. Engineering research 41.4.1.1. Deeper and narrower We will use another example: An engineer was attracted by a reputable university in the 1950s to teach coastal engineering. He was part of the ﬁrst generation of coastal engineering academics. We will call him GEN 1. He was an engineer hired from engineering practice and appointed into academe. We should call him Prof. GEN 1, P.E. He gave a broad, general engineering education to prepare his undergraduate students for the world “out there.” GEN 1 trained a total of six graduate students (GEN 2.1–2.6), having chosen them from the brightest undergraduate applicants. This unique treatment gave these students higher level technology and skills in addition to a general engineering education, so they could become specialists and leaders in their ﬁeld or perhaps teach the next generation of students. GEN 2.2–2.6 indeed became leaders in industry, engineering consulting, and government; GEN 2.1 decided to become a second-generation engineering

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professor. He trained a total of 28 graduate students of the next generation (GEN 3.1–3.28). GEN 3.8–3.28 went into industry, consulting, and government. GEN 3.1–3.7 elected to become academics and they supervised 30 graduate students each (GEN 4.1–4.210). GEN 4.31–4.210 went into industry, consulting, and government and GEN 4.1–4.30 chose to become academics. By now, there was an oversupply of potential coastal professors and GEN 4.1–4.30 had to endure that apparently inﬁnite holding tank called post-doctoral training — legitimized work camps constructed to hold, sort, hire, and reject candidates for university positions. This development is traced in Table 41.1. Research within an academic setting is essentially a reﬁnement process. Each generation digs deeper and works out the knowledge of the previous generation in greater detail. Because it is not possible to dig deep over a broad area, it is necessary to divide the research topics of the previous generation into smaller units. This results in a narrowing of the coverage by each researcher, as well as a deepening of the research subjects with each generation as shown in Fig. 41.7. This is reminiscent of the parallel development of the engineer technical expert silos discussed in Sec. 41.3.1, but the widths of the ﬁelds are much narrower in the research area. And surely the two developments feed on each other. Table 41.1. Exponential increase in engineers and professors. GEN

Engineers

Professors

1 2 3 4

1 5 24 203

1 2 9 39

Generation

Understanding

1

Broad

2

Narrower and Deeper

3

Narrower and Deeper

3.1 3.2 3.3 Fig. 41.7.

3.4 3.5 3.6

Evolution of knowledge with generations.

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Research in the sciences follows a moderni trajectory, which is based on the following social dynamic. At any one time there is a body of knowledge that is accepted to be true by consensus within the scientiﬁc community. Research consists of the search for new scientiﬁc principles by starting out from this scientiﬁc consensus. Once new hypotheses are proven, the research results are submitted to be published. They are essentially subjected to an initial process of truth-ﬁnding to see if a new scientiﬁc consensus can be built, supposedly on a higher plane of knowledge. This initial truth-ﬁnding step is normally entrusted to peer review, a process used extensively by the editorial boards of the scientiﬁc journals. Peer review determines what new articles will be published for the scientiﬁc community at large as a ﬁrst step in achieving a new general consensus on a topic. Thus, the “scientiﬁc method” builds up a body of scientiﬁc knowledge through research and consensus, which in turn is based on earlier scientiﬁc consensus. And the modern view is that this scientiﬁc process will eventually lead to truth. Engineering research at universities uses the above science research dynamic as a model; it follows a similar modern trajectory. The search for new engineering principles also attempts to move from one level of consensus to a next higher level of consensus.j In fact, it is diﬃcult to distinguish engineering research from science research, and much of what is called engineering research is essentially science research in that it focuses on analysis, rather than on design (synthesis). Engineering research in the universities is similar to science research because they are judged and funded by the same criteria. Funding for research is achieved through research proposals, which are judged and funded on the basis of what is purported to be another peer review process. But grants committees normally use the number of journal publications as the prime indicator of academic prowess. Only in that way can a committee simply crunch some (assumed to be hard) numbers to come up with a simple evaluation of the merits of the proposal. Any other criteria, such as outstanding engineering work performed, are normally considered to be too subjective and too time consuming to evaluate. This process also tacitly assumes that the quality of the publications is controlled by the journal editors and that all the authors listed in the chapter headers contributed equally. Is this really a thorough peer review, or does this simply transfer the value judgments from one review process to another? The workers on the research grants committees are indeed peers of the applicants. But that is only because these positions are poorly paid or not remunerated at all. Therefore, serving on these committees is only of interest to academic peers i Modern, a sociological term. A modern research path is based on the belief that with proper scientiﬁc method, given enough time and funding, we will eventually understand “the laws of nature” and be able to solve all the world’s problems. The postmodern reaction to this recognizes that the modern methods will not be able to reach these goals and that the modern premises are false: for example, that there are diﬀerent viewpoints, that there is no single voice of truth (pluralism), and that uncertainty will continue to play a large role, even a key role, in our process of discovery.1 j In view of the narrowing of research ﬁelds the “community” around each research subject becomes quite small and hence “consensus” is between a few individuals. This leaves much opportunity for shortcuts in the truth-ﬁnding process and for cronyism among the individuals involved.

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who can themselves beneﬁt from the system or have beneﬁted from it.k And certainly it is diﬃcult to ﬁnd peers from engineering practice, who are not rewarded by the process and who base their commitments on service to clients. Promotion and salary committees at universities normally function the same way. Although they have more time to use more subjective quality assessment methods, they also use the number of journal publications as the basic indicator of merit for academic appointment, tenure, promotion, and salary increase. Thus, there is essentially only one review process — the review process to determine if a paper should be published. And it was shown to have shortcomings. The other reviews simply use the results of the publication review process. The researcher with the most publications is almost automatically considered to be the most valued, is promoted, and receives the largest salary increases based on “merit.” This method of assessment for research grants, promotion, and salary increases puts faculty under great pressure to publish as many papers as possible. That is as true for engineering faculty as for the other academics. Thus, engineering professors are rewarded for producing papers, rather than for solving engineering problems. Such a system deepens understanding of engineering or scientiﬁc principles, the analysis side of engineering, but it does little to advance knowledge about design (synthesis). There is some encouraging movement in the right direction and there are exceptions. In Canada, for example, the national granting agency (NSERC) asks the applicant for: • The ﬁve most signiﬁcant (lifetime) contributions to the profession — not papers, but all work within the profession, the academic world, or social contexts of the engineering. • Other evidence of impact — engineering designs, reports, patents, etc. • Contributions to the training of highly qualiﬁed people — not only graduate students, post-docs, and undergraduate students, but also technicians and technical experts, other academics, and engineers from practice. Other similar improvements no doubt occur in other jurisdictions. There are some other concerns with the present research methodology. a. The drive to obtain longest possible list of publications (and hence the research funding and the promotions) has some undesirable side eﬀects: • It pays to work in a research niche where one can produce papers quickly. This encourages, for example, papers based on numerical modeling, rather than ﬁeld work and physical testing. • It pays to split a piece of work up into as many papers as possible, published serially and in diﬀerent journals and conference proceedings, rather than producing one valuable deﬁnitive paper at the end of a research program. Surely, this seriously diminishes the real value of a publication. • It pays to attach as many authors’ names as possible to each paper, so that as many people as possible can legitimately claim authorship and add the kA

temptation for nepotism?

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paper to their list of publications. Surely, this seriously diminishes the value of authorship. • It discourages venturing into really new, but uncharted areas because really new ideas do not ﬁt well into an accepted consensus pattern and because they may lead into blind alleys and produce “useless” research. None of the above practices are conducive to good research and, in fact, they are an expensive waste of everyone’s time. b. There are problems with research funding itself. For example, when large research grants support large research projects or research centers of excellence, they generate a feeling of ownership of the problem. The research group feels that no other research groups should be given much funding, since that would in their opinion dilute their eﬀort (and it might lead to competition). This results in political moves within in the research community to protect turf. Doidge8 describes this internecine warfare in brain research; Venter9 describes similar protectionist politics in connection with the mapping of the genome. These are examples from other disciplines, but surely this also occurs within the coastal research community. c. There are few papers describing practical applications, designs, and case studies in engineering journals. Practicing engineers have no incentive to write papers for engineering journals. It does not inﬂuence their funding, salaries, and promotions, as it does for academics. Their advancement is based on the quality of work they perform for clients and on hours of billable time. Since practicing engineers also have little incentive to join editorial boards and peer review exercises, there are few members from engineering practice on most peer review committees, which means there is little support for practical papers on such committees.l d. With the extreme pressure to publish results, the GEN 3 academics must participate in the positive feedback loop of publications → research funding → more publications → more research funding, eventually leading to tenure or promotion. Hence, they cannot aﬀord much time or interest to study and work on aspects of engineering that will provide a broader and more valuable education to their students, since this does not directly contribute to their research output. Nor can they aﬀord to spend time on research of practical engineering applications, which were GEN 1’s strength. e. Engineering education, research, and innovation has been forced into a science type of peer review straightjacket so that the very meaning of engineering (ingenuity) has been immobilized and innovative designs and their impacts are only discussed in largely hidden and often proprietary design reports, not in the peer reviewed literature, where it could beneﬁt other engineers. This is, of course, a description of central trends. Many encouraging changes are being made to this restrictive system. Many individuals refuse to follow the system slavishly. Many academics are dedicated to producing good and valuable research and to working very hard and equitably on peer review boards. And even more l Hence, research journals receive few practical papers to review, so they publish few practical papers. Therefore, few practicing engineers read the research literature and practice and research drift further apart.

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encouraging, many academics dedicate their lives to educate the students to make them competent engineers. 41.4.2. Teaching engineers Teaching engineers is essentially based on an extrapolation of ideals and education theories that are over 100 years old and are severely restricted by present day market concepts. Traditionally, the university hired competent practicing engineers and appointed them as professors, as GEN 1 above. This person “professed” his knowledge (always a he) to the students through the true and tried method of lectures. In fortunate cases, the lectures were adequately supported by enlightened and enlightening laboratory sessions and tutorials. The students would take notes and those notes became their handbooks as they went out into practice. There were also unfortunate instances where a professor simply rambled on with his “wisdom” and made no eﬀort to accommodate the individual students’ needs. Today this traditional lecturing system is being updated, but in most cases, the lecture format is still the main vehicle for knowledge transfer. And it is very eﬃcient. It permits the transfer of the greatest amount of information, in the least time on the part of the professors as well as the students. Lately, the method has been criticized and sometimes with good reason. But in fact, properly presented lectures backed up by good graphics, good handouts, good testing, laboratory work, and tutorials cannot be surpassed. The criticisms of lecturing are mostly based on cultural phenomena. The arguments are as follows. It is very diﬃcult for students to concentrate on a one-hour lecture because their attention span has been shortened through video, television, and the Internet. Further, today’s students are accustomed to being coached (taught by doing), rather than by listening and reading. And modern students are used to multitasking. Therefore, modern teaching methods should take advantage of these attributes.1 It was shown in Sec. 41.3 that engineering education needs to be updated to prepare the engineer to solve today’s problems in a contemporary fashion. In addition to thorough familiarity with technical details and facility with design, the engineer must posses a number of skills. Most important of these newly needed skills are communication and leadership. Unfortunately, the engineering educational system prefers to focus on the technical subjects, and it is good at that. Professors overﬂow with technical information, particularly when it is close to their speciﬁc research niche. So our engineering graduates come equipped with the latest technical knowledge, but often in a relatively narrow area. They may know very much about waves, but have little facility to apply this to design of structures, or to analysis of sediment transport. They may know much about design of structures, but do not really understand wave dynamics. They may know about the most sophisticated numerical modeling technology, but have little idea how to evaluate the model results. They often lack knowledge of complete areas of engineering involvement, such as infrastructure renewal, because they were not taught about infrastructure renewal, only about design of new structures. In general, engineering students have

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had little training in communicating their technical work to others (nontechnical people, or people in related disciplines). 41.4.2.1. Course example I was fortunate to be able to conduct an undergraduate coastal engineering course from 1968 to 2001. From the beginning, this course attempted to mirror engineering practice. The problems handed out each week were not simply sample design calculations — they were actual designs. The course consisted of two hours of lectures, three hours of tutorial, and countless hours of group work each week for 12 weeks. The tutorials were real tutoring sessions. The students were encouraged to ask any questions they had about the assignments and they would even get the ﬁnal answers from the tutors. But their questions were answered only after they answered the tutors’ question: “What do you think?” In other words, the students were expected to think thoroughly and clearly about the problem and about the missing pieces of information before they asked for help. But they could solve most of their problems by proper communication with the resource staﬀ. The tutors were graduate students in coastal engineering. During the ﬁrst lecture, the class (usually about 25 students) was divided into groups of three or four. These were the “engineering companies.” To assume ownership, the students were asked to think of an appropriate name for their company. They were also asked to appoint a CEO. The companies remained the same throughout the semester. This exposed the diﬀerences in individual approaches to problem solving within the group, necessitated accommodation of diﬀerent people, ideas and insights, and resulted in learning from each other. The company appointed a new CEO each week who was expected to show leadership by taking charge of the weekly project and conducting all formal communication. The ﬁrst assignment was also handed out on this ﬁrst day. It was typically, “for such-and-such site, prepare a proposal to design a small harbor.” The site and tasks of the assignment changed every year. In other years the students might be asked, for example, to determine the rates and extent of coastal erosion and mitigate the erosion. The proposal had to include cost estimates and timelines. It was due in one week, to be presented in written form to the professor and as a verbal, illustrated presentation by the CEO to the rest of the class (their peers) and to the tutors and the professor. Of course, none of the students knew anything about coastal engineering at this point and so to put together their “proposal” they were simply encouraged to use the library and their brand new course notes, and to ask questions.m In the last week of class, the companies presented their ﬁnished design, again in writing and orally by the CEO to the class. The diﬀerences between the initial proposal and the ﬁnal design formed excellent feedback for the students of what they learned in the course. The weeks in between the proposal and the ﬁnal design presentation were primarily ﬁlled with lectures and tutorials on technical design details that they needed to complete their overall design. Each week a short written m Now

of course the Internet would be a huge source of information.

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design report was expected, which was sometimes also presented orally. The periodic presentations to peers and the presentations at the beginning and the end were very good for the students, in that they functioned as students, teachers, and critics at diﬀerent times. The topics treated in the intermediate weeks were wave theory, wave statistics, wave calculations, long waves and water levels, wave transformation, design of coastal structures, coastal management, sediment transport, and physical and numerical modeling. The tutorials complemented the lectures. Two tutorials were two laboratory sessions. In one session the students used a large wave ﬂume to learn about waves, wave dynamics, laboratory methodology and equipment, and analysis of model results. In the second laboratory session they built a cross-section of their breakwater design and tested it to destruction. Further details about this course and the course material may be found in Kamphuis,1 Introduction to Coastal Engineering and Management. This basic coastal textbook is essentially an edited and somewhat extended version of the course notes. The textbook was written at the repeated request of past students and tutors. The book has been adopted in coastal engineering courses around the world and thus some of the above course philosophy is trickling into other coastal courses to add to similar innovation taking place in many diﬀerent ways in many schools. The approach in this course attempted to correct some of the misalignments of education and practice of that time. The success in meeting the course objectives varied from year to year, but the objectives always formed the framework in which the course was presented. It exposed the students to the working environment like they would encounter once they graduate and enter engineering practice. Some examples: they worked in groups, they were forced to work with the same colleagues and with their strengths and weaknesses, they needed to negotiate with these colleagues, they communicated their work through reports and presentations to a critical audience (their professors and their peers) and they had to make decisions based on incomplete information. All these involved listening. The students listened to the other members of their company, to their peers and professors in the critiques of their presentations, and to their professors and tutors in lectures, laboratory sessions, and tutorials. Not everyone worked well in this environment. Some students in their exit comments noted that this was the best course (or even the only course) that in their opinion came close to real engineering. Others ﬂatly stated that it was the worst course they had ever attended — the professor taught them nothing. If it were not for their own hard work and leaning on their colleagues, they would have never passed. Such a course has obvious advantages in that it teaches communication, leadership, and teamwork. But it also has two negative aspects: time and evaluation. There is no doubt that such a course requires much more time from everyone — the professors, the tutors, and the students. In an age when eﬃciency counts, it is difﬁcult to convince all the actors that the extra eﬀort is worth it, or even to ﬁnd the time required to teach and learn in this way. Professors are forced by the system to be very engaged in research as was seen in the previous section. Since research publication is the motor for promotion, etc., the professors prefer to spend less time in the classroom. The students have to work for other courses, which probably also

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call for more time. And the students are less prepared to spend long hours on their coursework than students 30 years ago. 41.4.2.2. Student evaluation The evaluation of individual students in a group work environment is problematic. Who did the work? Who is simply coasting? Does everyone in the group deserve the marks given for the projects? Are some bright and especially hardworking students not suﬃciently recognized or held back by the rest of their “company”? Can all this be solved by a strict individual examination at the end of the course? The subject of evaluation can be readily extended beyond this one course. We need to reconsider how and what to evaluate. In the present environment, universities are rewarded for the number of students they graduate and there is constant pressure to pass as many students out of ﬁnal year as possible. Failing students is frowned upon by university administrations. So if all students are expected to pass, exactly what and how should we evaluate? Perhaps this is a great opportunity to stop giving students marks and to concentrate on educating them. Let the market (the prospective employers) determine how successful they will be. The problems with that approach are threefold: (1) If university entrance conditions were stringent, it might be reasonable to assume that students who passed that restriction can be expected to pass, unless something goes drastically wrong. But university entrance conditions vary greatly and have been lowered in recent times. They are therefore no indication of eventual success (unless the university applies its own rigorous entrance screening). (2) Prospective employers expect easy numbers and academic transcripts on which to base their hiring decisions. (3) How are scholarships and bursaries to be decided?

41.5. The Future? 41.5.1. Integral education ASCE4 entitled The Vision for Civil Engineering in 2025 states the following about civil engineering education: The civil engineer is knowledgeable. He or she understands the theories, principles, and/or fundamentals of: — Mathematics, physics, chemistry, biology, mechanics, and materials, which are the foundation of engineering — Design of structures, facilities, and systems — Risk/uncertainty, such as risk identiﬁcation, data-based and knowledge-based types, and probability and statistics — Sustainability, including social, economic, and physical dimensions

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— Public policy and administration, including elements such as the political process, laws and regulations, and funding mechanisms — Business basics, such as legal forms of ownership, proﬁt, income statements and balance sheets, decision or engineering economics, and marketing — Social sciences, including economics, history, and sociology — Ethical behavior, including client conﬁdentiality, codes of ethics within and outside of engineering societies, anticorruption and the diﬀerences between legal requirements and ethical expectations, and the profession’s responsibility to hold paramount public health, safety, and welfare. Although all the points are important, some are traditionally viewed as primary (e.g., scientiﬁc foundations and design) and some are considered secondary (e.g., risk/uncertainty, sustainability, and ethics). The earlier sections of this chapter show that in a contemporary engineering environment we do this at our peril. Knowledge about risk/uncertainty and sustainability is essential, as is knowledge of public policy, business and the social sciences, ethical behavior, health, and safety.n The above list must not be viewed as a shopping list. It must represent diﬀerent facets of an integral knowledge diamond. Civil engineers also need skills beyond the basic technical education. ASCE4 also discusses the skill set needed by civil engineers: The civil engineer is skillful. He or she knows how to: — Apply basic engineering tools, such as statistical analysis, computer models, design codes and standards, and project monitoring methods — Learn about, assess, and master new technology to enhance individual and organizational eﬀectiveness and eﬃciency — Communicate with technical and nontechnical audiences, convincingly and with passion, through listening, speaking, writing, mathematics, and visuals — Collaborate on intra-disciplinary, cross-disciplinary, and multi-disciplinary traditional and virtual teams — Manage tasks, projects, and programs to provide expected deliverables while satisfying budget, schedule, and other constraints — Lead by formulating and articulating environmental, infrastructure, and other improvements and build consensus by practicing inclusiveness, empathy, compassion, persuasiveness, patience, and critical thinking. Unfortunately, such skills are often poorly taught and mostly considered as addons to the real engineering education. Thus, we normally teach and learn real engineering in a core curriculum (often consisting of the professors’ favorite subjects) and then we teach and learn skills in separate courses, such as: — — — —

elective courses; masters courses in management and business administration; periodic professional courses after graduation; professional in-house training after graduation.

n Risk,

uncertainty, and sustainability concerns are actually rapidly rising.1

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But it is very important that the skills be learned as part of the daily routine, both while doing the coursework as students and while working as engineers-intraining (and as practicing engineers). The skills must be seen as more facets on the same knowledge diamond. Only if the skills are taught integrally with the knowledge courses will skills become part of the engineering toolbox and not something that was taught in isolation and is viewed as belonging in a separate compartment. This integration of skills and knowledge is diﬃcult to accomplish in engineering classes where the professor and the students are driven by a love for innovation, implementation of exciting ideas, and use of mathematics.o The above analysis only covers strict minimum knowledge and skills requirements. We also need to return to the early 20th century model, where engineers were trained in the arts and humanities for a well-rounded education. Today’s engineers need a basic knowledge about, for example, history, philosophy, psychology, language and culture, music, and the arts. This arts and humanities education provides a greater balance. Students will have been exposed to important concepts such as truth, belief, and reality, and they will better understand the social context of their projects. They will also relate better to the stakeholders and their concerns. With respect to skills education: skills are often taught subjectively. For example, communication courses are often feel-good courses in which students must be encouraged. But skills courses should clearly also meet standards. For example, communication in university or engineering practice is more than show and tell. Typical student presentations are: “this is what we did; we used all the correct formulae and here are the fantastic results” or “we did not achieve our goals but we did the best we could.” Engineering students (as well as practicing engineers) cannot simply assume that their audience is in agreement because their reasoning includes the best technical knowledge available. They need to know the alternatives as well as their solution. They need to be able to explain that there are no ultimate solutions, only optimum solutions. They must be able to identify clearly what are the relevant parameters and boundary conditions within which they made the decisions. They must learn to argue their case for various audiences and make their presentations ﬁtting and convincing. Such communication should use, or at least refer to contemporary communication tools such as the Internet, numerical models, and computer games. It should introduce approaches to negotiation and development of win–win strategies, where necessary. Evaluation of presentations and reports needs to account for all these details, rather than simply giving an averaged feel-good, encouraging assessment. And communication is just one example of a whole suite of necessary skills to be so taught and evaluated. The above discussion indicates two basic scarcities: time and people. a. Time. A breakwater design calculation can be made in hours. Preparation of a presentation of relevant design parameters can take days. Preparation of case o The author noted consistently over many years that everyone was interested and paid attention when formulae were presented, but eyes quickly glazed over when design philosophy, coastal management issues, environmental impacts, etc. were discussed.

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studies needs weeks. Where will this additional time come from? To make time: • Engineering curricula must be reviewed once again. To do this, engineering practice and engineering education must cooperate closely and on an ongoing basis through committees, seminars, workshops, etc. This needs more than lists of good ideas from the engineering societies and accreditation boards. Roadmaps must be developed on how to implement the necessary changes within the practical boundary conditions such as posed by the global economy. These roadmaps must be presented to and argued with both engineering practice and education, who must both be involved in their implementation. The results of the workshops and seminars will need to be implemented quickly, rather than decades later. • Engineering curricula need to be extended. It will be diﬃcult to convince the government departments of education of this necessity and this again requires cooperation between engineering institutions and educational institutions. Above all, the revised and extended engineering curricula must emphasize that all the knowledge and skills are facets of one integral knowledge/skills diamond. b. People. Such reformed education needs special people. For example, it needs teachers with experience in two-way communication of information as it pertains to engineering projects. Where do these people come from? They must come, among others, from the engineers who have experience with this. 41.5.2. Cooperation The above sections show that much greater cooperation is needed between university and engineering practice. Who pays for this? Section 41.4 showed that the existing (mostly market-driven) emphasis by universities on production of papers discourages production of well-rounded engineers. Yet the universities have few alternatives. On the other side, engineering employers (engineering ﬁrms, governments at all levels, industry, innovation networks, etc.) compete in a global market, where engineering contracts are normally based primarily on cost and only secondarily on quality. To keep engineering costs down, the engineering staﬀ must work eﬃciently and many decisions revolve around “billable time.” Thus, engineering practice and education are both tied down by market requirements. Unless a win–win scenario can be produced, there will be no progress on ensuring better engineering education. Whether or not we agree with this domination by market parameters, it is a fact of life. Yet, in spite of the huge inﬂuence of market forces on our decisions, we must work toward a social dynamic within this market system that values attributes beyond simple $ costs. We must assert, for example, that producing good engineers takes priority over eking the last $ out of the system through maximizing billable time, or maximizing university income through building more eﬃcient research machines. This means it is time for engineers to take back the management of engineering from masters of business administration, accountants, and lawyers. On the academic side, engineers need to take back engineering education from the grip of university

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managers who slavishly follow a system driven by research publications. Such a revolution calls for well-developed leadership capabilities. The above is, however, mere saber-rattling, unless we can present innovative solutions. Where is the room to create a win–win situation? In a strict market economy, the only resort is to show that better education of engineers makes good business sense.p Universities realize that success of its engineering alumni in practice is what drives the reputation of their engineering schools. Excellent graduates bring prospective (particularly high ﬂying) employers to the universities. They will also generate more ﬁnancial support from engineering practice. But that message is lost somehow among the research and publication hype and must be delivered and reiterated loudly and clearly by the practicing civil engineers. And for the universities to feel rewarded for producing good engineers, they must be able to identify the results in $ returns. On the engineering side, surely it is no secret that better engineers produce better work, which means enhanced reputation, more work, and greater proﬁts for engineering ﬁrms, more public conﬁdence and lower costs for governments and industry, and greater innovation for knowledge networks. For example, just a few good engineers who can design and present a project properly can make the diﬀerence between obtaining project approvals on schedule, and endless delays and litigation. Such messages must be clearly stated and packaged for engineering management’s consumption, so that ﬁnancial calculations in the head oﬃces of engineering ﬁrms go far beyond simply keeping engineering costs minimum. There are a number of improvements that can be made through close cooperation between engineering practice and education. 41.5.2.1. Better engineering The meaning of better engineering is not clear. It must be carefully deﬁned by teams consisting of both practicing engineers and academics. Once deﬁned, it must be ushered into engineering practice and university together. This involves the workshops and seminars discussed in Sec. 41.5.1, but also needs direct contacts between policy-makers for civil engineering practice and for the universities. 41.5.2.2. Evaluation It is essential that the present systems of evaluation be modiﬁed and broadened. Evaluations to be reviewed include evaluation of: • • • •

professors primarily by number of publications; research quality primarily by number of publications; universities primarily by the quality of research; engineering contracts on the basis of $ cost;

p This is parallel to the struggle to convince business that green (concern for the environment) is better for the bottom line than only black (singular concern for $ proﬁts). That battle has been quite successful. Recently, business converts to a newly-minted green attitude are marketing their new green image with vigor because it is now perceived that it is worth $$.

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• engineers by billable hours; • graduates by marks and transcripts. This task is very diﬃcult, but time has come for this type of renewal in both engineering practice and education. 41.5.2.3. Interaction on the shop floor Engineering students, professors, and administrators, such as deans, must regularly spend time at practical engineering. Such professional internships and sabbaticals will: • Generate understanding of industry in the academics — most importantly that practical engineering requires an integral approach. • Help the practicing engineers understand what engineering schools do. • Facilitate direct knowledge transfer. The latest research results will be brought directly onto the industry shop ﬂoor; the latest practical engineering problems will be shared directly with the academics. The engineers will have opportunity to help formulate the research programming of the academic and the academics will have the opportunity to involve themselves in solving practical engineering problems. • Inﬂuence the type of research embarked on by the universities. • Inﬂuence the teaching in the engineering schools and the type of graduates the schools will produce. • Encourage the practicing engineers to take advantage of lifelong learning opportunities. • Encourage the academics to question if the science research model ﬁts engineering practice and encourage industry to become more involved in university research and in the politics of distributing research grants. • Strengthen ties between individual academics and engineers. Practicing engineers must also regularly spend time to attend and give lectures, seminars, and workshops at the universities, and to be involved in curriculum updating. Such educational leaves will: • Expose the students to real life problems, points of view, and solutions. • Expose engineers to today’s students and their environment. • Establish direct contacts between the students and industry. This would facilitate the job marketplace — job postings and hiring, arrangement of student work terms, etc. • Strengthen ties between individual academics and engineers. A valuable addition would be an oﬃcial system of cross-appointments, where engineers are appointed at the universities as “Engineer in Residence” or as “Visiting Engineering Professor,” with similar oﬃcial appointments of academics to industry. Engineering schools must develop more project courses in which students work on current projects brought into the school by practicing engineers and supervised

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by an academic and an engineer. Such engineering interactions must be established at the undergraduate, post graduate, and faculty levels. Finally, familiarity with industry needs to be initiated in the students through regular ﬁeld trips to actual industrial and construction sites. Much of this is already being done at various levels in many jurisdictions and many such activities have already existed for many years. These eﬀorts need to be encouraged, rather than curtailed because of funding or because other aspects of education are considered to be more important. Clearly, this will cost money to both the industry and the universities. Since both have to function within the global market framework, these are major challenges faced by both parties. But the alternative is not an option because the present poorly functioning system will deteriorate into a dysfunctional system. Unfortunately, it is easy to see the advantages of these interactions, but to obtain support for this cooperation, the intrinsic values must be translated into $ values. It must be shown that engineers who can communicate well and are aware of the latest research, and professors and students who have industrial contacts and experience can save millions of $, for example, because stakeholders will be better informed through them on the real issues and therefore ﬁnd a project less objectionable. It is easy to show that an engineer does not meet her quota of billable time because of an educational leave, but demonstrating the beneﬁt of better quality engineering graduates that come out of a revised curriculum, perhaps shaped as a result of someone’s professional sabbatical or educational leave, is very diﬃcult. Similarly, showing that a professor’s paper production has decreased because he spent more time in practice is easy. Demonstrating the beneﬁts to the professor’s expertise, stature, and quality of his graduates is much more diﬃcult. We are waiting with great anticipation for the first university that pays much more for professional sabbaticals than for research sabbaticals and bans stay-athome sabbaticals.q We are waiting for more engineering schools to include a closely supervised practical engineering year in their curricula. We are waiting for engineering ﬁrms to set up a ﬁnancial support system so students become familiar with their world “out there” through ﬁeld trips and can spend time learning practical engineering in their shop on a regular basis. 41.5.3. Thinking big Traditionally, civil engineers were master builders of infrastructure. They were cognizant of technical aspects, safety and health implications, etc. of their constructions. The latest vision of civil engineers4 sees the civil engineer as: Entrusted by society to create a sustainable world and enhance the global quality of life, civil engineers serve competently, collaboratively, and ethically as master: — planners, designers, constructors, and operators of society’s economic and social engine — the built environment; q And of course we need to encourage sabbaticals and leaves in educational systems and engineering practice in which they are absent.

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— stewards of the natural environment and its resources; — innovators and integrators of ideas and technology across the public, private, and academic sectors; — managers of risk and uncertainty caused by natural events, accidents, and other threats; — and leaders in discussions and decisions shaping public environmental and infrastructure policy. Not much has changed in the basic concept. But the mandate has broadened, mainly through concern and involvement of a much stronger socioeconomic presence in the design process. And there are large implications in this updated mandate: • Engineering education must be integral for engineers to acquit themselves of all these tasks eﬃciently. • Can engineers aﬀord to take upon themselves all these responsibilities in a litigious society, where any involvement in any aspect of design and construction exposes the engineer to being targeted by a voracious legal system? • Can engineers aﬀord to shrink away from these tasks, just because they will be exposing themselves to litigation? They are the technical experts and they are ethically bound to ensure that projects are safe and environmentally and socially responsible. As Fig. 41.5 shows, engineers must be actively involved in predesign, in the discussions of concepts and resilience, in the design of the PES (structures and how they ﬁt into their environment) and in the integration of the PES and the SES within which it must perform. Properly educated engineers can also uniquely function as facilitators in discussions with stakeholders, owners, and decision makers. Young engineers must be educated to be competent for all these varied and exciting tasks, rather than be trained to enter a market only as commodities with speciﬁc, limited technical expertise to be hired by management teams that have little knowledge of the technical aspect of the designs and the risks involved in building (or rebuilding) the projects. Since active involvement in the whole design process involves very high exposure to litigation, a vitally needed skill, in addition to the excellent list of skills in ASCE4 is a thorough understanding, ability, and conﬁdence to deal with the legal system. This involves due diligence and appropriate standards of care, and understanding risks of exposure to opportunistic (global) legal predators. This will mean long meetings with company lawyers to make sure that you, as engineer, understand the lawyers and to make sure that the lawyers have a clear idea about what is involved in engineering. For example, lawyers, who are only comfortable with 100.00% certainty, must be educated about risk and uncertainty in engineering and on how we deal with uncertainty. Engineers must also be able to function competently as expert witnesses. Unless civil engineers can think big and conﬁdently prepare for the risks that come with their work, civil engineering will reduce to menial servitude, rather than the exciting, innovative, and productive life that it was and can still be. This again clearly requires leadership skills.

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Engineers must also become involved in the legislative/political process. Instead of only reacting to legislation, such as environmental legislation, engineers should also be vitally involved in forming it. This point is also made by Galloway.5 Finally, the traditional trust in engineers is gone. As a result we now must explain and justify every step of our work in detail and in understandable language. But let us think big and regain that public trust by producing competent engineers and ethically responsible, socially acceptable, sustainable, environmental-friendly designs. 41.6. Conclusion We live and work in challenging times. The contemporary tasks facing engineers are very much more interesting than ingeniously putting together building materials in designs of structures. We must design PES that ﬁt integrally and function eﬀectively and sustainably within a very complex SES. Many designs are also concerned with infrastructure renewal. This is much more diﬃcult than starting with a clean slate to design new PES. What challenges and how interesting! But engineers must be motivated to see these very complex tasks as challenges and opportunities, rather than as extraneous drudgery. And engineers must be educated appropriately to be able to execute these very complex tasks. That involves major reorganizations of civil engineering practice and education, within very stringent global market constraints and within a short time frame. This can be done only through close cooperation and mutual respect between civil engineering practice and civil engineering education. The future? The future is ours to make. References 1. J. W. Kamphuis, Coastal engineering — Quo vadis?, Coastal Eng. 53, 133–140 (2006). 2. ASCE, Civil Engineering Body of Knowledge for the 21st Century (American Society of Civil Engineers Publications, Reston, VA, USA, 2004). 3. ASCE, Preparing the civil engineer for tomorrow by raising the bar, Civil Engineering (American Society of Civil Engineers Publications, Reston, VA, USA, 2007), pp. 64–71. 4. ASCE, The Vision for Civil Engineering in 2025 (American Society of Civil Engineers Publications, Reston, VA, USA, 2007). 5. P. D. Galloway, The 21st Century Engineer — A Proposal for Engineering Education Reform (American Society of Civil Engineers Publications, Reston, VA, USA, 2007). 6. M. Kurlanski, The Big Oyster (Ballantine Books, USA, 2006). 7. F. Mercks, Dutch coastal defense research, summary and conclusion, Internal Report, Rathenau Instituut, The Netherlands (2007). 8. N. Doidge, The Brain that Changes Itself (Viking Press, Penguin, USA, 2007). 9. J. C. Venter, A Life Decoded (Viking Press, Penguin, USA, 2007).

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