PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS
FURTHER TITLES IN THIS SERIES 1 J.L.MER0 THE MINERAL RESOURCES OF THE SEA 2 L.M.FOMlN THE DYNAMIC METHOD IN OCEANOGRAPHY 3 E.J.F.WOOD MICROBIOLOGY OF OCEANS AND ESTUARIES 4 G.NEUMANN OCEAN CURRENTS 5 N.G.JERLOV OPTICAL OCEANOGRAPHY 6 V.VACQUIER GEOMAGNETISM IN MARINE GEOLOGY 7 W.J. WALLACE -~ THE DEVELOPMENTSOF THE CHLORINITY/SALINITY CONCEPT IN OCEANOGRAPHY 8 E. LlSlTZlN SEA-LEVEL CHANGES 9 R.H.PARKER THE STUDY OF BENTHIC COMMUNITIES 10 J.C.J. NIHOUL (Editor) MODELLING OF MARINE SYSTEMS 1 1 0.1. MAMAYEV TEMPERATURE-SALINITY ANALYSIS OF WORLD OCEAN WATERS 12 E.J. FERGUSON WOOD and R.E. JOHANNES TR OPlCAL MAR INE PO LLUTION 13 E. STEEMANN NIELSEN MAR IN E PHOTOSYNTHESlS 14 N.G. JERLOV MARINE OPTICS 15 G.P. GLASBY MARINE MANGANESE DEPOSITS 16 V.M. KAMENKOVICH FUNDAMENTALS OF OCEAN DYNAMICS 17 R.A.GEYER SUBMERSIBLES AND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINEERING 18 J.W. CARUTHERS FUNDAMENTALS OF MARINE ACOUSTICS 19 J.C.J. NIHOUL (Editor) LENCE BOTTOM TU R BU ~.~~ 20 P.H. LEBLOND and L.A. MYSAK WAVES IN THE OCEAN 21 C.C. VON DER BORCH (Editor) SYNTHESIS OF DEEP-SEA DRILLING RESULTS IN THE INDIAN OCEAN 22 P. DEHLINGER MARINE GRAVITY 23 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF ESTUARIES AND FJORDS -_ -_ 24 F.T. BANNER, M.B. COLLINS and K.S. MASSIE (Editors) THE NORTH-WEST EUROPEAN SHELF SEAS: THE SEA BED AND THE SEA IN MOTION 25 J.C.J. NIHOUL (Editor) MAR I NE FOR ECASTl NG 26 H.G. RAMMING and 2 . KOWALIK NUMERICAL MODELLING MARINE HYDRODYNAMICS 27 R.A. GEYER (Editor) MARINE ENVIRONMENTAL POLLUTION 28 J.C.J. NIHOUL (Editor) MARINE TURBULENCE 29 M. WALDICHUK, G.B. KULLENBERG and M.J. ORREN (Editors) MARINE POLLUTANT TRANSFER PROCESSES 30 A. VOlPlO (Editor) THE BALTIC SEA 31 E.K. DUURSMA and R. DAWSON (Editors) MARINE ORGANIC CHEMISTRY 32 J.C.J. NIHOUL (Editor) ECOHYDRODYNAMICS 33 R. HEKlNlAN PETROLOGY OF THE OCEAN FLOOR 34 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF SEMI-ENCLOSED SEAS ~~
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Elsevier Oceanography Series, 35
PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS Edited by
B. JOHNS
Department of Meteorology, University of Reading, 2 Early Gate, Whiteknights, Reading RG6 2AU, England
ELSEVIER Amsterdam
- Oxford - New York
- Tokyo
7983
ELSEVIER SCIENCE PUBLISHERS B.V., Molenwerf 1, P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, NY 10017
Library of Congress Cataloging in Publication Data
Main e n t r y under t i t l e : Physical oceanography of c o a s t a l and shelf seas. ( E l s e v i e r oceanography s e r i e s ; 3 5 ) Includes b i b l i o g r a p h i c a l references and index. 1. Oceanography. 2. Ooasts. 3. Continental s h e l f . I. Johns, B. (Bryan) 11. Ceries. GC26.P46 i g @ 3 551.46 63-1662 ISBN 0-444-42153-X
ISBN 044442153-X (Vol. 35) ISBN 0 4 4 4 4 1623-4 (Series) 0 Elsevier Science Publishers B.V., 1983 All rights reserved. No part of this publication may be reproduced. stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publishers, Elsevier Science Publishers B.V., P.O. BOX330, 1000 AH Amsterdam, The Netherlands.
Printed in The Netherlands
V
LIST OF CONTRIBUTORS
Davies, A.G., Institute of Oceanographic Sciences, Crossway, Taunton, Somerset TAI 2DW, U.K Davies. A.M., Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside L43 7RA, U.K. Dawson, G.P., Department of Meteorology, University of Reading, 2 Earley Gate, Whiteknights. Reading RG6 2AU, U.K. Heaps. N.S., Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside L43 7RA. U.K. Howarth, M.J., Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside L43 7RA, U.K. Johns, B., Department of Meteorology, University of Reading, 2 Earley Gate, Whiteknights. Reading RG6 2AU, U.K. Jones, J.E., Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead. Merseyside L43 7RA. U.K. Kornar, P.D., School of Oceanography. Oregon State University, Corvallis. OR 97331. U.S.A. Pugh, D.T., Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead. Merseyside L43 7RA, U.K. Robinson, I.S., Department of Oceanography, University of Southampton, Southampton SOY 5NH. U.K. Soulsby, R.L., Institute of Oceanographic Sciences, Crossway, Taunton, Somerset TA 1 2DW. U.K.
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INTRODUCTION
A distinguishing characteristic of ocean dynamics in shallow coastal and shelf regions is the interaction that exists between seabed conditions and the overlying layers of water. The interaction takes place because of the strong influence of seabed topography and friction on the form of the shallow-water dynamics. The interaction is two-way because of the dynamically induced alterations in the seabed topography that result from sediment movement. In turn, these alterations lead to changes in the near-seabed currents and the frictional processes themselves. The scales involved range from those of high-frequency surface gravity wave propagation over beaches up to those of tidal- and wind-induced circulations in shallow shelf seas. A unitary structure is sought in this book by evaluating in each chapter the effect of seabed topography and friction on a range of scales of motion encountered in the coastal and shelf environment. Chapters 1 to 3 relate to the smaller-scale dynamics associated with surface-wave propagation over beaches. In Chapter 1, a preliminary account is given of a set of field observations made in order to acquire an understanding of the way in which sand movement occurs at the seabed as a result of irregular-wave induced motion. To assist in the interpretation of these observations, a theoretical treatment is given for the determination of the near-seabed oscillatory flow induced by the propagation of a train of surface waves over a rippled beach structure. Later sections consider the role of friction in the near-seabed boundary layer in a wave-induced flow together with its relationship to sedimenttransport processes. Chapter 2 examines the longshore current generated over a sloping beach by obliquely incident breaking waves in the surf zone. An explanation of this is given in terms of the radiation stress induced by the incoming waves. Here again, the process of seabed friction, with the addition of the horizontal turbulent mixing of momentum across the surf zone, plays a crucial role in the theoretical analysis. An important part of this chapter relates to a comparison of the theoretical predictions with the limited observations that are available. A subsequent section deals with the estimation of the rate of longshore sand transport and reviews a large number of littoral-drift measurements. Chapter 3 presents a treatment of the modelling of turbulence beneath waves using a turbulence-energy based closure scheme. The emphasis here is primarily on a consideration of the effectiveness of the type of empirical bottom-friction law described in Chapter 1 together with an evaluation of the dependence of the friction coefficient on the wave and beach parameters. Thus, this chapter provides some guidance in the selection of appropriate seabed frictional parameters for use in the estimation of sediment movement beneath waves and the determination of longshore-current structure. Chapters 4 to 7 relate to the tidal-scale dynamics existing farther off shore and
Vlll
over the adjacent shelf regions. In Chapter 4, a review is given of observational and analysis techniques together with some recent work on the tidal structure over the continental shelf of North-West Europe. The purpose of Chapter 5 is to study the principal features of the bottom boundary layer in terms of field observations made mainly near the seabed. The interpretation of these observations is aided by a comparison with the predictions of different boundary-layer models, the theories of which are also given in this chapter. This is a chapter containing much basic information; i t leads naturally into Chapter 6 where an account is given of a numerical model of shallow-water flow over topography. This employs the same type of turbulence-energy closure scheme as that used in Chapter 3. A series of numerical experiments is described relating to the steady (or quasi-steady) flow over different seabed topographies. The main purpose here is to evaluate the seabed stress and velocity profile. Such information is of primary importance to the sedimentologist when estimating bed-load transportation rates. Tidally induced residual circulations are examined in Chapter 7. These flows are generated by the non-linearity in the tidal dynamics and may be identified by averaging the tidal current over an interval of time in excess of the longest period present in the forcing tidal harmonics. The circulations contribute to the long-term distribution and transport of water propertie: and are of relevance to the oceanographer concerned with the dispersal of sedimeni suspensions. The treatment given here examines the generation of residual vorticity in tidal flows in the neighbourhood of bottom topographic features such as sandbanks and by the action of seabed friction in water of variable depth. The final chapters relate to the numerical modelling of the wind-induced circulation in the shallow shelf seas of North-West Europe. In Chapter 8, primary consideration is given to the wind-induced residual circulation in the North Sea. This is the flow pattern that remains after the purely tidal component has been subtracted from the combined meteorologically and tidally induced circulation. As in the case of the tidally induced residual circulation, the wind-induced residual circulation must also be expected to influence the dispersal of sediment suspensions in shallow shelf seas. The model described in Chapter 8 is three-dimensional in character and employs a turbulence closure determined by the larger-scale dynamics. A comparison is made between the predictions of the model and actual currents monitored by the deployment of current meters in the North Sea. Chapter 9 presents a formulation of a three-layered model which may be used to study the wind-induced flow in a stratified sea. A different fluid density and eddy viscosity is prescribed in each layer. Thus, the scheme may be applied to model currents in the surface, thermocline and bottom layers in a shallow shelf sea. ‘The externally specified forcing results from the surface wind stress, the atmospheric pressure gradient and the tide-generating forces. The mathematical framework developed in Chapter 9 is applied in Chapter 10 i n a series of numerical experiments designed to simulate flow conditions in a basin approximating the Celtic Sea. A primary purpose here is to attain an understanding of the inertial currents generated by the action of an imposed surface wind stress. A discussion is given of inertial currents observed in the Celtic Sea and these are
IX
interpreted in terms of results obtained in the numerical experiments. It will be noted that certain topics that might have been expected to appear in a book on coastal and shelf dynamics have not in fact been considered. These include continental-shelf waves, storm surges and coastal-upwelling processes. Their omission in no way implies a relegation of their importance in the general scenario of shelf dynamics. However, their inclusion, as well as partly duplicating surveys to be found elsewhere in the literature, would not fully conform with the objective of studying those processes that are dominated by the effect of both seabed topography and friction. B. JOHNS (Editor)
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XI
CONTENTS List of contributors . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER I . WAVE INTERACTIONS WITH RIPPLED SAND BEDS
V VII i
Introduction .............. ........................... Part I. Field observations . . . , . . , . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 11. The potential flow over a rippled bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........._....... Deep flow over an idealised bed profile . . . . . . . . Deep flow over natural bed profiles . . . . , , . , , . . . . . . . . . . . . . . . . . . . . . . . . . The propagation of surface waves over a rippled bed of infinite horizontal extent . . . Wave reflection by a rippled bed of limited horizontal extent Part 111. Some considerations of the wave boundary layer. . . . . . . . . . . . . . . . . . . . . . ......................... General comments . . . . . . . . . . . . . . . Sediment transport in a transitional wave boundary layer beneath irregular waves . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 21 29 39 39 51 62
CHAPTER 2 . NEARSHORE CURRENTS A N D SAND TRANSPORT ON BEACHES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................. Introduction . . . . . . . . . ... Wave-generated nearshore currents , , , , . . . . . . . . . . . . . . . . . Littoral drift . . . . . . . . . . . . . . . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. Summary . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . ................................. ........ .....
67 67 68 83 104 105
......................
....................
CHAPTER 3. TURBULENCE MODELLING BENEATH WAVES OVER BEACHES Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction ............ ............................... Wave-induced turbulent flow above a plane horizontal boundary . . . . . . . . . . . . . . . . Turbulent flow beneath wave shoreline over a sloping beach . . . . . . . . . Concludingremarks . . . . . . .... ................ ................ References . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 4. OBSERVATIONS O F TIDES OVER T H E CONTINENTAL SHELF O F NORTH-WEST EUROPE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sea-level measurements ............... ........... Current measurements . . . . . . . . . . . . . . . . . . . . Tidal analysis . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cotidal charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of shelf tides . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . .
i
4 10 10
105
Ill
Ill 111 Ill 120
132 133
135
135 139 139 155 157
170 178
XI1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i84 185
CHAPTER 5 . T H E BOTTOM BOUNDARY LAYER OF SHELF SEAS
189
................................. Introduction . . . . . . . General . . . . . . . . . . . . . . . . . . . . . . . . ........... Mean velocity and turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicting T~ ................................. ... Subdivision of the boundary layer . . . . . . . . . . . . . . . . . . . . . . . . Turbulence structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations . . . . . . . . . . . . . . . . ............................. Thebedlayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................
189 189 189 191 192 193 193 195 195
ndary layer . . . . . . . . . . . . . . . . . . . . . . ... The oscillatory boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillatory planetary flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stratified flow . . . . . . . . . . . . . . . . . . . . . . . . . ................... The depth-limited boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leading edge flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow over topography . . . . . . . . . . . . ........................ Turbulence spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General . . . .......................................... Surface layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. The outer part of the planetary boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . .................... The stably stratified surface layer . . . . . . . The stably stratified planetary boun ylayer . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidal variation . . . . . . . . . . . . . . ................................ Topographic variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... The inertial subrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dissipation range . . . . . . . . . . . . . . . .................... The bursting phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General . . . . . . . . . . . . . . . . . . . ............................ Variation of stress with current speed . . . . . . . . . . . . . . . . . . . . . Variation of stress with height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................... Variation of stress with bottom roughness . . Otherfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . .......................................... Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... Listofsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
196 197 197 197 199 200 203 204 204 204 211 217 226 232 240 241 245 245 245 247 247 248 249 250 250 251 251 251 254 256 256 256 258 260 260 262
CHAPTER 6 . A NUMERICAL MODEL O F SHALLOW-WATER FLOW OVER TOPOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
267
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261
.................
Rough turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The logarithmic layer .................................. The mean velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence . ................................... The seabed ro .................................. Von Karman’s constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The outer part of the marine boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...............................................
XI11
..... Formulation of models with one horizontal dimension Depth-averaged velocity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depth-averaged velocity and turbulent kinetic energy model . . . . . . . . . . . . . . . . . Depth-dependent velocity and turbulent kinetic energy model . . . . . . . . . . . . . . . . Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation of coordinates for depth-dependent model . . . . . . . . . . . . . . . . . . Numerical solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrostatic assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions of the depth-averaged models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physicaltests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depth-averaged model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . River Taw sandwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling two horizontal dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lateral boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the depth-averaged model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the three-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References .............................................
268 268 269 270 270 271 212 274 274 274 277 278 278 282 2x2 289 309 313 313 314 314 316 317 317 319 319
CHAPTER 7 .TIDALLY INDUCED RESIDUAL FLOWS . . . . . . . . . . . . . . . . . . .
321
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidally induced residual eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Headland eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circulation around parallel sandbanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............................. Definition of residual flows . . . . . . The depth-averaged equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residuals from the depth-averaged momentum equations . . . . . . . . . . . . . . . . . . . . . . The vorticity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some simple solutions for residual vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasi-Lagrangian solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eulerian solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
321 323 323 325 327 328 331 333 335 339 342 343 346 351 353 353 355
CHAPTER 8. COMPARISON OF COMPUTED A N D OBSERVED RESIDUAL CURRENTS D U R I N G JONSDAP '76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
357
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-dimensional shelf model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The meteorological data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind-induced circulation for the period 1-9 April 1976 . . . . . . . . . . . . . . . . . . . . . . .
357 357 359 363 364
.......
.......................
..........................................
...............................................
XIV Surface current to wind-speed ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
384 385 385 386
CHAPTER 9. DEVELOPMENT O F A THREE-LAYERED SPECTRAL MODEL FOR T H E MOTION O F A STRATIFIED SEA . 1. BASIC EQUATIONS . . . . . . .
387
................................ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical integration of the equations of motion . . . . . . . . . . . . . . . . . . Eigenvalues and eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The inverse transformation .. ................................ General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case of eddy viscosity uniform through the depth in each layer . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
387 387 388 391 393 394 397 397 400
CHAPTER 10. DEVELOPMENT O F A THREE-LAYERED SPECTRAL MODEL FOR T H E MOTION O F A STRATIFIED SEA . I1. EXPERIMENTS WITH A RECTANGULAR BASIN REPRESENTING T H E CELTIC SEA . . . . . . . . . . . .
401
Abstract . . . . . . . . . . . . . . . . . . . . . .................. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rectangular sea model . . . . .................. Radiation boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . Model with fine grid: time and space splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grid schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time variations of current and internal elevation . . . . . . . . . . . . . . . . . . . . . . . . . Effects of open-boundary radiation . . . . . . . . . . . . . . . . . . . . . . . . Effects of changes in eddy viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basin with one coastal boundary; effects of changes in the duration and magnitude ofthewindpulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inertial currents in the Celtic Sea . . . . . .................. Concluding summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
401 401 403 413 418 422 423 425 442 452
................................
457 461 462 464 464
SUBJECTINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
467
I
CHAPTER I
WAVE INTERACTIONS WITH RIPPLED SAND BEDS A.G. DAVIES
INTRODUCTION
The interaction between surface water waves and bottom sediments, both on the continental shelf and in shallow coastal waters, is a topic of considerable interest for physical oceanographers, coastal engineers and geologists. Although tidal currents are often a more persistent and predictable influence on such sediments, surface waves also play an important part in determining the patterns of sediment movement on the seabed. At a typical offshore location, the influence of surface waves may be of two rather different kinds. On the one hand, the general wave climate at that location, together with the local tidal currents, may give rise to a gradual migration of sediment in one preferred direction; it may be that this residual transport is at least qualitatively predictable on the timescale of many years. On the other hand, a local storm event at the location may have a more dramatic effect on the bottom sediments, but it is likely to be a far less predictable effect and one of importance on the much shorter timescale of a few days. Which of the two influences is the more important for the geologist will depend upon the details of the particular loiation in question, its likely exposure to storms and so on. However, it is the latter influence which is likely to be the more important for the coastal engineer concerned with the erosion of the seabed around structures, or with the design of coastal defences. An obvious first consideration in tackling the problem of sediment transport by waves is to identify the circumstances in which the waves can interact with the seabed. In general, whet1 swell waves travel in water of depth rather greater than the surface wavelength, the wave-induced motion is confined to the surface layers of the fluid. This motion is, to all intents and purposes, frictionless, such that the waves propagate without dissipating their energy (Lamb, 1932, Art. 348; Putnam and Johnson, 1949). As the waves travel on to the continental shelf and approach the shoreline, there comes a point at which the surface wavelength becomes greater than the local waterdepth. Here the waves start to “feel” the bottom, to the extent that they induce an oscillatory motion which extends right down to the seabed. As the water becomes still shallower, the motion near the bed generally becomes increasingly intense, and various interactions with the bed start to become important. One consequence of the existence of a wave-induced motion close to the seabed is that it gives rise to wave-energy dissipation in various ways; for instance, by bottom friction and by any associated sediment transport i f the bed is erodible, by bottom percolation if the bed is permeable, or by “bottom motion” if the bed comprises soft mud. In a natural situation, a decrease in the surface wave energy flux approaching
2
the shoreline may result from a combination of these mechanisms. There are further considerations which arise once the waves “feel” the bottom, which are not associated with wave-energy dissipation. One possibly important consideration is that, if there are undulations on the bed, incident wave energy may be scattered by the bottom topography. A further separate consideration is the effect of wave refraction in shallow water. Ultimately, any wave energy which succeeds in reaching the shoreline must be dissipated in the shallow breaker zone, o r be reflected back out to sea. It is clear from the above list of mechanisms that the problem of the interaction of surface waves with the seabed is a rather complicated one. Therefore, in order to make some progress in this chapter, we shall confine our attention to certain aspects which we shall decouple from the problem as a whole. For the most part, we shall concentrate on properties of wave-induced flows over sand beds which are rippled but impermeable. The properties examined are of two main types. Firstly, we shall consider frictionless aspects of the flow over bottom ripples; it is important to understand these aspects fully, since it is only in relation to an appropriate irrotational solution that frictional effects in the flow can be quantified. Secondly, we shall consider the boundary-layer aspects. In general, the wave boundary layer above the seabed may be characterised as a thin layer of intense velocity shear. Typically, the boundary layer is only a few centimetres thick, even when i t is fully turbulent and, if it is non-turbulent, i t may be only a few millinietres thick for typical swell wave periods. An intriguing but complicating factor in the determination of wave boundary-layer characteristics, a n d of energy-dissipation rates over sand beds, is the role of sand ripples in the problem. The common state of a natural sand bed in oscillatory flow is to be covered with ripples. These will form on a flat sand bed provided that the waves are capable of causing a to- and fro-motion of the surface grains, and they will persist provided that the bed shear stress does not become too large. The near-bed motions may be rather involved, particularly if the ripples are steep, in which case vortices may be formed and shed above them in each wave half-cycle. Essentially, when a n oscillatory flow above a rippled bed starts from rest in either direction, it initially follows the bottom contours and remains attached over the entire ripple profile. In sufficiently active flows however, there is a point in the wave cycle a t which the flow separates above the lee slope of the ripple and a vortex begins to roll up. The vortex remains trapped above the lee slope until the flow reverses, whereupon it is ejected into the flow, ultimately to break up and dissipate its energy. This process, which occurs in each wave half-cycle, is important for two main reasons. Firstly, the amount of wave energy which is dissipated by bottom friction’depends rather critically upon whether o r not the process of vortex formation and shedding occurs. Secondly, as far as the bed shear stress is concerned, the presence of vortices above the lee slopes of the ripples results in form drag on the rippled bed. This acts in addition to the surface shear stress, o r skin friction, on the individual bed-roughness elements. The partitioning of the bottom stress into skin friction and form drag is of fundamental importance in sediment-transport studies since, a t least as far as bed-load sediment motion is concerned, it is skin friction alone which is responsible for mobilising the bed material. Ultimately, however, the vortex shedding process
3
has an importance as one of the mechanisms for carrying the material into suspension. So not only are the details of the flow near the bed of great significance in connection with wave-energy dissipation, but they are important also in building up an accurate picture of the processes of sediment entrainment and transport. The brief discussion above indicates the complexity of the problem and the need for simplifying assumptions before any attempt can be made to interpret field data. It indicates also the considerable interest of the problems of surface-wave propagation over topography and of wave boundary-layer dynamics. Not surprisingly, these problems have been studied for many years, and much work has been carried out on the associated problems of sediment transport and ripple formation in oscillatory flow. For the most part, previous work has consisted of theoretical studies and laboratory experiments aimed at elucidating specific aspects of the interaction between water waves and the seabed. There have been surprisingly few field studies, however, and some considerations relevant to the interpretation of field data are the starting points of this chapter. We shall be concerned at the outset with some observations of a wave-induced flow near the seabed, and with some associated observations of sediment movement on the bed surface. In our interpretation of these field results we shall concentrate on the two main topics mentioned above; firstly, on the perturbing effects of sand ripples on the (irrotational) flow near the seabed and, secondly, on the nature of the wave boundary layer. Without an adequate quantitative understanding of these aspects of the problem, a correct interpretation of observations of the type described here is not possible. The discussion in this chapter is not confined to purely practical considerations however. In the development of procedures to achieve an understanding of the detailed points above, the initial problem is placed in a much broader framework. One result of particular interest arises from the treatment of surface wave propagation over ripples on the seabed, in which it is shown that sand ripples may reflect a significant proportion of the wave-energy incident upon them. We examine this topic in detail, since it has quite general implications for the evolution of a class of ripples on the seabed. The term “ripple” is used throughout this chapter to cover wavy bedforms of all sizes, from short wavelength sand ripples, perhaps a few centimetres long, to “sandwaves” in shallow water, perhaps many tens of metres long. The chapter is divided into three parts. In Part I, we consider a particular set of field results which enable the near-bed velocity field to be related to a set of observations of sediment transport on a bed of rippled sand. A prerequisite to a full understanding of the response of the bed to the wave-induced motion is a knowledge of the potential flow in the fluid domain. Thus, in Part 11, we isolate the problem of surface wave-induced potential flow over an impermeable rippled surface. Although we are concerned only with the idealised case of uniplanar non-separating flow in which the surface wave crests are assumed to be parallel to the ripple crests, we find that the results obtained are helpful in interpreting the field results in Part I. As mentioned above, we then extend the potential flow analysis to study wave reflection by sand ripples. and we consider the implications which arise from this for bedform growth. In Part 111, we turn our attention to the wave boundary layer. For the most part we treat relatively simple cases in which the irrotational “ripple problem” can be decoupled from the “boundary-layer problem”; this procedure is valid only i f the
A
flow is non-separating, and if the boundary-layer thickness is very much smaller than the ripple wavelength. We treat the cases of turbulent and non-turbulent boundary layers, and we discuss methods for the prediction of the bottom stress in both cases. Where it is relevant to do so, we relate these predictions to the field observations. Although, in Parts I1 and 111, we are motivated by the need to interpret the particular field observations described in Part I, the techniques employed and the results obtained have a far wider application than the narrow problem with which we start. In effect, the field observations are introduced as a vehicle for a rather general discussion of seaiment transport by waves.
PART I. FIELD OBSERVATIONS
The most extensively studied aspect of the problem of sediment transport by waves has been the determination of conditions at the threshold of sand motion. Many workers in the laboratory have attempted to identify critical sinusoidal waves of single frequency which are just capable of causing sediment motion on a flat bed. In general, such waves have been defined by their free-stream velocity amplitudes and periods, and the bed material by its size and density. The experimental results have then been expressed in the form of a critical value for the ratio of the forces disturbing the sand grains to the downward restoring forces tending to keep them on the bed. Some formulae of this kind were considered by Davies and Wilkinson (1979) in the context of field experiments which were carried out to study the threshold of sediment motion by waves. These experiments, which are the starting point in the present chapter, were carried out in order to achieve a detailed understanding of the circumstances in which sand motion occurs on the seabed as a result of irregular wave-induced motion. The particular aim of the experiments was to observe the sediment motion taking place on a very small area of the seabed (about 1 m2) and to make comparisons with threshold motion formulae from laboratory work. Full details of the field site, the equipment, the experimental procedure and the analysis of the data have been discussed by Davies et al. (1977) and Davies and Wilkinson (1979), and are not repeated at length here. Essentially, the procedure in the experiments was to measure the wave-induced velocity in the free stream flow using electromagnetic flowmeters, and to observe simultaneously the response of the sand bed with an underwater television system. The bed material was sampled, and the principal features of the bottom topography were measured. Thus sufficient information was obtained to enable comparisons with laboratory results for the threshold of motion. A field site was selected which was sheltered from both steady flows and tidal currents, and where also the effects of shoaling were small. The mean waterdepth was about 5 m, and the measured swell waves had energy predominantly in the range 11-12 s. The orbital motion caused by the waves was almost uniplanar and, when the bed was rippled, this plane was nearly perpendicular to the ripple crest lines. Two component velocity measurements were made in this vertical plane at a height of 1 m above the mean bed level. It was found that the rms vertical velocity was negligibly small compared with the horizontal velocity, and that the
5
swell waves (wavelength - 60 m) approximated long waves in shallow water. There was no turbulence at the measuring height which could be associated with the presence of the seabed below; in fact, the flow there appeared to be essentially frictionless. This is consistent with the field observations of Lukasik and Grosch ( I 963) who demonstrated the inviscid nature of a surface wave-induced flow at a height of 38 cm above the bed for wave periods from 8 to 13 s. I t is consistent also with the field experiments of Tunstall and Inman (1975), in which the irrotational nature of a wave-induced flow was demonstrated at distances greater than a few centimetres above a rippled sand bed. The bed in the present experiments either comprised natural relict ripples caused by some previous wave activity, having wavelength 85”cm and height 12 cm, or was flattened out in the vicinity of the measurement rig by divers. This flattening process resulted in the grain-size range of the potentially mobile material being the same as that of the material on the crests in the rippled bed case, namely 1.0-2.0 mm, with D,,= 1.4 mm. On the flattened bed, very occasional to- and fro-bed load motion of sand was observed and, when this motion occurred, it was widespread over the bed surface. In the case of the rippled bed, occasional to- and fro-bed load motion also occurred, but it was confined to the region of the crests. Above the lee slopes of the ripples, the flow was always non-separating. A detailed justification for the statement that the ripples were relict, and not in equilibrium with the measured waves, is given later. In this connection, Treloar and Abernethy (1978) concluded that there is a time lag in the adjustment of ripple parameters to changing wave conditions. In particular, they found that, after storms, ripple heights observed in the field were higher than heights predicted for equilibrium ripples from laboratory work. The ripples in the present experiments were of this type. One form of data analysis carried out by Davies and Wilkinson (1979) involved characterising each wave half-cycle in a record by its peak free-stream velocity value and presenting the results in the form of a histogram. The synchronous television records were then examined to determine which waves moved sediment, and which did not, and these subsets of the measured waves were identified on the histogram. Occurrences of sediment motion in this analysis were taken to include both the “incipient motion” of a few sand grains dislodged from their resting places, and the “general motion” of the entire bed surface in the field of view. Figure 1.1 shows two representative histograms which were first presented by Davies and Wilkinson (1979), and which illustrate some of the basic aspects of the problem discussed in the remainder of this chapter. Firstly, it may be noted that both outer histograms (including shaded and unshaded regions) have the same general form. I n particular, both are bimodal distributions with the most commonly occurring waves having velocity amplitudes of about 10 cm s-’, in both positive and negative directions. This type of distribution is expected for rather irregular sea waves; for sinusoidal waves in the laboratory, the distribution would comprise contributions only at the (single) velocity-amplitude value in both directions. Secondly, it may be noted that the shaded regions, indicating those waves which moved sediment, are symmetrically placed in the tails of the histograms. However, this is as far as the similarity between the two histograms goes, and there are distinct differences which we consider below. In general, histograms of the present type are valuable only i f the
(om),
6 Occurrence 80
n 60
lllil
- 40
-30
RIPPLED BED
j~
,I
u
u
Range
Translion
Occurrence
4 8o
n
FLAT BED
4 4 1
m
K O M A R X MILLER
MANOHAR
7
wave half-cycles are effectively independent of one another, to the extent that sediment motion in a given half-cycle does not depend upon the motion in previous half-cycles. Although this may be a slight oversimplification in the present experiments for reasons which have been discussed by Davies and Wilkinson (1979), there is good reason to proceed with an interpretation of the data on the basis of the histograms in Fig. 1.1, since the to- and fro-bed load motion was such that the grains were repeatedly starting and stopping. There was no persistence of grain motion from one half-cycle to the next. The resuIts in Fig. I.la for the rippled bed, for which the representative (zerocrossing) period of the waves which moved sediment was T = 9 s, indicate that sediment motion did not occur for velocity amplitudes Urn5 8 cm s- I ; in the range 8 5 Urn5 22 cm s-I, sediment motion was caused by a proportion of the recorded waves; and for fim 2 22 cm s - ’ , sediment motion was caused by all the recorded waves. The immediate point to note here is that there was not a uniquely defined threshold velocity amplitude at which sediment motion commenced, but a rather broad “transition range”, centred on Urn= 15 cm s - ’ , between the categories of no motion and total motion. The apparent randomness in the threshold condition is due largely to the fact that it is the bed shear stress, and not the free stream velocity, which governs the onset of sediment motion. In Part 111, a method is described for calculating the bed shear stress associated with a measured free stream velocity record of the present kind. It is shown that, for irregular waves, peak free stream velocities do not have a one-to-one correspondence with equivalent peak bed shear stresses, and that the variations involved are such as to produce a relatively better defined sediment threshold stress amplitude, than the threshold velocity amplitude discussed above. For the flat bed, the representative wave period was again T = 9 s, and the onset of sediment motion occurred at 0 ‘ = 30 cm s-’ (see Fig. 1.lb). Although few of the recorded waves achieved this peak velocity value, the flattening of the bed can be seen to have had the effect of almost completely inhibiting sediment motion. Insufficient waves achieved peak values in the range 30-50 cm s- to know whether a transition range would have been found in this case, but what is clear is that an apparent doubling (at least) of the measured threshold velocity amplitude resulted when the bed was flattened. This aspect of the problem is discussed in detail in
‘
Fig. 1.1. Histogram representation of the measured velocity data, in which the peak value of Urn achieved in each wave half-cycle is plotted against the number of half-cycles which had i6, falling within each incremental range. The full histogram indicates the distribution for all the waves in the record, and the shaded one is a subset of this indicating only those waves which moved sediment. Positive and negative velocities indicate the shoreward and seaward directions, respectively. Figure l . l a is based on measurements made above a rippled bed during a 110-min experimental run: here sediment motion occurred in the region of the ripple crests only. The transition ranges are indicated by the arrowed bars. Figure 1.1 b is based on measurements made above a flat bed of the same material during a 120-min run; here very occasional sediment motion occurred over the entire bed surface. The arrows show the possible lower bounds of the transition ranges. Also shown are some typical laboratory results for the threshold of sediment motion on a flat bed; these are based on the representative wave period 7’= 9 s, and are for the sand-grain sizes indicated.
8
Part I1 in terms of the irrotational flow over a rippled surface. In particular, the relationship between the measured “free stream” velocity and the near-bed velocity is considered, and the apparent discrepancy between the velocity amplitudes at the threshold of motion on the rippled and flat beds is explained. Essentially, it is shown that the near-bed potential velocity above a ripple crest is significantly greater than the measured free stream velocity. I t is argued in Part 111 that this causes an enhancement of the shear stress in the region of the crest compared with the shear stress which would occur on an equivalent flat sand bed. This is rather different from the effect on the threshold of motion attributed to ripples by Madsen and Grant (1976, pp. 27-28), who suggest that “the local value of bottom slope would influence threshold conditions, with movement being initiated at points where the bottom slope is steepest”. The fact that sediment motion in the present experiments occurred only in the region of the crests suggests that the latter effect was unimportant compared with the former. The results for the flat bed in Fig. I.lb may be compared directly with laboratory determinations of the threshold velocity amplitude. A comprehensive list of threshold motion formulae from laboratory studies has been provided by Silvester and Mogridge (1971). Here we take as examples the formulae of Bagnold (1946), Manohar (1955) and Komar and Miller (1975), which may be used to predict the threshold velocity amplitude of a sinusoidal wave above a flat sand bed; the formula of Komar and Miller is based upon a re-analysis of the data of five laboratory workers. The present calculations have been made for the grain size Dso = 1.4 mm, and also for the sizes 1 and 2 mm which were the effective lower and upper limits of the size range of the mobile sediment. In addition, the results included in Fig. 1.lb have been calculated for the single wave period T = 9 s; this is sufficient since each of the formulae is relatively insensitive to period, and Manohar’s is independent of it. While the three sets of laboratory results can be seen to be in fair agreement with one another, the field observations occurred at a slightly lower value of velocity amplitude than predicted, even allowing for some uncertainty as to which grain size in the mobile range was actually moving. This may be due in part to the fact that it is not possible to compare objectively the present threshold criterion with those adopted by the other workers; the fact that incipient sediment motions have been included in the shaded regions in the figure may have tended to give a low threshold value. Also, the extent of a possible transition region is unknown in the present case so that, if the midpoint of such a range was taken as an indication of the threshold velocity amplitude, a value rather larger than the lower limit of about 30 cm s - ’ would be appropriate. In this connection, Komar and Miller (1975) have pointed out that sediment threshold formulae from the laboratory, relating to sinusoidal waves above a flat sand bed, may be expected to give conservative results for the threshold velocity amplitude if the waves are irregular and the bed is not entirely smooth. They note, firstly, that the interactions of wave trains of slightly differing period will lead to the generation of high instantaneous velocities near the bed and, secondly, that “small protuberances” (as opposed to ripples) on the bed may cause sediment motion at lower velocities than suggested by the threshold formulae. Despite these considerations, it is reasonable to suggest on the basis of the present field results that critical velocity amplitude values in the field and in the laboratory are in tolerable
9
agreement, and that the velocity amplitude is a parameter which provides a first step towards the definition of conditions at the threshold of motion. We shall be concerned in what follows to develop some of the further steps in the argument, and to achieve an understanding of the reasons both for the apparent difference in results for the flat and rippled beds, and also for the transition ranges in the histograms. We proceed initially by assuming that it is legitimate to decouple the frictionless “ripple” problem from the frictional “ boundary-layer’’ problem. This step is justified if the boundary layer is very thin compared with the ripple wavelength, and if also the flow is “stable” in respect of the sand ripples. The justification for the first assumption in relation to the field observations above is given in Part 111. As far as the second assumption is concerned, Sleath ( 1975a) has demonstrated that “instability” occurs in oscillatory flow above a rippled bed of wavelength L if a , / L > 0.5, where the near bottom excursion amplitude A^, = U m / u , and u = 2 a / T . In other words, instability arises if the orbital excursion (22,) in the unperturbed flow is greater than the ripple wavelength. The term instability is used here to refer not to turbulence as such, but to rather more organised and repeatable disturbances in the flow, including flow separation above the lee slopes of the ripples. If flow separation occurs, active ripple development may take place, involving the scour and deposition of sand associated with vortex formation and shedding. In such circumstances, the oscillatory flow is said to be in “equilibrium” with the bed. I t may be recalled that, for the earlier results relating to the rippled bed, the critical velocity amplitude values defining the transition range were 8 and 22 cm s - ’ , with few wave half-cycles in the record having velocity amplitudes exceeding 30 cm s-’. Assuming that each wave half-cycle was sinusoidal with associated wave period of T = 9 s, estimated values of (A^,/L) for the respective velocity amplitudes are as follows: 0.13, 0.37 and 0.50. It follows, on the basis of Sleath’s criterion, that the flow was stable for almost all the wave half-cycles in the experiment. In fact, confirmation of this was provided by visual observations of the flow which showed no sign of separation above the lee slopes of the ripples, even for the largest of the measured waves which might possibly have been in equilibrium with the bedforms. This suggests that the bed was composed of relict (or fossil) ripples, formed by previous wave activity at the field site, and was not in equilibrium with the measured waves. The framework for analyzing the situation described above is shown in Fig. 1.2. A thin wave boundary layer is shown adjacent to the bed, above which is a layer of non-separating potential flow perturbed by the rippled surface. This gives way to the layer of unperturbed potential flow within which the velocity measurements described above were made. It was noted earlier that, at the measuring height, the flow was non-turbulent. We are therefore justified in assuming that the wave-induced motion in the main body of the flow was irrotational to a good first approximation. In practice, turbulence may be generated in the flow by other mechanisms than those associated with bottom friction, for instance by breaking surface waves. However, no such turbulence was recorded in the present experiments. In what follows, we aim to relate the measured velocity, U,(t), to the velocity U,(x,t ) , which is tangential to the bed and just outside the boundary layer, and then to deduce the motion within the boundary layer itself on the basis of the appropriate
10
-I
L
Fig. 1.2. Definition sketch. The decoupling of the “ripple” and nrating flow.
“
boundary-layer” problems in non-sep-
local value of U , ( x , 1 ) . Part I1 of this chapter is concerned with the potential flow aspect of the decoupled problem, and Part 111 with the boundary-layer aspect. The framework for analysis described above assumes that the water depth is rather greater than one ripple wavelength; if this is not so, the unperturbed potential layer will not arise. It should be appreciated also that the simple model takes no account of wider considerations, such as the possibly important effects on the flow of the permeability of the bed, of sediment motion and of drift velocities above the bed which may be associated with the early stages of ripple formation (Sleath, 1974a).
PART 11. T H E POTENTIAL FLOW OVER A RIPPLED BED
I . Deep flow over an idealised bed profile We shall consider initially some properties of “deep” flow over a rippled surface, since it is this case which is relevant to the interpretation of the field observations described in Part I. In particular, we shall set out to define the thickness of the layer of perturbed potential flow above an impermeable rippled surface, and also to determine the relationship between the velocity in the free stream (i.e. at infinity) and the velocity at the bed surface itself. These irrotational features of the flow must be understood before an interpretation of the field data can be attempted. In this section, results are obtained by three different analytical approaches, the first a linearised analysis, the second involving a conformal mapping of the fluid domain into a half-plane, and the third drawing on some arguments from classical potential theory. We shall be concerned only with ripples having smooth crests; a solution for the non-separating flow over ripples with sharp crests has been given by LonguetHiggins (1981, Sect. 4). The problem in each case is as depicted in Fig. 1.3. In the flow domain D we
sinusoidal bed
Fig. 1.3. Definition sketch. Two-dimensional, irrotational, deep flow above a bed which is rippled indefinitely in both the positive and negative directions of x.
satisfy Laplace's equation:
v2+=O
inD (1) where is the velocity potential. On the rippled surface C we satisfy the kinematical condition:
+
-+x
lX + +y
=0
on C
where the subscripts denote differentiation. We assume that the flow in D is steady, uniplanar and non-separating. The assumption of steady flow does not preclude the use of the results obtained in this section in cases in which the flow is unsteady since, for a deep flow, time may be introduced into the formulation as a parameter. However, in the unsteady oscillatory case, the assumption of non-separating flow will be valid, in practice, only if the orbital excursion of the free-stream oscillation is less than the ripple wavelength. We start with a result from linear perturbation theory (see Milne-Thomson, 1968, Sect. 15.40) for the flow bounded by a sinusoidal surface of small amplitude lying along the x-axis (Fig. 1.3). The unperturbed free-stream velocity in the + x direction is U and, to the order of approximation adopted, the bed [ y b= {(x)] is given by:
{( x ) = h cos( Ix + 8 )
(3) where b is the ripple amplitude, I ( = 277/L) is the ripple wavenumber and 6 is an arbitrary phase angle. By expanding the velccity potential as a power series (+ = + +2 + . . . ) and linearising the boundary condition 2 in the usual way, it may be shown that the perturbation, (Pz,to the velocity potential existing in the assumed absence of bottom undulations, namely = - U x , is such that:
+,
+,
+=+,++,=
- U x - Ubexp(-b)sin(Ix+8) (4) correct to second order. Discussion of the detailed way in which this result is obtained is deferred to Part 11, Sect. 3. For the present, it may be noted that eq. 4 is based upon a linearised kinematical boundary condition which is applied at the mean bed level ( y = 0), and that the horizontal ( u ) and vertical ( 0 )components of velocity are given by ( u , u ) = ( - &, -I$~).It follows that the horizontal velocities at the crest and trough positions on the bed surface, u,, and u , , , respectively, are given
12
by: ucr = U ( 1 5 b l ) , tr
and so depend directly on the ripple steepness ( h l ) . These are the extreme values of velocity in the flow, and the result for the crest is plotted in Fig. 1.4, The thickness of the layer of perturbed potential flow corresponding to eq. 4 may be defined in several ways, and we shall compare results based on two definitions here. Firstly (definition I), we shall assume that the height (gl) of the perturbed potential layer is the height at which the perturbation to the horizontal velocity falls to I % , or to 1056, of its value at the bed. The results on the basis of this definition are simply: Definition 1
i
1% rule: 10% rule:
exp( - j j l / )
= 0.01
exp( -,fi,/)
=
In terms of ripple wavelength L
=2
0.1
or or
P I /= 4.605 g l / =2.302
~ / 1 the , respective results from definition 1 are
9, ;= 0.733L ( 1 % rule), and 3, = 0.367L (10% rule). These results, which are plotted
in Fig. l S a , show that the thickness of the perturbed layer extends to a height of rather less than one ripple wavelength above the bed, and that significant velocity variations are confined to a layer of height rather less than one half of a wavelength. (Essentially the same conclusion is true for steady separating potential flow over a rippled bed, though we do not treat this case in detail here.) Secondly (definition 2 ) . we shall take the height p2 to be that height at which the velocity differs from the unperturbed value U by 0.01 U, or by 0.1 U . On this rather different definition, the height g2 is such that: Definition 2
i
100 61
0.01 U rule:
exp( $ 2 / )
0.1 Urule:
exp(g2/)= l o b /
=
Results calculated on this basis are plotted in Fig. 1Sb. It may be noted that for ripple steepnesses bf < 0.00318 and < 0.0318 T , for the 0.01 U and 0.1 U rules, respectively, heights j 2are unobtainable since the surface velocities do not achieve the specified critical values. For larger values of bl, the heights g2 increase rapidly and, ultimately, attain values which are comparable with those obtained on definition 1. In general, definition 1 is the more valuable in the design of experiments of the type described in Part I, since it enables the true free-stream velocity to be identified unambiguously. On the other hand, definition 2 is the more valuable one whenever absolute, rather than relative, velocity variations are of interest; for instance, definition 2 may be helpful in the context of the validation of current-meter data which is possibly contaminated by unwanted ripple-induced effects. We consider next an alternative, though related, method for obtaining near-bed velocities and perturbed-layer thicknesses. This method involves transforming the flow domain D in the physical or z-plane, into the upper half of the A-plane. The conformal mapping suggested by Taylor et al. (1976) is:
+ ib exp( i I A ) where z = x + iy and A = + ix. The bed [ y b= c(x)] is defined in the new coordi-
z
=
f (A )
=A
13
____
-
- .. -..-
Results b a s e d on eq
4
f o r a sinusoidal b e d
Results f r o m m e t h o d of Davies ( 1 9 7 9 ) for a sinusoidal b e d (crest position) Results b a s e d o n eq
7 l o r a b e d given b y eq 6 (crest position)
Results f r o m m e t h o d o f Davies (19791 for a b e d given b y eq 6 Ucr
U
Fig. 1.4. Norrnalised surface velocities 2t the ripple crest position, for both sinusoidal beds (eq. 3), and bed profiles given by eq. 6.
nates by
x = 0, so that:
.yh = h COS(
lt)
The ripples given by eq. 6 are symmetrical about their crests in the horizontal direction. For small (bf), x = t and so the bed is purely sinusoidal in the original coordinates. However, for increasing values of ( h l ) , the ripple crests become progressively more peaked, and the troughs flatter and longer. Such profiles are more realistic than sine waves as representations of natural wave-generated sand ripples. The velocity components in D are determined as follows. If the fluid motion in the transform plane (A-plane) is prescribed by the complex potential w( A), the velocity components at a point P in that plane are given by u p - i u p = -dw/dA. The velocity components at a corresponding point Q in the physical z-plane are then
14
I OFFlNITlON 1
54 61,
4-
3-
7 30'
2-
10% rule
1-
(a)
0-
e 5
4
3
/
/ 2
I
tb)
Fig. 1.5. The thickness of the layer of perturbed potential flow. 1.5a and b are for definitions 1 and 2, respectively. The curves are defined in the key in Fig. 1.4.
15
given by:
where the dash denotes differentation (see Milne-Thomson, 1968, Sect. 6.29). I t follows that the horizontal velocities at the crest and trough positions o n t h e bed. u,, and u t rrespectively, are given in relation to the unperturbed velocity U by: u,,=
(8)
u(1 f bl)-’
tr
(see Fig. 1.4). It may be noted that, as ( b l ) increases, the velocity at the crest increases very much more rapidly than that for a purely sinusoidal ripple given by eq. 5. The thickness of the perturbed layer may be calculated from eq. 7 according to definition 1 or 2. In terms of the new coordinates:
Definition 1
i
1% rule:
exp( - 4 i / )
= O.O1/(1
10% rule:
exp(-t,l)
= 0.1/(1
T 0.99 b l )
crest
T 0.9 b l ) (trough)
The results in Fig. 1.5 based upon the above definitions are for the ripple crest position only, and are presented in terms of the old coordinates with heights measured from y = 0. The perturbed heights given by definition 1, on both the 1 % and 10% rules, are rather smaller than the respective (constant) values for the purely sinusoidal bed, due to the much enhanced crest velocity in the present case as ( h l ) increases (Fig. 1.4). The heights given by definition 2 are in rather closer agreement with the results for a sinusoidal bed, but are always slightly greater since, at a given height above the crest and for a given ripple steepness, the horizontal velocity in the present case always exceeds the velocity for a sinusoidal bed. Clearly, the two definitions must produce perturbed heights which are equal when the bed velocity at the crest is twice the free stream value; in the present case, this occurs if (61) = 0.16 7r. The third and final approach which we consider briefly in this section is the method proposed by Davies (1979). The aim of this rather general method is to perturb a steady streaming motion (cf. G I in eq. 4) in the (x,y)-plane, by the introduction of a repeated pattern of discrete singularities, such that one of the streamlines of the resulting motion is distorted into any desired ripple shape. This approach may be used to predict the velocity field close to the rippled surface in relation to the unperturbed free-stream flow, as well as the thickness of the layer of perturbed potential flow. As in the two earlier methods, the flow is assumed to be two-dimensional, deep and non-separating, and the ripple profile is assumed to be repeated indefinitely in the positive and negative direction of x. The particular formulation discussed by Davies (1 979) assumes that the ripples are symmetrical
16
about their crests, though this assumption may be relaxed if required. The singularities superimposed on the basic streaming motion have been taken as doublets lying closely adjacent to one of the mean streamlines, and having strengths which depend upon the ripple feature being modelled. Each doublet introduced into each ripple wavelength permits a pair of coordinate values to be specified on the ripple surface. The singularities all being on one side of the streamline defining the ripple surface in the (x, y)-plane, the potential on the other side gives the unique solution of the problem. In this approach, the only possible inaccuracy is that the distorted streamline may not accurately represent the desired ripple shape between specified coordinate points. However, this difficulty may be overcome by a suitable choice for the spacing between the row of doublets and the prescribed profile. For brevity, the solution of Davies (1979) is not quoted here, and the reader is referred to the original paper for the details of the method. Essentially, it has been found that a good representation for a ripple profile can be obtained by taking sixteen specified points, equally spaced on the surface in each wavelength. The strengths of the sixteen associated doublets may then be found by solving a set of linear simultaneous equations by, for instance, a matrix inversion procedure. N o simple analytical reductions of the general result (cf. eq. 5 or 8) are possible in this case. However, the solution for the velocity field is readily computed and, in order to demonstrate the good agreement which exists between results obtained by this method and results from the two earlier methods, some comparisons are included in Figs. 1.4 and 1.5 for the idealised cases of a sinusoidal bed, and a bed given by eq. 6. I t may be seen that results from the present method tend to give slightly higher values for the perturbed layer thickness in the case of a sinusoidal bed, and slightly lower values for crest velocity than given by either of the two earlier methods. These variations appear to arise because the choice of sixteen specified points per ripple wavelength is insufficient to define the ripple profile adequately in the region of ii crest which is strongly peaked, and not because of any inherent differences in the analytical methods employed. Fortunately, the variations are not too worrying in practice since they become substantial only for h l z 0 . 1 5 ~ .and this is greater than the steepness of many naturally occurring features. Allen (1968) quotes the range o f steepness of natural “small-scale’’ ripples ( L < 60 cm) as 0 . 0 5 ~< hl < 0 . 2 ~ and . of “large-scale’’ ripples ( L > 60 cm) as 0.0171 < b l < 0 . 1 ~ We . continue the discussion of the application of this method in the section which follows. 2. Deep flow over naturul bed profiles
Natural wave-generated sand ripples have crests which are more peaked, and troughs which are longer and flatter, than a sine wave and, for this reason, eq. 6 provides a helpful initial represeritation. In general, however, natural ripples have more complicated profiles than those given by eq. 6, and a knowledge of the flow over such irregular features is required if bottom erosion and sediment transport are to be adequately understood. In this section, we start with results obtained by the method of Davies (1979) for the flow over a typical natural sand ripple. We then show how results for natural profiles can be obtained by superimposing solutions From linear theory.
17
i -06
I ~ 0 5
4
~ 0 4
*
-03
4 ~ 0 2
*
-01
4 "
0
iX
" i d
(b)
0
, , , 0
I,/,/rn,A,,
0 5
j,,"&,
Norrn,:lised r i v p l e heiqht 0 3n
I I
;
1 .Q
,
I
,
I
15
,
I
,
I
2
*
U __
U
Fig. 1.6. Non-separating potential flow above a natural bed profile. 1.6a shows the streamlines (given by J. = constant), and 1.6b shows the associated vertical profiles of normalised horizontal velocity above the crest and trough positions.
The first example is concerned with a typical natural wave-generated feature observed on the seabed in about 5 m depth of water. The original ripple profile is shown in detail as the full line in the lower part of the later Fig. 1.7b, and it was representative of the ripples on its up-wave and down-wave sides. The ripple wavelength was L = 81 cm, so that L / h = 0.16 where h is the depth, and its overall
steepness was 0 . 1 5 ~ The . modified profile for which calculations have been made by the method of Davies (1979) differs from this original profile to the extent that the slight asymmetry about the ripple crest in Fig. 1.7b has been removed. In other words, the streamline corresponding to the bed surface has been distorted into a shape which has been assumed symmetrical about the ripple crest. This streamline (the zero streamline $ = 0) passes through the sixteen prescribed coordinate points shown as crosses in Fig. 1.6a, with the sixteen singularities giving rise to the distorted streamline all lying below $ = O . The flow above $ = O displays the expected features, namely that the flow returns to its unperturbed state with increasing height above the bed, and that the streamlines converge over the crest and diverge over the troughs. The consequence of this may be seen in Fig. 1.6b where vertical profiles of horizontal velocity are shown for the crest and trough positions. At the crest the surface velocity is 1.89 U, where U is the unperturbed free-stream velocity, and at the trough the suiface velocity is 0.55 U . Thus there is a variation of 3.4 to 1 in the surface velocity from crest to trough, which is rather greater than the variation of 2.6 to 1 found for sinusoidal ripples of the same steepness ( b l = 0 . 1 5 ~ ) Above . the bed, and the profiles in Fig. 1.6b tend quite rapidly to the unperturbed velocity value 17, the perturbed layer thickness may be identified according to either of the earlier definitions. In particular, at the crest: Definition 1
1% rule: 10% rule:
PI[=
4.90 2.76
Definition 2
0.01 lJ rule: 0.1 U rule:
jj2i=4.78 jjri=2.70
where I= 277/L, and where the results are with respect to the mid-height between crest and trough. Evidently, the thicknesses are similar on the two definitions as a result of the almost doubling of the unperturbed velocity at the crest and it may be concluded, on the basis of either definition, that any measurements made at a height of one ripple wavelength above the mean bed level are far above the perturbed layer, and any at a height of half a ripple wavelength are only influenced a little by the presence of the undulations. This is a generally valid conclusion, and i t is one which confirms that in the present experiments, in which measurements were made at a height greater than one wavelength above the bed, the flow was “deep” in respect of the ripples. Further profiles of the kind shown in Fig. 1.6b have been presented by Davies (1979). The second complementary approach which we consider here involves a generalization of the earlier result based on linear theory. We now consider a more complicated bed than eq. 3, defined in Fourier series form as follows:
where h, and 8, are the amplitude and phase angle of the qth harmonic constituent, respectively. Even though, in any particular case, the coefficients in the series (i.e. constituent ripple amplitudes) may be small for the higher harmonics, their effect on the flow near the bed may be large since, as seen earlier in eq. 5 , the effect of ripples depends not on their amplitudes, but on their steepnesses. So the choice of N in eq. 9 must be governed by the steepnesses of the various harmonic constituents, such that
19
+
the steepnesses of the ( N I)th, and all higher harmonics, are negligibly small compared with the steepnesses of the lower harmonics. By a simple extension of ey. 4, the solution & expressing the interaction of +, with is such that:
&= +, +
N
C&
= - Ux -
U
c b, exp( - 4 / y ) sin(
q/x
+ 8,)
q= 1
In the special case of a symmetrical ripple with 8, given from eq. 10 by: u,,/u=
1+
=0
( 4 = I , N ) , the crest velocity is
N
c qb,i q= 1
So the sum of the steepnesses of the constituent harmonics enables the departure of the surface velocity at the crest (x = 0) to be calculated in relation to the unperturbed velocity U. The results in Fig. 1.7 are for two asymmetrical ripples and have been obtained by adopting the choice N = 8 in eq. 9. This is thought to give a good compromise between the preservation of the important features of the ripple profiles, and the avoidance of concentrating on fine details of little general interest. Such details have been further eliminated in the present examples by the following smoothing procedure. Firstly, the profiles shown in the lower parts of Fig. 1.7a and b have been represented by thirty-two equally spaced points per ripple wavelength. Next, each set of coordinate points has been Fourier analysed using a Fast Fourier Transform routine to determine the amplitudes (b,), and phase angles (a,), up to the sixteenth harmonic. The smoothing procedure has then merely involved truncating each series at the eighth harmonic. The resulting series can be seen in Fig. 1.7 to correspond to simulated profiles which represent each ripple quite adequately. In each case, the fundamental ( q = 1) makes the major contribution, as expected. The contributions from the second harmonic ( 4 = 2) are also substantial, while the contributions from the remaining harmonics are relatively small and generally diminish, both in terms of amplitude and steepness, as q increases. In the upper parts of Fig. 1.7a and b the horizontal component of surface velocity is plotted in relation to the unperturbed velocity U for each point on the bed surface. ‘The horizontal, rather than the tangential, component has been plotted for simplicity, since the two curves are very similar for all x. The horizontal velocities fall rapidly, and generally quite regularly, with distance from the crest position and, although the curves contain quite marked local variations, these have no general significance. The peak-normalised velocities at the crests are 1.62 U and 1.86 U in Fig. 1.7a and b, respectively. The overall trends in the velocity results are much the same as results for the equivalent symmetricai profiles presented by Davies (198Oa, Part I) for which the peak-velocity values at the crests are 1.62 U and 1.81 U, respectively. This indicates that the fine detail of a ripple profile does not greatly influence the important peak-velocity result at the crest. In Fig. 1.7a there is a secondary ripple crest in the primary trough. This causes a local bed velocity maximum of 1.22 U, which is well above the unperturbed value. A similar local velocity maximum of value 1.03 U occurs in Fig. 1.7b above a small local bottom
20
2--
(a)
I
I
I
p ex
Fig. 1.7. The upper curve in each diagram shows the normalised peak value of horizontal velocity amplitude (U, = u / U ) at the bed surface, over a complete wavelength of the rippled bed. The original, asymmetrical, natural profile is shown in the lower part of each diagram together with the simulated profile, which has been obtained with N = 8. The profiles are drawn without vertical exaggeration. Ripple wavelengths: 112 and 81 cm in 1.7a and b, respectively.
undulation. It might be noted finally that both streamlines, and vertical profiles of horizontal velocity for any point on the bed, are readily obtained from eq. 10 and that, from such profiles, perturbed layer thicknesses may be evaluated. In practice, similar results for thickness are found by the present method and by the method of Davies (1979) discussed earlier in this section. The deep flow results described above have important implications for sediment transport studies. Firstly, we have seen that surface velocities, on natural ripples in non-separating flow, fall from maxima on the crests to minima in the troughs. This fall is generally quite rapid and regular with distance from the crest position, and it helps to explain why, in certain circumstances, sediment may be observed moving as
21
bed load in the region of the ripple crests, but not in the troughs (Davies and Wilkinson, 1979). Evidently, the sediment threshold stress is exceeded only on the crests, as a result of convergences and divergences of streamlines of the type shown in Fig. 1.6. For field workers, there is at present a need for theory of the type described above, on account of the uncertainty involved in any direct experimental measurement of wave-induced velocities right down to the bed surface. Secondly, in performing experiments in the field or the laboratory to study the sediment transport problem in detail, it is often important to make velocity measurements at heights above the bed which are sufficient to be outside the immediate influence of bottom undulations. In the example in Fig. 1.6b, the theory predicts that the largest variations in potential velocity occur very close to the bed (within the bottom 10 cm roughly). More generally, as noted earlier, a height of one ripple wavelength above the mean bed level is well outside the perturbed layer, while a height of half a ripple wavelength is only just within it, the flow being influenced very little by the bottom undulations. These heights should be taken into account in experimental design; for example, they should be considered in the general question of the positioning of current meters in the sea. 3. The propugation of surface waves over u rippled bed of infinite horizontul extent
We now extend the earlier results, based on linear theory, to consider the interaction of progressive surface waves with ripples on the seabed. The aim of this section is to show, firstly, how the velocity field over a prescribed rippled bed can be determined from the surface deformation associated with an incident wave; secondly, how perturbations to the velocity field caused by the ripples may be found either close to the bed, or throughout the full waterdepth, depending upon the ratio of depth to ripple wavelength; and, thirdly, how incident wave energy may be reflected by a rippled bed structure and, hence, how a coupling may exist between wave reflection and ripple growth. We assume that the water is of constant mean depth, that the bed is impermeable and rippled indefinitely in the direction of surface wave travel, that the surface wave crests are parallel to the ripple crests, and that the flow is non-separating. We seek a steady-state solution of this two-dimensional problem by expanding the velocity potential as a series in powers of a small parameter, which is later identified with ratios of the various length scales in the problem. In this approach, the bottom undulations are regarded as small perturbations on a plane surface, the bottom boundary condition being linearised in the usual way. Hence, from the condition that the component of fluid velocity normal to the bed must vanish on the boundary, the interaction between the (first order) flow, which would be present without the boundary perturbations, and the perturbations themselves, i s treated as a new source of (second order) fluid motion situated on the plane surface. Although the problem tackled in this section is physically unrealistic in that the number of ripples on the bed is taken to be infinite, the present solution provides a simple way of calculating the flow field over a variety of ripple profiles, given the mean water depth, and the surface wave period and amplitude. One major restriction on the solution, however, is that the bed wavelength must not equal approximately half the surface wavelength.
22
I
D
Fig. 1.8. Definition sketch. Two-dimensional, irrotational, free-surface flow above a bed which is rippled indefinitely in both the positive and negative directions o f x.
The problem is depicted in Fig. 1.8, in which the departure of the water surface from its mean level is q ( x , t ) , and that of the bed from its mean level is {(x). The governing equation for the velocity potential +(x, y , I ) is eq. 1, where D is now as defined in Fig. 1.8, and the full boundary conditions are as follows:
+ q t + = 0 on C , (kinematical condition) on C,(pressure condition) g-q - +, + +( +: + +$) = 0 -+xqx
+y
(11)
(12)
and on C, (kinematical condition) -+%Cx + +y = 0 (cf. q.2 ) , where g is gravity. We establish a basic hierarchy in each of +, q and { as follows:
+ = a+, + a2+, + . . ., q = “9, + a*q2+ . . . , l
=
a{, + a*i2+ . . .,
in which a is a small parameter; and we satisfy the boundary conditions 11-13 at the mean water surface and bed levels, y = h and y = 0, respectively, by the introduction of Taylor expansions. Thus we treat the original non-linear problem as a series of linear problems, the first to order a , the second to order a’, and so on. The complete details of this procedure are given by Davies (1980a, Part 1, Sect. 3). Equation 4 provides an example of a solution obtained on the same basis for the deep flow over a rippled bed. The problem to order a is as follows:
v2+,= 0 aq I __a+ I +-=O
ay
at
a+ 1
g q ,--=o
at
in
O , < y , < h , -c~<x
fromeq. 11, from eq. 12,
(14)
23
and so:
a+ 1
29
ony=h;
g--+L=O JY at2
ony=0 The problem to order a’ is made up of two separable parts. One is well known (see, for example, Peregrine, 1972) as Stokes’ theory to the second order of approximation, and it concerns the steepening of the surface wave crests and the flattening of the troughs, above a bed which is flat. We shall not be concerned with this part of the solution, but only with that which expresses the interaction between the first order motion, O( a ) , and the undulations, O( a ) , on the bed. The problem to order a’ is therefore taken as follows: V2G2= 0
in
O
}
gq2---=0 dat +2
i
(17)
onv=h
Here the boundary condition at the free surface is the same as in the problem O( a ) . while in eq. 19 the effect of the rippled bed is evident. We consider the interaction of surface waves of amplitude u , frequency u and wavenumber k , travelling in the + x direction and defined to order a by: 77, = a s i n ( k x - a t ) (20) with a bed comprising sinusoidal ripples of amplitude h and wavenumber /. and defined by:
=bcos(/x+6) (21) where 6 is an arbitrary phase angle (cf. eq. 3). The solution of eq. 14, which satisfies eqs. 15, 16 and 20, is simply: ua cash( k ~COS( ) kx - a t ) = k sinh( k h ) in which a is related to h and k by the dispersion relation: u
* = gk tanh kh
(23)
The solution of the problem O(a2),that is the solution of eq. 17 satisfying eqs. 18, 19, 21 and 22, is: baa
“= 2 sinh( k h )
{ - A ( / + k , .,y) cos[(/+ k ) x - at + 61
+ A ( / - k , 0 , y ) cos[ ( 1 - k ) x + at + s] }
24
in which the attenuation term is: A(r,s,y) =
gr cosh[ r ( y - h ) ] + s 2 sinh[ r ( y - h ) ] s2cosh(rh)-grsinh(rh)
Like eq. 22, eq. 24 is associated with a deformation of the free surface, which may be found from eq. 18. However, since the velocity potential (eq. 24) is attenuated upwards from the bed, q 2 will generally make only a small contribution to the surface elevation. For ripples of short wavelength ( I >> k ) , the attenuation terms in eq. 24 reduce to cosh[l(y - h)]/sinh(Ih), which may be contrasted with the downward attenuation term in eq. 22. Equation 24 can be seen to comprise two waves with wavenumbers which are the sum and difference of those of the incident waves and the bottom undulations. The sum wave is always in the onwards transmitted direction, whereas the difference wave may travel in either direction and is back reflected if the incident wavelength is sufficiently long. At the changeover point ( I = k ) , the difference wave has infinite wavelength and the energy flux associated with it vanishes. It may be noted also that the velocity potential (eq. 24) is singular when:
-
u 2 cosh[ ( I t- k ) h ] = g ( I
k k ) sinh[ ( I f k ) h ]
that is when I = k 2k. The physically admissible case is I = 2k, in the neighbourhood of which the solution must break down. The reasonant interaction implied by this singularity is apparent in the work of McGoldrick (1968), and is found quite commonly in related fields. For example, Rhines and Bretherton (1973) have found such a resonance in their study of planetary waves over a sinusoidally undulating ocean floor. In a wider context, the phenomenon is analogous to Bragg reflection of X-rays from a crystal plane. It might be noted also that it is analogous to the singularity identified by Milne-Thomson (1968, Sect. 15.43), at Froude number equal to one, in the problem of steady free-surface flow over an infinitely rippled bed. We discuss the implications of the infinite resonance for sediment transport later in this section. There are certain physical limitations on the first order solution which arise mainly as a result of the terms which are dropped in linearising the boundary Conditions; these limitations relate only to the potential flow analysis, and not to the validity of that analysis in terms of a total approach to the problem of surface wave motion over a rippled bed, including the effects of friction and so on. I f the bed is flat, the limitations are well known (Peregrine, 1972) and may be expressed: clk << 1. a / h << 1 and a / k 2 h ' << 8/3. Complementary conditions arise if the bed is rippled. In particular, as a result of the terms dropped in linearising the boundary condition 16, the limitations bI<< 1 and bk << 1 arise. In respect of the first condition on the ripple steepness, it has been shown by Davies (1980a) that, for a deep flow, the effect on the results of linearising the bottom boundary condition is small if hl < 0 . 1 ~ As . far as the second condition is concerned, it may be shown that an even more stringent limitation on bk arises as a result of the requirement of the method of solution that << 1. From eqs. 22 and 24, this may be expressed as follows:
1
bk 2
-A ( I
k , 0 ,y
1
<< 1
in 0 < y
25
0
05
1
15
u "'rn, h-15rn kh=O 7 5
be-0 I n Lh=?n .Y-
h
h=15rn kh=O 7 5 bP-O I n
Bh=O 3 n
Fig. 1.9. Vertical profiles of the amplitude of the normalised horizontal velocity above both crest and trough positions for (a) ripples in deep flow, (b) megaripples in intermediate flow and (c) sandwaves in shallow flow. The dashed curves indicate the results for a flat bed, that is the solution 0 (a).
26
This condition implies that the analysis is invalid if k >> 1. Furthermore, it suggests a restriction on the ratio of the ripple amplitude to the waterdepth; but this is expected since it is well known that, for large ( h / h ) , surface waves behave non-linearly (Jolas, 1960; Longuet-Higgins, 1977). The above limitations on the solution O( a ) apply in almost unmodified form to the solution 0 ( a 2 ) . This has been confirmed by Davies (1980a) by extending the analysis to 0(cr'). Finally, we note that the small parameter a itself, which was introduced to set up a basic hierarchy of terms in the solution, is connected with the smallness of ak, a / h , a / k 2 h 3 .hl, hk (or more strictly (hkA().This is evident from consideration of the solution 0 ( a 2 ) ,and is confirmed by the extension of the results to O(a3).It is not necessary, or possible, to identify a with any one of the above parameters, and it is not a requirement of the method of solution that this be done. The velocity components ( u , u ) = (-GX, -I+.,) may be calculated from eys. 22 and 24, and three contrasting examples of profiles of horizontal velocity amplitude are shown in Fig. 1.9. Each graph in Fig. 1.9 is based on the complete solution to order a*, that is the velocity amplitude associated with (+, + +2), for both ripple crest and trough positions. Where the profile of velocity amplitude, 0(cr), differs from the extreme profiles for the crest and trough, the (flat bed) result is shown by a dashed curve. The results have been normalised with respect to the peak horizontal velocity amplitude at the free surface i n the solution 0 ( a ) , namely U,,,,,,= g a k / a . The example in Fig. 1.9a is for typical sinusoidal ripples in deep flow; the waterdepth is 5 m, with kh = 0.5, h / = 0.177 and /h = JOT, and the associated surface wave period may be shown from eq. 23 to be 9.3 s. The slight downward attenuation of the flat bed profile, U , / U , , , ~ is ~ , governed by the parameter ( k h ) . The remaining part of the solution, u7/UnlilX,is attenuated upwards, and its effect in the present example is significant in only the bottom fifth of the depth. Clearly, the presence of the ripples has a major influence on the flow velocities near the bed. causing the ratio of the velocity amplitudes at the crest and trough positions on the bed itself to be 1.92 : 1. In Fig. 1.9b a contrasting example is presented relating to bed features having the dimensions of typical megaripples in water of depth which is equal to the ripple wavelength; in particular, h = IS m, kh = 0.75. hl= 0.177 and /h = 277, and the surface wave period is 11.2 s. Now the velocity perturbation due to the ripples extends throughout the lower half of the flow, though not up to the free surface itself. In the final example, in Fig. 1.9c, results are presented for bedforms with the dimensions of typical sandwaves in water which is relatively shallow compared with the ripple wavelength: in particular, h = 15 m, kh = 0.75, b / = 0 . 1 ~ and Ih = 0.377. and the wave period is again 11.2 s. In this case the velocity perturbation extends up t o the free surface on account of the relatively small value of ( I h ) . The dependence of the results in the three examples upon ( I h ) is expected on the basis of the earlier results for the thickness of the layer of perturbed potential flow. Under certain circumstances, the presence of the free surface causes relative increases in velocity over the ripple crests, and relative decreases in the troughs, compared with equivalent deep flow results. Although these variations are generally small, they may exceed 10% of the equivalent deep flow velocity amplitude for certain combinations of surface wave and ripple parameters (Davies, 1980a, Part I, Sect. 3.7). In general, variations of this kind appear to be negligible if /h 2 T, and (possibly) substantial
21
only if Ih 5 0 . 2 5 ~ However, . each case must be treated on its merits, firstly, since the largest variations tend to occur for parameter settings at which the theory starts to become questionable and, secondly, since the singularity in the solution at I = 2 k has a substantial and unwanted effect on the results in its neighbourhood. Finally, in none of the examples in Fig. 1.9 are the earlier limitations on the solution violated. This suggests that the limitations on the theory do not prevent its use over wide and practically important ranges of both surface wave and seabed parameters. We commented earlier that, in the neighbourhood of the singularity at I = 2 k , the solution must break down since 1+11 < 1+21, which violates !he assumed hierarchy of terms in the solution. Despite this, it is still interesting to consider the wave energy fluxes associated with +, and +2 near the singularity. The energy flux associated with is always in the positive direction of x, and is independent of ( k / / ) . However, as we noted earlier in connection with the terms of the right hand side of eq. 24, while the sum wave ( - cos[(l+ k)x - at + S]} has an energy flux always in the positive x-direction, the difference wave ( - cos[(/ - k ) x + at + S]} has an energy flux which is in the negative x-direction if I > k , and in the positive x-direction if I < k . I t follows that, at the singular point I = 2 k itself, the solution corresponds to a negatively travelling wave of infinite amplitude. Although this is physically unrealistic it implies that, where the solution remains valid in the neighbourhood of I = 2 k , there is a strong reflection of incident wave energy, such that the superimposition of 7 ,and v2 gives rise to a partially progressing and partially standing wave structure in the surface elevation. It might be noted that these are progressing and standing waves in the usual sense, since the attenuation expression in the numerator of the cosh(ky). Thus the important second term of eq. 24 reduces at I = 2 k to in the neighbourhood of I = 2 k , attenuation of +2 is essentially the same as that of namely a downward attenuation from the free surface, rather than the upward attenuation found generally for GI. The infinite resonance at I = 2k is associated with the only singularity in the solution to second order. However, if the analysis is extended to third order (see Davies, 1980a), further singularities may be identified, as well as the possibility of a feedback of energy into the main motion G I from the interaction of +>with ll.The rather complicated picture which is presented by the third order solution suggests that, while the solution to second order probably yields the most important aspects of the resonant interaction mechanism for practical purposes, it may miss other aspects. In other words, the present method of solution by small parameter expansion may be incapable of fully resolving the resonant interaction between the free surface and the bed. This is discussed further by Davies (1982a). Despite this, we shall assume in what follows that the solution to second order contains the essential features of the resonant interaction between the free surface and the bed for practical purposes. Depending upon whether 12 2 k or 2 2 k , distinct differences arise in the velocity field throughout the flow as a result of the T-phase shift in +2 at I = 2 k . If we assume for simplicity that the ripples give rise to complete reflection, such that the waves above them are purely standing waves, then, in terms of the horizontal velocity to 0 ( a 2 ) ,the general picture to emerge is of a weak horizontal velocity fluctuation above the ripple crests if 12 2 k , and a strong fluctuation if I2 2 k . Conversely,
+,
+,
-
28
above the troughs, there is a strong horizontal fluctuation if I s 2 k , and a weak fluctuation if 12 2 k . Clearly, there are sedimentological consequences of this for cases in which the bed is erodible. These relate to the question of whether an initially small periodic disturbance of the bed, with I = 2 k , is likely to grow, or be destroyed, as a result of the resonant interaction mechanism. On general intuitive grounds, i t can be argued that beneath the antinodes of a standing surface wave, where the lowest velocity amplitudes are found close to the bed, sedimentation rather than erosion is expected, as a result of the general tendency for accumulation of material to occur on relatively undisturbed parts of the bed. Conversely, beneath the nodes, where there is greatest horizontal activity near the bed, erosion rather than sedimentation is expected. So, in the context of the present analysis, sedimentation is expected to occur on the ripple crests and erosion in the troughs, causing ripples with I = 2 k to grow, only if 12 2 k . Conveisely, erosion is expected to occur on the crests and sedimentation in the troughs, causing the ripples to be destroyed. i f 12 2 k . Evidently, the conditions for the growth and destruction of initially small bed features are rather finely balanced near the resonance point, and i t follows that a small change in surface wavelength, in relation to the rather less readily changed bed wavelength, might have dramatic consequences in terms of the bed features, if that change involves crossing the singularity. A slightly more involved discussion of the phenomenon of possible ripple growth has been given by Davies (19XOa. Part I, Sect. 7), in terms of the residual mass-transport cells which exist under standing waves. Such an explanation has been put forward also by Nielsen (1979, Sect. 6) to account for his experimental observations of ripple growth beneath standing waves. However, this line of argument results in a rather less clear conclusion than the intuitive argument above. Finally, it should be emphasised that the resonance at / = 2k is infinite because the bed features have been assumed to be of infinite horizontal extent. Although the critical condition I = 2k still takes on a physical importance i f the ripples occupy only a finite region of the bed, the singularity found in the present solution +p2gives way to a finite resonance, as shown in Sect. 4. Consequently, while the infinite resonance might look somewhat damaging to the usefulness of the solution, this is not too worrying in practice. Certainly, the velocity field can be calculated on the present, relatively simple, basis if I t 2 k . In this section, we have shown how near-bed velocities over sinusoidal ripples may be determined in relation to prescribed sinusoidal incident surface waves. Since the theory used is linear, the principle of superposition holds and so, for more complex bottom topography, the final result for the velocity field may be built u p by superimposing solutions for the interaction of each significant Fourier component of the incident surface wave train, with each significant component of the ripple profile. In addition, we have suggested a mechanism for the formation of ripples as a result of a resonant interaction between the surface waves and the seabed. Heathershaw (1982) has carried out a laboratory study of the coupling which exists between wave reflection and ripple generation. However, no equivalent field experiments appear yet to have been conducted. In general, if the tidal range at a field site is small, such that the wavenumber k remains fairly constant for waves of given frequency, then bedforms may grow as a result of the resonant interaction mechanism. However, in locations with a fairly high tidal range where k may change
29
considerably during a tidal cycle, the possibility of finding ripples having / = 2 k is a little remote, since the bedforms would probably have insufficient time to adjust their wavenumber accordingly. The mechanism discussed above should not be confused with the well documented mechanisms of formation associated with ripples of generally rather short wavelength. Such features are commonly formed in deep flow by a process of scour and deposition, associated with vortex shedding from the lee slopes o f the ripples in each wave half-cycle. The present theory, which does not allow for flow separation, provides no direct information about this important phenomenon. However, as noted earlier, flow separation appears to occur only in cases in which the orbital excursion exceeds the equilibrium ripple wavelength. Consequently, separation is expected only if the surface wave height is sufficiently large. For the typical example of ripples in deep flow in Fig. 1.9a. separation is expected on the above criterion if the surface wave height ( 2 u ) exceeds 0.52 m, ( u k > 0.026); for the remaining examples in Figs. 1.9b and c, the linear theory is violated, in respect of uk and u / h , well before separation occurs. 4. Wave reflection by a rippled bed of limited horizontal extent
One of the general implications of the wave-reflection mechanism identified in the previous section is that, if the surface wavenumber ( k ) is equal to approximately one half the bed wavenumber ( I ) , wave reflection may tend to reduce gradually the height of an incident swell wave, and thereby reduce its potential for erosion at the seabed. However, the theory presented in section 3 is unsuitable for quantifying incident, reflected and transmitted wave-energy fluxes, since the bedforms have been assumed to be of infinite horizontal extent. In this section, we treat the case of sinusoidal surface waves incident upon a patch of sinusoidal ripples of limited horizontal extent, in order to examine how wave reflection depends upon the surface wave and ripple parameters; in particular, how it depends upon both the quotient ( k / / ) and the number of ripples in the patch. The problem tackled is for a strictly two-dimensional situation. Although, in principle, the analysis may be extended to the general three-dimensional case, the present problem is one of particular interest to workers in the field of sediment transport since, firstly, natural sand ripples are usually formed with crests which are parallel to the surface wave crests and, secondly, if their wavelengths are long, ripple profiles are sinusoidal to a reasonable first approximation. It is assumed again that the flow above the bed is non-separating and, as noted earlier, this restricts the analysis to ripples with relatively long wavelengths; in particular, to ripples with wavelengths which are greater than the orbital excursion of the wave-induced oscillatory flow near the bed. We study this problem on the same basis of linear perturbation theory as in Section 3, that is we seek the perturbation potential, O( a’), arising from the interaction of a prescribed first order motion, O(a), with a region of bedforms, O(a). The reflected and transmitted waves are then determined approximately by obtaining asymptotic solutions for the perturbation potential well away from the region of bedforms, on both the up-wave and down-wave sides. The problem is physically well posed to the extent that the radiation, or Sommerfeld, condition can be applied to ensure that the only waves in the perturbation solution are outgoing waves from the region of
30
disturbance on the bed; this condition is an indispensable requirement in obtaining a unique solution of physical interest of the steady-state problem. It might be added, finally, that the argument in this section is presented in rather more detail than those in the earlier sections, since the method is somewhat different from that which has been previously published by the author (Davies, 1980a, 1982b). Previous solutions for wave reflection by obstacles on the seabed, or by depth variations, have usually been restricted to simple topographies. This has often resulted from the need for a knowledge of the mapping of the fluid domain into some simpler domain for which the potential problem can be solved (e.g. Kreisel, 1949; Roseau, 1952; Fitz-Gerald, 1977). However, approximate analytical solutions of the boundary-value problem sought on a different basis have also been restricted generally to simple topographic variations, such as step changes in the depth (e.g. Newman, 196Sa; Miles, 1967). As a result of the convenience of a simple fluid domain, there have been rather few previous studies of surface wave reflection by undulating seabeds. However, two important features of the approximate solution presented here have been found in previous work. Firstly, it has been shown (e.g. Kreisel, 1949) that, at a few depths from an obstacle on the seabed, the potential is very nearly a superimposition of simple wave trains, that is of “propagating” modes. Secondly, it has been found that the reflection coefficient for an obstacle of rectangular cross-section on an otherwise flat bed is oscillatory in respect of the ratio of the width of the obstacle to the surface wavelength. The oscillatory nature of the reflection coefficient has been shown by Kreisel (1949), Newman (196Sb). Mei and Black (1969) and Fitz-Gerald (1977), and for long surface waves by Jeffreys (1944). In the present context, the width of the obstacle may be identified with the length of the ripple patch. Another relevant study has been carried out by Long (1973) who has considered the general problem of an arbitrary spectrum of surface waves propagating over an arbitrary spectrum of bottom perturbations, and has determined the time rate of change of the directional spectrum as a result of bottom scattering. Long’s analysis differs from that discussed in this section, since it is restricted to situations in which variations in the statistics of the surface wave and bottom perturbation fields take place on scales which are large compared to a surface wavelength, from which i t follows that the region of bedforms must be large. By comparison, in the cases of greatest physical interest in the present section, variations in mean quantities may take place on length scales which are of the order of a few surface wavelengths only. and so the region of bedforms may be of very limited extent. In fact, i f this is not so, the assumptions on which the analysis is based are likely to be violated. Both the present theory, and that of Long, predict a resonance in backscatter for surface wavelengths which are twice the bed wavelength. In Long’s theory the resonance is sharp due to the large size of the bedform patch whereas, in the present theory, the resonance is broad due to the limited extent of the patch, so that wave reflection results from bed wavelengths which are not exactly half the surface wavelength. As the patch length is increased, the resonance becomes progressively sharper and, ultimately, as the patch length tends to infinity, the resonance becomes infinite also (as found in Section 3). The problem is depicted in Fig. 1.10 in which the bedforms may be seen to
31
Y
water surface
seabed
Fig. 1.10. Definition sketch. Two-dimensional, irrotational, free-surface flow above a hed which is rippled only in the range L , < x G L , , and which is flat elsewhere.
occupy the region L , < x
< L,,
such that:
-co<x
L , <x < L, L,<x
+ rpt,
in-h
on Y = 0
and: -@',{,+{@vv+rpy=O
on Y = - h
(cf. eqs. 17-19). Since the vertical velocity perturbation at Y = - h occurs in L , < x < L, only, we may rewrite eq. 28 in the form:
where: V ( x , t ) = -Ox(x, -h,t).{,+@yu(x, -h,t).{
The effects of this disturbance o n the fluid as a whole are described by eqs. 26 and 27. We shall be concerned here only with the asymptotic behaviour of cp as x A co; by virtue of the radiation condition we require a solution which corresponds to outgoing waves in these limits.
-
If we prescribe CD as a periodic function of time and seek a steady-state solution of eqs. 26, 27 and 29, we find, for reasons which are well known, that this solution is indeterminate. We therefore employ the familiar device, described by Lamb ( 1932, art. 242), of introducing into the problem a small amount of friction proportional to the relative velocity. Although the coefficient of friction is set ultimately to zero, the device ensures the convergence of the integrals arising in the analysis. I t should be noted that the introduction of linear friction is essentially a mathematical device which does not accurately represent the way in which dissipation occurs in the flow in reality. It enters the formulation by way of a change in the surface boundary condition, such that eq. 27 is replaced by: on Y=O
gcp,+cp,,+pcp,=O
(27a)
in which p ( > 0) is the coefficient of friction. We now assume that 'p and its first and second derivatives tend to zero as 1x1+ 00, in such a way that Fourier transforms exist in x. Thus, eqs. 26, 27a and 29 become:
'
Gyy - t 2 @ = 0 g@,,
+ I$,, + pGt
(30)
in - h < Y < O , -co < ( < co =0
on Y = 0
(31)
The solution of eq. 30 is:
@((, Y . t ) =k((,t ) cosh((Y)
+ b ( ( ,t ) sinh((Y)
and we make this solution specific to the case of waves of frequency u by taking:
k((,t)=A(()exp(iat),B((,t)=B(()exp(iut)
and A ( ( , t ) = i l ( ( ) e x p ( i u t )
The solution I$ which satisfies eqs. 31 and 32 is then:
[ g( cosh( ( Y ) + ( u 2 i p u ) sinh( ( Y)] @ ( tY,, f > = ~ ~ [ ( ~ ' - i p u ) ( c o s h ( ( h ) -sinh((h)] g(~ -
m,
t)
Upon taking the inverse transform, the velocity potential q ( x , Y , t ) is given by:
We obtain our final result from eq. 33 by contour integration using the residue theorem. We note initially that the linear-friction term has the effect of displacing
33
certain singularities of eq. 33 off the &axis into the upper and lower half-planes. Therefore, we proceed, firstly, by taking $, to be the real part of a complex variable A = [ i x and, secondly, by replacing the range of integration - M < 5 < M in eq. 33, by integration around a closed contour consisting of the portion ( -r0. ro) of the [-axis and a semi-circle centred at the origin and having radius ro. The semi-circle must be taken in either the upper or lower A-half plane to ensure only outgoing waves from the region of bed disturbance as x + M , that is that the radiation condition is satisfied. In the limit r, 00, the required range of integration is recovered, since the semi-circle makes a zero contribution. For p > 0, the outgoing waves are damped as 1x1increases, but as p + 0 an oscillatory solution is obtained as 1x1 00. We shall be concerned here only with the solutions in the two asymptotic & M. limits x In terms of the new variable A, singularities of the integrand of eq. 33 arise at positions X = A given by:
+
--$
-+
-+
,
( u 2 - ipa) cosh( A,h)
-
gA, sinh( X,h)
=0
(34)
Since p is small, these positions must lie close to the positions A = A,, corresponding to p = 0 itself. Furthermore, since we shall later identify @(x, Y, t ) with $, given by eq. 22, we note that the dispersion relation (eq. 23) still applies in the present problem, and it follows that the positions A = A,, are given by:
A, tanh( X O h )= k tanh( k h ) = u2/g
(35)
This equation has two solutions on the real axis of A, namely A, = E,, = fk . and an infinite number of solutions on the imaginary axis of X given by A, = i x o , where x,, satisfies: - x o tan(xoh)
=
a2/I:
For non-zero p( > 0), each of these poles undergoes a small displacement from its reference position A = A,. In particular, the solution of eq. 34 may be shown to be such that the pole at A,, = k undergoes a small negative imaginary displacement into the fourth quadrant, the pole at A, = - k undergoes a small positive imaginary displacement into the second quadrant, while each of the poles on the imaginary axis of A undergoes a small real displacement. This is the desired effect of the introduction of the friction term. We are now in a position to choose our contours for the evaluation of the integral in eq. 33. We require, firstly, that any transients in x decay in the limits x + f M , and our choice of semi-circular contour of radius ro is therefore governed by the term exp( - i t x ) . For the calculation of the asymptotic value of the integral as x + - 00, the chosen contour must contain only poles which lie in the upper half-plane if the solution is to converge; and, for x -+ 00, the contour must contain only poles in the lower half-plane. Clearly, as r, + 00, a particular contour will contain all the poles in either the upper or lower half-plane. From eq. 33, the residue R A Pof the typical simple pole at A = A, may be expressed by:
+
+
R
=-A,
1 [gA,cosh(X,Y)+ 27
( u 2 - i p u ) sinh(A,,Y)]
gx P
2cosh(A,h) 2Aph + sinh(2A,,h)
34
In the asymptotic limit x + - M, for which the contour is taken in the upper half plane, the final result for the velocity potential is obtained from the residue of the pole at h p ,= - k ipk, ( k , , > 0) since, in the further limit p + 0, this term gives rise to an outgoing wave from the region of bed disturbance, that is to a “propagating mode”. The infinite number of poles close to the imaginary axis of h in the upper half plane give rise to “non-propagating modes”, which determine, at least in part, the wave field in the immediate vicinity of the region of bed disturbance. As p + 0. these non-propagating modes decay exponentially in x, and therefore are not considered here further. Finally, since the contribution to the integral from the semi-circular part of the contour may be shown to be zero as ro + M (Davies, 1980a), we arrive at the final result, as x + - 00:
+
rp(x, Y, t ) = 2 . r r i R ^ r , l ~ = o = i S 2 ( Y ) . A I E _. e_x, p [ i ( o t + k x ) ]
(36)
where:
Q(Y)=
2cosh[k(Y+h)] 2kh + sinh(2kh)
In the asymptotic limit x + + 00 for which the contour is taken in the lower half plane, the final result for the outgoing wave is obtained from the residue of the pole atX,,,= +k-ipk,(k,>O). Hence,asx+ +M: g ) ( x , Y , t ) = -2.rriRxp,l~=,,=zn(Y).Al,=,. e x p [ i ( a t - k x ) ]
(37)
To proceed any further it is necessary to specify @(x, Y , t ) and Y , ( x ) , and hence to determine V ( x , t ) and A(&). We shall consider here one special case in which x, Y , t ) , the unperturbed velocity potential, is given by eq. 22 and corresponds to a surface wave with elevation given by eq. 20, travelling in the positive x-direction. We prescribe the bed in eq. 25 as: Yh(x)= h sin(Ix + 6 )
in L , < x < L,
(38)
where 6 is an arbitrary phase angle. For continuity of bed elevation we take:
L,=(-nn-6)//
andL,=(rna-6)//
+
where n and rn are integers, so that there are ( 1 7 m)/2 ripples of wavenumber / in the patch. It follows from eq. 29 that V ( x , t ) is given by the real part of: ~ ( xt ,) = C , [ i / c o s ( / x + S ) + k s i n ( / x + 6 ) ] e x p [ i ( a t - k x ) ]
(39)
where: C’* = gukh/a cash( k h ) Hence, from eqs. 32, 36 and 39, the perturbation potential is given, in the limit + - w , by the real part of:
x
Similarly, from eqs. 32, 37 and 39, rp is given, in the limit
.Y +
+ 00,
by the real part
of: cp(x, Y , t ) = C , ~ ( Y ) ( k / l ) [ ( - I ) " ' - ( - l ) " ] i e ~ p [ ~ ( ~ f - k k x ) ]
(41 )
It may be noted that the outgoing waves in both eqs. 40 and 41 are properly attenuated with depth. In the particular case in which there is a n integral number of ripple wavelengths in the patch L , < x < L,, such that rn = n and 6 = 0, we obtain the results. as
x+
-- 0 3 :
where L = m.rr//. Here there is no disturbance in the perturbation solution o n the down-wave side of the patch, but there is a reflected disturbance on the up-wave side. The size of the reflected wave in relation to the incident wave can be assessed from the following ratio of the amplitudes of the velocity potentials: Amplitudecp(x, Y , t ) l , Amplitude @
.-
2.
-
u R_ u 2kh
2hk
+ sinh(2kh) . H ( 2 k / / )
(44)
where: H(2k//)
=
( - 1)"'(2k/l)
sin(2 krn.ir/l) (2k/l)'-
1
and in which a and u R denote the amplitudes of the incident and reflected waves respectively. [Note that u R may be positive or negative on account of the term sin(2kL).] A striking feature of this result is that H ( 2 k / l ) is oscillatory in the ratio of the overall length of the ripple patch 2 L ( = 2m71//) to the surface wavelength. This property is illustrated in Fig. 1.1 1 where H ( 2 k / / ) is plotted for rn = 1-4. We note also the importance of the critical condition 2 k / / = I , that is where the bed wavelength is half the surface wavelength. Since H( 1 ) = m77/2 the amplitude of the reflected wave increases in proportion to the number of ripples in the patch at this critical value and, ultimately, as rn + co the condition of infinite resonance described in Section 3 is attained. For finite rn, the function H(2k/l) does not achieve its peak value at 2 k / / = 1, but at a value 2 k / / 2 I ; in particular, the maxima for m = 1-4 occur at 2 k / l = 1.126, 1,036, 1.0165 and 1.009, respectively. More generally, where H = 0 the amplitude of the reflected wave is zero, and the incident wave is able to propagate over the ripple patch without reflection. O n the other hand, where H has its turning values, local maxima occur in the amount of wave energy reflected. It follows that, if a spectrum of surface waves is incident on a patch of ripples of wavenumber I, significant reflections of wave energy may be expected to occur in the neighbourhood of preferred values of k. In practice, this may cause a selective attenuation of the spectrum on the down-wave side of the ripples, compared with the incident wave spectrum. An important implication of eq. 44 is that it may take relatively few ripples to
36
6. KEY ~
rn=1
_ _rn= .2
5-
- _ _ _ in- 3 rn= 4
4-
3-
2-
1-
0-
-1-
Fig. 1.1 1. The response curves H ( 2 k / / ) for
M =
1 , 2, 3 and 4.
give rise in practice to a substantial reflected wave, at least if the surface and ripple wavenumbers are such that 2 k / l = 1. Typical results for the peak reflected wave amplitude, C,i are shown in Fig. 1.12, in which h,/a is plotted against b / h for the range rn = 1-10, and for ripple steepness hl= a/20. The curves show the peak values of the reflected wave amplitude which arise as (2k/l) is varied, but rn, hl and h / h remain fixed. The range of ( b / h ) has been taken as (0, 0.4) which is likely to encompass any naturally occurring features; strictly, ( b / h ) must be small for the theory to be valid (see Section 3), and for b / h 2 0.2, for which the curves are dashed, the results should be treated with caution. It should be emphasised also that the results in Fig. 1.12 are not expected to be accurate if ( h R / u ) is large, for reasons which are discussed shortly. In fact, if & , / a 2 0.5, for which the curves are also dashed, the reflected wave-energy flux is likely to be overestimated by at least 10%. Despite this qualification, it may be seen from Fig. 1.12 how few ripples may be needed to produce the important limiting result of total wave reflection (CiR/u= 1). As expected from Fig. 1.1 1 , peak values of b , / ~ are found where 2 k / l = 1; in fact, no peak value exceeds the value at 2 k / l = 1 itself by more than 0.01 for the curves
37 ? ,
4
1c
/I
/
/
/
09
Dt
0i
Of
0:
04
11.-
n:
Fig. 1.12. Results for the peak reflected wave amplitude ( h R ) obtained as ( 2 k / / ) is varied. The results are plotted as functions of h / h with m = 1-10. for steepness h / = 7r/20. The results should be treated with caution, firstly, for h / h 2 0.2 since the scaling assumptions start to break down in this region and. secondly, for h R / u 2 0.5 since here the reflected energy flux is likely to be overestimated by at least 10%. In both of these regions the curves are dashed.
plotted. More generally, it may be shown that, for given h / h and m , the larger the ripple wavelength the greater is the size of the reflected wave. Heathershaw (1982) has confirmed in the laboratory the existence of the resonance near 2 k / / = I , and has obtained good agreement with results of the type shown in Fig. 1.12. The case of a non-integral number of ripple wavelengths in the patch ( m = n + 1, 6 = 0) has been considered by Davies (1980a). From eq. 41, the transmitted wave is non-zero in this case and, when combined with a, produces small changes in the amplitude and phase of the waves on the down-wave side of the ripple patch, even in cases of zero reflection. An equivalent conclusion has been reached by Newman (1965b). Once again, the reflected wave in this case is resonant at 2 k / f =: 1, and a set of response curves, like those in Fig. 1.11, is readily obtained. There are certain physical limitations on the solution obtained in this section. For the most part, these are the same as those quoted in Section 3, relating to ratios of the various length scales in the problem. In addition, however, there now arises a condition relating to the wave energy fluxes in the component parts of the solution. In particular, in the assumed absence of energy dissipation, it is required that the incident wave-energy flux is balanced by the sum of the reflected and transmitted energy fluxes. In eqs. 38, 40 and 41, this balance is not established; in fact, eq. 40
38
should be viewed as providing an upper bound on the size of the reflected wave. The reason for the energy imbalance is that the linearised analysis does not permit any attenuation of the incident waves as they travel over the region of topography (f.,< x < L , ) , causing the predicted reflected wave in the perturbation solution to be overestimated and the transmitted wave to be very small or zero. In practice. if the reflected wave is non-zero, there must be a progressive attenuation of the incident waves in L , 5 x 5 L,, though the true result may be a compounded effect of several reflections (see Newman, 196%). In order to overcome this difficulty. at least in cases in which 2 k / l = 1, Davies (1980a) has presented an ad-hoc iterative procedure for recovering an energy balance in the solution, and thereby establishing more accurate predictions for the magnitudes of the reflected and transmitted waves, than given by eqs. 40 and 41. In respect of these equations, and in the extreme case of a prediction of total wave-energy reflection ( a R= u ) , i t has been found from this iterative procedure that the size of the reflected wave given by eq. 40 is overestimated by 25%. This is, of course, the worst possible case and, for smaller reflected waves in the perturbation solution, the overestimates are considerably smaller. Despite its shortcomings, however, the strength of the present method, which is strictly valid only for relatively small reflected and transmitted waves (IT/ << l@l), is in providing an estimate for the size of the reflected wave in the first place. I t has been shown here that for surface waves incident upon sinusoidal bedforms of finite horizontal extent, maximum reflection occurs at surface wavenumbers for which 2 k / l = 1. This result has certain consequences for sediment transport, at least if the bed on the up-wave side of the ripple patch is erodible. The existence o f a partially standing wave structure in - 00 < x < L , , caused by wave reflection, may deform this part of the bed into ripples having the preferred bed wavenumber 1 =: 2 k . As was pointed out in Section 3, such ripples are expected to have troughs and crests which are tied to the nodes and antinodes of surface elevation. Although we have sought no detailed information about the flow in the immediate vicinity of the original ripples in L , < .x < L , , it can be shown that the new bedforms should be a simple continuation of the initial pattern into - co < x < L , and, in fact, this has been confirmed in the laboratory by Heathershaw (1982). In a more general case in which a spectrum of waves is incident on a region of bedforms of wavenumber I , only those waves having k = 1/2 will be strongly reflected by the topography. Again, these reflected waves may cause the formation of new bed features of wavenumber I in - 00 < x < L , and, as these grow, wave reflection will become increasingly concentrated near the point in the spectrum where k = 1/2. This will lead, in turn, to a more pronounced standing wave structure in - 00 < x < L , , and a more rapid development of the new ripples until, finally, an equilibrium bottom profile is achieved. In other words, there is here a suggestion of a possible coupling between ripple growth on an erodible bed and the reflection of incident wave energy. N o mechanism exists for the formation of new bed features in the region L , < x < 00 by surface wave-seabed interaction. I t might be noted finally that the results presented in Sections 3 and 4 are entirely complementary, In fact, the results given in Section 3 can be obtained by examining the properties of the wave field over the region of ripples in L , < x < L , , though the solution is complicated by additional terms if this is done. The separation of the
39
problem into solutions for the “near field” over the ripples in Section 3, and for the “far field” well away from the ripples in Section 4, has been made simply for convenience.
PART 111. SOME CONSIDERATIONS OF THE WAVE BOUNDARY LAYER
1. General comments
Two central considerations in previous studies of the wave boundary layer have been wave-energy dissipation and sediment transport. The former topic has been of particular interest to coastal engineers, and the latter to both engineers and geologists. In order to obtain an adequate understanding of the near-bed processes relevant to these topics, it is necessary to subdivide the rather complicated problem of wave boundary-layer dynamics into manageable parts. Two obvious features o f interest are the wave boundary-layer thickness and the bottom stress under waves. But to make progress in quantifying these and other features, we need at the outset a proper understanding of the classification of wave boundary-layer flow; that is whether a boundary layer is laminar, transitional or turbulent. In addition, we need to understand the complex role of sand ripples in the problem. Furthermore, we need to extend the comparatively simple relationships which exist for sinusoidal flows, to relationships for the more complicated wave-induced flows which are found in nature. In this section, we shall examine these topics and, in the following section, we shall return to the field observations described in Part I, and show how some progress can be made towards interpreting the sediment threshold motion results in Fig. 1.1. It is generally accepted that the velocities induced by low waves above a locally flat region of seabed may be estimated from small amplitude wave theory and, if there are undulations on the bed, the method given in Part 11, Section 3, may be employed to determine the irrotational details of the fluid motion. In the context of field studies in which a spectrum of waves is normally present, we are interested in only those wave components which produce significant motions close to the seabed; in practice, only the components having wavelengths rather larger than the waterdepth “feel” the bottom and, therefore, give rise to the bottom processes of interest in this section. One initial assumption, which underlies almost all studies of the wave boundary layer beneath such surface waves, is that the surface wavelength is large compared with that of the bed roughness (Sleath, 1974b). This condition is usually well satisfied in practice, and it permits the free-stream flow outside the boundary layer to be treated as spatially uniform to a good first approximation. In the notation of Part 11, the assumption that the non-linear convective terms may be dropped from the boundary-layer equations in respect of the surface waves is justified if the wave steepness is such that ( a k )<< sinh(kh); i f any surface wave components have kh 2 7, no measurable bottom motion will be induced in any case. It is generally assumed also that the thickness of the oscillatory boundary layer which develops at the seabed is small compared with the waterdepth. This is normally the case for wind waves and swell, though it is not necessarily so for longer
40
period tidal waves. In effect, this assumption places an upper limit on the length of surface waves, and Jonsson (1978, Sect. 4.1) presents a criterion to be satisfied in this respect as 4na/kh2 < 20 or 30. This condition, which restricts the boundary-layer thickness to a value equal to less than one twentieth of the waterdepth, is well satisfied by the waves in the field experiments described in Part I. A final assumption, which is usually made by coastal engineers, is that the wave boundary layer is rough turbulent. While this is certainly true in situations of engineering importance involving, perhaps, the design of offshore structures or coastal defences, it is by no means always true of the wave boundary layers which exist in shallow waters during the long spells of relatively calm conditions between storm events. In fact, we show in the next section that sediment motion is possible even in boundary layers which are non-turbulent, and such boundary layers are therefore of interest to the physical oceanographer and the geologist. An initial difficulty which arises in studying wave boundary layers in the field is that almost all the available laboratory information on the subject is related to sinusoidal free-stream flows, whereas natural wave-induced flows tend to be rather irregular. This may give rise to some uncertainty in classifying wave boundary layers in the field, since the behaviour of the flow in one half-cycle may depend upon previous half-cycles. This is not a problem for laminar or transitional boundary layers, which may be classified on a half-cycle by half-cycle basis on the assumption of the independence of consecutive wave half-cycles. However, for turbulent boundary layers, this assumption is likely to be valid only if little or no turbulence energy generated in one half-cycle persists until the next half-cycle. In respect of sediment movement, it follows that, while the assumption of the independence of consecutive cycles may be reasonable for a to- and fro-bed-load action in which grain motion both starts and stops within each half-cycle, it is almost certainly invalid if there is a suspended load which persists from one half-cycle to the next. In order to make a boundary-layer classification, it is necessary to determine the wave Reynolds number and the “relative roughness” of the bed. The wave Reynolds number is defined by RE = fimAm/v, where ilk, is the free-stream orbital velocity amplitude, Y is the kinematic viscosity, and = Um/u is the near-bottom excursion amplitude, in which u is the wave frequency. The relative roughness is defined by A,/k, where k,, the equivalent roughness of the bed surface, is determined by the sand-grain size and any ripples present on the bed. For this reason, k , may vary in time, as ripples are formed or eroded by changing wave conditions. In the classification of wave boundary layers, the most basic distinction is between laminar boundary layers, the properties of which depend only on R E , and turbulent boundary layers, the properties of which depend upon both R E and A , / k , . Turbulent boundary layers are further classified as follows. I f the bed roughness is small compared with the thickness of the viscous sublayer, such that the roughness does not extend into the turbulent part of the flow, then the boundary is said to be hydrodynamically smooth. In this limiting case, the properties of the boundary layer depend only on R E , and are independent of i , / k , . However, if the bed roughness is large compared with the thickness of the viscous sublayer, the boundary is said to be hydrodynamically rough, and the properties of the boundary layer depend only on a , / k , , and are independent of R E . Between the laminar and turbulent regimes lies the so-called transitional flow regime.
A,
41
Jonsson (1967) first presented a delineation of flow regimes in terms of R E and . i m / k s , based upon the knowledge available at the time. His conclusions have been modified i n certain respects by subsequent experimental results, and a more recent delineation of regimes has been given by Kamphuis (1975). The best available knowledge at present has been summarized by Jonsson (1978, Sect. 3) (see. later. Fig. 1.14). For smooth beds, Jonsson suggests that the flow is laminar if R E 5 lo4, transitional if lo4 5 R E 5 3 X lo’, and (smooth) turbulent if R E 2 3 X lo’, “give or take a factor of two” at the interval ends. In fact, for design purposes, Jonsson suggests the value R E = lo’ for transition to turbulence. However, there are wide variations in suggested transitional regimes for smooth beds. For example, Collins (1963) obtained the value R E = 1.28 x lo4 for transition to turbulence, though it has since been argued by Sleath (l974b) and others that his experimental observations of “turbulence” were probably of essentially laminar flow phenomena. Collins (1963) also carried out a theoretical stability analysis based upon linear theory, which suggested transition to turbulence at R E = 231. This result has been re-evaluated by Von Kerczek and Davis (1972), who have also proposed limits for guaranteed stability against two-dimensional and three-dimensional disturbances in the flow as R E < 757 and R E < 181, respectively. Kajiura (1968) has calculated the values R E = 6.25 X l o2 and 4.23 X l o 5 for the lower and upper ends of the transitional range respectively, while Kamphuis’ ( 1 975) experimental results suggest values equal to 3 X 10’ and 6 x lo’, respectively. The nature of transitional flow is quite complicated (Merkli and Thomann, 1975; Hino et al., 1976; Ramaprian and Mueller, 1980), and more detailed comments on this subject are included in the next section. For rough beds, the situation is complicated by the need for a knowledge of the “relative roughness” of the bed ( a , / k , ) . For a flat smoothed sand bed, the equivalent roughness k , may be taken as the sediment grain diameter ( k , = D ) (Madsen and Grant, 1976; Hsiao and Shemdin, 1979). For a “not so well smoothed bed”, Nielsen (1979) has suggested the use of k , = 2.50, which is similar to Kamphuis’ (1975) recommendation of k , = 2 Dg0,in which the definition adopted is that 90% of the grains are finer than Dg0. However, in the general case of a rippled sand bed, it is rather more difficult to estimate the roughness. Putnam and Johnson (1949) noted the importance of sand ripples in this context, and Zhukovets (1963) suggested that the equivalent roughness of a rippled sand bed might be taken as D sinh([( Hr + D ) / D ] ’ I 4 } ,where H , is the ripple height, which is of the order of the grain size ( 0 )in the absence of ripples. Jonsson (1967) suggested the equivalent roughness k , = 4Hr, which has been assumed by Tunstall and Inman (1975) and has been recommended also by Hsiao and Shemdin (1979). Nielsen (1979) has also suggested that k , should be of the order of magnitude of the ripple height, and quotes the empirical formula:
from which the relative roughness can be obtained, in which L is the ripple wavelength. In suggesting empirical relationships for the ratios H , / L and Hr/a,, Nielsen distinguishes between the cases of regular waves in the laboratory and
42
irregular waves in nature on the grounds that, in the field, ripples tend to be less steep than under laboratory conditions. A rather different relationship for relative roughness has been proposed by Vitale (1 979) namely: k,
~--
2D9,+0.01 H ,
2,
a,
'This expression, which has been obtained from a re-analysis of the laboratory data of Carstens et al. (1969), places rather less emphasis on the ripple height than those quoted above. More recently, the same data set has been utilised by Grant and Madsen (1982) in formulating an expression for the roughness of a mobile sand bed in oscillatory flow. The roughness has been divided into two contributions, one of which arises from form drag on the ripples on the bed and which is a function of boundary geometry, and the other which is associated with near-bed sediment transport and which is a function of the skin friction component of the bottom stress. The form drag contribution, when expressed in terms of relative roughness, is very similar to the result of Nielsen quoted above, the empirical constant 25 being replaced by 27.7. Once the relative roughness of the bed is known, transition criteria may be employed to classify the flow. Jonsson (1978) has suggested the following formulae as practical lower limits for the rough turbulent regime: R E = lo4
a
if 1
and:
The first of these approximate rules is for very rough walls, and is based upon Sleath's (1 974b) criterion for "fully developed mixing", which Jonsson has transformed to:
'This is similar to an equivalent criterion proposed by Jonsson (1967), and is in good agreement with the results of Zhukovets (1963). The second approximate rule is for less rough walls, and is based upon the criteria of Kajiura (1968) and Kamphuis (1975). Kajiura's suggested formula is: RE=2000
(Ak, j -
and this appears later in Fig. 1.14 together with formula of Sleath (1974b) to delineate the rough turbulent regime. Having classified the oscillatory flow according to the above criteria, we are in a position to estimate the wave boundary-layer thickness. Jonsson (1967, 1978) and Jonsson and Carlsen (1976) have defined the boundary-layer thickness as the
43
distance ( 6 , ) between the bed and the lowest level at which the velocity equals the free-stream velocity, when this latter velocity is maximum. On this definition, Jonsson (1967) has presented a graph of (6,/Am) as a function of R E and ( a , / k , ) . Essentially, in laminar and smooth turbulent flows, (S,/a,) is a function of R E only and, in rough turbulent flows, it is a function of ( a , / k , ) only. In particular, for purely sinusoidal laminar flows above a smooth bed, 8, may be determined analytically and is given by:
For smooth turbulent flow, Jonsson has suggested the empirical expression:
6, -_- 0.0465 -~
"' _ _ JRE
and, for rough turbulent flow, the general rule:
306,
306,
-- l o g l o k ,
k,
=
a,
1.2-
k,
(45)
Jonsson and Carlsen (1976) have plotted ( S , / k , ) against ( a , / k , ) over the range 10
for smooth turbulent flow, and: u 1 - = - 1n(30$) 24,
K
for rough turbulent flow, where u ( y , t ) is the horizontal velocity, y is the upward vertical direction, t is the time, K is the Von Karman constant and u. is the shear velocity, which is defined by u , = J7o/p where T~ is the bed shear stress and p is the fluid density. The equations above are in the forms quoted by Yalin (1972, Ch. 2), in which the experimental constants have been determined from the laboratory work of Nikuradse, and they may be seen to incorporate the molecular viscosity and the bed roughness, respectively, as expected in the light of the earlier discussion. On general arguments, Smith (1977) has suggested a total boundary-layer thickness of the order
44
of (C./a), where C, is the maximum shear velocity in the wave cycle, and he has suggested further that the logarithmic layer has a thickness of the order of 0.02. (C,/o). For typical wind waves, it follows that turbulent boundary layers are only a few centimetres thick, with logarithmic regions of the order of 1 mm thick. In the field, Lukasik and Grosch (1963) have demonstrated directly the existence of a thin laminar wave boundary layer, by the departure of their observations from the predictions of potential theory. However, the wave boundary layer is generally so thin that controlled field observations within it are difficult to make. A related consideration which has attracted much attention from coastal engineers has been the subject of drag coefficients beneath waves, since their accurate determination is an essential step in calculating wave energy dissipation rates. Various definitions of the drag coefficient have been adopted (see Vitale, 1979), but perhaps the most common involves Jonsson's (1967) friction factor, f,, which is defined by the equation:
which relates the peak bed shear stress (fo) in a (sinusoidal) wave cycle to the peak free-stream velocity It should be noted, firstly, that this definition includes a factor (i)which is generally not present in the definition used in tidal work and, secondly, that the two peak values (f0 and d o not occur simultaneously. For laminar flow above a smooth bed, the bottom stress leads the free-stream velocity by 45". However, in the rough turbulent regime, the phase difference is rather .less than this, and decreases with increasing ( i m / k s ) ; for ( a , / k , ) = lo2 and lo', the phase differences are expected to be approximately 29" and 11 ", respectively (Jonsson, 1967). Such values have been confirmed experimentally by Jonsson and Carlsen (1976) who have reported phase differences of 30" and 25" for A , / k , = 28.4 and 124, respectively. Jonsson (1967) and, more recently, Kamphuis (1975) have presented graphs in which f, is plotted as a function of R E and ( A,/k,). These graphs show that f, has the same general behaviour as does the boundary-layer thickness; in particular, in the laminar and smooth turbulent regimes, f, is a function of R E only and, in the rough turbulent regime, it is a function of (a,/k,) only. Rather little is known about the behaviour of f, in the transitional regime, though Zhukovet's (1963) measurements suggest that it displays a complicated dependence on the Reynolds number. In the laminar regime, f, may be determined analytically and is given by:
(urn).
om)
In the smooth turbulent regime, Jonsson has proposed the empirical equation:
and, in the rough turbulent regime, thz equation:
45
Comparable equations have been proposed by Kajiura ( 1 968) and Kamphuis ( 1 975), and the latter equation, which implies decreasing f, for increasing ( a , / k , ) , has been confirmed experimentally by Jonsson and Carlsen ( 1 976). Equation 47 should not be used for small values of relative roughness, however; for ( a , / k , ) < 1.57, Jonsson (1978, Sect. 5 ) has suggested that the constant value f, = 0.30 should be used, which is consistent with Bagnold’s (1946) measurements. This is similar to the value f , = 0.25 for ( a , / k , ) < 1.67 proposed by Kajuira (1968), and tof, = 0.23 for ( d , / k , ) < 1 proposed by Grant and Madsen (1982). The range of low values of ( A , / k , ) for which f, is constant is relevant in practiie if the bed is rippled, and it is shown later that it is associated with relatively inactive flows for which vortex shedding from the bed is probably not fully developed. On the basis of a method of universal velocity distributions, Jonsson (1978, Sect. 7) has derived a relationship between f, and the ratio of boundary-layer thickness to equivalent roughness, namely: 0.0605
This is similar to the equation which results from the elimination of (A^,/k,) from eys. 45 and 47. Vitale (1979) has p r o m d a further relationship between friction factor and relative roughness in the rough turbulent regime, and has noted discrepancies between values obtained for mobile sand beds, and for the fixed sand beds of Bagnold (1946), Jonsson and Carlsen (1976) and Kamphuis (1975). More recently, Grant and Madsen (1982) have suggested a procedure for the determination of the friction factor for a mobile sand bed, using a relative roughness based upon both the form drag on the ripples on the bed, and also the near-bed sediment transport. The wave friction factor has also been discussed by Knight (1978) and Nielsen (1979). If the wave friction factor, and hence the amplitude of the bed shear stress, can be estimated, it is possible to proceed to some considerations of wave energy dissipation by bottom friction. For sinusoidal waves, the energy dissipation rate, o r the “mean specific energy loss” E per unit area of the bed, has been expressed by Jonsson and Carlsen (1976) as:
where the overbar denotes time averaging, and in which f, has been assumed to be constant during the wave period. Although this latter assumption is not strictly true, i t is a good approximation for rough turbulent boundary layers; for laminar boundary layers f , should be replaced in the above equation by 0.833 f, (see Jonsson, 1967). The subject of wave energy dissipation is rather broader than the single consideration of dissipation by bottom friction however, as has been pointed out by Shemdin et al. (1978) and Hsiao and Shemdin (1979). These authors have assessed various field determinations of drag coefficients based on observations of wave energy dissipation rates, and have discussed in detail various additional dissipation mechanisms that may arise in the field, such as bottom percolation for
46
permeable beds and “bottom motion” for beds comprising soft mud (see also Macpherson, 1980). The relative importance of the various mechanisms at a site is strongly dependent upon both the wave and bottom sediment properties. The above authors have concluded that percolation is most effective in coarse sand ( D> 0.5 mm), and that bottom friction is generally dominant when the bed is composed of fine sand ( D= 0.1-0.4 mm). As far as bottom friction is concerned, they emphasise that the friction coefficient is critically dependent upon the presence or absence of sand ripples on the bed, and that it may vary over two orders of magnitude on this account. In fact, they suggest that this may have given rise to the wide variations in wave-energy decay rates at the field sites which t E y have considered. However, Grant and Madsen (1982) have suggested more recently that such variations may be attributable to the effects of near-bed sediment transport. As a result of the work of Putnam and Johnson (1949) and others, i t has been widely assumed that wave friction factors of the order of lo-’ are appropriate in field applications. However, more recent determinations of the friction factor have produced much higher values. For example, Iwagaki and Kakinuma (1967) carried out field studies on a number of Japanese coasts, and obtained values off, from 0.02 to 2.32, with a median value of 0.18. Van Ieperen (1975) computed friction coefficients from wave data obtained at Melkbosstrand, S.A., and obtained values in the range 0.12-0.20. However, in neither of the above studies was the nature of the bed surface reported, and so it is difficult to identify the dissipation mechanisms at work. Treloar and Abernethy (1978) have used prototype wave measurements and hydraulic model data to determine wave friction factors for Botany Bay, Australia, and have obtained values in the range 0.006
41
tance of wave scattering in the present context has been discussed by Shemdin et al. (1978), and the mechanism has been examined by Long (1973), who has found it to be potentially important and strongly dependent on the spectrum of bottom irregularities. A persistent theme throughout the discussion so far has been the role of ripples in the determination of the equivalent bed roughness and, hence, friction factors and wave energy dissipation rates. While a reasonable working understanding of their role has been achieved for practical purposes, a detailed understanding of conditions for their occurrence, and of the wave energy bound up in the vortices formed above them, is only starting to emerge. We shall therefore consider here briefly some of the detailed properties of the interaction between oscillatory flows and rippled sand beds and, thus, attempt to explain the significance of some of the empirical formulae cited earlier. A comprehensive review of the nature of sand ripples has been given by Nielsen (1979, Sect. 5 ) , and his main conclusions are as follows. Firstly, ripples are present when the non-dimensional shear stress 0‘ is between 0.045 and 1.0, where 0’ = .i;/p(S - l)gD, in which S is the relative density of the sediment and ?(; is the peak bed shear stress in the wave cycle related to skin friction (as distinct from the total peak shear stress f0).In particular for 0‘ 2 0.045. the flow is unable to move sediment; for 0.045 2 0’ 2 0.20, sediment motion occurs and ripples form with steepness ( H , / L ) which is independent of 0‘ and equal to about 0.32 tan ‘p,,, where 9)” is the angle of repose of the bed material; for 0.2 2 0’ 5 1.0, the ripple steepness tends to zero with increasing 0’; and, for 0‘ 2 1.0, sand transport rates are very large and the ripples vanish. Under laboratory conditions, Nielsen suggests that the steepness may be calculated from:
H -IT=0.182 - 0 . 2 4 0 ” - 5 L
The ripple wavelength L itself is a complicated function of the flow and sediment , (see also Grant and parameters, but is of the same order of magnitude as 2 Madsen, 1982). Finally, as noted earlier, Nielsen has found that the properties of ripples in the field are strongly influenced by irregularities of the surface waves, the ripples being shorter and flatter than under regular waves. Detailed flow measurements over ripples have been made in the laboratory by Sleath (1975a). For A^,/L 2 0.5, he has observed “random instability” induced by the ripples, but no such instability for smaller values of this parameter. The instability is independent of transition produced by the granular roughness of the bed, and its nature is rather different from that of turbulence in the usual sense. In fact, Sleath argues that transition to turbulence associated with ripples is a gradual process, and that a clearly turbulent flow occurs at values of a,/L many times greater than that at which the first signs of instability appear. The situation is further complicated by the fact that any occurrence of sediment transport tends to inhibit the turbulence somewhat. As far as equilibrium ripple dimensions are concerned, Sleath finds that A,/L usually lies between 0.5 and 5 for natural ripples, and that values of steepness ( H , / L ) are commonly between 0.1 and 0.2. This is in broad agreement with the conclusion of Tunstall and Inman (1975) that ripples achieve equilibrium when 2.75 < A,/H, < 22.
48
As far as drag coefficients associated with rippled beds are concerned, the results of Bagnold (1946) for a fixed bed suggest that the drag coefficient is constant if a , / L < 1, and that it decreases like (a,/L)p3’4 for larger values of a,/L. Bagnold’s result for A,/L< 1 has been recast by Jonsson in the form (quoted earlier) f , = 0.30 for a , / k , < 1.57, on the assumption that k , = 4H, and that the representative ripple steepness H , / L = 1/27~.In the light of Sleath’s criterion for the onset of “random instability” in the flow (namely a , / L 2 0.5), it can be seen that the range A,/L < 1 encompasses both stable and slightly unstable oscillatory flows. Evidently, the decreases in f , which occur for a,/L > 1 are associated with a greater degree ‘of instability in the flow and, ultimately, with a fully developed process of vortex formation and shedding from the bed. Longuet-Higgins (1981) has explained the decrease in terms of ripple troughs becoming filled with vorticity, such that “for very long strokes (i.e. large a , / L ) an almost steady flow is achieved on each stroke”. For movable beds, Longuet-Higgins has noted, on the basis of the data of Carstens et al. (1969), that the drag coefficient is relatively large at a,/L = 0.75, and decreases for larger values of this ratio. He has suggested also that there is a consistent dependence of the drag coefficient on the ripple steepness. Further comparisons based upon the laboratory data of Bagnold (1946) and Carstens et al. (1969) have been made by Tunstall and Inman (1975), who have shown that the experimental observations are accounted for by values off, in the range 0.09-0.5, the actual value depending upon a,/L and decreasing as this ratio increases. This confirms that the drag coefficient for a rippled bed is at least one order of magnitude greater than that for a flat sand bed. If the ripples vanish in active flow conditions (0’ 2 I .O), the drag coefficient is expected to decrease somewhat. However, on account of sediment transport in the near-bed layer, i t is not expected to fall to a value appropriate to a flat immobile bed of, say, f, = 0.01-0.02 (see Grant and Madsen, 1982). This emphasises the need for reliable information about the nature of the bed i f wave energy dissipation rates in the field are to be quantified properly. In this connection, attempts have been made to determine how much wave energy is bound up in the vortices which are formed above. and shed from. a rippled bed. compared with the wave energy which is dissipated by bottom shear, sediment transport and other processes. For example Tunstall and Inman ( 1975) have calculated that shed vortices dissipate only about 7% of the total energy lost due to bottom effects and Vitale (1979), on the basis of his definition of the equivalent bed roughness, has concluded that the percentage contribution to bottom resistance by ripples is 25-34%. However, on the basis of an essentially inviscid theory, LonguetHiggins (1981) has determined drag coefficients which indicate that vortex shedding has a far greater relative importance than has been suggested by these workers. The detailed properties of the oscillatory flow above ripples have been studied theoretically by Sleath (197%) and Tunstall and Inman ( 1975) for smooth crested ripples, and by Longuet-Higgins (1981) for sharp crested ripples. Sleath (1975b) has assumed that the flow remains essentially laminar at all times, and his numerical solution is applicable for low wave Reynolds numbers. Tunstall and Inman (1975) have based their arguments on a unidirectional flow model, in which they assume that the vortices above the lee slopes of the ripples each comprise a rotational inner core and an outer potential region. They have used laboratory data to determine the
49
vortex size, the circulation and the radius of the vortex core and, hence, they have deduced the energy associated with each vortex. A rather different model involving the shedding of vortices in oscillatory flow has been proposed by Longuet-Higgins (198 l), in which discrete vortices are assumed to be generated at the sharp ripple crests with strengths related to the rate at which vorticity is shed from the boundary layer, as given by Prandtl’s rule. The rotational core of each vortex is assumed to expand with time, and i t is shown that, ultimately, combinations of vortices may escape from the bed as vortex pairs. The horizontal force on the bottom is expressed in terms of the vortices in the flow at any instant. We end this review with some comments relating to models of the turbulent oscillatory boundary layer. Such models have generally been related to flat rough beds so that, if ripples are present, their effect on the flow is apparent only through the specification of the equivalent bed roughness. The free-stream flow u = Urn(t ) is normally assumed to be deep and horizontally uniform, arid the linearized governing equation for motion in the turbulent boundary layer is written in the form:
where u( y, r ) is the horizontal velocity, y is the upward vertical direction and t is the time. For rough turbulent flow, the bottom boundary condition is normally taken as u = 0 on y = k , / 3 0 , consistent with eq. 46, and the eddy viscosity v, is defined in the usual way by T / P = v, du/dy. Kajiura (1968) has subdivided the oscillatory boundary layer into inner, overlap and outer layers by analogy with steady flow, and has assumed a different, but steady, form of eddy viscosity in each layer. Although this approach has its limitations, since it assumes an average state of turbulence over the wave period, its predictions are in reasonable agreement with Jonsson’s experimental results, in respect of both velocity amplitude and phase angle. A rather simpler description of the oscillatory boundary layer has been proposed by Brevik (1981), whose two-layer model incorporates the overlap and outer layers only. He also has made comparisons with Jonsson’s experimental results. In fact, Jonsson (1978), on the basis of his laboratory results and also by analogy with steady flow, has demonstrated certain universal velocity and phase relations in oscillatory rough turbulent flow. I n particular, he has proposed universal wall and defect velocity distributions, and has demonstrated the existence of a logarithmic overlap layer. In the “wall layer” adjacent to the bed, in which velocities scale on u, and lengths scale on the bed roughness k , , he proposes the logarithmic velocity distribution given by eq. 46, with K = 0.4. In the “defect layer” away from the immediate influence of the bed, in which velocities may be scaled on the maximum defect velocity and lengths on the boundary-layer thickness, Jonsson has also demonstrated the existence of a logarithmic velocity distribution. This distribution is evident at least in the inner part of the defect layer, and it confirms the existence of a logarithmic “overlap layer”. Kajiura (1968) has estimated that an overlap layer occurs if a,/k, 2 30, the thickness of which increases with increasing a,/k,. The existence of the overlap layer not only has enabled Jonsson to propose eq. 48, but it also permits the prediction of the phase shift between i’, and the maximum shear stress at the wall
(%I.
50
Further studies of oscillatory boundary-layer flow based upon the assumption of a steady eddy viscosity have been made by Johns (1969) and Smith (1977). As required in the “law of the wall” region, Smith has assumed an eddy viscosity which varies linearly with height above the bed. Thus he has obtained analytical results for velocity profiles in the turbulent wave boundary layer, and also results for the amplitude and phase angle of the non-dimensional boundary shear stress, from which wave drag coefficients are simply obtained. In fact, Smith’s model is a simplification to just one layer, namely the overlap layer, of the two-layer model of Brevik ( I 98 l), mentioned above. Bakker (1 975) and Johns (1 975) have adopted more realistic time varying formulations for the eddy viscosity, based upon representations of the Reynolds stress in terms of Prandtl’s mixing length hypothesis. Bakker has adopted v, = K * Y ~ ( ~ u /where ~ Y ~ K, is the Von Karman constant, throughout the flow. Johns has used essentially the same form near the bed, but in his free surface model, in which he allows the boundary layer to occupy half the waterdepth, he takes a viscosity which behaves like l J u / d y l in the upper layers. By integrating the boundary-layer equations numerically, Johns has obtained a logarithmic velocity profile above the bed, the thickness of which depends upon the phase, amplitude and frequency of the wave. More recently, Johns (1977) has used finite difference methods to solve the momentum and turbulent-energy equations for oscillatory boundary-layer flow. In particular, he has calculated the bed shear stress, and has derived an expression for the wave drag coefficient in terms of the bottom roughness. While this approach is more complicated, it is rather more satisfactory physically than the earlier mixing length approach. As far as irregular waves in the field are concerned, Smith (1977) has argued that it is probably reasonable as a first approximation to assume that each wave acts independently, and to assume further that the maximum shear velocity for each wave can be used to scale the eddy viscosity over the wave period. On a general argument, he shows that the peak representative eddy viscosity in a wave cycle is of the order ( i i * ) l / o .The objection to assuming the independence of consecutive wave cycles in modelling the flow in the boundary layer beneath irregular waves is the same as that which was discussed earlier in relation to the classification of wave boundary layers. While the assumption is likely to be valid for laminar and transitional boundary layers, its validity for turbulent boundary layers depends ultimately upon the amount of turbulence energy which persists from one half-cycle to the next. The selective review above has concentrated on certain hydrodynamical aspects of wave-induced flows. No detailed discussion of sediment-transport mechanisms has heen included, nor has the important situation in which the near-bed motions comprise contributions from both waves and currents been examined. In the latter case, a thick boundary layer is generally caused by the currents and a thin one by the waves; typically, for reversing tidal currents on the continental shelf, the tidal boundary-layer thickness might be of the order of 100 m, with a logarithmic layer of the order of a few metres, which is three or more orders of magnitude thicker than the wave boundary layers discussed earlier. It follows that turbulence generated by waves affects a rather different region of the flow from that which is affected by turbulence generated by currents. The interaction between the two turbulent motions in this general case has been discussed by Smith (1977), in connection with the
51
modelling of sediment transport processes on the continental shelf. More generally, despite the wide variety of previous studies of oscillatory boundary-layer dynamics and sediment transport by waves, much remains to be understood about the detailed properties of the flow near the bed. Although it may be argued that a reasonable working knowledge of many important features of the wave boundary layer now exists-for example, there appears to be general agreement about flow classification and the magnitudes of wave drag coefficients-there is an inadequate knowledge of other aspects, particularly those concerning sediment movement. For the field worker, these uncertainties are compounded by the fact that laboratory results have often been obtained for idealised situations in which the waves are sinusoidal, the bed is flat and the sand is of a single size. In real situations in which none of these conditions is satisfied, the use of laboratory results can be questionable. I n the next section, we see what progress can be made towards gaining a basic understanding of a particular set of field observations. Since the case discussed is a relatively uncomplicated one involving non-separating flow and bed load motion only, it serves to emphasise the difficulty which arises in treating more complicated cases in which, for example, the flow is separating and there is a suspended-sediment load.
2. Sediment trunsport in u trunsitionul wuve boundury luyer beneath irregulur
WNU~S
We shall be concerned in this section with an interpretation of the field results described in Part I. In the light of the comments made in the previous section, we shall be concerned with the nature of the flow in the boundary layer beneath the irregular measured waves, and with the calculation of the bed shear stress o n both the rippled and flat sand beds discussed earlier. For the interpretation of the present field results, we adopt an approach which assumes an “almost-laminar” boundarylayer structure, in which we attempt to extrapolate certain well-accepted ideas about laminar boundary layers into the transitional regime. It was noted in Part I that, when the bed was rippled in the field experiments, flow separation did not occur above the lee slopes of the ripples, even for the waves with the largest velocity amplitudes. Therefore, on the initial assumption (justified shortly) that the boundary layer was thin in relation to the ripple wavelength, we start by taking the calculated potential velocity, U7(x, r ) , at each point on the rippled surface as the “free-stream” velocity above an assumed flat sand bed (Figs. 1.2 and 1.13); this surface velocity may be determined in relation to the measured velocity U,(t), by one of the methods in Part 11, Section 2, as U,(x, t ) = C(x)U%(t). Clearly, for a flat bed, we take C( x) = 1 in this equation. However, for the rippled
F’lg. 1.13. Definition sketch. The “boundary-layer” part of the decoupled problem depicted in Fig. 1.2.
52
bed discussed in Part I, it may be shown that the enhancement of surface velocity at the crest was such that U , ( x , t ) = U,,,,(t) -- 1.9 U,(t). We use this result later. In order to justify fully the decoupling of the problem (Figs. 1.2 and 1.13). we follow Sleath (1975a, paragraph 16) in arguing initially that, for high values of the quotient L/S, ( L = ripple wavelength, 6, = boundary-layer thickness), the velocity distribution above a typical point on a rippled bed is nearly the same as that above a flat bed, provided that the (inviscid) free-stream velocity in the latter case is set equal to U T ( x , t ) . Essentially, the restriction on L/S, permits the non-linear convective terms, which arise in the boundary-layer equations in respect of the ripples, to be neglected. Now, if the wave Reynolds number is sufficiently small that the boundary layer is non-turbulent, we may estimate the order of magnitude of the where v is the kinematic viscosity of the boundary-layer thickness as 6, = water and u is the wave frequency; hence, for the field results in Part 1 for which 85 cm, v = 0.0131 cm2 s - ‘ and T = 2 n / o = 9 s, we obtain L/S, = 439. Therefore, if it can be demonstrated from the field data that the Reynolds number was sufficiently small for this value to be appropriate, the assumption of a boundary layer which was thin compared with the ripple wavelength is justified. In order to determine the nature of the flow in the boundary layer, we treat the rather irregular measured waves as if they were effectively sinusoidal and capable of being classified on the basis of the criteria discussed in the previous section. Furthermore, having argued above that, for a rippled bed, the velocity distribution in a non-turbulent boundary layer is nearly the same as that above a flat bed, we expect transition to turbulence to occur in the same way as for a flat bed, provided that the properly corrected surface potential velocity is taken as the velocity outside the boundary layer. As pointed out in the previous section, in making a flow classification it is necessary to calculate, firstly, Jonsson’s (1967) wave Reynolds number, defined for the present purpose by R E = a,A,/v where 2, = fiT/o and, secondly. the “relative roughness” of the bed A T / k , , where k , may be taken as the representative grain diameter D if the bed is flat. For a field experiment of the present type for which ranges of U T ,A , and D can be estimated, a region may be identified on a graph.of RE‘ against relative roughness to classify the measured waves. Such a region is shown and, in in Fig. 1.14, This relates to the ripple crest position (i.e. ,?i = particular, to the same set of data as presented in Fig. 1.la. The shaded area in Fig. 1.14 encompasses all those waves whch moved sediment in Fig. ].la; in the calculations a range of possible mobile sediment sizes, namely D = 1-2 mm (D,, = 1.4 mm), has been taken to allow for the natural mixture of grain sizes present. The waves for which no sediment motion was observed were associated generally with small values of RE, lying in the region marked “no motion”. Also shown in Fig. 1.14 are Jonsson’s ( 1967, 1978) recommended flow-classification criteria for laminar, transitional and turbulent, wave boundary-layer flows. It may be seen that, in the present experiments for the rippled bed, the flow was transitional according to these criteria in almost all the wave cycles recorded, fully justifying the assumption of a non-turbulent boundary layer. The same conclusion has been reached for the flat So, in order to interpret the present field bed results in Fig. l.lb, for which U, = observations, we appeal to previously published results for the transitional regime and, in particular, we make use of the results of Sleath (1970).
\/2v/a,
z=
k,,,,)
am.
53
SMOOTH'
LAMINAR
a , D
/
/
/
/
/
TRANSITIONAL
, 10'
JTC
,,
, , / ,
, ,,,,,,I , , , ,,,,,,,
1 o4
105
10'.
, ,
;1
, ,,,,,
10'
0,K T V
Fig. 1.14. Classification of flow regimes. The shaded area encompasses all the waves which moved sediment in the present field experiments on the rippled bed (2.53 X lo4 < R E i3.55 X lo5; grain \ i x I < D < 2 mm). Solid lines = Jonsson's (1978) recommended criteria for the boundaries between transitional and both laminar and turbulent flows. The two curves delineating the rough turbulent regimc are due to Sleath (1974b) for very rough walls, and Kajiura (1968) for less rough walls. Dashed line = Jonsson's (1967) earlier criterion for the boundary between transitional and rough turbulent flows. Dash-dot line = Jonsson's (1967) earlier criterion for the boundary between transitional and laminar flows.
Sleath has argued that the usual ideas of turbulence are inappropriate in considering flows which are not fully rough turbulent, and has based his approach upon a development of the usual arguments for laminar boundary-layer flow above a flat plate. For a deep oscillatory flow above a flat bed of sand, he has shown experimentally that departures from the familiar Stokes' shear wave solution occur. However, he has established that these departures have an ordered nature involving, firstly, the formation of flow-separation zones (i.e. vortices) at the level of the surface grains on the bed and, secondly, the periodic shedding of these vortices from the bed in each wave half-cycle. For monochromatic waves and a flat sand bed, for which the = Urn,Sleath has proposed the following criterion for surface velocity amplitude the onset of this process:
cl
U lD Z > 115 US*
Since those swell waves in the present experiment which moved sediment on the rippled bed had values of this parameter in the estimated range ( 1 17, 440), it seems likely that the vortex shedding process was important in the present case. Further experimental studies of the transitional wave boundary layer have been carried out by Merkli and Thomann (1975), Hino et al. (1976) and Ramaprian and Mueller (1980). Merkli and Thomann (1975) examined oscillatory flow in a smooth pipe and
54
observed turbulence occurring in the form of “periodic bursts”, followed by the relaminarisation of the boundary layer near to flow reversal in each cycle. They observed this phenomenon at R E = 4 X lo4, and suggested that it may occur in a range of Reynolds numbers 3.6 X lo3 < RE < 1.81 X lo5, consistent with results of a quasi-steady stability theory. For R E > 1.81 X lo5 they suggested that turbulent flow occurs throughout the wave cycle, though they made no observations to support this. Hino et al. (1976) performed experiments in a smooth pipe also, and identified a - 1.51 X l o 5 and A, > 1.6, where A, regime of “conditional turbulence” for R E 2 = i d / & and d is the pipe diameter. In this regime, they observed turbulence only in the decelerating phase of the wave cycle while, in the accelerating phase, a laminar-like flow was recovered. Finally, Ramaprian and Mueller (1980) have presented experimental results relating to a sinusoidal free-stream flow above a smooth bed in a U-tube. In a single experiment at RE = 7.1 x lo4, they observed random fluctuations in the boundary layer during the wave cycle, indicating that the flow was transitional. The maximum perturbations in velocity were found at a height of about 0.4 6, above the bed. They found that, both in respect of phase and amplitude, the velocity distributions in the boundary layer were not too different, in the mean sense, from theoretical predictions for laminar flows. This lends some support to the argument which follows. Despite the rather complicated nature of transitional oscillatory flow above a sand bed, the method which we adopt here involves modelling the boundary layer on the basis of a linear governing equation, and then correcting the results obtained to allow for the vortex shedding process described by Sleath (1970). The general aim of the procedure is to transform data sets of measured free-stream velocity, containing a spectrum of wave components, into equivalent “data-sets’’ of deduced bed shear stress. The specific aims are to examine the relationship between the free-stream velocity and the bottom stress in corresponding irregular wave half-cycles, and to examine also velocity amplitude and phase relationships in order to help explain certain aspects of the observations of sediment motion described in Part I. The problem is tackled by making a simple empirical modification to solutions of the linear equation:
where y is the upward direction normal to the bed, such that u ( x , y , t ) is the component of the velocity in the x-direction, locally tangential to the bed and locally uniform in x(cf. u( y , t ) in eq. 49). The stress T acting in the x-direction on a plane parallel to the bed is given by: r = p,-
dU
JY where p w = pv and p is the density of water. The bottom boundary is assumed at this stage to be the smooth plane y = 0 upon which the condition: u=O ony=0 is satisfied, and the motion in the inviscid region is taken as: u = UT(x, t )
asy
-+ 00
(52) (53)
55
This problem for the calculation of the bottom stress is of a standard boundary value type, and we consider below two methods of solution. Firstly, however, we deal with the question of the empirical modification to the solution of eqs. 50-53 to allow for the process of vortex formation and shedding above a rough bed. In a time-averaged sense, this process involves a thickening of the wave boundary layer. Hence, Sleath (1970) and Keiller and Sleath (1976) have accounted for it by assuming an increased value for the shear wavelength in the solution of eq. 50, namely X S , where X i s an experimentally determined constant, as opposed to 6. for a smooth bed. For a smooth flat bed and a monochromatic wave train, Sleath has shown experimentally that X = 1 as expected; but for permeable and impermeable beds of coarse sand (& = 1.13 mm), values have been found in the range 1 < X < 1.8. Such values give good agreement with velocity amplitudes measured at various heights in the boundary layer. From his own data, and also that of Kalkanis (1957, 1964), which was obtained in rather more active conditions. Sleath has proposed the following general behaviour for X:
For the purpose of calculating the stress near the bed, the following construction may be put upon X . In the first place, for waves of single frequency, the replacement of 6, by XS, is equivalent to replacing the dynamic viscosity p, by a steady effective viscosity X 2 p , in eq. 51. Assuming that the instantaneous shear d u / a y in eq. 51 can be retained in unmodified form, the instantaneous stress is then given by X’p,. d u / a y . This procedure has been followed by Davies (1980b) in interpreting the field data in Part I, with a representative value of X , assumed for simplicity to be constant throughout the experiments, and calculated from eq. 54 for the waves which moved sediment as X = 2.1. It remains only to determine the velocity field u ( x , y , t ) from eq. 50, subject to eqs. 52 and 53, and one obvious approach is to seek a solution based on a Fourier decomposition of the measured velocity data. If this data consists of a time series of 2 M equally spaced values, a curve passing through each of the data points can be expressed in the Fourier series form: M
M
where w = 2m/TR and T, is the record length. The coefficients in this series may be obtained, for example, by using a Fast Fourier Transform routine. For each constituent harmonic of the series, a solution of eq. 50 is readily obtained; in particular, the horizontal velocity in the boundary layer is given for the qth harmonic by:
where S,
=
\ / 2 v / q w . The corresponding component of stress,
rq, is
then given by
56
scaling the result from eq. 51 by X 2 and, eq. 50 being linear, the complete solution for the stress is obtained by superimposing the solutions T ~ q, = 0 to M . This procedure has been carried out on the velocity data sets obtained in the field experiments described in Part I, and some typical results are shown in Fig. 1.15 in which a portion of the horizontal velocity data [U,(t)J measured at a height of y = 1 m. (>> 6,) above the rippled sand bed is considered. Solutions for the stress in the boundary layer at the ripple crest position are shown for the “notional bed level” ( y = 0), and for two levels close to the bed ( y = 1 and 2 mm). These predictions of stress are based on the estimates C ( x ) = 1.9 for the crest and X = 2.1, such that C X 2 = 8. The ‘unsteadiness of the flow limits the thickness of the boundary layer, and this results in a very rapid attenuation of the magnitude of the stress with increasing height above the bed. Also, temporal phase shifts take place with height; the stress T leads Urn in phase very close to the bed, but this phase lead becomes small even at the height y = 2 mm. Also, due to the irregular nature of the velocity record and the fact that the bed shear stress depends upon the time history of the free-stream velocity, there is not quite a one-to-one correspondence between the peak stress and the peak velocity in each of the two half-cycles shown. ( I t should be noted here that the raw velocity data was reduced to having a zero mean ( A , , = 0 in eq. 5 5 ) , and that the same is therefore true of the deduced records of stress.) The peak values of velocity Urnin the two half-cycles are - 27.9 and 15.3 cm s- and the corresponding peak values of stress at, for example, y = 0 are - 20.2 and + 12.6 dynes cm-2. The (dimensional) quotients of the respective pairs of velocity and
+
’,
Fig. 1.15. A typical portion of measured horizontal velocity data, and corresponding sets of deduced shear stress “data” for three trial bed levels with C X 2 = 8. The bars on the horizontal axis indicate the time5 between which sediment motion occurred; the direction of this motion, shoreward ( + ) o r seaward ( - ) , corresponded to that of Urn and 7.
stress values are 1.38 and 1.22, and we see shortly how the existence of substantial differences of this type, systematically throughout the measured records, helps to explain to some extent the existence of the transition ranges in the peak velocity histogram in Fig. 1.la. Also indicated in Fig. 1.15 are the times between which sediment motion occurred in this typical portion of record. Particularly in the second half-cycle plotted, it can be seen that the instant of the onset of sediment motion is far more readily explicable in terms of the behaviour of T at the level y = 1 mm, than in terms of U, or T at the levels y = 0 or 2.0 mm. This indicates that considering stress at an “effective bed level” slightly abovey = 0 may be necessary to explain the sediment threshold results. We return to this point later. Before doing so, however, we consider briefly an alternative method for calculating u and T in the boundary layer. The solution of eq. 50 is now by an integral transform method, and it may be written in the form:
see Carslaw and Jaeger (1959, Sect. 14.2). This solution satisfies the boundary conditions (eqs. 52 and 53) and the initial condition: u ( x , y , 0 ) = C(X)U,(O)
The uncorrected stress
T(X,
y , t ) is given from eq. 51 by:
where:
and, in the present application, the final estimate for the stress is obtained by scaling the values from eq. 56 by X 2 . From a numerical integration of eq. 56, it is possible to assess the importance of the time history of Urn(t’) in t’ < t , in determining the stress T at time t . In practice, the weighting function W ( y , t , t’) is such that good estimates of stress can be obtained-that is, good agreement can be achieved with the earlier Fourier series method-by integrating in the restricted range ( t - t ) < t‘ 6 t. where t is rather less than one wave period. If, for example, U r n ( [ )is prescribed as a sine wave with the typical swell wave period of 10 s, and if the stress is evaluated at the level y = 1 mm, then for > 6 s it may be shown that there is agreement throughout the wave cycle between the two calculated time series of stress to within 5% of the amplitude of the stress. If i 2 4 s, the agreement is still within 10%. As far as the predicted peak stresses at y = 1 mm are concerned, agreement is to within 3% if r^ ;a 5 s, and to within 8% if 1 >, 4 s. I n other words, for swell wave frequencies, the stress at time t depends effectively upon the time history of only the preceding few seconds of the motion in the free-stream flow. This has been found also when Urn(t ) has been identified with the rather more irregular measured velocity data obtained in the field. Closer to the bed than y = 1 mm, the timescales 1 equivalent to those
58
quoted above are smaller but, at the base level ( y = 0) itself, a difficulty arises in evaluating the stress by the integral transform method, due to the non-uniform convergence of the integrand of eq. 56 as y + 0, t‘ + t. Nevertheless, for y-levels of interest in the present context, the integral in eq. 56 can be evaluated without undue numerical difficulty. So, not only does eq. 56 provide an independent check on the earlier Fourier series method, but it also permits an interesting insight into the importance of the time history of the free-stream velocity in determining the bottom stress. The remaining unresolved question, raised by the results for the bottom stress in Fig. 1.15, is as follows: If the “almost-laminar” theory is to be applied, then at what level above the notional bed level y = 0 should the shear stress be evaluated? In other words, can an “effective bed level” be determined at which the stress “acts” on the surface layer of sand grains? On a purely intuitive basis, this level might be expected to be of the same order of magnitude as, but perhaps rather less than, the median grain diameter. This question has been the subject of a detailed investigation by Davies (1980b), and an attempt has been made to answer it with reference to equivalent data sets of measured velocity and deduced shear stress of the type shown for the short portion of data in Fig. 1.15. A direct experimental determination of the level with reference to velocity profiles was not possible, since no velocity measurements were made in the field experiments across the very thin boundary layer. Instead, a somewhat less direct empirical approach has been adopted involving sets of instantaneous values of measured velocity, U m ( t ) ,and deduced stress, T , at the onset of sediment motion, such as may be determined from the left ends of the sediment motion bars on the time axis in Fig. 1.15. The threshold stress values have been used in an optimisation scheme to minimise the scatter in sets of such values as the bed level is varied. The optimum level which minimises the scatter has been taken as the “effective bed level”. For the rippled bed data in Part I. the optimum level has been found by this approach to bey,,,, = 0.93 mm, which is approximately two thirds of the median grain diameter (& = 1.4 mm). This is not only a physically sensible result and one which is consistent with the earlier expectation based on Fig. 1.15 that y,,,, = 1 mm, but it is also one which is associated with a sharp reduction in the relative amount of scatter in the results at the onset of sediment motion, when instantaneous velocity thresholds based o n C(,i I ) are compared with deduced stress thresholds. In fact, over all the available instantaneous threshold velocity and threshold stress data for the rippled bed, a reduction in the “relative variance” for stress at y =y,,,,, compared with velocity, of about 62% has been found, where the “relative variance” has been defined as the ratio of the variance of the set of instantaneous threshold stress (or velocity) values to the square of the mean of those values. This reduction is substantial and is thought to fully justify the present approach. The general, and expected, implication of the result is that it is the near-bed velocity field which explains the forces on the bed, and not the free stream velocity. While the analysis outlined above relating to instantaneous threshold motion conditions produces a dramatic reduction in variance when bottom stress rather than free-stream velocity is considered, a similar, though less marked, improvement in the definition of threshold conditions is found in histograms (cf. Fig. 1.1) based upon
Occurrence
t
t
RIPPLED BED
8o
(a)
-20
0
-10
U
2o
10
dynes cm-:
U
Transition
Ranqe
Occurrence
80
6o
'
FLAT BED
7 1
(b) y
=
ycrlt= 0.93m
Fig. 1.16. Histogram representation of the deduced shear stress at the effective bed level y = 0.93 rnm. Results for the rippled bed, corresponding to Fig. ].la, are shown in 1.16a (CX' = 8). The transition ranges are indicated by the arrowed bars. Results for the flat bed, corresponding to Fig. l.lb, are shown in 1.16b. The arrows show the possible lower bounds of the transition ranges.
60
the deduced stress at Y,,,~ = 0.93 mm. Of particular relevance in this context are the transition ranges in the histograms, in which only a fraction of the waves achieving particular values of velocity amplitude (Fig. 1.1), or deduced shear stress (Fig. 1.16), moved sediment. Possible sedimentological reasons for the existence of the transition ranges in the velocity histograms have been discussed by Davies and Wilkinson ( 1979). However, such explanations cannot realistically account for the substantial widths of the transition ranges, and a satisfactory understanding of the problem as a whole, leading to reductions in the widths, can be expected to be achieved only in terms of a correct modelling of the bed shear stress. Since, in practice, there is not a one-to-one correspondence between peak stresses and peak velocities in corresponding wave half-cycles, as found in Fig. 1.15, histogram results based on measured velocity data may be expected to differ from those based on shear stress “data” calculated at the effective bed level y =yCri,.In Fig. 1.16, histogram results based upon the deduced shear stress are presented for the same rippled and flat bed data sets as in Fig. 1.1. In Fig. 1.16a for the rippled bed, the transition range widths are indicated by the arrowed bars and, as in Fig. l.la, certain possible rogue values at the extremities of the main ranges have been disregarded. Ratios of the widths of the transition ranges to the sum of the widths of the transition ranges and the “ n o motion” ranges have been calculated for Figs. 1.la and 1.16a, and the results averaged for the positive and negative directions of motion for each histogram. By comparing these ratios, it has been found that there is a 19% reduction in the transition range width if the bottom stress at y = y c r i ,is considered rather than the measured free-stream velocity. However, despite this improvement, the transition range widths in Fig. 1.16a are still substantial and unfortunately, for sedimentological and observational reasons, it is probably unrealistic to hope to eliminate them entirely by further refining the physical arguments. One possible reason for their persistence is that the extension of the calculation of stress beyond the time of the onset of sediment motion in any given wave half-cycle may cause the peak stress to be in error on account of the sediment movement itself (Sleath, 1975a, paragraph 30). Their persistence may be due also to the oversimplified way in which the factor X given by eq. 54 has been determined, and applied, in the present study. As far as the flat bed results in Fig. 1.16b are concerned, we find that, as in Fig. l.lb, we cannot arrive at a clear conclusion concerning the transition range width, on account of the lack of a significant number of occurrences of sediment motion. However, since C = 1 in this case (as opposed to C = 1.9 for the rippled bed), it may be noted that there is broad agreement as to the sediment threshold stress (7-8 dynes cm-’) in Fig. 1.16a and b, and so the apparent discrepancy between the threshold values in Fig. 1.la and b is satisfactorily resolved. I t has been suggested by Davies (1980b) that, for the coarse sand sizes in the present experiments, the mechanisms of initial sediment entrainment are similar both in these experiments in unsteady flow, and in experiments such as have produced the usual critical Shields curve for grain motion in steady flow in the laboratory. This suggestion has been made partly on the basis of some simple considerations of the ratio of the laminar sublayer thickness to the sediment grain size at the threshold of sediment motion in the steady turbulent flow case, and partly on the basis of the known effects of unsteadiness in the flow on the threshold of
61
motion. On the latter point, it has been shown by Madsen and Grant (1976), and Nielsen (1979), that inertia forces acting on the surface layer of sand grains in an oscillatory flow are unimportant, and that the empirical Shields criterion is applicable as a quite general criterion for the initiation of motion in oscillatory flow. provided the bottom stress is properly evaluated. In practice, Madsen and Grant have used Jonsson’s (1967) friction factor to calculate the bed shear stress and, on this basis, they have replotted the experimental sediment threshold data of Bagnold (1946), Manohar (1955) and others, on a modified Shields diagram. Similar comparisons have been made by Komar and Miller (1975) and, in both cases, the agreement between experimental results obtained in unsteady flow, and Shields curve for steady flow, is good. It may be argued, therefore, that it is legitimate to compare sediment threshold stresses calculated on the basis described earlier, with threshold stresses determined from steady-flow experiments in the laboratory. For the median grain size ( D5, = 1.4 mm), the critical threshold stress amplitude obtained from the modified Shields diagram presented by Madsen and Grant is ?(, = 7.9 dynes cm-’, which is in close agreement with the results in Fig. 1.16. (Since the total bed shear stress is equal here to the skin friction +(;, we may express this value in non-dimensional form as 8’ = +(i/p( S - 1)gD = 0.034, which is in order of magnitude agreement with the value given by Nielsen (1979) for the threshold of motion of 8’ = 0.045.) As might be anticipated from the results in Fig. 1.15, such agreement is not found for histograms based on choices of effective bed level differing substantially from the optimum value y = yc,,,. For the grain sizes at the extremities of the observed range ( D = 1.0 and 2.0 mm), the equivalent values of peak stress from Shields diagram are f0= 5.5 and 12.95 dynes c m P 2 ( 8 ’ = 0.033 and 0.039), respectively. I t might be thought that the excellent agreement above for the D,, size is somewhat fortuitous. However, further support for the present approach may be obtained by comparing bottom-stress values calculated by the present method with values calculated on the basis of Jonsson’s (1967) friction factor. If the rather irregular waves in the measured records are assumed to be sinusoidal and are characterised by their velocity amplitudes and periods in successive wave half-cycles, it is possible to estimate bed shear stress amplitudes on a wave by wave basis, and then to compare critical stress values for the onset of sediment motion with threshold values obtained from Shields curve. I t was shown earlier that the flow in the present experiments was in an intermediate condition between laminar and both “smooth” and “rough” turbulent flow states. An unfortunate consequence of this is that there arises considerable uncertainty as to appropriate values for the friction factor, f w , due to the lack of laboratory measurements in this regime. However, from the graph presented by Jonsson (1967) on which curves have been sketched from a knowledge of the limiting values of f w , estimates of the friction factor have been made, and hence estimates of the amplitude of the bed shear stress ?,; = i p f , f i ? have been obtained. For the case of the rippled bed the predicted stresses at the lower and upper limits of the transition range are 2.1 and 11.4 dynes cm--2 (8’ = 0.009 and 0.049) respectively, based upon an equivalent bed roughness equal to the median grain size D,, = 1.4 mm such that f, = 0.018 and 0.013, respectively. For a representative value in the middle of the transition range, namely fim= 15 cm s -
+,
’
62
(fi,. = ficrc,t = 28.5 cm s- I), an estimate of
6.1 dynes cm-* (8’ = 0.026) is obtained for the D50 size ( f , = 0.015), while the values 5.7 and 6.7 dynes c m p z (8’ = 0.034 and 0.020) are obtained for D = 1.0 mm (f, = 0.014) and D = 2.0 mm (f, = 0.0165), respectively. Since these values purport to correspond to the threshold motion condition, it is relevant to compare them with the values from Shields curve of 5.5, 7.9 and 12.95 dynes cmp2, quoted above for the grain sizes 1.0, 1.4 and 2.0 mm, respectively. The agreement here is reasonable, at least for the smaller sizes. So, despite the uncertainty involved in the calculation, and the crude assumption that the waves were sinusoidal and, therefore, had almost a one-to-one correspondence between peak velocity and peak stress in corresponding wave half-cycles, it is evident that use of Jonsson’s friction factor enables some progress to be made in interpreting the present field results. In conclusion, we note that the approach described in this section relies upon the flow being non-separating and upon the legitimacy of the decoupling of the “ripple” and “ boundary-layer’’ problems. Ultimately, the physical argument underpinning it is that sediment threshold motion conditions should be sought on a deterministic basis, and that uncertainties in these conditions should be treated on a stochastic basis only as a matter of last resort. Although the approach appears to be well justified by the general improvement in the definition of the critical conditions at the threshold of sediment motion when bottom stress is examined rather than the velocity in the free-stream flow, it is not yet clear whether the detailed procedure described in this section is generally applicable in the transitional boundary-layer regime, due to the insufficient number of grain sizes and wave periods tested. The topic requires further study. Also it may be argued, with some justification, that the procedure is not of very great practical importance, due to the low values of wave Reynolds number associated with the field observations. It is possibly true, as suggested by Smith (1977), that on the timescale of years most sediment transport at a typical offshore site takes place as a result of a few extreme storm events. However, we have been able to demonstrate here that, even at low wave Reynolds numbers, sediment motion occurs on a bed of coarse sand, and the situation described is therefore one of considerable interest to physical oceanographers and geologists. Clearly, cases of greater practical importance will generally be more complicated and require different analysis techniques. Even if such techniques presently exist, the relatively simple exercise described above serves to emphasize how involved an analysis of this kind might become. In short, despite our fairly sophisticated present understanding of certain aspects of both oscillatory boundary-layer flow and sediment transport by waves, much basic research remains to be carried out in order to build up a complete understanding of these phenomena.
REFERENCES Allen, J.R.L., 1968. Current Ripples. Their Relation to Patterns of Water and Sediment Motion. North-Holland, Amsterdam, 433 pp. Bagnold, R.A., 1946. Motion of waves in shallow water. Interaction between waves and sand bottoms. Proc. R. Soc. London, Ser. A, 187: 1-18.
63 Bakker, W.T., 1975. Sand concentration in an oscillatory flow. Proc. 14th Coastal Engineering Conference, Copenhagen, Ch. 66, pp. 1129-1 148. Brevik, I., 1981. Oscillatory rough turbulent boundary layers. Proc. Am. Soc. Civ. Eng. J. Waterway. Port. Coastal and Ocean Division, 107 (WW3): 175-188. Carslaw, H.S. and Jaeger, J.C., 1959. Conduction of Heat in Solids (2nd ed.). Oxford University Press, Oxford, 520 pp. Carstens, M.R., Neilson, F.M. and Altinbilek, H.D., 1969. Bed forms generated in the laboratory under an oscillatory flow: Analytical and experimental study. U S . Army Corps Eng., Coastal Eng. Res. Cent., Tech. Mem., 28. 105 pp. Collins. J.I., 1963. Inception of turbulence at the bed under periodic gravity waves. J. Geophys. Res.. 68: 6007-6014. Davies, A.G., 1979. The potential flow over ripples on the seabed. J. Mar. Res., 37: 743-759. Davies. A.G., 1980a. Some interactions between surface water waves and ripples and dunes on the seabed. Inst. of Oceanographic Sciences, Rep. 108, 134 pp. Davies, A.G.. 1980b. Field observations of the threshold of sand motion in a transitional wave boundary layer. Coastal Eng., 4: 23-46. Davies, A.G., 1982a. O n the interaction between surface waves and undulations o n the seabed. J. Mar. Res.. 40: 331-368. Davies, A.G., l982h. The reflection of wave energy by undulations on the seabed. Dyn. Atmos. Oceans. 6: 207-232. Davies. A.G. and Wilkinson, R.H.. 1979. Sediment motion caused by surface water waves. Proc. 16th Coastal Engineering Conference, Hamburg, Ch. 94, pp. 1577- 1595. Davies, A.G., Frederiksen, N.A. and Wilkinson, R.H., 1977. The movement of non-cohesive sediment hy surface water waves. Part 2: Experimental Study. Inst. of Oceanographic Sciences. Rep. 46. 80 pp. Fitz-Gerald, G.F., 1976. The reflexion of plane gravity waves travelling in water of variable depth. Philos. Trans. R. Soc. London, Ser. A, 284: 49-89. Grant, W.D. and Madsen, 0,s.. 1982. Movable bed roughness in unsteady oscillatory flow. J .
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Kalkanis, G., 1964. Transportation of bed material due to wave action. U S . Army Corps Eng., Coastal Eng. Res. Cent., Tech. Mem., 2, 68 pp. Kamphuis. J.W., 1975. Friction factor under oscillatory waves. Proc. Am. Soc. Civ. Eng., J. Waterway, Port. Coastal Ocean Div., 101 (WW2): 135-144. Keiller. D.C. and Sleath, J.F.A., 1976. Velocity measurements close to a rough plate oscillating in its own plane. J . Fluid Mech., 73: 673-691. Knight, D.W., 1978. Review of oscillatory boundary layer flow. Proc. Am. SOC.Civ. Eng., J. Hydraul. Div.. 104 (HY6): 839-855. Komar. P.D. and Miller, M.C., 1975. Sediment threshold under oscillatory waves. Proc. 14th Coastal Engineering Conference, Copenhagen, Ch. 44. pp. 756-775. Kreisel, G., 1949. Surface Waves. Q. Appl. Math,, 7: 21-44. Lamb, H., 1932. Hydrodynamics (6th ed.). Cambridge University Press, Cambridge. 738 pp. Long, R.B., 1973. Scattering of surface waves by an irregular bottom. J . Geophys. Res., 78: 7861-7870. Longuet-Higgins, M.S., 1977. The mean forces exerted by waves on floating or submerged bodies with applications to sand bars and wave power machines. Proc. R. Soc. London, Ser. A. 352: 463-480. Longuet-Higgins, M.S.. 1981. Oscillating flow over steep sand ripples. J. Fluid Mech., 107: 1-35. Lukasik, S.J. and Grosch, C.E., 1963. Pressure-velocity correlations in ocean swell. J. Geophys. Res., 68: 5689-5699. Macpherson, H.. 1980. The attenuation of water waves over a non-rigid bed. J. Fluid Mech.. 97: 721-742. Madsen, 0 , s . and Grant, W.D.. 1976. Sediment transport in the coastal environment. M. I. T.. Dept. of Civil Engineering, Rep. 209, 105 pp. Manohar. M., 1955. Mechanics of bottom sediment movement due to wave action. U.S. Army Corps Eng., Beach Erosion Board, Tech. Mem., 75, 121 pp. McColdrick, L.F., 1968. Long waves over wavy bottoms. University of Chicago. Dept. of Geophysical Sciences, O.N.R. Ocean Science and Technology Group, Tech. Rep. I . 67 pp. Mei, C.C. and Black, J.L., 1969. Scattering of surface waves by rectangular obstacles in waters of finite depth. J. Fluid Mech., 38: 499-511. Merkli, P. and Thomann, H.. 1975. Transition to turbulence in oscillating pipe flow. J . Fluid Mech., 68: 567-575. Miles, J.W., 1967. Surface-wave scattering matrix for a shelf. J. Fluid Mech.. 28: 755-767. Milne-Thomson, L.M., 1968. Theoretical Hydrodynamics (5th ed.). MacMillan. London, 743 pp. Newman, J.N., 1965a. Propagation of water waves over an infinite step. J. Fluid Mech., 23: 399-415. Newman, J.N., 1965b. Propagation of water waves past long two-dimensional ohstacles. J. Fluid Mech.. 23: 23-29. Nielsen, P.. 1979. Some basic concepts of wave sediment transport. Technical University o f Denmark. Inst. of Hydrodynamics and Hydraulic Engineering, Ser. Pap. 20. 160 pp. Peregrine, D.H., 1972. Equations for water waves and the approximations behind them. In: R.E. Meyer (Editor), Waves on Beaches. Academic Press, New York, N.Y., pp. 95-121. Putnam. J.A. and Johnson, J.W., 1949. The dissipation of wave energy by bottom friction. Trans. Am. Geophys. Union, 30: 67-74. Ramaprian, B.R. and Mueller. A., 1980. Transitional periodic boundary layer study. Proceedings of the American Society of Civil Engineers. J. Hydraul. Div., 106 (HY12): 1959-1971. Rhines, P. and Bretherton, F., 1973. Topographic Rossby waves in a rough-bottomed ocean. J. Fluid Mech., 61: 583-607. Roseau, M., 1950. Contribution a la theorie des ondes liquides de gravite en profondeur variable. Publications Scientifiques et Techniques du Ministere de I’Air. Paris. No. 275. Shemdin, 0.. Hasselmann, K.. Hsiao, S.V. and Herterich, K., 1978. in: A. Favre and K. Hasselmann (Editors), Turbulent Fluxes through the Sea Surface, Wave Dynamics and Prediction. Plenum. New York, N.Y., pp. 347-372. Silvester. R. and Mogridge. G.R.. 1971. Reach of waves to the bed of the continental shelf. Proc. 12th Coastal Engineering Conference, Washington, D.C.. Ch. 40, pp. 65 1-668. Sleath, J.F.A., 1970. Velocity measurements close to the bed in a wave tank. J . Fluid Mech.. 42: 1 11-123. Sleath. J.F.A., 1974a. Mass transport over a rough bed. J. Mar. Res., 32: 13-24. Sleath. J.F.A., 1974b. Stability of laminar flow at seabed. Proc. Am. Soc. Civ. Eng., J. Waterways, Harbours Coastal Eng. Div., 100 (WW2): 105-122.
65 Sleath, J.F.A., 1975a. Transition in oscillatory flow over rippled beds. Proc. Inst. Civ. Eng., 59: 309--322. Sleath, J.F.A., 1975b. A contribution to the study of vortex ripples. J. Hydraul. Res.. 13: 315-328. Smith. J.D.. 1977. Modeling of sediment transport on continental shelves. In: E.D. Goldberg. I.N. McCave, J.J. O’Brien and J.H. Steele (Editors). The Sea. Vol. 6. Wiley-Interscience. New Y o r k . N.Y., pp. 539-577. Taylor, P.A.. Gent, P.R. and Keen. J.M.. 1976. Some numerical solutions for turbulent boundary layer flow above fixed. rough. wavy surfaces. Geophys. J. R. Astron. Soc.. 44: 177-201. Treloar, P.D.. and Abernethy, C.L., 1978. Determination of a bed friction factor for Botany Bay. Australia. Coastal Eng.. 2 : 1-20. Tunstall, E.B. and Inrnan. D.L., 1975. Vortex generation by oscillatory flow over rippled surfoces. .I. Geophys. Res.. 80: 3475-3484. Van Ieperen, M.P., 1975. The bottom friction of the sea-bed o f f Melkbosstrand. South Africa: A comparison of a quadratic with a linear friction model. Dtsch. Hydrogr. Zeitschr.. 28: 72-XX. Vitale, P., 1979. Sand bed friction factors for oscillatory flows. Proc. Am. Soc. Civ. Eng., J. Waterway. Port, Coastal Ocean Div.. 105 (WW3): 229-245. Von Kerczek. C. and Davis, S.H.,1972. The stability of oscillatory Stokes layers. Stud. Appl. Math.. Ll(3): 239-252. Yalin, M.S.. 1972. Mechanics of sediment transport. Pergamon Press. New York. N.Y.. 290 pp. Zhukovets, A.M., 1963. The influence of bottom roughness on wave motion in a sh;\llow body o f water. Bull. (Izv) Acad. Sci. U.S.S.R., Geophys. Ser., 10: 943-948. (Translated from Genphys. Ser.. 10: I56 1- I 570.)
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61
CHAPTER 2
NEARSHORE CURRENTS AND SAND TRANSPORT ON BEACHES PAUL D. KOMAR
ABSTRACT When waves break at an angle to a beach they generate a longshore current flowing parallel to the hhoreline and confined largely to the nearshore zone. This current. in turn, interacts with the waves to produce a longshore sand transport. This chapter reviews the theoretical analyses of formation of these currents and sand transport, and examines the data available to test the resulting relationships. The theoretical analyses have proceeded well beyond our existing data, and field data on the distributions of longshore currents and sand-transport rates across the surf and breaker zones are particularly scarce. Our basic inability to obtain sufficiently accurate measurements of sand-transport rates to test the theoretical relationships is a particular problem. It is also difficult to arrive at definite conclusions concerning the relative proportions of bed load and suspended load comprising the sand transport along beaches, although the existing measurements suggest that bed load is quantitatively more significant.
1NTROD U CTION
The longshore movement of beach sand becomes apparent and poses a potential problem whenever this natural movement is blocked by the construction of jetties, breakwaters or groins. Such structures act as dams to the littoral drift, causing a build-up of the beach on the updrift side, and at times serious erosion in the downdrift direction. Erosion problems of this nature were the initial motivation for the study of sand transport on beaches and the attempts to predict the rate of sand movement from the wave and beach conditions. An understanding of sediment transport on beaches is also necessary for t h e analysis of the formation of geomorphic features such as sand spits and barrier islands, in examinations of tidal-inlet processes, and in understanding the configuration of the beach profile or the irregularities in the shoreline along the length of the beach (such as cuspate shorelines). Important to the sand-transport processes are the water currents in the nearshore zone, particularly the longshore currents generated by waves breaking obliquely to the shoreline. In addition to this type of current, there may also be a cell-circulation system consisting of rip currents and their associated longshore feeder currents. This review will concentrate primarily on the longshore currents due to an oblique wave approach and the resulting sand transport. Only a comparatively brief discussion will be made of the origin of the cell circulation, which tends to dominate when the waves break nearly parallel to the shoreline. The generation of rip currents is reviewed elsewhere (Komar, 1976a, pp. 168-182), as are the effects of the cell circulation on the shoreline configuration (Komar, in press). The present review will first examine the generation of longshore currents by an
68
oblique wave approach and then turn to a consideration of the sand transport rate which results from the interaction of this current, the waves, and the sediment substrate. In each case the review will summarize briefly the theoretical analyses of the processes and then turn to an examination of the available data which can be employed to test the theoretical predictive relationships. As will be seen, however, there are many deficiencies in our data, both for the currents and the sand transport. Topics such as the modes of sediment transport on beaches, bed load versus suspension transport, and the relative transport rates of different grain sizes will also be examined.
WAVE-GENERATED NEARSHORE CURRENTS
It is well known that whenever waves arrive obliquely to a coastline and break at an angle to the beach they generate a longshore current flowing parallel to the shoreline. This current is confined to the nearshore, rapidly decreasing in velocity beyond the breaker zone, and is clearly wave-induced and cannot be attributed to ocean currents or tides. Such longshore currents can be generated in wave basins as well as observed on ocean beaches. Several theories have been proposed to account for these wave-generated longshore currents. The early theories have been reviewed by Galvin (1967) and Komar (1976a, pp. 183-188). Since these early approaches are for the most part theoretically unsatisfactory and do not provide reasonable predictions, we will not consider them here. Contemporary theories on the formation of longshore currents originate with the papers by Bowen ( 1969a), Longuet-Higgins (1 970a, b) and Thornton ( 197I). All three of these investigators employed the concept of radiation stress (Longuet-Higgins and Stewart, 1964) to describe the flux of momentum associated with the incoming waves, this being the driving force of the longshore current. These three investigators, at least initially, worked on this problem without the knowledge of one another’s efforts. It is particularly interesting to compare their respective approaches which differ considerably in formulations of the frictional drag on the current and the horizontal mixing across the surf-zone width. Of the three investigations, that of Longuet-Higgins (1970~1,b) leads to the simplest solutions and has been the main point of departure for most subsequent studies. Therefore, this review will concentrate mainly on his solution, pointing out where appropriate how the solutions of Bowen (1969a) and Thornton (1971) differ. I f E is the wave-energy density and a the angle the wave crests make with the shoreline, then the longshore component of the radiation stress is: Sxy= En sin a cos a
(1)
where n is the ratio of the wave group and phase velocities ( n = 1/2 in deep water and 1 in shallow water). If x is positive in the onshore direction and y is parallel to the shoreline, then S,, is the onshore flux of longshore ( y-directed) momentum due to the waves. Sxrof eq. 1 can be written as the product of the wave-energy flux per unit shoreline length, (ECn)cos a, where C is the phase velocity and therefore Cn is the group velocity, times sin a / C which is constant (Snell’s law). If wave friction is
69
negligible in deep water, then the energy flux is constant and i t follows that SKY= constant, Thus, with a wave train arriving from deep water. S,, reaches the beach relatively unabated and is only finally expended once the waves reach the nearshore and break. The dissipation of S,, in the nearshore serves as the basis for the generation of longshore currents in all of the analyses of Bowen (1969a). Longuet-Higgins (1970a, b) and Thornton (1971). Note that when cy = 0. according to eq. 1 S,, = 0 and there will be no driving force. Assuming t h e longshore currents are steady and constant in the longshore direction, the y-component of the equations of motion reduces to:
where ( R , ) is the time-averaged frictional drag due to the longshore current of velocity v ( x ) , h ( x ) is the still-water depth and q ( x ) is the wave set-up or set-down. a change in the mean water level due to the presence of the waves (Longuet-Higgins and Stewart, 1964; Bowen et al., 1968). The set-up and set-down are produced by the S,, component of the radiation stress, the onshore component of the momentum flux, and the $( x) profile is analyzed with the x component of the equations of motion which contains a aS,,/ax term. The last term on the left of eq. 2 represents the effect of horizontal mixing across the width of the surf zone, where p c is the eddy coefficient. Its main effect is to produce a smooth and continuous profile of the longshore current velocity across the nearshore zone, and will be discussed later in this section when the solutions for this complete profile are examined. From eq. 2 it is apparent that in the longshore direction there will be a aS,,/d.x “thrust” which generates the longshore current. Sxr being the local y-directed radiation stress component, a momentum flux, its local dissipation as the waves proceed across the nearshore in the x direction, aSXy/ax,must be balanced by an opposing force also acting in they direction. Physically what occurs is the generation of a longshore current which is then opposed by the frictional drag ( R y ) . This is analogous to the aS,,/ax in the onshore direction being opposed and balanced by the pressure gradient associated with the wave set-up, a seaward slope in the mean water surface. Following the approach of Longuet-Higgins (1970a), a solution of eq. 2 can be obtained first by neglecting the horizontal mixing term ( p , = 0) to obtain simple equations for the longshore current. Later this term is retained in order to examine the details of the longshore current profile. Neglecting horizontal mixing, the solution obtained depends on the form given to the frictional drag term, ( R , ) . Considering the stress as that due to the combined motions of the waves and longshore current, Longuet-Higgins obtained:
2 (RY)=; C,PU,V
(3)
where u, is the maximum horizontal orbital velocity of the waves, given by linear shallow-water theory as: u, = ”
2
g( ij
+h)]
(4)
'70
and:
Y=-
H
(T +h )
where H is the wave height. Numerous observations indicate that y is nearly constant between 0.6 and 1.2 within the breaker and surf zones, and is taken to be constant in all of the derivations. C, of eq. 3 is a drag coefficient which Longuet-Higgins evaluates from empirical curves based on uniform flow over a rough horizontal plate (Prandtl, 1952). Thornton (1971) also arrived at a relationship for ( R , ) equivalent to eq. 3 but evaluated the drag coefficient from the curves of Jonsson (1967) for oscillatory wave motions over a horizontal bottom. In contrast, Bowen (1969a) employed a frictional drag term proportional to the longshore current velocity v alone rather than to u,z, as in eq. 3. As pointed out by Longuet-Higgins (1970b), this seemingly simpler form adopted by Bowen actually leads to more complicated final solutions. All three investigators, Bowen (1969a), Longuet-Higgins (1970a, b) and Thornton (1971), follow the same procedure in evaluating the dS,,/dx term in eq. 2. Taking:
1 E=-PRH~ 8 n = 1, and making the approximation cos a = 1 (i.e., assuming small breaker angles), eqs. 1, 4 and 5 yield:
as,,
5
ax
4
__=-
p u i t r n sin a
(7)
where m = - d h / d x is the beach slope, taken to be constant, and { is a constant factor which results from the inclusion of the wave set-up and is given by:
1
[=-
1 +-3 Y 2 8 Employing eq. 7 for aS,,/dx longshore current velocity: 5.ir rn v = -{-un,
8
c,
. sin a
and eq. 3 for ( R , ) , solution of eq. 2 yields the
(9)
This very simple relationship, obtained by Longuet-Higgins (1970a) (but without including the 5 set-up factor), implies that for constant values of rn and C,, the longshore current is proportional to u , sin a, the longshore component of the wave orbital velocity. Neither Bowen (1969a) nor Thornton ( 1 971) obtained comparable simple expressions for the generated longshore current, their analyses from the beginning being directed toward the complete velocity profile and so did not initially neglect the horizontal mixing as did Longuet-Higgins (1970a). Komar and Inman (1970) obtained a longshore current relationship similar to that of eq. 9, but deduced on quite different grounds. As will be reviewed later in this chapter, Komar and Inman were examining two seemingly independent relationships for evaluating the sand transport along a beach (eqs. 19 and 22). The
71
success of both relationships in predictions of the littoral drift implied their equivalence, and a simple simultaneous solution of the two yielded: 0 , = 2.7u,, sin a h cos a h
(10)
where 6,is the longshore current at the mid-surf position, half way between the breaker line and the shoreline. Typically this mid-surf position is where in practice longshore currents are measured, using either dye tracers or floats. This value is also approximately the average value of the longshore current, both because the methods of measurement tend to produce an averaging and because, as will be seen shortly when we examine the entire velocity profile, the velocity at the mid-surf position corresponds closely to the average. Originally Komar and Inman (1970) did not include the cob a h factor in eq. 10, but it is now included following the later analysis of Komar (1975). I t should be recognized that in his derivation, Longuet-Higgins (1970a) could just as easily have fixed cos a at its value at the breaker zone rather than assuming cos a = 1, so that it would be present in eq. 9 as well. Komar and Inman (1970) also tested eq. 10 with the available field data on longshore currents generated by an oblique wave approach, and found that it agreed very closely with the observations. Komar (1975) continued this comparison, including the laboratory data as well as the field data. As shown by Komar (1979), utilizing eqs. 4 and 5 , eq. 10 can be simplified to: 1/2 . sin a , cos a , ?i= 1 .17( g H , ) (11) I t is seen in Fig. 2.la that this relationship agrees with the laboratory data as well as the field data. This comparison includes all of the data compiled by Galvin and Nelson (1967) except the field data of Inman and Quinn (1952) which certainly contains a large component due to cell circulation with rip currents, and except the laboratory data of Galvin and Eagleson ( 1 965). As shown by Komar (1 979, fig. 3), the data of Galvin and Eagleson form three distinct linear trends having different slopes depending on the magnitude of the breaker angle. In that it does this for all of the available theories, not just for eq. 11, and because it is the only data set that does not agree with eq. 11, it was concluded that the fault lay with the data of Galvin and E
,
j
I
I
I
K O M A R ANQ
T
I
!NMAN
I
I
1 1 . 1
I
I
I
(1970)
P u l n o r n . Munk and T r a y l o r (1949) 0 G o l w n or,tl S o v a q e (1966) 0 Komar 0°C l n m a n (I9701 Laboralory I)oto + P u i n o m . Munk a n d T r a y l o r (1949) x S a v i l l e (1950) .r B r e b n e r 0.-d Komphuis (1963)
~~
-.
-
-
-
-
L 20
40
-
60
-
L
U 80
-
-
U
100
120
i
L--140
m ( g H , ) 1 ‘ 2 s m a b c o s a , ,c m s-’
Fig. 2.1. a. Comparison of eq. 11, modified from eq. 10 by Komar (1979). with laboratory and field measurements of longshore currents due to an oblique wave approach. b. The same data sets used to test eq. 12 from CERC (1973). A comparison of the results demonstrates the mistaken beach-slope dependence of the CERC equation.
13
might be expected that eqs. 10 and 11 will not be appropriate under such conditions. The impetus for Komar (1979) to alter eq. 10 to the simpler but equivalent form of eq. 11 was to allow a comparison with the longshore-current formula recommended in the Coastal Engineering Research Center’s Shore Protection Manual (CERC, 1973). Starting with the solution of Longuet-Higgins (1970a), eq. 9 modified with eqs. 4 and 5 so that Z; depended on (gH,,)’’’ as in eq. 11, CERC assumed a constant drag coefficient C, = 0.01 so that Longuet-Higgins’ solutions reduced to: 1/2 . u , = 41.4 m ( g H , ) sin a,, cos a h (12) A comparison with eq. 11 based on the formula of Komar and Inman (1970) reveals that eq. 12 from CERC (1973) differs only by the inclusion of the beach slope, n7. Figure 2.1B tests eq. 12 with the same data utilized in Fig. 2.lA to examine eq. 1 1 . The conclusion is unmistakable; the longshore current is not directly proportional to the beach slope as stated in the CERC formula, eq. 12. Komar and Inman (1970) had pointed out the apparent lack of dependence of the longshore current on the beach slope. A comparison of their eq. 10, derived from two sand-transport relationships, with eq. 9 derived from radiation stress considerations, leads to the implication that:
-m-
-
C,
constant
The approximate constancy of this ratio might at first seem reasonable from the expected behavior of the drag coefficient and beach slope with increasing sediment grain size of the beach material (Komar, 1971a). As the grain size increases, i t is well-known that the beach slope increases. In that the grain size is also the main bottom roughness factor on beaches (where ripples are generally absent), the increase in grain size also causes an increase in C,.. Komar (1971a) suggested that such simultaneous increases in m and C, must lead to eq. 13, at least as a first approximation. However, as pointed out by Huntley (1976), this argument does n o t explain the laboratory results where most beaches are solid, with a generally small bottom roughness that is independent of the beach slope. Huntley instead explains eq. 13 in terms of the beach-slope effects on the turbulence generated by the wave-breaking processes. An increase in beach slope will produce a narrower surf zone over which turbulent breaking occurs, and hence a greater level of turbulence per unit surface area. This increased horizontal mixing results i n a decreased longshore current and hence a greater apparent C, value. In his study, Huntley (1976) made field measurements of the mean longshore current and horizontal velocity fluctuations, employing a two-component electromagnetic flowmeter. His measurements permitted a direct evaluation of C, and the results were found to be consistent with eq. 13, but were at least a factor or two smaller than expected for Reynolds numbers and bottom roughness appropriate for the beach. Huntley interpreted his results in terms of a three-layered model which will be described later. As discussed above, the longshore current velocity given by eqs. 10 and 1 1 refers to the magnitude at approximately the mid-surf position. In the analysis of Longuet-Higgins (19704, eq. 9 is actually the longshore-current velocity at the
14
breaker line, but in that model, which ignores horizontal turbulent mixing, the velocity at the mid-surf position is exactly one-half that of eq. 9. Without mixing, the velocity profile across the surf zone is one with a maximum at the breaker line, decreasing linearly to zero at the shoreline. The velocity is also zero beyond the breaker zone since Sxy= constant so that aS,,/dx = 0, leading to an abrupt discontinuity at the breaker line. Bowen ( 1 969a), Thornton ( 197I ) and Longuet-Higgins ( 1970b) all include horizontal mixing in their analyses so as to obtain more realistic velocity profiles. This involves retention of the third term on the left side of eq. 2. It is generally envisaged that horizontal eddies, originating primarily from the wave-breaking processes, will produce a horizontal transfer of momentum. Although the three studies retain this mixing term, their evaluations of it are considerably different. Bowen (1969a) takes p, = constant for the horizontal eddy coefficient in eq. 2. p e has dimensions of p L U . where L is a typical length scale and U is a velocity. Drawing an analogy with the benthic boundary layer, Longuet-Higgins (1970b) took L a x, arguing that the eddy dimensions cannot possibly be greater than the distance to the shoreline and will tend to be larger with increasing distance x from the shoreline. Taking U a [ g ( T + h ) ] ’ I 2 ,Longuet-Higgins obtained: pe=Npx[g(7Jl+h)J”2
(14)
where N is a dimensionless constant with the probable limits 0 < N < 0.016: the data comparison placed its value at approximately N = 0.005. Madsen et al. (1978) and Kraus and Sasaki (1979) take p, a xu, which is seen to be equivalent to eq. 14 i f u,,, is given by eq. 4 from linear wave theory. In a more detailed analysis of surf-zone turbulence, Battjes (1975) has argued that the characteristic size of the eddies is limited by the local waterdepth, not by the horizontal distance x to the shoreline [ L a (7 + A ) ] . He further evaluates the velocity scale, U , from the local rate of wave-energy dissipation per unit area. For a uniformly-sloping beach this leads to a relationship that is similar to eq. 14 but also includes the beach slope to the 4/3-rds power as a factor. On the basis of a comparison with laboratory data, Madsen et al. (1978) agreed that there should be a beachslope dependence in p,. This was also suggested by the field measurements described by Huntley (1976), but his simultaneous evaluations of the drag coefficient indicated a more complex model in which the lateral mixing resulting from the turbulence of wave breaking is not distributed throughout the water column but instead is limited to the upper part. As indicated earlier, such a layered model would help explain the results leading to eq. 13. If one has a non-planar beach, then the distinction between Battjes’ (1975) analysis of p e and that of eq. 14 from LonguetHiggins (1970b) is much greater. If the layered model of Huntley (1976) is correct, then a still more complex evaluation of pe is required. Thornton (1971) and Jonsson et al. (1975) evaluated p, on the basis of a different rationale, placing it in terms of the wave orbital motions. The length-scale dependence of p, is thereby taken as the horizontal orbital diameter and the velocity scale is the orbital velocity. This yields a maximum pe at the breaker line, decreasing both onshore to zero at the shoreline and in the offshore direction. In the Longuet-Higgins (1970b) model, eq. 14, p, continues to increase indefinitely beyond the breaker
75
zone. Thornton’s form for pe does require that eq. 2 be solved numerically, which he does (including beach slope irregularities), comparing the results to laboratory measurements of longshore-current profiles by Galvin and Eagleson ( 1965) and one set of field measurements by Ingle (1966). Battjes (1975) has criticized this approach of using the parameters of the orbital motions for characterizing the horizontal mixing, pointing out that the main source of the turbulence energy in the surf zone is the intense dissipation of wave energy by breaking. H e further points o u t that the contribution of the orbital motions is directly accounted for through the radiation stresses. Employing eq. 14 for pe and eqs. 3 and 7 for the other terms, solution of eq. 2 by Longuet-Higgins ( 1970b) yielded (again adding in the wave set-up effects): (l5a) where X = x / X , , X , being the distance from the shoreline to the breaker zone, V = o / o ~where: ~
where h , is the depth at the breaker zone, ( 15c)
B , =- P r - 1 A PI - P r B,=- P I - 1 A PI - P 2
where y and 5 are given by eqs. 5 and 8. The solution of eq. 15 yields a family of longshore-current profiles, Fig. 2.2, one for each value of P given by eq. 15e. P is seen to be a non-dimensional parameter representing the relative importance of the horizontal mixing as indicated by the value of N . When there is n o horizontal eddy mixing, then P = 0 and the saw-tooth distribution is obtained as before with a discontinuity a t the breaker line (Fig. 2.2). With increasing horizontal mixing, reflected in an increasing value for P , the profile becomes smooth a n d more realistic, the maximum in the profile shifting toward the shoreline afid also decreasing in magnitude. Inclusion of the horizontal mixing also
76
I .o
<
0.5
>
x
= x/x,
Fig. 2.2. Non-dimensional longshore current velocity profiles given by eq. 15 obtained by Longuet-Higgins (1970b). Each profile corresponds to a value of P as given by eq. 15e.
couples the water outside the breaker zone to the flow within the surf zone. now producing a longshore current outside the surf zone with no discontinuity at the breaker line. Longuet-Higgins (1970b) compared these solutions to the laboratory measurements of longshore-current profiles obtained by Calvin and Eagleson (1965). concluding that P lies in the range 0.1-0.4. But as we have seen, the Calvin and Eagleson (1965) data are suspect because of their lack of agreement with the other d a t a sets, so the validity of this comparison is uncertain (as are the other studies that have principally relied on the profile measurements of Calvin and Eagleson). As pointed out earlier, o u r existing measurements of longshore currents have been obtained almost entirely at the mid-surf position, corresponding to the X = 0.5 position in the solution of eq. 15. This data already has been employed to establish eqs. 10 and 11, so i t is apparent that we could at least ensure that the solution of eq. 15 agrees with that data by forcing it to agree with eqs. 10 and 1 1 at X = 0.5. This was done by Komar (1976b), obtaining a relationship for m / C , as a function of P . Once a reasonable value of P has been selected to define the overall shape of the profile as seen in Fig. 2.2, this relationship provides a value for m / C , and thus C, for a known beach slope. If this value of C, is used in computations with eq. 15 the value of the current at X = 0.5 will agree with eqs. 10 and 1 1 and hence with the available data. Examples are shown in Fig. 2.3 for two values of P but with the same wave and beach conditions. Also of interest is that according to the relationship of Komar (1976b), the ratio m / C , is relatively insensitive to P , lending further support for the approximation of eq. 13. Jonsson et al. (1975) and Kraus and Sasaki (1979) have
77 I
I
1
I
I
I
1
I
I
I
i
-
1
1
x/x, Fig. 2.3. Examples of longshore current profiles for extreme values of P in its expected range, calculated by the method of Komar (1976) wherein the selection of the drag coefficient C, ensures that the magnitude of the current at the mid-surf position agrees with eq. 10 as shown.
also shown in their analyses that the longshore current profiles are relatively insensitive to P . According to Huntley (1976), this may result from the way p L cis formulated as a function of the local waterdepth rather than being independent of depth as suggested by his layered model. In addition, Huntley indicates that as employed by Komar (1976b), C, is not a proper drag coefficient but instead is being used as an empirical coefficient whose value yields the correct mid-surf longshorecurrent velocity. All of the analyses of Longuet-Higgins (1970a, b), whether leading to the simple eq. 9 or the complete profile given by eq. 15, were based on the assumptions that cosa = 1 and U/U, < 1, i.e. the angle of wave approach is small and the magnitude of the generated longshore current is small in comparison with the wave-orbital velocities. The validity of the latter assumption can be evaluated approximately from eq. 10, which indicates that one must have a,, < 24" in order to ensure that u , / u , , , < 1. Breaker angles are usually much smaller than this, especially in the field, so the assumption would appear to be justified. In that eq. 10 agrees with the laboratory data even up to breaker angles approaching 45", this 24" cut-off may even be conservative. The desire to eliminate these assumptions made by Longuet-Higgins (1970a, b) has been the impetus for many of the subsequent theoretical analyses. These assumptions were particularly important to Longuet-Higgins' derivation of eq. 3 for the frictional drag, so many of the later modifications have centered on more refined analyses of drag that do not require these assumptions. Analyzing the drag in terms of the combined motions of the waves plus the longshore current, but neglecting lateral mixing, Liu and Dalrymple (1978) obtained solutions comparable to eq. 9
78
from Longuet-Higgins but far more complex. As indicated above, their solutions may have to be employed at breaker angles greater than 24" to 45". Madsen et al. (1978) and Kraus and Sasaki (1979) analyzed the bottom stress in a similar way to Liu and Dalrymple (1978), but included the lateral mixing (with pc a xu,), obtaining analytical solutions. The interesting feature of the Madsen et al. analysis is that they compared their solution to the data, obtaining an expression for the drag coefficient that approaches the friction factors of Jonsson (1977) when 0 << u,,, and the drag coefficients from a Moody-diagram when 0 >> u,,,. The analysis of Kraus and Sasaki (1979) is particularly interesting in that they compare their solutions to new, detailed measurements of longshore-current profiles obtained in the laboratory and field. The comparison with the laboratory data of Mizuguchi et al. (1978) is shown in Fig. 2.4. The comparison between theory and data inside the surf zone is seen to be very good. Unlike the data of Calvin and Eagleson (1965) employed by others in their comparisons, this data of Mizuguchi et al. agrees with eqs. 10 and 11 at the mid-surf position, and hence with the other data sets. The agreement outside the breaker zone is poorer, but as noted by Kraus and Sasaki, this may be due to inaccuracies in measuring the low longshore-current velocities in the presence of the wave field. Figure 2.5 shows a comparison with one set of field measurements obtained by Kraus and Sasaki. Here the agreement is very good in the outer surf and breaker zones, and in the offshore. This portion of the velocity profile is controlled by the primary breaker zone. The poorer agreement in the inner surf zone results from the breakdown of the planar-beach assumption in the analysis, and that the wave-height decrease is no longer linear. Instead, the step-type profile produced a zone of secondary breakers at approximately 20 m from the shoreline.
'I\
I/, '.---I 04
X
i
,
06
,
08
10
12
.
14
~
-
-
16 - -1 8 - _ _2 0
Fig. 2.4. Laboratory data of Mizuguchi et al. (1978) compared with the theoretical longshore current profiles obtained by Kraus and Sasaki (1979) and Longuet-Higgins (1970b). (From Kraus and Sasaki, 1979.)
79 L
L
a,
r
‘ 2 1
7
B r e a k e r zone
0
5
al
10
15
Offshore
>
20 25 30 35 distance, X (rn)
40
45
3
q
10
0 6
01.
0 2 n 0
02
OL
06
0 8
1 0
1 2
1I.
X Fig. 2.5. Measurements by Kraus and Sasaki (1979) of longshore currents and beach profile at Urahama Beach, Japan (a) and a comparison with their analytical solution for the current profile (b). (From Kraus and Sasaki, 1979.)
The above analyses assume regular waves of uniform height, breaking at a well-defined, fixed location. However, irregular waves such as are usually found in nature produce a zone of breakers, the larger the wave height the deeper the waterdepth in which it breaks. This will in turn affect the wave set-up and the distribution of the longshore current. Battjes (1973) has examined this effect, utilizing a wave-by-wave description of the irregular waves. Although his attention was directed mainly toward the wave set-up, Battjes also made calculations of longshore-current distributions. The lateral momentum exchange due to turbulence was neglected ( p e = 0), but in spite of this, smooth velocity profiles were obtained similar in appearance to those in Fig. 2.2. This result demonstrates that irregular waves breaking in a range of waterdepths will have much the same effect as lateral mixing and in the field it may be difficult to separate out the two influences. The
80
natural variability of wave periods and directions would also have to be considered in field situations. As has been shown by Battjes (1972), the angular distribution, i f not narrow, can strongly affect the radiation stress which provides the driving force for the longshore current, estimating that if the waves are treated as i f they are long-crested the total longshore thrust could be overestimated by as much as 100%. The analytical solutions of Longuet-Higgins (1970b) and others assume a uniform beach slope. The numerical solutions, such as those of Thornton ( 1 971) and Jonsson et al. (1975) do allow for an irregular beach profile. Symonds and Huntley (1980) obtained solutions, shown in Fig. 2.6, over an idealized bar-trough topography. I t is seen that the longshore current concentrates over the bar with a minimum over the trough, the velocity in the trough increasing with increasing P as it is driven mainly by the horizontal mixing. Symonds and Huntley obtained some field measurements under such conditions, indicating P = 0.1-0.4, a more exact estimation being precluded because of equipment failure. More commonly, observations show the maximum longshore current to be in the trough rather than over the bar. Symonds and Huntley show in their analyses that this is produced by longshore pressure gradients due to longshore variations in the wave set-up (which will be examined later) or to tidal currents. All of the solutions of Symonds and Huntley are numerical. McDougal and Hudspeth (1983) have obtained analytical solutions for beach profiles in which the still-water depth is proportional to xZ/', which approximates many beach profiles. This produces considerable changes in the longshore current profiles from those obtained by Longuet-Higgins (1970b), fig. 2.3, shifting the maximum current to the shoreline position ( X = 0). Common to the above theoretical analyses and data comparisons are averaging procedures both in time and depth, either of the conservative equations o r of the
1
+ a w 0
: 40
40
80
120
160
zoo
220
OFFSHORE DIS.TANCE (rn)
Fig. 2.6. Solutions for longshore current profiles over an idealized bar-trough topography obtained by Symonds and Huntley (1980). Each profile is for a fixed P value given by eq. 15e. the higher the value of P the greater the lateral mixing. (After Symonds and Huntley, 1980.)
81
measurements. The theoretical analyses seek steady, depth-averaged solutions, neglecting any time variations in the longshore current. The data employed above tend to represent the f l ~ wnear the water surface and usually involve a n averaging of several separate current measurements which are in turn derived from a Lagrangian approach of following dye or a float over a time span of several wave periods. However, some investigators have noted large time and spatial variations. especially in the field (Putman et al., 1949; Inman and Quinn, 1952; Harrison, 1968). Wood and Meadows (1975) and Meadows (1977) in particular have investigated these fluctuations, utilizing a fixed array of three current meters spaced vertically through the waterdepth. They noted large fluctuations in the longshore current with velocities reaching as much as 150% above the mean. Some of the unsteadiness of course arose from the longshore component of the wave-orbital velocities, but they concluded that the fluctuations were much too large to be completely accounted for by this. Velocity spectra showed significant energy at 78.8 s, leading Meadows (1977) to conclude that much of the unsteadiness resulted from edge waves. Variations of the mean currents with depth were found to be small, supporting the vertical averaging approach of the theoretical analyses and the use of near-surface current measurements to represent the entire flow. Much of the shore-parallel spatial variability of the longshore currents can be attributed to longshore variations in wave heights and set-up, important to the formation of the cell circulation with rip currents. In that the cell circulation can migrate in the longshore direction (Komar a n d Inman, 1970). i t may also be responsible in part for time fluctuations at a fixed position. In some circumstances the cell circulation only acts to “contaminate” the longshore currents due to an oblique wave approach, producing some longshore variability. At other times, mainly when breaker angles are zero or small, the cell circulation tends to dominate. Guza a n d Thornton (1979) also noted considerable variability in the longshore currents measured at Torrey Pines Beach, California, the variation occurring both at a fixed surf-zone location and spatially, even with n o obvious rip currents. They suggested that non-linear terms and local short-term variations in alongshore breaker heights are important in the equations of motion, and that free “eddy” motions may also be present in the surf zone. The principal purpose of their study was an attempt to correlate the longshore current velocity to direct measurements of S,, obtained with pressure transducers in 10 ms waterdepth. However, 17-min averages of S , , and the longshore currents both showed considerable variations with little or no correlation between the two. They point out that due to such variations, considerable temporal and spatial averaging will generally be required to obtain a representative picture of longshore currents, especially when comparisons are to be made with the theoretical analyses. A complete review of the origin of the cell circulation is beyond the scope o f this chapter, requiring a lengthy discussion of the literature. In brief summary, Bowen (1969b) a n d Bowen and Inman (1969) have demonstrated that a longshore gradient of the wave set-up, g d f j / d y ; is important in the generation of the cell circulation. T h e longshore currents flow parallel to shore from positions of high waves and set-up to positions of low waves and set-up, where the currents turn seaward as rip currents. The longshore variations in wave heights and set-up can be produced by
x2
wave refraction or by edge wave-swell wave interactions. The most general condition is where the longshore currents are generated by a combination of waves breaking obliquely at the shore together with longshore variations in wave heights. O’Rourke and LeBlond (1972) applied radiation stress concepts to determine the longshore currents in a semicircular bay under such conditions, including longshore variations in wave-breaker angles in their analysis. Keeley and Bowen (1977) measured longshore currents along more than 1 km length of beach and analyzed them in terms of longshore variations in breaker angles and heights. In their particular set o f measurements the longshore current was dominated by the oblique wave approach and given by a relationship such as eqs. 10 or 11, with the dHJ3.y and d a h / d . v terms together contributing approximately 10% to the overall current strength. Superimposed upon these large-scale currents were regular, small-scale circulation cells, probably caused by edge waves. Keeley (1977) demonstrated a correspondence between the patterns of the large-scale longshore currents and the development of a large-spaced cuspated shoreline. Komar (1971 b) has analyzed a peculiar situation which developed in a wave-basin study of cuspate shorelines. In these tests a condition was achieved wherein the oblique wave approach to the cusp flanks was opposed and balanced by a longshore variation in the wave breaker heights such that no longshore current and hence no sand transport occurred. Longshore gradients in wave breaker heights and set-up can also occur in the semi-protected areas such as in the lee of a breakwater. Gourley (1975. 1977) has made use of this to arrange laboratory wave-basin studies of non-uniform longshore currents. A number of studies have attempted to apply multiregression analyses to the prediction of longshore currents (Harrison and Krunibein, 1964; Brebner and Kamphuis, 1965; Harrison et a]., 1965; Sonu et al., 1967; Harrison, 1968; Allen, 1974; Nummedal and Finley, 1979). This approach is reviewed in Komar (1976a. pp. 196-197) and it is concluded that such empirical analyses yield results which are probably not applicable to beaches other than those upon which they are based. The various formulae also differ as to the relative importance of the various factors involved. Most of the empirical equations find the expected correlations with wave heights and angles of wave approach (but d o not include i l H , / i ? ~factors). Of most interest are the analyses which include the direct effects of the coastal winds as this factor is not included in the analyses already considered in this review. I n particular, Numrnedal and Finley (1979) found in an analysis of data from the South Carolina coast that the longshore component of the wind velocity accounted for most of the observed variance in the longshore-current velocity. This may have resulted in part from their use of the visual LEO observations, but does demonstrate the importance of the coastal winds in producing longshore currents, a factor that should not be ignored as is generally the case. The past decade has seen a considerable increase in our understanding of nearshore current generation, especially those formed by an oblique wave approach. We are now in the position of making reasonable predictions of this current from the known wave conditions, at least of the velocity at the mid-surf position, approximately the mean flow. Much remains to be learned about the distribution of the currents across the width of the surf zone. Although the theoretical analyses have
83
advanced considerably and yield reasonable-looking results, there remains a diversity of models on the best way to evaluate the frictional stress and parameterize the lateral mixing. And only initial attempts have been made to examine the effects of beach topography (non-planar beaches). But the main constraint to further progress IS certainly our almost total lack of quality field data on longshore currents, especially of the complete velocity profile.
LIITORAL DRIFT
When waves break obliquely to the shoreline, they produce a transport of sediment along the beach, the total volume of which is termed the littoral drift, the rate at which it moves being referred to as the littoral transport rate or longshore sand-transport rate. In that it is this sediment movement that is blocked by jetties, i t is natural that coastal engineers were the first to attempt to evaluate its volume at various coastal sites and to relate it to the causative wave conditions. It is also logical that the initial attempts would utilize the blockage by jetties to obtain measurements of volumes of littoral drift. Some thirty years ago, the Los Angeles District of the U.S. Army Corps of Engineers demonstrated an approximate correlation between the littoral drift rate and hindcast wave data (Eaton, 1951). This correlation involved a comparison of the littoral drift to the expression: P , = ( ECn),
Sin
(Y,
COS (Yh
(16)
where the wave-energy flux, ( E C n ) , , and angle a h are both evaluated at the breaker zone. The rationale behind the sin a, cos a, transformation is that ( E C n ) , alone i s the wave-energy flux per unit wave-crest length, (ECn),coscu, then places it on the basis of a unit shoreline length, and multiplication by sincu, yields the longshore component. Such a derivation is presented in Galvin and W a l e (1977). Following such an analysis, P , has commonly been referred to as “the longshore component of the wave-energy flux”. Longuet-Higgins (1972, p. 210) has taken exception to this terminology, pointing out that since ( E C n ) is a vector rather than a second-order tensor, the longshore component would be ( E C n ) , sincu,. He prefers to write P,as its equivalent S,,C,, “the product of two physically meaningful quantities” (C, is the phase velocity of the breaking waves). This suggestion has been followed by Inman et al. (1980), while others such as Bruno and Gable (1977) have attempted to relate the sand-transport rate to Sxy alone. Still others have developed a new terminology for P , while retaining it as a parameter: “energy-flux factor” (Galvin and Vitale, 1977); “ P , parameter” (Komar, 1976a). Although the L.A. District found a correlation between the longshore sand-transport rate and P,,the results were too scattered to warrant adoption of any empirical relationship. Subsequent studies have employed a variety of techniques to obtain the necessary quantitative data on sand transport and causative waves and currents. These are categorized in Table 2.1. Noda (197 1) provides a discussion of the various techniques, and the several studies listed in Table 2.1 describe the systems used in their particular investigations.
84 TABLE 2.1 Selected studies relevant to quantitative evaluations of sediment transport on beaches I. Blockage by breakwaters or jetties
Watts (1953a); Caldwell (1956); Bruno and Gable (1977); Bruno et al. (1981) II. Sediment tracers A. Fluorescent tracers: Russel (1960); Ingle (1966); Yasso (1966); Boon (1969); Komar and Inman (1970); Knoth and Nummedal(1978); Wang and Chang (1979); Hattori and Suzuki (1979); Duane and James (1980); Inman et al. (1980); Kraus et al. (1982) B. Radioactive tracers: Courtois and Monaco (1969); Duane (1970) 111. Mechanical bed-load trups Thornton (1973); Lee (1975); Sawaragi and Deguchi (1979) I V. Suspension meusurements
A. Pumping: Beach Erosion Board (1933); Watts (195311); Fairchild (1973. 1977); Coakley et al. (1979) B. Suspension volume samplers (“traps”): Fukusima and Kashiwamura (1959); Hom-ma and Horikawa (1963); Hom-ma et al. (1965); Kana (1978, 1979); Inman et al. (1980) C. Light scatter or transmission: Brenninkmeyer (1975, 1976); Thornton and Morris (1978); Leonard and Brenninkmeyer (1979)
The earliest studies to obtain quantitative measurements of the littoral drift as a function of the wave conditions utilized the deposition-erosion patterns in the vicinity of jetties to evaluate the littoral drift. Watts (1953a) was the first to do so. He determined the sand transport rate from the quantities of sand a bypassing plant had to pump past the jetties at South Lake Worth Inlet, Florida. His wave data came from a conventional pressure transducer and measurements of breaker angles were obtained from sightings from atop a tall building adjacent to the beach. Although the data were extremely scattered, they did permit the formulation of the first empirical relationship between the longshore sand transport rate and P,. In a similar study, Caldwell (1956) obtained additional data, the littoral drift being estimated from the rate at which an artificial accumulation of sand was eroded from the beach at the down-drift side of the jetties at Anaheim Bay, California. Waves were again measured with a pressure transducer and the angle of breaking determined by hindcasting rather than being measured directly. Combining his data with that of Watts, Caldwell obtained a revised empirical relationship. The studies of Watts (1953a) and Caldwell (1956) certainly represent pioneering attempts to obtain the data required to ectablish relationships for evaluation of the littoral drift. However, Greer and Madsen (1979) provide a detailed criticism of the studies, concluding that these data sets should not be employed in modern-day relationships for evaluating the littoral drift. The only subsequent study that has employed blockage by jetties to obtain data on littoral drift rates to compare with P , is that of Bruno and Gable (1977) and Bruno et al. (1981). Their measurements were obtained at Channel Harbor, Cali-
fornia, a site selected due to the combination of jetties at the harbor entrance and an offshore detached breakwater, together forming a littoral barrier. However, this arrangement probably introduces systematic errors in that the longshore currents are produced by the combined effects of an oblique wave approach and a longshore variation in wave-breaker heights. The prototype jetty-breakwater system at Channel Harbor is in fact very similar to the laboratory arrangement in the study of Gourley (1975, 1977) of such currents. The existence of these currents would enhance the entrapment of sand in the lee of the breakwater even when the waves are otherwise breaking parallel to shore so that P , = 0. The data of Bruno and Gable (1977, fig. 10) indicate the presence of a significant accumulation rate as P , becomes small and approaches zero. The littoral drift quantities were determined by repeated surveys of sand accumulating at the jetty-breakwater system, the survey intervals being one to three months. As presented in Bruno and Gable (1977), the data indicate transport rates nearly twice those obtained by other studies. The reanalysis of the data by Bruno et al. (1981) showed that this resulted from their reliance o n the visual LEO wave data with its considerable inaccuracies in estimating wave breaker heights and angles. As presented in Bruno et al. (1981), the data are consistent with the other data sets to be discussed later. During part of their study the waves were measured with a pair of wave gauges, and this data shows the best agreement with the other data (see later, Fig. 2.8). Improved visual measurements were utilized when wave-gauge measurements were not available. Although on average that data is also consistent with the other data sets, it shows a great deal of scatter and so provides little help in establishing a relationship between the littoral drift rate and P , . There are numerous problems associated with the use of jetties and breakwaters for evaluating littoral drift rates, some of which have already been pointed out. One of the foremost is the long-term nature of the measurements, it usually taking a month or longer for sufficient sand to accumulate to make its volume measurement meaningful. But during that time the wave conditions would most likely vary considerably, perhaps even producing a reversal in the drift direction. Such problems with the use of jetties led to other techniques that permit shorter-term measurements. In the 1960’s the use of sand tracers for determining sand movements became very popular. In nearshore studies these were almost exclusively fluorescent sand tracers rather than radioactive tracers, the latter offering obvious problems when used on public beaches. Ingle (1966), Yasso (1966) and Teleki (1966) all provide summaries of the techniques for tagging sand with a fluorescent color and some discussion of methods of use in the nearshore. Ingle (1966) in particular provides a good example of their application on beaches, conducting experiments on several California beaches. He compared his measurements of average tracer grain advection rates with various wave parameters, including P , , and with the grain diameter. Unfortunately, all of his data are extremely scattered, possibly because he obtained his grid samples for determining the tracer distribution by utilizing 3 X 3-in. cards coated with Vaseline. Such a technique obtains only the surface layer of sand grains, which is probably not sufficiently representative in that most of the tracer is buried within the beach. Due to its scatter and because no direct measurements were made of the thickness of movement,
necessary for the calculation of the volume transport rate, Ingle’s data have not been used in subsequent correlations between the sand-transport rate and the wave conditions. Komar and Inman (1970) also utilized sand tracers to measure longshore sandtransport rates. Two beaches were involved in the study, El Moreno Beach, a coarse-grained sand (D,,= 600 p ) beach on the Gulf of California, and Silver Strand Beach, a typical medium-grained sand (175 p ) Pacific Ocean beach near San Iliego, California. Figure 2.7 shows an example of a tracer distribution obtained at €31 Moreno, determined by sampling on a grid, in this example four hours after tracer injection. Unlike the study of Ingle (1966), the grid samples were collected with a volume sampler that penetrated down into the beach face, obtaining a large sample. Such a contour diagram permitted an evaluation of the mean longshore transport distance of tracer movement in those four hours, and hence a measure of the mean advection velocity, U , . This was converted into a volume transport rate, Q ? ,by multiplying by the surf-zone width, X , , and by the thickness of the layer of
EL MORENO. BAJA CALIFORNIA, MEXICO I I OCTOBER 1966 TIME
4 0 HOURS (LOW TIDE)
CONCENTRATIONS IN TRACER GRAINS/KILOGRAM
600
-
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EXAGGERATION
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5 X LONGSHORE FT
Ftg. 2.7. Distribution of fluorescent sand tracer on a beach after four hours of transport, having been injected in the position shown and the concentrations determined after transport by sampling on the grid shown (From Komar and Inman, 1970.)
87
sand moving along the beach, 6 :
Q, = UiXhb This thickness was determined by measuring the depth to which the sand tracer is buried within the beachface during transport, and by employing the method of King (1951) which involves the burial of a column of dyed sand and then noting the thickness of the upper portion that is cut away by the transport process. Komar and Inman (1970) measured the simultaneous wave conditions with an array of digital wave staffs which permitted accurate evaluations of the breaker angles as well as wave energies and periods. Komar and Inman (1970) based their empirical relationship on their field data together with that of Watts (1953a) and Caldwell (1956). The littoral drift formula presented in the Coastal Engineering Research Center’s Shore Protection Manual (CERC, 1973) also relies on these three data sets. They are shown plotted in Fig. 2.8 as PI of eq. 16 versus the sand transport expressed as the immersed-weight sandtransport rate, I , , given by: 1, = ( ~ -s P IP’Q,
(18)
where p, and p are the sand and water densities, respectively, and a‘ is a factor such that a’Q, is the volume of solid.sand alone, eliminating the pore space of Q, ( u ’ can usually be taken as 0.6). The use of the immersed-weight sand-transport rate rather than the volume transport rate, Qs, results from the sediment-transport considerations of Bagnold (1963, 1966) as applied to beaches by Inman and Bagnold (1963) and Inman and Frautschy (1966). The advantage of using I , rather than Q, is that the immersed weight takes into consideration the density of the sediment grains so that the resulting relationship will be applicable to sediment of any density, not just quartz sand beaches. An additional advantage is that I , and PI have the same units so the proportionality coefficient is dimensionless. Fitting a straight line to the field data, shown in Fig. 2.8, Komar and Inman (1970) obtained the relationship: I , = 0.77 PI (19) The disadvantage of using I , is that most people cannot envisage immersed-weight transport rates, and Q,, the volume transport rate, is the quantity required by engineers since it is the accumulation rate blocked by a jetty or breakwater. However, eq. 19 can be used directly to obtain such derivative relationships between Q, and PI of the form:
Q, = kPi
(20) with the value of k depending on the respective units of Q, and PI and on the density of the sediment comprising the beach. For quartz-sand transport and Q, measured in units m3 day-’ and P I in W m-I, eq. 19 yields the simple formula:
Q, = 6.8 PI (21) For the case of PI measured in units ft-lbs s - ’ f t - ’ of beach front and Q, in yds3 yr-’, one obtains k = 15 X l o 3 in eq. 20. These units correspond to the littoral drift formula presented in the Shore Protection Manual (CERC, 1973), but the value obtained here is almost exactly twice that given in CERC. The reason for this
88 10'
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Fig. 2.8. The available measurements of sand-transport rates on beaches compared with the wave conditions expressed as P , of eq. 16. The sand-transport rate is expressed as either the immersed-weight transport rate, I , , or as the volume transport rate, Q,, with the two empirical relationships shown.
difference is that in our evaluation of P, from eq. 16 we have employed the root-mean-square wave height, that wave height which gives the correct wave-energy evaluation. The CERC (1973) formula is set up to use the significant wave height which is a factor 1.418 larger than the rms value (Longuet-Higgins, 1952) so that the calculated wave energy would be a factor (1.418)*= 2 greater. Therefore, i f the significant wave height is employed rather than the rms wave height, the k values given above must be divided by 2. And it is then seen that the CERC (1973) formula is basically equivalent to eq. 19 from Komar and Inman (1970). The greatest uncertainty in the tracer method employed by Komar and Inman (1970) for measuring sand-transport rates is the determination of the thickness of sand movement, b, of eq. 17. As indicated above, two principal methods were employed in that study, the depth to which the tracer is buried within the beachface during the transport process, and the removal of the upper portion of a column of dyed sand inserted into the beachface. The latter method, devised by King (1951), can only be used on portions of the beach exposed at low tide. From the compilation of measured values in Table 2.2, it is seen that an order-of-magnitude range has been reported in the literature. King (1951) conducted the first thorough study of this parameter, making measurements on four beaches in Great Britain representing a range in wave-energy levels and beach sand size. She found a good correlation
89 TABLE 2.2 Measurements of the thickness of sediment movement Investigator
Method Tracer burial
Rods
X
King(1951) Komar and Inman (1970)
Results Sand column
X
X X
Williams (1971) Knoth and Nummedal(1978) Wang and Chang (1979) Gaughan (1979) Greenwood et al. ( 1 979) Inman et al. (1980)
X X X
Kraus et al. (1982)
X
-
X
X X
0.5-4 cm: fine-sand beaches 0.2-4.3 cm: coarse sand 2-6 cm: Silver Strand 6- 10 cm: El Moreno >z 4 cm 10 cm 1/8 inch (0.32 cm) 0.2-3.2 cm 8-24 cm 1-9 cm: maximal burial depth 0-5 cm: weighted mean burial depth 3.8-7.5 cm
X
between the thickness of sand disturbance and the breaker height, and also found that the coarser the sand size the thicker the layer of movement. The values measured by Komar and Inman (1970) generally agree with the results of King (195 1). Considering the generally higher wave-energy levels at Silver Strand Beach than at the study sites of King, the measured 2-6-cm thicknesses are in good agreement. The 6-10-cm range measured at El Moreno Beach is in basic agreement with King's measurements at Druridge, the large thicknesses of movement being accounted for by the coarse grain sizes of those beaches. On the other hand, in careful measurements of tracer burial depths on a Pacific Ocean beach. Gaughan (1979) obtained smaller values for h, ranging up to a maximum of 3 cm but averaging only 0.5 cm during the summer months and 1.1 cm during the winter. Greenwood et al. (1979) measured the depth of disturbance with 0.5 cm diameter rods driven into the sediment, each rod containing a loose-fitting washer that is free to fall to the sediment interface. Burial of the washers indicates the thickness o f sediment movement, and net elevation changes can also be monitored from the rod length. They found good agreement between their rod measurements and depth of tracer burial. At the site of their study on a Lake Ontario beach, both methods gave values in the range 8-24 cm. Further confirmation was provided by box cores revealing cross-stratification which indicated that most of the movement was an offshore shift of sand in the pronounced bar-trough topography of the study site. Inman et al. (1980) have particularly focused their attention on the evaluation of the thickness of movement during their measurements of sand-transport rates conducted as part of the Nearshore Sediment Transport Study (Seymour and Duane, 1979). They interpret the appropriate h for eq. 17 in three different ways: ( I ) as the maximum depth to which tracer is buried in the cores; (2) as a concentration-weighted mean depth; and (3) as the level in the core where the tracer concentration reaches 1 grain g - ' or less. Their data are shown in Fig. 2.8 plotted according to these three interpretations. Although the differences in plotting positions d o not appear large.
90
this is because the graph is log-log; the different interpretations of h result in as much as a factor 4.5 difference in the calculated Q , sand-transport rates. In their recent study of sand transport on beaches, Kraus et al. (1982) obtained a large number of core samples in order to investigate the distribution of the thickness across the nearshore and its variation through time. They found that most of the tracer was typically contained in the upper 6 cm, but grains were frequently found to depths as great as 20 cm. On the basis of analyses of many cores, they define a homogeneous layer where the sand tracer has been thoroughly mixed, and at greater depths a zone of tracer which is less frequently mixed vertically and probably spends a considerably smaller fraction of its time in transport. They take the thickness of this homogeneous layer as the value of h in the calculation of littoral drift rates. Of particular interest are the on-offshore distributions of tracer burial depths determined by Kraus et al. (1982). In general the pattern is one of maximum depths attained near the breaker line and in the swash zone, with lower and fairly uniform depths throughout the surf zone. Many of the distributions show an absolute minimum just shoreward of the breaker line. Typically the depths of burial average 3 to 4 cm in the surf zone, increasing to 6 to 8 cm in the breaker zone. It is apparent from this that the use of an “average” value of h in the calculation of the littoral drift is overly simplified. The results also suggest that much of the variability of reported values. listed in Table 2.2, result from measurements having been obtained at different points within this on-offshore distribution. Although an understanding of the variability of the thickness of sand movement on beaches is emerging, still further studies are required, especially due to its critical part in the evaluation of the littoral drift. In addition to making littoral drift measurements by sampling on a grid such as in Fig. 2.7, Knoth and Nummedal(1978) and Inman et al. (1980) also measured the sand-advection velocity, U , , by repeated sampling at a fixed longshore distance from the injection site. In this method the mean time of tracer passage is determined from the concentration versus time measurements, and that mean time is divided into the fixed longshore distance to yield U,. This method still requires a separate determination of the thickness of movement, b. Knoth and Nummedal report that their time series showed a series of concentration peaks rather than uniform changes through time, opening to question the evaluation of U , by this method. The resulting values are also often found to be significantly different than the results from the spatial grid method, generally being larger, sometimes by as much as a factor 2 larger. The data of Knoth and Nummedal plotted in Fig. 2.8 are primarily based on the spatial grid evaluation of U, as in the study of Komar and Inman (1970). Still another approach for measuring the littoral drift is the so-called tracer dilution method, first developed by Russell (1960). This involves a continuous or quasi-continuous injection of tracer at a known, fixed rate and then measuring its concentration at some distance down-transport, the distance and time being sufficiently great to insure complete mixing of the tracer and the normal beach sand. The resulting equilibrium tracer concentration depends on the amount of dilution by the littoral drift so its measurement provides an indirect determination of the drift rate. This method is generally hampered by requiring several hours or even days of continuous tracer injection before equilibrium conditions are achieved, but as shown
91
in the study of Duane and James (1980), a series of concentration measurements through time usually asymptotically approach an equilibrium value which can then be used in the drift evaluation. The advantage of the dilution method over the grid-sampling approach is that no measurement of the thickness of movement is required. Russell (1960) used the dilution method to measure the drift of shingle along the coast of England. Duane and James (1980) performed one measurement at Pt. Mugu, California, with the purpose of demonstrating the potential of this method on sand beaches. Their one measurement shows excellent agreement with the other data in Fig. 2.8 and with the relationship of eq. 19. Bed-load sediment traps have seen relatively little use on beaches for measuring the littoral drift. Thornton (1973) was the first to utilize them, obtaining measurements on the coast of Florida. His objective was to investigate the distribution of the littoral drift, which will be discussed later, and his data are inappropriate for the determination of the total littoral drift. Working on a Lake Michigan beach, Lee (1975) employed an array of box-type bed-load samplers in an attempt to determine the littoral drift. His data analysis indicated that the transport rates are on average only 36% of the rates predicted by eq. 19, explaining this result as due to the relative deficiency of sand found on the beach studied, gravel being abundant. However, Lee made a systematic error in his calculations of P, from his measured wave data, employing the deep-water phase velocity equation for the calculation of C,. My reanalysis of his data indicates that on average this makes his P,values a factor 2.52 too large, the factor ranging 1.44-5.03 depending on the relative values of the wave period and breaker height. Making this 2.52 factor correction, it is seen in Fig. 2.8 that Lee's data now agree reasonably well with the other data sets although they still show somewhat lower transport rates. As seen in Fig. 2.8, the existing field data on longshore sand-transport rates indicate that a reasonable correlation exists with P, of eq. 16 leading to empirical relationships such as those of eqs. 19 and 21. Although there is a fair amount of data scatter, both internally to any one data set and systematically between different sets, the scatter is comparable to that for sand transport in rivers. It is apparent that the 0.77 coefficient of eq. 19 first obtained by Komar and Inman (1970) has a high degree of uncertainty. The data given in Fig. 2.8 are derived from several coastal sites with a range of wave conditions and beach sand grain sizes [600 p (Komar and Inman, 1960) to 200 p (Duane and James, 1980)l. The complete ranges of applicability of eqs. 19 and 21 are presently unknown. Comparatively little work has been done on shingle and gravel movement along beaches, although Russell's (1960) investigation of shingle transport on a beach near Rye, England, represented the first application of fluorescent tracer techniques to quantitatively determine rates of longshore transport. Several subsequent studies have investigated the rates of longshore movement of individual shingle clasts, particularly examining their selective transport rates according to particle shape. The study of Hattori and Suzuki (1979) on the coast of Japan found that the tracer gravels moved at rates of 2 to 3 m day-' under normal sea conditions, while under storms the rate reached as much as 400 m day-'. They found no strong indications of selective transport according to shape, although there was significant progressive attrition and impact breakage of the tracer clasts. Hattori and Suzuki found a good
92
correlation between the mean longshore clast velocity and P I . If it is assumed that the movement has a thickness of one clast diameter, then their U , a P I relationship becomes equivalent to I , = 0.2P,; as expected, the quantity of transport is smaller than on sandy beaches. In addition to the field studies discussed above, there have been several laboratory investigations of littoral drift (Krumbein, 1944; Saville, 1950; Shay and Johnson, 1951, Johnson, 1953; Sauvage and Vincent, 1954; Savage, 1959, 1962; Kamphuis and Readshaw, 1979). In most cases these involved wave-basin tests where waves are made to break at some angle to the shoreline and the resulting littoral drift is trapped at the downdrift end of the beach. It is found (Savage, 1959, fig. 4; Komar and Inman, 1970, fig. 5; Das, 1973) that these laboratory results indicate much lower sand-transport rates than do the field measurements at comparable PI values. The straight line of eq. 19 actually appears to form an upper limit to the plotting of the laboratory data. Bagnold (1966) has shown for sand transport in rivers that the immersed-weight sand-transport rate is proportional to the available power of the flowing water with a constant proportionality factor only when the transport conditions are fully developed (sheet sand movement under river flood flow). Under lower flow regimes where the bottom is rippled, the sand-transport rate is less and the proportionality coefficient is lower and no longer constant. A similar effect might explain the relationship of the laboratory measurements of littoral drift as compared with the field data. In the field data of Fig. 2.8 the transport conditions were generally fully developed, the transport occurring mainly as flat-bed sheet movement with ripple marks being absent or confined to zones of low transport. I n the laboratory tests ripple marks are present and are out of scale with respect to prototype beaches, the ripples commonly forming an appreciable proportion of the total waterdepth in the surf zone, appearing more comparable to scaled prototype longshore bars. Such scaling problems make it questionable whether laboratory tests of littoral drift have much relevance to prototype beaches. In this respect the study of Saville ( 1950) is particularly intriguing; although Saville's longshore current measurements agree closely with eq. 1 1, his sand-transport measurements demonstrate a decreasing littoral drift with increasing P I . There appears to be no problem in scaling the nearshore currents, but there is in the resulting sand transport. Various laboratory studies have demonstrated a dependence of the sand-transport rate on the deep-water wave steepness, H o / L o , the maximum transport occurring in the range H,/L, = 0.01-0.03 (Saville, 1950; Shay and Johnson, 1951; Johnson, 1953). This steepness corresponds approximately to the critical value for the on-offshore shift of sand from the beach berm to offshore bars, the change between what are sometimes called the summer and winter profiles. This suggests that the rate of longshore sand transport increases at a time when there is a maximum on-offshore shift of sand modifying the beach profile, a very likely dependence. Unfortunately, the field data have not been obtained over a sufficient range of wave steepness values and with sufficient accuracy to demonstrate such a dependence. The laboratory tests also permit measurements under larger breaker angles than generally observed in the field. However, there has been only rough agreement as to what breaker angle yields the maximum sand-transport rate: a h= 30" (Johnson, 1953), 43" (Shay and Johnson, 1951), and 53" (Sauvage and Vincent, 1954).
93
Relationships such as eqs. 19 and 21 predict a maximum transport at a,, = 45" since PI of eq. 16 is a maximum at that angle. Both Bruno and Gable (1977) and Kamphuis and Readshaw (1979) have attempted to relate the sand-transport rate directly to the radiation stress, S,,, of eq. 1 rather than to PI. However, S,, is a maximum at a deepwater angle of wave approach of 45" and without significant bottom drag Sxyremains constant during shoaling, even if refraction proceeds to the point where a,, = 1". This conflicts with the apparent maximum in the sand transport at a breaker angle of 45" rather than a deep-water angle of 45". Therefore, i t would appear that Sxyalone should not be the primary factor in the analysis of littoral drift. Of interest, as we have already seen, the longshore current velocity is also a maximum at ah = 45" (eq. 11). Until now our analysis of the littoral drift has been almost purely empirical, based on plots of I , or Q , versus P , leading to relationships such as eqs. 19 and 21. Some attempts have been made at devising models that lead to the derivation of eq. 19. Komar (1971a) shows that a direct proportionality between the sand transport rate and PI results for the zig-zag sediment path in swash transport. Dean (1973) has devised a formula similar to eq. 19 based on considerations of the portion of the wave-energy flux that is dissipated by settling grains in suspension. However, Dean's model predicts that the proportionality coefficient, K = I /PI = 0.77, should not be a constant but instead will depend on the grain settling velocity, wave height and beach slope. Komar (1975, fig. 3) tested this prediction with the available data and found no dependence of K on these parameters. As will be demonstrated below, it appears that eqs. 19 and 21 result from a more basic sand-transport relationship together with the particular mode of generation of longshore currents by an oblique wave approach. Early workers such as Grant (1943) stressed that the littoral sand transport results from the combined effects of the nearshore waves and currents, the waves placing the sand in motion and the longshore currents producing a net sand advection. Such a model was given a mathematical framework by Bagnold (1963) and applied specifically to littoral drift evaluation by Inman and Bagnold (1963). Their analysis yielded:
where V ,is again the longshore current velocity, in practice measured at the mid-surf position, and u, is the maximum horizontal orbital velocity of the waves as given by eq. 4 evaluated at the breaker zone (the wave set-up, Tj, is neglected). The ratio (ECn),/u, is in effect proportional to the stress exerted by the waves which places the sand in motion but with no net transport, the longshore current, GI, producing the net longshore movement. K' is a dimensionless coefficient which must be determined empirically. It is apparent that much more thought about the processes producing the littoral drift has gone into the formulation of eq. 22 than in eq. 19. Komar and Inman (1970) utilized their littoral drift measurements to make the first test of eq. 22, obtaining direct measurements of U , for this purpose. The results are shown in Fig. 2.9, the best-fit straight line yielding K' = 0.28 as the coefficient in eq. 22.
94 lo3r-
r
UI
-z q Y
I I 1 1 1 1 , , , I I I I 11111 -- BAGNOLD MODEL K o m a r a n d I n m a n (1970)
-
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El Moreno B e a c h
0 Silver S t r a n d B e a c h
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Fig. 2.9. The data of Komar and Inman (1970) and Kraus et al. (1982) testing the Bagnold model of eq. 22, yielding K ‘ = 0.28 for this relationship which yields the immersed-weight sand transport rate. Alternatively, the relationship can be expressed as eq. 23 for the volume transport rate, Q,.
Komar and Inman (1970) therefore found that their data agree with both eqs. 19 and 22, concluding that eq. 19 is successful because the longshore current, U , , generated by an oblique wave approach, is given by the relationship of eq. 10. Equation 10 actually was obtained first by the simultaneous solution of eqs. 19 and 22 and was only later shown to fit the available longshore current data and to be derivable from radiation stress principles. Equation 22 is viewed as the more fundamental of the two sand-transport relationships in that it should apply whatever the origin of the longshore current U , used in eq. 22: tidal-generated, the currents of the cell circulation, wind-generated, as well as from an oblique wave approach. I n the special case where 6,is due only to an oblique wave approach and hence given by eqs. 10 or 11, then eq. 19 becomes applicable to the evaluation of the longshore sand-transport rate because of its equivalence to eq. 22. In addition to its more general applicability, another advantage of eq. 22 over eq. 19 is that it is usually easier to measure the longshore current, U , , than the wave-breaker angle needed in the evaluation of PI. In this connection, Walton (1980) provides a method by which a measure of the longshore current at any position within the surf zone can be utilized to evaluate the sand-transport rate, the measurement not having to be at the mid-surf position as required by eq. 22. However, Walton’s approach assumes the applicability of Longuet-Higgins’ ( 1970b) longshore current solution, and can therefore be used only when the longshore current and sand transport are due entirely to an oblique wave approach. As with the case of eq. 19, the general eq. 22 for the immersed-weight transport
95
rate can be modified to forms that directly yield the volume transport rate, Q,. As an example, for quartz-density sand one obtains: Q, = 2.5( ECn),-
*/
urn
(23)
where Q, has units of m3 day-' and the wave and current parameters are in mks units (W m-'1. Subsequent measurements to test eqs. 22 and 23 have been obtained by Kraus et al. (1982) and Wang and Chang (1979). Kraus et al. collected their data on the coast of Japan, two beaches being involved in the study, Ajigaura and Oarai. Although the two locations have similar beach characteristics, Oarai differed in that the experiment was performed in the sheltered region of a breakwater where the longshore current results from the combined effects of obliquely-incident waves and a longshore variation in wave heights, much as in the study of Gourley (1975, 1977) already discussed. This case is of particular interest in that under such conditions the model' of Bagnold, eq. 22, must be. utilized, relationships to P, such as eq. 19 being inapplicable. The littoral drift rates were determined with sand tracers, and their measurements of the thickness of movement were discussed earlier in this chapter. The three measurements obtained by Kraus et al. (1982) are plotted in Fig. 2.9 and are seen to agree very well with the previous data of Komar and Inman (1970) in establishing eq. 22, Especially note that the one measurement from Oarai obtained in, close proximity to a breakwater shows nearly perfect agreement, demonstrating that eq. 22 is applicable to conditions where the longshore current is not simply generated by an oblique wave approach alone. Wang and Chang (1979) obtained their data along the bayshore of a barrier island in the Gulf of Mexico. The wave heights ranged only 9-15 cm due to the small fetch of the lagoon, so the conditions were more comparable to wave basins than to open ocean beaches. Sand tracer was used to determine the transport rates. Again of particular interest, the longshore currents were not due entirely to an oblique wave approach, but were in part generated by local winds and tides. Wang and Chang (1979) found good agreement between their measurements and eq. 22, but obtained a coefficient K' = 0.18, approximately 36% lower than the 0.28 value for open coasts obtained by Komar and Inman (1970). This lower K' coefficient may result from the difference in energy levels of the beaches under study; the low-energy bayside beach versus the high-energy open coast. The problem may also be in the evaluation of the thickness of sand movement, already discussed, Wang and Chang using b = 1/8th in. (0.32 cm) in eq. 17; only a slightly higher value would produce agreement with the results of Komar and Inman (1970). Other possibilities are: (1) the tracer distributions of Wang and Chang are dominated by diffusion rather than advection, making their centroid determinations more uncertain; and (2) the distributions are strongly controlled by the beach topography, the transverse bars present collecting tracer in their troughs with less on the crests. In spite of the apparent systematic difference with the data of Komar and Inman (1970), the results of Wang and Chang still demonstrate the applicability of eq. 22 to conditions where the longshore currents are not completely wave induced. The discussion until now has been concerned with the total littoral drift in the
96
nearshore, basically the longshore movement of sand between the shoreline and outer edge of the breaker zone. Our attention now turns to the distribution of that transport, comparable to the distribution of the longshore current already examined. Here again there is a paucity of data, what little data exists having been obtained with bed-load traps. The model of Bagnold (1963) has been utilized in three studies to analyze the sand-transport distribution (Thornton, 1973; Komar, 1976b, 1977a; McDougal and Hudspeth, in press). Here the local stress, T ( x ) , exerted by the waves or combined waves and longshore current, places the sand in motion and the local longshore current, u( x), produces the longshore transport. Thornton ( 1973) applied such an analysis to the transport both inside and outside the surf zone, differing only in the method for analyzing the available power. Inside the surf zone this is given by a( E C n ) / a x and Thornton's transport formula becomes:
where 4, is the volume transport rate per unit surf-zone width, and B, is a dimensionless coefficient that must be determined empirically; the other parameters are as defined before. Thornton (1973) obtained field data from Fernandina Beach, Florida, in order to evaluate B, in eq. 24. The transport rate, q,, was measured using a series of bed-load traps placed in a line across the nearshore. The scheme could therefore potentially
I
1
I
I '
F
I \\
9
60
.L
2o 0
-
-
c
f I-
n. w n
& MWL
0 10 15
700
600
DISTANCE
500
400
300
FROM BASELINE ( f t )
Fig. 2.10. The distribution of the longshore sand transport measured by Thornton (1973) at Fernandina Beach, Florida, by bed-load traps in the positions shown. The dashed lines are based on the theoretical relationships, that inside the surf zone obtained from eq. 24 with B, = 0.08. (From Thornton, 1973.)
97
determine the distribution of the littoral drift rather than only the total quantity as had the previous studies. However, nearly all of Thornton’s traps fell outside the breaker zone with at most one or two within the surf zone. Figure 2.10 shows one example that includes measurements from the breaker zone and just to its shoreward side, indicating a maximum in the longshore transport in the breaker zone. Fernandina,Beach has pronounced bars and troughs, and all of the data show stronger transport over the bars than in the troughs. The dashed lines of Fig. 2.10 are the theoretical predictions, that inside the surf zone evaluated from eq. 24 with B, = 0.08, a value based on all of Thornton’s measurements inside the surf zone. The dashed line of Fig. 2.10 outside the breaker zone is given by the comparable formula for that region. Komar (1976b, 1977a) utilized the Bagnold (1963) model to analyze the distribution of the littoral drift which was then “calibrated” to yield the total transport. I , , as predicted by eq. 19. From the Bagnold model, the local immersed-weight sand transport rate per unit surf-zone width, i(x), is given by: i(x ) =
7Tkl ~
4
(0.5f) p y 2h( x ) 1) ( x )
where f is the drag coefficient for oscillatory wave motions according to Jonsson (1967) and Kamphuis (1975), and h and 0 are the local waterdepth and longshore current, both functions of the offshore distance x from the shoreline. k , is a dimensionless proportionality factor whose value is determined by the “calibration” process. This involves integration of i ( x ) across the surf zone according to: I , = i x h i (x ) d x
where X , is the surf-zone width. In this procedure the total transport I , is given by eq. 19 and the longshore current distribution, ~ ( x )is, given by the Longuet-Higgins (1970b) solution of eq. 15. This “calibration” demonstrates that the k , value to be used in eq. 25 to yield the correct total transport is a function only of the drag coefficient, C,, and P of eq. 15e which controls the longshore current distribution (Fig. 2.2). As discussed earlier, the value of C , is also fixed by the selection of P in such a way that the longshore current distribution agrees with eq. 1 1 at the mid-surf position. P is now seen to also control the sand-transport distribution. An example is shown in Fig. 2.1 1 of a longshore current distribution and resulting sand-transport distribution calculated by this “calibration” approach of Komar ( 1976b, 1977a). The calculated sand-transport distribution is truncated at the breaker zone since its derivation is based on the assumption that the total drift, I , , as given by eq. 19, is confined to the nearshore. The analysis predicts that the maximum i n the sand transport will be shoreward of the breaker zone where the stress due to the waves is a maximum, and seaward of the maximum of the longshore current distribution. This position results from the sand transport being due to the combined effects o f the waves and currents. The above analysis assumes a planar beach profile. McDougal and Hudspeth (in press) conduct a similar analysis for a concave-up, x’/’-dependent beach profile, generally a more realistic profile. Similar to their results on the longshore current
98 H, = 100 c m ‘&= 10” m = 0.100
I25
I
,.*
s a n d transport rate, s
E IOOU r u)
10
75
-5 u)
-z 50 al
0
a u) c
e
+
25 u u) U
0 10
05
x /x, Fig. 2.1 I . An example of calculated longshore current and sand-transport distributions determined by the approach of Komar (1977a). The longshore current distribution agrees with eq. 1 1 at the mid-surf position ( x / X ,= 0.5) and thus with most of the available current measurements. And the sand-transport distribution is “calibrated” so that its integration yields the total transport rate as given by eqs. 19 and 21. (From Komar. 1977a.)
distribution, such a profile causes the maximum in the sand-transport distribution to shift closer to shore than in the case of a planar beach. Sawaragi and Deguchi (1979) have obtained the completest field and laboratory measurements of the distribution of sand transport across the surf zone. They utilized circular bed-load traps which enable them to evaluate the on-offshore sand movements as well as the longshore transport. Figure 2.12. shows one example from the laboratory. In this example the maximum in the sand transport occurs at approximately x / X , = 0.6, while the maximum in the longshore current distribution is at x / X , = 0.4. These results are relatively independent of the wave steepness, H , , / L , , but change with sand-grain size. With finer sand the principal maximum in the sand transport shifts closer to the breaker zone ( x / X , = 0.8) with some tendency for a small secondary maximum at x / X , = 0.2. Sawaragi and Deguchi (1979) analyzed their sand-transport measurements in a non-dimensional format, arriving at:
= 23F4.’
where:
F=
r
-
rC
(Ps - P>g4,
for F < 0.3
(27b)
YY I
I
I
I
I
I
I
I
I
Fig. 2.12. Laboratory measurements of the distributions of the longshore current and the sand-transport rate showing the positioning of the maximum in the sand transport between the maximum of the longshore current and the breaker zone where the stress exerted by the waves is a maximum. (Based on data of Sawaragi and Deguchi, 1979.)
where Ds0 is the median of the grain-size distribution, r is the bottom stress exerted by the waves and T~ is the critical stress for sediment motion. In that both u and T vary across the surf zone, eq. 27 yields the sediment-transport distribution as shown in Fig. 2.12, which can then be integrated to yield the total transport. Equation 27 is a version of the Kalinske-Brown formula for sediment transport in rivers, adapted for use on beaches, the principal adaptation being that the stress, T . must be evaluated so as to include both wave and current effects. Iwagaki and Sawaragi ( 1962) had earlier attempted to apply a modified Kalinske-Brown formula for estimating the littoral drift, to my knowledge the first to try to modify a river-based equation. The approach of Bijker (1969, 1971) for evaluating the littoral drift also employs a modified river-based sediment-transport relationship, in his case utilizing the bed-load formula of Frijlink together with Einstein’s method for evaluating the suspended-load transport. Again, the principal adaptation to beaches is through the evaluation of the bottom stress. Bijker also undertook laboratory measurements of the littoral drift, the results of which are similar to those of Sawaragi and Deguchi ( 1979). However, Bijker’s measured distributions are considerably different from those he calculated theoretically. Formulations such as eq. 27 from Sawaragi and Deguchi (1979) and those of Bijker (1969, 1971) can be employed to examine time-variations in the sediment transport. Madsen and Grant (1977) similarly adapt the Einstein-Brown sediment-transport equations to time-varying sand movements under combined waves and unidirectional currents, and their methodology can be used in the nearshore. Whichever approach is employed, the bottom stress T under the combined oscillatory motions of the waves and any superimposed currents becomes dependent on time and the resulting sand transport also varies with time.
100
Bowen (1980) and Bailard and Inman (1981) employ the approach of Bagnold (1963, 1966) to an analysis of the instantaneous sand transport in the nearshore, including both on-offshore movements as well as longshore movements. An important inclusion is the effect of the local bottom slope on the sand-transport rate as it permits the analysis of the formation of longshore bars and other inshore topography. For example, employing the equations as given by Bowen (1980), Holman and Bowen (1982) have developed numerical models which simulate the growth of crescentic bars and oblique bars respectively under standing and progressive edge waves. Beach sand consists of a range of grain sizes, and it is of interest to determine the differential rates of longshore movement of these various size fractions. The most obvious approach to obtaining such data is by tagging the several size fractions with different fluorescent colors, and this approach has been used in a number of studies. However, the majority of these studies mainly focused on the on-offshore movement of the different size fractions, examining whether they approach some equilibrium “null point” position on the beach profile. Few of the studies examined the longshore movements in anything more than a qualitative manner. Yasso (1965) tagged four size classes of sand from the beach at Sandy Hook, New Jersey, injected it into the beach face, and then repeatedly sampled at a fixed position 30.5 m along the shore. Based on the arrival times of two grain-size fractions of different colors at his sampling position, Yasso concluded that the finer grains moved alongshore at the faster rate. However, the first arrival of sand tracer at the fixed sampling location depends more on the relative sand diffusion rates than on the advection rates, so that the interpretation is not unequivocal. Other studies, such as those of Ingle (1966, pp. 93-loo), sampled on a grid such as in the example of Fig. 2.7, and the tracer-distribution patterns can give an indication of selective longshore transport by grain size as well as on-offshore movements. Ingle divided the grain-size distribution into two fractions, a coarse half and fine half, tagging them with different colors, so that the grid sampling shows two distributions. Most of Ingle’s attention was centered on the on-offshore movements, like other studies finding that the coarser fraction moves offshore into the breaker zone while the finer fraction remains in the mid- to inner-surf zone. The grid patterns as well as rates of removal from the injection sites indicate faster longshore transport rates for the finer fraction, so that Ingle’s results appear to be in agreement with Yasso’s (1965). In their experiment at Pt. Mugu, California, already discussed, Duane and James (1980) also analyzed their data in terms of the contributions of the various grain sizes to the total volume transport rate. They found that the maximum volume occurs in the size range 0.15-0.18 mm, decreasing to both finer and coarser sizes. Reinterpreted in terms of velocities rather than volume transport rates, their results indicate faster longshore transport rates for the finer fractions, and so also agree with Yasso (1965) and Ingle (1966). Komar (1977b) found just the opposite result based on measurements at El Moreno Beach, Mexico. A somewhat different approach was also employed in that study. The grid samples that served as the basis for the tracer-distribution plot of Fig. 2.7 were sieved into their several size fractions and the numbers of tracer grains
101
counted in each. In this way transport patterns were obtained for each of the individual grain-size fractions even though only one tracer color was employed. The results showed that the coarsest sand grains found within the beach, the fraction centered at 1.19 mm, moved alongshore the fastest with a mean advection rate of 0.31 cm This rate is about four times faster than grains of diameter 0.30 mm. These results were interpreted along the lines of a model for selective sorting first proposed by Evans (1939), based on the movements of coal fragments and weighted balls along Lake Michigan beaches. According to this model, the finer sand grains swash high up the beachface where they move alongshore more slowly than the coarser grains that remain near the breaker zone and are transported by stronger longshore currents. This might also in part explain the contrasting results between the measurements of Komar (1977b), where swash transport dominated, versus the results found by Yasso (1965), Ingle (1966) and Duane and James (1980) on finer sand beaches with wide surf zones. Komar (1977b) also interpreted his selective transport results as implying that bed-load transport is more important than suspension transport, at least o n El Moreno Beach. Komar (1978) carried this analysis another step, examining the furthest longshore displacements of tracer grains found in the grid sampling, and hence the maximum advection rate, Urnax, of tracer on the beach. This was compared with the longshore current, U , , the reasoning being that if U,,,,/U, is small, indicating the sand lags well behind the transporting current, then there cannot have been a substantial amount of suspension transport. At El Moreno Beach U,,,,,/U, ranged 0.025-0.23 and ranged 0.0033-0.013 where ?? is the average tracer-advection rate. These values indicate a considerable lag of the sand movement. At Silver Strand Beach, San Diego, the Urn,,/U, values are more uncertain, appearing to be on the order of 0.3-0.6, with z/U, in the range 0.0095-0.0125, still indicating considerable lag with the implication that suspension transport is small. However, there is considerable uncertainty and differences of opinion as to the relative importance of bed-load versus suspension transport on beaches. Direct measurements of suspension concentrations have been obtained by pumping large quantities of water containing the suspended sand (Beach Erosion Board, 1933; Watts, 195313; Fairchild, 1973, 1977), with suspension “traps” that sample a small amount of water from some position within the surf zone (Fukushima and Kashiwamura, 1959; Hom-ma and Horikawa, 1963; Hom-ma et al., 1965; Kana, 1976, 1978, 1979; Inman et al., 1980), and by light scattering or attenuation (Brenninkmeyer, 1975, 1976; Thornton and Morris, 1978; Leonard and Brenninkmeyer, 1979). The advantage of the pumping approach is that large quantities can be processed, lending more confidence that the samples are representative of concentrations found in the surf; the disadvantages are that only large samples can be obtained so one cannot investigate time-variations in sediment concentrations, and the sampling must be done from a pier which may influence the results. The earliest published information on the distribution of suspended sand across the beach was derived from this method (Beach Erosion Board, 1933). It was found that the highest concentrations occurred at the breaker line and at the base of the swash zone, both being areas of high turbulence. Watts (1953b) and Fairchild (1973, 1977) found s
C
’
.
z/G,
102
similar distributions, also showing that the amount of sand in suspension is related to the wave height or energy. However, their data are extremely scattered and show only broad trends. The suspension “traps” usually consist of a vertical array of three or four sample collectors and so can be used to examine the vertical distribution of suspended sediments, and can be positioned at any location across the surf zone. They can also be triggered as a wave crest or trough passes so that phase relationships with the waves can be examined. Fukushima and Kashiwamura (1959), Hom-ma and Horikawa (1963) and Hom-ma et al. (1965) used suspended samplers made of bamboo poles to examine the vertical distribution of suspended sediments. They found that the suspended sediment in the swash zone and breaker zone can be fairly evenly distributed over the vertical due to the high degree of turbulence at those locations. Kana (1978, 1979) undertook a study on the coast of South Carolina. employing suspension samplers described in Kana ( 1 976). As shown in Fig. 2.13, he found the expected upward decrease in concentration with elevation above the bottom, and also found systematic differences depending on whether the waves were breaking by plunging or spilling, the plunging breakers throwing considerably more sand into suspension due to their more intense interaction with the bottom. One unexpected result, under the moderate wave energies prevailing during the experiments ( H b < 1.5 m), was that the concentration of suspended sediment decreased with increasing wave height. Under plunging waves the maximum concentrations were found approximately 3-5 m shoreward of the breakpoint, while under spilling waves the concentration was more uniform across the entire surf zone width. Inman et al. (1980) also report on suspension concentration measurements obtained during the NSTS experiments.
SUSPENDED SEDIMENT DISTRIBUTION I N BREAKING WAVES, PRICE INLET, S.C
Ly 2
I
0
I
0.05
I I I l l
0.1
I
I
I
I
0.5
1 1 1 1
1.0
I
I
I
1
1 1 1 1
5.0
10.0
I
I
1
50.0
SUSPENDED- SEDIMENT CONCENTRATION (g I-’ )
L g . 2.13. Vertical distributions of suspended-sediment concentrations showing an approximately exponential decrease with distance above the bottom with higher concentrations prevailing under plunging breakers than under spilling. (From Kana, 1978.)
103
Brenninkmeyer (1975, 1976) and Leonard and Brenninkmeyer (1979) employed opacity measuring devices, termed almometers, to determine instantaneous and continuous concentrations of suspended sediment across the surf zone. This device is limited to concentrations greater than 10 g of sediment per liter of water, and so can detect only the occurrences of relatively high concentrations, termed “sand fountains”. Utilizing spectra of his measurements, Brenninkmeyer found that during ‘‘ normal” conditions the suspended sediment movement is a low-frequency event, most of the movement centered at frequencies less than 0.25 Hz, with the relatively high-frequency component of the waves contributing little to the amount of total sediment transported. In contrast, Leonard and Brenninkmeyer found that under storm conditions higher-frequency sand movement is more common, but is still not controlled by the prevailing wave and swell periods. It was found that sand is rarely thrown into suspension in the breaker zone at high concentrations detectable by the almometer, and is highest in the transition zone at the base of the swash zone where the return swash collides with incoming wave bores. Thornton and Morris (1978) also measured sediment concentrations optically (as well as pumping), and were able to detect much lower concentrations, their experiments at Torry Pines Beach, California, yielding mean concentrations which ranged between 0.05 and 0.32 g of sand per liter of water. Spectral analyses were performed on the concentration measurements as well as on simultaneous current measurements obtained with an electromagnetic current meter. I t was found that the peaks of the suspended-sediment spectra occur at approximately twice the peak frequency in the velocity spectra, indicating two or more suspension “events” per wave period. Similar to the results of Kana (1978) and Inman et al. (1980), suspended-sand concentrations decreased exponentially with height above the bottom, there being some correlation between the total concentrations and the mean bed stress exerted by the waves. Measurements of suspended sediment concentrations can be used to calculate longshore transport rates of suspended load. If C is the average volume concentration of sand in suspension, assuming that this sand is transported by the longshore current V,, the longshore flux of suspended sediment is: suspension flux
= CU, A
(28)
where A is the total cross-sectional area of the nearshore region from the shoreline through the breaker zone (Dean, 1973; Galvin, 1973). This is of course only an approximate estimate, a more refined evaluation involving the integration of c(x. z, t)v(x, z, t) across the surf-zone area. Employing a relationship such as eq. 28 and utilizing his measurements of suspended-sediment concentrations, Kana (1978) estimated daily longshore transport rates near Price Inlet, South Carolina. Finding a close correspondence between the longshore sediment transport evaluated in this way with the “total” transport evaluated with relationships such as eqs. 19 and 21, Kana concluded that the suspended load accounts for the major portion of sand transport in the nearshore. In a similar analysis, Komar (1978) arrived at just the opposite conclusion. Utilizing eq. 28 to evaluate the suspension transport, eq. 10 for the evaluation of V,, and then dividing by I , of eq. 19, the ratio of suspension transport to total transport
104
is given by: Z,(suspension) I , ( total)
-
7.0: P, Ym
-
P
P
In deriving this relationship, whenever an approximation or assumption had to be made, it was made in favor of the computation of the suspended load at the expense of the proportion due to bed-load transport. Therefore, the relationship should systematically over-estimate the suspension transport. Employing the measured values of sediment concentration C determined by several studies, Komar (1978) found that eq. 29 yielded ratios in the range 0.066-0.26, indicating that suspension transport is comparatively low, less than about 25% of the total transport rate, the bed load forming the remaining 75% or more. This analysis of course involves many simplifications and inherent inaccuracies, so that the estimated percentages of suspended load must be viewed as highly uncertain, the calculations having been performed only for the purpose of giving a rough indication of the relative importance of bed-load versus suspension transport. However. the values are supported by the more complete data set and refined analysis of Inman et al. (1980). Based on their simultaneous measurements of the total transport rate with sand tracers and the amount of sand in suspension determined with suspension " traps", their analysis led them to conclude that suspension transport accounts for only approximately 10 to 20%, the exact value depending on the interpretation of the thickness of sediment movement used in the calculation of the total transport rate. Although more investigations are obviously required, the balance of the available evidence indicates that bed-load transport on beaches is quantitatively more important than suspension transport. The reason for this may be that any sediment that is sufficiently fine to remain in suspension for long periods of time will tend to be lost offshore through rip currents or by offshore diffusion.
SUMMARY
As is to be expected, our understanding of nearshore currents is at a more advanced state than our knowledge of sediment-transport processes on beaches. We are now able to make reasonable predictions of longshore current velocities under an oblique wave approach, but our predictions of the concomitant littoral drift have a high degree of uncertainty. This situation is perhaps to be expected as a similar condition exists in rivers. The standard formulae for predicting sand transport rates on beaches could easily be off by a factor of 2, even on beaches with relatively simple configurations lacking pronounced topography. This results from our basic inability to make accurate measurements of sand transport on beaches, each of the several methods employed having certain application problems leading to large uncertainties in the final evaluation. These inaccuracies have not allowed us to examine the expected dependencies of the sand transport on breaker types (plunging versus spilling) and on the wave steepness, nor to arrive at definite conclusions concerning the relative proportions of bed-load versus suspension transport. And nearly all of our existing data are for the total sand-transport rate, with almost no
105
measurements of the distribution across the nearshore zone. Thus, although considerable progress has been achieved since 1953 when G. Watts first attempted to relate sand-accumulation rates at a jetty to the wave conditions, many basic questions concerning sand transport on beaches remain unanswered.
ACKNOWLEDGEMENTS
I would like to thank Rob Holman, William McDougal and Farouk Abdel-Aal for their discussions of the many topics covered in this chapter and for their useful reviews of the manuscript. This review was undertaken with support from the Office of Naval Research, Geography Branch, under contract NR 388- 168.
REFECRENCES Allen. J.R.L., 1974. Empirical models of longshore currents. Geogr. Annal., Ser. A. 56: 237-240. Bagnold. R.A., 1963. Mechanics of marine sedimentation. In: M.N. Hill (Editor). The Sea. Wiley-lnterscience, New York. N.Y., pp. 507-582. Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. U.S. Geol. Surv.. Prof. Pap., 442-1, 37 pp. Bailard, J.A. and Inman, D.L., 1981. An energetics bedload model for a plane sloping beach: local transport. J . Geophys. Res., 86: 2035-2043. Balsillie, J.H., 1975. Surf observations and longshore current prediction. U S . Army Corps Eng., Coastal Eng. Res. Cent., Tech. Memo. 58. 39 pp. Battjes, J.A., 1972. Radiation stresses in short-crested waves. J. Mar. Res.. 30: 56-64. Battjes, J.A.. 1973. Set-up Due to Irregular Waves. Proc. 13th Conf. Coastal Eng., pp. 1993-2004. Battjes, J.A.. 1975. A Note on Modeling of Turbulence in the Surf-zone. Proc. Symp o n Modeling Techniques, San Francisco, Calif., pp. 1050- 1061. Beach Erosion Board, 1933. Interim Report. U.S. Army Beach Erosion Board, Office of Chief Engineer, 100 pp. Bijker. E.W., 1969. Littoral Drift as Function of Waves and Current. Proc. 1 Ith Conf. on Coastal Eng.. pp. 415-433. Bijker. E.W., 1971. Longshore transport computations. J. Waterways. Harbors Coastal Eng. Div.. ASCE (WW4) pp. 687-701. Boon, J.D., 1969. Quantitative analysis of beach sand movement: Virginia Reach. Virginia. Sedinientology. 13: 85-103. Bowen, A.J., 1969a. The generation of longshore currents on a plane beach. J. Mar. Re$., 27: 206-215. Bowen, A.J.. 1969b. Rip currents, I . Theoretical investigations. J. Geophys. Res.. 74: 5467-5478. Bowen, A.J., 1980. Simple models of nearshore sedimentation; beach profiles and longshore bars. I n : S.B. McCann (Editor). The Coastline of Canada. Geol. Surv. Can., pp. 1 - 1 I . Bowen, A.J. and Inman. D.L. (1969). Rip currents. 2. Laboratory and field observations. J. Geophys. Res.. 74: 5479-5490. Bowen, A.J., Inman, D.L. and Simmons, V.P., 1968. Wave ‘set-down’ and ‘set-up’. J. Geophys. Res.. 73: 2569-2577. Brebner, A. and Kamphuis, J.W., 1965. Model Tests on the Relationship Between Deep-water Wave Characteristics and Longshore Currents. Proc. 9th Conf. Coastal Eng.. pp. 191-196. Brenninkmeyer, B.M., 1975. Mode and Period of Sand Transport in the Surf Zone. Proc. 14th Conf. o n Coastal Eng., Copenhagen, 1974. pp. 812-827. Brenninkmeyer, B.M., 1976. In situ measurements of rapidly fluctuating, high sediment concentrations. Mar. Geol. 20: 117- 128. Bruno. R.O. and Gable, C.G., 1977. Longshore Transport at a Total Littoral Barrier. Proc. 15th Conf. on Coastal Eng., pp. 1203- 1222.
106 Bruno, R.O., Dean, R.G.. Gable, C.G. and Walton, T.L., 1981. Longshore sand transport study at Channel Islands Harbor, California. U.S. Army Corps Eng.. Coastal Eng. Res. Cent., Tech. Pap.. 81-2. 48 pp. Caldwell, J.M., 1956. Wave action and sand movement near Anaheim Bay. California. U.S. Army Beach Erosion Board, Tech. Memo., 68, 21 pp. CERC, 1973. Shore Protection Manual. Coastal Eng. Res. Cent., U.S. Army Corps Eng., U.S. Govt. Printing Office, Washington, D.C. Coakley, J.P., Savile, H A , Pedrosa, N.N. and Larocque, M., 1979. Sled System for Profiling Suspended Littoral Drift. Proc. 16th Conf. on Coastal Eng., pp. 1764-1775. Courtois, G . and Monaco, A,, 1969. Radioactive methods for the quantitative determination of the coastal drift rate. Mar. Geol., 7: 183-206. Dalrymple, R.A., Eubanks, R.A. and Birkemeier, W.A.. 1977. Wave-induced circulation in shallow basins. J. Waterway, Port. Coastal Ocean Div., ASCE, 103 (WWI): p. 117-135. Das. M.M.. 1973. Suspended Sediment and Longshore Sediment Transport Data Review. Proc. 13th Conf. on Coastal Eng.. pp. 1027-1048. Dean. R.G., 1973. Heuristic Model of Sand Transport in the Surf Zone. Conf. on Eng. Dynamics in the Surf Zone, Sydney, N.S.W., 7 pp. Duane, D.B., 1970. Tracing sand movement in the littoral zone: Progress in the radioisotope sand tracer (RIST) study. U.S. Army Corps Eng.. Coastal Eng. Res. Cent., MIX. Pap., 4-70, 46 pp. Duane, D.B. and James, W.R., 1980. Littoral transport in the surf znne elucidated by an Eulerian sediment tracer experiment. J. Sediment. Petrol.. S O : 929-942. Eaton, R.O.. 1951. Littoral Processes on Sandy Coasts. Proc. 1st Conf. on Coastal Eng.. pp. 140-154. Evans. O.F.. 1939. Sorting and transportation of material in the swash and backwash. J. Sediment. Petrol.. 9: 28-31. Fairchild, J.C., 1973. Longshore Transport of Suspended Sediment. Proc. 13th Conf. on Coastal Eng.. pp. 1069-1088. Fairchild, J.C.. 1977. Suspended sediment in the littoral zone at Vetnor, New Yersey, and Nags Head, North Carolina. U.S. Army Corps Eng., Coastal Eng. Res. Cent.. Tech. Pap.. 77-5, 97 pp. Fukushima, H. and Kashiwamura, M., 1959. Field investigation of suspended sediment by the use of bamboo samplers. Coastal Eng. Japan, 2: 53-58. Calvin, C.J., 1967. Longshore current velocity: A review of theory and data. Rev. Geophys.. 5 : 287-304. Calvin, C.J., 1973. A Gross Longshore Transport Formula. Proc. 13th Conf. o n Coastal Eng., pp. 953-970. Calvin. C.J. and Eagleson. P.S., 1965. Experimental study of longshore currents on a plane beach. U.S. Army Corps Eng., CERC Tech. Memo, 10, 80 pp. Calvin, C.J. and Nelson, R.A., 1967. Compilation of longshore current data. U S . Army Coastal Eng. Res. Cent., Misc. Pap., 2-67, 19 pp. Calvin, C . and Vitale, P., 1977. Longshore Transport Prediction-SPM 1973 equation. Proc. 15th Conf. on Coastal Eng., pp. 1133-1148. Gaughan, M.K., 1979. Depth of Disturbance of Sand in Surf Zones. Proc. 16th Conf. on Coastal Eng.. pp. 15 13- 1530. Gourley, M.R., 1975. Wave Set-up and Wave Generated Currents in the Lee of a Breakwater or Headland. Proc. 14th Conf. Coastal Eng., Copenhagen, 1974, pp. 1976-1995. Gourley, M.R., 1977. Non-uniform Alongshore Currents. Proc. 15th Conf. Coastal Eng., pp. 701-720. Grant, US., 1943. Waves as a transporting agent. Am. J. Sci. 241: 117- 123. Greenwood, B., Hale, P.B. and Mittler, P.R.. 1979. Sediment Flux Determination in the Nearshore Zone: Prototype Measurements. Workshop on Instrumentation for Currents and Sediments in the Nearshore Zone, Natl. Res. Council, Ottawa, Ont., pp. 99- 119. Greer, M.N. and Madsen, 0,s.. 1979. Longshore Sediment Transport Data: a Review. Proc. 16th Conf. on Coastal Eng., pp. 1563-1576. Guza, R.T. and Thornton, E.B., 1979. Variability of Longshore Currents. Proc. 16th Conf. on Coastal Eng., pp. 756-775. Harrison, W., 1968. Empirical equation for longshore current velocity. J. Geophys. Res., 73: 6929-6936. Harrison, W. and Krumbein, W.C., 1964. Interactions of the beach-ocean-atmosphere system at Virginia Beach, Virginia. U.S. Army Coastal Eng. Res. Cent., Tech. Memo, 7, 102 pp.
107 Harrison, W., Pore, N.A. and Tuck, D.R., 1965. Predictor equations for beach processes and responses. J. Geophys. Res., 79: 6103-6109. Hattori, M. and Suzuki, T., 1979. Field Experiment on Beach Gravel Transport. Proc. 16th Conf. on Coastal Eng., pp. 1688-1704. Holman, R.A. and Bowen, A.J., 1982. Bars, bumps and holes: Models for the generation of complex beach topography. J. Geophys. Res., 87: C I , 457-468. Horn-ma, M. and Horikawa. K., 1963. Suspended Sediment Due to Wave Action. Proc. 8th Conf. on Coastal Eng., pp. 168-193. Horn-ma, M., Horikawa, K. and Kajima, R., 1965. A study of suspended sediment due to wave action. Coastal Eng. Japan, 3: 101-122. Huntley, D.A., 1976. Lateral and Bottom Forces on Longshore Currents. Proc. 15th Conf. on Coastal Eng., pp. 645-659. Ingle, J.C., 1966. The Movement of Beach Sand. Elsevier, New York, N.Y., 221 pp. Inman, D.L. and Bagnold, R.A., 1963. Littoral Processes. In: M.N. Hill (Editor), The Sea. Wiley-Interscience, New York, N.Y., pp, 529-553. Inman, D.L. and Frautschy, J.D., 1966. Littoral processes and the development of shorelines. Coastal Eng., ASCE, pp. 5 1 1-536. Inman, D.L. and Quinn, W.H., 1952. Currents in the Surf Zone. Proc. 2nd Conf. on Coastal Eng., pp. 24-36. Inman, D.L., Zampol, J.A.. White, T.E., Hanes, D.M., Waldorf, B.W. and Kastens, K.A., 1980. Field Measurements of Sand Motion in the Surf Zone. Proc. 17th Conf. on Coastal Eng., pp. 1215-1234. Iwagaki, Y . and Sawaragi, T., 1962. A new method for estimation of the rate of littoral sand drift. Coastal Eng. Japan, 5: 67-79. Johnson, J.W., 1953. Sand Transport by Littoral Currents. Proc. 5th Hydraul. Conf.. Univ. Iowa Studies Eng., Bull., 34: 89- 109. Jonsson. I.G., 1967. Wave Boundary Layers and Friction Factors. Proc. 10th Conf. on Coastal Eng., pp. 127-148. Jonsson, I.G., Skovgaard. 0. and Jacobsen, T.S., 1975. Computation of Longshore Currents. Proc. 14th Conf. Coastal Eng., Copenhagen, 1974. pp. 699-714. Kamphuis, J.W., 1975. Friction factors under oscillatory waves. J. Waterways. Harbors Coastal Eng. Div., ASCE, WW2, pp. 135-144. Kamphuis, J.W. and Readshaw, J.S., 1979. A Model Study of Alongshore Sediment Transport Rate. Proc. 16th Conf. on Coastal Eng., pp. 1656-1674. Kana. T.W., 1976. A new apparatus for collecting simultaneous water samples in the surf zone. J. Sediment. Petrol., 46: 1031-1034. Kana, T.W., 1978. Suspended sediment transport at Price Inlet, S.C. Coastal Sediments '77, pp. 366-382. Kana, T.W., 1979. Surf Zone Measurements of Suspended Sediment. Proc. 16th Conf. on Coastal Eng., pp. 1725-1743. Keeley, J.R., 1977. Nearshore currents and beach topography, Martinque Beach, Nova Scotia. Can. J. Earth Sci., 14: 1906-1915. Keeley, J.R. and Bowen. A.J.. 1977. Longshore variations in longshore currents. Can. J. Earth Sci.. 14: 1897-1905. King, C.A.M., 1951. Depth of disturbance of sand on sea beaches by waves. J. Sediment. Petrol., 21: 131- 140. Knoth, J.S. and Nummedal, D., 1978. Longshore sediment transport using fluorescent tracer. Coastal Sediments '77, ASCE, pp. 383-398. Komar, P.D., 1971a. The mechanics of sand transport on beaches. J. Geophys. Res., 76: 713-721. Komar, P.D., 1971b. Nearshore cell circulation and the formation of giant cusps. Geol. Soc. Am. Bull., 82: 2643-2650. Komar, P.D., 1975. Nearshore currents: generation by obliquely incident waves and longshore variations in breaker heights. in: J. Hails and A. Carr (Editors), Nearshore Sediment Dynamics and Sedimentation. Wiley, London, pp. 17-45. Komar, P.D., 1976a. Beach Processes and Sedimentation. Prentice-Hall, Englewood Cliffs, N.J., 429 pp. Komar, P.D., 1976b. Longshore currents and sand transport on beaches. Ocean Eng. 111, ASCE, pp. 333-354.
1 ox
Komar, P.D., 1977a. Beach sand transport: Distribution and total drift. J. Waterway, Port, Coastal and Ocean Div., ASCE, 103 (WW2): 225-239. Komar, P.D., 1977b. Selective longshore transport rates of different grainsize fractions within a beach. J. Sediment. Petrol., 47: 1444-1453. Komar, P.D., 1978. The relative significance of suspension versus bed-load o n beaches. J. Sediment. Petrol., 48: 921-932. Komar, P.D., 1979. Beach-slope dependence of longshore currents. J. Waterway. Port. Coastal and Ocean Div., ASCE (WW4): 460-464. Komar, P.D., in press. Rhythmic shoreline features and their origins. In: R. Gardner. J. Pitman and H. Scoging (Editors), Large-Scale Geomorphology. Oxford Univ. Press, London. Komar, P.D. and Inman, D.L., 1970. Longshore sand transport on beaches. J. Geophys. Res.. 75: 5914-5927. Kraus, N.C. and Sasaki, T.O., 1979. Effect of wave angle and lateral mixing on the longshore current. Coastal Eng. Japan, 22: 59-74. (also Mar. Sci. Commun., 1979, 5: 91-126). Kraus, N.C., Farinato, R.S. and Horikawa, K., 1982. Field experiments on longshore sand transport in the surf zone. Coastal Eng. Japan, 24. Krumbein, W.C., 1944. Shore currents and sand movement on a model beach. U.S. Army Corps Eng., Beach Erosion Board, Tech. Memo, 7, 25 pp. Lee, K.K., 1975. Longshore currents and sediment transport in west shore of Lake Michigan. Water Resour. Res., 1 1 : 1029- 1032. Leonard, J.E. and Brenninkmeyer, B.M. (1979). Storm Induced Periodicities of Suspended Sand Movement. Proc. 16th Conf. on Coastal Eng., pp. 1744-1763. Liu, P., L-F. and Dalrymple, R.A., 1978. Bottom frictional stresses and longshore currents due to waves with large angles of incidence. J. Mar. Res., 36: 357-375. Longuet-Higgins, M.S., 1952. On the statistical distribution of the height of sea waves. J. Mar. Res., 1 1 : 245-266. Longuet-Higgins, M.S., 1970a. Longshore currents generated by obliquely incident waves, 1. J. Geophys. Res., 75: 6778-6789. Longuet-Higgins, M.S., 1970b. Longshore currents generated by obliquely incident sea waves, 2. J. Geophys. Res., 75: 6790-6801. Longuet-Higgins, M.S., 1972. Recent progress in the study of longshore currents. In: R.E. Meyer (Editor), Waves on Beaches. Academic Press, New York, N.Y.. pp. 203-248. Longuet-Higgins, M.S. and Stewart, R.W., 1964. Radiation stress in water waves, a physical discussion with applications. Deep-sea Res., 11: 529-563. Madsen, 0,s. and Grant, W.D., 1977. Quantitative Description of Sediment Transport by Waves. Proc. 15th Conf. on Coastal Eng., pp. 1093-1 112. Madsen, O.S., Ostendorf, D.W. and Reyman, A.S., 1978. A Longshore Current Model. Proc. Coastal Zone ’78, ASCE, pp. 2332-2341. McDougal, W.G. and Hudspeth, R.T., 1983. Wave setup/setdown and longshore current on non-planar beaches. Coastal Eng., 7: 103-117. McDougal, W.G. and Hudspeth, R.T., 1983. Longshore sediment transport on non-planar beaches. Coastal Eng., 7: 119-131. Meadows, G.A., 1977. Time Dependent Fluctuations in Longshore Currents. Proc. 15th Conf. on Coastal Eng., pp. 660-680. Mizuguchi, M., Oshima, Y. and Horikawa, K., 1978. Laboratory Experiments on Longshore Currents. Proc. 25th Conf. on Coastal Eng. in Japan (in Japanese). Noda, E.K., 1971. State-of-the-art of littoral drift measurements. Shore Beach, 39: 35-41. Nummedal, D. and Finley, R.J., 1979. Wind-generated Longshore Currents. Proc. 16th Conf. on Coastal Eng., pp. 1428-1438. O’Rourke, J.C. and LeBlond, P.H., 1972. Longshore currents in a semicircular bay. J. Geophys. Res., 77. Prandtl, L., 1952. Essentials of Fluid Dynamics. Haffner, New York, N.Y., 425 pp. Putnam, J.A., Munk, W.H. and Traylor, M.A., 1949. The predictions of longshore currents. Trans. Am. Geophys. Union, 30: 337-345. Russel, R.C.H., 1960. Use of Fluorescent Tracers for the Measurement of Littoral Drift. Proc. 7th Conf. on Coastal Eng., pp. 418-444.
109 Sauvage. M.G. and Vincent, M.G., 1954. Transport Littoral Formation de FlPches et de Tomholos. Proc. 5th Conf. on Coastal Eng., pp. 296-328. Savage, R.P., 1959. Laboratory study o f the effect o f groins in the rate of littoral transport. U.S. Army Corps Eng., Beach Erosion Board, Tech. Memo.. 114, 55 ppSavage, R.P., 1962. Laboratory determination of littoral transport rates. J . Waterways Harhors Div.. ASCE (WW2) 3 138: 69-92. Saville Jr., T., 1950. Model study of sand transport along an infinitely long. straight beach. Trans. Am. Geophys. Union, 3 I : 555-565. Sawaragi, T. and Deguchi, I., 1979. Distribution of sediment transport rate across a surf zone. Proc. 16th Conf. on Coastal Eng., pp. 1596-1613. Seymour, R.J. and Duane, D.B., 1979. The Nearshore Sediment Transport Study. Proc. 16th C'onf. on Coastal Eng., pp. 1555- 1562. Shay, E.A. and Johnson, J.W., 1951. Model studies on the movement of sand transported hy wave action along a straight beach. Inst. of Eng. Res., Univ. of California. Berkeley, Calif., Ser. 14, 5% pp. Sonu, C.J., McCloy, J.M. and McArthur, D.S., 1967. Longshore Currents and Nearshore Topographies. Proc. 10th Conf. on Coastal Eng., pp. 524-549. Symonds, G . and Huntley, D.A., 1980. Waves and Currents over Nearshore Bar Systems. Canadian Coastal Conference 1980, Proc. Natl. Res. Council. Canada, pp. 64-78. Teleki, P.G., 1966. Fluorescent sand tracers. J. Sediment. Petrol., 36: 468-485. Thornton, E.B., 1971. Variations of Longshore Current across the Surf Zone. Proc. 12th Conf. on Coastal Eng., pp. 291-308. Thornton, E.B., 1973. Distribution of Sediment Transport across the Surf Zone. Proc. 13th Conf. on Coastal Eng., pp. 1049-1068. Thornton, E.B. and Morris, W.D., 1978. Suspended Sediments Measured within the Surf Zone. Coastal Sediments '77, ASCE, pp. 655-668. Walton, T.E., 1980. Littoral sand transport from longshore currents. Am. Soc. Civ. Eng.. J . Waterway, Port, Coastal Ocean Div., 106 (WW4): 483-487. Wang, Y-H. and Chang, T.H., 1979. Littoral Drift along Bayshore of a Barrier Island. Proc. 16th Conf. on Coastal Eng., pp. 1614-1625. Watts, G.M., 1953a. A study of sand movement at South Lake Worth Inlet, Florida. U S . Army Corps Eng., Beach Erosion Board, Tech. Memo, 42, 24 pp. Watts, G.M., 1953b. Development and field test of a sampler for suspended sediment In wave action. U S . Army Corps Eng., Beach Erosion Board, Tech. Memo. 34. 41 pp. Williams, A.T., 1971. An analysis of some factors involved in the depth of disturbance of beach smd by waves. Mar. Geol., 1 1 : 145-158. Wood, W.L. and Meadows, G.A., 1975. Unsteadiness in longshore currents. Geophys. Res. Lett., 2: 503-505. Yasso, W.E., 1965. Fluorescent tracer particle determination of the size-velocity relation for foreshore sediment transport, Sandy Hook, New Jersey. J. Sediment. Petrol., 35: 989-993. Yasso, W.E., 1966. Formulation and use of fluorescent tracer coatings in sediment transport studies. Sedimentology, 6: 287-301.
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CHAPTER 3
TURBULENCE MODELLING BENEATH WAVES OVER BEACHES B. JOHNS
ABSTRACT An account is given of two numerical models for the simulation of turbulent processes beneath surface waves in the near-shore environment. Special attention is given to the induced bottom shear stress and the effectiveness of representing this in terms of an empirically based quadratic law.
INTRODUCTION
The determination of the bottom boundary stress beneath waves and currents is a fundamental problem faced by the sedimentologist investigating processes in the near-shore environment. Invariably, the flow is turbulent and the bottom stress is supported by the Reynolds stress in the system. In practical applications, the bottom stress is frequently represented by an empirically based law involving a friction coefficient or factor. The friction-factor approach has been extensively investigated with a view to relating the empirical coefficient to bottom roughness conditions and the wave parameters. These studies have usually been carried out in the laboratory where measurements are made beneath regular surface waves generated in a flume. An account of experimental results obtained from the Danish oscillating water tunnel is given by Jonsson and Carlsen (1976). In the near-shore oceanographic environment, surface-wave propagation is complicated by beach topography, variable wave input parameters and wave-breaking processes in the surf zone. A knowledge of the distribution of the bottom stress and its dependence on these complicating factors is an essential requirement for the calculation of longshore currents (Longuet-Higgins, 1970). The primary purpose of this study is to investigate the nature of the bottom stress beneath both regular non-breaking waves and the bore-like propagation in the surf zone. The theoretical procedure used follows that given by Johns (1977, 1980) and Johns and Jefferson ( 1980).
WAVE-INDUCED TURBULENT FLOW ABOVE A PLANE HORIZONTAL BOUNDARY
A two-dimensional situation is considered in which Ox and Oz are respectively horizontal and vertical axes. The origin, 0, is located within the equilibrium level of the free surface and, in the presence of a surface wave motion, the instantaneous position of the free surface is given by z = [(x, t ) . We consider the wave-induced turbulent flow above a plane horizontal boundary
112
situated at z {
=
u
COS(
= - h.
kx
-
This flow is generated by a progressive surface wave given by:
at)
(1)
We prescribe a horizontal main-stream velocity, U , at the outer limit of a turbulent layer, of thickness 6 ( << h ) , adjacent to z = - h by writing:
U = U,
COS(
kx - a t )
Hence, by using small-amplitude inviscid wave theory, it follows that: aa
u,=-
sinh k h a ' = gk tanh kh T h e boundary-layer equation for the balance of momentum in the region - h + S has the form:
In eq. 4, the overbar denotes a Reynolds average and a prime a turbulent departure from that average. The vertical velocity. W,in the boundary layer is related to U by the equation of continuity for an incompressible homogeneous fluid:
A fundamental problem in using eq. 4 to calculate the boundary-layer flow relates to the treatment of the vertical transfer term. This term is frequently parameterised in terms of the gradient of the Reynolds-averaged velocity by writing:
where K is a vertical transfer coefficient. This coefficient is then made the object of various hypotheses in order to make eq. 4 a basis for practical boundary-layer calculations. Reference here may be made to the contribution of Kajiura (1968) Bakker (1975) and Johns (1975). In our present boundary-layer treatment we shall use a closure hypothesis in eq. 4 based upon that described by Launder and Spalding (1972) for a non-reversing flow situation. Hence, o u r application of this procedure t o a high-frequency reversing flow may possibly be viewed with some concern. Nevertheless, our parameterisation is still based upon a gradient transfer law (eq. 6) (which is the real potential weakness) as used by other workers and its distinguishing feature is solely the way in which we determine the value of K . This is done by introducing a subsidiary equation for the Reynolds-averaged turbulent energy density, which satisfies:
z,
._
aE
a
at
ax
--
a az
--
-+-(uE)+-(wE)=(-u'w'
~
(7)
T h e terms o n the left-hand side of eq. 7 represent the local time-rate of change of the turbulence energy and the advection of turbulence energy by the Reynolds-averaged motion, respectively. The first term on the right-hand side of eq. 7 represents the
113
work done by the Reynolds stress against the Reynolds-averaged motion and is thus a production term of turbulence energy. The second term simulates the vertical redistribution of turbulence energy by the turbulent motion and the last term is a sink term representing the loss of turbulence energy by dissipation. Thus, we see that eq. 7 models changes occurring in the turbulence energy resulting from non-local (in both space and time) physical processes. In the case of local equilibrium conditions, eq. 7 reduces to a balance between production and dissipation in which:
( - Z q z = aii c In our closure scheme, we relate K in eq. 6 to the turbulent energy density by a dimensionally based relation of the form:
K
=
(E'/2
(9)
where I is a length representative of the vertical mixing scale and ,'I2 is a measure of the turbulent velocity scale. Hence, the representation eq. 9 implies that the vertical exchange coefficient used in the parameterisation of turbulent momentum transfer is determined by a non-local condition in both space and time. This contrasts with other applications of the gradient transfer law eq. 6 in wave problems and implies that the Reynolds stress in our model is a function of non-local conditions. Before applying the theory, it is necessary to parameterise the term in eq. 7 representing the redistribution of turbulence energy by the turbulence itself. This can be done most easily by assuming that the redistribution follows a gradient transfer law with the same exchange coefficient as that for momentum exchange. Thus, we write:
Finally, the dissipation term in eq. 7 must be parameterised in order to fully close the problem. This is again done on the basis of dimensional reasoning and we write: (11)
= E_3/2//1
.
where I' is a length scale representative of the dissipation process. By considering the balance of turbulence energy adjacent to z = - h , where production tends to equal dissipation (Townsend, 1956), eq. 8 yields the shear-stress, T , in the form
Our formulation is therefore equivalent near z length theory provided that: 13/'- [K(z+h)14
where
K
asz-
=
- h to the classical Prandtl mixing
-h
(13)
is Von Karman's constant 0.4. We therefore take:
- C1'4K( Z + h ) , [' - C-
3'4K(
Z
+h)
as Z
--f
-h
(14)
I14
where c is an empirical constant whose value is recommended by Launder and Spalding (1 972) as 0.08. Although eq. 14 holds strictly only in the limit as z -+ - h we shall, for the present, assume that eq. 14 applies through the boundary-layer region. This assumption will be relaxed in later generalisations of our modelling procedure. Introducing these results into eqs. 4 and 7, the Reynolds-averaged flow conditions in the boundary layer are determined from:
a ax
aii + -c2 at
a aZ
+ -.uw
=
and:
where: K=C”~K(Z+~),!?”~ (..1/4,!?3/2 € =
K(z+h)
Equations 15-17 clearly have a singular behaviour at z = - h and the lower boundary conditions must therefore be applied at some level z = - h + 6,) where 6, << 6. The physical basis for this procedure is provided by Jonsson and Carlsen (1976) who found evidence of a logarithmic velocity profile beneath waves and, for - h < z < - h 6,,, we write:
+
where z , is the roughness length of the underlying elements. Accordingly, as a lower boundary condition on U ,we take:
’The corresponding value of W may then be deduced from eq. 5 which is readily found to yield: W
In(:)
=
[
6, In(:)
+SO - z o l
at z
= -h
+ 6,
An appropriate boundary condition on E at z = - h + S,, may be found by identifying eq. 18 with a constant stress layer profile. The validity of this procedure will be considered in a later section of this study. Combining eq. 18 with eys. 6, 8 and 17 in the limiting case as z h + z o , we find that: E = K 2 c p l/2V2 for - h + z O < z < - h + S o (21)
+
which, in particular, leads to: h + So
(22)
I15
This simply states that there is no diffusive flux of turbulence energy from the boundary layer into the constant stress layer region. Boundary conditions at the outer edge of the turbulent layer present no problem. Firstly, we require that the boundary-layer flow merge into the imposed oscillatory main-stream and secondly that there is no diffusive flux of turbulence energy between the boundary-layer region and the non-turbulent main-stream flow. These requirements lead to: -
u = u, cos(kx - U t ) ) -
i
dE _ -0 dZ
atz= -h+S
For the purpose of describing the numerical experiments carried o u t with the turbulent boundary-layer model, it is convenient to non-dimensionalise the governing equations by writing U
i = -( z u,
t
+ h s, ) -
= at
E= U L E €
= u:ui
Equations 15, 16 and 5 then become:
(27) aii aa -+-=o I32
ai
In these equations, the important non-dimensional parameter fixing the amplitude of the main-stream flow is:
116
I t may be expressed in terms of the surface-wave parameters by using eq. 3 and then leads to: a
=
ak cosech kh
(30) Thus, with typical near-shore data for which h = 2.5 m, u = I rad s- and a = 26 cm, it follows that k h = 0.53 and a = 0.1. As will be seen from the numerical experiments, the value of a plays a crucial role in determining the nature of the dynamical processes in the turbulent shear-wave boundary layer. The method of numerical solution of the model equations was described fully by Johns (1977) and it is not necessary to repeat this here. It suffices to say that a finite difference grid was used in the 2 direction whilst the terms in eqs. 26 and 27 involving horizontal derivatives were processed numerically in terms of a pseudospectral technique. The first numerical experiments are designed to determine the dependence of the boundary layer response on a and i, and, to this end, we consider 0.075 < a 6 0.125, ( ~ i , = 3 X l O ~ ~ , a ~ , = 2 x I 0 ~ ~ a n d a S = O . l .wTi ht huhs=, 2 . 5 m , u = l r a d s - ’ and u = 26 cm it follows that zo = 1.44 mm, 6, = 9 mm and 6 = 47 cm. In order to interpret the solution, it is convenient to give the truncated spectral representation of ti developed by Johns (1977). This has the form:
’
3
ti = (ti(?)>
+C
[ A c ~o s ( p x ) +B,, sin(px)]
(31)
p= I
where: x=i-t (32) We are interested in the leading term in eq. 31 which represents the Eulerian residual flow induced in the boundary layer by the wave motion. The variation of this quantity through the boundary layer is given for cy = 0.075, 0.1 and 0.125 in Fig. 3.1. This steady streaming motion is of relevance to the net movement of near-bottom sediment suspensions and, for the values of a considered, we note that the streaming velocity reaches its maximum value at a level in the boundary layer that is independent of a. Its value then satisfies: ( ~ 2 = ) (~0 . ~3 0~) ~ +~O( a 2 )
(33)
and reduces to zero through an outer layer that thickens as a +. 0. An increase in the value of a is seen to compress the steady streaming into a concentrated jet immediately adjacent to i = O . It is noteworthy that the steady streaming derived from eq. 33 is less than the corresponding value deduced by Longuet-Higgins (1953) in the case of a laminar boundary layer which is then given by: ( ~ 2 = ) (0.75)a ~ ~ ~
+ O(a3)
(34)
The dependence of the boundary layer streaming on the roughness length of the underlying bottom elements has been investigated by considering a = 0.1 with 5 x l o p 4 < i, < 6 x lop3. The distribution of ( 6 ) through the boundary layer is given in Fig. 3.2. From this, it is seen that the maximum streaming velocity is only weakly dependent on the bottom roughness but that its value at the outer edge of the
1 I7
1
2
0
3
Fig. 3.1. Boundary-layer profile of induced streaming: a = 0.075, 0.1, 0.125; or?,
1.
L. I
\*.
A
z
0.
lo2
Fig.3.2.AsinFig.3.1 e x c e p t n = 0 . 1 , 5 ~ 1 0 - ~ ~ i , ~ 6 ~ 1 0 - ~ .
=3X
118
logarithmic layer does show a significant variation with 2,. In fact, the streaming velocity tends to increase with a decrease in 2,. Equally relevant to the processes of bed load and suspended-sediment transport is the bottom boundary stress resulting from the turbulent shear-wave dynamics. This is readily determined from the boundary-layer solution and we may express the bottom stress, T ~ in, the form:
[ K g )
-rh=
P
z=
(35)
-h+8,
which, by virtue of eq. 8, yields: -rh - c'/* 2 PU,
[ 2sgn( ti)]
;=,
T h e variation of T~ with the phase x for various 2, is given in Fig. 3.3 with a = 0.1. 1t is noteworthy that although the primary dependence of T,, is on a harmonic involving cos x there is also a n important contribution from the odd harmonic sin 3x. This important result is a consequence of the non-linearity of the problem a n d is not reproduced by any simple empirically based law. It is also apparent that the amplitude of the bottom stress is strongly dependent on the roughness length of the bottom elements and that this will be crucial in determining the bottom stress beneath waves in the near-shore zone. A frequently used empirical representation of rh in terms of the orbital wave velocity beyond the boundary layer has the form: rl,
=
(37)
cfPlulu
0.02
0.01
+b 9 ub 0
- 0.01
- 0 02
0
n/ 2
3n/2
lT
2
X
Fig. 3.3. Variation of bottom stress with phase
x: a=0.1;
5X
do <6X
119
which, on using eq. 2, leads to: (38) where C, is a friction coefficient. The Fourier expansion of eq. 38 yields an expression of the form: 00
7b 2 = c,
c
a2p+1
cos[ (2P +
1)xl
(39)
p=o
PU,
where a , = 8 / 3 ~ .This leads only to even harmonics, the time-averaged value of T,, being zero. The variation implied by eq. 37 is therefore incompatible with the important contribution from sin 3x that is apparent in the model value of T,,. Nevertheless, a value for the friction coefficient can be derived that leads to the best approximation of the bottom stress by choosing C, so as to minimise the value of
This then leads to:
lZT(
T h / f u ~ ) l c o sXIcos
c,=
x dX (40)
~ 2 v c o s 4dxx
In the case of a=0.1 and i,= l o p 3 we find that C,= 5.52 x lo-?. With m, u = 1 rad s - ' the non-dimensional parameter setting corresponds to ( I = 26 cm, u, = 47 cm s - ' and z,) = 0.5 mm. The reconstruction of the bottom stress using this value of C, in eq. 37 is given in Fig. 3.4. Apart from the already noted failure of eq. 37 to reproduce the variation in T~ resulting from sin 3x it will also be observed that the peak bottom stress occurs before the peak surface elevation. h
= 2.5
X
Fig. 3.4. As in Fig. 3.3 except a = 0.1, 2 , hroken line from eq. 38 with C , = 5.52X
=
The solid line is the boundary-layer calculation. The
120
Evidently, the peak bottom stress occurs in front of a wave crest. Additionally, in contrast with eq. 37, the model bottom stress has a non-zero mean value over a wave period given by: 'b ~2
2.26 X lo-'
P'd m
This has the same sense as the wave-induced streaming motion and is in the direction of wave propagation. In spite of the obvious shortcomings in using eq. 37 to approximate the bottom stress obtained from the model, it is useful to determine the dependence of the optimised friction coefficient on the roughness parameter 2, since, from Fig. 3.3, we see that the bottom stress has a crucial dependence on this quantity. This has been done with a = 0.1 for 5 x l o p 4< 2, < 6 x lo-' and the resulting values of C, have been fitted by a least squares method to a power-law representation which then yields:
C, = 0.74( 2,)0'7' Thus, for the range of roughness lengths considered here, the friction coefficient varies in value between 3.35 x lo-' and 1.96 x Using eq. 24, we see that eq. 42 may be written:
1
0 71
C, = 0.74(?
(43)
Hence, the friction coefficient is increased by an increase in the bottom roughness and is decreased by either an increase in the amplitude of the main-stream velocity or an increase in the wave period.
TIJRBULENT FLOW BENEATH WAVES APPROACHING A SHORELINE OVER A SLOPING BFACH
In the previous section, we considered the turbulent flow induced beneath regular sinusoidal waves in water of uniform depth. In the near-shore environment, a train of incoming surface waves only has this regular sinusoidal structure when it is uninfluenced by the shallow-water effects that become increasingly important over a sloping beach. Under these conditions, the incoming waves cease to have a regular sinusoidal form, begin to steepen and approach a breaking situation. After this, they propagate through the surf zone in the form of a sequence of periodic bores. These are characterised by a highly turbulent face and a relatively smooth non-turbulent back. Our earlier discussion of turbulent flow beneath regular waves clearly does not apply to this phase of the propagation process in which there is an important generation of turbulence at the face of each bore and which must be expected to influence the distribution of bottom stress. In the present section, we generalise the use of the turbulence-energy equation to simulate this additional process. The same two-dimensional situation is considered as on p. 11 1 but now the origin 0 is located at the seaward extremity of the analysis region and the equilibrium
121
position of the shoreline is at x = L. We consider the case of a plane-sloping beach in which the equilibrium depth, h , is given by:
);
h ( x ) = h o (1 -
(44)
As described by Johns (1980), the Reynolds-averaged velocity (U,W)then satisfies:
aii + u aii + w aii at ax az
-
7
- g-
a{ ax
a ( -a ( -+&) + ax
aZ
(45)
where the pressure has been assumed hydrostatic and, in comparison with eq. 4, we include an extra term to represent the effect of the horizontal turbulent exchange of momentum. Ultimately, this term will be parameterised so as to simulate the horizontal redistribution of momentum at the face of each of a sequence of propagating bores. Equation 5 may be integrated vertically to yield: at ax - h -."/'iidt=O
and eq. 7 for the Reynolds-averaged turbulence energy density is modified to:
aE -aE -aE -+ u-+ w-= at ax az
aii ( - P )aii- + (-rn)-+ @ -E ax az
(47)
In eq. 47, the first term on the right-hand side is a source term representing the gain of turbulence energy by production at the face of a bore. This exactly balances the extraction of energy implied by the second term on the right-hand side of eq. 45. represents the redistribution of turbulence energy in the horizontal and vertical by the action of the turbulence and is a generalisation of the corresponding vertical redistribution term in eq. 7. As in Johns (1978) and Johns a d Jefferson (1980), it is convenient to introduce the coordinate transformation: u = -z + h
l+h
so that the beach and free surface correspond to u = 0 and u = 1, respectively. With x, u and t as independent variables, eqs. 45 and 47 may be transformed to yield:
aii + -u- aii + 0-aii = -g- 8
-
ax
at
au
ax
-
a -( - d 2 ) + - - f J , U ~ 2 + u z m ) ] (49) ( - -d a2 )uI+ a[ ( au ax dU
and: -
aii aE -aE + w- aE = ( - aii - + uu q - + [ - ( uxu'2+ u z r n ) ] - + @ - € at ax au ax au
where: w
= ut
+ uu, + wu,
and the subscripts denote - differentiations. The quantity - p ( u , ~ ' ~ + u Z m ) is a measure of the turbulent flux of momentum across, and normal to, the isopleths u = constant which, in particular, we take to be
I22 __
zero at the free surface u = 1. Similarly, - P U ' ~ represents the turbulent flux of momentum in the horizontal resulting from production at the face of a bore. Each of these quantities must be parameterised in terms of the Reynolds-averaged flow. This is not as straightforward as on p. 112 owing to the sharp curvature of the isopleths near the face of the bore and, in the present model, we simply generalise the procedure to:
Thus, if a, = 0, eq. 52 becomes equivalent to eq. 6. Further, we parameterise the effect of horizontal turbulent transfer by writing:
where N is a horizontal exchange coefficient and, on the - basis of order of magnitude u comparison with estimates, we are completely justified in neglecting ( ~ ' ~ ) r 3 a , / din g( a{/&) provided that << g ( h + {). Finally, in eq. 50, we assume that the turbulent redistribution again follows a gradient transfer law with the same proportionality coefficient as for momentum exchange and write:
Introducing these representations into eqs. 49 and 50 and denoting the total depth
{ + h by H we obtain:
From eq. 46 we obtain:
(57) and find that o is determined diagnostically from:
[:
Hw=u-
H L'iidu]
-
$[H/OOiidu]
The parameterisation of K and c in terms of the turbulence energy follows that given on p. 113 and we again write: K
= c1/41El/2
where now the length scale I is determined from a local similarity hypothesis. This
123
leads to: -K(
I=
P/f)
~
I=
KZ~,
at u = 0
Since, in contrast with p. 113, we are no longer dealing with a boundary-layer region, eq. 60 appears to be a more appropriate way of representing the length scale and is simply a generalisation of Von Karman theory. There is, however, some evidence that it may over-represent the mixing near the free surface and that this point may require attention in subsequent developments of the theory. The parameterisation of N involves a further introduction of empiricism and has no counterpart on p. 113. The horizontal exchange coefficient is represented by writing: N
(61)
= yLE‘/’
where yL ( y < 1) is a horizontal mixing scale and El/’ is again the turbulent velocity scale. Clearly, y must be assigned a value in the numerical experiments and a suitable criterion governing its choice should be established. I t will be noted that the horizontal production term in eq. 56 contributes throughout the depth of the water except in so far as this is influenced by the depth structure of E. I t may, in fact, be more realistic to incorporate an attenuation with depth into N in order to restrict the contribution of N(&/dx)’ to the region adjacent to the face of the bore. However, this would require the introduction of further empiricism which is not considered desirable at this stage. The boundary conditions to accompany eqs. 55 and 56 are that there is no fluid slippage at the beach, no transfer of momentum across the free surface and no transfer of turbulence energy across the bed or free surface. Additionally, we have to satisfy the kinematical condition at the beach and free surface. These requirements lead to: u=O o=O
atu=O atu=Oandu= 1
& -=0
atu=1
aE -=0
atu=Oandu=l
audU
The scaling procedure used by Johns (1980) shows that the solution of these equations depends on the parameters zo/h,, y, h,/L and on the quantities describing the wave input at the seaward end of the analysis region. In our experiments, we prescribe a regular oscillatory wave input (with period t P) given by: H=h,+osin(
atx=O
Additionally, then, the oscillatory response in 0 < x < L depends on an amplitude factor a / h , and a non-dimensional wave period tp( g h 0 ) ” * / L . The solution of the equations is obtained numerically by a finite-difference
I24
Fig. 3.5. (a) Contours of equal values of 1 0 3 E / ( g h 0 ) beneath the waves with horizontal production included at t = 0; (b) As in a, except that t = tD/2.
125
discretisation of the x, u and t variables with various refinements described by Johns (1980). Further details are not given here but it is appropriate to comment o n a frequently used method of treating developing discontinuities in the solution of the shallow water equations. This depends on the use of an artificial viscosity (Lax and Wendroff, 1960) and implies, of couise, a loss of energy from the system. This approach may be contrasted with the present formulation which also implies a self-regulatory energy extraction from the Reynolds-averaged flow which is greatest in regions where the horizontal gradient term &/ax has its maximal effect. The distinguishing feature of our approach, however, is that this energy is not discarded but is retained in a turbulence energy budget. It is therefore available for the determination of the transfer coefficients in our parameterisation of the Reynolds stress terms. Thus, the procedure is not designed solely to secure the computational stability of the numerical scheme (although, of course, it has this necessary effect) but is based on a conceptual view of the physical problem. The first experiments are concerned with the distribution of turbulence energy beneath the waves and, in particular, with the effect of including the horizontal production term N ( d i i / a x ) ’ in eq. 56. In these experiments, we take y = 0.1, h,/L = 0.025, z o / h 0 = 0.002, a / h o = 0.16 and a non-dimensional wave period 0.5. This basic parameter setting, which will also be used subsequently unless otherwise stated, corresponds to a typical near-shore dimensional setting in which h , = 2.5 m, L = 100 m, z o = 5 mm, a = 40 cm and t , = 10.1 s. With horizontal production included in eq. 56, Fig. 3.5a and b show contours of equal values of 103E/(gh0) at the instants t = 0 and t = t , / 2 , respectively. Superimposed on these is the profile of the free surface from which the relatively steep front of each wave is apparent. Also evident as the waves approach the shoreline is the decrease in wave amplitude and the wave set-up that occurs. Associated with the face of each wave, we note the contribution made by the horizontal production term and how this leads to a marked strong horizontal gradient in the turbulence energy. This process generates a distribution of turbulence energy that spreads downwards and backwards to affect the turbulence intensity adjacent to the beach. In order to evaluate the full effect of the horizontal production term, it is informative to repeat the above calculations with N ( & / ~ x )removed ~ from eq. 56. The horizontal mixing term is still retained in eq. 55 which implies a drain of energy from the Reynolds-averaged flow. However, this energy is not retained in the turbulence energy budget and is not therefore available to contribute to the value of the exchange coefficients in eqs. 59 and 61. In particular, the value of E in eq. 61 will be reduced with a corresponding reduction in the value of N . The effect of this will be to reduce the strength of the horizontal mixing of momentum in eq. 55 and therefore to lessen the smoothing of the surface profile the effect of which is apparent in Fig. 3.5a and b. That this is in fact the case may be seen from Fig. 3.6a and b. The surface profile is now sharper with steeper wave fronts. Additionally, there is some evidence of subsidiary wavelets behind the crest of each of the breakers. With horizontal production included in the turbulent energy balance the increased horizontal mixing smoothes out these wavelets. In a sense, then, the inclusion of the horizontal production term in eq. 56 is equivalent to an increase in
126
Fig. 3.6. (a) As in Fig. 3.5a except that horizontal production is excluded; (b) As in Fig. 3.5b except that horizontal production is excluded.
127
the horizontal mixing scale, y L . However, changes in the horizontal mixing scale cannot directly influence the value of the vertical exchange coefficient and, from this point of view, the retention of the horizontal production term forms an essential part of the way in which the bottom shear layer is reproduced in the model. Further conclusions to be drawn from Fig. 6a and b relate to the distribution of contours giving equal values of 10’ E / ( g h , ) . In contrast with the case when horizontal production is included, maximum values of the turbulence intensity now occur in the shear layer adjacent to the beach. Moreover, the magnitude of E is increased and this is a consequence of the increased value of the surface wave height resulting from the decrease in horizontal mixing and the correspondingly reduced attenuation of the waves as they approach the shoreline. The contrasting results obtained in these two cases indicate the importance of retaining the energy extracted from the Reynolds-averaged flow in a turbulence energy budget. They also underline the need to fix the value of y on some basis so as to produce the correct wave attenuation in the model. The distribution of turbulence energy determined in these calculations immediately yields the bottom stress through the relation:
E,,
is the turbulence energy at a = O . where velocity, u,, is given by:
Equivalently, the bottom friction
and the variation of this may be compared with that of the depth-averaged velocity, given by:
ii,, -
u,
1 I
=
iida
The point of this comparison is to determine the potential effectiveness of an empirical friction law similar to eq. 37 for regular sinusoidal surface waves. Such a law has the form:
In fact, the model calculations show that Urn is indistinguishable from the wave orbital velocity at the outer edge of a shear layer just above the beach. Consequently, eq. 67 is the exact analogue of eq. 37. With the basic parameter setting, the temporal variations of u * / ( g h 0 ) ’ / * and ii,/(ghO)1/2 during a wave cycle at a position corresponding to x = (0.59)L are given in Fig. 3.7. We note that the sense of the friction velocity changes abruptly during the wave cycle. The change occurs somewhat in advance of a flow reversal and is almost discontinuous. To some extent, a similar behaviour can be seen in Fig. 3.3 for regular sinusoidal waves but in the present case this feature is more marked. It may, perhaps, be an indication of a deficiency in the use of a gradient transfer law for momentum exchange in a high-frequency reversing flow situation. Although
128
0
+
I
05
075-0
t/tp
I
- 0 05
-0 I
Fig. 3.7. Variation of u * / ( g h o ) ’ / 2 and U , , , / ( ~ / I ~ ) ’ / ~at x = (0.59)L during a wave cycle. The continuous line is the friction velocity. The broken line is the depth-averaged velocity.
“memory” is built into the parameterisation by the way in which the vertical exchange coefficient is related to the turbulence energy, it may also be the case that the shear stress is not a strictly local function (in space and time) of the vertical velocity gradient. In spite of these comments, the variation of the bottom friction velocity is seen to be approximately in phase with that of the depth-averaged velocity and this suggests that eq. 67 may yield a useful approximation. The friction coefficient in eq. 67 is given by:
and, in general, this will have a temporal variation during a wave cycle. Following eq. 40, however, we may again define an optimised and temporally invariant form by:
Using eq. 69 it is then possible to compute C, from the model and to use this in eq. 67. The resulting approximation to the bottom stress may then be compared with the model value as computed from eq. 64. In Fig. 3.8, we give the variation of 10’ ~ ~ / ( p g h during ,~) a wave cycle at x = (0.59)L as determined from eqs. 69 and 64. We note that the quadratic friction
129
Fig. 3.8. Variation of 1 0 3 7 , / ( p g h , ) at x = (0.59)L during a wave cycle. Continuous line is from eq. 64. The broken line is from eq. 67 with C, given by eq. 69.
law underestimates the magnitude of the on-shore bottom stress and overestimates the magnitude of the off-shore stress. In relation to the peak values of the stress, the under- and over-estimates are, respectively, of order 19 and 35% of the model value. We also observe how the quadratic friction law smoothes out the already noted virtually discontinuous behaviour of the model bottom stress at instants of flow reversal thus leading to further percentage errors at these times during the wave cycle. The calculation of C, from eq. 69 has been carried out with the fixed value of z o / h O given in the basic parameter setting. In order to determine variations in the friction coefficient with changes in the roughness, it is therefore necessary to repeat this calculation with a systematic variation in z o / h O , the other parameters in the setting being unchanged. This has been done for 2.5 x l o p 4 < zo/lzO < 2 X lo-’ and, for x = (0.59)L, the results may be fitted to a power law which then gives:
1
0.374
C, = 0.326/
Equation 70 leads to the extreme values 1.47 x lo-’ and 3.19 x lo-’ and yields the same qualitative behaviour for C, as does eq. 43 for regular sinusoidal waves; an increase in the bottom roughness again increases the value of the friction coefficient. Quantitatively, however, eq. 70 indicates a lower rate of increase of C, with z o in comparison with eq. 43. We therefore see that the value of the friction coefficient in the quadratic bottom stress law will be dependent on the type of wave motion that is
130
being considered. Additionally, we also find that C, varies with distance from the shoreline and that the coefficient and index in eq. 70 are changed if a power law is used to represent C, at different positions. In fact, for z o / h o = 2.5 X l o p 4 we find that C, varies over the analysis region between values of 1.10 X 10 and 1.66 X 10 In a similar manner, we have determined the dependence of C, on t , by using the basic parameter setting but with a variation of the non-dimensional period in the range defined by 0.5 < rp( g h O ) ' / 2 / L6 2.0. Considering again the point x = (0.59)L, the coefficient may be fitted to a power law in which:
(7,= 0.0167
[
L fP(
0 791
(71)
Rh 0 11 / 2 1
which yields the extreme values 2.89 X Hence, as in the case and 0.96 X of regular sinusoidal waves, the friction coefficient decreases with an increase in the wave period. Quantitatively, eq. 71 implies a slightly greatei rate of decrease in the friction coefficient with the wave period than does eq. 43 which again illustrates the problem dependent nature of C,. The qualitative decrease of C, with an increase in t , may be explained by noting that the bottom stress is determined by the turbulence energy density. As the wave period increases, the wave input into the analysis region becomes less frequent and, at a fixed position, turbulence will be generated less frequently by the action of the production terms in eq. 56. Consequently, in the intervening period. the background
Fig. 3.9. Variation of 1 0 3 ~ h / ( p g h , , )beneath the waves at f broken line is from eq. 67 with C, given by eq. 69.
= 0.
Contmlous line is from eq. 64. The
131
of turbulence persisting from one wave cycle to the next will have a longer period of time in which to decay, thus reducing the value of the bottom stress. For a fixed input amplitude, however, the depth-averaged velocity will be effectively independent of the wave period. This tendency is then reflected in a reduced value of the friction coefficient. The effectiveness of the quadratic law in representing the bottom stress in the interval 0 < x < L may be assessed by evaluating C, from eq. 69 as a function of x. The bottom stress as calculated from eq. 67 may then be compared with the model evaluation based on eq. 64. With the basic parameter setting, the results of this comparison are given in Fig. 3.9. In this, 103~,/pghois evaluated at the commencement of a wave cycle for the interval 0 < x < L . The surface elevation is superimposed on the same diagram which enables the occurrence of the peak stresses to be related to the form of the surface profile. I t is again noted that the quadratic law tends to underestimate the onshore bottom stress, especially beneath the face of a wave. It is also noteworthy that the peak value of the onshore bottom stress always occurs in front of the wave crest. Beneath the sloping back of the wave, the quadratic law representation compares well with the model value although some deviation occurs in the neighbourhood of a flow reversal where an abrupt change in the sign of the model stress is again apparent. Peak values of the offshore stress occur beneath the wave troughs and their magnitude tends to be overestimated by the quadratic law. A frequently used technique to derive the bottom stress depends on the assumption of a constant stress layer near-bottom velocity profile in which:
By using measured values of the near-bottom velocity, it is possible to fit this to a relation having the form eq. 72 and then to deduce the value of T,, = pu*lu*l. This procedure has been used by Jonsson and Carlsen (1976) and its successful application clearly depends upon the existence of a constant stress logarithmic layer. It is therefore informative to use the results computed on p. 127 to investigate the near-bottom velocity profile and to compare this with a constant stress layer logarithmic profile. We note first that the numerical model indicates an almost uniform velocity structure through the depth. The only significant variation in the velocity structure occurs in a thm shear layer, having a thickness of about (0.05)h0, adjacent to the beach. Effectively, then, the depth-averaged velocity, U , , is equivalent to the wave orbital velocity at the outer limit of the shear layer. Consequently, eq. 67 may be looked upon as being a parameterisation of the bottom stress in terms of the wave orbital velocity at the outer edge of the near-bottom shear layer. Conceptually, it is therefore a physically equivalent representation of the bottom stress to eq. 37. In order to evaluate the effectiveness of the constant stress profile hypothesis with I u * l = c ' / ~ E L /we ~ , have used the basic parameter setting to make a comparison of eq. 72 with the model profile at x = ( O . 5 9 ) L but with a variation in zo/h,. The variation of U / ( g h o ) ' / 2in the shear layer is given in Fig. 3.10a and b at two instants during the wave cycle for z o / h , = 0.0005 and 0.002, respectively. From t h s compari-
132
-0 08
0
U / (gh,)
0.1 112
Fig. 3.10.a. Variation of U / ( g h o ) ' ' 2 at x = (0.59)L at f = 0 and f = f , / 2 with z o / h , , = 0.0005. The continuous line is from the model calculation. The broken line is from eq. 72 with u* = c ' / ~ E A /b. ~ As . in a, except that z o / h , = 0.002.
son, it is noted that the constant stress layer hypothesis becomes less tenable with a decrease in z o / h o . This result is not easy to explain in terms of the physics of the system but it may suggest a need for caution in using a constant stress layer hypothesis to derive a lower boundary condition or in using a procedure based on eq. 72 to estimate the bottom stress.
CONCLUDING REMARKS
By considering the relatively simple and idealised models of surface-wave propagation, it has been shown that the resulting induced flow field leads to a complicated distribution of bottom friction. Even in the first case of regular sinusoidal waves, the instantaneous bottom friction is found to be strongly influenced by non-linear processes and, for this reason, cannot be exactly represented by any straightforward empirically based law. An approximate representation of the bottom stress in terms of a quadratic friction law leads to a friction coefficient that is dependent on the
133
wave amplitude and frequency and the bottom roughness conditions, In the second model, relating to propagation through the surf zone in the near-shore environment, the numerical experiments show the bottom stress to be an even more complicated functional of the wave dynamics. It is found that turbulence production at the face of each surface disturbance significantly influences the bottom stress and that this must therefore be taken into account. Again, the instantaneous bottom stress cannot be exactly represented in terms of a quadratic law. An attempt to do so yields a friction coefficient with an even more complicated dependence on the wave and roughness parameters in comparison with the case of regular sinusoidal waves. Further complications can be expected in the case of wave propagation over irregular beach topographies and for wave trains that are obliquely incident to the shoreline.
REFERENCES Bakker, W.T., 1975. Sand Concentration in an Oscillatory Flow. Proc. 14th Conf. Coastal Eng., Copenhagen, 1974, 11: 1129-1 148. Johns, B., 1975. The form of the velocity profile in a turbulent shear wave boundary layer. J. Ceophys. Res., 80: 5109-5112. Johns, B., 1977. Residual flow and boundary shear stress in the turbulent bottom layer beneath waves. J. Phys. Oceanogr., 7: 733-738. Johns, B., 1978. The modeling of tidal flow in a channel using a turbulence energy closure scheme. J. Phys. Oceanogr., 8: 1042-1049. Johns, B., 1980. The modelling of the approach of bores to a shoreline. Coastal Eng., 3: 207-219. Johns, B. and Jefferson, R.J., 1980. The numerical modeling of surface wave propagation in the surf zone. J. Phys. Oceanogr., 10: 1061-1069. Jonsson, I.G. and Carlsen, N.A., 1976. Experimental and theoretical investigations in an oscillatory rough turbulent layer. J. Hydraul. Res., 14: 45-60. Kajiura, K., 1968. A model of the bottom boundary layer in water waves. Bull. Earthquake Res. Inst., Univ. Tokyo, 46: 75-123. Launder, B.E. and Spalding, D.B., 1972. Mathematical Models of Turbulence. Academic Press, New York, N.Y., 169 pp. Lax, P. and Wendroff, B., 1960. Systems of conservation laws. Commun. Pure Appl. Math., 13: 217-237. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. SOC.London, Ser. A, 245: 535-581. Longuet-Higgins, M.S., 1970. Longshore currents generated by obliquely incident sea waves, 1. J. Geophys. Res., 75: 6778-6789. Townsend, A.A., 1956. The Structure of Turbulent Shear Flow. Cambridge University Press, Cambridge.
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I35
CHAPTER 4
OBSERVATIONS OF TIDES OVER THE CONTINENTAL SHELF OF NORTH-WEST EUROPE M.J. HOWARTH AND D.T. PUGH
INTRODUCTION
The aim of this chapter is to describe the observation, analysis, synthesis and presentation of the tidal movements of the waters of the North-West European Shelf seas. These water movements are both vertical and horizontal: vertical displacements of the water surface with approximately twice-daily period are familiar to all shore-based observers, but to the mariner the twice-daily variations in the horizontal movements, called tidal streams, are equally important. Although there is a long history of observations of tides, both for practical and for scientific purposes, the relatively recent advent of electronic technology has enabled a systematic study with a detail which would previously have been impossible. Variations in sea level and currents may now be measured and automatically recorded over periods of months, at any position on the continental shelf; furthermore, these observations can be processed and analysed by computer both quickly and accurately. Not only are these detailed descriptions of tides now possible, but they are also necessary, in order to meet the greater accuracy required by mariners, coastal engineers, and others who plan how the characteristics and resources of the sea may be best protected and utilised. The physical forces which set the sea in motion extend over the whole spectrum of frequencies. Short-period wind-generated waves are the most obvious to the casual observer: the associated level changes and currents are beyond the scope of this chapter, but it must be remembered that they form a high-frequency background noise which must be averaged in the process of measuring the variations of level and currents at lower frequencies. The drag of the winds on the sea surface also produces low-frequency changes in level and currents for the period from hours to days over whch they act. Meteorological effects also include the raising and lowering of sea level in inverse response to changes in atmospheric pressure. Over longer periods, from weeks to years, changes in weather patterns and climate result in changes of residual circulation and long-term changes of sea level. For example, present sea levels are generally some 40 m higher than those towards the end of the last ice age, 10,000 yrs ago. Locally, these changes are influenced by epeirogenic effects-defined as vertical land movements of regional extent having geological origins. These very low-frequency effects are also beyond the scope of t h s chapter, but form a background against which we measure our main concern: variations due to tidal forces. Although any definition of tides must be somewhat arbitrary, such a definition is
136
a necessary first step in the development of this chapter. We define tides as periodic vertical or horizontal movements which have a coherent amplitude and phase relationship to some periodic geophysical force. In our case we consider the movement of water, but movements of the atmosphere and of solid earth may also be tidal. The dominant geophysical forcing function is the variation in the gravitational field on the surface of the Earth, caused by the regular movements of the Earth-Moon and Sun-Earth systems. Movements due to these are termed gravitational tides to distinguish them from movements induced by regular meteorological forcing. The latter are termed meteorological or radiational tides because they occur at periods directly linked to the solar day. Arguably, seasonal changes in levels and in the circulation of sea water over an annual period are also regular and hence tidal: they will also be considered in our discussions. A further group of regular motions which will be considered, and which owe their origins to the major astronomical tides, are the shallow-water or non-linear tides, which are usually harmonics of the major marine tides, and which are generated by non-linear hydrodynamic effects in shallow water. The relative importance of each physical input is best summarised by a table of variance, which may be generated by spectral or harmonic analysis. Table 4.1 shows such a distribution, generated from a lunar month of elevation and current observations at an offshore structure in the North Sea (Pugh and Vassie, 1976). Elevation, being a scalar quantity, has only one parameter, but currents, being vector quantities, usually measured as speed and direction, must be represented by two variables. The usual convention is to resolve an instantaneous, measured, current into a north-south and an east-west component, as has been done here. Variance, which is computed as the square of the displacement of a measured quantity from the mean, is therefore also proportional to the power in the system which may be associated with each input. For elevations, and for the dominant north-south component of current, the proportion of the total energy which is tidal exceeds 97%. Shallow-water tides are more important for currents than for elevations. The weaker east-west currents have proportionally more non-tidal energy, but the total of their non-tidal energy is comparable with that in the north-south component. Tidal energy is split
TABLE 4.1 The distribution of variance in the current and elevation measurements at the Inner Dowsing. day 91/72 to 119/72 Variance (total)
Tidal (5%)
Residual ( % )
Astronomical
Shallow-water
elevations
2.3 12 m2
99 (semidiurnal = 97.7)
0.2
0.8
East-west current component
0.0283 m2 s-’
83.5 (semidiurnal = 82.5)
5.0
11.5
North-south current component
0.3029 m2 s-’
97.4 (semidiurnal = 96.7)
1.7
0.9
137
among several frequency bands, which have periods of approximately one day or integral fractions thereof. In all three variables, the tidal movements of semidiurnal periods contain a dominant proportion of the total energy. This is characteristic of most tidal phenomena. However, the relative proportions shown here will vary from place to place, and from month to month, depending upon the local tidal conditions and the importance of meteorological storms. Practical people concerned with the sea must have noted basic patterns which they transmitted verbally to each other from earliest times: it has even been suggested (implausibly) that Stonehenge could have been an early tide predicting system (Beach, 1977). In addition, possible connections between religion and the tides, which were once thought to provide a tangible terrestrial key to the workings of the universe, encouraged an interest from those more philosophically inclined, including such scientific giants as Galileo and Newton *. The physical mechanism responsible for tide generation, in terms of the gravitational forces of the Moon and Sun, was established by Newton in his Principia. However, there had been many earlier theories which sought to explain the two main observed patterns: that there are two tidal high waters during each lunar day, and that maximum tidal ranges occur a day or two after new or full moon. Even during the Dark Ages the Venerable Bede was able to describe how the time of high tide progressed, gradually occurring later from north to south along the east coast of Britain. However, the first systematic programme to observe tidal movements away from the coast seems to have been Edmond Halley’s survey in the English Channel during the summer of 1701. The objectives, for what may be considered the first oceanographic cruise, were (Halley, 1701): “. .. to use all possible diligence in observing the Course of the tides in the Channel1 of England as well in the mid seas as on both Shores, a d t o inform your self of the precise times of High and Low Water; of the sett and strength of the Flood and Ebb and how many feet it flows in as many places as may suffice to describe the whole. And where there are irregular or half Tides to be more than ordinarily curious in observing them.”
Exactly how Halley proposed to measure the times of high and low water offshore is not known, but as his earlier works included a paper on how to measure the depth of water using air compression and a syphon tube he may have had something like this in mind. As a result of his survey, Halley published the first chart of tidal streams, which was generally used for the next hundred years, and which showed the times of maximum eastward flow, relative to the time of local high water. Modern charts published by the Hydrographic Department show hourly vectors of tidal flow relative to the time of high water at Dover, which may be determined from published annual tables. The first chart representing the behaviour of tidal levels was tentatively constructed by Thomas Young (1807), who is more widely known for developing the wave theory of light. Young proposed the construction of charts displaying “contem-
* A valuable account of “Newton’s work on the theory of the tides” is given by: J. Proudman, 1927. Isaac Newton. Memorial Volume. Mathematical Association, London, pp. 87-95.
138
porary” lines, joining places which have high water simultaneously-we now call these cotrdal lines. Interestingly, Young acknowledged that his ideas on the wave theory of light were partly influenced by Newton’s description of the tides in the Gulf of Tongking. The first attempt to add information on tidal range to such charts was probably made by Whewell(1836), who drew as many lines parallel to the coast as the whole tidal range in yards. Whewell also made the controversial proposal that there were offshore points of zero tidal range from which cotidal lines radiated, called arnphidrornic points; however, the existence of one such area was soon confirmed by Captain Hewett who, in 1840, measured offshore levels by continual depth measurements from a moored boat over a tidal period, in the southern North Sea. Figure 4.3 shows a chart of the extent of mid-nineteenth century ideas on British tidal phenomena from an anonymous publication by the Society for Promoting Christian Knowledge, 1857. It indicates areas of progressive and stationary tidal waves and nodes, but avoids presenting cotidal lines. Although the hydrodynamic connection between sea-levei changes and currents had been previously recognised as a necessary condition to be satisfied by correct tidal charts, the formal mathematical connection was not made until Proudman and Doodson (1924) published their chart of the principal lunar semidiurnal tide (M,) for the North Sea. Their fomulation made possible the production of more accurate charts offshore, by dynamically combining coastal level measurements with offshore current measurements, to give directions and gradients to cotidal and corange lines and surfaces. A fuller description of the development of cotidal maps is given by Marmer (1928), and anyone interested in the history of tidal science should read the excellent account in Deacon (1971). Throughout history, the dual motivation of scientific interest and practical need is apparent. Today, the most elaborate tidal charts covering the continental shelf of North-West Europe and of many tidal constituents (Oberkommando der Kriegsmarine, 1942; Great Britain Hydrographic Department, 1940; Sager and Sammler, 1975, Ministry of Agriculture Fisheries and Food, 1981) are not adequate to meet either requirement, and are gradually being superseded by derailed maps of local regions (Pugh and Vassie, 1976; Robinson, 1979; Prandle, 1080). Eventually the Institute of Oceanographic Sciences intends to publish a coherent series of maps of the whole area, making use of the long periods of offshore level and current measurements which are now routinely collected (Fig. 4.4 for example). The basic techniques for the development of these charts are elaborated in this chapter. Where measurements are sparse, it is useful to refer to analytical, or more commonly, numerical computer models of shelf tides, which can be particularly reliable for the major ( M 2 ) constituent. In turn the development of numerical models requires adequate boundary conditions and internal checks, which may be obtained best from reliable tidal charts. This need is strongest where near-shore schemes such as, for example, the Wash Water Storage Scheme, and the Severn Estuary Barrage Scheme are being evaluated. Hydrographic surveys of offshore areas must be corrected for tidal changes in sea level, and this is usually done from tidal charts. Survey accuracies of 0.10 m and better are now required both for the operation of deep-draught vessels in channel approaches to ports, and in the
139
planning of oil and gas production platforms. The hydrocarbon industry also needs to know the strength of the tidal currents in which pipe lines are to be laid, and the tidal contribution to extremes of both elevations and currents for which structures must be designed. Scientifically there is a strong interest in the tidal observation, not only for their own sake, but also because tides are responsible, principally through non-linear processes, for a component of residual water circulation, for sediment movement patterns, coastal erosion and deposition, movements of fish larvae, and for the dissipation of energy from the astronomical Earth-Moon system. Because non-linear processes are involved, there is a growing interest in charts of higher harmonics of the fundamental astronomical tides, as well as in charts of the fundamental tides themselves.
OBSERVATION TECHNIQUES
Neither Captain Hewett's laborious depth plumbings, nor the measurements of currents from an anchored boat-traditionally by counting the paying out rate of knots in a line attached to a streamed log-could be sustained for much longer than one or two tidal cycles; the cost becomes prohibitive, and, more fundamentally, the accuracy of the measurements is limited by boat motions and timing errors. When planning a measurement programme to chart the tides in a particular area, several design factors must be considered. Where should measurements be made? For how long? To what accuracy? Is datum stability needed for the coastal levels? Finally, can these requirements be met within acceptable cost limits? Fortunately, the range of instruments now available extends from relatively cheap, inaccurate, short-term meters, to the accurate, long-term, internally recording, and inevitably expensive, equipment sold on a specialised international market. For rough local surveys the former may suffice, but for proper regional tidal studies the latter are essential. Good tidal analyses normally need a minimum of one lunar month of observations, accurate to 0.01 or 0.05 m s - ' , with samples taken at least once an hour. For elevations, coastal measurements extending over a year or longer are both desirable and relatively easy to obtain. Sea-level measurements
Any measuring system has a series of basic components. Initially there must be some sensible physical parameter which can be detected by a transducer. The signal from the transducer is then transmitted to a recorder where a permanent registration is made. Descriptions of these several aspects elsewhere in the literature, include the Hydrographic Society publication (1976), articles by Lennon (1976), Pugh (1976) and Rae (1976a), the Canada Marine Sciences Directorate (1979), and Cartwright et al. (1980a). Although there remain problems of instrument development, particularly with respect to datum stability for offshore gauges, technology is ahead of science, as a result of substantial progress in the former since the mid-1960s. Here we summarise the fundamental principles adopted and present some new comparative data on the performance of stilling wells and pneumatic coastal systems.
140
A laboratory based physicist would almost certainly express surprise that measurements of a level to 0.01 m, or in some cases 0.10 m presented any problems, since measurements several orders of magnitude more accurate are routine. Our peculiar difficulties arise because sea levels are measured in a hostile environment, where the surface is moving at speeds of decimetres per second due to waves. The major problem lies in averaging these wave levels to obtain the value required, without introducing errors. The simplest method is to read a graduated tide pole or a connected flight of tide poles at intervals, averaging out the waves subjectively. In calm conditions an accuracy of 0.02 m is feasible, but in the presence of large waves, even 0.10 m becomes difficult. One improvement is to strap a transparent (perspex) tube to the pole, connecting to the sea through a narrow bore tube. Level fluctuations in the tube are stilled relative to the external wave oscillations, making accurate reading easier, particularly if the tube water is coloured with a dye. However, the sheer tedium of reading, particularly at night, makes poles unsuitable for long-term measurements. Since their introduction (Palmer, 1831), the vast majority of permanent coastal tide gauges have consisted of an automatic chart recorder operated by a float which moves vertically in a well, the well being connected to the sea through a relatively small orifice or narrow pipe, so that the external wave oscillations are damped. Such systems have functioned reliably for many years. However, there are a number of fundamental problems associated with stilling well levels. Where tidal predictions are based on an analysis of levels in a particular well, errors will go undetected if predictions are also checked against levels in the same well. Two such errors among several discussed by Lennon (197 l), are associated with water-density differences during the tidal cycle and with draw-down due to flow past the well. A major limitation is the generally non-linear response of the well-to-sea connection in the presence of external waves.
-
v(H)
effective inlet arm #p well cma sectional area Aw
Fig. 4.1. Basic stilling-well configuration and notation used for the development of the stilling-well equation.
141
The theoretical aspects of stilling well behaviour have been treated by Noye (1974). The behaviour of a system is described by developing a stilling well equation. Using the notation of Fig. 4.1: excess of head outside well H
= h,,-h,
average influx velocity over area A ; by continuity: Aw(
%1
= A;V(
(1)
= v( H
)
(2)
H)
and substituting from eqs. 1 and 2 in 3 :
(3)
~
dH dt
A; + -v( A,
H )=
dh dt
(4)
Acceleration effects through the orifice are assumed negligible; the effective orifice area A; will be less than the measured area A,, because of the contraction of the water stream as it passes through the inlet. Noye distinguishes between wells with a simple orifice, where the flow is controlled by energy considerations, which have non-linear properties, and wells with a long pipe connection, where the flow is controlled by friction, which have linear theoretical properties. Well with a pipe connection Suppose the pipe length is L, and the kinematic coefficient of viscosity of sea water is v. For fully developed laminar flow, standard theory gives:
where A , is the pipe cross sectional area. Substituting eq. 5 into 4 gives a simple equation with an exact solution for an external harmonic water-level variation, h , = a sin(wt):
h,
= a,a
sin( w t - E )
(6)
where (Yw=(l
6 =
+?J*)-”*
arctan ?J
and
1
rvLpAw gAt
‘The term in brackets is a function of the well parameters and has dimensions of time and is often called the well time constant. Expression 6 shows that the well levels fluctuate with an amplitude proportional to the external wave amplitude, and with a phase lag. For a given well, the attenuation factor and the phase lag depend only on frequency, not on the external wave amplitude, and so the system is linear. For laminar flow two conditions must be satisfied: the pipe length must be long compared with the diameter (L, > 100 D, is adequate), and the velocity in the pipe
I42
must not be sufficiently high to generate turbulent flow. Theoretically the Reynolds number Dp~v~/v must not exceed 2000. We shall return to the practical application of this theory later. Well with a circular orifice In this case the velocity of flow into the well, from Toricelli’s theorem, is:
(7) because of the relationship between potential and kinetic energy. Substituting into eq. 4 gives: v = (2gh)’/2
dH dt
+ CIHI’/’
-
sign( H ) =
dh dt
where:
For a circular orifice, A ; = yA, = 0.6 A , , by experiment. There is no exact analytical solution to eq. 8: the system has a non-linear response to external waves and so the amplitude of the fluctuations is important. A single harmonic wave input gives an output with a fundamental oscillation of the same frequency, but distorted by varying amounts of higher harmonics. Also, the principle of superposition of solutions no longer holds-we cannot compute the responses to individual waves and add them. The equation may be solved numerically in terms of a non-dimensional parame-
”
0
0.5
1.0
1-5
wove amplitude (m)
Fig. 4.2. Curves for computing the dimensionless frequency p, as a function of wave amplitude and wave period. Orifice area is 1% of well area.
..'. I...
-4 I...
. . I
1.11-
us.
I&*.
N.11.
k' .
*.IS.
I "..S
I.',.
_.... M.W
"..,i
.
*.a,-
._,.-
".I.-
147
0
8
I
3
2
4
5 6 78910
p
20
30
0
40 50
parameter from Fig 4 2
Fig. 4.5. Theoretical amplitude and phase response for a non-linear well, as a function of the dimenhionless frequency ,B (see Fig. 4.2).
A commonly adopted value of A , / A , is 0.01 (i.e. D,/D, = 0.1). For wells of this configuration, and a single harmonic external wave, /3 may be read from Fig. 4.2, and the corresponding attenuation and phase lag then read from Fig. 4.5. For example, a wave of period 10 s, and amplitude 0.7 m has an attenuation factor 0.05 and a phase lag of 87". At tidal frequencies the amplitude response is given approximately by: a
=
1 - 0.64p4
(9)
and the phase lag:
w
e=-- *
(10)
37.r
The difficulties with such stilling wells become apparent when accurate measurements are required. It is not possible to relate the energy spectrum of the well measurements to that of the water-level variations in the open sea, and some of the high-order tidal constituents in harmonic analyses could be due to well effects and may not have real oceanographic causes. There may also be systematic differences in the mean level inside and outside the well related to the wave amplitude; in this case changes in wave amplitude would appear as low-frequency changes of level in the well. This set-down effect may be quantified by numerically solving the integral equation:
Y[
,A$( H ) d t
=0
(11)
where the integral is taken over a period 7,long compared with the wave period, but short compared with the low-frequency changes being measured, v( H ) is the rate of flow into or out of the well. The form of the function v( H ) can give non-zero mean values of H : for a linear connection, for example as represented by eq. 5, the mean value of H is zero, but for a non-linear connection this is not so. Noye (1974) shows, for example, a well set-down of 0.10 m for an 8-s wave of amplitude 1.5 m, where the waterdepth is 3.0 m. Another cause of asymmetry, and hence non-zero mean H ,
I48
i s the different value of y + for flow into a n d out of the well, likely for certajn configurations: for example, for a coned well entrance. No experimental investigation of the magnitude of these effects in a practical stilling well system has yet been published; however, although such wave-related low-frequency level changes would he significant for precise studies of meteorologically driven regional circulation (e.g. Thompson, 1981), they are less likely to contribute significant errors at tidal frequencies. From the above theoretical arguments, one should design stilling wells with pipe connections which satisfy conditions for laminar flow. Even theoretically, however, these conditions are difficult to satisfy unless either a n unrealistically long pipe is specified or alternatively the well has a time constant of several minutes, which is not acceptable for operational gauges. Figure 4.6 shows the theoretical length of 0.025 m diameter circular pipe for optimum wave filtering, various well diameters and a time constant of 40 s. Also shown are the semidiurnal tidal amplitudes which are permissible if theoretical laminar flow conditions are not to be violated. As can be seen, these restrictions may be important for a large diameter well where large tidal ranges are to be measured. It is also necessary to consider whether the flow observed in practice is actually laminar, even when flow rates are less than indicated by the critical Reynolds number. Experiments a t Holyhead using 1.05 facilities (D.J. Brookfield, pers. commun., 1978) suggest that because of other influences, for example pipe roughness, laminar flow may be elusive in practice. Figure 4.7 shows the observed differences between well level and sea level at different rates of change of sea level over a 10-day period. The line A - B represents the theoretical behaviour of the pipe well under conditions of laminar flow. However, even for Reynolds numbers less than 2000, the observed performance does not follow the laminar theory. A cautionary approach to the practical design of linear (pipe) wells is obviously necessary. A n alternative approach is to measure the sea level as a pressure at some fixed point below the surface. The basic hydrostatic relationship gives:
P
= PA
10.
10
09
-
08
-
-15
07
-
-20
-
-06E &O t
5
z04-
803 -
-
+ pgh
-
$02 01 -
MAXIMUM SEMIMURNAL - 5 0 TIDAL -60 AMPLITUDE(m) FOR LAMINAR CRITERIA
149 %EIGHT DIFFERENCE mm (well level-sea level) 03
well diameter 0 9 m d e t pipe length 4 8 m
diameter 0 0254m
M)
00
I
..
. . . ... . : . . ' . . ._>: :.
. i,'
-03
-0 2
-
Theaeticoi
,...?..->....
.
.
Orn,"O,
M
Re-2000 :.
flow regime
Re=Z@X
RATE OF FLOW
1
I
-01. .{,,a:
,
ii
IN PPE
mm P'
,
I
..
.I .:: I
,. .. ,.,
100
.
. ...
.
,
.
300
400
500
Fig. 4.7. Observed differences between well level and sea level plotted against rate of change o f sea level. Holyhead, March 1978. The continuous line A - B shows the theoretical height difference for a linear response. which should apply for R , < 2000. Clearly even in this region the response is not as theoretically predicted .
where P is the measured pressure, PA is the atmospheric pressure o n the water surface, p is the water density, g is gravitational acceleration and h is the depth of water to be measured. Robert Hooke and Edmund Halley used this method to measure waterdepths in the seventeenth century. For coastal measurements an underwater electrical transducer may be used, but for reasons of datum stability and in order to have only expendable equipment in the vulnerable underwater areas. pneumatic systems are usually preferred by I.O.S. Pressure measurina and recording
and flow control Constriction f o r low pass filtering may be inserted here GO5
supply
T
internal
bomd
Orifice diameter 3mm
Cylindrical pressure-point open at bottom
Fig. 4.8. Parameters for a basic pneumatic bubbling system for tube lengths less than 200 ni. For longcr lengths and large tidal ranges, special design is necessary.
150
The simplest system consists of a partially inflated tyre or “floppy bag” with a tube connection to an above-water chart recorder. Better performance and datum stability are obtained with pneumatic bubbler systems, such as that illustrated in Fig. 4.8. Compressed air (or other gas) from a cylinder is reduced in pressure through one or two valves so that there is a small steady flow down a connecting tube to escape through an orifice in an underwater cannister or “pressure point”. At this underwater outlet, for low rates of gas escape, the gas pressure is equal to the water-head pressure. This is also the pressure recorded ashore, apart from a small increment to provide the pressure head which drives the air along the tube. For tube lengths in excess of 200 m, careful design is necessary (Pugh, 1972, 1976), but for the majority of installations the system illustrated will be accurate pneumatically to within 0.01 m water-head equivalent, Because the viscosity of gas is much less than that of water, laminar flow conditions are easily satisfied. If the pressure is measured using a differential transducer which responds to the difference between the system pressure and atmospheric pressure, then only the water-head pressure is recorded. The variations in density during a tidal cycle must be considered, but these are usually not important, or they are easily allowed for. The underwater pressure point is designed to minimise errors due to waves; datum level is at the orifice, which must not be less than 3 mm diameter if the effects of surface tension on excess pressure within the escaping bubbles are not to exceed 0.01 m water-head equivalent (1 mb of pressure). Within I.O.S. three types of commercial recorder have been used with bubbler systems. Neyrtec chart recorders which measure pressure using a mercury manometer, although only accurate to 0.02 m if used carefully, have proved reliable. However, for long-term installations, including some permanent stations on the British A-Class Network, either Aanderaa differential loggers (Browell and Pugh, 1977) or Ott loggers are used. The former, which, during normal operation, record pressures on magnetic tape at 10-min. intervals for 4 months without attention, are favoured for most research applications, but in some cases, operation requirements make the chart display of the Ott instrument more suitable. In all three cases datum stability is maintained at 0.01 m over several weeks. Although the trend is now away from stilling wells towards more versatile systems such as the bubbler gauge, the majority of coastal measurements available for constructing tidal charts were made using stilling wells. Recently two sets of critical comparisons of the field performance of the two types have been completed. In the first, the results of parallel tidal analyses of three simultaneous but independent measurements of sea level over a month, at Holyhead (D.J. Brookfield, pers. commun., 1978) were compared. Table 4.2, which lists the results of this comparison, shows that in this case, at least for tidal data, and for the systems described, measurement differences are small enough to make variability from non-instrument sources the major uncertainty. The second comparative test was made at Newlyn, between the permanent Ordnance Survey orifice stilling well gauge, which has been in operation since 1915, and a bubbler system with an Aanderaa logger (Pugh, 1981a). The difference between the two gauges over a year of recording had a variance of 20.3 cm2 and a standard deviation of 4.5 cm; 37% of the difference in the variance was directly
151 TABLE 4.2 Comparison of harmonic tidal constants obtained from three different tide gauge systems making simultaneous 28-day measurements at Holyhead
Well No. I . 0.90 m diameter 0.064 m diameter cone orifice Well No. 2. 0.90 m diameter 0.038 m diameter side orifice Differential pneumatic with Aanderaa logger
3.150
1.802
290.7
0.600
327.8
0.035
28.4
3.148
1.8015
291.0
0.600
328.1
0.035
29.3
-
1.803
290.8
0.599
327.8
0.035
28.6
attributable to a phase lag of 1.3" (2.7 min) in the M, tidal constituent, due to the very large well area, 1.765 m2, and the small orifice, 3.2 cm diameter. The theoretical lag for a non-linear well, using eq. 10, is 1.2", which is a satisfying agreement. Other tidal constituents, particularly S, and N,, will also contribute to the difference variance. Another interesting difference between the two series of measurements showed in the spectral analyses of the non-tidal residuals. Table 4.3 summarises the distribution of residual variance in the various tidal bands. Over the whole spectrum the well gauge had 5.5 cm2 more residual variance than the pneumatic gauge. However, 2.26 cm2 of this extra variance is in the semidiurnal band. The likely explanations for this non-oceanographic variance input, is intermittent partial blockage of the well orifice, and also timing errors due to accuracy limitations of chart recording. Some of the cusped non-tidal energy reported by Munk and Cartwright (1966) in the semidiurnal tidal band at Newlyn, must therefore be instrumental. The Newlyn well is not typical of normal permanent well installations
TABLE 4.3 Newlyn-non-tidal residual variance distribution
Band (cpd)
Pneumatic Aanderaa (cm')
Well (ern')
0 0.00 + 0.48 1 0.8 + 1.1 2 1.8 + 2.1 3 2.7+ 3.2 4 3.6 + 4.2 6 5.4 + 6.3
237.2 2.32 2.18 0.34 2.58 0.13
237.3 3.34 4.44 0.50 2.86 0.32
Total
263.6269.1
152
for two reasons. Firstly, the well is much larger than the normal 0.2 m (0.5 m diameter) cross-sectional area, which would make errors due to partial blocking more serious; secondly, the sympathetic attention given to the gauge recorder by the permanent operator is exceptional, and other unmanned gauges are therefore much more likely to suffer chart recorder errors. On balance the pneumatic gauge and Aanderaa recorder are favoured, but although small differences in level are important for non-tidal studies, the tidal constituents determined by parallel analyses, as shown in Table 4.4, agree very well. For studying tidal dynamics the results from either type of gauge could be used. Although the advances in sea-level instrumentation have enabled a far more detailed coastal coverage, the real advances came with the development of the recording open-sea or offshore tide gauge (often abbreviated to OSTG) in the decade from 1965 to 1975. This development culminated with an international comparative exercise of both oceanic and continental shelf gauges (SCOR, 1975). Rae (1976b) gives a summary of the problems which were overcome during this development period. Further developments in the design of gauges, particularly gauges for oceanic deployment, are described by Cartwright et al. (1980a). The latest deep-sea gauge developed by I.O.S. is shown in Fig. 4.9. I t is capable of over a year of pressure measurements in depths to 4000 m. For shelf measurements recent technical developments have been towards cheaper more compact systems with high datum stability. The three basic components of these bottom-pressure measuring systems are: (a)
TABLE 4.4 Newlyn sea-level gauge tests: 1 year comparative harmonic analysis (l61/78 Pneumatic Aanderaa
0.0410686 1 .O 158958 13.9430356 15.04 10686 27.9682084 28.4397295 28.9841 042 30.0 30.0821373 3 1.015858 43.4761563 57.9682084 58.9841042 86.9523127
Standard deviationVariance
+
140/79)
Well gauge
H(m)
go
H(m)
go
0.1 176 0.0 120 0.0528 0.0628 0.0483 0.3226 1.7069 0.5665 0.1641 0.0182 0.01 14 0.1 1 I6 0.0723 0.0093
288.6 326.1 340.4 110.5 169.3 1 14.2 133.7 171.5 177.2 28.5 32.6 166.3 218.4 329.6
0.1087 0.01 37 0.05 12 0.0625 0.0507 0.3256 1.7067 0.5627 0.1614 0.0 I66 0.01 15 0.1088 0.0696 0.0082
186.6 336.4 340.8 1 1 1.6 166.7 115.9 135.0 178.9 177.3 20.4 29.8 169.6 218.1 334.7
Res.
1.360 m 0.1329 m
1.359 0.1352
Obs. Res.
1.849 m2 0.01766 m2
1.847 0.01 827
Obs.
153
Fig. 4.9. A Mark-IV I.O.S. tide gauge prepared for deployment.
the ability to place and recover a gauge from the seabed; (b) a transducer with sufficient stability and sensitivity to respond to bottom-pressure changes; and (c) a recorder with sufficient capacity to log values over at least 30 days. For gauge deployment and recovery, acoustic pop-up command techniques are used for ocean stations, and for many shelf stations. In shallower water a buoyed system is usually preferred. Fortunately, aerospace requirements have resulted in the development of suitable compact loggers having a low power consumption, which are capable of recording more than a year of 15-min values. Aerospace requirements have also
154
produced improved transducers; the two critical factors are datum stability over long periods, and a low sensitivity of the transducer to changes in ambient temperature. Originally capacitance plate transducers and vibrotrons were used. Modern instruments favour either strain gaugers or quartz crystals. Strain-gauge sensors have proved reliable and relatively cheap, but the most consistent temperature-independent results have been obtained from commercially available quartz-crystal transducers. Pairs of quartz-crystal sensors have been deployed at two sites in the Indian Ocean, separated by 470 km. One site was on the Somali Shelf at a depth of 526 m (12) and the other site was in deep water at 3613 m (11). Comparison of the tidal results from each of the two sensors at these stations is made in Table 4.5. For both stations the agreement is remarkably good; stable phases are even obtained for the diminutive M, and M, constituents, for which the amplitudes are around 3 mm. In the case of the deep gauge, 3 mm requires an accuracy of better than of the total pressure signal. Figure 4.10 shows the pressure variations in the four sensors at low frequencies. The standard deviation in the residuals, shown in Table 4.5, is due to low-frequency changes in bottom pressure which are coherent on and off the shelf, and have periods longer than a day. The origin of these variations is unknown, and again the conclusion is that the instrumention now leads the understanding of the oceanographic processes. Although the techniques have been developed to a level where tidal constituents and even lower-frequency variability can be determined, stability over months or years is not yet satisfactory, and long-period tides cannot be determined offshore. Preparation and deployment of the instruments is a skilled and intricate activity, if the required level of reliability is to be obtained. Inevitably, when this attention is
TABLE 4.5 Comparison between harmonic constituents from paired quartz transducers at two sites in the Indian Ocean I1:4" 1 4 " 52" 52'E; 3613 m Days 145-174, 1979 Sensor2262 Sensor2291 Standard deviation (linear trend removed) (m) Temperature coefficient (mb/"C)
0.014
0.018
10.2
24.4
H(m)
go
H(m)
0.144
354.5 351.4 34.2 75.9 297.8 299.5
0.145 0.260 0.325
0.259 0.324 0.160 0.003 0.003
0.160
0.003 0.003
go 354.5 351.3 34.2 75.5 300.4' 297.7
12: 7" 10" 49' 49'E; 526 m Days 149-178, 1979 Sensor 662 Sensor2622 0.013
0.0 12
0.0
1.o
H(m) 0.148 0.268 0.292 0.148 0.003 0.003
go
353.6 351.3 34.7 758 298.5 305.6
H(m) 0.148 0.268 0.292 0.149 0.003 0.003
go
353.7 351.2 34.7 75.5 297.2 304.6
155
1
I 0
'
L
i 2
1 rnbar
'
i " 4
i 6
'
i 8
'
i 10
'
i 12
'
i
I
I
I
I
14
16
18
20
22
I 24
Time (days)
2 6 May 1979
Fig. 4.10. Low-frequency variations of bottom pressure in mb from two sets of paired quartz sensors deployed at two Indian Ocean stations, May/June 1979. Station separation was 470 km.
compounded with the cost of suitable ships for deployment and recovery, measurements of tides offshore are expensive. However, the results are vital for a proper understanding of shelf tidal behaviour. Current measurements
The field of ocean currents is a three-dimensional vector which varies in three space dimensions and in time. For tidal currents the vertical component can be neglected since it is orders of magnitude smaller than the horizontal component (except, perhaps, near the sea floor if the bottom slope is appreciable). Also, the vertical variation of the horizontal component of the tidal current is small so that often it can be realistically eliminated by considering depth mean currents. However, the vertical variations are significant when considering the processes involved in energy loss, which causes the near-bottom currents to be weaker than the near-surface currents and also to lead the near-surface currents. Even with these simplifications, the tidal current field is still far more complex than the elevation field-a scalar which varies in two space dimensions and in time. In addition, measuring currents in the sea is difficult since not only must the instrument operate reliably and accurately in harsh conditions but also the method of deployment of the instrument must not affect the measurements.
156
Semidiurnal tidal currents predominate in the continental shelf seas of North-West Europe. In most areas their amplitude is large and easily measurable. This will be reflected in this section, which considers only measurement techniques suitable for tidal currents in shelf seas. As for all tidal measurements, records at least a year long are necessary before an independent analysis can be calculated. However, because of the difficulty and expense of taking current observations such record lengths are rare (far rarer than for elevations). For shorter record lengths some of the significant tidal constituents can only be calculated by assuming relationships, either from theory or from other year-long measurements, usually of elevation, which may or may not be applicable. In particular the relationships break down near an amphidrome, where elevations are small and change rapidly in space and currents are large, or near an anti-amphidrome. The shorter the record, the more assumptions that are required. However, 6 months, 1 month and 1 fortnight-the period of the spring/neap cycle, are useful record lengths. The shortest possible record length is one tidal cycle (12.5 h) but observations over 25 h are better since tidal variations with a period of a day are usually significant. If short period observations are made it is prudent to measure both at springs and neaps to determine the spring/neap cycle better. Measurements should be recorded at least once an hour in order to determine the higher harmonics, often important in shallow seas. Earliest measurements of tidal currents were probably made by timing the passage of a piece of wood (the “log”) floating past an anchored ship. By 1900 more accurate instruments had been invented which could measure at depth and these were developed up to the mid-1960’s (Bohnecke, 1955; Neumann, 1968). Nearly all were deployed from an anchored ship. They measured speed either by counting the revolutions of a rotor/propeller/cup-anemometer in a given time or by measuring the force of the current (proportional to the square of its speed) directly on a plate or via the inclinations of a pendulum. Direction was measured either internally by a compass or estimated from the ship’s heading and compass. The data could be recorded internally (usually on a dial or by balls in pockets) or transmitted to the ship. The observations were in general of short duration; notable exceptions being those from light vessels which were made two or three times a day (two early tidal analyses of these are by Doodson, 1930, and by Proudman, 1939). Most of the data shown on nautical charts and in tidal stream atlases has been obtained by these methods applied over a few tidal cycles. The major problems arise from the need for a ship. The records are short and expensive to obtain; since 1945, permanently moored light vessels have become rarer. The motion of the ship either swinging at anchor (preferably it should be anchored-fore and aft) or from waves will corrupt the data. The need to anchor the ship makes working in deep water difficult. Finally, the presence of the ship will alter the flow of the water and the magnetic field in the vicinity of the ship. These problems are reduced if the current meter records internally for at least a month and so can be deployed beneath a moored buoy. Such instruments were first produced in quantity in the mid 1960’s (Richardson et al., 1963) and have generated a large increase in tidal current data. Different recording methods were tried- ink on paper, paper-tape, photographic-but these have been superseded by magnetic tape because of its reliability and because the data can be transferred easily to digital
157
computers where most processing is done. A technique now being developed uses a solid state electronic memory, with no moving parts. The meter’s and the mooring’s response to surface waves is still a major problem. It is reduced if the buoy is moored beneath the surface and the meters attached to the taut line. This prohibits measurement near the surface and in shallow water, which is usually not a serious restriction for tidal currents. A vector averaging sampling scheme improves the meter’s response-in it speed and direction are sampled frequently, 0(1) Hz, east and north components calculated and the average values (for example over 10 min) recorded (Weller and Davis, 1980). Most meters have had vanes of various sizes and rotors (suspect in the presence of surface waves) or propellers (Pitt, 1980) and have had problems with fouling and obstruction of the sensors. New methods of speed measurement with no moving parts, to reduce this problem, are being developed- for instance acoustic, both doppler shift and time of travel, and electro-magnetic, since seawater is an electrical conductor. The only alternative to the compass for direction measurement seems to be the flux-gate magnetometer. Moorings present two further problems. Firstly, energy can be extracted from the dominant semi-diurnal motion to contaminate either the mean flow or the higher tidal harmonics (by for instance mooring inclination to the current). Secondly, on reliability, moorings are vulnerable to fishing and other vessels. Moorings are eliminated by taking measurements from the shore, for instance using telephone cables or by radar backscatter. Seawater is an electrical conductor so that as it moves through the Earth’s magnetic field a potential is generated which, across a strait, can be measured via a telephone cable and from this the water transport through the strait may be calculated (Robinson, 1976). A technique still under development is to calculate the surface currents in cells from the doppler shift in the backscatter of H.F. radar (Frisch and Weber, 1980).
TIDAL ANALYSIS
The data explosion which has followed the development of recording sea-level gauges, current meters and other oceanographic instrumentation, has led to the establishment of national and international data banks (e.g. Jones and Sankey, 1980). Whatever the extent of the banked data, the subsequent need is to reduce the profusion of numbers to a few significant values. Godin (1972) eloquently describes this process of data analysis, as the extraction of the soul or quintessence of the record. In tidal analysis the aim is to produce significant time-stable values which may then be used for tidal prediction, and which, if possible, may be related physically to the processes of tide generation, and have some regional coherence. Analysis of elevations, a scalar quantity, will obviously be easier than analysis of currents, which are vectors. From our original definition of tides, it is clear that the form of analysis must relate marine observations to astronomical arguments. The simplest and oldest technique is to relate the time of local semidiurnal high water to the time of lunar transit-this time interval is known as the local establishment. The age of the tide is a term applied to the time interval between new or full moon and the maximum
158
(spring) range. There are many other such terms which have been found useful to mariners (International Hydrographic Organisation, 1974). Modern analyses are more elaborate, and may be categorised either as harmonic analyses or response analyses. These two techniques will both be described in more detail; although their basic approaches to the representation of the relationships with astronomical variables are quite different, the resulting parameters are related to each other. Common to both is the expectancy that these parameters have temporal stability; hence the term tidal constants. Implicit also, in the use of this term, is the assumption that if a sufficiently long series of elevations or currents is available at a site, then a true value for each constant would be obtained. In practice measurements extend over finite periods, often a year, a month, or even a few days, and so the results analysing these finite data can only approximate the true constants: the longer the period available for analysis, the better will be the approach to these true values. Harmonic Analysis starts with the assumption that the astronomical forcing terms can be adequately represented by a finite number of harmonic terms each having different angular speeds q,. In general terms (Cartwright, 1977) un = a o , + bo,
+ cwJ + dw, + em, + fa6
where a, b, c, d , e and f are integer coefficients, and the angular speeds astronomically defined: 27r/o,
= mean
27r/w2
=
sidereal month
2m/w3
=
tropical year
(13) w
are
lunar day
27r/w, = 8.85 yrs,
variation in the lunar perigee
18.61 yrs, regression of the lunar node
27r/o,
=
27r/w,
= 21,000
yrs, variations in the solar perigee (perihelion)
The relative values of the amplitudes of each astronomical forcing harmonic A n , may be determined by laborious algebraic expansion (Doodson, 1921) or by Fourier analysis of an extended time series derived from the ephemerides, the lunar and solar coordinates (Cartwright and Taylor, 1971). The values of a are said to define the tidal species, a = 0 for long period tides, a = 1 for diurnal tides and a = 2 for semidiurnal tides. Energy in these species is modulated by harmonic terms involving w 2 , w 3 , etc. For w 2 , b varies from -5 to + 5 and defines the group within each species. Within each group the value of w 3 , which also varies from - 5 to + 5, is said to define the constituent. From analysis of a year of data, only harmonic constituents which differ by one unit in w3 are separable. The major semidiurnal lunar consituent M, has a coding (200 def), whereas the solar term is (220 def). In harmonic analysis we fit a tidal function:
and minimise Z R z ( r ) ,the square of the residual difference between the model and the observed values, where:
R(f)=O(t)-T(t)
1SY
and O ( t ) is the observed value at time f . In the model for T ( t ) ,Z,, is the mean sea level, and H , and g, are the amplitude and phase lag of the astronomical forcing of the nth constituent, which are determined by the analysis. The summation is taken over a finite number of harmonic constituents, depending on the length and quality of the observed data. Typically, for a year of data n = 60. The choice of constituents is based largely on the relative amplitudes A , in the expansion of the astronomical forcing, but the values of A , are not involved in the computations. The factors,f,, and u , are an attempt to incorporate modulations due to terms in w4, w5 and w6, particularly in u s ,by theoretical time varying factors in the astronomical arguments. These are the undetermined def terms in our coding examples. V,, adjusts the phases to allow for astronomical conditions at the time origin of the data. The least squares fitting is now performed quickly by computer matrix inversion (Murray, 1964, 1965); earlier versions of the harmonic method used tabulated filters to enable manual determination of the constituents (see, e.g., Doodson, 1954). In practice only a limited number of consituents contain most of the tidal energy. Traditionally these have been allocated alphameric labels, and numerical suffixes to show the species to which they belong. For many sites the following are important: Period
Long-period species
Ssa 2w, Mmw,Mf 2w,
Angular speed (degrees h 182.6 days 27.57 13.66
w4
,
28.82 hrs 24.06 23.94
Diurnal species 0, w - o2 p, @ o - % Kl W O + % Semidiurnal species
+
N2 2 ~- ,w 2 w4 12.66 M2 2Wl 12.42 s 2 2wo 12.00 11.97 K , 2(w0+o,)
Source I)
0.082' h-l solar semi-annual 0.545 lunar monthly 1.098 lunar semi-monthly 13.943 14.959 15.041
lunar diurnal solar diurnal luni-solar diurnal
28.440 28.984 30.000 30.082
lunar elliptic principal lunar principal solar luni-solar semidiurnal
27r/wo is the mean solar day, and o o =o ,+ o2+ wi. I t is therefore not an independent frequency from those previously listed, but it is included in the above table for clarity. The major terms over the continental shelf of North-West Europe, as in most of the oceans, are the semidiurnal M, and S, constituents. However. as the tides progress into shallow water, bottom friction, depth limitations and advective effects generate additional constituents due to non-linear processes. Energy is transferred to higher species, particularly to the fourth, sixth and higher species of even-order. Additional terms are also generated in the zero, first and second species.
160
Most important among these are: 2(u, - w l ) 14.77 days
Long-period species
Msf
Semidiurnal species
2SM, 2u, - w I 11.60 hrs 2MS2 2 ~- ,0, 12.87
Fourthdiurnal species M, MS, Sixthdiurnal species M,
4u, 2(w,
6u,
+ wl)
6.21 6.10 4.14
1.016 interaction of M, and S,
31.016 interaction of M, and S, 27.968 interaction of M, and S, 57.968 first harmonic of M, 58.984 interaction of M, and S, 86.952 second harmonic of M,
Selection-of the values of a,, to be included in an analysis is sometimes thought of as a black art. However, certain basic rules exist. In general the longer the period of data the greater the number of constituents which may be included. One criterion often quoted is due to Rayleigh, whereby only constituents which are separated by at least a complete period from other constituents, over the data length available, should be included. Thus, to separate M, from S, requires 360"/(30.0 - 28.98) h, or 14.77 days of data; to separate S, from K,, 182.6 days are required. This minimum period necessary to separate a pair of constituents is sometimes called their synodic period. Where instrument noise and background meteorological signal are low, Munk and Hasselmann (1964) and Godin (1972, pp. 141- 145) have argued that the Rayleigh criterion is unnecessarily restrictive. In practice, the Rayleigh criterion is a good guide for tidal analyses of data from the North-West European Shelf, but finer resolution is feasible in ideal conditions such as tropical oceanic sites. When choosing which terms to include, scrutiny of the analysis from a nearby reference station is helpful, and where the data length is too short to separate two important constituents (S, and K , is the obvious example), it is usual to relate the amplitude and phase of the weaker, by a ratio and phase lag, to the amplitude and phase of the stronger. If a local reference station is not available, then their relationships in the astronomical forcing function (the equilibrium tide), amplitude ratio ( A n l / A n 2 ) and the equilibirum phase lag, may be used. In the case of S, and the weaker K,, the amplitude ratio is then 0.27, and there is zero phase lag. Table 4.6 shows a set of 27 primary constituents, the 8 related constituents with their equilibrium relationships. which are often used at 10s Bidston for analyses of 29 days of data around the area of the North-West European Shelf. For a year, sets of 60 or 100 constituents are normally used. Amin (1976) used 325 constituents in an analysis of 19 years of data from Southend. Finally, when selecting constituents for inclusion the Nyquist criterion must be observed, whereby no term having a period less than the sampling interval may be resolved. Where hourly digitisation of data is practised, the shortest period obtainable is 2 h, so that resolution of M,, would just be possible; in practice this is not a severe restriction except in very shallow water, when more rapid digitisation is necessary. As described, harmonic analysis may be applied to any scalar quantity and its application to elevations (or bottom pressures) is straightforward. Currents, however, are strictly water movement in three dimensions. Although vertical movements are rarely considered, the movement in two dimensions, being a vector quantity. needs special treatment. Currents are usually measured as a speed q ( f ) and a direction
161 TABLE 4.6 A basic set of harmonic constituents and related constituents for the analysis of 29 days of tidal data from the North-West European Shelf Speed (" h r - ' )
Major constituents
zo
0.0 0.5443747 I .0158958 13,3986609 13.9430356 14.4920521 15.0410686 15.5854433 16. I39 I017 27.9682084 28.4397295 28.984 I042 29.5284789 30.0000000 3 1.0158958 42.9271398 43.4761563 44.025 1729 57.4238337 57.9682084 58.4397295 58.9841042 86.4079380 86.9523127 87.4238337 87.9682084 88.9841042
Mm Msf
QI 01
MI Kl Jl
00 I M2 N2 M2 L2 s2
2SM2 MO, M3 MK, MN4 M4 SN, MS4 2MN, M6
MSN, 2MS6 2SM,
mean level non- linear spring-neap lunar diurnal lunar-solar diurnal
lunar perigee principal lunar semidiurnal principal solar semidiurnal
Reference Constituent
Related constituents
14.9178647 14.9589314 15.0821353 15.1232059 27.8953548 28.5 12583I 29.9589333 30.082 I373
Kl Kl
Kl K, N2 N2
52 52
Equilibrium
relationship
a*
P*
0.019 0.33 I 0.008 0.0 14 0.133 0.194 0.059 0.272
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
related amplitude/reference amplitude phase/reference phase
*a
=
*,B
= related
f?( t ) . Speed may be harmonically analysed, but direction, which may cycle through 360" back to zero, may not. The usual technique is to resolve current components into two orthogonal directions, usually north V ( t ) and east U ( t ) , and to perform analyses on each scalar component separately. This method yields four parameters
162
( H , and g, of U and V ) at each tidal frequency. Alternatively, (Taylor, 1921; Godin, 1972, pp. 145-148; Gonella, 1973) the currents may be resolved into two components which rotate in opposite senses. In complex number flotation, the current C ( t ) is given by: N
C
C(r)=a,(t)+
a,exp(iu,t)
(15)
K= -N
Note that negative and positive values of K are included in the summation. The K th constituent is given by: CK(t)=a-Kexp(-iuKr)+aKexp(iu,!)
(16)
and represents a periodic function with period 27r/uK. It corresponds to an ellipse in the XY plane, having major and minor axes: M K = IaKI
+k-KI
mK=laKI-
k-KI
a K and a - , are complex anticlockwise and clockwise components each having a
constant amplitude and an initial direction at some specified time origin. Again, four parameters represent each tidal constituent in the current. This particular type of representation is useful in revealing rotational characteristics of current vectors, which are not obvious from analyses of U and V components. For example if la, I or is zero, then the ellipse point sweeps a circle. If l a K [- la-,[ is zero the current is rectilinear; if la,l> the sense of tidal constituent ellipse rotation is anticlockwise, but if l aK/< la-Kl the rotation is clockwise. The constituents are usually determined as in the scalar case, by a least-squares fitting procedure. The technique of analysing currents into rotary components is particularly valuable when studying inertial currents. These have a non-tidal origin, being generated by storms and maintained by a balance between Coriolis and centrifugal accelerations (see the account by N.S. Heaps in Chapter 9). Their period is given by 2w, sin+, where is the latitude and so for the North-West European Shelf seas lies between 16 and 14 h. They appear as a peak at approximately the inertial period in the residual, non-tidal, spectrum, but only in the clockwise component since they have negative vorticity. Inertial currents are observed in the Celtic Sea and North Sea, mainly above the thermocline and in summer months. Speeds of 0.15 m s - ’ have been observed in the Celtic Sea, corresponding to a length scale of 1.3 kin. Their decay is slow, taking about a week. The basic ideas involved in Response Analysis are common to many activities: a system, sometimes called a “black box” is subjected to a stimulus or input, and yields an output. The characteristics of the system may be evaluated by comparing the input and output functions. The mathematical background has been extensively developed in electrical and communication engineering, but application to ocean tides was pioneered by Munk and Cartwright (1966), with extension to the area we consider by Cartwright, (1968). The gravitational potential due to the Moon and Sun is computed as a time series, and considered to be the input or forcing function. The observed levels, pressures or currents are the output function. The relationship between these is the response of the oceanic “black box”. Thus the astronomical
+
I63
processes and the oceanographic processes are explicitly separated in this treatment. Because we know that physically the local tides are generated by the tidal forces acting over a wide area, it is better to take a broadly representative value for the gravitational potential input. Munk and Cartwright (1966) use an expansion of the potential at the surface of the earth in complex spherical harmonics: M
P ( w , t )=
M
C C
g[a:(t)U:(u)
+~,w>Y?(w]
n = O m=O
where 8, X and t are the colatitude, east longitude and time variables, respectively. a: and b," are the real and imaginary parts of a complex time-varying coefficient C,"(t), computed directly from formulae for the lunar and solar motions. ( U , i V , ) represents the variation in the potential over the Earth's surface. In practice the values of n and m which need be included are very few. The degree, n , is normally adequately represented by integer 2, and the order, m , requires values 0, 1, 2 and perhaps 3. These orders correspond with the long period, diurnal, semidiurnal and terdiurnal species, respectively. One advantage of response analysis is the facility to input other forcing functions in addition to the gravitational potential. Munk and Cartwright (1966) introduced the idea of a radiation potential, which varies with the radiant solar energy incident on a unit surface in unit time. The physical way in which this radiant energy couples with the ocean to produce tidal movements is the "black box" response. The principal radiational input occurs at annual, diurnal (exactly 24 h) and semidiurnal (exactly 12 h) periods, the latter at the same period as the harmonic gravitational S, tide. The tides are represented as the weighted sum of past values of each input potential :
+
=/
00
T (t )
w ( .)P( t
-
7)d7
0
where W ( T ) are the weights to be attached to the P values at time ( t - 7 ) . For physical systems it is reasonable to expect w to become small as t becomes large. In discrete form it has been found that two increments of 7 of 48 h give satisfactory results. Thus: T(r)=W ( O ) P ( O ) + W ( ~ ) P ( ~ - T , ) + W ( ~ ) P ( ~ - T ~ )
where 7, and 7, are 48 and 96 h, respectively. The response weights have a physical meaning in that they represent the residual present response of the system to a unit impulse 7 hours previously. The complex admittance of the ocean to these inputs is given by the Fourier transform of the weights: W ( w ) = L M w ( 7 ) exp( - Z O T )
d7
or in dscrete terms, with only two lags: W ( w ) = w ( O ) + w ( l ) exp(-iw,,)+w(2) exp(-iwT,)
In practice there are mathematical advantages in using both past and future values of the potential. An alternative way of deriving the admittance at a site is to evaluate the
164
frequency spectrum of the input and output signals by Fourier transformation: 00
P ( w ) = / - _ E ( t ) exp(-iwt) d t
O ( r ) exp(-iwt) d t
O ( W ) = / - ~ m
and to determine W ( w ) :
Admittance curves of amplitude and phase lag are obtained only for those parts o f the spectrum where there is input signal, that is, within the range of the tidal species. Where only a short period of observational data is available, it is convenient to first evaluate the admittance of the station relative to predictions or observations at a more permanent local reference station. These may be converted into admittances relative to the gravitational and radiational potential, using the known admittances Amplitude response
u
o 02r
DIURNAL
on elevotion
R
I20
Qj
0l1
20
I
0 80
0 90
I00
0
I10
Frequency (cpd) Amplitude response
u
SEMlClURNAL
Phase log of curent
on
8 0 10-
R
180
I 90
2 00
Frequency (cpd) Fig. 4.1 la. Responses for U currents (see 4.1 Ib).
2 10
elevation
I6f Amplitude response
o'm[
V
(b)
DIURNAL Phase lag of current
on elevation
>
e
0.20 R
Frequency (cpd) Amplitude response
...I R
Frequency (cph) Fig. 4.1 Ib. Responses for V currents. Phases and amplitudes of current components relative to elevations as a function of frequency in the diurnal and semi-diurnal tidal bands. Data are from the Inner Dowsing Light Tower, where the semidiurnal tides propagate from north to south, to a first approximation as a progressive Kelvin wave.
of the reference station but are more usually converted into harmonic constants. This is equivalent to relating amplitudes and phases for unresolvable constituents in harmonic analysis (Cartwright et al., 1969, 1980). An additional refinement is necessary when using the response method in shallow water (in the same way that the harmonic method includes many additional frequencies): recognising that the variable depth in shallow water is an additional effect, higher powers of the first order computed tides are used as an additional input to the system; additional weights, whch usually have a zero or very short lag, are computed. A first fit to the data, without interaction, is needed to give this additional input. The response method has been extensively applied to elevation data from oceanic and continental shelf stations, but there are few detailed reports of its application to the analysis of shelf currents (Cartwright et al., 1980a). Figure 4.1 1 shows a slightly different aDDlication in which the current components are amplitude and phase
166
related to the elevations. For example, the phase lags may be used to determine the relative importance of progressive and standing wave elements at different frequencies. A progressive wave will have currents which lag the elevation by zero or 180", whereas a purely standing wave would have 90" or 270" lag. When the admittances at a station have been computed, it is a simple matter to compute the equivalent harmonic constituents from the response at that frequency. Conversely, approximate admittances may be obtained from the harmonic constituents by plotting both the phase differences, and the ratio between the amplitudes in the observed and astronomical series, against frequency. The admittance which emerges shows that the harmonic constants contain systematic features. One advantage of the response method is that it allows for a smoothness of ocean response and so is able to fit the observed data with satisfyingly fewer parameters. To the oceanographer, the concept of a "black box" process is appealing, as it isolates the ocean system and enables it to be considered according to hydrodynamic laws. Nevertheless, it must be admitted that much remains to be done in the physical interpretation of these response functions in terms of ocean characteristics. In particular, tidal charts of response parameters are not being constructed: even where elaborate response analyses are made, the final presentation is in the form of charts of the major harmonic constituents (Cartwright et al., 1980a). Mariners accept charts of harmonic constituents because they have clearly defined units of length and time; for scientists too, charts of harmonic constituents may be interpreted hydrodynamically in terms of Kelvin wave propagation and amphidrome distribution. Applications of the least-squares criterion to the fitting of tidal parameters is only possible using computers. Formerly, elaborate systems of filtering unbroken data were necessary. However, using the least-squares condition, fits may be made -simultaneouslyto several spans of data from a site: at the times when there are gaps in the data, no f i t is attempted. An interesting application of this data blocking facility occurs where measurements of level extend over only part of the tidal range -for example, where a gauge can only be installed at a site which dries at low water. Theoretically it is possible to fit a harmonic tidal function only to the observed, upper levels. A recent empirical investigation of the stability of the tidal parameters determined from abbreviated data was made by progressively removing observations from the lower levels of a completely recorded range at Steep Holme in the Severn Estuary. Surprisingly the major constituents in this case could be extracted from series where as little as half of the tidal range was available (Evans and Pugh, 1982). We began this discussion of analysis techniques by invoking the expectancy that parameters determined by tidal analysis should have temporal stability, while recognising that secular changes of tidal constituents have been reported (Cartwright, 1972). Nevertheless, analyses of individual months of tidal data from the same location invariably show variations about some mean value. Table 4.7 (Pugh and Vassie, 1976) summarises the range and variability of eight important harmonic constituents. In general, elevations are less variable than currents; greatest percentage variability is found in the weaker constituents and current components. For the M, constituent the standard deviations for the elvations, V-component and weaker U-component of current are, respectively, 1.5, 8.1 and 14% of the mean values of constituent amplitude. Variability in the constituents from year to year is
167 TABLE 4.7 Stability of Inner Dowsing tidal constituents for elevation and currents from monthly analyses ~
H
g
SD
mean
max
min
mean
SD
114.0 153.1 232.2
7.3 75.2 13.3
123.9 272.0 248.1
98.3 22.7 199.6
max
min
~
5 U 0
5 U 0
5 U 0
5 U
u
5 U 0
s U
u
5 U 0
5 U
u
0.177 0.006 0.045
0.0 12 0.002 0.008
0.194 0.009 0.061
0.149 0.002 0.032
0.149 0.007 0.040
0.0 12 0.004 0.008
0.171 0.0 13 0.05 1
0. I40 0.002 0.03 1
285.3 209.2 45.8
6.0 99.0 7.6
298.5 325.2 55.7
274.1 41.2 31.4
0.376 0.038 0.130
0.012 0.018 0.03 1
0.392 0.086 0.191
0.349 0.024 0.085
139.4 215.5 278.5
3.4 30.6 22.5
144.2 269.7 328.9
133.6 162.0 252.1
1.969 0.200 0.693
0.029 0.028 0.056
2.01 1 0.244 0.767
1.919 0.151 0.592
162.2 231.1 304.0
1 .o
10.6 11.3
163.5 250.4 322.3
160.5 216.5 290.1
0.682 0.064 0.237
0.010 0.018 0.025
0.698 0.101 0.270
0.667 0.039 0.201
208.5 289.9 345.6
1.8 13.8 13.1
210.6 3 12.0 7.3
204. I 269.3 331.2
0.035 0.033 0.065
0.004 0.02 1 0.0 12
0.041 0.066 0.084
0.028 0.002 0.039
242.3 107.3 30.1
8.9 65.3 20.0
256.0 240.7 65.2
229.9 22.0 5.1
0.037 0.02 1 0.057
0.009 0.015 0.017
0.048 0.049 0.086
0.022 0.007 0.032
297.8 155.8 79.4
17.1 62.7 12.1
314.7 273.6 101.3
25 1.4 54.9 61.3
0.032 0.02 I 0.026
0.003 0.01 I 0.008
0.038 0.046 0.038
0.028 0.009 0.01 1
272.6 138.4 93.3
3.4 37.7 35.3
277.8 200.4 151.1
266.6 90.5 50.8
-
SD = standard deviation.
5 = elevation in metres. u = east-west
components of current in m s - I .
u = north-south components of current in m s -
I.
Phases are in degrees relative to the equilibrium tide at Greenwich.
TABLE 4.8 Stability of Lerwick sea-level analyses from annual analyses. 1959%I967 (8 years, 1963 excluded)
H
g
mean
SD
max
min
mean
SD
max
min
0.100 0.079 0.035 0.079 0.076 0.583 0.21 1 0.0 15
0.030 0.019 0.013 0.002 0.001 0.003 0.00 1 0.001
0.157 0.114 0.06 1 0.082 0.078 0.586 0.213 0.017
0.046 0.052 0.023 0.077 0.074 0.577 0.209 0.014
225.2 249.8 31.3 164.2 312.2 346.7 282.9
10.4 101.3 1.9 1.0 0.6 0.9 4.7
241.9 292.1 33.3 165.7 312.7 347.5 291.1
207.6 2.9 28.8 162.6 311.0 345.0 175.8
168
much less, but not negligible. Table 4.3 (J.M. Vassie, pers. commun., 1982) summarises the annual variability of major constituents at Lerwick over eight separate years. For M, the amplitude variation has a standard deviation of 0.5% of the mean value. Reasons for the variability include analysis limitations due to the non-tidal energy at tidal frequencies, instrumental changes, and real oceanographic modulations of the tidal behaviour. The different approaches of the harmonic and response methods lead to slightly different rules for estimating theoretical standard errors (Munk and Cartwright, 1966). In the case of a harmonic tidal analysis we make use of the fact that the standard error due to a random background noise of variance N 2 in the elemental band around a constituent frequency is: standard error in amplitude, H
standard error in phase g =
N
=-
Jz
N ~
H f i radians' For a time series of length T, the elemental band has a frequency span l / T . However, better stability is achieved by integrating the noise level over a series of elemental bands, for example the whole of the diurnal or semidiurnal frequency band might be included. If the non-tidal residual variance in the averaging band, of width d F is S 2 , then the noise density is S2/dF. The value of N 2 to be used to calculate standard errors is given by:
',
Integrating over a whole tidal band for S2will tend to underestimate N because i t assumes a uniform noise density across the band, whereas the noise background in practice rises around the major tidal lines-for this reason some workers prefer to use the residual variance in the elemental band of the constituent, rather than the average residual variance over a complete tidal band. In either case, a spectral analysis of the residual variance is required to determine its frequency distribution. Also, in both cases, when applied to shelf tides, the standard errors computed are significantly less than the observed variations in constituent amplitudes and phases. Some variability is undoubtedly instrumental-for example, a chart recorder cannot normally be set to better than + 2 min, equivalent to a phase error of I " in the semidiurnal constituents. The values quoted in Tables 4.7 and 4.8 are for instruments with clockwork timing, and the standard deviations would be reduced with modern quartz-controlled timing. Nevertheless, particularly for monthly analyses, there remains a real oceanographic variability, which is not due to variations in the astronomic forcing functions, as these variations persist, no matter how fully these forces are represented. Corkan (1934) identified a seasonal modulation to the M, constituent, which has subsequently been investigated by Cartwright (1968), Pugh and Vassie (1976), Amin (1982) and Baker and Alcock (1982). In the harmonic analysis of a year of data, this
169
modulation may be represented by two additional constituents:
,
M A 28.94304" h -
'
MB, 29.02527" h - ' which have amplitudes of approximately 1 to 2% of the M, amplitude, in the North Sea. Of these modulations, only about 0.7% is directly due to astronomical effects. These modulations appear to be a shelf phenomenon as their effects in the proximity of the shelf edge, for example, at Lerwick, are much smaller. Because of their seasonal nature, they can be expected to change slightly from year to year. Although these modulations are of intrinsic oceanographic interest, they present a problem when seeking undistorted harmonic constituents from 29 days or less of data at a site. One possible palliative is to compare the results with a 29 day analysis of simultaneous data from a nearby standard port, for which a more reliable value of the constituents has been determined from analysis of a long period of data. The constituents at the site may then be empirically amplitude and phase adjusted according to the ratio and phase differences of the short and long-term constituents at the standard port. An example of how this type of adjustment can yield a coherent regional pattern from a series of non-simultaneous data is given by Alcock and Pugh (1980) where amplitude adjustments of up to 4% in M, were necessary. The increased variability of the M, constituent. in shallow water suggests that a non-linear process of interaction with surges is responsible. Because of the cubic law of energy dissipation, more energy is lost from tide and surge together than would be lost from their propagating separately. Since surges are bigger in winter, the amplitude values of M, would be seasonally reduced at that time, and they would have a maximum amplitude during the quieter summer months, as reported by Cartwright (1968). On the basis of this hypothesis, the answer to the question of whether the undistorted summer tidal parameters or the parameters averaged over a year represent the true local value of M, must be a matter of opinion. Another non-gravitational tidal feature, the radiational S, tide, may be computed as the response to the radiational input potential as defined by Munk and Cartwright (1966). The radiation potential has a large S, constituent. The physical nature of the coupling between the solar radiation and the oceans is not well established; in tropical regions the semidiurnal cycle in atmospheric pressure, which has an amplitude of 1.25 mbars and a maximum value at 1000 and 2200 local time, would produce the observed S, radiational tides, given a similar ocean response to both gravitational and pressure forcing (Pugh, 1979). Theoretically the ratio of the radiational to the gravitational S, tide would be 0.076, and the radiational tide would lag by 120" (4 h). Away from the equator, the amplitude of the atmospheric pressure tide reduces rapidly, but paradoxically, the relative amplitude of the radiational tide increases. Zetler (1971) found an average ratio of 0.16 for the east and west coasts of the United States. On the North-West Europe shelf, Cartwright (1968) found an average ratio of 0.18, with phases averaging 129" (231" in his convention). Amin (1982) reports values between 0.10 and 0.17 on the west coast of Britain with average phase lags of 143". Our unpublished analyses of Celtic Sea and English Channel tides have shown a mean ratio of 0.15 with a mean phase lag of
I70
155". The ratio at Portland was anomalously high at 0.42, implying some local influence on the S, tides, perhaps because of the difficulty of computing response curves in the vicinity of an amphidrome, or perhaps because of a different pattern of generation and propagation. It is not clear to what extent these large radiational S, tides are shelf phenomena, but recent analysis of several months of deep-sea pressure measurements in the North Atlantic has given similar high ratios (J.M. Vassie, pers. commun., 1982). If the linear gravitational tides due to astronomical forcing, as computed by the response technique, are evaluated at harmonic frequencies, and compared with the total observed tide obtained by harmonic analysis, the observed tides are usually slightly smaller and lag by a few minutes. For example, a recent one-year harmonic analysis of the actual M, tide at Newlyn gave an amplitude and a phase of 1.700 m and 135.5", compared with the linear gravitational response tide of 1.872 m and 131.3". If the response to the triple interaction tide is added to the linear gravitational tide, the combined M, tide, 1.728 m and 134.7", is very close to the total observed tide. At Newlyn this M, non-linear tide has an amplitude of 0.179 m and a phase of 276". Elsewhere in the Celtic Sea and Channel the triple interaction M, tide averages 0.095 of the linear gravitational tide, with a mean phase lag of 145". Physically these non-linear processes may be interpreted as a reduction in the tidal amplitudes and delays in their time of arrival, due to the finite amplitude of the incoming tidal wave itself. Tidal analysis is not as difficult or as mysterious as many oceanographers suppose. Provided that certain basic rules relating to the data length and to the number of independent parameters demanded are followed, good results are obtainable. The main fault usually committed by inexperienced analysts is to ask for too much from too little data. Nevertheless, extraction of the optimum tidal information from a record, for example by exploiting the complementary aspects of different analysis techniques, necessarily requires some further experience and informed judgement.
COTIDAL CHARTS
Analysis is the process of reducing the amount of data to a comprehensible quantity-a harmonic tidal analysis in most cases reduces 1 yr's hourly observations (8760) to a set of between 60 and 120 pairs of numbers, the amplitudes and phases of the constituents. About 500 such analyses, most based on a data length of less than 1 yr, of which over 100 are for offshore sites, are now available for the continental shelf seas of North-West Europe. T h s tidal information can be rendered comprehensible by presenting it in cotidal charts, contour maps either of the phase and amplitude for each constituent or of the timing and range for the tide as a whole. The cotidal charts for the constituents within each species-diurnal, semidiurnal, etc.-are similar because within the species the frequencies are closely spaced, leading to similar ocean responses if the processes governing them are the same. The detailed differences between the charts for constituents within a species contain further information on the fine tuning of the responses and of the tidal processes.
171
The species themselves are widely spaced and may be generated by different dynamics-the diurnal and semidiurnals, from forced waves transmitted into the area, with wavelength proportional to their period; others, especially the higher harmonics, are dominated by generation within the area by non-linear processes. (The direct astronomic forcing of tides within North-West European Shelf seas is small compared with their co-oscillation with the tides in the Atlantic Ocean.) Hence much of the dynamics can be presented in a few charts-here for three constituents 0,,M , and M,. The charts are based on observations of elevation or seabed pressure at the places shown in Fig. 4.4 *. The most obvious gap in the observations is in the central and eastern area of the North Sea, leading to considerable uncertainty in drawing the charts for this area. In drawing the charts, more reliability has been placed on analyses based on longer data sets, since the errors in these analyses should be smaller. What is gained by a parallel knowledge of tidal currents? The tidal wave dynamics can be better determined, in some circumstances more information is available for the drawing of cotidal charts and energy budgets can be estimated. The phase difference between elevations and currents indicates the closeness of the wave to being progressive and transmitting energy, when the currents and elevations are in phase, or to being standing and not transmitting energy, when the currents and elevations are in quadrature (for an example, Pugh and Vassie, 1976). Standing waves are the combination of two progressive waves with the same frequency travelling in opposite directions, usually an incident and a reflected wave, involving nodes where, for the elevations, there are large spatial phase gradients and small amplitudes and one quarter wavelength away anti-nodes where, for the elevations, the phase is approximately constant and the amplitude large. The current wave is displaced one quarter wavelength with respect to the elevation wave. Proudman and Doodson (1924) and Defant (1961) showed that tidal current information will supplement elevation information to give the slopes of the phase and amplitude contours. Robinson (1979) further discusses the technique, as applied to the Irish and Celtic Seas. The slopes are obtained via the depth mean equations of motion (see, later, eq. 21) for each frequency and contain contributions from acceleration, Coriolis, bottom friction and advective terms. Robinson tabulated each contribution for 0,,M,, M, and M,. In all cases the acceleration and Coriolis terms were largest, with the contribution of bottom friction being about 10%. For the sixth diurnals it should be greater than this since the forcing term for this species is derived from quadratic bottom friction (Gallagher and Munk, 1971). The advective term is more difficult to calculate since it involves spatial gradients and an appropriate distribution of measurements. For the diurnal and semidiurnal frequencies it was small and could be neglected (as it was originally by Proudman and Doodson, 1924, and Doodson and Corkan, 1932, when they drew M, cotidal charts of the seas arouhd the British Isles). For M, however, the advective contribution is important since it is a forcing term for the constituent, arising from M, through terms in the velocity squared. * Figure 4.4 is placed on pp.
145-146.
172
Tides in the Atlantic Ocean occur significantly in three species: diurnal, semidiurnal and terdiurnal. The largest amplitudes, both in the Atlantic and on the shelf, are in the semidiurnal species, and arise from the gravitational attraction of the Moon and the Sun. In the astronomically forced tide the major constituents are in amplitude order M,, S,, N,, K , with ratios 1 : 0.46 :0.19 :0.13. The M, amplitude at the shelf edge is about Im, (Cartwright et al., 1980a). The cotidal chart for M, is shown in this chapter since it has the largest amplitude and is easiest to determine. Any significant differences with the other constituents will be mentioned; these usually occur near amphidromes. The diurnal species is the next most significant, arising from the Moon’s and Sun’s declination, with principal constituents 0, and K , . At the shelf edge their amplitudes are approximately 0.07 and 0.08 m, respectively, much less than for the principal semidiurnal constituents. The chart for 0, is presented since K , is the more difficult to determine because two other relatively large constituents, PI and S , , are close to it. (PI has an equilibrium amplitude of one third of K,’s and 6 months data is needed to split them; whilst S, is not a significant tidal constituent, meteorologically derived energy is present at this, daily, period which can contaminate the estimate for K,-a year’s data is necessary to separate them.) The amplitudes of the third diurnal constituents are small, the largest, M,, is 0.01 m at the shelf edge. In the Malin Shelf Sea there is a local resonance, causing M, amplitudes to exceed 0.05 m at the mouth of the Clyde. However, although a cotidal chart is not shown here, one for this species is useful since it shows the seas’ response to forcing at another frequency (e.g. Robinson, 1979; Cartwright et al., 1980a). All the constituents in the fourth diurnal and higher species (and a few in the semidiurnal) are generated by non-linear, shallow water processes. Above the sixth diurnals the generation and significance of the constituents is usually localised to a particular estuary away from which they do not propagate. Cotidal charts of these can only be drawn in detail for each estuary. In addition, all tidal waves are long waves in shallow water, since their wavelength is much greater and their amplitude much less than the waterdepth, h. The approximate phase speed is J&, so that in water 50 m deep the M, wavelength (TJ& is 1000 km and the M, wavelength is 330 km. The distribution of the former can fairly easily be determined with existing observations, but the scale of the latter, M,, is too small for an adequate determination in most areas. Also, non-linearities in the response of the measuring system can spuriously transfer energy to the higher harmonics in the observations. Hence, not only is the high-frequency signal smaller in amplitude and wavelength, but also the uncertainty in th- measurements is larger. Therefore, for the higher-frequency constituents, a cotidal chart for M, only is presented here and in detail only for the region south of 53.5”N, where observations are sufficient. The cotidal charts for the North-West European Shelf seas for the constituents 0 , ,M, and M, are shown in Figs. 4.12-4.14. Since the forcing for 0, and M, is similar- the tides in the Atlantic Ocean-we shall discuss their charts before considering that for M,, which is only generated within the shelf seas. The Atlantic Ocean tides are described in Cartwright et al. (1980a). From their figs. 1 l a and 12a, the Atlantic Ocean cotidal charts for 0, and M,, it can be seen that the shelf sea cotidal charts connect smoothly with the Atlantic Ocean charts. Both 0, and M,
173
Fig. 4.12. Cotidal chart for 0 , ;dashed line: elevations (in cm), solid line: phases.
progress northward along the shelf edge, M, taking 5 hrs to travel from the Celtic Sea to the Shetlands. 0, is not as simple, however, since its phase does not increase linearly along the shelf edge but increases and decreases several times; this will be discussed further below. On the shelf the M, tide propagates through the Celtic Sea into the English Channel and Southern North Sea, into the Bristol Channel and into the St. George’s Channel and Irish Sea, through the Malin Shelf Sea, North Channel and into the Irish Sea and anticlockwise round the North Sea. The responses of the English Channel and Irish Sea are very similar, the wave taking about 7 h to travel from the
174
55"
48"
Fig. 4.13. Cotidal chart for M,; dashed line: elevations (in m), solid line: phases.
shelf edge to Liverpool and half an hour longer to reach Dover. The wave which travels along the English Channel to the southern North Sea arrives one cycle earlier than the wave which travels the greater distance via the northern North Sea. The latter travels round the North Sea from the Shetlands to Norway in three complete cycles, by which time most of its energy is dissipated and so its amplitude is small. The amplitude of the wave increases towards the right of the sense of progression, because of Coriolis acceleration. The wave is standing, its phase changes little over a large area, in the Bristol Channel and in the Irish Sea, where two tidal waves meet, one via the St. George's Channel and one via the North Channel.
175
61N
55"
48"
Fig. 4.14. Cotidal chart for M,; south of 53.5"N; dashed line: elevations (in cm), solid line: phases. 'The sense of progression of the wave along the coast is indicated.
Amphidromes, regions of zero tidal elevation about which the tidal wave rotates, occur in the North Channel and in the southern and eastern North Sea. Degenerate amphidronies, where the tidal elevation zero would hypothetically occur inland, since the convergence of the co-phase lines is toward an inland point, appear to be in southeast Ireland, to the north of the English Channel and in southern Norway. Since the amphidrome to the west of Denmark and the degenerate amphidrome in Norway are poorly defined by the observations some uncertainty is attached to their
116
positions. This also holds for the 0, amphidrome off Norway. The degenerate amphidromes north of the English Channel and in southeast Ireland are associated with nodal “lines” between the Isle of Wight and Cherbourg and across the St. George’s Channel so that the M, tide in the Celtic Sea is out of phase with that in the eastern English Channel and in the Irish Sea. I t will be shown below that amphidromes occur where an incident and a reflected tidal wave meet and are spaced half a wavelength apart. Because of dissipation, the reflected wave will
60”N
155”
50”N
Fig. 4.15. Map of mean spring near-surface tidal current amplitudes in cm s - ’ (after Howarth. 1982).
177
contain less energy than the incident wave and so the amphidrome occurs closer to the reflected wave than the incident. That this is not so in the southern North Sea-the amphidrome is closer to the English than the Dutch shores-indicates a significant energy flux through the Dover Straits, of the order of the energy dissipated in the Southern Bight. Maximum M, amplitudes, over 4 m, occur in the Gulf of St. Malo and the Bristol Channel; an amplification about four times from the shelf edge. Amplitudes over 2 rn occur in the eastern Irish Sea, eastern English Channel and Dover Straits, where there is an anti-amphidrome (similar to an anti-node) and near the Wash. Amplitudes are small, less than 0.5 m in the eastern North Sea, away from the German Bight. A fuller description of the M, tide is given in Huntley (1980). A contour map of mean spring (M, + S,) near surface current amplitudes is shown in Fig. 4.15. The map is based on Sager and Sammler (1975), supplemented by analyses of recording current meter observations (Howarth, 1982). Not shown are the small scale variations in current amplitudes associated with headlands and islands which can lead to large local currents. For both currents and elevations the ratio of S, to M, amplitudes is about 0.35 in most places, only differing from this value near current or elevation amphidromes, respectively. Where the M, wave is progressive the current, v, is proportional to the elevation, {:
where h is the waterdepth. Near elevation amphidromes the currents have local maxima, c,f. nodes, particularly evident in the North Channel, St. Georges Channel and between the Isle of Wight and Cherbourg. The most obvious local minima occur in the Irish Sea, to the east and west of the Isle of Man, where the M, wave travelling northwards through the St. Georges Channel meets the wave travelling southwards through the North Channel, and also to the south of Cork. These are about one quarter wavelength away from elevation amphidromes. Clearly the constrictions of topography also influence currents, for instance the high currents in the Dover Straits. Comparing the M, cotidal chart with previous charts drawn from observations (see Introduction), with those based on numerical model computations (for instance, 2-dimensional, Flather, 1976; Pingree and Griffiths, 1981b; 3-dimensional, Davies and Fumes, 1980) and with those generated by physical models (Chabert d’Hieres and le Provost, 1978) shows small but significant differences. There are larger differences between Fig. 4.13 and Admiralty Chart no. 5058 in the Celtic Sea, the Bristol Channel and the North Channel. In the Celtic Sea the M, wave arrives earlier near the shore than in the middle (see also Robinson, 1979) whereas Chart 5058 indicates the opposite. Figure 4.13 also shows the M, wave progressing northeastwards along the Cornish peninsula towards the Bristol Channel whereas Chart 5058 indicates the wave travels eastward. Finally, the amphidrome in the North Channel in Fig. 4.13 is presented as degenerate, in Islay, in Chart 5058. The 0, cotidal chart, Fig. 4.12, shows broad similarities, making due allowance for the more than doubling of wavelength, with the M, chart, Fig. 4.13, in the North
178
Sea but is different for the western coastal seas. The 0, wave progresses anticlockwise around the North Sea in one cycle with an amphidrome very close to the shore of southern Norway. Comparing the 0, and M, charts for the North Sea the difference in their wavelengths is greater than their frequencies warrant. Shelf amplification for the diurnals is less than for the semidiurnals. For example, the maximum 0, amplitude for the North-West European Shelf seas occurs in the Wash, 0.2 m, an amplification of only three times the shelf edge value. In the western coastal seas there is a degenerate amphidrome in southwest Ireland and an amphidrome west of the Dover Straits. Throughout the Celtic Sea, Bristol Channel and English Channel and in the Irish Sea the wave is standing. In contrast to M,, there is little amplification in the Bristol Channel and Gulf of St. Malo. There is an anti-amphidrome in the Southern Bight of the North Sea. Fewer 0, cotidal charts have been published-Robinson (1979) for the Irish Sea, Prandle (1980) for the southern North Sea, Pugh and Vassie (1976) for the Dowsing region and the German Naval Charts, Oberkommando der Kriegsmarine ( 1942): For the English Channel, charts based on a physical model are published in Chabert d'Hieres and le Provost (1978). No numerical model charts for 0, have been published. Agreement is reasonable apart from the German Naval charts where i t is poor, for instance these do not show an amphidrome near the Dover Straits. The drawing of cotidal charts for the higher harmonics is more difficult. Figure 4.14 shows the M, cotidal chart south of 53.5"N; north of 53.5"N there are too few observations, particularly offshore, to determine the cotidal charts but the direction of propagation along the coasts has been indicated. North of 53.5"N the M , amplitude is small, less than 0.1 m. except in the eastern Irish Sea where an amplitude of 0.2 m occurs near Liverpool. M, is clearly generated within the shelf seas since its amplitude at the shelf edge falls to about 0.01 m and its phase is directed towards the ocean. Maximum amplitudes, about 0.4 m, occur in localised areas-in the Severn Estuary, Wash and Gulf of St. Malo, but note that in the Thames Estuary, for instance, there is little amplification. The largest area of high amplitude, up to 0.3 m, is in the Dover Straits, where there is an anti-amphidrome. Over the rest of the shelf seas the amplitude is less than 0.1 m. Amphidromes occur off southeast Ireland, in the Southern North Sea and twice in the English Channel. This chart can be compared with similar charts published in the same papers as for 0,.In addition there is one numerical model of M, (Pingree and Griffiths, 1979). Again agreement is reasonable except with the 1942 German Naval Charts.
DYNAMICS OF SHELF TIDES
In the previous section we described the observed 0,,M, and M, cotidal charts; in this section we will elucidate some of their dynamics. Much is linear; most of the diurnal and semidiurnal distribution can be modelled by Kelvin waves forced by the ocean tides, although continental shelf waves contribute to the diurnal distribution along the shelf edge. Non-linear dynamics are significant for energy dissipation and for the generation of higher harmonics. The motion of tidal waves in most seas can be described in terms of Kelvin waves
17Y
(Lamb, 1932, Art. 208). A Kelvin wave is a “long wave in shallow water” propagating in a channel on the surface of a rotating Earth. The Coriolis parameter, ,f, is taken to be constant, implying the channel extends over only a short range of latitude. The deflecting force arising from the Earth’s rotation is balanced by an elevation gradient (the transverse velocity is taken to be zero) leading to an exponential decrease in amplitude away from the coast. The wave propagates in one direction only: in the northern hemisphere with the shore on its right. I t was originally calculated for an infinitely long uniform straight channel with constant depth, h , for which if a boundary is at y = 0, considering y > 0: 1= 1”exp( - f , / c ) COS( K X - a t ) Zl=-{
v
=
R c
(18)
0 (by assumption)
The phase speed c = O / K = dgh and u/( are the same as for a non-rotating channel, see eq. 17. Miles (1972) has shown that the reduction in depth over a continental shelf and small changes in the coastline have little effect on the wave’s phase speed (and hence wavelength). A Kelvin wave incident in a rectangular gulf is perfectly reflected by the gulf‘s head, provided the gulf is not too wide (of the order of a wavelength at the latitude of North-West Europe) (Taylor, 1921). Close to the gulf‘s head the reflected wave is not a pure Kelvin wave but the perturbations (Poincare waves) decay to zero in a distance from the gulf‘s head of the order of the gulf‘s breadth. The combination of the incident and reflected waves leads to a system of amphidromes positioned half a wavelength apart along the centre line of the gulf-analogous to a standing wave i n the non-rotating case. Near an amphidrome the elevation amplitude is small and its phase varies rapidly in space whilst the current amplitude is large and its phase constant. One quarter of a wavelength away, the reverse holds. In the body of the gulf the tidal streams are rectilinear parallel to the shore, whereas near the head they are elliptic. In a shallow rectangular gulf friction progressively reduces the amplitude of the wave, making the reflected wave weaker than the incident. The amphidromes are then displaced from the centre-line of the gulf towards the shore closer to the reflected wave. This is by a greater distance the farther the amphidromes are from the head of the gulf (Rienecker and Teubner, 1980). If the incident and reflected Kelvin waves have amplitudes lo and ale, respectively, in the vicinity of the amphidrome ( a < l), the amphidrome is moved transversely a distance -(Jgh In a ) / ( 2 f ) (Pugh, 1981b). For large frictional losses this leads to degenerate amphidromes, postulated inland. For Kelvin waves near to the shore the amplitude contours will be parallel to the shore and the phase contours perpendicular to it. If friction is significant both sets of contours are rotated clockwise of this. The superposition of two Kelvin waves of equal amplitude travelling along a channel in opposite directions yields amphidromic points-points of no elevation response about which the wave uniformly rotates (Proudman, 1953, art. 130). This, essentially linear, Kelvin wave dynamics forced from the shelf edge can describe most of the features in the 0, and M, cotidal charts, Figs. 4.12 and 4.13.
180
If the dynamics were truly linear, cotidal charts for all the constituents in the same species with the same forcing would look similar. The wavelengths of the constituents differ slightly, causing the positions of the amphidromes to differ by a few kilometres longitudinally. However, amphidromes for S, and N, are also displaced laterally, to the right of the corresponding M, amphidrome, indicating proportionally more energy is lost at these frequencies than at M, (Pingree and Griffiths, 1981a,b). There is a conceptual difficulty in considering energy losses and single constituent dynamics when non-linear dissipation is significant. Considering the time, rather than the frequency domain, the position of the semidiurnal amphidrome in southeast Ireland, determined daily, moves laterally by over 70 km during the spring-neap cycle, being farthest from the axis at springs (Pugh, 1981b) corresponding to a minimum reflection coefficient a. This again indicates non-linear dissipation with 19 times more energy dissipated at spring tides compared with neaps. Away from amphidromes the S,/M, amplitude ratio is about 0.35, in common with the Atlantic but significantly different from the astronomic forcing ratio of 0.46. The phase difference (S, - M,) is 30-40”, indicating that spring tides occur 1.5 to 2 days after New or Full Moon (the “Age of the Tide”). The N,/M, amplitude ratio is about 0.2 and the phase difference is 20-30”. Near amphidromes the amplitude ratios and phase differences vary greatly, since each constituent’s amphidrome is in a slightly different position. Another aspect of non-linear dissipation is that in general these amplitude ratios decrease slowly and phase differences increase slowly as the distance increases away from the forcing, as proportionally more energy is extracted from the weaker constituents. The amplitude of the tide at the coast is determined by the degree of amplification across the shelf and by dissipation. To first order the tide can be modelled by the superposition of two progressive waves travelling in opposite directions with exponential decay proportional to distance (Redfield, 1958, 1978). If reflection at the coast is perfect the wave there is standing. The further away from the reflecting coast the more progressive the wave becomes, to supply the energy which is dissipated. This implies both a phase progression towards the coast and that the phase differencebetween currents and elevations changes in space. Dissipation, however, is only significant in shallow water, less than O(20) m, and may be neglected for tidal dynamics for most of a shelf sea (e.g. Clarke and Battisti, 1981). [Linear analytic models, including the Earth’s rotation, of the tide propagating from a deep sea into a continental shelf sea are given in Clarke and Battisti (1981) and Huthnance (1980) and for barotropic tidal currents Battisti and Clarke (1982).] There are two causes of amplification: changes in geometry and resonance. Geometric amplification occurs through decreasing water depth, h , or decreasing estuary breadth, b. I t does not depend on the wave’s frequency; the amplitude of a progressive wave, assuming no energy loss, is proportional to K 4 h - , . Amplification by resonance is frequency dependent and applicable to trapped or standing waves. Here the wave’s amplitude varies in space, between nodes and anti-nodes. but the nearer the wave’s frequency is to the natural frequency of the sea, the greater will be its amplitude at an anti-node. Resonance will not occur in a heavily damped system but where it does occur energy dissipation will locally be large since the currents will be fast.
181
The largest M, amplitudes occur in the Bristol Channel and Gulf of St. Malo, where values four times those at the shelf edge are observed. Clearly resonance is involved, since 0, amplitudes show no such amplification. Two forms of resonance are possible, 1/4 wavelength transverse to the shelf edge (“organ pipe”) and 1/2 wavelength parallel to the shelf edge, the wave being trapped by the depth discontinuity and the Earth’s rotation. To first order the Earth’s rotation can be neglected in “organ pipe” resonance- the predicted amplitude will be slightly over-estimated (Clarke and Battisti, 1981) and for the Celtic Sea/Bristol Channel the resonant period will be over-estimated by about 5% (Huthnance, 1980). Studies based on a one-dimensional model (Fong and Heaps, 1978) and based on observations (Heath, 1981) suggest that the 1/4 wavelength resonant period of the Celtic Sea is about 11 h and of the Celtic Sea and Bristol Channel combined is slightly longer than 12 h, close to but less than the period of M,. The resonant period of the Bristol Channel, to the east of Lundy, is between 7 and 9 h, and the semidiurnal tides in i t can be realistically modelled by an analytic one-dimensiona! wedge-shaped model with friction, but without rotation (Robinson, 1980). The model again shows that non-linear dissipation is important when considering other constituents besides M *, in that proportionately more energy is supplied and dissipated during spring tides than would be expected from a consideration of M, alone. If both elevations and currents are known energy fluxes and budgets can be calculated. Mean energy fluxes, like signal variance but unlike dissipation, can be calculated separately for each constituent and summed to give the total mean energy flux (Pugh and Vassie, 1976). Energy budgets were first discussed by Taylor (1919), for the Irish Sea, and more recently by Garrett (1975), who pointed out that Taylor had omitted a term in the mean energy equation (term 2 below). The full equation for a sea area, G, with an open boundary, M , is:
1
2
3
4
where n is an outward unit normal to an element d s of the open boundary, dA an element of sea area and the overbar denotes the time average. h is the waterdepth below mean sea level, v the depth mean horizontal velocity vector, { the elevation, 2, is the height of an equipotential surface above its mean position and F is the bottom friction force. lEcontains three effects: direct astronomic tidal forcing and the movement of the solid Earth both due to Earth tides and tidal loading. Terms 1 and 4 are usually the largest, representing the apparent energy flux of the tidal wave and its dissipation by bottom friction. Terms 2 and 3 are corrections to these and are usually of opposite sign since the scale of 5, is much greater than the size of the sea area, so that their sum is small. For a single tidal constituent:
l= loCOS( 01- g s ) ; v = [u,,cos(ot - g u ) ,vo cos(ot - g,)] and the mean energy flux through unit area due to it is:
Robinson (1979) has calculated the terms in eq. 19 for the Irish and Celtic Seas for
182
M,, S, and 0,. He showed terms 2 and 3 were small, but made a measurable contribution to the energy budget and that term 4 is difficult to estimate since it probably depends on lvI3 and hence relies heavily on its accurate determinations in regions with high velocity. The flux of energy into the Celtic Sea (about 190 GW due to M,) is about three times that into the North Sea through its northern entrance. About a half of the former is dissipated in the Celtic Sea, Bristol Channel and Irish Sea and about a half in the English Channel. Only about one tenth passes through the Dover Straits but this represents one quarter of the total energy flux into the North Sea. Most of the energy in the North Sea is dissipated in the shallow Southern and German Bight areas. The energy fluxes through the North Channel and Skaggerak are small. (Flather, 1976; Cartwright et al., 1980a). The 0, cotidal chart, Fig. 4.12, shows that the propagation of the diurnal wave onto the shelf is not as simple as for the semidiurnal wave (c.f. Fig. 4.13). Kelvin waves with any frequency can occur but at frequencies lower than the local inertial frequency (1.53 cpd at 50"N) continental shelf waves also occur (Mysak, 1980). These are waves trapped along a depth discontinuity, the shelf edge or the coast, with their restoring force arising from changes in potential vorticity as fluid elements are displaced up or down the slope. They propagate in the same direction as Kelvin waves, their characteristics being determined by the depth profile (e.g. Caldwell et al., 1972). In the ocean their amplitude decays to zero rapidly and on the shelf, compared with Kelvin waves, they have shorter wavelengths, larger speed, elevation ratios and elliptic currents. In the region of St. Kilda a Kelvin wave combined with a shelf wave with comparable magnitudes at the shelf edge can model the 0, and K , elevation variability and their large currents ( - 0.08 m s-', a significant proportion of the M, current), although the origin of the shelf wave is not known (Cartwright et al., 1980b). For most of the shelf seas the amplitude of M, is comparable with that of 0 , ,in the range 0.05-0.10 m, occasionally exceeding this in shallow water. The amplitude of M, is smaller, everywhere less than 0.1 m, except near the Isle of Wight where it reaches 0.2 m. Higher harmonics than M, are only significant in estuaries. The fourth, sixth and higher diurnal species are generated by non-linearities in the seas' response, caused by shallow water and friction. Considering the generation of higher harmonics in more detail the depth mean equations of motions are, for momentum: aV
-+v~vv+252xv=-gv5--at 1
2
3
4
and for continuity:
1
2
3
where the depth mean velocity: v = - / P1 h+l
u(z)dz -h
c, +
5
vlvl + A v 2 v 6
183
Terms 5 and 6 in eq. 21 represent dissipation by bottom friction and the transfer of momentum by horizontal mixing, respectively, with a mixing coefficient A . In shelf seas term 6 is small in comparison with the others and can be neglected. The non-linear terms in the equations are, in the momentum eq. 21, term 2, advection, resulting from spatial gradients in the velocity, and term 5, bottom friction and, in the continuity eq. 22, term 3 important where the elevation amplitude is significant compared with the waterdepth, h. The higher frequencies generated by advection and shallow water are even harmonics since components with the same frequency are multiplied together. The bottom friction term generates both even and odd harmonics-the even harmonics arise from the ( h [) denominator and the odd from v(v1, for instance:
+
8
8 cosot + __ cos3wt + . . . i , if v 15a
i
= ivo cosol
(For a fuller analysis see Gallagher and Munk, 1971.) Hence, each of the terms contributes to the generation of M, from M,, but only bottom friction significantly for M,. (This implies that dissipation will be large where M, is large.) In a progressive wave the current amplitude is proportional to the elevation amplitude and hence the M, amplitude will be proportional to the square of the M, amplitude. This implies that at springs the fourth diurnals will be proportionately greater than at neaps (through the combination of M, with MS,). Pingree and Maddock (1978) used a numerical model to compute the importance of each of the forcing terms to the M, tide in the English Channel, showing that there the most important was the shallow-water term in the equation of continuity and the least important was the bottom friction term. The advection term could be important locally, for instance near Cap de la Hague, near Cherbourg, where the direction of fast tidal streams is changing. Just as important for the overall M, prediction as internal generation was the progression into the Channel of externally generated M,. M, is also generated locally in a trapped form, by small scale (of the order of the tidal excursion) topographic features: sandbanks (Huthnance, 1973). headlands and islands. The dynamics associated with these features have been much studied recently, mainly in terms of vorticity and mainly to predict mean currents, another aspect of non-linear dynamics (e.g. Chapter 7). The effects of the higher harmonics are apparent as distortions to the shape of the tidal wave. These can be significant if the peak ebb and flood currents have different amplitudes which is only caused by even harmonics (for M,, these are M,, M,, etc., but not Mh). For M, and M, the inequality occurs if the phase of M, minus twice the phase of M, equals 0, +a, + 2 a , etc., (e.g. Howarth, 1982). Then not only will there be large differences between the peak ebb and flood bottom stresses but also a mean bottom stress, both significant for sediment transport (Pingree and Griffiths, 1979). For elevations the relation (phase of M, minus twice the phase of M,) is often observed to be 0, +a, f 2 ~ etc., , if the M, wave is standing and +a/2. 3a/2, etc., if it is progressive. The theoretical conditions for these phase relations are discussed by Heath (1980) who showed that in the generation of M, the influence of friction differed from that of shallow-water continuity and advection.
I84
For progressive waves without frictional generation both elevations and currents tended to the quoted phase relation. For standing waves the elevation and current phase relations tended to differ by a/2, with the quoted relation holding for elevations if frictional generation dominated and for currents if the other two terms dominated. Maximum differences between peak flood and ebb flow were associated with regions of strong frictional generation of M, if the M, wave was progressive and with regions of M, generation by advection or shallow-water continuity if the M, wave was standing. In the main, tidal currents show little vertical structure over most of the water column, so that the often used concept of depth mean currents is justified in tidal dynamics. Structure occurs because of energy loss by bottom and internal friction. The frictional forces are greatest near the seabed, creating a boundary layer in which the velocity decreases to zero and the phase of the velocity is in advance of the depth mean value (typically for M, by 5-10'). Since the vertical structure depends on the wave's frequency, o, with less structure at higher frequencies, there will be differences in the structure of the clockwise, effective frequency ( o- f ) , and anticlockwise, effective frequency (o+f), components of current. At M, these differences are large since ( w + f ) / ( o - f ) is approximately 12 for north European latitudes, with the clockwise component exhibiting more vertical structure. Hence, the anticlockwise component becomes more important towards the seabed and the sense of rotation of the ellipse can be reversed. For further discussion see Prandle (1 982) and Soulsby (Chapter 5 ) . DISCUSSION
Our aim has been to summarise the present condition of empirical knowledge of the tides of the North-West European continental shelf. It is not possible to draw final conclusions. We have described the results of progress made over a very active decade, but much remains to be done by applying the new techniques of measurement and analysis. The following list of possible developments over the next decade is not exhaustive, but might make interesting reading in ten years' time. The techniques for measuring bottom-pressure tidal variations on the shelf are well established and need little improvement. A few measurements offshore, extending over a year or longer would be useful. Satellite altimetry may allow more uniform coverage. Coastal gauges which use bubbler or similar techniques should replace the traditional stilling-well configurations. Although current meters are probably adequate at present for measuring the dominant (M2, 0 , )tidal currents, uncertainties in measuring the higher harmonics and the vertical structure of tidal currents near the seabed need to be reduced. Longer period current measurements are needed to confirm the relationships used when analysing short periods of data (for example the K,/S, relationships) and for identifying currents due to radiational tides. More measurements are needed, particularly near amphidromes. Further analysis work is needed on the separation of linear and non-linear tides. The techniques for tidal analysis and prediction in very shallow water such as the Severn Estuary are still inadequate. Methods which rely more on the physics rather than the spectroscopy of the situation may be more successful. The physical reasons
I85
for the annual and monthly variations in the major tidal constituents need to be clarified. Sufficient current observations exist, although many are in need of analysis, to enable the drawing of contour maps of the ellipse parameters for significant constituents. Combinations of currents with elevations will enable better cotidal charts to be drawn from existing data. However, their combination is not strictly valid where there is energy being fed into a wave, so that charts of higher harmonics such as M, and M, will require special consideration. Definitive charts of diurnal constituents in the vicinity of the shelf edge will be of special interest. A distinction should be made between processes which generate and radiate non-linear energy, and those where the non-linear effects are locally confined. Energy sinks for both linear and non-linear tides need to be more carefully defined. The dynamical interpretation of cotidal charts will need to develop to account for new features, particularly in the case of the diurnal and higher harmonics charts. Where non-linear processes are dominant, studies of variations through time and their physics are an alternative to elaborate spectroscopy, Numerical models already predict satisfactory cotidal charts where the tidal dynamics are predominantly linear, and also in some non-linear cases. The higher harmonic charts and the vertical structure of tidal currents provide sensitive tests for models since these are determined by non-linear dynamics and by the parameterisation of friction. Numerical models can then be used to elucidate some of the tidal dynamics and calculate total energy budgets. The implications of tidal dynamics, particularly the asymmetry of tidal flows, as represented by their higher harmonics, for sediment transport and biological activities need further investigation. Whatever progress is made, it is appropriate to remember Hillaire Belloc’s bleak prognosis: “Indeed, I think that as we go on piling measurements upon measurements, and making one instrument after another more and more perfect to extend our knowledge of material things, the sea will always continue to escape us.” I t is a demanding challenge.
REFERENCES Alcock, G.A. and Pugh D.T., 1980. Observations of tides in the Severn Estuary and Bristol Channel. Institute of Oceanographic Sciences, Rep. 112, 57 pp. (unpubl.). Amin. M., 1976. The fine resolution of tidal harmonics. Geophys. J. R. Astron. SOC.,44:293-310. Amin, M., 1982. On analysis and prediction of tides on the west coast of Great Britain. Geophys. J. R. Astron. Soc.. 68: 57-78. Anonymous, 1857. The Tides. Society for Promoting Christian Knowledge, London, 189 pp. Baker, T.F. and Alcock, G.A., 1983. On the time variation of ocean tides. Paper presented at the Nineth Int. Symposium on Earth Tides, New York City, 17-22 August 1981. Battisti, D.S. and Clarke, A.J., 1982. A simple method for estimating barotropic tidal currents on continental margins with specific application to the M, tide off the Atlantic and Pacific coasts of the United States. J. Phys. Oceanogr., 12: 8-16. Beach, A.D., 1977. Stonehenge I and lunar dynamics. Nature (London). 265: 17-21. Belloc, H., 1925. The Cruise of the ‘Nona’. Constable, London. Bohnecke, G., 1955. The principles of measuring currents. Publ. Sci.. Association d’oceanographie Physique, UGGI, No. 14, 23 pp.
186 Browell, A. and Pugh, D.T., 1977. Field tests of the Aanderaa pressure logger with pneumatic tide gauges, and the design of associated pneumatic control circuits. Institute of Oceanographic Sciences, Rep. 37, I 1 pp. (unpuhl.). Caldwell, D.R., Cutchin, D.L. and Longuet-Higgins, M.S., 1972. Some model experiments on continental shelf waves. J. Mar. Res., 30: 39-55. Canada Marine Sciences Directorate, 1979. Tidal Workshop. Manuscript Rep. Ser. 52, 196 pp. Cartwright, D.E., 1968. A unified analysis of tides and surges round north and east Britain. Philos. Trans. R. SOC.London, Ser. A, 263: 1-55. Cartwright, D.E., 1972. Secular changes in the oceanic tides at Brest 1711-1936. Geophys. J. R. Astron. SOC.,30: 433-449. Cartwright, D.E., 1977. Physics of the oceans and atmospheres. Lectures presented at an international course, 1975, International Centre for Theoretical Physics, Trieste, 2: 740-763. Cartwright, D.E. and Taylor R.J., 1971. New computations of the tide generating potential. Geophys. J. R. Astron. SOC.,23: 45-74. Cartwright, D.E., Munk, W.H. and Zetler, B.D. 1969. Pelagic tidal measurements: a suggested procedure for analysis. Trans. Am. Geophys. Union, 50: 472-477. Cartwright, D.E., Edden, A.C.. Spencer, R. and Vassie, J.M., 1980a. The tides of the Northeast Atlantic Ocean. Philos. Trans. R. SOC.London, Ser. A, 298: 87- 139. Cartwright, D.E., Huthnance, J.M., Spencer, R. and Vassie, J.M., 1980h. On the St. Kilda shelf tidal regime. Deep-sea Res. 27A: 61-70. Chabert d’Hieres, G., and Le Provost, C., 1978. Atlas des composantes harmoniques de la maree dans la Manche. Annal. Hydrogr. 6: 5-36. Clarke, A.J. and Battisti, D.S., 1981. The effect of continental shelves on tides. Deep-sea Res.. 28A: 665-682. Corkan, R.H., 1934. An annual perturbation in the range of tide. Proc. R. SOC.London, Ser. A, 144: 537-559. Davies. A.M. and Fumes. G.K., 1980. Observed and computed M, tidal currents in the North Sea. J. Phys. Oceanogr., 10: 237-257. Deacon, M.. 1971. Scientists and the Sea, 1650-1900. Academic Press, London. 445 pp. Defant, A,, 1961. Physical Oceanography. Vol. 2. Pergamon Press, Oxford, 598 pp. Doodson, A.T., 1921. The harmonic development of the tide generating potential. Proc. R. SOC.London, Ser. A, 100: 305-329. Doodson, A.T., 1930. Current observations at Horn’s Rev, Varne and Smiths Knoll in the years 1922 and 1923. J. Cons. Int. Explor. Mer, 5: 22-32. Doodson, A.T., 1954. The analysis of tidal observations for 29 days. Int. Hydrogr. Rev., 31: 63-92. Doodson, A.T. and Corkan, R.H., 1932. The principal constituent of the tides in the English and Irish Channels. Philos. Trans. R. SOC.London, Ser. A, 231: 29-53. Evans. J.J. and Pugh, D.T.. 1982. Analysing clipped sea-level records for harmonic tidal cbnstituents. Int. Hydrogr. Rev., 59: 115- 122. Flather, R.A., 1976. A tidal model of the North-West European continental shelf. Mem. SOC.R. SCI. Liege, 10: 141-164. Fong, S.W. and Heaps, N.S., 1978. Note on quarter-wave tidal resonance in the Bristol Channel. Institute of Oceanographic Sciences, Rep. 63, 15 pp. (unpubl.) Frisch, A.S. and Weber, B.L., 1980. A new technique for measuring tidal currents by using a two-site H F doppler radar system. J. Geophys. Res. 85: 485-493. Gallagher, B.S. and Munk, W.H., 1971. Tides in shallow water: Spectroscopy. Tellus, 23: 346-363. Garrett, C., 1975. Tides in gulfs. Deep-sea Res., 22: 23-35. Godin, G., 1972. The Analysis of Tides. Liverpool University Press, Liverpool. 264 pp. Gonella, J.A., 1973. Some statistical characteristics of ocean motions from current measurements. Mem. SOC.R. Sci. Liege, 4: 139- 149. Great Britain Hydrographic Department, 1940. Atlas of Tides and Tidal Streams, British Islands and Adjacent Waters (2nd ed.). Hydrographer of the Navy, London. Halley, E., 1701. Letter to Josiah Burchett, Secretary to the Admiralty. In: E.F. MacPike (Editor), Correspondence and Papers of Edmond Halley. Taylor and Francis. London, pp. I 17- 1 18.
187 Heath, R.A., 1980. Phase relations between the over- and fundamental-tides. Dtsch. Hydrogr. Z. 33: 177- 19 1. Heath, R.A., 1981. Resonant period and Q of the Celtic Sea and Bristol Channel. Estuarine. Coastal Shelf Sci., 12: 291-302. Howarth, M.J., 1982. Tidal currents of the continental shelf. In: A.H. Stride (Editor), Offshore Tidal Sands. Chapman and Hall, London, pp. 10-26. Huntley. D.A., 1980. Tides on the North-West European continental shelf. In: F.T. Banner, M.B. Collins and K.S. Massie (Editors), The North-West European Shelf Seas: the Sea Bed and Sea in Motion. Vol. 11. Elsevier, Amsterdam, pp. 301-351. Huthnance, J.M., 1973. Tidal current asymmetries over the Norfolk Sandbanks. Estuarine Coastal Mar. Sci., 1 : 89-99. Huthnance, J.M., 1980. On shelf-sea “resonance” with application to Brazilian M, tides. Deep-sea Res. 27A: 347-366. Hydrographic Society, 1976. Proceedings of Symposium on Tide Recording. Hydrographic Society, Spec. Publ., 4, 201 pp. International Hydrographic Organisation, 1974. Hydrographic Dictionary, Part 1 (3rd ed.). I.H.O., 32, 370 pp. Jones, M.T. and Sankey T., 1980, The MIAS oceanographic database-an integrated databasejdata dictionary system. In: G.J. Baker (Editor), Database Achievements, A.P. Publications and British Computer Society, London, pp. 69-95. Lamb, H., 1932. Hydrodynamics (6th ed.). Cambridge University Press, Cambridge, 738 pp. Lemon, G.W., 1971. Sea level instrumentation, its limitations and the optimisation of the performance of conventional gauges in Great Britain. Int. Hydrogr. Rev. 48: 129-147. Lemon, G.W., 1976. National network to monitor sea level; and the Committee o n Tide Gauges. Dock Harbour Auth., 57: 53-54. Marmer, H.A., 1928. On cotidal maps. Hydrogr. Rev., 5: 195-205. Miles, J.W., 1972. Kelvin waves on oceanic boundaries. J. Fluid Mech., 55: 113-127. Ministry of Agriculture, Fisheries and Food, 1981. Atlas of the Seas around the British Isles. M.A.F.F., Lowestoft, 100 pp. Munk, W.H. and Cartwright, D.E., 1966. Tidal spectroscopy and prediction. Philos. Trans. R. SOC. London, Ser. A, 259: 533-581. Munk, W.H. and Hasselmann, K., 1964, Super-resolution of tides. In: Studies on Oceanography, Tokyo, pp. 339-344. Murray, M.T., 1964. A general method for the analysis of hourly heights of the tide. Int. Hydrogr. Rev., 41: 91-101. Murray, M.T., 1965. Optimisation processes in tidal analysis. Int. Hydrogr. Rev., 42: 73-81. Mysak, L.A., 1980. Recent advances in shelf wave dynamics. Rev. Geophys. Space Phys., 18: 21 1-241. Neumann, G., 1968. Ocean Currents. Elsevier, Amsterdam, 352 pp. Noye, B.J., 1974. Tide-well systems. J. Mar. Res., 32: 129-194. Oberkommando der Kriegsmarine, 1942. Karten der harmonischen Gezeitkonstanten fur das Gebeit der Nordsee. Marineobservatorium Wilhelmshafen, Ausgabe A, Nr 2752 Hrsg (Bearb.). Palmer, H.R., 1831. Description of graphical register of tides and winds. Philos. Trans. R. SOC.London, 121: 209-213. Pingree, R.D. and Griffiths, D.K., 1979. Sand transport paths around the British Isles resulting from M, and M, tidal interactions. J. Mar. Biol. Assoc. U. K., 59: 497-513. Pingree, R.D. and Griffiths, D.K.. 1981a, S, tidal simulations on the North-West European shelf. J. Mar. Biol. Assoc. U. K., 61: 609-616. Pingree, R.D. and Griffiths, D.K. 1981b. The N, tide and semidiurnal amphidromes around the British Isles. J. Mar. Biol. Assoc. U. K., 61: 617-625. Pingree, R.D. and Maddock, L., 1978. The M, tide in the English Channel derived from a non-linear numerical model of the M, tide. Deep-sea Res., 25: 53-63. Pitt, E.G., 1980. The measurement of ocean waves and currents. J. SOC.Underwater Technol., 6: 4-12. Prandle, D., 1980. Co-tidal charts for the Southern North Sea. Dtsch. Hydrogr. Z., 33: 68-81. Prandle, D., 1982. The vertical structure of tidal currents and other oscillatory flows. Geophys. Astrophys. Fluid Dyn., 22: 24-49.
188 Proudman, J., 1939. On the currents in the North Channel of the Irish Sea. Mon. Not. R. Astron. Soc.. Geophys. Suppl., 4: 387-403. Proudman, J.. 1953. Dynamical Oceanography. Methuen, London, 409 pp. Proudman, J. and Doodson, A.T., 1924. The principal constituent of the tides of the North Sea. Philos. Trans. R. Soc. London, Ser. A, 224 ges. Int. Hydrogr. Rev. 49: 71-97. Pugh, D.T., 1972. The physics of pneumat Pugh, D.T., 1976. Methods of measuring sea level. Dock Harbour Auth. 57: 54-57. Pugh, D.T., 1979. Sea levels at Aldabra atoll, Mombasa and Mahe, western equatorial Indian Ocean. related to tides, meteorology and ocean circulation. Deep-sea Res. 26: 237-258. Pugh, D.T., 1981a. Comparative tests of sea level data from the Newlyn tide well and an Aanderazi pneumatic system. Institute of Oceanographic Sciences, Rep. 119. 15 pp. (unpubl.) Pugh, D.T., 1981b. Tidal amphidrome movement and energy dissipation in the Irish Sea. Geophys. J. R. Astron. SOC.,67: 515-527. Pugh, D.T. and Vassie, J.M., 1976. Tide and surge propagation off-shore in the Dowsing region o f the North Sea. Dtsch. Hydrogr. Z., 29: 163-213. Rae, J.B., 1976a. Offshore measurements of tides and sea level. Dock Harbour Auth., 57: 57-58. Rae, J.B., 1976b. The design of instrumentation for the measurement of tides offshore. In: R. Brittcm (Editor), Symposium on Tide Recording, Hydrographic Society, London, pp. I 1 I - 133. Redfield, A.C., 1958. The influence of the continental shelf on the tides of the Atlantic coast o f the United States. J. Mar. Res., 17: 432-448. Redfield, A.C., 1978. The tide in coastal waters. J. Mar. Res., 36: 255-294. Richardson, W.S., Stimson, P.B. and Wilkins, C.H., 1963. Current measurements from moored buoys. Deep-sea Res., 10: 369-398. Rienecker, M.M. and Teubner, M.D., 1980. A note on frictional effects in Taylor's problem. J . Mar. Res.. 38: 183-191. Robinson, I.S., 1976. A theoretical analysis of the use of submarine cables as electromagnetic oceanographic flowmeters. Philos. Trans. R. Soc. London, Ser. A, 280: 355-396. Robinson, I.S., 1979. The tidal dynamics of the Irish and Celtic Seas. Geophys. J. R. Astron. Soc., 56: 159-197. Robinson, I.S., 1980. Tides in the Bristol Channel-an analytic wedge model with friction. Geophys. J. R. Astron. Soc., 62: 77-95. Sager, G. and Sammler, R., 1975. Atlas der Gezeitenstrome fur die Nordsee, den Kana1 und die Irische See (3rd ed.) Seehydrographischer Dienst. D.D.R., 8736, 58 pp. SCOR, 1975. An intercomparison of open sea tidal pressure sensors. Report of SCOR Working Group 27 (Tides of the open sea). UNESCO Tech. Pap. in Mar. Sci., 21, 67 pp. Taylor, G.I., 1919. Tidal friction in the Irish Sea. Philos. Trans. R. SOC.London, Ser. A. 220: 1-93. 'Taylor, G.I., 1921. Tidal oscillations in gulfs and rectangular basins. Proc. London Math. Soc.. Ser. 2. 20: 148- 181. Thompson, K.R., 1981. The response of southern North Sea elevations to oceanographical and meteorological forcing. Estuarine Coastal Shelf Sci., 13: 287-302. Weller, R.A. and Davis, R.E., 1980. A vector measuring current meter. Deep-sea Res. 27A: 565-582. Whewell, W., 1836. On the results of an extensive system of tide observations made on the coasts of Europe and America in June 1835. Philos. Trans. R. Soc. London, 126: 289-341. Young, T., 1807. A course of lectures on natural philosophy and the mechanical arts, Volume I . Johnson, London, 796 pp. Zetler, B.D., 1971. Radiational ocean tides along the coasts of the United States. J. Phys. Oceanogr., 1: 34-38.
189
CHAPTER 5
THE BOTTOM BOUNDARY LAYER OF SHELF SEAS R.L.. SOULSBY
1. INTRODUCTION
l.l. General
Tidal currents on the continental shelf are influenced to a greater or lesser extent by the effect of friction at the seabed. In deep water with slow currents, the boundary layer in which the frictional forces act occupies a relatively thin region near the bed, whereas, in shallow water with faster currents, it may occupy the entire waterdepth and dominate the tidal dynamics. Similar behaviour is seen under steady currents and meteorologically induced currents. In the sea the bottom boundary layer is practically always turbulent, so that the frictional forces are transferred by turbulent processes. Correspondingly the transfer of scalar properties such as heat, salt or pollutants is dominated by turbulent diffusion. In addition, the stresses produced at the bed control the movement of the seabed sediments, which may be further enhanced by surface-wave motion. However, we will not be directly concerned with either diffusion of scalars or sediment transport here, except as they affect the water motions. The chapter describes the principal features of the vertical current structure and the turbulence properties observed in the various kinds of bottom boundary layer encountered in shelf seas. The approach is primarily observational, though a basic theoretical framework is introduced to enable the measurements to be fitted into a generalisable pattern. Spectra of turbulence and the “bursting phenomenon” are also discussed, in as far as they aid the understanding of boundary-layer processes. 1.1’. Mean velocity and turbulence
The profile of the time mean velocity u ( z ) as a function of the height z above the bed is of interest in many applications. The overbar denotes a mean of the quantity over a suitable time interval, typically 10 min in the tidal context. For convenience, however, we will generally omit the overbar on U , while retaining it on turbulence quantities. The velocity generally increases with z from zero at the bed to its value at the water surface, if the boundary layer occupies the entire waterdepth (Fig. 5.la), or to the free stream velocity U, at the edge of the boundary layer in water which is deeper than the boundary-layer thickness (Fig. 5.lb). In the former case the boundary-layer thickness is equal to the waterdepth h , while in the latter case we define a boundary-layer thickness 6, based on mean velocity such that U( S,) 2 Urn. In the laboratory the usual definition is U(S,)= 0.99Urn, but in the sea it is not
190
usually possible to use such a precise definition. The displacement thickness and momentum thickness (see Hinze, 1975, p. 595) are also commonly used measures in the laboratory, but are not so widely used in the sea, perhaps because of the difficulty of defining them in reversing flows. If there is no boundary layer at the water surface due to wind stress, and no shear due to, for example, topography or density currents, then the velocity is constant with height for aL,< z < h , and is equal to the frictionless free-stream value U'. The turbulence properties of the water velocity are also important in many applications. We take coordinate axes with the positive x-axis along the direction of the bed shear stress, the z-axis perpendicular to the bed and the y-axis orthogonal to x and y in the right handed sense. Then the instantaneous velocity components are U + u, V + u and W + w in the x, y and z directions, respectively, with u , 0 and w being the turbulent fluctuations about the means _ _U , V and W. The quantities which and the Reynolds shear are of most general interest are the variances u 2 , u2 and 2, stresses - puw, - p z and - p;. Here the water density is p, and the overbars again indicate time averages over, typically, 10 min. The choice of averaging time, and also the rate at which measurements are taken, is discussed in Soulsby (1980). The term - p G represents the rate of turbulent transfer of x-momentum across the x-J plane, and hence, apart from very near the bed where viscous stresses are important, is , which the water at one level exerts a frictional force equal to the shear stress T ~ by on the level beneath it. Similarly the terms -p& and - p z represent the shear stress T~~ in they direction acting across the x-y plane, and the shear stress T~~ in the x direction acting across the x-z plane, respectively. Note that - puw also represents the shear stress in the z direction acting across the y - z plane, but this is a less helpful and -p; are less description; similarly the alternative descriptions of -p& helpful. The shear stress T~~ acting across successively lower planes in the flow ultimately acts on the sea bed. If the bed shear stress, 70, exceeds a threshold value it may move the bottom sediment. It is convenient to define a friction velocity ZL in terms of the bed shear stress, according to the relation: 2
p u * = 7"
Because the dynamics near the bed are dominated by T ( ] , it is usual to scale both mean and turbulent velocities by u , here. As a general rule we find that u., a,,,a, and a,, where a,, = (u2)"' etc, are all of the same order of magnitude, with: u,, > U" > a, > u*
(2)
and u,/U, = 0.05, typically. Much of the rest of the chapter is devoted to examining the forms of the vertical profiles of U , a,,,a,, a,, - p G , - p G and - p% under different conditions in the sea. The kinetic energy of the turbulence per unit volume is given by:
The energy is mainly due to large eddies, whose dimensions are comparable with the measuring height z . These break down into successively smaller eddies until the
eddies are small enough for molecular viscosity to take effect. The energy is then dissipated into heat at a rate c per unit mass per unit time. This mainly occurs at eddy sizes which are comparable with the Kolmogorov length scale: 1,
= y3/4c-
‘/4
(4)
where Y is the kinematic viscosity. Very few measurements have been made in the sea with sufficiently small instruments to be able to resolve these motions (one exception being those of Grant et al., 1962), but many experiments in the laboratory and the atmosphere have succeeded in measuring them. I . 3. Predicting
ro
It it often necessary to be able to estimate ro in terms of more readily measurable or predictable quantities such as U,, the depth-averaged velocity fi, or a measured velocity at one height. Unfortunately, the system of equations of motion is not closed when turbulence is included, so that it is impossible to relate the turbulence quantities directly to the mean flow, and assumptions must be made. The simplest approach is to use a quadratic friction law of the form: 70
= PC,U
2 7
where U is an available velocity. This form is widely used in depth-averaged numerical models of the tidal dynamics of geographical areas, in which case U is the appropriate velocity. The drag coefficient in this case is usually taken as constant over the area. It may either be specified, with C , = 0.0025 being a popular figure. o r used as a free parameter to tune the model to fit observations. The results are usually good enough to give tidal elevations quite accurately, but they are less successful for predicting the geographical distribution of T ~ )for sediment-transport purposes. In practice the measured value of C , will vary with waterdepth, seabed composition, the phase of the tide (because T~ is not quite in phase with and in addition the direction of T~ may be different to that of due to veering of U with z . The variation with waterdepth and seabed composition can be overcome to some extent by using the form of C , suggested by Dawson et al. in Chapter 6. Current measurements for use in sediment-transport prediction are usually made at a height z = 100 cm, and the corresponding drag coefficient C,,,,, is somewhat larger than C , based on fi. Phase and veering changes are very small in the bottom metre, so T~ can be more accurately predicted by U,,, than by U . The variation of C,,, with seabed composition is later given in Table 5.4. A better approximation is to relate the shear stress to the shear by an eddy viscosity assumption of the form:
o),
Unlike molecular viscosity, the eddy viscosity K , , varies with position and with the flow in a way which has to be assumed. The simplest assumption is to take i t as constant in space and time, and models using this assumption can predict some of
192
the observed behaviour of many kinds of boundary layers. A rather better approximation is to use the form: K,,
(7)
= KU.Z
where K is an empirical constant whose value is discussed in section 3. This form of K , , reproduces the near-bed velocity profile much more accurately than does K , , = constant. A great variety of more sophisticated forms for K , , have been used, including those used in other chapters of this book, but it is not proposed to review them here. 1.4. Szihdivision of the boundary layer
Generally speaking, the turbulent energy and the shear stress decay from a maximum value at or near the bed to zero at the outer edge of the boundary layer, at heights which we will denote 6, and S,, respectively (Fig. 5.1). It is not necessarily the case that 6, = 6, = 6,. Weatherly and Martin (1978) found, using a sophisticated one-dimensional numerical model of the bottom boundary layer in a deep flow, that in the absence of density stratification 6, was nearly three times as large as 6,-,. In laboratory flows the edge of a turbulent boundary layer forms a distinct, but highly
I::
OIJTER LAYER
---t-l
LOGARITHMIC LAYFR
W R I A C b LAYFR
Current speed
\
BED LAYEH
Turbulent kinetic energy
.f
(bl
Q
--_
A
z
z
~-~~
7-
NUN-TURBIJLEN1 ‘I
1
//
TURBULENT
OUTER LAYER
SURFACE LAYER HED LAYER
Current speed
Turbulent kinetic energy
Fig. 5.1. Schematic illustrations of the subdivision of the boundary layer. (a) For a boundary layer which occupies the entire waterdepth; (b) for water which is deeper than the boundary-layer thickness The shear-stress profile is qualitatively similar to that for energy, but tends to zero at the surface in a. The various layers are not drawn t o scale.
193
contorted, interface between the turbulent and non-turbulent flow. While the position of the interface could in principle be used as a measure of boundary layer thickness, it is rather inaccessible in the sea. Nonetheless, it seems physically realistic to define the boundary layer as being the region in which the turbulent energy and shear stress are non-zero so that 6, or ST are preferred to a,, as a measure of boundary-layer thickness. However, it is observationally easier to measure S,,, so this has more often been presented in the literature. Where it is either not known or not important which version is used we will just use the symbol 6. The boundary layer can be subdivided into a number of layers (Fig. 5.1), which have received different names in different branches of turbulence research. The subdivision into three layers given below is thus not the only one which could be used, but seems to be convenient for marine work. Very near to the bed is a layer in which either the bed is sufficiently smooth that the effect of molecular viscosity dominates the dynamics, or the presence of roughness elements causes horizontal variations in the profiles of velocity and turbulence around and just above them. The term bed layer can be used to cover both types. In the sea the bed layer is typically a few centimetres thick. Above this is the logarithmic layer, in which neither the details of the bed nor the nature of the free-stream flow affect the local dynamics, and the velocity and turbulence profiles take particularly simple and universal forms. I t generally extends to a height of a few metres in the sea. Above this again is the outer luyer, in which the velocity and turbulence profiles depend strongly on the nature of the free-stream flow, and are thus not universal. In the sea there are a variety of different kinds of free-stream flow which must be considered separately. For example, the tidal oscillation, the earth’s rotation, and vertical density gradients all modify the forms of the velocity and turbulence profiles in distinctive ways, and may also occur in combination with each other. The outer layer extends to height 6, which is typically some tens of metres in the sea.
I . 5. Turbulence structure As well as examining the 10 min averaged quantities, we wish to look at the details of what goes to make up the averages. Spectra of the turbulent velocities reveal how the energy and stress are distributed with respect to the frequency or wavelength of the motions. They also demonstrate the cascade of energy towards smaller eddy sizes and ultimately to dissipation. In addition, the time series of the uw product is examined to distinguish those features of the flow which are responsible for the turbulent momentum exchange. I . 6. Observations
The aim of the chapter is to describe the features of the boundary layer as far as is possible in terms of experimental observations made near the seabed. However, where this is not possible reference is made to measurements in rivers, in the atmosphere, beneath drifting pack-ice, and in the laboratory, where the boundary layers are expected to be dynamically similar to the marine case. Many of the examples are drawn from the literature, and further examples can be found in the
TABLE 5.1 Description of the stations worked in Start Bay and Weymouth Bay Start Bay Stn. 1
Start Bay Stn. 2
Start Bay Stn. M
Weymouth Bay
50' 14.3" 3" 38.4'W Sand. median diameter = 230 p m (2.109), std. dev. = 0.419. Probably rippled, possibly occasionally suspended.
50" 34.5" 2" 20.5'W Flat immobile sandy gravel, median diameter = 220 pm, (2.25+). std. dev. = 2.1+.
Instruments Heights, z (cm)
50" 14.3" 3" 37.9'W Rippled sand, median diameter = 300 p m (l.75+) std. dev. = 0.359, occasionally in suspension. Dunes 50- 100 cm high, 7-10 m wavelength. 2 x EMCM 30, 140
50' 15.6" 3" 34.1'W Immobile very coarse sand and shell, median diameter = 1.2 m m ( - 0.229). std. dev. = 1.4+. No ripples or larger features. 2 orthogonal EMCM Mean = 65
5 X Marconi C/M 150,250.700, 1000, 1300
2 x EMCM 30, 140
Mean waterdepth, h (m) Roughness length, zo (cm) R =h/r, 2/h
14 0.35 4~ 103 0.021, 0.10
42 0.12 3.5 x lo4 0.015
19
27 0.040 6.8 X lo4 0.01 I . 0.052
2/20
86,400
540
Latitude Longitude Nature of bottom
1 1.9X 10' 0.08. 0.13, 0.37. 0.53, 0.68 150. 250, 700, 1000, 1300
750. 3500
reviews of the benthic boundary layer by Wimbush and Munk (1970) and Bowden ( 1978).
In several instances examples are presented from our own observations made off the south coast of England, in Start Bay and in Weymouth Bay. The positions and descriptions of the various stations are given in Table 5.1. Turbulence measurements were made using electromagnetic current meters (EMCM) with Colnbrook sensing heads mounted at various heights within the bottom 2 m. Details of the experimental set-up and analysis techniques are described in Soulsby (1980, 1981) and Soulsby and Dyer (1981). Velocity profiles within the bottom 2 m were obtained from up to 6 Braystoke propeller current meters mounted on the same frame as the EMCMs. A string of 6 Marconi current meters was deployed for 9 days at a site in Start Bay to give profiles of current speed and direction throughout the water column. Background information on temperature and salinity profiles and suspended sediment concentration was obtained regularly, and the data sets referred to were all obtained when there was little wave activity. 2. THE BED LAYER
2. I . Classification The flow in the bed layer can be classified into smooth, rough and transitional hydrodynamic roughness regimes using the criteria established by the experiments of Nikuradse (1933) on flow in pipes internally roughened with sand: smooth turbulent: u,d/u < 3.5 transitional: 3.5 < u,d/v < 68
(8)
rough turbulent: u,d/v > 68 The roughness Reynolds number u,d/v is based on the friction velocity u*, the grain diameter d of the seabed sediment, and the kinematic viscosity v. Measurements by Sternberg (1968, 1970) showed that a similar classification obtains in the sea, provided that the height of the roughness elements replaces d if the bed is not flat. His results indicated values of 5.5 and 165 for the critical values of u.d/v. All three roughness regimes are common in the sea. A crude indication of the roughness regime of a site at the time of peak velocity can be obtained from a knowledge of the nature of the seabed alone, because over large areas of the sea the sediment distribution is in approximate equilibrium with the shear-stress distribution. Thus d and u. are interrelated, and v does not vary greatly in coastal waters. Warwick and Uncles (1980) found a close correlation between the peak shear stress of the M, tide and the observed bed type. The values of T,, are summarised in Table 5.2, and the deduced roughness regime for each bed type given. If the bed is rippled, it is the ripples which form the major roughness element rather than the grains, so using d in the roughness Reynolds number is inappropriate. If there is a mixture of grain sizes, the effective roughness may be very different from the median grain diameter, as the fine grains may fill the gaps between the large grains to give a relatively smooth surface.
I96 TABLE 5.2 An indication of the likely roughness regime of a site with an equilibrium sediment distribution. at the time of peak velocity, based on bottom type alone. The association of U. and bottom type is drawn from Wanvick and Uncles (1980). The grain diameter d is inferred from their description of the bottom type; the smooth sand is assumed to be unrippled, and the sandwaves rippled having the Nikuradse equivalent grain size quoted. The kinematic viscosity is taken as Y = 0.014 cm2 s- I (IO"C, 35%0)
Assumed d (cm)
Bottom type Smooth sand or mud in bays Smooth sand Sandwaves Smooth gravel Rock
0.006 0.03 15
1.5 > 30
Associated u. (cm s--I )
u.d/v
Hydrodynamic roughness
1.2 2.2 2.8 3.6 4.6
0.5 5 3000 400 >lo4
Smooth Transitional Rough Rough Rough
The smooth and rough turbulent cases are discussed below, but as the transitional regime has been less well studied it is not included. 2.2. Smooth turbulent flow
The bed layer in smooth flow corresponds to the viscous sublayer, in which viscous stresses are comparable with the Reynolds stress. The total shear stress is the sum of these, given by: dU dz
r x 7= pv-
-
-
puw
(9)
The total stress is more or less constant with z through the sublayer, and equal to the bed shear stress T~ = pu?. Re-arrangement of eq. 9 thus gives: dd uz - l ( uv ? + u w ) As the bed is approached the Reynolds stress tends to zero, as the w component in term and integrating with respect to z particular is inhibited. Neglecting the subject to the condition U = 0 at z = 0, yields the velocity profile:
uw
ulz U(z)=Profiles having this form were measured in smooth turbulent flow at the seabed by Chriss and Caldwell(l983) using a hot thermistor current meter. They found a linear profile, corresponding to eq. 11, extending to a height, typically, of about 1 cm (Fig 5.2). In terms of the viscous length scale v/u,, the sublayer thickness varied between 8 and 20 v/u, for different data sets. In most cases the thickness was appreciably thicker than the traditional laboratory value of about 12 v/u,; in addition, the variation from run to run appeared to be real, rather than due to experimental error, but no external cause could be firmly identified. Thus the traditional laboratory findings may need to be modified for use in the sea. The Reynolds stress - p G increases towards the top of the viscous sublayer.
197
-1 5
2
i
0 (cm
s-')
Fig. 5.2. The velocity profile within and above the viscous sublayer. (Reprinted from Chriss and Caldwell. 1983, copyrighted by the American Geophysical Union.)
Inspection of eq. 10 shows that this causes the velocity to progressively decrease with z below the value given by eq. 1 I , as can be seen in Fig. 5.2. In the smooth regime any roughness elements are submerged in the viscous sublayer and do not appreciably affect the flow. 2.3. Rough turbulent flow In the rough regime the roughness elements project well beyond the (hypothetical) viscous sublayer. It is their interaction with the flow that determines the velocity and turbulence profiles, with viscosity being unimportant. The details of the profiles vary with horizontal position over the roughness elements, which act as topography at this scale. The thickness of the rough bed layer can be taken as the height at which the horizontal variations die out. In Chapter 1, Davies shows that for potential flow over sand ripples this has happened at a height of one ripple wavelength. Mulhearn and Finnegan (1978) found that the mean velocity over a bed of randomly arranged stones in a wind tunnel exhibited horizontal variations up to a height corresponding to the average spacing of the stones. Their turbulence profiles, - however, exhibited variations to about twice this height, with - p E and w 2 increasing, and u' decreasing, with height. It is noteworthy that it is the spacing, rather than the height, of the roughness elements which principally determines the thickness of the bed layer. 3. T H E LOGARITHMIC LAYER
3.1. The mean velocity
In most boundary layers there is a range of heights for which the height z is simultaneously too great for the details of the geometry of the bed to affect the flow,
198
a n d too small for the flow here to be influenced directly by the free-stream velocity Urn or the boundary-layer thickness 6. A particularly revealing derivation of the velocity profile in this region was given by Jackson (198 1) and is summarized here. Above the bed layer it is postulated that the flow depends mainly on the bed shear stress, characterized by u*, and only weakly on the geometry of the bed. Thus where this is true the mean velocity can be written:
where H is the height of the roughness elements, and I , , I,, etc. are other lengths describing the roughness geometry. Heights are referenced to the displacement height z,above a n arbitrary origin of z. In the outer layer the free-stream velocity Urn and boundary-layer thickness 6 become important. but the bed geometry is not, and the velocity can be written:
If there is a range of z in which eqs. 12 and 13 are both valid simultaneously, then equating dU/dz from each and multiplying by ( z - z , ) gives:
By separation of variables, both sides are independent of (z - zr)and equal to a constant A . Integration of the 1.h.s. of eq. 14 gives:
Thus, eq. 12 can be written: u,
u=-K
z- z , I n ( 7 )
The numerical value of Von Karman's constant K must be determined experimentally, as discussed in section 3.4, and so must the functional form of the roughness length zO,as discussed in section 3.3. Jackson (1981) showed that the reference height z, for a rough bed is the level at which -roappears to act, and for many bed geometries is approximately equal to 0.7H. Most measurements in the sea are made a t heights much greater than z,,so it is usual to neglect this and write eq. 16 in the simpler form:
u=-UK.
In(:)
199 1000 53
5 4
100
(cm)
10
1
0
I
I
I
I
I
I
10
20
30
40
50
€0
u
001
’ Current velocity
(cm 5’)
Fig. 5.3. The velocity profile within the logarithmic region, averaged over 30 min at Stn. I . Start Bay.
Fig. 5.4. Schematic illustration of a slightly curved velocity profile (arbitrary units). The true velocity profile has been measured at 7 heights which are in fact outside the logarithmic region indicated by the lower part o f the solid line. A regression line (dashed) passed through them gives an apparently good fit. but if treated as a logarithmic profile would overestimate u. by 50% and z,, by a factor o f 10. The curvature may be caused by bedforrns. density stratification, acceleration. etc.
Velocity profiles having the logarithmic form, eq. 18, have been measured under a great variety of experimental conditions in the sea. An example is shown in Fig. 5.2, where the upper curve depicts a regression line of the form of eq. 18 passed by Chriss and Caldwell (1983) through their velocity measurements on the Oregon shelf for heights in the range 1-59 cm. Another example, taken in the bottom 2 m at Stn. 1 in Start Bay, is shown in Fig. 5.3 withthe z-axis plotted logarithmically. Profiles of this form measured near the seabed can be used to obtain u. (and ) the slope, and z o from the intercept, of a regression line of U on hence T ~ from In(z). The logarithmic profile has become so well established in the literature that i t is tempting to f i t it to any near-bed velocity profile. However, this can be misleading, as a slight curvature in the profile, due perhaps to topography, acceleration or stratification, can easily be missed, which can cause the apparent u , and z,) to be very different from their true values (Fig. 5.4). Estimation of the errors in u+and zo obtained from a log profile is discussed by Wilkinson (1983). 3.2. Turbulence
Within a few metres of the seabed, occupying the region known in the meteorological literature as the surface layer, the decrease with height of the shear stress is
200
TABLE 5.3 Ratios of turbulence quantities in the atmospheric and marine surface layers. For the marine measurements u. was taken as ( - ~ ) ’ ” at the measuring height. Our measurements are averaged over heights between 30 and 140 cm, those of Heathershaw (1979) at 100 and 150 cm, and those of Bowden and Ferguson (1980) between SO and 210 cm Source
Site
a,,/u.
av/u.
U,.,/U.
r/E
Ariel and Nadezhina ( 1976)
Atmosphere Start Bay, Stn. 1 Start Bay, Stn. 2 Weymouth Bay Irish Sea Irish Sea
2.4 2.6 2.3 2.4 2.9 2.6
1.9
1.2 1.4 1.2 1.2 1.s 1.2
0.19
Our measurements Heathershaw (1979) Bowden and Ferguson (1980)
~
I .8 ~
1.8
~
0.20 ~
~
0.18
only a small fraction of ro. Thus to within the errors of measurement the Reynolds stress - p z is more or less constant here. Bowden and Ferguson (1980), for example, found no significant difference between Reynolds stresses measured simultaneously at heights of 50, 100 and 200 cm above the bed of the Irish Sea. The direction -of _the stress - is also constant with height ( - p G = 0). In addition, the variances u 2 , v 2 and w 2 are almost constant with height and proportional to u?. The ratios uu/ur, U J U , and u,/u, from atmospheric and laboratory experiments by many authors were compared by Ariel and Nadezhina (1976) and their means are quoted in Table 5.3, together with our values in Start and Weymouth Bays, and Heathershaw’s (1979) and Bowden and Ferguson’s (1980) values from the Irish Sea. The agreement is quite good. The ratio of r x 7to the turbulent kinetic energy E is also shown. The value of this ratio is often assumed to be constant under widely varying conditions. However, observations by Gordon and Dohne (1973) in the Choptank Estuary, U.S.A., and Heathershaw (1979) in the Irish Sea, both indicated that r x 7 / E became smaller at times near to slack water. This may be due to E containing rectified “noise”, e.g. surface-wave motion, but could alternatively be due to turbulent energy remaining at the time of flow reversal. The height to which the logarithmic layer extends is often quoted as being about 0.18. However, this is very much an order-of-magnitude estimate, as in practice it depends strongly on the free-stream conditions. The logarithmic layer becomes progressively thinner as the effects of unsteadiness, rotation, stratification, etc., become more marked. The height to which the turbulence quantities remain approximately constant also depends on the free-stream conditions. If the flow has no vertical density stratification, the logarithmic layer may extend well beyond the surface layer. If there is strong density stratification, however, only the lowest part of the surface layer will exhibit a truly logarithmic velocity profile. Expressions for these heights under different conditions are discussed further in section 4. 3.3. The seabed roughness length
An estimate of the bed shear stress ro often has to be made from a current-velocity measurement made at a single height, most commonly that at 1 m above the bed,
20 I
U , , , . This can be done either by using the quadratic friction law, eq. 5 , or the logarithmic velocity profile, eq. 18. In the former case it is necessary to know the value of the drag coefficient (C,,,, if the velocity is U,,,), and in the latter, the value of zo. The two are directly related via eq. 18 by:
The two methods are thus mathematically equivalent, but the logarithmic profile method has the advantages that it can be applied to measurements made at any height within the logarithmic layer, and that zo can be predicted directly from the bed surface geometry by an equation of the form of eq. 17. An appropriate value of z o or Ciao can be assigned provided the nature of the underlying seabed is known. Measurements of zo from logarithmic profiles by many authors over a variety of substrates were collated by Heathershaw (198 1). An extended and reworked version of h s results is shown as Table 5.4, which shows the geometric mean value of z,), and the corresponding ClOO, for various categories of seabed type. Also shown is a factor which expresses the amount of variation found in zo for each category; multiplying and dividing the mean by the variation factor corresponds to one standard deviation either side of the mean. Mixtures of grain sizes have relatively small values of zo, as the fine grains fill the spaces between the coarse grains. However, they also have large variation factors, because the degree of filling which can take place will vary with the relative proportions of the different grain sizes. The large mean zo for rippled sand is due to form drag on the ripples, whch is the major source of roughness felt at heights greater than a ripple wavelength. Values of C,,, vary through a factor 4 from the smoothest to the roughest substrate, illustrating the dangers of using a constant drag coefficient for all substrates. TABLE 5.4 Typical values of the roughness length zo and the drag coefficient C,,,,, for different bottom types. The data is drawn from 18 sources, 13 of which are cited in the table by Heathershaw (1981) on which the present one is based. The remainder are: Lesht (1979), the values given in Table 5.1, and unpublished values measured by the Institute of Oceanographic Sciences, Taunton. The geometric mean z o for each category is quoted, a s this most accurately reflects the logarithmic way in which L,, is measured. The standard deviation of In( z,) is In(variation factor). The quoted “number of observations” takes no account of the amount of data comprising an individual observation; nor was any account taken of the prevailing hydrodynamic roughness regime Bottom type
20
Mud Mud/sand Silt/sand Sand (unrippled) Sand (rippled) Sand/shell Sand/gravel Mud/sand/gravel Gravel
0.02 0.07 0.005 0.04 0.6 0.03 0.03 0.03 0.3
(cm)
Variation factor
c,oo
-
0.0022 0.0030 0.00 16 0.0026 0.006 1 0.0024 0.0024 0.0024 0.0047
4.1 2.0 1.3 4.5 6.7 3.0 1.6
NO. of observations
202
Table 5.4, in effect, describes the functional dependence of z,, on bed geometry introduced in eq. 17. The function depends on u , H / v and hence, if H represents the grain diameter d , on the hydrodynamic roughness regime. For smooth turbulent flow, laboratory experiments show that: zo =
V ~
9u*
There is not yet sufficient evidence to confirm this expression for the sea; for example, the measurements by Chriss and Caldwell ( 1 983) described earlier gave values of zo ranging from v/4u* to v/272u,. The experimental values of z , ) in Table 5.4 were taken from the literature without any classification into hydrodynamic roughness regimes, and thus should be viewed as being representative of the “typical” flow over each substrate (see Table 5.3), with some being smooth, some transitional, and some rough. For rough turbulent flow, the roughened pipe flew experiments of Nikuradse (1933) indicated that: z,) = d/30
(21) while Kamphuis (1974), from measurements in a rectangular channel roughened with gravel of various sizes, obtained a value 15 for the divisor. This is a particularly simple form of eq. 17, in which the only relevant dimension of the bed geometry is the grain diameter. The values of z,, quoted in Table 5.4 are all considerably larger than would be predicted by eq. 21 with either divisor, suggesting that naturally settled sediments may have a less even distribution than has been used in laboratory studies. A quite general form of eq. 17 was given by Wooding et al. (1973) based on both laboratory and atmospheric data. When applied to a rippled seabed with ripple height H and wavelength X their expression simplifies to:
Equation 22 can be tested against the measurements made by Soulsby et al. (1983) over 300 p m sand at Stn. 1 in Start Bay. A light shadowing technique used to measure the ripple shape showed that X = 20 cm and H = 3 cm, for which eq. 22 predicts zo = 0.4 cm. This is in acceptable agreement with the value z ~= , 0.7 cm obtained from simultaneously measured velocity profiles, in spite of eq. 22 being used outside its quoted range of applicability. Measurements of z,, made by Smith and McLean (1977) well above sandwaves in the Columbia River yielded z,, = 1.5 cm for separated flow over features with X = 82 m, H = 3.2 m compared with a predicted value of 6.6 cm from eq. 22; and zo = 6.6 cm for unseparated flow over features with X = 96 m, H = 2.1 m compared with a value of 1.9 cm from eq. 22. Thus eq. 22 can predict the order of magnitude of z,,, which is usually good enough, as i t is generally the logarithm of z,, which appears in expressions. Typical dimensions of current generated ripples are h = IOOOd, with a steepness of H/X = 1/7 (Yalin, 1977). Substituting these into eq. 22 allows z o for rippled sand to be expressed directly in terms of grain diameter as: z , , = 19d
(23)
203
The value z , = 0.6 cm given in Table 5.4 thus corresponds, using eq. 23, to 320 p m sand. If sediment is in suspension this can further modify z,,. Smith and McLean (1977) found that their field measurements fitted the expression: p a , ( u,z - u.J 2
zo =
(24)
d P, - P )
where pu?, is the threshold shear stress for bedload movement, p, is the sediment density, and the constant a , = 26.3. Dyer (1980) also found good agreement with eq. 24 with a , = 26.3 at low sediment-transport rates. At higher rates the picture became complicated, because the shape of the ripples was varying with the tide, and the corresponding change in z o was comparable with that due to sediment suspension. It is commonly found that a hierarchy of bedforms i s present in areas of strong sediment transport. Thus large sandwaves ( A - 100 m) may have dunes ( A - 10 m) on their backs, with ripples ( A - 30 cm) in turn on their backs. Smith and McLean (1977) showed that each class of bedform acts as topography at heights z << A , but at heights z 2 A it acts as roughness. Thus in the lowest few centimetres the sand grains act as roughness with an appropriate zo and u,. Above this, the ripples produce a larger zo and u,, which now additionally contains the form drag of the ripples. Higher up, the dunes provide a still larger zo and u., and so on. This gives rise to segmented velocity profiles. For the purpose of sediment-transport prediction it i s the shear stress acting on the grains alone (skin friction) which is relevant. Thus if estimates of T()have been made from profiles measured more than a few centimetres above the bed, any form drag due to ripples or larger features must be subtracted before inserting the shear stress into a (flat-bed) sediment transport formula. 3.4. Von Karman’s constunt
Before u* can be estimated from a measured velocity profile it is necessary to assign a value to K . It can be obtained from eq. 18 using velocity profile data together with a simultaneous independent value of T(). Laboratory experimenters have generally obtained K = 0.40 or 0.41, while the experiments of Businger et al. TABLE 5.5 Measured values of Von Karman’s constant
K
Source
Site
K
Many authors Businger et al. (1971) Schotz and Panofsky (1980) Charnock (1959) Smith and McLean (1977) Chriss and Caldwell ( 1983) Soulsby and Dyer (1981) Soulsby and Dyer ( 1 98 1)
Laboratory Atmosphere Atmosphere North Wales coast Columbia River Oregon Continental Shelf Start Bay, Stn. 1 Weymouth Bay
0.40-0.4 I 0.35 0.35 0.39. 0.46 0.38 0.43 0.400 0.392
204
(1971) and Schotz and Panofsky (1980) have led atmospheric workers to favour K = 0.35. Relatively few measurements of K have been made in marine boundary layers, but a summary of these (Table 5.5) shows closer agreement with the laboratory than with the atmospheric figure. Thus a good general purpose marine value appears to be K = 0.40.
4. THE OUTER PART OF THE MARINE BOUNDARY LAYER
4.1. General
Seven types of outer layer are identified here, based on an extension of the classification given by Bowden (1978). In practice, of course, a boundary layer may combine features of several of these idealised types. A relatively simple theoretical basis is presented for some of the types, in order to put the experimental observations into context. Fuller theoretical treatments are given by Sverdrup (1927), Long (198 I ) and Prandle (1982), with varying degrees of sophistication in the eddy-viscosity distribution used. 4.2. The plunetury boundury luyer
The influence of the earth's rotation on a steady, uniform, unbounded, unstratified deep flow causes the current and shear stress to veer progressively clockwise (in the northern hemisphere) with increasing distance from the bed. At the outer edge of the boundary layer they merge with the geostrophic flow, in which friction is absent and the driving pressure gradient is balanced by the Coriolis force. A planetary boundary layer does not grow in space or time as most other boundary layers do. It has been extensively studied in the atmosphere, and will occur in the sea well away from coasts, beneath currents which are steady over times comparable with the inertial period, f - ' . Here f = 2 D sin Ic, is the Coriolis parameter, D the earth's angular frequency of rotation, and Ic, the latitude. For simplicity we will assume that 0. In the southern hemisphere the magnitude we are in the northern hemisphere,!> will be the same, but the direction of veering will be reversed. Current structure
The equations of motion for the horizontal velocities U and V in the x and y directions, respectively, driven by a steady surface slope in the x direction, are:
where { is the water surface elevation, and g the acceleration due to gravity. Note that the convention for the direction of the x-axis is different from that used in section 1. Eddy-viscosity assumptions of the form of eq. 6 are introduced for r X 1and
205 ry,,
and it is further assumed that K , ,
= K Y L ,so
that eqs. 25 and 26 become:
We define a complex velocity vector Q = U + iV, and a complex slope S = d { / d x noting that a { / a y = 0 in the present problem. Then multiplying eq. 27 by i and adding it to eq. 28 gives:
+ id{/ay,
At the bed, taken as z =zo, the boundary condition is Q = 0. Outside the boundary layer the frictional term in eq. 29 is negligible, so that the geostrophic current Q, is given by: Q,=-Sig
f
Note that because S is real, Q, is entirely imaginary, so that U, = 0 and the current is directed along the positive y-axis in the northern hemisphere, i f d { / d x > 0. The upper boundary condition is then Q + Q , as z + 03. If K,, is taken as constant with height, the solution of eq. 29 with its boundary conditions is the familiar Ekman (1905) spiral. However, we will choose to use the more realistic eddy viscosity distribution given by eq. 7, K , , = K U . Z . The bed shear can be obtained from the solution of eq. 29, and the value stress 7, = ( r x , + i-ryL)II=z,, of u* to be used in eq. 7 is given by I ~ ~ / p 1 ' /Equation ~. 7 agrees well with measurements in the lower part of the boundary layer, but in reality K , , reaches a maximum at a height between about 6,/3 and S , / 2 , and then decays with z to its molecular value at z = 6,. However, eq. 7 provides a good compromise between realism and simplicity, and thus is adequate for the present purpose. The solution of eq. 29 with this eddy-viscosity distribution and the same boundary conditions was given in a different context by Smith (1977), and in our notation is:
+ ker *to + kei2to
1
ker [ kei to- kei [ ker to
ker 5 ker to kei 6 kei to
+z(
+
ker2,$, k e i 2 [ ,
The Kelvin functions ker and kei are tabulated by Abramowitz and Stegun (1964). The solution is given in terms of the non-dimensional variables 2( fz/~u.)'/~and to= 2( fz,/~u,)'/~. In this form the shape of the velocity profile depends only on one combination of the input parameters, namely to.This combination is more often written as the surface Rossby number, Ro, = u./fzo = 4/~((:. The height z is scaled by the length u./f, so the boundary-layer thickness and logarithmic layer thickness will also be proportional to u./f, as well as possibly depending on t,,(or Ro,). The normalised speed A , = ~ Q ~ / ~ Qand m ~the , direction relative to the geostrophic flow direction G U = tan-'[Im(Q)/Re(Q)], are plotted on semilogarithmic axes in Fig. 5.5. Here, Re and Im represent the real and imaginary parts respectively of a
<=
206
-.
I_
Y
N
c
d (1
N )rJ,
5
-4
10 -
-10-z
-3
lo6-
Co=10
~.,=10-~
4,
Fig. 5.5. The normalised speed A , , and flow direction for different values of the roughness parameter to.
+,,. obtained from the linear eddy-viscosity inodcl.
207
Ibl
35 7 (m)
30
25
20
15
I0
5
0
F,'ig. 5.6. a. The variation of the speed IUI and bearing B,, with depth through [he boundary layer fornied beneath drifting Arctic pack-ice. with z measured vertically downwards from the underside of the ice. I h t a of McPhee and Smith (1976). b. The modulus 171 and bearing 0, o f the shear stress from the same experiment.
+(,
complex number, and the convention is that a positive value of corresponds to an anticlockwise rotation. Four values of 6, and hence Ro, are used to cover the range likely to be encountered in shelf seas, as well as values which will be needed later in the discussion of oscillatory, and oscillatory-planetary. boundary layers. Assuming K = 0.40, the values of Ro, which correspond to the selected values of 5,) are: 0 - 10-1, R ~=, 1 0 7 ; to= 1 0 - 2 , R ~ =, 105; = 10-3. R ~ )=, 1 0 7 ; 5,) = 10-4, RO, = 10'. The speed A , is proportional to In E2, and hence In z , from the bed up to a height of about E 2 = 1 or, for K = 0.40, I = O.lu,/f. There is a slight dependence on ((,, with the thickness being progressively less than the above figure as 5,) decreases. Above this, as E 2 (and hence z ) increases A,, increases more slowly than In z , overshoots the geostrophic value slightly, and then tends to the geostrophic value .Au = 1 . The direction is more or less constant from the bed up to a height of about E 2 = l o - ' or z = O.Olu./f. The flow near the bed is directed anticlockwise of the geostrophic direction by an amount which increases with to.Above this, as 6' (or z ) increases, the direction veers progressively clockwise to attain the geostrophic direction, with a very slight overshoot. The height 6, at which both the speed and and equal direction attain their geostrophic values is more or less independent of to about E 2 = 10 or 8 , = u./f. An expression for the total amount of veering was derived without recourse to assumptions about an eddy-viscosity distribution by Csanady ( 1967), who postulated that near the bed a logarithmic region should exist, while well away from the bed the profile should obey a velocity defect law. Matching these profiles in an overlap
+"
208
region yielded the expression for the angle S+ between the directions of geostrophic flow Q,: sin( S+)
7,
and the
Au. =
~
KIQJ
The constant A has to be determined experimentally, and Tennekes (1973) recommends A = 5. The same arguments showed that the thickness of the boundary layer is:
S"
Cu*/f (33) where C = 0.3 is a typical value obtained in atmospheric experiments. This is rather smaller than the value C = 1 indicated by the linear eddy-viscosity model, possibly =
because the model overestimates K,, in the upper half of the boundary layer. Probably the most complete set of measurements made to date in a planetary boundary layer are those made by McPhee and Smith (1976) in the boundary layer beneath drifting Arctic pack-ice. This comprised a well-mixed layer 35 m thick of nearly neutral stability, and measurements of the three velocity components were made at each of eight levels throughout it. The speed IUI and azimuth O,, of the mean current (Fig. 5.6a) have qualitatively similar profiles to Fig. 5.5, noting that Fig. 5.6a is plotted on linear axes, and that the convention for the compass bearings 8, is that a positive bearing corresponds to a clockwise rotation, i.e. the opposite of The speed becomes more or less constant beyond z = 12 m, but veering extends out to z = 32 m, corresponding to C = 0.44 in eq. 33 with u. = 1.0 cm s- I . f = 1.4 X l o p 4 s-I. There is little veering between the ice surface and z = 16 m, but appreciable veering between 16 and 32 m, giving a total veering of 24" top to bottom. This would correspond to A = 3.8 in eq. 32 with u* = 1.0 cm s p l and IQ, I = 23.6 cm s-I. There were, however, non-negligible effects due to the topography of the underside of the ice, which caused the profile to differ from its horizontally homogeneous form.
+,.
Turbulence The shear-stress profile can also be obtained from the linear eddy-viscosity model. The eddy-viscosity assumption, eq. 6, gives:
Inserting the derivative of eq. 31 with respect to z,together with eq. 7 for K,,, into eq. 34 yields: 7=
pKu*Qmt
2( ker25, + k e i 2 t , )
[-
(ker'tker[,+ kei'tkert,)
+ i( ker't kei t o- kei't ker to)]
(35)
where ker' and kei' are the first derivatives with respect to 5 of ker and kei, respectively. The non-dimensional amplitude A T= I T ~ / ~ ~ K and U * the Q ~direction ~,
209
70
60
50
40
30
-. Y 3 N
20 , u ICF
10
7
0 5 0"
Fig. 5.7. The non-dimensional magnitude A , of the shear stress, and its direction & obtained from the linear eddy-viscosity model, for different values of the roughness parameter 5".
J
I
I
, , I
I
I
I
I I I I I
I
I
I 1 l l l l
I
I
1
I
l
1
l
I
-0
10
10 0
=
2
r -
- --
III
-
0
w
0"
-
-
lo4=
lo5,
,, ,
I , ,
I
I
I
I
I
I l l [
I
r
10
10
1
1
1
1
'
1
lo-'
50
Fig. 5.8. The drag coefficient C , ( t 0 ) and the parameter Z,,, as functions of
(
to
I
2 10
cbT = tan---’[Im(7)/Re(T)Jrelative to the direction of the geostrophic current are plotted on linear axes in Fig. 5.7. The value of AT at the bed is:
-K-
where C , by 1
c,,(to1 =
(36)
ICY2
= u?/IQ,1*
K
is the geostrophic drag coefficient. From eq. 35 C,, is given
2t:
4 ( k e r 2 t ,+ k e i ’ t o ) 2
+ kei’t,kei t,l)2
[(ker’tokerto
+ ( ker’tokei t o- kei’<,ker 6,) 1’1 and is plotted against 5, in Fig. 5.8. The value to C, = 6.2 x lo-’ for to= C , = 0.46 x and 5, contain the relatively inaccessible quantity calculate in a practical problem is the quantity Zo,
(37) of C,, increases with to*from for to= 10 I . The parameters 6 u.. A much easier parameter to defined by:
which contains instead the geostrophic current speed IQ,I. The plot of against 6,) i n Fig. 5.8 allows toand C,, and hence 6, to be derived from the readily estimated quantities f , z o and IQ,1. Referring back to Fig. 5.7, the non-dimensional stress A , decreases rapidly with height just above the bed, reaching 90% of its bed value at a height of t 2= 0.4 for all the t ovalues. Taking this to indicate the top of the “constant stress” surface layer gives a predicted surface layer thickness of O.O4u,/f. This compares with an expression given by Busch (1973), which when combined with eq. 32, predicts a surface layer height of 0.008u./f. The height at which the stress is negligible compared with its surface value is more or less independent of toand equal to about &* = 50. This corresponds to 6,= 5u,/f, a value somewhat larger than a,,, as discussed in section 1. However, the model’s overestimation of K , , at large heights will exaggerate the value of 6, even more than that of 6,. The value of GT at the bed is the same as that of +,,; thus the current and stress are aligned at the bed as expected. The stress veers strongly clockwise with height, executing about 3/4 of a revolution between the bed and z = S,. Above z = ST the direction has little meaning, as the magnitude of the stress is very small. The veering is almost independent of to for all z, apart from a small constant difference necessary to make GT = GU at the bed. Because the stress veers more rapidly than the current, it is generally directed clockwise of the current at the same height, though the angular difference will be more than 180” for z >, 0.56,. Measured profiles of the magnitude 17) and bearing 0, of the Reynolds stress obtained under the Arctic pack-ice by McPhee and Smith (1976) are presented in Fig. 5.6b, where again 0, has the opposite rotation convention to G7.The features are
21 I 0.6 -
0.5-
+
X
0.4 -
-2
+
X
x
3
0.3-
+
X
+
X
0.2 -
+
X
X
0-
X I
I
I
1
2
3
-
I 4
LIZ/";
+ ++
X
0.1-
1 5
1
-
2
" /u,
2
2
Fig. 5.9. The variation with depth of the turbulent variances through the boundary layer formed beneath drifting Arctic pack-ice, with z measured vertically downwards from the underside of the ice. Data of McPhee and Smith (1976).
qualitatively similar to those indicated by the model in Fig. 5.7. A marked decrease in 171, accompanied by strong veering in O,, occurs just below the ice. The stress tends to zero at z = 35 m, corresponding to 8, = 0.49u,/'. This is only slightly thicker than 6, in this case, because the boundary layer was capped by a density discontinuity, with the water outside it being stably stratified. The direction 0, veers progressively clockwise with increasing z , through about 150" between the surface and the edge of the boundary layer. The features in the central portion of the profiles were again thought to be associated with- the _ large-scale topography of the underside of the ice. Profiles of the variances, u2, o2 and 7 normalised by u:, measured McPhee -in and Smith's (1976) experiment are presented in Fig. 5.9. At the surface u 2 > u2 > w 2 , and all three components decrease with z to the outer edge of the boundary layer at 6,: = 0.49u,/f. The deviations from a smooth decrease with z are again probably due to the topographic effects mentioned above. 4.3. The oscillatory boundary layer
Currents on the continental shelf, whether they be tidally or meteorologically driven, are generally unsteady. There is thus rarely time for a steady planetary boundary layer to develop fully. In the tidal case the currents are driven by an oscillatory surface slope of fairly regular, though not necessarily sinusoidal, form, while meteorological forcing causes unsteady currents of a less regular nature. Restricting our attention initially to oscillatory forcing, we consider an unstratified, uniform, rectilinear flow. This occurs in the sea in channels and near to coastlines, because the effect of the Coriolis force is nearly balanced at each instant by a transverse pressure gradient to maintain no flow normal to the coast. As we will see in the next section this is only approximately true. Strictly speaking, the effects of the earths rotation can only be neglected if u >>f,where u is the angular frequency of oscillation; but for tidal motions u is generally similar in magnitude to f.
212
However, it is useful to examine the purely oscillatory boundary layer, partly because its existence is commonly assumed in the modelling of tidal estuaries, and partly because it makes a helpful prelude to the study of the oscillatory planetary boundary layer. In a very shallow flow the inertial forces are negligible compared with the friction, which balances the surface slope at each instant giving a quasi-steady flow. When the flow is deep, however, the inertial forces become comparable with the frictional forces well away from the bed, causing the currents here to lag behind those near the bed. In sufficiently deep water the boundary layer is analogous to that formed under surface waves, the only difference being that the period is much greater. Current structure The equation of motion for the velocity U(z, t) in a deep, uniform, unstratified flow over a flat bed driven by a periodic surface slope in the x direction is:
For a sinusoidal time-dependence of angular frequency u, the velocity and surface slope can be written as U = Re[U exp(iut)] and a.(/ax = Re[S exp(iut)], where U and S are the complex amplitudes of U and d { / d x , respectively. Substituting the time-dependence into eq. 39, together with the eddy-viscosity assumption eq. 6, gives:
The condition of n o slip at the bed is U = 0 at z = z o . Outside the boundary layer the frictional term in eq. 40 is negligible, so the oscillatory free-stream velocity amplitude here is given by: ( I , =
ig -s U
The upper boundary condition is then U + Urnas z -+ co. The form of eq. 40 and its boundary conditions is mathematically identical to that of eq. 29 and its boundary conditions. Thus a close analogy exists between the planetary and oscillatory boundary layers, and if the same form for K,, is used in both, then the solutions for the velocities and stresses will be the same. However, the interpretation is different. The complex notation in the planetary case serves to combine the x and y directions, while in the oscillatory case it combines the in-phase and out-of-phase parts of the motion. In the planetary case IQI is the speed and +L, the direction of the current relative to the geostrophic flow direction, while in the oscillatory case the equivalent quantities IU I and represent respectively the amplitude of the velocity, and its phase relative to the free-stream flow. An anticlockwise rotation (#,, > 0) in the planetary case corresponds to a phase lead in the oscillatory case. A wide variety of forms for K , , have been used. Most of them have kept K , , constant in time, as this maintains eq. 40 linear in exp(zat) and avoids the generation of higher harmonics. The results of a laboratory experiment on the
+"
213
oscillatory boundary layer by Jonsson and Carlsen (1976) showed that K , , was in fact not only time-varying but was actually negative at certain phases of the oscillation. However, this can only be modelled using quite sophisticated techniques. The simplest assumption, that K,, is constant with z as well as in time, leads to the Stokes shear-wave solution given in Lamb (1975). Various more elaborate analytical representations are reviewed by Knight (1 978). We will use a time-independent form of K,,, for the reasons given above, with a linear z dependence of the form of eq. 7 as used in the planetary boundary layer. A common choice for the velocity scale in K,, is u,,,, = 17,,/pI'/', where /,TI is the amplitude of the sinusoidally varying bed shear stress. The eddy viscosity is thus given by: K,,
(42)
= KU*,,Z
The solution for the oscillatory boundary layer is then given directly by eq. 31, substituting u for f, U for Q,and Urnfor Q,. The interpretation of Fig. 5.5 in terms of the vertical variation in velocity amplitude and phase is as given above. The parameter tothat determines the form of the profiles can alternatively be written as the ratio u,,/uzo = 4/~,$:, which fulfils the same sort of role as the surface Rossby number. Instantaneous velocity profiles at each phase can be constructed from the amplitude and phase. Figure 5.10 shows an example for numerical values of u / IP&l -10
0
10 206"
324"342'360"
10000
10
1000
1
100
10
I
5'
z(cm) 10
10
1
10
0.1
1 0 ~ ~
-30
-20
- 10
0
10
20
30
Velocity(cm s ~ ' )
Fig. 5.10. Instantaneous velocity profiles through half a cycle of the linear eddy-viscosity model simulation of oscillatory flow. Both dimensionless and dimensional axes are presented, the latter corresponding to u = 1.40X s-I, zo = 0.09 cm, and u., = 1.26 cm s - ' . The profiles move from left to right in time, separated by a phase interval of 8O, or about 17 min in the dimensional units.
214
u = 1.40 x l o p 4 s - ' (the M, tide), zg=0.09 cm, a n d u*,= 1.26 cm s - ' , which U,l = 30 cm s . I , corresponds to to= l o p 2 . The free-stream velocity amplitude is l a n d the corresponding drag coefficient is C, = 1.8 x l o p 3 .The free-stream velocity lags behind the bed shear stress and the bottom current by 9". or about 19 min. The boundary layer is about 90 m thick, corresponding as in the planetary case to t 2 =10, or 6, = u * J u . The logarithmic layer varies from about 50 cm thick (0.006u,,/u) near to slack water, to about 20 m thick (0.2u.,/a) at peak currents. The choice of u*,,as the velocity scale used in K , , , eq. 7, gives optimum accuracy a t the time of peak.current, and hence should yield a reasonably accurate value for the maximum of q,,as might be required for sediment-transport predictions, for example. However, at all other times in the tidal cycle it will be an overestimate of K x , . If the eddy viscosity is governed at all times by the instantaneous value of the sinusoidally varying bed shear stress, then the tidally averaged value of K , , is based on U . , ( C O S ' / ~ U ~ ) , where () denotes a tidal average. This gives K , , = ~(0.763~,,,,);. As an experimental test of this expression we examined the data of Jonsson and Carlsen (1976, test No. I), who measured velocities at a range of phases, for each of a range of heights above the artificially roughened bed of a tunnel in which water was oscillated sinusoidally. We averaged their tabulated values of K , , through the complete cycle for each height z, and plotted the results against u*,,,z. where u*,,,was obtained from the measured bed shear stress. Near to the bed the curve is linear. and follows the relation:
K,,
=
K(o.75U*,)Z
(43)
where we have assumed K = 0.40. This expression would be more suitable than eq. 42 in calculations of mean tidal energy dissipation, for example. We turn now to non-sinusoidal time-variations in velocity, as might be encountered at coastal sites with complicated tidal curves, or in the open sea for meteorological induced currents. An expression for the velocity profile which is applicable t o a general time dependence was derived by Soulsby a n d Dyer (1981). Rather than making a n eddy-viscosity assumption, they used dimensional arguments which closely followed those of Monin and Obukhov that led to the form of the velocity profile in a stratified medium (Monin and Yaglom, 1971). An accelerating flow can be characterised by the rate of change, ir,, of the friction velocity, and this can be combined with U. to give a characteristic length scale:
T h e modulus sign is introduced to ensure that if the flow is accelerating in either the + x or - x direction then A > 0, while if i t is decelerating then A < 0. A Taylor expansion of the functional form for 6'U/az leads, after integration with respect t o :, to a velocity profile of the form:
The constant y in eq. 45 has to be determined experimentally. Velocity profile measurements in the bottom 2 m in Start and Weymouth Bays gave y = 0.04.
21s
Comparison with values obtained from analysis of Jonsson and Carlsen’s (1976) laboratory profiles suggested that y may be a slowly decreasing function o f 8 / z ( , . Equation 45 tends to the usual logarithmic form, eq. 18, near the bed. For very small values of z/A the flow is effectively steady; Soulsby and Dyer (1981) arbitrarily chose Iz/AI < 0.005 as defining near-steady flow. The validity of eq. 45 is limited to heights for which z i * is approximately constant over a time interval equal to z / y u , . This restricts its range of applicability if the time dependence is sinusoidal. I t can be seen from eq. 42 that when the flow is accelerating ( A > 0) the velocity is less than the logarithmic value at that height by an amount which increases with z. This is qualitatively true at heights greater than those for which eq. 45 is valid, and can be sufficient to give rise to a midwater maximum in the profile. Conversely, when the flow is decelerating ( A < 0) the velocity is greater than the logarithmic value by an amount which increases with z . As an example of field measurements in a n oscillatory boundary layer we turn to those from the string of current meters deployed at Stn. M in Start Bay (see Table 5.1). The velocity at each height was phase-averaged over eight tidal cycles spanning
\
z (m)-
Decelerating
- - - 02 *
~7
10 *
8-
*
6-
Accelerating
1~~~~~
0
4-
*
.
2-
* 1
,
-q/////f; 80 1
a
0
5 20 3 0
0
2
4
6
8
10
12
Hours after High Slack Water
Fig. 5.1 I . a. Half-hour averaged profiles of current speed taken over an ensemble average ebb tide at Stn. M in Start Bay. The number above each profile denotes the time in hours after High Slack Water. Profiles 7, 8 and 9 are plotted as the negative of the speed for clarity, and to facilitate comparison with Fig. 5.10. b. The ensemble average current speed at a height of 13 m over a complete tidal cycle.
216
spring tides, to produce a representative semidiurnal tidal cycle with a correspondingly small random fluctuation. Vertical profiles of current speeds, each representing a 30-min average from the representative tidal cycle, are shown in Fig. 5.1 la, with every other profile omitted for clarity. The current displays a very non-sinusoidal time dependence at this site (Fig. 5.11b), with the ebb tide having a much longer duration than the flood because of the formation of an eddy off the adjacent headland. The measured profiles were made in a limited depth of water (20 m), and the lowest current meter was placed at z = 1.5 m. Nonetheless, the profiles in Fig. 5.1 la bear a qualitative resemblance to the central portion of the theoretical curves in Fig. 5.10. They also exhibit midwater maxima in the accelerating profiles, and convex-downwards curves in the lower portions of the decelerating profiles, as predicted by eq. 45. Logarithmic profiles passed through the readings of the meters at 1.5 and 2.5 m indicated that z o = 1 cm and u*,,, = 4 cm s - I . Turhulerice Shear-stress profiles for the oscillatory boundary layer are obtained from the model of the planetary boundary layer by making substitutions similar to those described above for mean velocity profiles. The non-dimensional amplitude and the phase lag of the shear stress are then given by Fig. 5.7. The phase changes rapidly with height, to the extent that in the upper part of the boundary layer the friction acts in the same direction as the surface slope. This gives rise to the existence of velocities exceeding the free stream velocity, the “overshoot” commented on earlier. Measurements of the turbulence characteristics of a tidal estuary were made at a
22
T
DECELERATING E B B TIDE
DECELERATING FLOOD TIDE
METERS ABOVE BOTTOM
Fig. 5.12. Distribution of turbulent intensity with depth at different phases of the tide in the Choptank Estuary. Circles indicate au/U; triangles indicate a,/U. (Reprinted from Gordon and Dohne. 1973, J. Geophys. Res., 78: 1971-1977, copyrighted by the American Geophysical Union.)
I
0
1
2
3
4
I
I
I
5
Fig. 5.13. The variation of shear stress with depth in a tidal flow at half-hourly intervals referenced to high water, measured off the North Wales coast. Ordinate: ( h - z ) / h ; abscissa: T ~ (dyne ~ . cm-2)-in o u r notation. (After Bowden et al., 1959, reprinted with permission from the Geophysical Journal o f the Royal Astronomical Society.)
number of heights by Gordon and Dohne (1973). Profiles of a,/U and uw/U (Fig. 5.12) show larger values near the bed on the decelerating phase of the tide than on the accelerating phase. Gordon (1975) further demonstrated that near to midwater there was an appreciable phase lag of both the turbulent kinetic energy and the Reynolds stress relative to the current speed at the same height. Bowden et al. (1959) deduced the shear-stress distribution from measurements of the current at various heights above the bed off the coast of North Wales (Fig. 5.13). These also showed strong phase changes with depth, with the phase of the stress high up in the water column lagging well behind that near the bed. Direct measurements very close to the bed, such as our own and those of Bowden and Ferguson (1980), do not display a detectable phase shift in either the stress or the variance, as it is expected to be very small here. 4.4. Oscillatory planetary flow
Under conditions in which both the oscillatory and rotational aspects of the flow are important the behaviour of the boundary layer becomes rather complicated. Both the phase and direction of the current vary with height, giving rise to time-dependent veering. This case is discussed at some length here, because i t is the most widely distributed form of boundary layer on the continental shelf. Current structure The theory of a tidal oscillation on the rotating earth has been dealt with, among others, by Sverdrup (1927), Bowden et al. (1959) and Prandle (1982). The present derivation is similar to that used by Prandle. The equations of motion for the U ( z , t ) and V( z , t ) components of velocity in the x and y directions, respectively, in a deep, uniform, unstratified flow over a flat bed driven by time-varying surface slopes in
218
both the x and t h e y directions are:
dU
The shear stresses T~~ and T~~ have been modelled using the same eddy viscosity assumptions that were made in section 4.2. For a sinusoidal oscillation of angular frequency a the velocity components can be written as:
+ h, sin at V = a , cos at + h, sin at
U = a,, cos at
(48)
(49)
As in section 4.2, the velocity vector ( U , V ) is written in complex notation as Q = U + iV. At this point it is convenient to make use of the device introduced by Thorade in 1928 (see Defant, 1961) of dividing the velocity into rotary components by expanding cos a t as [exp(iat) exp( - iut)]/2, and sin at as [exp(iat ) exp( -iat)]/2i in eqs. 48 and 49. The complex velocity Q ( t )can then be written as:
+
Q=R+tR_
(50)
where:
R + = I[( a ,
+ b,) + i ( a ,
-
b,)] exp(iat)
and:
R - = i [ ( a u- h,) + i ( a , + h,)] exp( - i o t )
(52)
The quantity R + represents a velocity vector of constant magnitude whose direction rotates anticlockwise with frequency a, when viewed from above. The quantity R - likewise represents a velocity with a constant magnitude (generally different from that of R + ) which rotates clockwise with frequency u. The two motions combine via eq. 50 to give a velocity vector which describes an ellipse. The properties of the tidal ellipse can be derived from the clockwise and anticlockwise components, as illustrated by Prandle (1982). In particular we note that the maximum current attained, corresponding to the semi-major axis U' of the ellipse. is given by: (53) Ua= IR+I + 1R-I and the minimum current, corresponding to the semi-minor axis Uh is given by: uh
=IR+I-
IR-I
(54)
If lR+l> IR-1, then U , is positive and the net motion rotates anticlockwise. If 1R-I > lR+l, then U , is negative and, as R - dominates, the net motion rotates clockwise. If the phases of the anticlockwise and clockwise components relative to an arbitrary time origin are and +-, respectively, the phase of the net motion is given by:
++
+
= (k-++)/2
(55)
219
The direction (relative to the x-axis) of the maximum current is given by:
0 = (+++ +-)/2
(56)
Adding eq. 46 to i times eq. 47, and using the complex notation Q gives:
where the surface-slope terms have been combined in complex form as S. The slope is subdivided into anticlockwise (S,) and clockwise ( S - ) rotary components in the same manner as the velocities. Equation 57 can be separated into the independent anticlockwise and clockwise components, which after re-arrangement give, respectively:
and:
At the bed the boundary condition is R , layer the frictionless current is given by:
=R
=0
at z = zo. Outside the boundary
The upper boundary condition is then R , + R , , and R - - + R , - as z co.The form of both eq. 58 and eq. 59 is mathematically identical with that of eq. 29 and also with eq. 40. The boundary conditions are also identical to the previous cases. Thus a close analogy exists with both the pure planetary and the pure oscillatory boundary layers. It is important to note, however, that while the quantity (a + f ) is always positive in the northern hemisphere, the quantity (a - f ) may be positive or negative. To keep the discussion simple, but still reasonably typical, we choose again to consider only the northern hemisphere, but in addition to limit the latitude to values which keep ( a - f ) positive. For semidiurnal tides this means latitudes south of 74"N, and for diurnal tides, latitudes south of 28"N. With this restriction the factor multiplying iR- on the left-hand side of eq. 59 is negative, whereas in eqs. 29, 40 and 58 i t is positive. Equation 59 is analogous to a steady planetary boundary layer in the southern hemisphere; thus the amplitudes of the clockwise component of the motion are obtainable from the solution given in section 4.2, but the directions are the negative of those obtained previously. Solutions for the case of an eddy viscosity constant in space and time were obtained by Sverdrup (1927) and discussed further by Prandle (1982). However, we will concentrate on using the linear eddy-viscosity distribution used in the previous cases, namely K , , = K U , , , Z . The velocity scale chosen is the maximum value u*,,, --$
220 [al
(bl
1"
R
Fig. 5.14. Schematic illustration of current hodographs of: (a) the anticlockwise rotary component, R +: and (b) the clockwise component, R - , of the oscillatory planetary tidal flow. The shape of the individual hodographs is constant as they rotate, but the shape of the net current hodograph which is their sum varies periodically.
attained by the friction velocity during a cycle. As in the oscillatory boundary layer case, this will model the flow near the maximum velocity well, but a smaller value would be more accurate on average over a tidal cycle, and would give a better estimate of the tidal energy dissipation. Thus the anticlockwise component of velocity R , can be obtained from eq. 31 by substituting (a +f) for f , R , for Q, R , + for Q m + ,S , for S, and u*,,, for u,. The clockwise component is similarly obtained from eq. 3 1 by substituting ( u - f ) for f . R - for Q , R , - for Q m , S - for S , u e m for u , , and reversing the sense of rotation of the direction +". For the anticlockwise component, GU is positive at the bed and decreases to zero with height. From Fig. 5.14 we see that this corresponds to a velocity vector which spirals clockwise upwards. As the rotation of the velocity vector with time is anticlockwise, the velocity high up therefore lags that at the bed. For the clockwise is negative at the bed and decreases to zero with height, correspondcomponent, ing to a velocity vector which spirals anticlockwise upwards. As the velocity vector rotates clockwise with time, the velocity high up again lags that at the bed. Thus, irrespective of the relative sizes of the anticlockwise and clockwise components, it is always the case that the current progressively lags that at the bed with increasing height. The relative contributions from R , and R - can be obtained from eq. 60. For a semidiurnal tide in temperate latitudes, typically ( u + f )/( u - f ) = O( lo), so, for equal sizes of the rotary slope components S , and S - , the clockwise velocity amplitude IR, - 1 outside the boundary layer will be much larger than the anticlockwise amplitude IR,+I. Near to a coast where the currents are constrained to run approximately rectilinearly, IR,+I = IR,-I (as U , = 0, eq. 54), so that the amplitude of the anticlockwise rotating component of slope IS+Imust be much greater than the
+"
22 1
-60"
55"
N
50'
Fig. 5.15. The ratio of the anticlockwise to the clockwise rotary component of depth-average velocity, predicted by R. Flather's numerical model of the mean spring tide in the seas around the British Isles. The model is based on a grid with spacing 20' of latitude by 30' of longitude, extending to the shelf edge. Shaded regions have a net clockwise rotation of the current vector. Contours at 0.1. 0.2, 0.5, 1.0, 2. 5 and 10.
clockwise, IS- I. The ratio of the anticlockwise to the clockwise component of velocity can be obtained in a depth-averaged sense if the semi-major and semi-minor axes of the tidal ellipse are known; for example from a depth-averaged numerical model. Equations 53 and 54 then give the depth-average ratio:
where a circumflex denotes a depth-averaged quantity. Values of
0,and fib,kindly
222
140
120
100
-
80
E
1 1
' c c2
60
40
20
0 02
04
06
Normallsed amplitude
08
6
10
a
x)
Phase
Fig. 5.16. Example of the variation with height of: (a) the normalised amplitudes: and ( h ) the phases o f s I, i,,= 0.09 cnl and s- I. u = 1 . 4 0 ~ the rotary components of motion, for / =1.19X u.,, = 1.26 cm s I. Note that the negative of the phase of the clockwise component is presented.
provided by R. Flather from his depth-averaged numerical model of the mean spring tide on the continental shelf around the British Isles (similar to the model described by Flather, 1976), were used to compute the geographical distribution of the ratio lR+l/lk-l, (Fig. 5.15). Rectilinear flow (ratio = 1 ) is found in many offshore regions, as well as near coastlines. Roughly equal areas of the sea are occupied by anticlockwise (ratio > 1 ) a n d clockwise (ratio < I , shown shaded) rotating motions. Prandle (1982) showed that the areas in which the rotation i s anticlockwise are those i n which the co-range lines around a n amphidromic point are roughly equally spaced. T h e ratio varies geographically from about 90% anticlockwise motion (ratio = 9) t o about 90% clockwise motion (ratio = 0.1). An example is shown in Fig. 5.16 of the amplitudes and phases of the rotary components at latitude 55"N ( f = 1.19 X l o p 4 s - ' ) for a semidiurnal tide ( a = 1.40 x lop4 s p I ) , with z ~=, 0.09 cm and u.,,, = 1.26 cm s - I . The values of a, z,, and u*,,, chosen are the same as those used in the example of oscillatory motion, Fig. 5.10. However, in the present example a linear scale in i has been used, to better display the relative thicknesses of the two rotary boundary layers. Only the lowest I50 m has been shown, as this is where most of the variation takes place. but neither the amplitude nor the phase of the clockwise component have quite attained their free-stream values by this height. The velocities were re-dimensionalised by use of eq. 64, derived later.
223
At a given true height z, the non-dimensional height 5 2[( u + / ) z / K u * , , , ] for the anticlockwise component is much greater than that for the clockwise component, (. = 2[(u - ~ ) z / K u * ~ , ] so , that the true vertical scale of the anticlockwise velocity structure is compressed relative to the clockwise. The boundary-layer thicknesses of the anticlockwise and clockwise components are given, by analogy with eq. 33, respectively, by: + =
A value for the constant C is derived from measurements in section 4.6; it is assumed that the same value of C applies to both 8 and 8-. T w o independent additive boundary layers thus exist, with that for the anticlockwise component being much thinner than that for the clockwise component. All the other features of the velocity structure also have a smaller vertical extent for the anticlockwise than the clockwise component. Thus there is a thin logarithmic layer near the bed rotating anticlockwise, superimposed on a thicker logarithmic layer rotating clockwise. The value of E,, is always larger for the anticlockwise component, so that, from Fig. 5.5, the total phase lag through the boundary layer is larger than for the clockwise component. The actual net velocity structure depends on the relative amounts of anticlockwise and clockwise motion present at a particular site, and hence on the eccentricity of the free-stream tidal ellipse. We follow Prandle (1982) by examining three cases: a rectilinear free-stream flow, an anticlockwise rotating free-stream ellipse of eccentricity 0.5, and a clockwise rotating free-stream ellipse of eccentricity -0.5, with the same numerical values as in Fig. 5.16. +
Cose I : Rectilinear free-streurnflow. Equation 54 shows that we require IR,+l = IR,-J to obtain I / , = 0 in the free-stream flow. This is the case illustrated in Fig. 5.16. The variation with height of the semi-major axis U , , and eccentricity U,/U,, of the tidal current ellipse is obtained by combining R , and R - at the same height, using eqs. 53 and 54 (Fig. 5.17a). The value of U , converges on the free-stream value at a height of about 80 m. The current does not remain rectilinear down to the bed. as lJh/Ul takes a small positive value within the boundary layer, corresponding to a narrow, anticlockwise rotating ellipse, whose eccentricity increases downwards to a value of about 0.13 at the bed. The direction @ of the major axis, a n d phase + of the time of maximum current are obtained from eqs. 56 and 55 (Fig. 5.17b). The time of maximum current is t,,, = +/u, and thus the current well away from the bed lags that near the bed, as before. The orientation of the major axis a t the bed is anticlockwise of that in the free stream. It veers clockwise with increasing height, reaching its most clockwise orientation at 50 m, then veers slowly anticlockwise to the free-stream value at a greater height. However, the deviations are very small; the maximum difference between orientations is less than 4". A t a height of 100 m both @ and + are within 2" of their free-stream value.
z
m
z m
140
140
120
120
100
100
I 80
80
60
60
40
40
20
20
0
20
10
U,
cm s
-8
30
-6
-4
U,
cm s - l
-2
L1 2
1
uh I u a
02
z
04
m 140
[C)
120
100
80
60
40
a
20
0
20
10 U,
cm s-’
30
40
Fig. 5.17. The variation with height of the parameters defining the tidal ellipse, for the same numerical values as in Fig. 5.16. (a) The semi-major axis Ua, and eccentricity Ub/Ua, for case (1): rectilinear free-stream flow: (b) the direction of the major axis, and phase 9 of the time of maximum current, for cases ( I ) . (2) and (3); (c) U, and U b / U a for case (2): anti-clockwise free-stream ellipse of eccentricity 0.5; (d) U, and U,,/Ua for case (3): clockwise free-stream ellipse of eccentricity -0.5. Dashed lines show the values of Uaw.
225
This example applies to a rectilinear free-stream current far from a coast. I t may be contrasted with the example of purely oscillatory flow, discussed in section 4.3, having the same values of u, zo and ZI~,,.In the pure oscillatory flow it was assumed that the presence of an adjacent coastline constrained the flow to be rectilinear, and co-linear, at all depths. The velocity amplitude converged on the free-stream value at a height of about 30 m, giving a rather thinner boundary layer than the oscillatory-planetary solution. In reality, however, neither the oscillatory nor the oscillatory-planetary solution is strictly valid near to a coastline or in a narrow channel. The neglect of the advective terms and the vertical component of velocity in the equations of motion is no longer justified. A secondary flow due to Coriolis presumably establishes a surface gradient normal to the coast which in turns gives rise to downwelling (upwelling) on a coast to the right (left) of the current. Near the bed the current speed is small, the Coriolis force is insufficient to balance the transverse surface gradient, and hence a return flow away from (towards) a coast o n the right (left) of the current occurs near the bed. When the tide reverses all the other motions will reverse, but with a time lag relative to the main current. Case 2: Anticlockwise free-stream ellipse, U h m / U a m= 0.5. From eqs. 53 and 54 we find that the relative sizes of the rotary components are given by IR,+I = 31Rm . I. Figure 5 . 1 7 ~shows that U, converges on its free-stream value at a height of about 40 m. The boundary-layer thickness based on U, is thinner than in case (1), because of the dominance of the anticlockwise component. The eccentricity U,,/U2 increases slowly downwards to a value of about 0.6 at the bed. Thus the motion is anticlockwise at all heights and the ellipse is more open near the bed than in the free stream. The variation of @I and @ does not depend on the relative sizes of lRm+land IR,-I, and hence is identical with that for case 1, Fig. 5.17b. Case 3: Clockwisefree-stream ellipse, Ubm/lJdm= - 0.5. From eqs. 53 and 54 we find that the relative sizes of the rotary components are given by IR,+l= IRm-I/3. Figure 5.17d shows that lJa converges on its free-stream value at a height of about 120 m. The boundary layer based on velocity is thicker than in case ( 1 ) this time, because the clockwise component is dominant. The eccentricity is negative everywhere, and lU,,/U,l decreases downwards. Thus the motion is clockwise at all heights and the ellipse is narrower near the bed than in the free stream. However, for some values of toand U,,,/Ua, it is possible to obtain a clockwise rotation in the free stream with an anticlockwise rotation at the bed. The equivalent is not true if the free-stream rotation is anticlockwise. Again, the variation of @I and @ is given by Fig. 5.17b. Velocity profiles of this type were measured by Pingree and Griffiths (1974) in 188 m of water on the edge of the continental shelf SW of Lands End at heights between 2 and 98 m. A temperature profile showed that the bottom 100 m was well-mixed, and above this there was a sharp temperature interface. This has the effect of restricting boundary-layer growth, as will be discussed in section 4.6. Rotary component analysis showed that about 80% of the kinetic energy occurred at the semidiurnal frequency, and of this about 95% was in the clockwise component. The eccentricity of the tidal ellipse was about U,,/lJa = -0.6 at all heights. The profile of the semidiurnal currents was logarithmic up to 33.5 m, and there was no
226
detectable phase difference or veering up to that height. Overall the phase at 98 m lagged about 13" (i.e. 26 min) behind that near the bed, though this was attributed to the thermocline descending semidiurnally below 98 m. There was some scatter in the direction of the major axis of the ellipse, but the general indication is a veering o f about 5"-8" clockwise between 2 and 98 m. Further examples taken off the South Coast of England are given by Pingree and Griffiths ( 1 977). Harvey and Vincent (1977) working in the southern North Sea found a time-averaged veering of about 8" clockwise between heights of 0.58 and 2.96 m and about 6.5" of veering between 2.96 and 10 m, indicating that veering was appreciable both within and above the logarithmic layer. Turbulence
As in the pure planetary and pure oscillatory cases, the theoretical form of the shear-stress distribution can be obtained from the linear eddy-viscosity model. The shear stress 7 is divided into rotary components in the same way as the velocity, each having a constant amplitude and rotating respectively anticlockwise and clockwise. The anticlockwise component 7, is given by eq. 35, with u , replaced by u ~ , , ,IQ, , I by IR,+I, 5 by 5, and toby to+. The clockwise component 7 - is given similarly. The two components combine vectorially to give an ellipse whose major axis is given at the bed by: PU*ni[CY2(to+)IRm+I + c Y 2 ( t 0 - ) ~ ~ m - ~ ]
(63) where C D ( t o )is as defined by eq. 37 and plotted in Fig. 5.8. As )-ro)n,;ix = pu:,,,, eq. 63 can be written in terms of the maximum (U,,) and minimum (Uh,) currents outside the boundary layer as:
1 7 0 1 m a x=
+
(64) Thus if u*, and the eccentricity U,,,/Uam of the ellipse are known, as well as u, f and z o , then Uam and Uhmcan be obtained from Fig. 5.8 and eq. 64. If U , , and Uhm are known instead, then uI, can be obtained iteratively by first assuming values for C,(to+) and C,(to-) of say 0.0025, and calculating an initial value for u~,,,,from and to-,and hence improved values for C n ( & o +and ) which initial values of to+ C,(to-) can be obtained. The value of C,(t,,+) is always larger than that of C,(<,-), because C, increases with to,and to+> to-.In the examples, cases 1, 2 and 3, quoted earlier, the values were to+= 0.0136 and 0.00387, for which CD(tol)= 0.0020 and CD(to-)= 0.0012. The height at which the stress vanishes, ST, will be greater than that based on velocity, S,, as was the case for the planetary and oscillatory boundary layers. Additionally, the contribution from the clockwise component extends to a much greater height than that from the anticlockwise component, because of the different and to-relative to z . scalings of to+ There does not appear to be any experimental evidence of the turbulence structure throughout the boundary layer in this type of flow. The stress measurements of Bowden et al. (1959) mentioned above under the heading of oscillatory flow were in fact made under these conditions, but as they were made near a coast the size of the lateral component of stress was not significantly different from zero. 'In>=
t[cl!!2(tO+)(uam
uhm)+c~'2(60-)(u,,
- uhm)]
[,,..=
221
4.5. Stratified flow
Any of the preceding types of boundary layer may also occur in water which has an initial density stratification. In the more usual case of stable stratification, the turbulence in the boundary layer mixes the water near the bed so that a well-mixed layer is formed with a sharp density interface between it and the non-turbulent water above. This inhibits the mixing process, and the boundary layer will be thinner than that which would be formed in stratified water, as was shown by the numerical model experiments of Weatherly and Martin (1978). They also found that when the boundary layer was capped by a density interface the various measures of its thickness (au, S,, S,, etc.) were more nearly equal than in the unstratified case. The stratification may be caused by vertical gradients of temperature, salinity or suspended sediment. The interface provides an easily accessible measure of the boundary-layer thickness; this is made use of in section 4.6. Current structure In the atmospheric boundary-layer, temperature stratification, producing both positive and negative stability, is of dominant importance, and consequently its effect on the velocity profile has been thoroughly studied. A number of stability parameters exist. The gradient Richardson number:
Ri=
-
R
apo/az
Po ( a u / a z
1’
is the most easily measured. However the flux Richardson number: -
where the mean and fluctuating parts of the density are denoted in this section by p(, and p, is the most easily interpreted in terms of the relative roles of buoyancy and shear. The term represents a vertical flux of density, such as a flux of heat, salt or sediment. A commonly used quantity in expressions for the velocity profile is the Monin-Obukhov length:
The ratio z / L then forms a third stability parameter. The three parameters can be related to each other as, for example, Ri = R i ( z / L ) , using experimental data (e.g. Businger et al., 1971). For a stably stratified ( L > 0) flow, dimensional arguments (see, e.g., Monin and Yaglom, 1971, Ch. 4) show that the velocity profile is given by: (68)
The constant p has been evaluated by a number of experimenters in the atmosphere, a widely accepted value being j3 = 4.7 (Businger et al., 1971). The stability is commonly defined as near-neutral if l R i l < 0.03.
228
11'
w
Fig. 5.18. Infra-red satellite images of the surface temperature of the seas around the British Isles. Darker tones depict warmer water, and lighter tones represent colder regions. Thermal fronts can be identified as the junctions between dark and light regions. (Reprinted from Pingree and Griffiths, 1978, J. Geophys. Res., 83: 4615-4622, copyrighted by the American Geophysical Union.)
The maximum value of z / L in water of depth h is h / L . The values of g, K and po in eq. 67 are practically constant, and the heat flux due to solar radiation, which is generally the main contributor to G, is also roughly constant over a restricted area and season. Thus the overall stability of the water column can be characterised by
229
Fig. 5.19. Predicted positions of frontal boundaries from a numerical model, S = 1.5 (heavy line). Hdtching represents stratified conditions during the summer months ( S > 2), and regions that are well mixed throughout the year are indicated by stippling ( S > I). The notation S on the figure is equivalent to S,,<; in the text. (Reprinted from Pingree and Griffiths, 1978, as for Fig. 5.18).
h / u ? , or, if the drag coefficient is also assumed to be roughly constant, by the quantity h / f i 3 , where is the depth-averaged current speed (Simpson and Hunter, 1974). Pingree and Griffiths (1978) defined a similar parameter Spc;= log,,[h/C,(~')], where h and U are in cgs units, the drag coefficient C , = 0.0025, and the angle brackets () denote a tidal average. They calculated Spcifrom values of
230
ii obtained from their numerical model of the M, tide on the shelf seas around the British Isles, and concluded from the known positions of fronts between well-mixed a n d thermally stratified sea areas that a critical value of SPcifor summer conditions around Britain was S,, = 1.5. The predicted positions of the fronts agreed well with those observed by infra-red satellite imagery (Fig. 5.18). and also with shipborne surface-temperature measurements. The geographical distribution of SPc,(Fig. 5.19) shows that in summer, stratified water (SP, > 1.5) occurs in the northern North Sea, in a small area of the Irish Sea west of the Isle of Man, and in a broad band landward of the shelf edge, including the Celtic Sea. The water is well mixed throughout its depth (SpG< 1.5) in the southern North Sea, the English Channel, the rest of the Irish Sea, the North Channel, and a few areas bordering the Scottish coast. In the former areas, therefore, the velocity and turbulence structure will be modified due to stratification in the summer months. In the winter (say December to April) the whole region is approximately isothermal from surface to bottom, and therefore can be taken to be of near-neutral stability for boundary-layer purposes. Stratification by suspended sediment In the lower part of the atmosphere, the heat flux is roughly constant with height, so that L (eq. 67) is also roughly constant. The stability parameter z / L thus increases linearly with height, so that there is a layer near the ground which is of near-neutral stability, whilst above some critical height the air is significantly stratified. The critical height can be defined as that at which lz/LI = 0.03, by making a slight modification to the previously quoted criterion for near-neutrality. The same behaviour is presumably true of thermal or saline stratification in the sea. However, it is not true if suspended sediment is the cause of the stratification, as the density flux due to the sediment decreases strongly with height so that L is a function of 2 . Thus the height dependence of z / L is less straightforward than it was in the case of heat flux. Moreover, the density flux increases with the shear stress, so that i t is not known at the outset whether an increase in shear stress will increase o r decrease the stability parameter z / L . To investigate this we have constructed a theoretical expression for z / L based on existing expressions for the suspended sediment concentration profile. A paper giving details of the derivation is being prepared, and we present only the conclusions here. For a deep, steady, uniform flow over a flat, unrippled bed of non-cohesive sediment, z / L can be written in terms of the height z , the friction velocity 21,. the threshold shear stress pu?, for bed-load transport, and the settling velocity '.L; of the grains, as:
Here, y , = 1.56 x l o p 3 is an empirical constant evaluated by Smith and McLean (1977), p, is the density of the sediment, z o is the value of the roughness length when sediment is suspended given by eq. 24, and h = w , / K u * . The values of w, and u*, can be obtained from known experimental functions of the grain diameter d. Thus assuming quartz grains in water at 2OoC, the stability can be calculated as a function of z , d and ue only.
23 1
4
Grain diameter (pm)
Fig. 5.20. The gravitational stability of the water column when non-cohesive sediment is in suspeiihion over an unrippled equilibrium bed. The curve z ~ , / , ! , ( z ~ ~=) 0.03 is obtained from eq. 69, the curve 1 1 . = id., is taken from the threshold curve presented by Miller et al. (1977). and h = W , / K U . is based on K = 0.40 and the settling velocity curve presented by Shepard (1963). The curve h = 3 is taken as the threshold of suspension (Bagnold, 1966).
Equation 69 shows that the height dependence of z / L is such that i f ti > 1. z / L decreases with z , but if h < I , z j L increases with z . Thus i f the position o f the bed is taken to be a t z = z 0 , then i f z J L ( z , , ) > 0.03 and h < I . the entire water column is stably stratified, hut if h > 1, a height will be reached at which z / L = 0.03 and above this the water column is near-neutral. I f z , , / L ( z , , )< 0.03 urzd ti > 1, the entire water column is near-neutral, hut if h < 1, a height will be reached at which z / L = 0.03 and above this the water column is stably stratified. The curve i 0 / L (z ~ ,=) 0.03 as ii function of u , and d is plotted in Fig. 5.20. together with the curve h = 1, and curves delineating the thresholds of bed-load and suspended transport. Figure 5.20 indicates at a glance whether or not suspended sediment will cause significant stratification of the water column, for a particular u* and d . For silt-sized material ( d < 63 p m ) appreciable stratification will tend to occur only in the upper part of the water column, with the stratification extending t o the bed only rarely. This is confirmed by the measurements made by Parker et al. (1980) in the Bristol Channel, where despite the concentration o f suspended silt exceeding 4 g I - ' near the bed, the value of Ri was greatest at about mid-water, and only exceeded 0.03 occasionally. If sand (63 < d < 2000 p m ) is i n suspension it will generally cause appreciable stratification, either in a layer above the bed, o r throughout the water column. Only for a very restricted range of u , and d will suspended sediment cause n o appreciable stratification. The expression for z / L , eq. 69, relies on assumptions which are only valid for 0 < z / L 2 0.03. This is sufficient for delineating the critical conditions as in Fig. 5.20. Forms for the velocity and concentration profiles when stratification is established were derived by Taylor and Dyer ( 1 977), Smith and McLean ( 1977) and Adams and Weatherly (1981). The latter authors showed. using a numerical model
232
of the sediment-stratified boundary layer, that under some circumstances the slope dU/d(ln z ) of the velocity profile is constant in z, but greater than the unstratified value by a n amount which increases with the sediment concentration. In the past this has been interpreted as a decrease in K , but it is more in line with modern thinking to keep K a constant a n d modify the expression for the velocity profile.
Turbulence Stable stratification has the effect of reducing vertical mixing because vertically moving parcels of water have to act against the buoyant-forces. One would intuitively suppose that stable stratification would thus reduce w 2 but leave u’ and 7 relatively unaffected. However, measurements in the atmosphere quoted by Busch _ _ (1973) show that, on the contrary, the ratio w2/u’ increases from its neutral value of about 0.25 to about 0.36 at z / L = 0.03 and maintains this value for all larger z / L . in,ternal waves with almost This could be due to the presence of non-breaking circular particle orbits which are increasing w 2 and u7 by similar amounts.Internal waves have w out ofphase with u so that they contribute nothing to - U W , thus increasing the ratio w2/( - &); this is also observed (Businger, 1973). I t is very difficult to distinguish internal wave motion from turbulence when both are present. If the internal waves ure breaking they make an appreciable contribution to the mixing process. 4.6. The depth-limited boundary luyer
It is often the case in rivers and shallow seas that the depth of water is less than the thickness which the boundary layer would otherwise attain. The growth of the boundary layer is thus restricted and the whole depth of water is turbulent. The general features of the individual types of boundary layer already discussed, such as veering and phase lags, are still present, but restricted in magnitude. The distribution of boundary-iayer thickness in the seus uround the British Isles The expressions for the thickness of the anticlockwise and clockwise boundary layers, eq. 62, can be compared to the waterdepth h to give a n indication of whether o r not the flow is depth-limited. For the anticlockwise component:
6, _ -
cu,,
(a+f)h
/I
=C
Ck’2 A +
where:
Similarly we define:
233
60‘
550 N
50”
Fig. 5.21. The distribution of the parameter A + in the seas around the British Isles, indicating the ratio of the boundary-layer thickness of the anticlockwise rotary component of motion, to the waterdepth. Equation 71 was evaluated at each grid square of Flather’s numerical model (see (Fig. 5.15). According to the linear eddy-viscosity model: for A + < 8 an “infinite depth” assumption is valid; for A + < 20 the boundary layer based on current speed is thinner than the waterdepth; for A + z 200 the flow is quasi-steady. Contours at 5 , 8, 20, SO, 100, 200, 500. 1000 and 2000.
As an example, we have calculated the geographical distribution of A + and A _ for the semidiurnal tide ( u = 1.40X lop4 s-’) over the shelf seas around the British Isles. The distribution of the maximum depth-average current tfl, was obtained from the numerical model, developed by R. Flather, of the mean spring tide in this sea area which was mentioned earlier. To interpret the contours of A + and A- (Fig. 5.21 and 5.22) we need to assign values to C and C , in eq. 70. The value of C , will be different for the two rotary components as discussed earlier; however, for consistency with Flather’s model we will put them both equal to his value of 0.0025. The
234
-60'
55' -N
- 50"
Fig. 5.22. As for Fig. 5.21, but showing the clockwise parameter A - (eq. 7 2 ) .
choice of a value for C depends on which feature of the boundary layer one is interested in. Based on the results of the linear eddy-viscosity model of section 4.2, the height at which both the speed and direction of the currents attain their geostrophic values is given by C = 1. Thus, 6 , / h = 0.05A, so that the contour A = 20 delimits the areas in which 6, < h. On this basis, 6- exceeds the waterdepth over most of the continental shelf, and 6, also exceeds the waterdepth over substantial areas of the shelf. However, it must be remembered that the linear eddy-viscosity assumption greatly overestimates K , , at large z , and hence the model produces an artificially large value of C. A more realistic model would produce a smaller C , and thus a larger contour on the A distributions would delimit the areas in which 6, < h.
235
I t might seem reasonable to argue that, as the anticlockwise boundary layer is always nested within the clockwise one, the effective height at which the total velocity attains its free-stream value is given by the larger value, namely 6 . However, we should also take account of the relative sizes of JR+Iand IR I. This is effected by taking the depth-averaged ratio l k + l / [ k I ,whose distribution was plotted in Fig. 5.15, and defining a weighted mean boundary layer thickness 6 such that 6 is made up of 6, and 6 - in the ratio lk+l:1k-l :
Substituting for lk+l and 1k-l from eqs. 53 and 54; for 6, and 6 and putting u,, = c;/'fid,gives:
from eq. 62;
Rather than rely on the value of C obtained from the linear eddy-viscosity model, which tends to overestimate 6, we turn to field measurements to assign an empirical value. Pingree a n d Griffiths (1977) made precision temperature profiles at a number of stations o n the shelf to the southwest of England, which revealed a bottom mixed layer whose thickness increased from 38 m near the Cornish coast to 103 m near the shelf edge. The measurements were made in May, when stratification is relatively mild, so the mixed layer would correspond to a bottom boundary layer which is only slightly thinner than that obtaining under neutral conditions. The measured mixedlayer thicknesses are given in Table 5.6, together with the corresponding values of C, obtained by applying eq. 74 a t each station with Flather's values of fidand U,. The mean value of C for the six stations is 0.075. Using this value in eq. 74. with C', = 0.0025, gives: (75)
Values of
Ua and
U , from Flather's numerical model were used in eq. 75 to
TABLE 5.6 Values of the constant C in eq. 74 obtained from comparison with mixed-layer thicknesses measured at 6 stations by Pingree and Griffiths (1977). The velocities ha and fi,, were obtained from R. Flather's depth-averaged numerical model for mean spring tides. The mean of the 6 values is C = 0.075 Station
Mixed-layer thickness (m)
7906 7907 7908 7909 7910 7913
38 43 46
75 103 103
60 73 67 68 76 74
-8 19 - 31 - 38 - 47 - 49 -
1.11 1.11 1.10
1.10 1.09 1.09
0.059 0.05 1 0.05 1 0.083 0.101 0.103
236 I
I
I
I
60"
559
N
50'
Fig. 5.23. The distrihution of boundary-layer thickness 6 around the British Isles. calculated from eq. 75 and Flather's numerical model (see Fig. 5.15). Regions in which 6 exceeds the waterdepth are shaded. Contours at 20, 40, 60, 80, 100 and 120 m.
calculate the geographical distribution of 6 for the semidiurnal tide in the shelf seas around the British Isles (Fig. 5.23). The geographical distribution of the depth below the water surface of the top of the boundary layer, given by h - 6, is presented in Fig. 5.24. Negative values indicate those regions in which the potential value of 6 exceeds the waterdepth, so that the boundary layer is depth-limited; in Fig. 5.23 these regions are shaded. Figure 5.23 shows that in the eastern part of the English Channel the boundary layer is depth-limited, but in the western English Channel and Southwest Approaches it occupies just the bottom 60 to 80 m. The thickness decreases northwards in the Celtic Sea to about 20 m off the southern coast of Ireland. In the southern Irish Sea 6 > h , but both west and east of the Isle of Man 6
231
60"
550
N
50'
Fig. 5.24. The distribution of h - S in the seas around the British Isles, showing the depth below the water surface of the top of the bottom boundary layer. The calculation of S was as for Fig. 5.23. Contours at - 100, -60, -40, - 20, 0 , 20, 40, 60, 100 and 200 m. Negative values indicate that the predicted S exceeds the waterdepth. The zero contour gives the positions where thermal fronts should develop in summer.
thins to less than 20 m. On the shelf west of Ireland and Scotland 6 is typically about 20 m, generally thinning offshore, and increasing northwards to over 80 m near Shetland. In the North Sea 6 increases southwestwards from less than 10 m off the Norwegian coast to a potential value of over 100 m off the East Anglian coast, with the whole of the southern North Sea being depth-limited. The distribution of 6 in the North Sea shown in Fig. 5.23 is rather different to that presented by Kraav (1969), who used a height-invariant K , , model and took 6 to be determined by the anticlockwise rotary component only. The observed position of thermal fronts, Fig.
238
5.18, supports the present distribution. Regions in which the boundary layer occupies the entire depth occur also around Lands End, between Scotland and Northern Ireland, around Orkney and Shetland and off Peterhead. The boundaries of the shaded regions, within which 6 > h , correspond closely to the lines indicated as fronts by means of the stratification parameter Spci(Fig. 5.19), and identified as such by infra-red satellite imagery (Fig. 5.18) (Pingree and Griffiths, 1978). These are the lines along which the mixing associated with the turbulent boundary-layer outcrops at the water surface. In order for mixing to take place throughout the water column it is necessary not only for there to be sufficient turbulent energy available to overcome the potential energy associated with the thermal stratification, but also for the energy, which is generated near the seabed, to be capable of reaching the water surface. The former condition depends on whether S,, >< 1.5, while the latter depends on whether 6 >< h. Both h / 6 and Spc;increase with h and decrease with Ua, so that their distribution might be expected to have broad similarities. However, the difference between the two parameters is sufficient that one would not expect the positions of contours of h / S = 1 and S p c i= 1.5 to correspond closely. Nonetheless, comparing Figs. 5.19 and 5.24, it appears that they d o so in this particular sea area. In the regions in which mixing does not reach the surface, the water will be well-mixed, and hence isothermal, from the bed to the top of the boundary layer only. Above this, extending to the water surface, the effects of bottom-generated mixing will not be felt, and in summer a thermocline may develop. Measurements of the depth of the lower limit of the thermocline in July in the North Sea, compiled and plotted by Tomczak and Goedecke (1964), show fair agreement with the predicted depth of the top of the mixed layer (Fig. 5.24) as far north as about 57"N. North of this, particularly around the Norwegian coast, the measured thermocline depth is much less than that of the predicted mixed layer, which may indicate that there is insufficient mixing near the surface to permit the thermocline to penetrate deeper. The interface separating the well-mixed water from the weakly mixed upper water, contoured in Figs. 5.23 and 5.24, will correspond in the summer months to the subsurface position of fronts. High concentrations of phytoplankton are found at an interface, because both nutrients mixed from the seabed, and sufficient light, are found there simultaneously (Pingree et al., 1975). Wind mixing is, of course, also a n important factor in determining thermal structure. Although the terms boundary layer and mixed layer have been used interchangeably above, it is likely that the thickness of the mixed layer (based on thermal structure) is determined by the preceding spring tide, and to a large extent maintains its thickness through the subsequent neap tide. Figures 5.23 and 5.24 are based on mean spring tidal currents; on neap tides the boundary layer (based on velocity structure) would be thinner, occupying the lower part of the mixed layer. The calibration of C was based on the mixed-layer thickness in early summer, and hence Figs. 5.23 and 5.24 apply to that season. In winter, under isothermal conditions, 6 may be rather thicker, whereas in late summer, when stratification is well established. 6 will be reduced in thickness.
239
Current structure
To investigate the effect of a restricted waterdepth on the velocity profile, we turn again to the linear eddy-viscosity model. The upper boundary condition of U + U, as z 00 is n o longer appropriate; instead we take the condition of zero shear stress at the free surface, T = 0 at z = h. This assumes that there is no wind stress. The total veering (or phase difference), I%$,,, from surface to bottom was calculated for various values of to,as a function of th= 2 ( f h / ~ u , ) ’ / ’ for various values of t(,(Fig. 5.25). For Eh > 5 , the value of is about equal to the deep-flow value for all t,,. Noting that A = 4 / ( ~ C h ” t i ) , the value = 5 corresponds to A = 8. T h u s in the areas with A < 8 in Figs. 5.21 and 5.22 the “infinite depth” approximation is valid, whilst in the areas with A > 8 a depth-limited model must be applied. In terms of the linear eddy-viscosity model, the clockwise component is depth-limited everywhere on the shelf, and the anticlockwise component is also depth-limited everywhere except for the outer shelf and off the Norwegian coast. The value of a$(, becomes progressively smaller as E,, decreases from 5. For t,,< 1, the phase difference &+,,.is less than 2” for all t o< l o - ’ . Thus one can consider the flow to be quasi-steady i n areas where A > 200, with the water surface slope being balanced at all times by friction, a n d inertial and Coriolis effects being negligible. From Figs. 5.21 and 5.22, the clockwise component is quasi-steady over large areas of the shelf, and hence has negligible phase changes and veering from seabed to surface. This is true for the anticlockwise component only in a few isolated regions. Again a more sophisticated model might yield different critical values of A. The logarithmic layer may occupy a large portion of the depth, as happens in -j
<,,
Fig. 5.25. The total veering, a@,, from surface to bottom, given by the depth-limited linear eddy-viscosity model as a function of the non-dimensional waterdepth th. and for various values of the roughness parameter 6”.
240
channel flow. The example of oscillatory flow given in Fig. 5.1 1 was depth-limited. The velocity profiles near to maximum current, from 5 to 7 hours after high slack water, are logarithmic up to about 7-10 m, or 35-50% of the depth. Turbulence The effect of a restricted waterdepth on turbulence profiles is similar to that described for velocity profiles, namely a reduction in the magnitude of the phase lag, or veering, top to bottom as the waterdepth is reduced. If the water is sufficiently shallow that the flow is quasi-steady, the shear-stress profile takes a particularly simple form. Thus if the time-dependence and Coriolis terms can be neglected, and the stress at the surface is zero (no wind), the equation of motion for unstratified flow over a uniform bed can be vertically integrated to give the shear-stress profile: r ( z ) = r ( , ( l-
h
The bed shear stress balances the surface slope quasi-steadily. and is given by:
Practical examples of turbulence in a depth-limited flow have already been given, as the measurements of both Bowden et al. (1959) (Fig. 5.13). and Gordon and Dohne (1973) (Fig. 5.12), were made in shallow water. 4.7. Leading edge flow
When a steady non-turbulent flow passes over a flat plate a boundary layer forms which thickens progressively with distance from the leading edge of the plate. This has been extensively studied in laboratory experiments and is discussed in detail by Hinze (1975). In the sea we do not generally find a leading edge in this sense, but when the flow encounters an abrupt change in seabed roughness, such as passing from mud to gravel, an internal boundary layer may develop in a similar way. While direct evidence of this in the sea does not appear to be available, it has been observed in the atmosphere. Current structure An example is shown in Fig. 5.26 of the velocity profiles, bed shear stress, and internal boundary-layer thickness over smooth-rough and rough-smooth changes, drawn from the atmospheric data of Bradley (1968). Downstream of a change of roughness the bed shear stress adjusts rapidly to its new value, and a logarithmic profile which corresponds to the new u. and zo develops. The height 8, at which this merges with the upstream, unmodified profiles grows downstream, and several expressions relating it to the distance x downstream of the roughness change have been proposed. One of the most successful is that of Jackson (1976):
in which zg is the larger of the two roughness lengths involved. The transition
24 1
between the new and old logarithmic profiles at z = 6, is usually quite abrupt. In this context the terms smooth and rough are used in a relative sense, rather than to indicate the hydrodynamic regime. Turbulence Mulhearn (1978), in a wind-tunnel study of the internal boundary layer downstream of an abrupt change from a very rough to a smooth surface, found that the Reynolds stress increased almost linearly with z to the upstream constant stress value at a height of about 2 6,/3. Over the entire downstream fetch studied, corresponding to 5000 times the upstream z o , there was only a slight decrease from the upstream profiles of 2 and 7.The horizontal variance 2 decreased more rapidly, though an equilibrium profile was still not attained. Thus it seems that the turbulence characteristics only attain equilibrium at a very large distance downstream of a rough to smooth bed change.
- 10
~. 8
9
I
I
I
1
1
5
0.6
07
08
0.9
1.0
10 11
1
"
'"297
20
10 x [rn)
Fig. 5.26. Measurements of air flow over smooth-rough and rough-smooth surface changes. The curves are constructed from the data presented by Bradley (1968). The smooth surface was tarmac ( z o = 0.002 cm), and the rough surface consisted of a mat of wire spikes ( z o = 0.25 cm). a Adjustment of velocity profile caused by smooth-rough change. Numbers against profiles refer to the stations shown in d. b, Adjustment of velocity profile caused by rough-smooth change. Profile 1 is also shown. for reference. c, The variation with position relative to the roughness changes, of the ratio of the measured local bed shear stress to the far upwind value. d, The extents of the smooth and rough surfaces, and the positions of the measuring stations. The dashed lines show the growth of the internal boundary layers, whose heights were taken as the upper limit of the linear portion of each curve in a and b.
242
If the upstream flow is non-turbulent (the true leading edge flow case), the shear stress and variances all decrease smoothly from a maximum value at or near the bed to zero at the edge of the boundary layer. Downstream of a smooth to rough bed change the profiles are qualitatively similar to the above, except that they decrease only to their upstream values at the edge of the internal boundary layer. 4.8. Flow over topography
All the types of boundary layer considered so far have applied to flow over a flat, horizontally homogeneous bed. A frequent cause of non-uniformity of the flow is the presence of sedimentary bedforms, ranging in size from ripples to sand banks. For 2-D bedforms, continuity ensures that the current speed is faster over the crests than over the troughs. The flow is thus spatially accelerated on the upstream flank and decelerated on the downstream flank.
Current structure Departures of the velocity profile from that found over a flat bed with the same local bed shear stress are primarily due to advection of the upstream current structure, An estimate of the magnitude of this can be obtained by comparing the advective terms in the equations of motion with the frictional term. The equation for the U component of motion in a shallow flow can be written as:
The terms WaU/az and a z / a x are usually much smaller than U a U / a x and pp'dr,,/dz, respectively. The term U a U / a x can be estimated over a wavy boundary of wavelength X and height H from the potential flow solution given by Milne'Thomson (1968). The departure from the free-stream velocity U, which occurs at the crest is aHU,/X, while at the trough it is -7rHU,/A, so that: - Utrough) u-au = um(ucrest
X/2
ax
4.irHUi
-~ -
X2 The friction term can be approximated in shallow water by:
where h is the mean waterdepth and C , the drag coefficient. The topography will have an appreciable effect on the velocity profile if the ratio of these terms exceeds 10% say:
243
Putting C,,
= 0.0025,
the criterion that topography is important becomes:
I? H -> 2 x 10-5 h2 To test condition 83, we refer to field data and numerical model results. Measurements by McLean and Smith (1979) over sandwaves in the Columbia River, [J.S.A. ( H = 2 m, h = 100 m, h = 15 m, h H / h 2 = 3 X 10~-') showed strong topographic effects in both the mean flow and the turbulence characteristics. The velocity profiles measured by Dyer (1971) over sandwaves in the Solent, southern England ( H = 7 m, h = I20 m, h = 10 m, hH/h2 = 5 X l o p 3 ) also showed topographically associated departures from the logarithmic form (Fig. 5.27). I n contrast the numerical model of flow over a low sandbank ( H = 5 m, A = 6 km, h = 10 m. hH/h2 = 1.4 X lo-'), described in Chapter 6 of this book, yielded velocity profiles which were everywhere close to the local logarithmic form. Thus, condition 83 appears t o he ;I good guide to the importance o r otherwise of topography. For further discussion of the form of the velocity profile over topography the reader is referred to Chapter 6.
Velocity ( m s-')
w
L
0
100
200
300
400
Metres
Fig. 5.27. Velocity profiles measured over sandwaves in the Solent. The numbers against the profiles correspond to the station positions shown on the sandwave cross-section shown below them (flow from left to right). Values of zo (cm) and u. (cm s - I ) shown against each profile were calculated from the lowest two readings in each case. (After Dyer, 1971, reprinted with permission of the Geophysical Journal of the Royal Astronomical Society.)
244
..
.
214cm
25307
-
50.47:
.
.
.
.
'
,
,
.. ... '
,
f ' :
,
, '
:
.
,
"
'-214cm
'
I 35cm
35cm
&
\
-..
q
z] x
x
m
m
I
L
2.50 _--/
I
0.'l2
O . L 0 : 5 <
Oi62 0'75 0.87
1.00
Fig. 5.28. The distributions of the Reynolds normal and shear stresses over an ensemble-averaged sandwave. Individual data points are shown for the highest measuring height, together with a fitted curve: only the fitted curves are shown for the other three measuring heights. The shape of the sandwave is shown beneath each plot. The notation p u " is equivalent to p G 2 in the text, and so on. The velocity components are calculated with x directed along the local mean streamline. (Reprinted from McLean and Smith, 1979, J. Geophys. Res., 84: 7791-7808, copyrighted by the American Geophysical Union.)
Turbulence The pressure gradients and spatial accelerations set up over topography cause the turbulence structure to vary with position. The measurements of the turbulence field over sandwaves (Fig. -5.28) made by McLean and Smith (1979) showed that the maximum value of u 2 along the streamline corresponding to z = 35 cm occurred above the crest, but at greater heights the maximum was advected progressively
245 -
downstream until at z = 214 cm it was situated over the trough.The behaviour of u2, w2and p z did not fit this pattern, but it was apparent that w 2 had a very similar spatial distribution to - p G . It was found that the value of - p G , averaged along a streamline over one wavelength of the topography, increased with height. This is indicative of the exchange which takes place between the pressure distribution near the bed which is felt as form drag, and the Reynolds stress which must incorporate this at higher levels. Downstream of the crest of a bed feature the bed shear stress is reduced. If the minimum value of T,, drops to zero, a region of separated flow forms in the lee of the crest, causing a near-bed upstream flow between the crest and the separation point. Separation often occurs in the lee of ripples, dunes and sandwaves, but generally does not over a sandbank. Further discussion of the Reynolds stress distribution is given in Chapter 6.
5. TURBULENCE SPECTRA
5.1. General
The frequency spectrum of the turbulent velocity components is useful in many contexts. It indicates the range of frequencies which make important contributions to the turbulent motion, and hence the frequency response necessary for turbulence measuring instruments. The power-law dependence of the high-frequency portion reveals which scales of the motion can be considered as isotropic. It also permits calculation of the dissipation rate of the turbulent kinetic energy, and hence, as the turbulent energy is derived from the mean motion, the dissipation of tidal energy. The typical velocity fluctuation occurring at a particular length or time scale can be obtained from the spectrum. This is required for calculating the yawing forces acting on subsurface buoys or submersibles. Turbulence measurements are nearly always made as a time series at a fixed point, so that the measured spectrum is primarily a frequency spectrum. However, according to Taylor’s hypothesis, the turbulence pattern is advected past the sensor by the mean current much more rapidly than it is developing temporally. The measured fluctuations therefore correspond more closely to the spatial than to the temporal variation of the velocity. For this reason turbulence spectra are generally presented as wavenumber spectra, with the wavenumber k b e k g related to the . wavenumber spectrum Ecya(k ) of a velocity frequency n by k = 2 ~ n / U ( z ) The component a is obtained from the frequency spectrum Saa(M ) from the relation:
A similar expression applies to the cospectrum E a p ( k )of two components a and ,8. Experiments in the atmosphere (Webster and Burling, 1981) have shown that Taylor’s hypothesis is strictly valid only for wavenumbers for which k z > 3 [ n > 0.5U( z ) / z ] , but it is nevertheless common practice to apply the transformation, eq. 84, at all n. A wide range of frequencies contribute to the variances, so turbulence
246
spectra are usually presented against a logarithmic frequency scale. The distribution of energy is most easily distinguished in this case if the wavenumber weighted spectrum is plotted, which shows equal energy contributions as equal areas. 5.2. Surface layer
In both the atmosphere and the sea the vast majority of measurements of spectra have been made near the bed. In this region, extending up to say 0.1 6, the shape of the spectrum of a particular velocity component is found to be constant within the energy-containing range of k if the wavenumber is scaled by the measuring height, i.e. if the abscissa is taken as k* = kz. The shape varies with the degree of stratification, and hence depends on the stability parameter z / L (see section 4.5). This surface-layer scaling is well established in the atmosphere (Kainial et al., 1972). and has been found to extend, at least in the case of near-neutral stability, to spectra measured in marine surface layers (Soulsby, 1977). The wavenumber weighted spectra k E U , ( k ) / ? , k E w w ( k ) / 7 and k E u m ( k ) / G ,are plotted with surface layer scaling in Fig. 5.29. Each spectrum is the mean of about 200 individual 12-min spectra measured at the various sites and heights listed in Table 5.1. The averaging was performed after applying the k* scaling to the individual spectra, and it is seen that the variability as indicated by the standard error bars is quite small. All the spectra have significant contributions from wavenumbers covering a range of 3 to 4 decades. The wavenumbers contributing to kE,,,,(k ) are smaller than those contributing to kE,,( k ) , i.e. the horizontal motions are larger (along the flow direction) than the vertical motions, while those contributing to kE,,,( k ) lie between them.
Fig. 5.29. Average wave-number weighted, normaliaed velocity spectra and cospectrum. Each spectrum is the average of about 200 individual 12-min spectra taken at the heights and sites shown in Table 5.1. Standard error bars are shown at three values of k'.
247
Recent research, including a closer study of the individual spectra which went into Fig. 5.29, indicates that scaling k with z is only approximately correct, even very near the bed. Panofsky (1973) suggested replacing z by the wavelength A, at which k E , , " ( k ) is a maximum, and found that A,/z decreased with z. We use a similar approach here for spectra outside the surface layer. However, instead of using A,,, as the scale length, we introduce a quantity z , defined by: zx = X,/A,
(85)
The constant of proportionality A , is chosen to give z , 4 z as z + 0, so that the usual surface layer scaling is recovered at the seabed. The constant A , , determined from experimental values quoted by Pasquill (1972) and Busch (1973) takes a value of about A, = 3, which compares with a value A , = 2.7 for the position of the peak of k E , , ( k ) in our Fig. 5.29. The definition of k* is then modified to: k,* = k z ,
(86)
This permits Fig. 5.29 to be used also in the outer part of the boundary layer and for stabilities other than neutral, provided the dependence o f z, on z / 6 and z / L is known. 5.3. The outer part of the planetury boundary layer For evidence of the behaviour of z , outside the surface layer we turn again to the measurements made by McPhee and Smith (1976) under the Arctic pack-ice. They found that their w spectra were of similar shape through almost the whole of the boundary layer beneath the ice, though their position on the k* axis shifted rightwards with increasing z . They tabulated the variation of A; through the boundary layer, where A',, is the wavelength at which each spectrum first matched a straight chord with -2/3 slope, i.e. a somewhat smaller value than A,n. Their data has been used here to produce the plot of z x / z against z / 6 shown in Fig. 5.30 with assumptions: ( 1) that the average boundary-layer thickness 6 through their experiment was 29 m; (2) that A, = 2A',,, (estimated from their fig. 16); and (3) that A, = 3 . The straight line shown in Fig. 5.30 and given by: Z
_ zx -b5 Z
is a reasonable fit to the data, and obeys z
--j
z , as z
-+
0.
5.4. The stably stratified surface layer
Only stable stratification will be considered here, as the shape of unstably stratified spectra is modified by buoyant energy production. The variation of A ,,, with stability in the atmospheric surface layer has been the subject of a number of studies, those up to 1972 being reviewed by Pasquill (1972) and Busch (1973). Kaimal et al. (1972) working in the atmospheric surface layer showed that as the stability increased, the shapes of their velocity spectra and cospectra did not change appreciably, but their position on the k* axis all shifted progressively rightwards by similar amounts. Wamser and Muller (1977) found that their own data for the
248
Fig. 5.30. The variation of zA through a boundary layer. Values of zA were calculated from the data of McPhee and Smith (1976), and the value of S for their experiment was taken as 29 m.
atmospheric surface layer fitted the linear relationship: Z -
Am
Z
=0.4+ L
(88)
Provided we take A , = 2.5, which is well within the range of experimental values presently available, eq. 88 can be written as: =A _ -
z
1 1+2.5z/L
5.5. The stably stratified planetary boundary layer A limited amount of work has been done on the variation of A,,, in the stable atmospheric boundary layer outside the surface region (Wamser and Muller, 1977: Caughey, 1977) which has shown that, as would be expected, z,/z decreases still further for a combination of both height and stability. A simple way of combining eqs. 87 and 89 based on the assumption that the effects of height and stability act independently is to put:
z, _ z
1 -z/6 1 +2.5z/L
Wamser and Muller (1977, fig:7) presented a plot of z / X , , , ( l J ) against z / L , where A,,(U) is the wavelength at which the u spectrum has its peak, but as they do not quote a value of 6 for their work it is not possible to test eq. 90 directly. However, if it is assumed that A,,(U)= 15z, (cf. Busch, 1973, A,,(U)=20z,: our Fig. 5.29, A,,,(U) = lot,) the value of 6 can be chosen so as to make eq. 90 a reasonable f i t to
249
Fig. 5.31. Test of eq. 72 against the atmospheric data of Wamser m d Muller (1977). L was calculated at a height of SO m. It was assumed that A,,,(U) = 15 ti,and 6 = 335 m.
their data. Figure 5.31 is a reproduction of their fig. 7 but with the lines fitted by the original authors omitted, and lines corresponding to our eq. 90 for 6 = 335 m inserted. The fit is seen to be adequate, suggesting that eq. 90 can be used to give at least rough estimates of velocity spectra throughout neutral and stably stratified boundary layers if the abscissa of Fig. 5.29 is taken as k,* instead of k * . 5.6. Tidul variation
We can investigate the tidal dependence of the turbulence spectrum by examining a contour plot of the spectra from Start Bay Stn. 1 through part of a tidal cycle. An example of nS,,(n), the frequency spectrum of w ,is shown plotted against n in Fig. 5.32a, while the tidal variation in current speed is shown in Fig. 5.32~.The overall energy level rises and decays as the current speed increases and decreases. There is additionally a shift of the whole spectrum to lower frequencies as the current decreases from 0348 onwards as would be expected if surface layer scaling is obeyed in a quasi-steady fashion. To test this, the normalised spectrum k E , , (k ) / T is shown plotted against k* in Fig. 5.32b. I t is apparent from the nearly parallel contours that with this scaling the form of the spectrum does not change appreciably with tidal phase. Thus it appears that near to the bed the spectral content of the turbulence adjusts rapidly to the ambient conditions. Measurements of E , , ( k ) , E , , ( k ) and E , , ( k ) in the bottom 210 cni of the Irish Sea by Bowden and Ferguson (1980) revealed no variation in the form of the spectra with the phase of the tide. On the other hand, measurements by Anwar (1981) at about mid-depth in a laboratory flume showed that while k E , , ( k ) / T was relatively independent of acceleration for all k*, the small k* part of kE,,(k)/' u was lower for accelerating than for steady or decelerating flow, and the small k * part of k E , , ( k ) / G was higher for decelerating than for steady or accelerating flow. Thus i t is possible that well above the bed some non-steady dependence on tidal phase occurs.
250
=
0'
I
I
I
I
i
0000
0100
0200
0300
0400
Time (EST)
J
7 - 8 Sept 1975
Fig. 5.32. Contour plots of the vertical velocity spectrum for 2 = 140 cm at Start Bay, Stn. 1 through part of an ebb tide. (a) / 7 S W W ( nversus ) n and time; (b) k E , , ( / , ) / w * versus k * and time; (c) variation in mean velocity at z = 140 cm with time. ~
5.7. Topographic variation McLean and Smith (1979) measured turbulence spectra at five heights at each of a number of positions over sandwaves 2 m high and 96 m long. At a given height they were unable to discern any systematic differences between spectra measured above the crest and above the trough. Thus although, as mentioned earlier, the mean velocities, variances and covariances do show departures from a quasi-local behaviour over these sandwaves, the spectral content of the turbulence adjusts rapidly to the ambient conditions. However, it is possible that the spectrum might also exhibit a spatial lag over very short, steep bed features.
25 I
5.8. The inertiul subrunge At wavenumbers much larger than those at which turbulent energy is produced. but much smaller than those at which it is dissipated, similarity arguments show that the energy spectrum should take the Kolmogorov form:
where E is the energy-dissipation rate. The constant A , is empirically found to be about 0.5, and, because the motion is isotropic in this range of k , i t can be shown that A , = A , = 4 A J 3 . Measured spectra are generally found to fit this form at large k when plotted on log-log axes (e.g. Heathershaw, 1979; Bowden and Ferguson, 1980; and our own spectra), and eq. 91 can be used as a means of estimating E. Spectra measured at Stn. 1 in Start Bay (see Table 5.1) on a spring tide gave values o f ~ = 5 . 6e r g g - ' s-' at z = 3 0 cm, and E = 1.8 e r g g - ' s - ' at z = 140 cm. Heathershaw (1979) obtained values of € = O h 3 erg g - ' s - ' at z = 100 cm and c = 0.57 erg g - ' s - ' at z = 150 cm from spectra measured in the Irish Sea. 5.9. The dissipution runge
Viscous dissipation of the turbulent energy takes place at very large k , for which k l , = O( l ) , where I, is the Kolmogorov length scale defined by eq. 4. In the tidal - l o - ' cm typically. This portion of the spectruni bottom boundary layer I , = scales not with k z , but with k l , . Various theoretical forms for the spectrum within the dissipation range of k have been proposed. They are generally written in terms of the three-dimensional spectrum E3(k 3 ) , representing the energy contributed by a spherical shell of wavenumbers in the vector k-space, where k , = Ikj, and ipl,"E,(k,) d k , = E . A widely accepted form for the spectrum is that obtained by Pao (1965):
E3(k 3 )= A 3 ~ 2 / 3 k ; 5exp[ / 3 - $ A 3 (klK)4'3] Equation 92 was found to agree well with high-frequency spectra measured with a hot-film probe in the sea by Grant et al. (1962), for a value of the constant A , = 1.7, which corresponds to A = 0.56. The one-dimensional spectra can be obtained from eq. 92, but, as this involves an integral which can only be performed numerically, they cannot be written in algebraic form.
6. T H E BURSTING PHENOMENON
6.1. Generul
So far we have discussed the behaviour of only the mean value o f the Reynolds stress - p G over an interval of 10 min or more. We will now look in more detail at the way in which it is made up, by examining the instant by instant behaviour of the uw product. A typical example (Fig. 5.33), taken at Stn. 1 in Start Bay, shows that uw occurs in intermittent events whose amplitudes are many times the long-term
252
uw c m 2 2 400
BURST
A
A
SWEEP
D-DEC
r
I
UP-ACC
1
v
Time (sl 0
180
-400
400
360
180
A A
A M
-400
400 r
w
v
360
540
-400
400r
1
I "
V
720
540
1 -400
1
A
A
A
A
A
A
A A
i
Fig. 5.33. An example ol'the un' product for a 12-min portion of Run 513 at := 140 c n ~ The . 85 events which make up 90% 0 1 -E,are bhadrd black and classified by quadrant. For thih record U(140)= 51.3 cm s - ' and -=(140) = 15.7 cn? s - ? .
253
average &, each lasting several seconds, with quiescent periods in between. A similar behaviour has been observed in laboratory flows of many kinds (e.g. Corino and Brodkey, 1969; Willmarth and Lu, 1972; Comte-Bellot et al., 1978), and also i n the sea (e.g. Gordon, 1974; Heathershaw, 1974). Most laboratory studies have been made over hydrodynamically smooth boundaries, but Grass (197 1) showed that the intermittent events occur over hydrodynamically transitional and rough boundaries also. Laboratory studies have fallen into two categories: flow visualisation, in which the motion of near-bottom fluid is photographed using a tracer to delineate the motions; and velocity measurements, usually with hot-wire anemometers, which are more closely analogous to the marine measurements. Few studies have been made using both techniques simultaneously, and those that have done so found i t difficult to correlate the visible coherent structure with a measured velocity signature (Offen and Kline, 1973). There is considerable disagreement about the processes taking place, partly because different investigators have used different criteria to identify the events. However, the commonest approach is to divide the motions into quadrants in u-M’ space, identifying each quadrant with a type of event. Various names have been given to them; we will use those shown in Fig. 5.34. Bursts are sporadic, but organised, ejections of near bed fluid upwards ( w > 0) into the flow. As they originate near the bed their velocity is lower ( u < 0) than that of the surrounding fluid at the measuring height, and hence uw is negative during a burst. Sweeps carry high-velocity ( u > 0) upper fluid downwards ( w < 0) to the bed, and hence also make a negative uw contribution. Up-accelerations ( u > 0, w > 0) and down-decelerations ( u < 0, w < 0) are weaker events making positive uw contributions. It is generally agreed that bursts make the greatest contribution to the net stress, and hence the whole process has become known collectively as the bursting phenomenon. Sweeps also make a large contribution, while up-accs and d-decs are less important, so that the net is negative, corresponding to a positive Reynolds stress. Most laboratory studies have concentrated on motions at heights less than a few times the viscous sublayer thickness. While some workers maintain that bursts move
Fig. 5.34. Classification of bursting events in u - w space, as bursts, sweeps, up-accelerations and down-drcslrralions.
254
outwards in a continuous motion from the bed to the outer edge of the boundary layer, where they are manifested as turbulent bulges (Kim et al., 1971), some maintain that it is a two-stage process with the sublayer bursts being triggered by separate events in the outer layer, which themselves cause the turbulent bulges (Praturi and Brodkey, 1978). Others suggest that the former happens at low Reynolds number and the latter at higher Reynolds number (Falco, 1974). Typical marine Reynolds numbers are very large (U,-,h/v = lo7 for the record shown in Fig. 5.33, where U,-, is the mean velocity 5 m below the surface), so it is unlikely that bursts originating at the bed will reach the measuring height ( z = 140 cm in Fig. 5.33) in a single continuous motion. It has been proposed that a cycle of events occur in the sublayer, with a burst being followed by a sweep (Corino and Brodkey, 1969). There is no evidence of this in Fig. 5.33, so possibly the events lose this cyclic behaviour well above the bed. Indeed, for the record shown in Fig. 5.33, there is a tendency for bursts and d-decs to occur together in groups of about 10-20 events, and sweeps and up-accs to occur in similar groups, though it is not yet known whether this is generally true. There is also laboratory evidence of a roll-like structure associated with the bursting process whose diameter is about 6 times the viscous sublayer thickness (Blackwelder and Eckelmann, 1979). A ‘similar structure is found on the scale of the entire boundary layer (Tennekes, 1973). In spite of the uncertainty in knowing how to relate laboratory to field observations of the phenomenon, it is clear that some organised process in turbulent flows is responsible for producing a negative correlation between u and w , and that the events exemplified in Fig. 5.33 are the major agents involved, so it is on this basis that we will investigate them. A definition of the events which is closely in accord with this approach is that used by Gordon and Witting (1977) and this is the scheme we have used. A long record (12 min in our case) of uw was taken and the largest single value of IuwI found. The uw time series was followed backwards and forwards in time from the time of peak luwl until uw had decayed to 1/10 of the peak value; all the data points included in this time interval were taken to comprise the first event. This was identified by quadrant as a burst, sweep, etc., and its duration, stress contribution and amplitude (defined as the mean of uw over the event) calculated. The record was searched for the next largest luwl of the remaining points to find the second event, and the process continued until the cumulative & due to the identified events was 90% of the total for the record. Events identified in this way are shown in Fig. 5.33. In this case 90% of the stress was accounted for in only 26% of the time. Some of the principal features of the Reynolds stress can be examined in terms of bursting. The variation in the stress with current speed, with bed roughness, and with height must be associated with variations in the bursting activity. But, for example, is the increased stress which accompanies an increase in current speed caused by a greater number of events, by longer-lasting events. by larger amplitude events, or by a reduction in the relative contribution from d-decs and up-accs? Measurements from three distinct sites and flow conditions have been analysed to study this:
( 1 ) Run 509-Weymouth bile mud and shell.
Bay: Approximately steady flow over smooth immo-
255
60
60
40
40
20
20
0
0
ap 150
15 0
'0 100 wzl
100
Ll
Ll;
5:c
50
50 0
Time (BST) 11 ADril 1976
Fig. 5.35. Statistics of bursting for Run 616 at z
= 65
cm
( 2 ) Run 513-Start Bay, Stn. 1: Approximately steady flow over rough rippled mobile fine sand. (3) Run 616-Start Bay, Stn. 2 : Decelerating flow over fairly rough immobile coarse sand and shell. Values of the measuring heights, roughness lengths and waterdepths are given in Table 5.1.
256
6.2. Variation of stress with current speed
The data from Run 616 (Fig. 5.35) exhibits a large variation in current speed. The numbers of bursts and sweeps occurring in each 12-min record are similar to each other, and decrease from about 50 to 10 as the near-surface velocity U,-5 decreases from 122 to 23 cm s-’. The number of up-accs and d-decs stays roughly constant at about 10 per 12-min record. The durations of all types of event increase with decreasing current speed, with bursts and sweeps having longer durations than up-accs and d-decs. The amplitude of all the events are similar and decrease with l.lps. Bursts and sweeps each contribute about 50% to the net with up-accs and d-decs both contributing about - 5% (the net & due to identified events is 90% of the total see earlier). These fractions are more or less independent of U,-,, except for the last three records when wave activity may have affected the figures. Thus the increase in Reynolds stress with current speed is caused principally by an increase in the amplitude of the events, with an increase in their duration being offset by a reduction in their number. The amplitude is found to be approximately proportional to L(%5, giving rise to a quadratic friction law. A similar behaviour could be seen in the accelerating and decelerating portions of Run 509 and 5 13.
z,
z,
6.3. Variation of stress with height
Table 5.7 shows mean statistics of the four kinds of events at heights of 30 and 140 cm for a 2.2-h period of Run 509 and a 4.2-h period of Run 513 when velocities were approximately steady. Standard errors (not shown) were also calculated and allowed the significance of differences between the mean values to be assessed. The kinematic stress at 140 cm is significantly larger than that at 30 cm for Run 509, but the stresses are equal for Run 513. This is unexpected, as the stress might be expected to increase upwards over the dune topography of Run 513 as found by Smith and McLean (1977), but be constant or decrease with height over the flat bed of Run 509. However, we will take the results at face value and examine how the stresses are made up in terms of the events. The number, duration, amplitude and stress contribution of bursts are all similar to those for sweeps (though not in all cases significantly so), and those for up-accs are similar to those for d-decs. The number of all events decreases with height, while their duration increases. for both Run 509 and 513. However, while the amplitude and stress contribution of all events decreases with height for Run 513, they remain almost constant for Run 509. The difference between the two runs may be due to the topography in Run 5 13, or it may indicate that while changes in amplitude and stress contribution occur for the small z / z g of Run 513, they become constant at the larger z / z o of Run 509.
-z
6.4. Variation of stress with bottom roughness
The figures in Table 5.7 can also be used to assess the difference in bursting activity between the smooth bed of Run 509 and the rough bed of Run 513. I t is not possible to assign completely differences in bursting between Runs 509 and 5 13 to the difference in z o , as while z is the same in both runs the ratios z / z , , and z / h are
TABLE 5.7 Mean values of the bursting statistics for Runs 509 and 513 Run
509 513 509 513
I
U
(cm)
(cms-’)
30 30 140 140
36.7 40.0 44.7 54.7
- uw
Number/( 12 min)
B 5.6 12.3 7.2 12.3
55 68 36 47
S
56 71 34 41
U
21 35 12 15
Duration
D 20 34 11
13
B
1.6 1.2 2.6 1.8
(5)
S 1.4 1.0 2.7 1.8
U 0.8 0.7 1.7 1.2
D
0.9 0.9 1.6 1.2
Amplitude (cm’ s C 2 )
Stress contribution (%)
B
B
S
U
D
58 75 54 61
53 71 53 50
-9 -26 -9 -11
-11 -29 -8 -10
26 87 29 73
S 26 96 29 68
U 24 93 26 67
D 25 86 25 65
258
not, nor is the near-surface velocity U,- the same. Bearing this in mind, we note in Table 5.7 that the number of all events is greater over the rough bed of Run 513 than over the smooth bed of Run 509, their duration is shorter, and their stress contributions greater. However, the most striking difference is that the amplitude of all the events is much greater over the rougher bed, and by an amount which greatly exceeds that due solely to the difference in U5+5. 6.5. Other fbctors
In summary, the variations in number, and in duration, of events with current speed, height or bed roughness tend to counteract each other so as to give a roughly constant fraction of the time being occupied by events. This is about 10-13% of the time for bursts and sweeps. and 2-4% for up-accs and d-decs, under all conditions. For the remaining 75% or so of the time the signal is comparatively quiescent. The relative contribution from the different events varies with height and roughness, but not with current speed. The different events all have about the same amplitude, and this increases with increased roughness and as the square of the current speed. McLean and Smith (1979) made turbulence measurements at 5 levels between 10 and 215 cm above the bed, from which they produced uw time series. Lagged cross-correlations of uw at pairs of different heights showed little correlation, indicating that the bursting events, if present, were of limited vertical extent. As each measurement level was at roughly twice the height of the level below it, this means that the vertical extent of events is 2 0 . 5 ~This . compares with a streamwise extent, obtained from our data by converting the durations in Table 5.7 via Taylor’s hypothesis, of about 0.5-2z for bursts and sweeps. The weakness of the vertical cross-correlation means that caution is necessary when interpreting the detailed structure of the bed shear stress from measurements of uw made some distance above the bed. The motions which convey momentum vertically, giving rise to the Reynolds stress, may also convey scalar quantities such as heat, salinity or sediment. Antonia (1977) found that bursts and sweeps of heat flux in the atmosphere occurred simultaneously with bursts and sweeps of momentum flux, with sweeps providing the major contribution to both horizontal and vertical heat fluxes. Grass (1974) observed fine sand being carried upward by bursts in single continuous motions through almost the entire boundary layer of his relatively low Reynolds number laboratory experiment. Our own observations of a sandy sea bed using an underwater television camera confirm that the sediment is carried upward in intermittent swirls, though it remains to be shown that these correspond to bursts.
7. CONCLUSIONS
In this chapter we have examined a number of idealised types of bottom boundary layer. However, nature is rarely simple, and in the sea the actual observed boundary layer will generally be the resultant of several different effects acting together. Thus the oscillatory-planetary boundary layer in a shallow sea may be
259
depth-limited, it will frequently exhibit significant thermal stratification effects, i t may additionally be modified by sandwaves or sandbanks, and for non-uniform sediment distributions it may also develop an internal boundary layer. Even at carefully chosen sites with simple bathymetry and uniform sediments it is not always easy to compare theory with fact. For example, the compasses in a string of conventional current meters need very careful intercalibration if they are to detect vertical veering which amounts to only a few degrees from one current meter to the next, and is often time-dependent as well. Nevertheless, as far as can be judged, the boundary layer beneath tidal and other currents on the continental shelf appears to exhibit many similarities with boundary layers in the atmosphere and the laboratory. However, it would be presumptious to suppose that every facet of the marine boundary layer is an identical analogue of those found in better known boundary layers. Each feature has to be checked as carefully as is possible against measurements in the sea, and while a qualitative correspondence with the expected behaviour is generally found, there may also be quantitative differences. Thus, for example, while a viscous sublayer similar to that found in the laboratory has been shown to exist under suitable conditions at the seabed, its effect on the logarithmic part of the velocity profile as measured by the roughness length appears to be different in the sea. On the other hand, the slope of the logarithmic profile in the sea, as indicated by Von Karman’s constant, is apparently much the same as in the laboratory, but is different from the currently accepted atmospheric value. Differences such as these could arise if apparent constants are in reality weak functions of, say, the Reynolds number or the surface Rossby number of the flow. Generally speaking these numbers, expressing essentially the ratio of the boundary layer thickness to the thickness of the bed layer, are rather smaller in the marine boundary layer than in the atmospheric, but are much larger than in laboratory flows. Theoretical treatments of the marine bottom boundary layer are quite well advanced, with many approaches being available which are more sophisticated than those given here. Most of them have been applied to isolated and specific cases, rather than forming the basis of a general theoretical framework, but should in principle be capable of generalisation. By contrast measurements in the sea with which to test the theory are relatively scarce. Because field experiments by their nature lack the control of laboratory experiments, results from many experiments at different sites, and made with different instruments are needed before a typical pattern emerges. It is apparent that while there are ample field observations of some aspects of bottom boundary layer flow, there are also conspicuous gaps. Over the last ten or fifteen years a relatively large number of measurements of mean velocity profiles, as well as a lesser number of turbulence measurements, has been made in the height range 10-200 cm. Nonetheless, the catalogue of roughness lengths (Table 5.4) shows only one measurement made over mud, and none over a rocky bottom. Outside that height-range measurements are much scarcer. Below a height of 10 cm measurements are only just starting to be made, and some of the results obtained will need confirmation from other sites before they can be considered to be generally true. At heights above 2 m quite a few measurements of mean velocity have been made, though mostly at a rather limited number of discrete heights and not made
260
simultaneously with near bed measurements. Only a very limited number of measurements of profiles of turbulence extending above 2 m have been made, and to my knowledge none at all to heights exceeding 10 m. This must be due partly to the difficulty of providing a stable mounting for the instruments well above the bed. Measurements of turbulence profiles would also help to improve empirical expressions for the scaling of the spectral distribution of turbulent energy, such as eq. 90. A number of measurements of the dependence on density stratification exist, but rather fewer of the dependence on height. An added relevance of such expressions is that the wavelength of the peak of the w spectrum is thought to govern the mixing length, used in turbulence modelling. Finally, although considerable effort has gone into the study of the bursting phenomenon in its own right, especially in the laboratory, i t still remains for a practical and usable method to be proposed which will incorporate the processes involved into a theoretical framework for the bottom boundary layer.
8. ACKNOWLEDGEMENTS
I wish to thank my colleagues for their useful discussions, and Drs. S.R. McLean and D. Prandle for their constructive comments on this topic. Much of the programming and data handling was performed by Ms. H.L. King, Mrs. B. Wainwright and Mrs. L. Ellett. These are thanked, as are Drs. R. Flather and R.D. Pingree who kindly made available their numerical model results and field data respectively. The cooperation of the Marine Biological Association of the U.K. and the captain, officers and crew of the RV'"Sarsia", in the collection of the new data, are gratefully acknowledged. Part of the new work was supported financially by the U.K. Department of the Environment.
9. LIST OF SYMBOLS
Symbol
Meaning Fourier coefficients dimensionless constants non-dimensional current speed, shear stress, respectively Kolmogorov constants for the one-dimensional u , 0 and w spectra, and the three-dimensional spectrum, respectively exponent in the sediment-concentration profile [ = w , / K u * ] dimensionless constant drag coefficients: C,,, applies to U,,, sediment grain diameter kinetic energy of turbulence per unit volume of water [ =
ip(U2 + 7 + 7)J
spectral distribution of a in terms of wavenumber
26 1
Coriolis parameter [ = 2 D sin $ 1 functions acceleration due to gravity waterdepth height of roughness elements, sand ripples, sandwaves, etc.
=J-1
L n
Q R,, R
imaginary part of complex quantity wavenumber in radians per unit length [ = 2 a n / U ] non-dimensional wavenumbers [ = k z , kz,] eddy viscosities [ = - %/( a W / a z + a W / d x ) , etc.] Kolmogorov length scale [ = v3/4c11/4] lengths describing bed roughness geometry (e.g. spacing of elements) Monin-Obukhov stability length frequency in cycles per unit time complex velocity [ = U + i V ] anticlockwise, clockwise rotary components of velocity, respectively gradient, flux Richardson numbers, respectively real part of complex quantity surface Rossby number [ = u . / f i o ] complex water-surface slope [ = d l / d x i d { / d y ] spectral distribution of (Y in terms of frequency time coordinate fluctuating part of velocity in x, y , z directions, respectively friction velocity [ = ( r 0 / p ) ' / * ] threshold friction velocity for sediment motion amplitude of oscillatory friction velocity rate of change of friction velocity [ = du,/dt] mean velocity in x , y , z directions, respectively semi-major, semi-minor axes of tidal ellipse, respectively mean velocity at I = 100 cm, 5 m below surface, respectively sediment-settling velocity (positive) right-handed Cartesian coordinate axes: unless indicated otherwise, the origin is at the sea bed, with x along the current direction, and z vertically upwards seabed roughness length displacement height a length-scale proportion to A, general turbulent variables dimensionless constants boundary-layer thickness (general) boundary-layer thickness based on U = U,, E = 0 and r = 0, respectively internal boundary-layer thickness angle between bed shear-stress direction and geostrophic flow direction
+
zo
non-dimensional boundary-layer thickness rate of dissipation of turbulent kinetic energy per unit mass of water water-surface elevation bearing of current, shear stress, respectively, expressed relative to north and with opposite sense of rotation to +", GT. Von Karman's constant wavelength of bedform wavelengths at which k E , , ( k ) , kE,,( k ) , respectively, have their maxima wavelength at which k E , , ( k ) first matches a chord with -2/3 slope length scale associated with acceleration [ = u,lu,l/u.] kinematic viscosity of water non-dimensional variables proportiona: to z ' 1 2 , zd/', h'", respectively non-dimensional group of readily estimated flow variables density of water mean, fluctuating part, respectively, of the density of stratified water or suspension density of sediment angular frequency of oscillatory tidal motion standard deviation of a [ = (2)'/*] shear stress in the x direction which is exerted by the water above a plane orthogonal to z , on that below it defined analogously to r x z complex shear stress [ = T~~ + i r y z ] bed shear stress [ = )T(,=,,,] direction (or phase) of current, shear stress, respectively relative to the x-axis (or t = 0) direction (relative to x-axis) of major axis of tidal ellipse latitude angular frequency of earth's rotation depth average of X value of X at a large distance above the seabed mean of X over, typically, 10 min mean of X over a tidal cycle anticlockwise rotary component of X clockwise rotary component of X
Abramowitz, M. and Stegun, LA., 1964. Handbook of Mathematical Functions. Dover, New York, N.Y., 1046 pp. Adams. C.E. and Weatherly, G.L., 1981. Some effects of suspended sediment stratification on an oceanic bottom boundary layer. J. Geophys. Res., 86: 4161-4172.
263 Antonia, R.A., 1977. Similarity of atmospheric Reynolds shear stress and heat flux calculations over a rough surface. Boundary-Layer Meteorol., 12: 35 1-364. Anwar, H.O., 1981. A study of the turbulence structure in a tidal flow. Estuarine Coastal Shelf Sci.. 13: 373-387. Ariel, N.Z. and Nadezhina, Y.D., 1976. Dimensionless turbulence characteristics under various stratification conditions. Izv. Atmos. Oceanic Phys., 12: 492-497. Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. Geol. Surv., Prof. Pap. 422-1, U.S. Govt. Printing Office, Washington, D.C.. 37 pp. Blackwelder, R.F. and Eckelmann, H., 1979. Streamwise vortices associated with the bursting phenomenon. J. Fluid Mech., 94: 577-594. Bowden, K.F., 1978. Physical problems of the benthic boundary layer. Geophys. Surv., 3: 255-296. Bowden, K.F. and Ferguson, S.R., 1980. Variations with height of the turbulence in a tidally-induced bottom boundary layer. In: J.C.J. Nihoul (Editor), Marine Turbulence. Elsevier, Amsterdam. pp. 259-286. Bowden, K.F., Fairbairn, L.A. and Hughes, P. 1959. The distribution of shearing stresses in a tidal current. Geophys. J. R. Astron. Soc., 2: 288-305. Bradley, E.F., 1968. A micrometeorological study of velocity profiles and surface drag in the region modified by a change in surface roughness. Q. J. R. Meteorol. Soc., 94: 361-379. Busch, N.E., 1973. On the mechanics of atmospheric turbulence. In: D.A. Haugen (Editor). Workshop on Micrometeorology. American Meteorological Society, Boston, Mass., pp. 1-65. Businger, J.A., 1973. Turbulent transfer in the atmospheric surface layer. In: D.A. Haugen (Editor). Workshop on Micrometeorology. American Meteorological Society, Boston, Mass., pp. 67- 100. Businger, J.A., Wyngaard, J.C., Izumi, Y. and Bradley, E.F., 1971. Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28: 181-189. Caughey, S.J.,1977. Boundary-layer turbulence spectra in stable conditions. Boundary-Layer Meteorol.. 11: 3-14. Charnock, H., 1959. Tidal friction from currents near the seabed. Geophys. J. R. Astron. Soc., 2: 215-221. Chriss, T.M. and Caldwell, D.R., 1983. Universal similarity and the thickness of the viscous sublayer at the ocean floor. J. Geophys. Res., in press. Comte-Bellot. G., Sabot, J. and Saleh, I., 1978. Detection of intermittent events maintaining Reynolds stress. In: Proceedings of the Conference on Dynamic Measurements in Unsteady Flows. Marseilles and Baltimore. Dynamic Flow Conference, Skovlunde, Denmark, pp. 2 13-229. Corino, E.R. and Brodkey, R.S., 1969. A visual investigation of the wall region in turbulent flow. J. Fluid Mech., 37: 1-30. Csanady, G.T., 1967. On the “resistance law” of a turbulent Ekman layer. J. Atmos. Sci., 24: 467-471. Defant, A., 1961. Physical Oceanography, Vol. 2. Pergamon. Oxford, 590 pp. Dyer, K.R., 1971. Current velocity profiles in a tidal channel. Geophys. J. R. Astron. Soc., 22: 153-161. Dyer, K.R., 1980. Velocity profiles over a rippled bed and the threshold of movement of sand. Estuarine Coastal Mar. Sci., 10: 181-199. Ekman, V.W.. 1905. On the influence of the Earth’s rotation on ocean currents. Ark. Mat., Astron. Fys., 2: 1-53. Falco, R.E., 1974. Some comments on turbulent boundary layer structure inferred from the movements of a passive contaminant. In: AIAA 12th Aerospace Sciences Meeting, Washington, D.C. Paper 74-99, American Institute of Aeronautics and Astronautics, New York, N.Y. 5 pp. Flather, R.A., 1976. A tidal model of the north-west European continental shelf. Mem. Soc. R. Sci. Liege, Ser. 6, 10: 141-164. Gordon, C.M., 1974. Intermittent momentum transport in a geophysical boundary layer. Nature, 248: 392-394. Gordon, C.M., 1975. Sediment entrainment and suspension in a turbulent tidal flow. Mar. Geol., 18: M57-M64. Gordon, C.M. and Dohne, C.F., 1973. Some observations of turbulent flow in a tidal estuary. J. Geophys. Res., 78: 1971-1978. Gordon, C.M. and Witting, J., 1977. Turbulent structure in a benthic boundary layer. In: J.C.J. Nihoul (Editor), Bottom Turbulence. Elsevier, Amsterdam, 306 pp.
264 Grant, H.L., Stewart, R.W. and Moilliet, A., 1962. Turbulence spectra from a tidal channel. J. Fluid Mech., 12: 241-263. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. J. Fluid Mech., 50: 233-255. Grass, A.J., 1974. Transport of fine sand on a flat bed: turbulence and suspension mechanics. In: Euromech 48, Proceedings of the Colloquium on Transport, Erosion and Deposition of Sediment in Turbulent Streams, Copenhagen. Institute of Hydrodynamics and Hydraulic Engineering, Lyngby, pp. 33-34. Harvey, J.G. and Vincent, C.E., 1977. Observations of shear in near-bed currents in the southern North Sea. Estuarine Coastal Mar. Sci., 5: 715-731. Heathershaw, A.D., 1974. “Bursting” phenomena in the sea. Nature, 248: 394-395. Heathershaw, A.D.. 1979. The turbulent structure of the bottom boundary layer in a tidal current. Geophys. J. R. Astron. Soc.. 58: 395-430. Heathershaw, A.D., I98 1. Comparisons of measured and predicted sediment transport rates in tidal currents. Mar. Geol., 42: 75-104. Hinze, J.O., 1975. Turbulence (2nd ed.). McGraw-Hill, New York, N.Y., 790 pp. Jackson, N.A., 1976. The propagation of modified flow downstream of a change of roughness. Q. J. R. Meteorol. SOC., 102: 775-779. Jackson, P.S.. 1981. On the displacement height in the logarithmic velocity profile. J. Fluid Mech.. 1 1 1 : IS-25. Jonsson, I.G. and Carlsen, N.A., 1976. Experimental and theoretical investigations in an oscillatory turbulent boundary layer. J. Hydraul. Res., 14: 45-60. Kaimal, J.C., Wyngaard, J.C., Izumi, Y . and Cote, O.R., 1972. Spectral characteristics of surface-layer turbulence. Q. J. R. Meteorol. Soc., 98: 563-589. Kamphuis, J.W., 1974. Determination of sand roughness for fixed beds. J. Hydraul. Res.. 12: 193-203. Kim, H.T., Kline, S.J. and Reynolds, W.C., 1971. The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech., 50: 133-160. Knight, D.W., 1978. Review of oscillatory boundary layer flow. J. Hydraul. Div., Proc. Am. Soc. Civ. Eng., 104: 839-855. Kraav, V.K.. 1969. Computation of the semidiurnal tide and turbulence parameters in the North Sea. Oceanology, 9: 332-341. Lamb, H., 1975. Hydrodynamics (6th ed.). Cambridge University Press, Cambridge. 738 pp. Lesht, B.M., 1979. Relationship between sediment resuspension and the statistical frequency distribution of bottom shear stress. Mar. Geol., 32: M19-M27. Long, C.E., 1981. A simple model for time-dependent stably stratified turbulent boundary layers. Dept. of Oceanography, Spec. Rep. 95, University of Seattle, Seattle. Wash.. 170 pp. McLean, S.R. and Smith, J.D., 1979. Turbulence measurements in the boundary layer over a sand wave field. J. Geophys. Res., 84: 7791-7807. McPhee, M.G. and Smith, J.D., 1976. Measurements of the turbulent boundary layer under pack ice. J. Phys. Oceanogr., 6: 696-7 1 1. Miller, M.C., McCave, I.N. and Komar, P.D.. 1977. The threshold of sediment motion under unidirectional currents. Sedimentology, 24: 507-527. Milne-Thomson, L.M., 1968. Theoretical Hydrodynamics (5th ed.). Macmillan. London. 743 pp. Monin, A.S. and Yaglom, A.M., 1971. Statistical fluid mechanics, Vol. 1. MIT Press, Cambridge. Mass., 769 pp. Mulhearn, P.J., 1978. A wind-tunnel boundary-layer study of the effects of a surface roughness change: rough to smooth. Boundary-Layer Meteorol., IS: 3-30. Mulhearn, P.J. and Finnegan, J.J., 1978. Turbulent flow over a very rough random surface. Boundary-Layer Meteorol., 15: 109-132. Nikuradse, J., 1933. Laws of flow in rough pipes. National Advisory Committee on Aeronautics, Tech. Mem. 1292, 60 pp. (English transl.) Offen, G.R. and Kline, S.J., 1973. Experiments on the velocity characteristics of “bursts” and o n the interactions between the inner and outer regions of a turbulent boundary layer. Dept. of Mechanical Engineering, Rep. MD-3 I. Stanford University. 229 pp.
265 Panofsky, H.A., 1973. Tower micrometeorology. In: D.A. Haugen (Editor), Workshop on Micrometeorology. American Meteorological Society, Boston, Mass., pp. 151- 176. Pao, Y-H., 1965. Structure of turbulent velocity and scalar fields at large wavenumbers. Phys. Fluids, 8: 1063- 1075. Parker, W.R.. Smith, T.J. and Kirby, R., 1980. Observation of density stratification due to suspended fine sediment. In: T. Carstens and T. McClimans (Editors), 2nd International Symposium on Stratified Flows. Tapir, Trondheim, pp. 955-966. Pasquill. F., 1972. Some aspects of boundary layer description. Q. J. R. Meteorol. Soc., 98: 469-494. Pingree. R.D. and Griffiths, D.K., 1974. The turbulent boundary layer on the continental shelf. Nature, 250: 720-722. Pingree, R.D. and Griffiths, D.K., 1977. The bottom mixed layer on the continental shelf. Estuarine, Coastal Mar. Sci., 5: 399-413. Pingree, R.D. and Griffiths. D.K., 1978. Tidal fronts on the shelf seas around the British Isles. J. Geophys. Res., 83: 4615-4622. Pingree, R.D., Pugh, P.R., Holligan, P.M. and Forster, G.R., 1975. Summer phytoplankton blooms and red tides along tidal fronts in the approaches to the English Channel. Nature, 258: 672-677. Prandle, D., 1982. The vertical structure of tidal currents. Geophys. Autrophys. Fluid Dyn.. 22: 29-49. Praturi, A.K. and Brodkey, R.S., 1978. A stereoscopic visual study of coherent structures in turbulent shear flow. J. Fluid Mech., 89: 251-271. Schotz. S. and Panofsky, H.A.. 1980. Wind characteristics at the Boulder Atmospheric Observatory. Boundary-Layer Meteorol., 19: 155- 164. Shepard, F.P., 1963. Submarine Geology (2nd ed.). Harper and Row. New York, N.Y. and John Weatherhill. Tokyo, 557 pp. Simpson, J.H. and Hunter, J.R.. 1974. Fronts in the Irish Sea. Nature, 250: 404-406. Smith, J.D., 1977. Modeling of sediment transport on continental shelves. In: E.D. Goldberg, 1.N. McCave, J.J. O'Brien and J.H. Steele (Editors), The Sea, Vol. 6 . Wiley-Interscience, New York, N.Y.. pp. 539-578. Smith, J.D. and McLean, S.R., 1977. Spatially averaged flow over a wavy surface. J. Geophys. Res.. 82: 1735- 1746. Soulsby, R.L., 1977. Similarity scaling of turbulence spectra in marine and atmospheric boundary layers. J. Phys. Oceanogr.. 7: 934-937. Soulsby, R.L., 1980. Selecting record length and digitization rate for near-bed turbulence measurements. J. Phys. Oceanogr., 10: 208-219. Soulsby, R.L., 1981. Measurements of the Reynolds stress components close to a marine sand bank. Mar. Geol.. 42: 35-47. Soulsby, R.L. and Dyer, K.R.. 1981. The form of the near-bed velocity profile in a tidally accelerating flow. J. Geophys. Res., 86: 8067-8074. Soulsby, R.L., Davies, A.G. and Wilkinson, R.H., 1983. The detailed processes of sediment transport by tidal currents and by surface waves. Institute of Oceanographic Sciences, Rep. no. 152. 80 pp. Sternberg, R.W., 1968. Friction factors in tidal channels with differing bed roughness. Mar. Geol.. 6: 243-260. Sternberg, R.W., 1970. Field measurements of the hydrodynamic roughness of the deep-sea boundary. Deep-sea Res., 17: 413-420. Sverdrup, H.U., 1927. Dynamic of tides on the North Siberian Shelf. Geofys. Publ., 4, 75 pp. Taylor, P.A. and Dyer, K.R., 1977. Theoretical models of flow near the bed and their implication for sediment transport. In: E.D. Goldberg, I.N. McCave, J.J. O'Brien and J.H. Steele (Editors), The Sea, Vol. 6, Wiley-Interscience, New York, N.Y., pp. 579-602. Tennekes, H., 1973. Similarity laws and scale relations in planetary boundary layers. In: D.A. Haugen (Editor), Workshop on Micrometeorology. American Meteorological Society, Boston, Mass., pp. 177-2 16. Tomczak, G. and Goedecke, E., 1964. Die thermische Schichtung der Nordsee auf Grund des mittleren Jahresganges der Temperatur in 1/2" and 1 Feldern. Dtsch. Hydrogr. Z., Reihe B, 8: 1- 182. Wamser, C. and Miiller, H., 1977. On the spectral scale of wind fluctuations within and above the surface layer. Q. J. R. Meteorol. Soc., 103: 721-730.
266 Warwick, R.M. and Uncles, R.J., 1980. Distribution of benthic macrofauna associations in the Bristol Channel in relation to tidal stress. Mar. Ecol.-Progr. Ser., 3: 97-103. Weatherly, G.L. and Martin, P.J., 1978. On the structure and dynamics of the oceanic bottom boundary layer. J. Phys. Oceanogr., 8: 557-570. Wehster. I.T. and Burling, R.W., 1981. A test of isotropy and Taylor’s hypothesis in the atmospheric boundary layer. Boundary-Layer Meteorol.. 20: 429-443. Wilkinson, R.H.. 1983. A method for evaluating statistical errors associated with logarithmic velocity profiles. Geomar. Lett. (in press). Willmarth, W.W. and Lu. S.S., 1972. Structure of the Reynolds stress near the wall. J. Fluid Mech., 55: 65-92. Wimbush, M. and Munk, W., 1970. The benthic boundary layer. In: A.E. Maxwell (Editor), The Sea, Vol. 4, Part I. Wiley-Interscience, New York, N.Y., 791 pp. Wooding, R.A., Bradley, E.F. and Marshall, J.K., 1973. Drag due to regular arrays ol roughness elements of varying geometry. Boundary-Layer Meteorol., 5: 285-308. Yalin, M.S., 1977. Mechanics of sediment transport (2nd ed.). Pergamon. Oxford, 298 pp.
261
CHAPTER 6
A NUMERICAL MODEL OF SHALLOW-WATER FLOW OVER TOPOGRAPHY G.P. DAWSON, B. JOHNS and R.L. SOULSBY
1. INTRODUCTION
The observation of the formation and movement of bottom topographical features in regions of shallow-water flow has provoked many studies of the mechanisms involved. Normally the problem is split into two distinct stages; one involves the investigation of the dynamical properties of the flow, such as shear stresses, velocity profiles and turbulence characteristics, and the other involves determining relationships between these dynamical properties and the actual sediment transport associated with them. Bagnold (1966) and Yalin (1963), amongst others, have proposed formulae which are widely used to determine bed load and suspended-sediment transport from shear stresses and velocity profiles. This chapter deals exclusively with the investigation into the dynamics of the non-separating flow over fixed topographical features in the expectation that the results may be used in conjunction with sediment-transport formulae to infer the likely movements and developments of these features. Previous studies of flow over topography have been made in the field (Smith and McLean, 1977), in the laboratory and analytically as well as with numerical models. Field observations of shallow-water flow over topography are few and far between, due to the difficulty of setting up instruments in such cases, and also due to the abundance of complicating factors, such as the tidal variations and the irregularity and three-dimensionality of the topography. Soulsby (1981) does, in fact, take measurements near a sandbank which illustrate veering of currents and shear stresses due to the three-dimensional effects of the topography. One of the earliest laboratory experiments was one carried out by Motzfeld (1937) who looked at pressure variations and streamlines over a sinusoidal train of hills. Smith (1969) made shear stress and velocity measurements for various flow depths, while Zilker et al. (1977) looked at various wave slopes. Zilker and Hanratty (1979) continued this investigation by looking at steeper waves where separation occurred. Smith’s experiments are particularly well tabulated, and therefore provide a good basis for a comparison with the present numerical model. Analytical studies, such as those by Benjamin (1959) of deep flow over a wavy boundary and Sykes (1980) of three-dimensional flow over a small hump are necessarily idealised. The case of shallow-water flow driven by a pressure gradient does not lend itself easily to analytical solution. Several numerical models have previously been developed which use a turbulence closure similar to the present model, the essential difference being the use of a linear
268
mixing length formulation rather than a similarity hypothesis. These are used by Taylor et al. (1976), Gent and Taylor (1976) and Taylor (1977). However, these are models of deep flow and are not driven by a pressure gradient, which leads to important dynamical differences in particular in the distribution of the bottom shear stress. Taylor and Dyer (1977) have applied a version of these to the case of near-bed flow. Richards and Taylor (198 1) again use a similar turbulence closure scheme for a model of shallow-water flow, but periodic boundary conditions are used, implying again that there is no net pressure gradient across each hill. They also examine the shear-stress distribution and amplitude for various wave slopes. The aims of the present study are: (a) T o develop and prove the accuracy of a two-dimensional model of shallow-water flow by comparison with laboratory and field d a d ; (b) to use the model to examine a much greater variety of flow parameters and topogr:phical parameters than has been done previously; (c) to investigate the accuracy of a c ,pth averaged model for predicting shear stress over topography; (d) to examine the relationship between the velocity profile and bottom shear stress for each parameter setting, with a view to using velocity measurements from the field to estimate bottom shear stress; and (e) to develop a three-dimensional model which can be used to examine three-dimensional aspects of flow over topography. To best achieve these aims the numerical model used is basically that developed by Johns (1978), with adaptations as described in section 2, and the extension to three dimensions in section 5.
2. FORMULATION OF MODELS WITH ONE HORIZONTAL DIMENSION
Cartesian coordinates are used to denote the horizontal and vertical axes by Ox and Oz, respectively. The origin 0 is situated in the equilibrium level of the free surface. The bottom topography and the elevation of the free surface above its equilibrium level are denoted by z = - b ( x ) and z = {(x, t ) , respectively. u ( x , z, t ) denotes the Reynolds averaged velocity in the x direction, and U ( x , t ) denotes its average throughout the depth. The equations of continuity and horizontal motion for a two-dimensional and hydrostatic case are: -as+ d [ / i ~ d ~ ] = O at at - h and : au ar + u- au + w-du = - g - a{ + -1 -
at
ax
az
ax
az
where g is the acceleration due to gravity, r is the horizontal shear stress, and p is the density of the fluid.
2.1. Depth-averaged velocity model If we average these equations with respect to z we arrive at the following
269
depth-averaged equations: s + - [ (a( + h ) i i ] at ax
=O
(3)
If we further assume that ( u z -
uz)= 0, then:
where rh is the horizontal bottom shear stress. If we now use a quadratic friction law to represent the bottom shear stress;
where C, is a coefficient of friction, then eq. 4 becomes:
au
-aii
at
ax
-+u-=
a5
ul ul
ax
(5 +h)
-g--C,---
This equation, together with eq. 3, forms the basis of a simple depth-averaged velocity model.
2.2.Depth-averaged velocity and turbulent kinetic energy model A more sophisticated approach involves considering also the production, transport and dissipation of turbulent kinetic energy (TKE),E, satisfying
where K , is the exchange coefficient and 6 is the dissipation of TKE. If we assume and ii( ?/p) then averaging with respect to z gives: that iiE
=a
=m,
In this equation, and in eq. 2, we propose that:
3= c'/ZE, P
where c = 0.08 and Eh denotes the TKE at the bottom: ~
E,
=aE
where (Y is an empirical constant, and:
where /3 is an empirical constant and K is Von Karman's constant. We then arrive at the following equations:
(9)
270
and:
These, together with eq. 3, form the basis of a depth-averaged velocity and TKE model. The constants a and ,8 may be determined from experimental data, or from a depth-dependent numerical model.
2.3. Depth-dependent velocity and TKE model If we now propose that: r --K P
au Mm
eq. 2 becomes:
-au + u - +au w - = - gau -+a[K at ax az ax
az
M
*] az
and eq. 7 becomes:
--+ aE
U-+ aE
ax
at
W -aE =
az
K
($!j'+:[K M
az
dz
Subsequently, we assume that K , = K , = K . To produce a second order turbulence closure we propose that: K = ,1/41~1/2
(17)
where I is a length representative of the vertical turbulent mixing scale. A similarity hypothesis of the type used by Johns (1978) gives:
I=
-K(
E'/*/I)
a [ E '/'/I]
,I =
K Z ~
at z
=
-h
-
az
where z o is the roughness length. 2.4. Boundary conditions
At x = L , the downstream end of the analysis region, a radiation boundary condition is used:
At the upstream end, x = 0, the velocity profile is specified. (In the depth-averaged models a depth-averaged velocity is specified.) A logarithmic variation of velocity based on the bottom roughness length is used in the depth-dependent
27 1
model:
u( 2) = U"
[
In l+Further boundary conditions for the depth-dependent modei are the surface kinematical requirement, no fluid slippage at the bottom, no applied surface stress, and no diffusive flux of turbulent energy across the boundaries:
a t z = -h
u=w=0
au -= dZ
atz={
0
dE-0
atz= -h,z={
dZ
2.5. Transformation of coordinates for depth-dependent nzodel The solution of these equations is facilitated by introducing a new vertical coordinate defined by:
A further transformation of this coordinate produces a fine resolution near u = 0 and u = 1: 0
+ 0" = 0" exPC
w1
(24)
where:
and uo << 1 is a disposable parameter. This scheme has advantages for the implementation of the numerical scheme described later. If we now write fl( t )= aa/(?5 and define ri = a({ + h ) , the final set of prognostic equations is:
aii + -(a
dt
dA
ua)
+-Pl -(d5'a
017)=
-g({
a{ + 1 i a +-ac + h ) -ax ( { + h ) * F % [ at]
212
a6 + a ( u 6 ) + -i a (W E ) = at ax P at
-
where:
cp = K / P w is obtained diagnostically from:
Boundary conditions 22 are then applied at -h.
5 = 0 and 5 = E,, instead of z
=
{ and
Z =
2.6. Numerical solution procedure
2.6. I . Depth-averaged models The horizontal coordinate is discretised according to: x=xl=(i-
1 ) A x i = 1 , 2 , ..., m
A x = L / ( m - 1)
The time coordinate is discretised according to: t = t p = p A t p = 1 , 2 , ...
I f we define a ( x , , t p ) = U P and we compute 11 at points i = I , 3,. . ., m points i = 2 , 4,. . ., m then the finite difference equations are:
(32) -
1 and
from eq. 3:
In the case of the depth-averaged TKE model, as well as eq. 33, we have: fromeq. 12:
from eq. 13:
5 at
213
2.6.2. Depth-dependent model If we now define u ( x , ,tJ,t p ) = u i , then:
from eq. 27 :
from eq. 28:
These equations are solved at appropriate points using a Gaussian solution procedure with the following exceptions: The momentum equation is not applied at i = 1, where the velocity profile is prescribed. At i = m - 1 the advective term is omitted. This is not so in the energy equation as the advection scheme uses a one-sided differencing procedure. The boundary conditions 22 are now:
274
Note that the diffusive flux conditions are applied at the half points, thus ensuring formal conservation in the numerical scheme. The vertical transformation 24, leading to a fine grid spacing near the bottom and the surface, ensures that the half points are not far removed from these boundaries. The equations are solved by time stepping from an initial state until friction in the system dissipates the transient response, and a steady-state solution remains.
3. TESTS OF T H E MODELS
3. I . Hydrostatic assumption A sufficient condition for the hydrostatic assumption to hold is that the length scale of the topography is several times greater than the depth of the water, i.e. that the shallow-water approximation is satisfied. However, it has been noted that the shallow-water equations are still appropriate in cases where the topography is of a similar scale to the depth of the water, e.g. Frost et al. (1974). The results of the comparison with the experimental results of Smith (1969) fall into this category and, although the theory may not be strictly applicable, the general agreement between the computed results and experimental data is the best evidence that we have for the realism of the model.
3.2. Numerical tests 3.2.1. Initial state and integration In each of the models a final solution was produced by time stepping from an initially specified set of fields. It was found that little or no computer time was saved by attempting to use initial fields designed to reproduce the final results as closely as possible, and that setting most fields to zero was sufficient. This is because the transients produced by almost any initial field took as long to dissipate as those produced from zero initial fields. Thus initial fields were generally set as:
u ( x , ‘$1= 0 o(x,
S) = 0
Hx) =0
(42)
E j x , ‘$1= 0 K j x , 6) = v where v is the molecular viscosity, but: u ( x = 0)= specified input profile and E ( x = 0) = TKE due to velocity profile. In the case of the depth-averaged model U(x = 0) is specified. In the depth-averaged velocity and TKE model E ( x = 0) is calculated from a balance between production and dissipation of TKE derived from a one-seventh power law velocity profile based on the specified U. In the depth-dependent models, both one-seventh power law and logarithmic law
215
velocity profiles were tried:
u ( 0 , z ) = U"
(44)
It was found that, in each case, upon reaching a steady-state solution, the model needed at least several horizontal grid increments to adjust to its equilibrium profile. For this reason the topography was never introduced before the ninth horizontal grid point. In cases where a greater adjustment region occurred, the model was allowed to approach a steady state, and then a profile from further downstream was substituted as the input profile, and the integration was continued until a steady state was reached. For the production runs, input profile 44 was used, and the associated energy profile was E(O,z)=-
I2
C ' P
au
[ ) -
az
(45)
3.2.2. Conservative operators To check that the depth-dependent model was numerically consistent and conservative, an evaluation of the depth-averaged quantities was carried out:
A small discrepancy was present. This was due to the boundary conditions being applied at the half levels above the bottom and below the surface of the water, as explained in section 2. Using 21 levels with uo = 0.0005, this discrepancy is calculated to be about 0.5% of the bottom shear stress term, and does not have a significant effect on the dynamical behaviour of the system. Taking this term into account, the balance of terms was exact. 3.2.3. Numericul truncation 3.2.3. I . Horizontal truncation. A sinusoidal hill was represented with resolutions of 4, 6 and 8 grid increments, respectively, and flow parameters corresponding to those of run (A)SH (see section 4). The computed bottom shear stresses from the last two runs agreed to within 2%, though the first run had fairly large departures from these. It may be concluded that six grid increments are sufficient in this case. As a further test, parameters corresponding to those of run (A)WB were used to compare 6 and 12 grid increment hills. The results displayed in Fig. 6.1 show that differences here were mainly less than 2.5%. It may be concluded that six grid increments are sufficient for the range of parameters used in this study.
216
>
0 A
X/L
Fig. 6.1. Bottom shear stress distribution. Solid curve: 12 grid increments per hill: crosses: 6 grid increments per hill.
r lo3-
1 31 levels (b)
I+$
lo2-
i
10 -
~
.6 -5 -4 % deviation from c s I profile
-3
-2
-1
0
Fig. 6.2. Percentage deviation from a constant stress logarithmic velocity profile. Curve u: 21-level model; curve b: 31-level model.
211
3.2.3.2. Vertical truncation. A total of 21 vertical levels were used in all the production runs presented. Results for the standard parameter settings were indistinguishable from those obtained with a 31-level model, except in the region just above, and very close to, the bottom. The 21-level model was observed to have a slight excess of shear stress at the second and third levels. This varied between 0.5 and 3% of the bottom shear stress depending on the parameters used. Comparison with an 11-level model showed that this feature became more noticeable as the resolution decreased, and is therefore likely to be due to truncation error. An associated feature was observed in the velocity profiles. Figure 6.2 shows that the shape of the velocity deviation profile is similar in each case, but also that there is a greater initial deviation from a logarithmic profile in the 21-level model. It might be assumed that this initial deviation would disappear entirely were a sufficient number of vertical levels to be used. All the results following take account of this by incorporating an offset zero for the velocity deviation profiles. The offset used is taken as the average velocity deviation between the third and fourth levels in the vertical. 3.2.4. Coordinate compression The spacing of levels in the vertical depends on the parameter a,. If a logarithmic profile is to be reproduced, then, ideally, for a rough wall flow: (47)
a, = z o / h
In practice, a, = t o / h , is sufficiently accurate for all variations of h used. When the flow is hydrodynamically smooth ( z , << v / u * ) , z o ceases to be important, and the velocity profile depends upon the friction velocity and the molecular viscosity. In this case we should choose: (48)
a, = Y / U * h
Again, in practice u, = v / u , , h , is sufficiently accurate, where u,, velocity upstream of the topography.
is the friction
3.2.5. Undetermined error The result of run (G)WB showed a confinement of the boundary layer, but numerical analysis showed that there was not a balance between the individual terms in the equations used. This was possibly due to a rounding off error in one of the variables. The results of this run have therefore not been used in this chapter. 3.3. Assumptions of the depth-averaged models
In deriving the depth-averaged TKE equation, two assumptions were made: -
uE= iiE and :
(49)
These assumptions were tested against the depth-dependent model using the param-
218
eters of run (A)SH. It was found that: -
uE= (0.91 k O.Ol)GE and:
7%U-
dZ
= (1.3
37
k 0.6)G-
dZ
The first assumption is quite good, but affects only the advection of energy, which is usually negligible. The second assumption is not satisfactory, even if the factor of 1.3 is incorporated into the model. This model appears to be less useful than the simple depth-averaged velocity model, and hence it was not used further. 3.4. Tests of conslants
c and K are empirically determined constants, and it is therefore important to test the model's sensitivity to them. Firstly the value of c was halved to 0.04 in a model using flow parameters as run (A)SH. The velocity field did not change by more than 2%. In fact, in an analytical solution of the linear problem where production and dissipation of TKE balance, c drops out from the determination of velocities altogether. One might expect a highly non-linear case to show greater changes, however. c = 0.08 is the value recommended by Launder and Spalding ( 1 972). The value of K was varied to 0.2 and 0.6. This caused velocities to change by up to a factor of 1.5 near the bottom, and TKE by up to a factor of 5. Bottom shear stresses also varied by a factor of 5. Clearly results are very sensitive to K . A general consensus gives K = 0.40 which is supported in particular by Soulsby and Dyer (1981). The viscosity v is dependent upon temperature; v = 1.4 x l o p 2 cm2 s - is apcm2 s - ' was used for the propriate for water at about 1O"C, though Y = 1.1 x laboratory simulation at about 20°C. In the rough turbulent case the value of v is not relevant, as the eddy viscosity is everywhere much greater. In the smooth wall case the value of v was varied to model different temperatures, but variations in the velocities and shear stresses were not significant.
'
3.5. Physical tests 3.5.1. Velocity and shear-stress profiles on a flat bed
Firstly the depth-dependent model was used to determine the shape of velocity and shear-stress profiles over a region with no topography, i.e. the only variation in depth is that due to the slope of the water surface. Assuming that the advective term is negligible, a balance occurs between the pressure gradient and the shear-stress term:
Integrating from z to
in the vertical:
219
This shows that the expected shear-stress profile is a linear decrease from its bottom value to zero at the surface. Including the advective term gives:
so:
We can conclude that any deviation from a linear shear-stress profile is due t o advection of velocity. The velocity profile may be determined theoretically by equating the shear stress with K . &/dz e.g. if the shear stress is constant:
and K
=c”~E’”/
u,K( z o -tz
+h )
(58)
Integrating with respect to z :
u*
(59) K
This is a constant stress logarithmic profile, with which we wish to compare the computed profiles. Figure 6.3 shows the computed shear-stress and velocity profiles over a flat bed with flow parameters as in run (A)SH, as well as the deviation of the computed velocity profile from a constant stress-logarithmic (c.s.1.) profile. Both profiles are consistent with expectations, the deviation from the c.s.1. velocity profile being associated with the linear decrease of shear stress. 3.5.2. Coles Wake Law A collection of physical measurements of boundary-layer flow have suggested the existence of a feature known as “Coles Wake Law” as described by Coles ( 1 956), whereby the velocities in the upper layers of the flow exceed the c.s.1. velocities:
where y is an empirical constant and w is a function of depth. Figure 6.4 shows a comparison of the computed profile with a “Coles Wake Law” profile. The discrepancy between the two profiles may be due to the treatment of the mixing length scale in the numerical model. The model proposes that the mixing length scale increases all the way up to the surface, whereas in reality it would be zxpected to decrease towards the top of the boundary layer. To test this hypothesis an artificial attenuation of the length scale was introduced:
280
loo( Velocity / deviation
l + f
=a
1oc
1c
1
-
8
4 U
u,
1 2 1 1 6 1 2 0 1 I -5 -4 -3 -2 % Deviation
I -1
0
Fig. 6.3. Computed velocity and shear-stress profiles above a flat bed.
-
-
Fig. 6.4. Computed velocity profile and velocity profile according to “Coles Wake Law”. Solid curve: computed profile; dashed curve: “Coles Wake Law” profile.
Fig. 6.5. As Fig. 6.4 with modification of computed length scale.
28 1
This is a linear attenuation above the mid depth. The computed profile was then in closer agreement with Coles Wake Law, as shown in Fig. 6.5. However, the shear-stress profile contained a discontinuous gradient, which is unrealistic. As the bottom shear stress was changed by less than 3%, it was decided not to use the attenuation of the length scale in production runs, but to bear in mind that the velocity profiles are not strictly correct in the upper layers of the flow. This is reasonable as the attention of this study is focussed mainly on the features near the bottom.
3.5.3. Comparison with experiment Two comparisons were made with experimental measurements using the depthdependent model, one with the laboratory experiment of Smith (1969), and the other with field measurements in the river Taw. As previously noted, the parameters used in Smith’s experiment are not well within the bounds of the hydrostatic assumption, but his set of results is the only one known to the authors which might feasibly be used to test the model. The detailed results of the comparison with laboratory experiment are contained in a manuscript by Johns et al. (1980). The model was modified to simulate smooth wall flow, and comparisons were made between the experimental and computed velocity profiles, and the bottom shear-stress distribution. Though there was some doubt about the measured velocity profiles, the computed bottom shear-stress distribution was both qualitatively and quantitatively similar to the measured one (see Fig. 6.6). A particularly interesting feature of the computed results was a periodic variation in the thickness of the laminar sublayer over the topography, a feature which would be impossible to measure with the experimental equipment used. Overall, the results of this comparison showed that
Fig. 6.6. Variation of bottom shear stress over the topography. Solid curve: computed profile; dashed curve: experimental profile.
282
the numerical model was successful in reproducing an experimental bottom shearstress distribution, and it is to be expected that the model should behave equally well in a field case. Another conclusion of the study is that the over-representation of mixing in the upper layers causes smoothing of the predicted velocity profiles. However, later experiments have shown that, even with an attenuation of the length scale (as described previously), the nature of the profiles did not change significantly. The only differences were minor details in the upper sections of the velocity profiles. A computation with field case data was carried out mainly for the purpose of supplementing the sparse physical measurements obtained, but this also provided a test for the model. Input to the model was the waterdepth (bottom topography) and the depth-averaged velocity at one point. Roughness length was estimated from a measurement of shear stress at 18 cm above the bed at this point. The model showed that the shear stress at the bottom was about 70% greater than that at 18 cm, and that the velocities calculated at 18 cm and 56 cm agreed to within about 5% of the measured velocities. On-site observations suggested that the shear stress at the bottom was greater than its value at 18 cm, thus confirming the trend of the model’s prediction. Detailed results of this computation are contained in section 4.
4. RESULTS
4.1. Depth-dependent model Two sets of experiments were performed initially-one using parameter settings associated with a single topographical feature (e.g. a sandbank), and one using parameter settings associated with a periodic bottom topography (e.g. sandwaves). Table 6.1 shows the parameter settings used in each case, and Fig. 6.7 illustrates their physical meanings. In future, runs with parameters relating to a single hill will be labelled “SH”, and those relating to a wavy boundary will be labelled “ WB”. 4. I . 1. Bed shear-stress distribution
Figures 6.8-6.1 1 show the bed shear-stress distribution for runs (A)SH, (A)WB, (J)WB and (K)WB, corresponding to a bank, sine waves, asymmetrical downstream waves and asymmetrical upstream waves. The position of maximum bed shear stress can be accounted for in each case by considering both the depth-averaged velocity and the non-linear velocity advection. The depth-averaged velocity has a maximum where the minimum depth occurs, due to continuity. The minimum depth occurs slightly downstream of the crest of the hill due to the slope of the water surface. Thus in cases where velocity advection is small, as in run (A)SH, the maximum bed shear stress occurs downstream of the crest. Note that this is solely a feature of shallow-water flow. The inclusion of the velocity advection term, however, causes a major change in the pressure gradient (slope of the free surface) and the combined effect is to cause the shear stress maximum to amplify and to move upstream of the depth-averaged velocity maximum. Increasing non-linearity of the flow results in an amplification of this effect. (This result is contrary to that reported by Richards and
283 TABLE 6.1 Parameters used for numerical modelling
Standard
0.5
Variations
0.75 0.1
lo-)
0.1
5 ~ 1 0 - ~ sinx
10-2
IO-~ 0.25 0.01 5x 5 X lo-‘
(sin x)”’ (sin x ) * Standard
0.3
Variations
0.75 0.1
1.5 X lo-’
0.1
5X10-4
1-cosx
0.1 lo-)
0.25 0.01
5x lo-’ 5X
asy. upstream asy. downstream ~
Blank spaces indicate “standard” parameters
Taylor, 1981, because of the absence of a free surface in their model.) This leads to the conclusion that, for a fixed set of flow parameters, a long (shallow-sloped) hill will have a shear stress maximum downstream of the crest, a shorter (steeper-sloped) one may have a maximum actually on the crest, and a still shorter (steeper-sloped) hill will have a maximum upstream of the crest. The energy advection term, however, does not appear to play a significant role in the distribution of shear stress for any of the parameters studied. 4.1.2. Velocity and shear-stress profiles Figures 6.13-6.16 show the velocity and shear-stress profiles at various positions over the same four topographies. Note that in run (A)SH the shear-stress profiles are almost linear, and the velocity profiles are almost identical to “flat-bed’’ profiles, implying that the non-linear advective terms are small. The WB runs, however. show considerable distortion of the stress profiles and associated deviations from the “flat-bed’’ velocity profiles. On the upstream side of the hill flow is under the influence of a strongly favourable pressure gradient, which causes an acceleration of
284 sin x
I-cos x
( Run
@J SH 1
(Run @ WB)
Asymmetrical downstream waves (Run @ WB)
t uo +
0.21 kJ*
h0
Asymmetrical upstream waves ( Run @ WB)
uo
+
T ho
0.2 1
b
G
Fig. 6.7. Illustration of meaning of parameters.
the flow, and the shear stress increases rapidly near the bottom. On the downstream side the pressure gradient is adverse, causing a marked deceleration of flow and a rapid decrease of shear stress near the bottom, though the shear stress in the mid-depths does not decrease so rapidly, thus producing a maximum in shear stress above the bottom. The velocity profiles show much smaller deviations in magnitude, not more than 20%from the c.s.1. profiles. They generally show increases on the downstream side of
285
Fig. 6.8. Bottom shear-stress distribution for run (A)SH. Run
d W0
25-
20-
1 5 ~ 1.o
Fig. 6.9. As Fig. 6.8 for run (A)WB. Run @ WB
Fig. 6.10. As Fig. 6.8 for run (J)WB.
286 Run 3.0
0 WB
1
0.0
Fig. 6.11. As Fig. 6.8 for run (K)WB
the topography, and decreases on the upstream side, relative to the c.s.1. profiles. This is quite easily interpreted in terms of the shear-stress profiles. Where the bottom shear stress is smaller than the shear stress in the mid-depths, the c.s.1. profile will be based on a small u,, and therefore predicts velocities smaller than the actual profile which is associated with the greater shear stress in the mid-depths. The argument is similar for a region where bottom stress is much greater than shear stress in the mid-depths. 35.0 -
30.0-
250-
L U
L
200 -
150 -
100-
50-
Fig. 6.12. Bottom shear-stress distribution for run (B)WB. Curve a: including all terms; curve h: excluding velocity advection term.
287
Fig. 6.13. Velocity and shear-stress profiles at various positions over the topography for run (A)SH
Run @ W B
Fig. 6.14. As Fig. 6.13 for run (A)WB.
288 Run 0 W B
looR-o
05
O0OO
10
20
I 1 Fig. 6.15. As Fig. 6.13 for run (J)WB.
4.1.3. Contour plots of percentage departure from c.s.1. profiles As velocity profiles are often calculated in terms of a constant stress layer, it is interesting to examine the departure of the velocity profiles from a constant stress ,.oO
0.5
1;
10 15 20
kFJ
0 000
10
Fig. 6.16. As Fig. 6.13 for run (K)WB.
Run
0
WB
289
logarithmic profile. The logarithmic profile is:
The percentage departure from this profile is:
This is the quantity which is contoured in Figs. 6.17-6.20 for the four different topographies. These plots may be compared with the velocity profiles in section 2. The information contained in them is similar, except that the contour plots show nothing about the c.s.1. profile itself. 4. I . 4. Contour plots of shear stress (r / pu i upstream)
Figures 6.21-6.24 illustrate the distribution of shear stress throughout the body of the fluid, scaled on the bottom shear stress at a point upstream of the topography. These contour plots exhibit the same information as the shear-stress profiles in section 2 in an alternative manner. It is interesting to note how the advection of the sheared velocity profile leads to an effect analogous to a "plume" of shear stress originating at the crest of each hill. 4.1.5. Contour plots of shear stress ( r / p u i local)
Figures 6.25-6.34 illustrate the distribution of shear stress throughout the body of the fluid, scaled on the local bottom shear stress. Plots for variations of parameters in the SH runs have not been presented as the profiles are all nearly linear, as in run (A)SH and (E)WB. The only non-linear feature discernible is the slight displacement with height of the maxima and minima of shear stress. 4.2. Depth-averaged model The depth-averaged model was tested against the depth-dependent model using fixed and space-dependent coefficients of friction. In each case studied, a constant value of the coefficient of friction was calculated from the depth-dependent model, the calculation being based on parameters upstream of the topography. Also in each case a power-law representation, suggested by Johns and Jefferson (1980), was used:
?)"
c,= c(
Note that this is almost numerically identical to an evaluation based on a full-depth logarithmic velocity profile:
[ z",
C , = C In--1
These, of course, relate only to models in which the boundary layer occupies the full depth of the fluid. The values of C and n in eq. 64 were optimised to produce the best overall fit of the bottom shear-stress profiles in the depth-averaged model to
290
Fig. 6.17. Contour plot of deviation from a constant stress logarithmic velocity profile for run (A)SH. Contours are at I , 2, 5, 10 and 20%. The numbers on the axes are arbitrary. The symbols denote positions of computational points. The crowding of these above the topography is a result of the locally increased resolution.
29 1
.0
2.@
Fig. 6.18. As Fig. 6.17 for run (A)WB.
4 .E
6.0
i
.a
292
Fig. 6.19. As Fig. 6.17 for run (QWB.
293
294
4 .a
7.a
Fig. 6.21. Contour plot of shear-stress throughout the body of the fluid scaled on the bottom shear-stress upstream of the topography for run (A)SH. Contours are at 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0%.
295
'I
l
t
l
x
x
i
t
I
I
I
/I
'\
1
_-
I
/ /
I
I
t I
i
f
I I
I t
I
f
I I
I
I
I
I
I
x 1 I
.%
J
.i:
Fig. 6.22. As Fig. 6.21 for run (A)WB.
4
.a
I
' ?.&I@
f
x
296
10.11
8.0
,
I
v
6 .a-
4
.a-
? .0
.0
0
?.0
Fig. 6.23. As Fig. 6.21 for run (J)WB.
4.0
6.a
7.0
291
'8.8-f
-I
I
I
I
I
I
1
Fig. 6.24. As Fig. 6.21 for run (K)WB.
'
:
I
z z z'z z z I
i';--;i
298
Fig. 6.25. Contour plot of shear stress throughout the body of the fluid scaled on the local bottom shear stress for run (A)SH. Contours are at 0.2, 0.4,0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8. 2.0. 2.5 and 5.0%.
299
IE.(
8.f
6.0
4.0
?.0
.I!
Fig. 6.26. As Fig. 6.25 for run (A)WB
300
Fig. 6.27. As Fig. 6.25 for run (C)WB.
i
i m
r"' tC-
302
2.0
Fig. 6.29. As Fig. 6.25 for run (E)WB.
4.0
6.0
7.0
303
304
I 1
1
~
1
1
1
I
8.0-
11
.R
? .0
Fig. 6.31. As Fig. 6.25 for run (H)WB.
4.0
6.0
7.0
,
305
.0
2 .a
Fig. 6.32. As Fig. 6.25 for run (1)WB.
4.0
6.0
7.0
306
Fig. 6.33. As Fig. 6.25 for run (J)WB.
307
.E
2.0
Fig. 6.34. As Fig. 6.25 for run (K)WB.
4.0
6.0
7.0
308
L
4.0
x
2D model C:190x10~2n=0208
+ +
~:461x10-~n-0323
o o
C1=385xlO-'
C f ' CC?,"
\
!
2 0-
2
10-
OOL
Fig. 6.35. Bottom shear-stress distribution of depth-dependent model (solid curve) and depth-averaged n = 0.323) for run (A)SH. n = 0.208; + : C = 4.61 x models (0:constant C,; x: C = 1 . 9 0 ~
those in the depth-dependent model. This was done using the values obtained from runs with different values of z o / h 0 , as this is the parameter upon which C,is most dependent. These values are: C = 1.90. lop2 n
= 0.208
(66) These constants were also optimised individually for run (A)SH. Here optimisation involved a calculation based on a point upstream of the topography, as well as at the
GO
Fig. 6.36. As Fig. 6.35 for run (D)WB.
309 TABLE 6.2 Deviation of bottom sheai stress calculated by the depth-averaged model from that calculated by the depth-dependent model. The constant C, had the same numerical value, 3.85 X lo@, for all runs % deviation from depth-dependent model
Constant C, Run (A)SH standard (B)SH u / h , = 0.75 (C)SH Q / h o = 0.1 (D)SH a / l = (E)SH ~ / l = (F)SH u n / 6 = 0.25
- 12
(G)SH un/& = 0.01 (H)SH z n / h n = 5 X lo-’ (1)SH z n / h , = 5 X (J)SH (sin x ) ” ~ (K)SH (sin x)’ Run (A)WB standard (B)WB a / h , = 0.75 (C)WB u / h o = 0.1 (D)WB a / l = l o - ’ (E)WB ~ / l = (F)WB u , / J g h , = 0.25 (H)WB z o / h , = 5 X lo-’ (1)WB z o / h 0 = 5 X (J)WB asy. upstream. (K)WB asy. downstream River Taw (3 hills)
- 19
- 24
-4
- 16 - 12 - 12
- 15 -
12
-21 - 21 - 12 - 33 -3 - 32 -
10
-
12
10 -8
-
- 12 - 14 -23 to -25
Power law C, -3 -9 -4 -9 -1 -2 -
11
+5
+3 -9 -9 -7 - 14
+3 - 30
-4 -8 +2 +3 -8 -9
- 16 to -21
crest of the hill. It can be seen that this produces an excellent fit on the rest of the curve (see Fig. 6.35). It would be possible to produce individually optimised values of C and n for each case study, but as the purpose of the investigation is to be able to apply the model to any set of parameters, it would be more useful to give generally optimised values. It is clear that as the hill becomes steeper ( a / l becomes larger), the depth-averaged model becomes less successful at reproducing the bottom shear-stress variation, both in amplitude and phase (see Fig. 6.36). This is because the depth-averaged model assumes a constant velocity profile. Also the coefficient of friction is based on a power law or logarithmic velocity profile. It is clear from previous results that the vertical structure deviates from this as a / l increases, so the depth-averaged model becomes dynamically incorrect in these cases. Table 6.2 summarises the comparison of the depth-averaged model bottom shear stress with the depth-dependent case. Maximum discrepancy usually occurs near the crest of the hill, so this point is used for the comparison. 4.3. River Taw sandwaves
Measurements obtained from the site were: (1) the waterdepth; (2) the configuration of the bottom topography; (3) the velocity at a height of 18 and 56 cm above
310
Fig. 6.37. As Fig. 6.35 for river Taw models.
the crest of the second sandwave in Fig. 6.37; (4) the shear stress at 18 cm; and (5) measurements of suspended-sediment transport. It should be noted that observations suggested that the flow may be separating in the lee of the hills. The flux of water, used as an input parameter to the model, was deduced from velocity and depth measurements. The bottom roughness length was estimated from the shearstress measurement at 18 cm by first running the model with a guessed roughness length, and then changing its value using the power-law eq. 64 to scale the shear stress at 18 cm to its measured value. The results of the model are summarised in Table 6.3 for the position at which the measurements were made. Note that the computed bottom shear stress is 70% greater than the shear stress at 18 cm. Figure 6.37 shows the distribution of bottom shear stress together with a comparison with a depth-averaged model as in section 4.2, illustrating that it gives very poor results in this case. Figures 6.38 and 6.39 show the shear-stress distribution and velocity profile deviations throughout the body of the flow.
TABLE 6.3 Velocities and shear stresses at experimental points as predicted by the numerical model with zo = 5 . 7 ~ 1 0 - ~m Height above bottom (cm)
56 18 0
Measured
Predicted )u (m s - ' )
~ / (m2 p s-*)
u
1.05 X
1 . 3 9 1~0 - 3
1.095 0.958
-
-
1.04 0.931 0
~ / (m2 p s
-
~
~
1 . 3 9 10-1 ~ 2 . 3 6 ~10-3
(m s - ' )
*
-
1
1
-
-
*
*
-
"
' "
*
*
I
I
31 1
312
h \
\;
313
5. MODELLING TWO HORIZONTAL DIMENSIONS
5.I . Formulation
T h s formulation of a three-dimensional model is simply an extension to a second horizontal dimension, of the depth-dependent model described in section 2. Cartesian coordinates denote the two horizontal axes and the vertical axis by O x , Oy and Oz, respectively. The equation of continuity is now:
The equations of momentum are:
as a(us) -+-+at
a(us)
ax
?Y
1
a(wc)
P
a'$
+---
f 6 = - g ( { + h ) - +al ax
l a as --z[ .a] (68) ({+q2 P
1
and:
al -+
+
( { + qP2
(69)
The turbulent kinetic energy equation is now:
aE+ a(UE)
-
ax
at
+-a(&)
+
+--K
P
a(&) a'$
-
( { + q 3 F( K
1
$)2
+ ( $)2]
The vertical component of motion is given by:
Boundary conditions are: aE u , 0, --- 0
at
at'$=O
au ag, x aE ag, =Oat'$=[, To test the lateral boundary conditions a depth-averaged model covering two horizontal dimensions was also developed. Averaging the depth-dependent equations and calling upon a quadratic friction law produces: a{ a@ a6 -+-+-=o at
ax a y
(73)
314
att -+at
d(Utt) ax
+--a(,,)J Y
a6 a(u6)+-a(-= Uu) -+at ax aY
Jl - c,u( u2+ s2)”2 f i = -g ( 5 + h ) ax
+$5= - g ( S . + h ) -Jl -cfU(U2+sy2 aY
(74)
(75)
5.2. Numerical solution procedure The horizontal grid scheme is identical for both the depth-averaged and the depth-dependent models. The horizontal coordinates are discretised according to: x = x i = ( i - ] ) A x i = l , 2 , ... m
y=y,=(j-
1 ) A y j = l , 2 , ... n
Ax=AL/(m-1)
(76)
Ay=BL/(n-1)
(77)
The vertical coordinate in the three-dimensional model is:
E = t k = ( k - ] ) A [ k = 1 , 2 , . . . p A ( = t , / ( p - 1)
(78)
The time coordinate is: t=t,=qAt
q = 1 , 2 ,. . .
(79)
The finite difference equations are similar to those described in section 2, the only essential difference being that the Coriolis term in the “ u ” equation is evaluated at the old time step, but in the “u” equation is evaluated at the new time step. Problems were encountered in both models when attempting to produce a steady state solution for large values of a / l (steep hills). It was determined that the instability was due to aliasing, and that a satisfactory steady state could be reached using a selective horizontal smoothing operator, as suggested by Shapiro (1970). The operator should ideally smooth all 2 grid increment waves, and leave all others unaffected. It was found that a simple five point operator produced a very similar field to a more selective one operating on 25 points. It was concluded that the simple operator is sufficient to eliminate the aliasing without altering the remaining field significantly. The operator used is: 1
= tzij
+ i[ z i + i +1 . j + 2,-
1.j
+ z,.j+ I + z1.j-
11
(80)
A one-dimensional version is used along the boundaries: z i = I2z
I
+ L4[ z i + i + “ ; - I 1
(81)
These were applied to the velocity and surface elevation fields at every fifth time step. A physical basis for the necessity of applying a smoothing operator is the occurrence of the dissipation of energy in horizontal turbulent eddies. Soulsby (1981) reports observations of correlations between u’ and u’ in the vicinity of a bank, confirming that these eddies exist. 5.3. Lateral boundary conditions
To model a coastal feature such as a sandbank, it would be appropriate to have one coastal boundary and three open ones. The depth-averaged model was used to
315
evaluate the effect of various boundary conditions on the three open boundaries with flow parameters as in run (A)SH. It was found impossible to use a simple radiation boundary condition on all three open sides. However, a satisfactory solution was reached by using a radiation
(b)
3
Shaded Region
IS
Topography
Flow is from Left to Right
Contours are of y - componeni
of v e l o c i t y
/////I/
3 Fig. 6.40. a-d. Contours of y component of velocity using a depth-averaged model for various boundary formulations (see text).
316
boundary condition on the open side perpendicular to the coastline, and putting d / @ = 0 for all variables on the remaining open sea boundary. The effectiveness of these boundary conditions was tested using the four situations illustrated in Fig. 6.40a-d. Diagram a shows the symmetrical case with two solid side walls. Diagram b shows the case where the analysis region is widened and the resultant asymmetry across the hill. Diagram c shows the case where the open-sea boundary condition is changed to a/+ = 0. The result is very similar to b. Diagram d shows the same open-sea boundary condition, but with the analysis region back to its original size. The features are the same as in c with only minor distortions of the pattern. It may be concluded that this open-sea boundary is a reasonable one provided that it is not used too close to the topography. This boundary condition should be satisfactory in the depth-dependent model also. 5.4. Results of the depth-averaged model
Figure 6.41 shows the bottom shear-stress distribution over an elongated bank, for which a / [ = in the direction of the main component of flow. This may be There is a distinct compared with Fig. 6.42 which is the same except that a / / = difference in the pattern. In the first case flow is deviated by up to about 7" around the corners of the bank, and there is reduced flow over the crest, resulting in a minimum in the shear stress near this point. In the second case the flow deviation is smaller, only up to about 4", and in this case there is a maximum in the shear stress over the bank. Clearly the shorter (steeper) hill deviates the flow less than the longer one.
Fig. 6.41. Bottom shear stress scaled on the bottom shear stress at X as predicted by a depth-averaged model for a / , =
317
Fig. 6.42. As Fig. 6.41 for a / / = lo-*.
5.5. Results of the three-dimensional model
Preliminary runs were performed to look at the vertical structure in the cases examined with the depth-averaged model. Veering in the vertical was found to be < 0.3' due to the topography with a / l = l o p 3 ,and the Coriolis force produced an additional veering of similar magnitude. The shear stress exhibited veering about twice as large as the current. With a / l = lo-* it is to be expected that veering due to the topography would be much greater, causing significant differences between the depth-averaged and the depth-dependent models.
6. CONCLUSIONS
The model developed appears to be satisfactory for the purposes of this study, the most important aim of which is to be able to predict bottom shear stresses accurately. Unfortunately not many data were available to test the model's predictions, but the model did compare favourably with the experimental results of Smith (1969). The parameters in this case may not be directly applicable to the model, as they stretch the hydrostatic assumption, but there is no reason to assume that the model should behave less well when used with other parameters. Comparison with the data from the river Taw showed that the model was behaving correctly qualitatively. A stringent test was not possible as the actual measurements were subject to fairly large observational error. The main body of results produced by running the depth-dependent model with a variety of parameter settings provides a comprehensive view of the flow dynamics in regions of simple bottom topography. The parameters used reflect the extremes that are likely to be observed in the coastal environment, with the exception of very steep
318
hills, for which the hydrostatic assumption does not hold, and for which separation may occur. It is apparent that a depth-averaged model can produce a reasonable prediction of shear stress on the condition that the topography has a very small slope ( 5 and that a spatially variable coefficient of friction is used. Depth-averaged models are clearly rather poor at predicting shear-stress distributions both in magnitude and phase when velocity advection is not negligible, in which case a depth-dependent model is necessary. As the direct measurement of shear stress at the bottom poses a difficult problem in terms of instrumentation, it would be very useful to assess the possibility of using indirect methods. Diagrams of deviation from constant stress logarithmic profiles provide the basis for deriving the bottom shear stress at a point, from two or more measurements of velocity above the point. However, it must be noted that the computed profiles are only likely to be accurate up to the mid-depths, as explained in section 3 , so velocity measurements must be made near to the bottom. Further, it may be shown through mathematical analysis that a small error in velocity measurements will lead to a very large error in the calculated shear stress. However, applying the correction will invariably produce a more accurate result than assuming that the velocity profile is everywhere logarithmic. Alternatively the diagrams of distribution of shear stress throughout the fluid provide the basis for predicting the bottom shear stress in terms of the shear stress measured somewhere above the bottom. It has clearly been feasible to produce a three-dimensional model of shallow-water flow, though results from this model are still only preliminary. It should be possible to determine, through further experimentation with the model, over which range of parameters the model predicts behaviour differing significantly from a depth-averaged model or a depth-dependent model with one horizontal dimension. In all these respects the aims of this study have been satisfactorily attained, though it would undoubtedly be desirable to be able to compare the model’s predictions with experimental results involving parameters well within the acceptable limits of the model’s assumptions. Measurements from a field case would be the most desirable, though possibly also the least reliable. Having performed experiments over a variety of parameters, it should be possible to use the results to estimate the likely flow dynamics for certain combinations of parameters not already considered. However, the added complication that an irregular topography might give (such as in the case of the River Taw) would cause any estimation to be purely speculative. It would be better, if possible, to actually use the model to simulate the situation in question. The three-dimensional model offers the most scope for future experimentation, as it is at a stage where it may be applied to a similar range of parameters as those used in the present study. Obviously there would be little point in using it where results are essentially similar to depth-averaged or two-dimensional depth-dependent cases, so a determination of which cases show significant three-dimensional effects deserves attention. It is likely, for example, that a bank which is elongated in the direction of flow will exhibit more interesting features than one which is elongated in the cross-wise direction, the flow over the middle of which is essentially two-dimensional. (Note that the two-dimensional model may be viewed as a cross-section
319
across such a bank where the edges are far enough away to have a negligible effect on the flow.) The results obtained so far indicate that a bank such as the Sizewell-Dunwich bank may be modelled reasonably well with a depth-averaged model covering two horizontal dimensions, as vertical veering is very small. A topographical feature on a shorter scale produces considerably more veering, however, and thus can only be modelled with any accuracy using the three-dimensional model.
ACKNOWLEDGEMENT
This work was done under a NERC contract with the Institute of Oceanographic Sciences, Taunton. G.P. Dawson acknowledges support as a research assistant under the terms of this contract.
REFERENCES Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. Geol. Surv., Prof. Pap. 422-1, U.S. Govt. Printing Office, Washington, D.C. Benjamin, T.B., 1959. Shearing flow over a wavy boundary. J. Fluid Mech., 6: 161-205. Coles, D., 1956. The law of the wake in the turbulent boundary layer. J. Fluid Mech., 1: 191-226. Frost, W., Maus, J.R. and Fichtl, G.H., 1974. A boundary layer analysis of atmospheric motion over a semi-elliptical surface obstruction. Boundary-Layer Meteorol., 7: 165- 184. Gent, P.R. and Taylor, P.A., 1976. A numerical model of the air flow above water waves. J. Fluid Mech., 77: 105-128. Johns, B., 1978. The modeling of tidal flow in a channel using a turbulence energy closure scheme. J. Phys. Oceanogr., 8: 1042-1049. Johns, B. and Jefferson, R.J., 1980. The numerical modeling of surface wave propagation in the surf zone. J. Phys. Oceanogr., 10: 1061-1069. Johns, B., Dawson, G.P. and Soulsby, R.L., 1980. A comparative study of numerical and laboratory simulations of turbulent flow over a wavy topography. (Unpubl.) Launder, B.E. and Spalding, D.B., 1972. Mathematical Models of Turbulence. Academic Press, New York, N.Y. Motzfeld, H., 1937. Die turbulente Stromung an welligen Wanden. 2. Angew. Math. Mech., 17: 193-212. Richards, K.J. and Taylor, P.A., 1981. A numerical model of flow over sandwaves in water of finite depth. Geophys. J. R. Astron. Soc., 65: 103-128. Shapiro, R., 1970. Smoothing, filtering and boundary effects. Rev. Geophys. Space Phys., 8: 359-387. Smith, J.D., 1969. Investigations of turbulent boundary layer and sediment transport phenomena as related to shallow marine environments, Part 2. USAEC Contract AT (45-1)-1752 Ref. A69-7, Dept. of Oceanography, University of Washington. Smith, J.D. and McLean, S.R., 1977. Spatially averaged flow over a wavy surface. J. Geophys. Res., 82: 1735-1746. Soulsby, R.L., 1981. Measurements of the Reynolds stress components close to a marine sandbank. Mar. Geol., 42: 35-47. Soulsby, R.L. and Dyer, K.R., 1981. The form of the near bed velocity profile in a tidally accelerating flow. J. Geophys. Res., 86: 8067-8074. Sykes, R.I., 1980. An asymptotic theory of incompressible turbulent boundary layer flow over a small hump. J. Fluid Mech., 101: 647-670. Taylor, P.A., 1977. Some numerical studies of surface boundary layer flow above gentle topography. Boundary-Layer Meteorol., 11: 439-465.
320 Taylor, P.A. and Dyer, K.R., 1977. Theoretical models of flow near the bed and their implication for sediment transport. In: E.D. Goldberg (Editor), The Sea, Vol. 6. Wiley-Interscience, New York, N.Y., pp. 579-601. Taylor, P.A., Gent, P.R. and Keen, J.M., 1976. Some numerical solutions for turbulent boundary layer flow above fixed rough wavy surfaces. Geophys. J. R. Astron. SOC.,44: 177-201. Yalin, M.S., 1963. Mechanics of Sediment Transport. Pergamon Press, New York, N.Y., 290 pp. Zilker, D.P. and Hanratty, T.J., 1979. Influence of the amplitude of a solid wave wall on a turbulent flow. Part 2. Separated flows. J. Fluid Mech., 90: 257-271. Zilker, D.P., Cook, G.N. and Hanratty, T.J., 1977. Influence of the amplitude of a solid wave wall on a turbulent flow. Part 1. Non-separated flows. J. Fluid Mech., 82: 29-51.
32 1
CHAPTER 7
TIDALLY INDUCED RESIDUAL FLOWS I.S. ROBINSON
1. INTRODUCTION
The dominant motion in most coastal seas around the world is oscillatory flow driven by the tides of the adjacent ocean basin. This is particularly true in the shelf seas of North-West Europe, where tidal streams are typically 1 m s - ’ or more. However, whilst a knowledge and understanding of the tidal dynamics of a sea is important, it is not sufficient to determine the transport of properties associated with the water, such as heat, dissolved salts, pollutants, suspended material, etc. Whilst the tidal mean distribution of these is strongly influenced by the tidal oscillatory flow acting to disperse their concentration by a “tidal diffusion” process, their tidal mean transport is controlled by the residual flow. Here we may think of the residual flow loosely as the non-oscillatory part of the total water flow; more precise definitions of residual velocity, or residual transport will be mentioned later. Typically in shelf seas the residual flow speeds may be one, or even two orders of magnitude less than the tidal streams. In consequence, they are difficult to measure since the residual signal of a current meter may be no greater than the noise level associated with the tidal signal, and there may be non-linear errors associated with current meters which generate false residual records, as discussed by Gould (1973), Howarth (1980) and Booth et al. (1978). Thus, even the most detailed charts of residual circulation based on Eulerian or Lagrangian flow-measurement techniques are of questionable accuracy, and it is important that a theoretical understanding of residual flows be obtained as a check on the reasonableness of the interpretation given to the observation of residuals. Similarly, the use of numerical finite difference models of tidal flows in coastal seas to predict residual circulation must be handled carefully if noise associated with the finite differencing scheme is not to be confused with genuine residual flow energy. Residual flows can be produced by wind drag on the sea surface, or driven by lateral density gradients due to non-uniform salinity or temperature distributions. However, they can also be generated by the tidal flow itself. It is the non-linear interactions of the oscillating tidal streams, leading to residual flows, which form the subject of this chapter. In recent years, several workers have begun to explore this subject (e.g. Zimmerman, 1981) and a summary of their results and a description of the present state of the art, are presented here. Such tidally driven residuals are important because they are persistent features, linked to the local bottom or coastal topography, fluctuating only with the strength of the semidiurnal tides over the regular spring-neap cycle. Therefore, even if they are considerably weaker than storm-driven residual wind drifts which occasionally occur, they can contribute more
322
significantly to the overall long-term distribution and transport of water properties than do the stronger, but intermittent and directionally inconsistent wind-driven flows. Even residual flows too small to detect in tidal current meter records may be important in this context, and the only way to determine these is by solving the theoretical problem of their generation by the tides. Emphasis will be placed in this chapter on those residual flows which form closed streamlines within the area being studied, i.e. residual circulation. Non-circulatory residual flows through a coastal sea may be driven in part by tidal interactions within the sea, but whether the irrotational “ tidal stresses” produced by non-linear tidal interactions are balanced by Coriolis forces, the friction of residual flows, or by sea surface slopes, depends on the conditions of mean surface elevation and residual flow imposed at the open boundaries of the region under study. Since these depend on conditions in the outside ocean, the problem is not soluble in isolation, either by an analytical approach or by numerical modelling. This limitation to the study of residual flows must be stressed, since it is sometimes ignored in the results of modelling exercises. The best that can be done theoretically is to predict what the residual tidal stresses are. Observations of mean sea-level slopes are then necessary to determine the through residuals, as demonstrated by Prandle (1978) in estimating the flow through the Straits of Dover. Models such as those of Pingree and Maddock (1977a) for the English Channel and Maier-Reimer (1977) for the North Sea, which do not specify residual boundary conditions, cannot strictly be used to estimate through flows, although the internal circulatory residuals predicted by them are mere meaningful. Nonetheless, the accurate prediction of residual circulation is of considerable value. In practical applications, such as the transport of coastally discharged pollutants away from the inshore region, a knowledge of the local residual eddy structure associated with coastal topography is more important than knowledge of the net through drift in the whole sea. Furthermore, the measurement of through residuals is normally achieved by placing a line of current meters across a sea area, e.g. from the Isle of Wight to the Cherbourg Peninsula, to measure flows through the English Channel. Assuming that local residuals can be accurately determined at each current-meter location, it must still be determined whether the residual flow observed is contributing to the through residual transport, or is merely part of a circulatory residual gyre. A knowledge of the magnitude and horizontal length scale of such gyres associated with local topographic features is therefore essential if current-meter rigs are to be spaced to prevent aliasing of the through residual by the gyres. The fact that circulatory residuals can be determined without baundary conditions, whereas through flows cannot, is reflected in the mathematical convenience of obtaining the curl of the equations of motion and examining tidal and residual vorticity. In this way the surface elevation is eliminated from the problem. Examination of the vorticity also proves to be a useful way of understanding in more depth how the residual generation mechanism operates, and this has been the approach of all attempts at an analytical expression of the problem. The invocation of vorticity arguments permits the discussion of several types of residual generation situations in a descriptive non-rigorous manner which is a helpful preliminary to the mathematical analysis.
323 2. TIDALLY INDUCED RESIDUAL EDDIES
Although it will be shown later that the generation mechanisms are essentially similar, it is possible to isolate topographically distinct situations in which residual eddies can be expected to be generated by tidal oscillations. a. Headland eddies Figure 7.1 shows the result of taking the mean over the tidal cycle of the hourly current values near Portland Bill in the English Channel, predicted by the Admiralty tidal stream atlas (Hydrographic Department) which is based on local observations. Strong residual eddies are clearly apparent. These eddies have been studied in some detail by Pingree and Maddock (1977a, b, 1979a), Pingree (1978) and Maddock and Pingree (1978) who are able to reproduce their velocity distribution in a numerical model, provided that non-linear advective acceleration terms are included. Tee (1976, 1977) has demonstrated the occurrence of similar residual eddies in a model of the Minas Channel in the Bay of Fundy, whilst Mardell and Pingree (1981) point out how higher tidal harmonics accompany the generation of the residual flows. An apparently obvious way of explaining this phenomenon is to suggest that the tidal flow has difficulty in following the coastline closely at a sharp corner, streamlines separate at the headland and over a tidal cycle there is a net flow of water offshore from the headland as in Fig. 7.2a. The return residual flow required to satisfy continuity creates the two gyres which are observed. However, such an explanation ignores the fact that the residual circulation pattern indicates the presence of residual vorticity which must be accounted for. The scale of the gyres almost certainly requires that sources of vorticity generation other than streamline separation from a headland must be operating. One such source is the effect of
$ J
c 1 m/s __+
1
2O3O'W
t
+ I
2020'
Fig. 7.1. Tidal mean of hourly stream values off Portland Bill
324
I
I
I
I
I
I
I I I I I I I
+
Fig. 7.2. Possible mechanisms for the generation of headland eddies.
bottom friction acting where the sea becomes shallower towards the coast. For the same velocity, a shallow water column experiences a greater effect from bottom friction than a deeper water column. Consequently tidal flow parallel to the coast will experience a torque as indicated in Fig. 7.2b which will produce vorticity within the flow. Since an equal and opposite effect occurs on the reverse tidal flow, a straight coastline will not generate a net residual vorticity. At headlands, the tendency of the streamlines to “overshoot” the sharp corner will result in positive vorticity being carried offshore to the east, and negative vorticity to the west of the promontory, where it will not be completely balanced by the return flow. For a smoothly curved headland the basic tidal flow will tend to be more symmetrical, and this mechanism will not generate residual vorticity very strongly. However, Fig. 7 . 2 ~indicates a different torque which acts to produce residual vorticity when the streamline pattern around the headland is almost the same for both tidal-stream directions. Irrespective of the depth, if the bottom friction is considered to be proportional to the second power of the tidal velocity, then where a current shear occurs the bottom friction exerts a net torque on the sea. Thus, even if the flow field approaching the headland were in constant-depth water, and therefore irrotational, and the streamline pattern around the headland was that of a potential flow, the bottom friction resulting from increased velocities inshore around the
325
d t idaI streamsc
P
J
C
Fig. 7.3. Residual eddies shed by an island in a tidal stream.
headland would exert a torque to produce vorticity in the flow leaving the headland. Thus positive vorticity would be carried to the east, and negative to the west. Because the vorticity in this mechanism is being generated only at the headland, it cannot be cancelled on the return flow, and two residual gyres remain off the headland. This seems to be the most satisfactory descriptive explanation of headland eddies, since it does not require a priori that flow separation, or even tidal asymmetry occurs. Of course once the residual vorticity is present, the tidal asymmetry is a consequence, and the mechanism of Fig. 7.2.b must also occur in the presence of sloping bottom topography. In the case of tidal flow around islands both mechanisms act as local vorticity sources to produce the characteristic four residual eddies indicated in Fig. 7.3 which have been studied by Pingree and Maddock (1979b). Actual flow separation, where a streamline detaches itself from a sharp promontory to create a shear layer as indicated in Fig. 7.2d is a small-scale phenomenon which probably does not influence the flow field significantiy on scales longer than a kilometre. Because shallow-sea dynamics are so strongly influenced by bottom friction, analogy with boundary-layer separation in laboratory fluid dynamics is misguided. Because horizontal pressure gradients must be strong in shallow-sea flows to overcome bottom friction, they tend to prevent flow separation occurring. In practice this means that where a flow starts to detach from a sharp promontory as the tidal stream increases during its cycle, the water level in the flowing part soon rises above that in the backwater. The pressure gradient across the shear layer produces acceleration round the corner and the streamline separation is suppressed. Where separation does occur, the shear layer will break down due to instabilities over a short distance, and a relatively small amount of vorticity will be diffused into the flow field by horizontal turbulent eddies. b. Circulation around parallel sandbanks Caston and Stride (1970) pointed out that in the vicinity of the long parallel sandbars off the Norfolk coast in the southern North Sea, there is a tendency for ebb and flood asymmetry to occur such that there is a mean clockwise circulation around the sandbanks. In this case the sandbanks are about 10 m below the surface, compared with a typical sea depth of 40 m. They are 2 km wide, about 9 km apart and several tens of kilometres long. Huthnance (1973) has developed a two-dimensional analytical representation of tidal currents which flow obliquely to the sandbars, which can account for these residual flows. As in the headland flow, however,
326
it is more illuminating to consider the vorticity mechanisms which operate in this situation. Provided there is a component of flow normal to the topographic axis, as fluid columns move on to the sandbar, they are squeezed and the tendency for potential vorticity to be conserved results in anticyclonic (clockwise in the northern hemisphere) vorticity being generated. Conversely as fluid columns move off the sandbars into deeper water, they spin up cyclonically. This occurs whichever direction the tide is flowing, so that the net effect over a tidal cycle is for negative (clockwise) vorticity to be found over the shallow area, and positive in the surrounding deep water, as illustrated in Fig. 7.4. This is discussed further by Zimmerman (1978a). The resulting current pattern must have the residual flows parallel to the sandbars as shown, which manifest themselves as asymmetric ebb and flood tides if the tide is almost parallel to the bar. Bottom friction also tends to generate vorticity because the same bottom frictional stress will have more effect on a shallow fluid column than a deeper column. Thus for the oblique flow (a) in Fig. 7.5 (upper part) a patch of fluid straddling the slope between shallow and deep water will experience a torque in the sense shown. This torque produces vorticity which is advected with the tide. Cyclonic vorticity is produced in the fluid as it moves onto the sandbar in the flood or the ebb phase of the tide, and anticyclonic as it moves off. Then in the mean, cyclonic vorticity is found in the fluid on the sandbar, and anticyclonic in the surrounding deep water, resulting in the residual flow parallel to the sandbar as indicated. However, if the tidal flow is oblique in the opposite direction (b) as indicated by Fig. 7.5 (lower part), the same argument results in the opposite direction of mean vorticity and residual flow. Adding the two mechanisms of friction and column-stretching, it will be noted that for case (a) the effects oppose, whilst for case (b) they reinforce each other. The fact that orientation (b) appears to be the normal occurrence, not only in the Norfolk sandbanks but also wherever such tidal sandbars occur (Zimmerman, 1981 ; Off, 1963), suggests that there is some feedback from the residual current system to the process of sandbar growth. Thus the preferred orientation in which the sandbars grow is such as to produce the stronger residual circulation. Further speculation
Fig. 7.4. Vorticity generation by column stretching and squeezing over a ridge.
327
' I
flood
deep
I,/S,,
shallow
I
ebb
I
1
mean
-
Fig. 7.5. Vorticity generation by differential friction over a ridge.
about the sediment-transport processes involved is beyond the scope of this chapter, but this interface between tidal and sediment dynamics promises to be a fertile research area as evidenced by the recent work of Huthnance (1982a, b) and Heathershaw and Hammond (1980). Similar inferences from the study of headland eddies have been made bv Pingree (1978) and Pingree and Maddock (1979al c. Basin eddies
Residual eddies due to a variety of causes are to be found in estuaries and tidal embayments. A particular type of basin eddy has been identified in hydraulic models by Yanagi (1976, 1978), modelled numerically by Oonishi (1977) and represented analytically by Yasuda (1980). The mechanism is illustrated in Fig. 7.6. It is assumed that the tidal oscillation in the basin will be of a standing wave form, so that tidal streams will decrease in magnitude towards the head of the basin. On the flood tide, the effect of bottom friction acting more strongly on the water column in the shallows on either side of the basin, and the effect of lateral eddy viscosity, is to exert a torque at the sides of the basin in the sense shown in Fig. 7.6a. As the velocity decreases towards the head of the basin, so the torque is reduced. Torque of an opposite sense acts during the ebb flow (Fig. 7.6b). The torque produces vorticity in the fluid, which will oscillate tidally as the sense of the torque oscillates. However, because of the gradient in the magnitude of the torque, stronger vorticity will be brought in on the flood than is carried out on the ebb. Over a tidal cycle, the mean result is for residual eddies to exist as shown in Fig. 7 . 6 ~ . Such explanations as these can give an intuitive understanding of how residual eddies may be generated by the tides. However, vorticity arguments can be decep-
328
Fig. 7.6. Residual eddy generation in a tidal basin.
tive, particularly when a mean circulation is sought from the asymmetry of a much faster oscillatory flow. Furthermore these heuristic arguments can give little indication of the magnitudes or scales of eddy currents to be expected. A more rigorous analytical approach is therefore required.
3. DEFINITION OF RESIDUAL FLOWS
Before setting out the problem analytically it is worthwhile to discuss what is intended by the term “residual”. In coastal oceanography, where tides are predominant, the term residual is used to refer to the non-tidal part of the motion. In the context of time series of sea-level elevation or velocity, the residual is defined as the difference between the observed and the tidally predicted parameter. This yields a time-varying residual, which should contain no energy at the discrete tidal frequencies. However, for theoretical considerations it is more convenient to define the residual as the time mean, so that with this definition the residual a K of a fluid
329
parameter a would be: aR
1
The choice of the averaging period T depends on the nature of the primary flow. In what follows the driving tide is considered to be the lunar semi-diurnal frequency M , , in which case T = 12.42 h. This is a point to note when theoretical results are being compared with observations, or being used to predict practical consequences. In what follows, non-linear interactions of the M, tide only are considered, whereas in practice there will be non-linear interactions between all the tidal harmonics present. This of course produces the higher, shallow-water tidal frequencies mentioned in Chapter 4, but also gives rise to fortnightly, monthly and longer period flows which may or may not be distinguished from the observed residual flow, depending on the length of data record available for analysis. If the effect of interactions with the diurnal, and overtide frequency bands is ignored, it is perhaps most helpful to think of tidally driven residuals being generated by a single dominant semidiurnal tide whose amplitude is being modulated by the changing phase relationship of M, to S, and the other semidiurnal tidal frequencies. Thus the residual, being related to the magnitude of the driving tide, will fluctuate over the fortnightly, monthly and longer period cycles, but can be related at a given tidal cycle to the present amplitude of the semidiurnal tide. Given a long enough time series of current-meter data, it may be possible to use this fact to determine the tidally driven residual by examining the fortnightly velocity fluctuations. Some of this will be due to the astronomically driven MSf tide which could be calculated. The rest will be due to tidal interactions and will be directly related to the tidally driven residual. If the residual is assumed to be proportional to the tidal stream generating it then the ratio of the amplitude of the fortnightly period residual to the mean should be that of the S, to the M, tidal-stream amplitude. Such an approach to the determination of residuals from current-meter data remains to be tested. The residual measured by a current-meter rig is an Eulerian residual velocity. In coastal seas which are often well-mixed vertically and whose motion is dominated by the barotropic tide, it is customary to neglect baroclinic motions and ignore any vertical structure of the flow by dealing with depth-averaged velocities. This enables the problem to be reduced to two dimensions. However, it is necessary to specify carefully what is to be defined as the residual depth-averaged velocity. In water of mean depth h , and tidal elevation 1,the instantaneous depth-averaged velocity at position x o is:
where z is measured vertically upwards from the mean free surface, and uh(z ) is the local horizontal component of velocity at height z . The residual of this, termed the Eulerian residual, is:
330
This corresponds to the residual velocity which would normally be obtained from an Eulerian current-meter rig at x,,. On the other hand, if the particular fluid column present at time to. at position x o , is followed during the period T its instantaneous velocity at time t will be approximately U (j,: iidt')Vii, and the residual velocity of such a column will be the time mean of this, termed the Lagrangian residual u L , i.e.:
+
The second term in eq. 2 is known as the Stokes drift velocity, as discussed by Longuet-Higgins (1969). In principle it can be calculated from a given oscillatory long-wave motion where V U is defined, but in those regions of interest where residuals are being generated by interaction with topography, V U is not simply defined and in practice the Stokes' drift may only be obtained readily from numerical models, e.g. Pingree and Maddock (1977a). Moreover, eq. 2 is an approximation, which is valid only for weak spatial gradients of U. Observations by Dooley (1974) and Ramster and Durance (1973) have demonstrated the inadequacy of eq. 2 when V U is large, and Zimmerman (1979) has analysed in detail the validity of the assumption, "Lagrange = Euler Stokes". It should be noted that the formulation of eq, 2 is strictly valid only where the horizontal velocity u,, is assumed to be independent of depth. Where there is a vertical structure of velocity (as is always the case because of bottom frictional drag) the true Lagrangian residual must be based on a three-dimensional, rather than a two-dimensional Taylor expansion about x o. Such a three-dimensional approach is beyond the scope of this discussion, but it may well play a part in the actual transport of fluid properties. There is as yet insufficient information about the vertical structure of tidal currents in the open sea to estimate the magnitude of the vertical structure effect, although its importance is known in rectilinear flow in weakly stratified estuaries, e.g. Bowden (1963). In practice, eq. 2 will indicate the average mean transport of particles initially in a given fluid column. The effect of vertical velocity structure, allied to vertical mixing, will be to enhance the diffusion of particles horizontally from the new position of the fluid column described by eq. 2. Pingree and Maddock (1977a) have pointed out that except in the special case of a rectilinear tidal flow, the Lagrangian residual velocity is not the same as the residual transport velocity uT. This is the residual velocity required to transport, withm the mean depth h , the same volume of water as is actually transported past the given position x o , i.e.:
+
Comparison with eq. 1 indicates: (4)
The residual transport velocity is the parameter relevant to the two-dimensional
33 1
finite difference modelling of the distribution of water properties in a coastal sea. In the following discussion, u E and uT only will be considered. 4. THE DEPTH-AVERAGED EQUATIONS OF MOTION
As already indicated, the dynamics of the tidal flow will be examined in two-dimensional form by considering the depth-averaged equations. This is not to say that the effects of the vertical structure on the depth-averaged flow is completely ignored, as will become apparent, but it does limit the usefulness of the analysis in situations such as Liverpool Bay where the freshwater runoff from the Lancashire coast contributes to a density-driven residual circulation which differs between the upper and lower parts of the water column (Ramster and Hill, 1969; Heaps, 1972). The assumption is implicit that the turbulent energy generated by the bottom friction of the dominant tidal flow is sufficient to mix both mass and momentum efficiently throughout the water column. Following the notation listed in the Appendix, the instantaneous local equation of continuity is:
Integrating vertically, and applying the kinematic free-surface conditions yields the depth-averaged equation of continuity:
% +v[(h+y)u]=O at or:
a y au av -+-+-=o at
(7)
ax a y
The local horizontal equation of motion is:
au, +u,. Vu,+w-+f(ii,Xuh)= at az aU h
-gv
where the hydrostatic pressure assumption has been made: p = p a + P d l - l’ -
(8) (9)
[’ is the equilibrium tide, which incorporates the effect of the tide-generating potential. To allow for earth tides, { must be considered relative to a geopotential
surface rather than a reference frame fixed to the seabed, but it is more convenient to incorporate this correction into the definition of {’. If density-driven flows were to be modelled, a term in vp must be included as in Huthnance (1981), but this is excluded here for the sake of clarity. The last term in eq. 8, represents the internal stresses. FI, is an internal stress acting in the j direction on a plane at right angles to the i axis ( i = x, y by z, j = x, y ) . Molecular viscosity is neglected in comparison with the turbulent Reynolds stresses. It is worth examining in some detail the depth integration of eq. 8 since it gives rise to some extra terms which can generate residuals from oscillatory flow. For the
332
total acceleration terms in the x wise momentum equation:
where:
The final ter n in eq. 10 expresses the x momentum defect involved in transforming the - three-diI iensional horizontal flow field into two dimensions. Terms such as [ u 2 - (U)’] and [&- ( U ) ( 5 ) ] will normally be non-zero if there is any vertical structure in the horizontal flow field, whether due to the bottom boundary friction, in which context the terms are included by Huthnance (1981), surface wind and wave stresses, density gradients or any other cause. However, it is only where there are horizontal changes in the structure, or horizontal gradients of total depth, that the terms contribute to the momentum equation. Since residual flows are generated in those regions where the depth variations are significant, the terms may well be quite important. Because they are in the form of a Reynolds stress term, Heaps (1978) in his development of the depth-averaged equations of motion, incorporates them with lateral turbulent eddy stresses into a coefficient of lateral eddy viscosity. However, they can in principle be measured much more easily than turbulent Reynolds stresses, given a sufficient vertical density of conventional current meters on an array of current-meter rigs. For this reason they are treated here as a separate term, although in the absence of sufficiently detailed data regarding the vertical structure of tidal currents their magnitude can only be approximately assessed. I t is worth noting here that these terms will contribute to the residual flow field in two ways. Firstly, they are quadratic terms, and thus generate a mean value from harmonic quantities even if the profile shape remains constant in time. Secondly, if the structure itself varies in form over the tidal cycle, as is quite likely, there will be a further contribution to the residual momentum. There is clearly scope for further research in this area based on field measurements of the vertical profiles of tidal streams in areas of seabed topographic features. Turning to the depth integration of the final, turbulent Reynolds stress term in eq. 8, considering the x component we obtain:
Here W, is the x component of wind stress at the surface, and the bottom stress has been assumed to have a quadratic dependence on the depth mean flow. k is the coefficient of bottom friction, normally given a value between 0.002 and 0.0026. The Reynolds stress terms due to horizontal turbulent momentum exchange across the vertical x, z and y , z planes have been represented by the final term where B is a coefficient of eddy viscosity appropriate to the two-dimensional depth-averaged situation. In fact, this strictly requires:
333
Turbulence theory, as in e.g. Proudman (1953) cannot formally justify the parameterisation of the whole of the right-hand side of eq. 13 in the left-hand side, but the approximation provides a reasonably simple closure for the two-dimensional depth averaged equations. In analytic solutions B is normally assumed to be constant, whilst in numerical models B may be permitted to vary with position according to some turbulence criterion (e.g. Maier-Reimer, 1977, chose a horizontal eddy viscosity proportional to the sea depth), but is usually considered to be two-dimensionally isotropic. Hence, the full depth-averaged equation of motion becomes:
au
-+uuvii+f(k,xii)+a= at
-gv
W
_ _k_Ulul + B V
’U (14)
The corresponding equation in two components in transport form is:
+--W,
kU4-
P
a
av
-+--
at
( h+
o2
+ B’v’U
uv + a- - v2 - +fU+ ( h + { ) a , = - g ( h + S)ay( a l+ a~ ( h + 5 )
ax ( h + { )
wy- kV4.+ P
( h + 5)’
+ B’v2V
Although a formal relation exists between B and B‘, its specification is of no practical importance since B and B‘ are chosen empirically anyway. What will be noted later is how the specification of a spatially varying B or B’ can contribute to the generation of vorticity.
5. RESIDUALS FROM T H E DEPTH-AVERAGED MOMENTUM EQUATIONS
Although there is more insight to be gained from obtaining the curl of eqs. 14 and 15, it is worth pausing at this stage to examine how the non-linear terms of the momentum equation give rise to residuals in a fundamentally oscillating system, and to consider how various authors have treated the problem in this way. Let us assume that the motion being described consists of a dominant (semidiurnal) tidal frequency and a residual only, thus simplifying the situation by ignoring higher and lower tidal harmonics, and the modulating effects of a spectrum of semidiurnal frequencies. For our purposes, the residual is defined as the time mean, as in section 3, rather than the remainder after the predicted tide has been removed. Hence: u=u,+u,
l = 51 + S E
334
where u l ,
represent the oscillatory motion at the tidal frequency:
27r T
(J=-
and u E ,etc. are defined as in eq. 1 . On substitution of eq. 16 into 14, the quadratic terms contain contributions to higher harmonics and the mean flow, as well as the fundamental frequency. At this stage it is convenient to introduce the assumption that luEl<< lull and IIEI<< I{,/ which enables linearisation of the problem to be performed. Thus on taking the time mean of eq. 14 we obtain an equation of the form:
+ B V 2 u E - Fu,
(17)
Here the circumflex indicates the time-averaging process. c is a term arising from the advective terms and F arises from the bottom friction term. Both depend on u I . Terms including products of u E and lEhave been ignored. This process produces a linear equation for the residual flow, which could be solved for a given set of boundary conditions constraining the residual motion. However, the setting up of eq. 17 depends on the solution of the principal harmonic motion both to supply the form of c and F, and also to define the driving terms u i i and - k u I Iu I I/( h + 5 ) . These driving terms are described by Nihoul and Ronday (1975) as the “tidal stress”. They argue that an appropriate method of solution for residual flows is to solve first for the tidal motion to obtain the tidal stress, and then separately to solve the linear residual eq. 17. Alternatively, it is possible to obtain uT from the solution of the residual transport equation which has similar terms to eq. 17 and a further “tidal stress” term of the form -g{,v({I - l’).Nihoul (1980) uses this method in a numerical model to calculate realistic residual flow patterns in the North Sea and shows that the spatial distribution of bottom friction and tidal stress results in there being some regions where energy is transferred to the mean flow and others where it is removed from the mean flow. The actual form of the terms indicated in eq. 17 depends on whether the tidal flow is rectilinear or not. Heaps (1978) has developed these equations in detail. He discusses their value particularly in relation to their use in numerical modelling. However, the full equations become extremely complex and therefore of little value in gaining a physical understanding of the mechanisms of residual generation. Analytical solutions by this method have been achieved only for simplified cases, e.g. Huthnance (1 973) for rectilinear tidal flow over a two-dimensional bottom topography with linear friction. Huthnance (1981), however, has been able to make further use of the process of linearising the time mean of the momentum equation. Under conditions of weak bottom friction (i.e. for a bottom friction decay time longer than a day) and where the tidal excursion is much less than a topographic length scale, he shows that a Lagrangian residual flow must lie along contours off/h. The strength of the residual circulation around closed f / h contours is weakly controlled by the bottom friction,
335
i.e. the strength of the bottom friction does not affect the residual, which adjusts so that the depth-distributed bottom stress vector has a time average of zero when integrated around the fluid circuit as it moves. Whilst this result is useful to understand how weak residuals may flow in the deeper parts of shelf seas, it is not appropriate to understanding the stronger residuals produced by short-length scale coastal and bottom topography in shallow seas where the tidal excursion is often longer than the topographic length scale. Moreover, Huthnance’s result appears to neglect bottom friction as a means of generating residuals, which is clearly a possibility from eq. 17 and will be shown in the next section to be increasingly important as shallower water is considered. Loder (1 980) has developed a similar approach to Huthnance in order to demonstrate an effective mechanism for the generation of the observed residual clockwise circulation around Georges Bank at the entrance to the Gulf of Maine.
6 . THE VORTICITY EQUATION
It is worthwhile to turn from the momentum equation to consider whether the vorticity balance can cast more light on the small-scale local residual circulations associated with topographic features in shallow water. Taking the curl of eq. 14 we obtain: aw -+
at
vx
[(f+t)k,xii]
+v X a =
_-
V X
W - k V X - “‘l h+l P(h + l )
+ Bv’w
+vBxv’ii
Using the continuity eq. 6 this may be rewritten:
+
V B X v2ll h+l
expressing the rate of change of the potential vorticity, ( w + f ) / ( h + .$) as a particular fluid column is followed. Equation 19 is of most value in considering barotropic motions of the deep ocean, where the source and sink terms of the right-hand side may be small apart from the curl of the windstress, so that potential vorticity is almost conserved and is a useful parameter with which to describe the flow. In shallow seas, however, vorticity is generated and destroyed quite strongly by the source and sink terms, e.g. the bottom-friction dissipation time constant may be only a few hours. Potential vorticity is clearly not conserved even over a tidal time scale, and it is more helpful to consider the relative vorticity of a fluid column. Then rearranging, we obtain:
336
Equation 20 points clearly to the different mechanisms which enable an irrotational tidal-flow pattern to generate vorticity. The strength of these terms in typical coastal sea situations is indicated in Table 7.1. The terms are oscillatory, and it is the amplitude of the tidally varying torque which is given. The term d f/dt ( = vaf/dy) is negligibly small in tidal, coastal sea applications where the latitudinal variation of planetary vorticity is insignificant over a tidal-excursion length scale. The first term, A, on the right-hand side of eq. 20 represents the spin up or down of a fluid column as it is stretched or squeezed to pass over topography, or to TABLE 7.1 Typical magnitudes of terms in the tidal vorticity equation Continental shelf
Coastal
Estuary
sea
Typical estimates of parameters controlling vorticity generation
Seadepth h,m Horizontal length scale of h L,,m Seabed slope v h Tidal elevation 5, m L,, m Horizontal length scale of 5 Surface slope due to tide 05 Tidal-stream amplitude ul. m s - ' Horizontal length scale of u , L,, m Coriolis parameter (mid-latitude) f . sTidal frequency u, s - ~
'
100
-
104+105 lor3 1 lo 6
10 lo3
50 lo4 +
2 lo5
-
lo-*
5 loh
lor
0.5
1
lo4 + 10'
lo4
5x10 1.5 5 x lo3
-
-
-
104
lo4
2x
10-5
10- l o
Horizontal length scale of w
~ , , m
Consequent vortrcrty growth rates of terms in equation A , ( -w- - i) -f- ( hd + 5 ) = f u I V h s-* 5 x i 0 - ~ O+ 5 x 1 0 r 9 h + { dr B, v x a SC2 3 x 10-'O
lo3
2~10-~-.2~10-~ lo-'
3x
2x10-'"
F, BV ' w G, v B x dw ~
(maximum of A
dt,,,
above) 1 dw
a=--
u dt
10-hw 5x10-14'5x10-'z
10-6w lo-''
2 x 10-sw
SC2
SC2
5x
2x
5~ 1 0 - 7
S-'
5 x 10-5
2x
5x10
SC2
v2U
-
?
G
337
UNIFORV
DEPTH
(d)
Fig. 7.7. Schematic representation of vorticity generation mechanisms: (a) potential vorticity conservation -column stretching and squeezing; (b) differential friction due to velocity shear (for quadratic friction law); (c) differential friction due to lateral depth gradients; (d) asymmetrical depth profile changes.
accommodate tidal height fluctuations (see Fig. 7.7a). The magnitude indicated in Table 7.1 assumes that w is of the same order or less thanf. Term B comes from the lateral variation in vertical structure. To indicate the possible magnitude of this term, in the absence of observed data, we can consider an extreme hypothetical case in which the flow profile changes from being constant with depth to parabolic, within 10 km, as indicated in Fig. 7.8, whilst U and h are unchanged. 1i.e. in the parabolic profile, uh(z) = 1.5h-'/2ii(z + h)I/' and -u ' = %(ii)*.] For a one-dimensional, constant depth case, eq. 11 gives a , = 8 / 8 x ( u 2 -
338
EE 10 km-
Fig. 7.8. Example of vertical profile change.
U 2 ) = 1.25 x 10p5U2.Assuming a 10-km length scale for a , transverse to the flow, yields v x a = 1.25 . 10p9U2.This is represented schematically in Fig. 7.7d. Term C is the surface wind torque. Apart from the weak interaction with tidal height, it is independent of the tidal regime and irrelevant to our discussion. It has been included thus far to enable comparison to be made with the tidally driven torques. The magnitudes given in Table 7.1 are derived from a steady wind stress of 0.1 N m-*, with zero wind stress curl, acting over a variable seadepth. It should be remembered that the wind-driven torque i s intermittent, whilst the tidal torque is regular. Term D is the purely dissipative part of the bottom friction. The rate at which the vorticity is dissipated is controlled by the amplitude of the tidal velocity. The dissipation time constant Tf = ( h {)/kliiI has also been included in Table 7.1. I t is less than a tidal cycle for most coastal sea conditions. This is an important factor when considering how the tidally oscillating vorticity can be rectified to create a residual vorticity field since it implies that vorticity generated in one tidal cycle does not significantly survive into the next cycle. It should be remembered, however, that T, is itself varying over a tidal cycle as lul oscillates. Term E also originates in bottom friction, but it acts to generate positive or negative vorticity. It may be further divided into two terms:
+
The first represents the torque produced by bottom friction in a shear flow as indicated in Fig. 7.7b. It occurs only because a quadratic friction formulation has been adopted and therefore the faster flow tends to be retarded more than the slower. The second term expresses the fact that for the same depth-averaged velocity, the bottom-friction force is the same, but its depth-distributed effect is greater in shallow than it is in deep water. Bottom slopes transverse to the flow therefore generate vorticity in the sense shown by Fig. 7 . 7 ~ . Term F is a diffusive term due to lateral eddy viscosity acting to redistribute vorticity such that steep gradients are smoothed out. For a vorticity field of characteristic length scale L,, the time scale associated with this dissipation process i s L i / B . Table 7.1 indicates that in shallow seas taking B as 100 m2 s p ' (Heaps, 1978), this is longer than T, for L, of order 10 km or more, in which case bottom friction controls the dissipation of vorticity. However, it becomes significantly less than a tidal cycle when L , is less than about 2 km, suggesting a lower limit on the
339
length scales to be expected in the residual vorticity field, and consequently the size of residual circulation gyres. Term G indicates that vorticity is generated when B varies in space. I t is hard to assess its importance in reality since the actual horizontal eddy viscosity appropriate to a particular sea is not known from observations. Indeed estimates of B tend to be based o n the value which needs to be used in numerical models to achieve realistic solutions. In some tidal models (e.g. Maier-Reimer, 1977) B is allowed to vary spatially, either to satisfy a turbulent energy or other physically based closure scheme or simply to achieve a n empirical fit of modelled tides to observations. The magnitude of vorticity generated by this mechanism is indicated in Table 7.1 for a variation of B amounting to 10 m2 s - ’ in 10 km. Thus numerical models which permit B to vary must generate this magnitude of vorticity, although it appears to be small compared with the other terms. The rates of vorticity generation in Table 7.1 are tidally oscillating quantities. so that the amplitude of the tidally oscillating vorticity may be estimated from the maximum value of dw/dt (the maximum of terms A, B, E and G) as shown in the bottom two lines of the table. If the frictional decay time constant is much less than a tidal period, as in the estuarine case, w is probably over-estimated. I n the continental and coastal sea cases, water column stretching and squeezing over topography is the most important mechanism for generating vorticity, but in the coastal sea the bottom friction mechanisms are comparable to it. In both these cases the magnitude of tidal vorticity is comparable withf, but increases in shallow water. It is worth pointing out that term B is nearly comparable with A, suggesting that the horizontal variation of vertical structure could be a source of vorticity generation, and may repay further study in tidal-stream observations programmes. In the estuarine situation it is not possible to assume that w +f=f. Indeed the estimate of w as obtained from the frictional generation term E suggests that w >>f. It is therefore not meaningful to estimate the magnitude of term A. Interpretation of eq. 20 is further complicated in estuaries when the tidal elevation becomes comparable with the waterdepth. Moreover, the strong friction ensures that the vorticity survives for only a very short time, and it is concluded that the use of the vorticity approach to understand residual flows is not appropriate for shallow estuaries. In the absence of any topographic features, i.e. ~h = 0, vorticity will still be generated weakly by the tidal motion because of O[ occurring in terms A and E. However, in shelf and coastal seas the resulting magnitude of o will be 2-3 orders of magnitude less than for the topographically induced cases, and this justifies the neglect of surface slope effects in what follows. Indeed the mechanisms by which residual vorticity is generated topographically would operate just as strongly if a free slip rigid lid surface condition were imposed on the ocean.
7. RESIDUAL VORTlClTY
The magnitudes of vorticity indicated in Table 7.1 are typical amplitudes of a tidally oscillating variable. Equation 20 can also be interpreted in terms of the generation of residual vorticity by considering its time average. Terms B, C and G
340
will be discarded although they may each contain a residual component. Its magnitude for B and G can only be speculation, but since these terms were small in Table 7.1 we may conclude that they do not constitute a major source of residual vorticity, and neglect them. The wind stress curl is beyond the scope of this study, and will not be considered further. Its residual effect is much less than its instantaneous torque since the wind stress is intermittent in magnitude and direction. I t may nonetheless be the dominant influence on non-tidal water movements, particularly in shallower seas. It is not, however, predictable and regular as is the residual torque due to tidal interaction which we are considering here. The major way in which residual vorticity arises is through the non-linear interactions represented in terms A and E of eq. 20. If the time mean of eq. 20 is
If 5 << h then the non-linear effects of { in the denominator of various terms can be ignored, yielding:
Here w has been split into a,,,the r e s i d u a i t y , and a , ,the oscillating vorticity. kIiilw/( h + { ) has been approximated to kliilwo/h which is approximately 2ku,oo/ah if ii is a nearly rectilinear oscillating tidal stream with amplitude u l . The product lUl w should not produce a mean, provided I u EI << u I .
,
Treating the bottom friction dissipation in this way is somewhat of an oversimplification. Hunter (1975) has shown how the linear friction coefficient appropriate to residuals aligned with the rectilinear tidal stream should be 4u,k/7r, and 2ulk/7r for transverse residuals. The choice of a scalar linear friction of 3uIk/7r, as suggested by Hunter, had we taken the mean of eq. 14 before proceeding to take the curl, would have led to a dissipation term of 3ku,w,/ah. The exact specification of an appropriate frictional dissipation of w,, will depend on the residual flow patterns of a particular case. In this section we are concerned only to establish the order of magnitude of the terms, and note that the use of an anisotropic mean frictional drag coefficient would not contribute to the vorticity generation terms. Similarly, provided luEl << u , , the product of IUI and U in the third term on the right of the above equations will not contribute to the mean at order u: or w l u l and may be neglected, leaving: -
w+f
u.vw---
or:
h
d (h dt
2kul + { ) = - __ w,, + BV 7rh
2 ~ , 1
34 1 TABLE 7.2 Typical magnitude of terms in the time mean of the vorticity equation
u.vw
-uw
Shelf seas
Coastal seas
2 . 5 1~ 0 - ~
4 x lo-'
S-2
2.5~10-'~+2.5~10-~
4x
S-2
5 x 10-13
4x IO~."
S-2
10-~~,,
3 x 10-'w,
S-2
10-6w0
SC2
Lh
SC2 s-2
W I -
-uVh h
wii
--
-2klii nh wff B V 'ao
L,
Typical magnitudes of these terms are given in Table 7.2. Those due to the tidal fluctuation of surface height involving seem to be much smaller than the others and can be neglected. Consequently, in the absence of topographic features, the residual vorticity due to the tidal wave alone is negligibly small. There remain two mechanisms of apparently comparable magnitude, which generate residual vorticity. The first (ti. v w ) comes from the advective acceleration in the momentum equation, and represents the way in which the oscillating current carries positive tidal vorticity in one direction and negative in the other, so that if there is a horizontal gradient of tidal vorticity in the tidal stream direction, a residual vorticity field is produced. This process is studied further in the next section. The other mechanism (o,ii. v h / h ) is a non-linearity in the production of tidal vorticity by the stretching and squeezing of water columns as they move over topographic features, which might lead to some mean vorticity. However, while Table 7.2 gives the magnitude of the non-linear terms. the magnitude of the resulting time mean depends on the relative phases of the oscillating quantities. Considering that in the circumstances where o,U. v h / h is likely to contribute t o the mean vorticity, the tidal vorticity w , will itself be due largely to the column stretching mechanism, it can be shown that in typical conditions w I and u are in quadrature and therefore in fact contribute nothing to the mean vorticity. Typically the tidal flow can be considered to be nearly rectilinear so that in the situation illustrated in Fig. 7.9 when the velocity is at its maximum positive, the fluid columns are at their mid-excursion position ( B ) , travelling into shallow water, and will possess their mean vorticity. As the velocity falls to zero the column is at the shallowest part of its excursion (C), and therefore possesses its minimum vorticity. Similarly at E , in the deeper part of its excursion, it possesses the maximum vorticity. Now there may be other contributions to w , from the bottom friction generating mechanisms which are in phase with ii, but it is to be expected from
342
Fig. 7.9. Sketch to illustrate how
II
and
w
are in quadrature for column stretching vorticity generation.
Table 7.1 that these would be small, so eq. 22 reduces at a first approximation to:
A similar result was obtained by Zimmerman (1978b) using a linear dependence of bottom friction on velocity. Equation 23 indicates that the magnitude of residual vorticity is determined by the balance between its generation by the advection of tidal vorticity by the oscillating flow, and its dissipation by bottom friction dependent o e tidal velocity and by turbulent eddy diffusion. Because the magnitude of 6 . v w is so dependent o n the relative phases, it is impossible to generalise from Table 7.2 about the magnitude of oo.Zimmerman (1978b) proceeded to solve eq. 23 by considering a fundamental Kelvin wave tidal flow field, perturbed by a statistically defined depth topography. His conclusion was that the maximum amplitude of residual vorticity occurred for topographic features of length scale comparable with the tidal excursion (see also Zimmerman, 1980). Such an approach gives valuable general results, but conceals the fundamental mechanism of residual vorticity generation which has been identified by eq. 23. To explore this we return to a simplified form of eq. 20 so that the generation of tidal vorticity and its advection to produce residual vorticity can be viewed together. 8. SOME SIMPLE SOLUTIONS FOR RESIDUAL VORTICITY
Whilst the order of magnitude estimates of residual vorticity indicated in Table 7.2 and the statistical conclusions of Zimmerman (1978b) are of some value, it would be useful in the planning and interpretation of current-meter data gathering exercises to be able to estimate the probable magnitude of a residual flow induced by a particular topographic feature. Bearing in mind the discussion of the previous section, it is reasonable to simplify eq. 20 to the form:
where smaller terms and the effect of tidal elevation have been ignored.
343
u. Quasi-Lagrungian solution
In many shallow-sea situations, the semidiurnal tidal current ellipses are very thin, and the tidal streams can be approximated to the rectilinear form U = u , cos at. If we choose curvilinear coordinates (s, n ) such that s is the distance measured along the tidal stream line, and n is the normal, eq. 24 can be written:
where it is assumed that over the limited sea area being studied the phase of the tidal current is uniform and that the residual velocity associated with the residual vorticity is small enough to be neglected in comparison with V ( s ) ,the tidal stream amplitude along s. Numerical solutions to this equation have been obtained by Robinson (1981). In order to demonstrate the part played by advection in the generation of residual vorticity, eq. 25 was solved in the Lagrangian sense, following a particular fluid column: dw dt
-=
rate of vorticity generation (in source region)-dissipation
rate
Fig. 7.10. Vorticity possessed by fluid elements during their tidal excursion. Unit vorticity is imparted to the element as it passes the source position in one direction and subtracted as it returns in the opposite direction. I - 4 correspond to fluid elements which encounter the source at different points along their trajectory.
344
-1
- 5/6
- 34 -93
Max time constant = 6 lunar hours time-varying vorticity source
(b)
Distance from vorticity source (tidal excursions)
Fig. 7.1 1. Isometric plot of local vorticity variation with space and time, in a region adjacent to a vorticity source: (a) unit vorticity imparted to all elements as they pass source; (b) vorticity imparted proportional to the velocity of fluid elements, with unit vorticity for maximum velocity.
345
Thus the vorticity possessed by a fluid column was evaluated numerically as it traced its tidal excursion out and back to its starting point. Figure 7.10 shows the time history of vorticity possessed by fluid columns encountering a point vorticity source at a particular location along their tidal excursion. The decay part of the cycle
-0
‘L
Fig. 7.12. Tidal mean vorticity distribution due to oscillatory flow past a vorticity source region. The time constant for dissipation is 10 h at the maximum tidal velocity: (a) point source; (b) and (c) distributed sources: (d) long source region.
346
undulates because of the tidal fluctuation in the decay time constant. From the vorticity of individual elements the distribution of vorticity in space and time associated with a point vorticity source can be obtained as shown in Fig. 7.1 1. This clearly reveals a fundamental feature of eq. 25, that vorticity cannot penetrate further than a tidal excursion beyond the point at which it is generated. Furthermore, it cannot penetrate normal to the tidal streamlines at all, except by eddy diffusion processes. This, therefore, sets a constraint on the area of sea over which a particular topographic feature can influence the residual circulation. Because the dissipation time scale is of the order of a tidal period, the residual velocity field, which is not included in eq. 25, cannot transport the vorticity out of the region of interest before it is almost completely dissipated. Figure 7.1 la shows the case where the rate of vorticity generation is proportional to the tidal velocity (i.e. the column stretching mechanism represented by the first term on the right-hand side of eq. 25). In this case each fluid column receives the same amount of vorticity as it passes the point source. Fig. 7.1 1b illustrates the vorticity distribution when the vorticity source rate is proportional to the velocity squared, as for the frictional generating mechanisms. In this case, fluid columns passing the (hypothetical) point source gain vorticity proportional to the instantaneous tidal velocity. In each case there is a large tidal mean component to the vorticity distribution, related in magnitude to the strength of the vorticity source terms. The residual corresponding to Fig. 7.11b is shown in Fig. 7.12a. Figures 12b, c and d show the distribution of residual vorticity for distributed vorticity source regions. Figure 12d corresponds to a vorticity source region which is long and uniform in the tidal stream direction. Physically this could be due to the sea floor sloping in a direction normal to the tidal stream, as in a long uniform canal of non-rectangular cross-section. In this case, although vorticity exists along the length of the generation region, only at the ends, where there is a gradient of the vorticity generation strength in the stream direction, is there any residual vorticity. This clearly illustrates that the mechanism for the generation of residual vorticity is the tidal advection of a gradient of tidal vorticity. h. Eulerian solution Equation 25 is also capable of solution in an Eulerian sense if some simplifications are made. I t can be rewritten: aw -
at
+ V c o s a t -aw dS
where: f h
A ( s ) = -V-
dh
8s
kV C ( s ) =7
= A ( s ) c o s a t + B ( s ) cosallcosatI- C(s)lcosat(w
347
It would be attractive to seek an analytical solution of this equation in the form: w=
+ Re( W, exp(iat) + W,
W,
exp(2iat)
+ . . .)
where W,, W,, . . . are complex amplitudes, functions of s. However, the non-linearities which generate the mean from the fundamental oscillation ensure that the solutions at different frequencies are coupled in such a way that a solution is not readily available by this route. Another approach is to relate the spatial variability of w to the spatial variability of the topographic and tidal-stream amplitude forcing terms, A ( s ) and B( s ) , since eq. 26 is linear in respect to s provided A ( s ) and B ( s ) are chosen to be linear functions, and provided the tidal velocity amplitude V in the advective term is considered uniform along s. (This may not be strictly true, particularly where the tidal excursion moves over topography, but any resulting non-linear dynamics which are neglected, would be small in comparison with the vorticity forcing terms.) Thus we let: A ( s ) = z A n l exp(-ik,s) m
B ( s ) = C B n exp( -ik,s) n
and assume w(s, t )=
1 exp(-ik,,,s)+ ; C B n Y n ( i ) e x p ( - i k n s )
1 ;CA,,,W,(t)
n
m
where k, and k,, are spatial wavenumbers. The damping term C ( s ) is taken for simplicity to be spatially uniform. This is adequate for the general conclusions which follow, but such an approximation would lead to errors in the numerical solution for a particular topography. For clarity the effect of A ( s ) and B ( s ) forcing terms can be considered separately. Thus when the column stretching mechanism A ( s ) alone is considered, introducing t ' = at;we have: dW, dt' 1.e.:
ikmV cos t'W, U
=
cos t' -
C Wm ~
U
lcos t'l
dW, + ( A , l c o s t ' l - i A , c o s t ' ) ~ ~ = c o s t ' dt'
where:
A
c
- _ = I - 0
kV ha
k V -~ A,= m - kn1E 0
2
where E is the tidal excursion of a fluid column. Equation 27 can be solved numerically for different values of A , and A,. Figure 7.13 illustrates typical results. With fairly high bottom friction effects, A , = 1 (corresponding to the friction when the tidal streams are 1 m s - ' and the depth is about 10-15 m), Figs. 13a-c show how the regime changes significantly between
348
4
tidai cycles
(bi
Fig. 7.13. Variation with time of real (left) and imaginary (right) parts of W , , : (a) A , = I , A, = 2; (c) A , = 1, A, = 4;(d) A , = 0.1, A, = I ; (e) A , = 0.5, A , = 1.8.
=
1. A,
=
I ; (h)
A,
A,
= 1 and A, = 4. In the former case, the real part of W, (which corresponds to the spatial distribution of vorticity in spatial phase with the forcing distribution) is strongly sinusoidal at the tidal period (about a zero mean), whilst the imaginary part (90" out of spatial phase with the forcing) is sinusoidal at twice the tidal frequency and has a non-zero mean. When A, = 1 the tidal excursion is 1 / 7 ~ times the topographic wavelength, whereas when A, = 4 (Fig. 7 . 1 3 ~ )the excursion is slightly greater than the topographic wavelength. The advection of vorticity therefore interacts more strongly with its generation and as can be seen the real and imaginary parts of W, both have an irregular waveform composed of higher frequencies, which repeats over the tidal cycle (real part) or half the cycle (imaginary part). The mean of the real part is still zero, and that of the imaginary is non-zero. In Fig. 7.13b the advective interaction is just beginning to show in the waveform. Reduction of frictional dissipation, A , , merely has the effect of increasing the amplitude of W, without affecting its time variation significantly (Fig. 7.13d and e). To illustrate the mechanism clearly, the significance of Fig. 7.13e in the spatial domain is shown in
349
Fig. 7.14. At a the velocity is flowing to the right, and the A , cos t' term (referred to as the forcing) is strongly positive. Hence the vorticity amplitude increases where the forcing is in phase with the vorticity and decreases where they are out of phase. The result is that the vorticity amplitude increases, but the waveform moves very little to the right. At c the tide is turning and the vorticity is at its maximum amplitude. At d although the tidal stream is now to the left, it has much less effect than the A, cos r' term which is now negative, causing a reduction in the vorticity amplitude. Between d and e the generation terms causes the vorticity waveform to shift about 60" to the right despite the leftward advective influence of the tidal stream, and e-h repeat a-d in reverse. Thus the vorticity pattern, although i t is moving slightly all the time, spends half of the cycle approximately 30" to the right and half 30" to the left of the mean position shown in Fig. 7.14i. The net result of the position shift and the amplitude variation is that the vorticity distributed in phase with A ( s ) has a tidal period oscillation with a zero time mean,
topography 1
. S
distribution of vortlclty generat ion term
Fig. 7.14. Variation of w in space and time for column stretching by tidal flow over a sinusoidally corrugated topography: (a)-(h) variation over a tidal cycle; ( i ) tidal mean distribution.
350
--. 0.25 ,
A?
Fig. 7.15. Variation of tidal mean vorticity with A, (the frictional*pation parameter) and A z (which expresses the ratio o f tidal excursion to topographic length scale). I ( W,,,)is the amplitude of the spatially oscillating time mean vorticity, generated by the column stretchlng mechanism.
whilst that 90" out of phase has a non-zero mean and experiences a half tidal period oscillation ( M 4 vorticity). The sinusoidal bottom topography associated with A ( s) is also drawn in Fig. 7.14 and it will be seen that the positive mean vorticity is associated with the deeper water and the negative with the shallower, in agreement with the discussion of section 2b.
02-
5
A, = 4.0
0
20
I
6.0
4.0
I
8.0
A2
Fig. 7.16. Variation with A , and A, of mean vorticity generated by the bottom friction mechanism.
35 1
In order to examine how the magnitude of the mean vorticity (i.e. the size of the time mean of the imaginary part of W , ) varies with A , and A,, Fig. 7.15 has been plotted. This clearly shows the residual vorticity increasing with decreasing A towards a limiting value at very low A , and peaking to a maximum for A, close to 2. In fact with lower friction the mean vorticity peaks at a slightly lower A,, and for high friction the peak is not defined although the maximum shifts to higher A,. Thus the maximum generation of residual vorticity is expected when the topographic length scale is about equal to the local tidal excursion (i.e. topographic wavelength 77E/2), in agreement with the results of Zimmerman (1978b). Figure 7.16 shows curves plotted as in Fig. 7.15, but for the case of Y, driven by the bottom friction generating mechanism B ( s ) , using a numerical solution of the equation:
,
(28) The result is very similar to the column stretching mechanism, but with slightly lower magnitudes of Y . Thus given the optimum length scales of vorticity forcing, the actual mean vorticity which results is wo = 0.45A/a or 0.4B/a, or less if the frictional dissipation is greater. Consequently the maximum residual vorticity we might expect to find in a shallow sea is given by:
in the case of the column stretching mechanism, and:
for the friction mechanism. If the dissipation is large, or the topographic length scale too small or large, then the mean vorticity will be considerably less than these values.
9. RESIDUAL CIRCULATION
Whilst consideration of residual vorticity provides a relatively simple conceptual model with which to explore the mechanism of tidally driven residuals, the interpretation of vorticity in terms of residual currents cannot be uniquely achieved without the specification of residual flow conditions around the perimeter of the region under study. In order to illustrate the significance of the vorticity results and to estimate the typical magnitudes of residual flows to be expected from given topographic features, we shall consider the influence of a simple circular bump or hollow in an otherwise plain topography, and a s s u m e that the residual currents decay to zero with distance from the feature. The distribution of residual vorticity is shown schematically in Fig. 7.17. The potential vorticity and bottom friction mechanisms must be treated separately, and the magnitudes of maximum vorticity to
352
plan v i e w
Q
side v
,
,
COLUMN MECHANISM STRETG-g+
.- _ *
FRICTION
MEC HANlSM
Fig. 7.17. Sketch of residual vorticity distribution and possible residual circulation patterns associated with a simple circular hollow topographic feature.
be expected in each case are estimated from the previous section as:
fv
Qf, = 0.45 - - = 0.45f( A h / h ) ah d s
(1)
k~ a Qf=0.4-a dn
(1) o y v
0.4kA - = - ( A h / h )
given that the peaks of Figs. 7.15 and 7.16 are being considered so that the size of the topographic bump is specified as approximately equal to the tidal excursion (i.e. A s = A n = E / 2 = V/o). In the friction term only the variation of h has been considered for simplicity (i.e. A ( V / h ) = ( - V / h ) ( A h / h ) . Q h is not dependent on V or h , but only on the proportionate depth change, whilst Qf varies with V / h . The ratio of G2,/Qf is j h / k V which confirms that the frictional generation of vorticity is most important in shallow water with strong tidal streams, e.g. for V = 1 m s - ' , h is required to be less than 20 m for Qf > Qh.
353
Figure 7.17 indicates the expected patterns of the residual circulations associated with the residual vorticity. Approximate estimates of the maximum velocities in each case suggest: ' h E
oh= -
8
o,=
QfE ~
8
(assuming the velocity is o around a circle of diameter E in which the vorticity is Q/2). Thus, for example, if the prevailing tidal streams are of magnitude 1 m S C ' ( E = 14 km), then o h = lO(Ah/h) cm s-I, so that a 40% depth change would yield residuals of 4 cm s-l. 10. CONCLUSIONS
The aim of this chapter has been to present the basic equations controlling the generation of residual flows through the rectification of oscillatory tidal currents by non-linear dynamics. The solutions presented have been simplified in order to demonstrate the basic mechanisms involved. The approximate order-of-magnitude estimates of residual vorticity and circulation suggest that significant residual flows can be generated by spin up and spin down of water columns moving over large depth variations provided the tidal streams are at least 1 m s-'. With increasing tidal currents, and depths of less than about 20 m, the vorticity generated by frictional effects becomes most important and residuals of 10 cm s - ' or more are to be expected from larger topographic features. However, the analytical approach which has been used becomes invalid when the residual flows predicted approach the same order as the tidal streams, and therefore cannot strictly be applied in the most energetic cases of residual flow generation such as headland eddies, or in shallow estuaries. Whilst the analytic approach helps to clarify the mechanisms involved, the lack of simple analytical solutions points towards the value of numerical solutions, both to confirm the broad conclusions drawn here, and also to solve the residual flow in real topographies. Numerical solutions will best be achieved using the primitive momentum equations rather than by using the vorticity equation, as here, because of the numerical problems of integrating from the vorticity field to obtain the velocity distribution. Further numerical experiments will be worthwhile since because of their permanence even small tidally driven residuals must be significant to the distribution and transport of water properties, and can be an important feature of the physical oceanography of shallow tidal seas. LIST OF SYMBOLS Am
Amplitude of vorticity generating term (column stretching mechanism) at spatial wavenumber k ,
354
Amplitude of vorticity generating term (bottom friction mechanism) at spatial wavenumber k , B horizontal eddy viscosity in the depth-averaged velocity equation B' horizontal eddy viscosity in the transport equations E tidal excursion ( = 2V/a) F a term in the mean momentum equation arising from non-linear friction terms internal stress acting in t h e j direction on a plane at right angles to the i direction L typical horizontal length scales (defined in Table 7.1) T tidal period = 12.42 h time constant for dissipation of momentum and vorticity by bottom r, friction horizontal volume transport rate in x direction U horizontal volume transport-rate vector U V horizontal volume transport rate in y direction V (in section 8) tidal-stream amplitude in the s direction wind stress W coefficient of vorticity generated by column stretching mechanism at Wm spatial wavenumber k , coefficient of vorticity generated by bottom friction mechanism at Yn spatial wavenumber k , a = a,, a y excess momentum term due to vertical current structure, in depth averaged-equation of motion term in the time mean momentum equation arising from non-linear C friction terms Coriolis parameter acceleration due to gravity local depth of sea below the tidal mean level quadratic bottom friction coefficient (normally 0.002) spatial wavenumbers of vorticity generating terms unit vector, z direction direction normal to (rectilinear) tidal-stream direction tidal-stream direction (for rectilinear tidal currents) time Eulerian velocity in x direction horizontal vector Eulerian velocity depth mean horizontal velocity vector Eulerian residual mean velocity Lagrangian residual mean velocity mean transport velocity velocity vector at tidal frequency Eulerian velocity, y direction typical residual velocity generated by column stretching mechanism typical residual velocity generated by bottom friction mechanism Eulerian velocity, z direction Bn
355
horizontal direction horizontal direction vertical direction surface displacement equilibrium tide surface displacement at tidal frequency residual elevation density of sea water radian frequency of the tide vorticity about a vertical axis mean vorticity vorticity at tidal frequency typical maximum vorticity generated by column stretching mechanism typical maximum vorticity generated by bottom friction mechanism horizontal operator d / d x , d / d y
REFERENCES Booth, D.A., Howarth, M.J., Durance, J.A. and Simpson, J.H.. 1978. A comparison of residual currents estimated with current meters and a parachute drogue in a shallow sea. Dtsch. Hydrogr. 2.. 31: 237-248. Bowden, K.F., 1963. The mixing processes in a tidal estuary. Int. J. Air Water Pollut., 7: 343-356. Caston. V.N.D. and Stride, A.H., 1970. Tidal sand movement between some linear sand banks in the North Sea off northeast Norfolk. Mar. Geol., 9: M38-M42. Dooley, H., 1974. A comparison of drogue and current meter measurements in shallow waters. Rapp. P. V . Reun. Cons. lnt. Explor. Mer, 167: 225-230. Gould, W.J.. 1973. Effects of non-linearities of current meter compasses. Deep-sea Res., 20: 423-427. Heaps, N.S., 1972. Estimation of density currents in the Liverpool Bay area of the Irish Sea. Geophys. J. R. Astron. Soc., 30: 373-380. Heaps. N.S., 1978. Linearized vertically integrated equations for residual circulation in coastal seas. Dtsch. Hydrogr. Z . , 31: 147-169. Heathershaw, A.D. and Hammond, F.D.C.. 1980. Secondary circulations near sand banks and in coastal emhayments. Dtsch. Hydrogr. Z., 33: 135-151. Howarth, M.J.. 1980. Intercomparison of current meters in fast tidal currents. Institute of Oceanographic Sciences, Rep. 94, 26 pp. Hunter, J.R., 1975. A note on quadratic friction in the presence of tides. Estuarine Coastal Mar. Sci., 3: 473-475. Huthnance, J.M., 1973. Tidal current asymmetries over the Norfolk Sandbanks. Estuarine Coastal Mar. Sci., 1 : 89-99. Huthnance, J.M., 1981. On mass transport generated by tides and long waves. J. Fluid Mech., 102: 367-388. Huthnance, J.M., 1982a. On one mechanism forming linear sand banks. Estuarine Coastal Shelf Sci., 14: 79-99. Huthnance, J.M., 1982b. On the formation of sand banks of finite extent. Estuarine Coastal Shelf Sci., 15: 277-299. Hydrographic Department, 1962. Approaches to Portland, Tidal Stream Atlas. Hydrographer of the Navy, Taunton. Loder, J.W., 1980. Topographic rectification of tidal currents on the sides of Georges Bank. J. Phys. Oceanogr., 10: 1399- 1416. Longuet-Higgins, M.S., 1969. On the transport of mass by time-varying ocean currents. Deep-sea Res., 16: 431-447.
356 Maddock, L. and Pingree, R.D., 1978. Numerical simulation of the Portland Tidal Eddies. Estuarine Coastal Mar. Sci., 6: 353-363. Maier-Reimer, E., 1977. Residual circulation in the North Sea due to the M,-tide and mean annual wind stress. Dtsch. Hydrogr. Z., 30: 69-80. Mardell, G.T. and Pingree, R.D., 1981. Half-wave rectification of tidal vorticity near headlands as determined from current meter measurements. Oceanol. Acta, 4: 63-68. Nihoul. J.C.J., 1980. Residual circulation, long waves and mesoscale eddies in the North Sea. Oceanol. Acta, 3: 309-316. Nihoul, J.C.J. and Ronday, F.C., 1975. The influence of tidal stress on the residual circulation. Tellus. 29: 484-490. Off, T., 1963. Rhythmic linear sand bodies caused by tidal currents. Bull. Am. Assoc. Pet. Geol., 47: 324-34 I. Oonishi, Y., 1977. A numerical study on the tidal residual flow. J. Oceanol. Soc. Jpn. 33: 207-218. Pingree, R.D., 1978. The formation of the shambles and other banks by tidal stirring of the seas. J. Mar. Biol. Assoc. U.K., 58: 211-226. Pingree, R.D. and Maddock, L., 1977a. Tidal residuals in the English Channel. J . Mar. Biol. Assoc. U.K., 57: 339-354. Pingree. R.D. and Maddock, L., 1977b. Tidal eddies and coastal discharge. J. Mar. Biol. Assoc. U.K.. 57: 869-875. Pingree, R.D. and Maddock, L., 1979a. The tidal physics of headland flows and offshore tidal hank formation. Mar. Geol., 32: 269-289. Pingree, R.D. and Maddock, L., 1979b. Tidal flow around an island with a regularly sloping bottom topography. J. Mar. Biol. Assoc. U.K., 59: 699-710. Prandle, D., 1978. Residual flows and elevations in the southern North Sea. Proc. R. Soc. London, Ser. A, 359: 189-228. Proudman. J., 1953. Dynamical Oceanography, Methuen, London, 409 pp. Ramster, J.W. and Durance, J.A., 1973. A comparison from the northern North Sea of the drift of a parachute drogue with estimates of Lagrangian drift calculated from data collected at a triangle of moored current meters. I.C.E.S. CM 1973/C:4, 8 pp. Ramster, J.W. and Hill, H.W., 1969. Current system in the Northern Irish Sea. Nature, 224: 59-61. Robinson. IS., 1981. Tidal vorticity and residual circulation. Deep-sea Res., 28: 195-212. Tee, K.T., 1976. Tide-induced residual current, a 2-D nonlinear numerical tidal model. J. Mar. Res., 34: 603-628. Tee, K.T., 1977. Tide-induced residual current-verification of a numerical model. J. Phys. Oceanogr., 7: 396-402. Yanagi, T., 1976. Fundamental study on the tidal residual circulation-I. J. Oceanol. Soc. Jpn. 32: 199-208. Yanagi, T., 1978. Fundamental study on the tidal residual circulation-11. J. Oceanol. SOC.Jpn. 34: 67-72. Yasuda, H., 1980. Generating mechanism of the tidal residual current due to the coastal boundary layer. J. Oceanol. SOC.Jpn, 35: 241-252. Zimmerman, J.T.F., 197%. Dispersion by tide-induced residual current vortices. In: J.C.J. Nihoul (Editor), Hydrodynamics of Estuaries and Fjords. Elsevier, Amsterdam, pp. 207-216. Zimmerman, J.T.F., 1978b. Topographic generation of residual circulation by oscillatory (tidal) currents. Geophys. Astrophys. Fluid Dyn., 1 1: 35-47. Zimmerman, J.T.F., 1979. On the Euler-Lagrangian transformation and the Stokes’ drift in the presence of oscillatory and residual currents. Deep-sea Res., 26: 505-520. Zimmerman, J.T.F., 1980. Vorticity transfer by tidal currents over an irregular topography. J. Mar. Res., 38: 601-630. Zimmerman, J.T.F., 1981. Dynamics, diffusion and geomorphological significance of tidal residual eddies. Nature. 290: 549-555.
357
CHAPTER 8
COMPARISON OF COMPUTED AND OBSERVED RESIDUAL CURRENTS DURING JONSDAP '76 A.M. DAVIES
ABSTRACT
A three-dimensional numerical model of the North-West European Shelf has been used to compute the wind-induced circulation on the shelf for the period April 1-9, 1976. The computed circulation pattern is in good agreement with observations taken at this time as part of the JONSDAP '76 exercise. The surface current to surface wind ratio was computed with the model. This ratio was found to vary extensively in space and time from near zero to up to 15%.A time-averaged value over the period varied spatially over the exposed areas of the North Sea from 3 to 7%, in reasonable agreement with an observed time and space averaged value in the North Sea of 4.2%.
1. INTRODUCTION
In a previous paper (Davies, 1979) the monthly mean wind-induced circulation of the North Sea and the North-West European Shelf, for the period mid-March to mid-April 1976, was computed. This month coincided with the in/out period of the JONSDAP '76 (Joint North Sea Data Acquisition Program, 1976) Experiment, during which the majority of the current measurements were made. A two-dimensional fine mesh North Sea model and a three-dimensional shelf model of coarser resolution were used in these calculations. The wind-induced circulation during the period was computed with both models using, as meteorological input, time-averaged wind stress and pressure-gradient fields. Consequently, the computed circulation was an average over the month and comparisons with instantaneous values of measured current were not possible. In this chapter the three-dimensional shelf model used previously (Davies, 1979) is again employed. However, in the calculations presented here (unlike in the previous computation), the eddy viscosity depends upon the square of the depth mean current, and consequently increases as the current increases. This dependence of eddy viscosity upon current has been used previously (Davies and Furnes, 1980) in computing the three-dimensional distribution of the M, tidal currents over the shelf. With this formulation of viscosity excellent agreement between computed and observed M, tidal elevations and currents was obtained. The three-dimensional model grid used in the present calculation covers the
358
6O0I
,
,Flex area
5O01
1oaw
0
10"E
Fig. 8.I. Three-dimensional shelf model: finite difference grid.
continental shelf (Fig. 8.1) and has a resolution of 1/3" latitude by 1/2" longitude. This is the same grid as that used by Flather (1976), extended to include the Skagerrak and Kattegat. The hydrodynamical equations include quadratic bottom friction but are otherwise linear. A radiation condition is applied along the model's open boundaries to allow disturbances from the interior of the model to propagate outwards. Boundary forcing is provided by M, tidal elevation and a meteorologically induced elevation computed using the hydrostatic approximation. The model was run with M, tidal input along the open boundaries for the period 0000h, 1 April-2400h 9 April, 1976, using hourly values of wind stress and pressure gradients, extracted from the 10-level atmospheric model employed in routine weather prediction at the U.K. Meteorological Office(Benwell et al., 1971). The pure tidal motion was also computed for the period, and this solution subtracted from that computed with tide and meteorological forcing to yield the meteorologically induced circulation. This method of computing the wind-induced circulation has been used previously (Davies and Flather, 1978). The period April 1-9, 1976 is ideal for comparing observed and computed meteorologically induced currents. During this time there was a comprehensive deployment of current meters in the North Sea for comparison purposes. Also the magnitude and direction of the wind field varied throughout the period, producing an inflow of water into the North Sea along the Norwegian coast on April 3, 1976
359
(Furnes and Saelen, 1977), followed by a substantial outflow through the Norwegian Trench on April 7, 1976. Residual sea-surface elevations of the order of 1 m occurred along the German coast, during this period. 2. THREE-DIMENSIONAL SHELF MODEL
The equations of continuity and motion for homogeneous water, neglecting non-linear terms and shear in the horizontal, may be written in polar coordinates as:
where we denote by: x, east-longitude and north-latitude, respectively the depth below the undisturbed surface Z t time elevation of the sea surface above the undisturbed depth E undisturbed depth of water h the density of sea water P the radius of the Earth R the Coriolis parameter = 2 w sin Y w the angular speed of the Earth's rotation the acceleration due to gravity g east-going and north-going components of current at depth z u, N a coefficient of vertical eddy viscosity P atmospheric pressure at the sea surface In order to solve eqs. 1, 2 and 3 for 6, u and v , appropriate boundary conditions at sea surface and seabed have to be specified. The surface conditions are:
+
+
-.( N g ) o
= F,
and:
where F, and G, denote the components of wind stress acting on the water in the x and directions, suffix 0 denoting evaluation at z = 0. Similarly at the seabed, z = h , postulating a quadratic law of bottom friction, gives:
+
360
where k is the coefficient of bottom friction, taken as constant. Davies and Furnes (1980) found that a value of k = 0.0050 was an appropriate value to use in a three-dimensional model, and this value has been used here. We now seek a solution of eqs. 1, 2 and 3 for 6, u , 2) subject to boundary conditions 4 and 5. Expanding the two components of velocity u and u in terms of m depth-dependent functions f,( z ) and horizontal-space and time-dependent coefficients A , ( x , t ) and B , ( x , t ) gives:
+,
+,
Using the Galerkin method in the vertical space domain eqs. 2 and 3 are multiplied by each of the basis functions F,, and integrated with respect to z over the interval 0-h. By integrating the term involving the vertical eddy viscosity, boundary conditions 4a, b and 5a, b can be included (Davies, 1979, 1980), giving:
and:
h av df, + Gs fl(0) -J N - -dz P dz d z -
(9)
For / = 1, 2,..., m. It is evident from eqs. 8 and 9 that the bottom and surface boundary conditions occur in these equations as products with f,(O) and f,(h), and therefore if these products are to be non-zero, thef, must be chosen such that:
f l( 0 )* 0 ,
andf, ( h ) * 0
(10)
Turning now to the choice of basis functions f,(z). In a series of numerical computations, Davies (1980) has shown that an expansion of 10 cosine functions is sufficient to accurately reproduce the depth variation of current, and has applied such an expansion to the computation of the monthly-mean wind-induced circulation of the North-West European Continental Shelf (Davies, 1979). An expansion of cosine functions has also been used by Davies and Furnes (1980) in computing M, tidal currents on the shelf. Here again cosine functions are used, with f, given by: Z
f, = cos a,h and a suitable choice for a, is: cu,=(r-l)r
f o r : r = l , 2 , ..., m
(11)
(12)
36 1
in whch case: =0
fr'(0)
and: fr'(h)=O
where: fr' = dfr/dz
It shouid be noted that this choice off, also satisfies condition 10. In the present calculation the eddy viscosity N was independent of the depth coordinate z. (This is not a restriction on the method presented here, since computations using the Galerkin method in which N has varied with z have been presented by Davies, 1980). For the case of N independent of z the fr given by eq. 11 are eigenfunctions of:
Integrating by parts the term involving N in eqs. 8 and 9, and using eqs. 15, 16, 13 and 14, and substituting expansions 6 and 7 into 8 and 9 we obtain, taking advantage of the orthogonality of the basis functions given by eq. 11 :
and:
f o r / = 1, 2, ... m where: m
'h=
c
r= 1
m
Arfr(h),oh=
c
r= 1
Brfr(h)
362
Similarly substituting expansions 6 and 7 into the continuity eq. 1 gives:
Before considering the solution of eqs. 17, 18 and 20, it is necessary to specify how N will vary with x,@ and t . Davies and Furnes (1980) accurately computed the distribution of the M, tidal currents over the shelf using a parameterisation of eddy viscosity of the form: U
with JI = s-’, and the dimensionless coefficient K = 2.0 X l o p 5 . This formulation of eddy viscosity is used in the present calculation. In eq. 21, U and U denote depth mean currents given by: m
m
U=
c Arar
r=
U=
1
c Bra,
r= 1
where:
Having specified the formulation of eddy viscosity the problem is to solve eqs. 17, 18 and 20 to find the time variations of [, A , and B, over the sea area, subject to initial and boundary conditions, given the changing time and spatial distributions of F,, G,, a P / & and aP/d+. Currents at any depth can then be calculated from the A , and B, using expansions 6 and 7. In order to solve these equations it is necessary to discretise in the horizontal and with time. This discretisation is accomplished using a staggered finite difference grid in the horizontal and a forward time stepping technique to advance the solution through time (see Davies, 1980, for details). Solutions are generated from a state of zero displacement and motion, expressed by,
&=O;A,=Br=O
a t t = O ( r = 1 , 2 , ... m)
(23)
Along a closed boundary the normal component of current is set to zero, for all t a 0, thus:
A , cos 4 + B, sin I) = O( r = 1, 2,. . . m)
(24)
where denotes the inclination of the normal to the direction of increasing x. Consider now conditions satisfied along the open boundaries. A radiation openboundary condition was applied along the edge of the continental shelf. This condition involves a prescribed relation between total normal component of depth mean current q and total elevation 6 given by:
4 = q T + q M + c/h(6-
tT
-t M )
(25)
where f M , the meteorologically induced sea-surface elevation, is determined from:
t d x , +, 4 = ( F - p ( x ,
$7
t))/pg
(26)
363
where is a mean atmospheric pressure taken to be 1012 mbars, and P(x,+, t ) is the atmospheric pressure at the sea surface at point x, 9 on the model’s open boundary at time t . In eq. 25, c = (gh)”’; the normal component of the current due to meteorological influence q M is set to zero, and the tidal part of the normal current qr arising from the M, component of the tide is determined from: qT = Q
M ~‘OS[
uM,t
+
‘M,
- YM,]
(27)
The change in sea-surface elevation arising from the M, tide is given by: [T = H M 2 ‘OS[
‘M,l
+
‘M,
- gM,]
(28)
In eqs. 27 and 28, uM2 denotes speed, VM, the phase of the equilibrium constituent at time t = 0 at Greenwich, QM2the amplitude of the normal component of depth mean M, tidal current, and yM2the phase of that current. Also, H M 2and gM2denote amplitude and phase of the M, tidal elevation. The M, tidal terms H M 2 , RM,, Q M 2 and yM2 along the open boundaries of the model were those derived previously (Flather, 1976) in computing the M, tide on the continental shelf; based on the observations of Cartwright (1976). From eq. 27 the depth mean current along the open boundaries of the model can be determined. However, in order to close the problem, i t is necessary to make an assumption about the contribution of each term in expansions 6 and 7 to this depth mean current. In the absence of any detailed knowledge of the current’s vertical structure along the shelf edge, the assumption was made that only the first term in each expansion contributed to the current at the shelf edge. Consequently along the model’s open boundary: A,=B,=O
( r = 2 , 3 , ... m)
(29)
and from eqs. 22 and 25 we obtain:
where q , and q, are, respectively, the u and v components of depth mean current given by eq. 25.
3. THE METEOROLOGICAL DATA
In order to integrate eqs. 17, 18 and 20 forward through time, the meteorological forcing functions, namely the two components of wind stress F, and G,, and the pressure gradients d P / d x and d P / d $ , have to be evaluated at each grid point of the sea model, throughout the period of interest. These forcing functions were calculated from hourly values of the geopotential height of the 1000 mbar surface extracted from the twice daily routine weather forecasts produced by the 10-level numerical model of the atmosphere (Benwell et al., 1971) at the Meteorological Office, Bracknell. Using the geostrophic balance equations, the geostrophic wind was calculated from the geopotential height data. Surface winds were determined from the geo-
364
strophic winds using the formula of Hasse and Wagner (1971) with a cross-isobar angle (the angle between the directions of surface and geostrophic wind) of 20”. The wind stress at the sea surface was computed using a quadratic law, with the components of wind stress 4, G, given by:
where C , the drag coefficient varies with wind speed, thus:
1
0.565 for: W f 5 -0.12 + 0.137 W for: 5 < W G 19.22 2.513 for: W > 19.22 where W , the surface wind speed is in metres per second, and pa the density of air. Details of the derivation of the wind stress and pressure gradients from hourly values of the geopotential height data extracted from the 10-level atmospheric model have been given previously (Davies and Flather, 1978), and will not be repeated here. C,
X
lo3=
4. WIND-INDUCED CIRCULATION FOR THE PERIOD 1-9 APRIL 1976
The model was run with M, tidal input along the open boundaries for the period 0000h, 1 April-2400h 9 April, 1976. The meteorological input to the model consisting of hourly values of wind stress and pressure gradients computed as described in the previous section, together with meteorologically induced elevations on the open boundary computed using the hydrostatic approximation (see eq. 26). The pure tidal motion was also determined for the period, and this solution subtracted from that computed with tide and meteorological forcing to yield the meteorologically induced circulation. This method of computing wind-induced circulation has been used previously (Davies and Flather, 1978). The wind-induced circulation during the period April 1-9, 1976 is particularly appropriate for verifying a three-dimensional numerical model, as during this period the wind fields over the North Sea fluctuated in both magnitude and direction. The complex nature of this wind-induced circulation provides an ideal test of the dynamic response of the three-dimensional shelf model to the imposed meteorological forcing. Also, during this period there was a comprehensive deployment of current meters in the North Sea, against which model results could be compared. The wind field over the North Sea changed from a light north-westerly air-stream on April 1-2, 1976, to a stronger south to south-westerly air flow on April 3, 1976, associated with the passage of a depression to the south of Iceland. At 0600h, April 3, 1976, strong southerly winds off the west coast of Scotland of the order of 35 knots (Fig. 8.2a), induced surface currents in the sea to the west of Scotland of over 100 cm s-I, (Fig. 8.3a), although the magnitude of the depth mean currents in this area were of the order of 20 to 40 cm s-’ (Fig. 8.3b). Surface currents of the order of 60 cm s-’ are also evident to the south of Ireland (Fig. 8.3a), associated with the south-westerly winds of up to 20 knots in this region. A northerly surface flow into the Irish Sea is apparent from Fig. 8.3a, driven by the southerly winds. At this time
365
Fig. 8.2a and b. Caption on p. 367.
366 IC )
361
Fig. 8.2. Weather charts for April 3-6, 1976.
the wind field over the North Sea was light and surface currents, on average, were below 10 cm s-'. Transport of water northward through the Norwegian Trench at latitude 59" 20" was also computed with the model, and this has been compared with transports (Fig. 8.4) computed from current observations made along this line, by Furnes and Saelen (1977). It is evident from Fig. 8.4 that during April 3, 1976 the computed transport became southerly, with a maximum value of 0.4 Sv (1 Sv = lo6 m3 s-I). Using an idealised model of the North Sea, Davies and Heaps (1980) demonstrated that a southerly transport in the Norwegian Trench occurred when there were southerly winds over the North Sea. In their idealised model the North Sea was represented by a rectangular basin of constant depth 65 m, open at its northern end, and the Norwegian Trench was included as a deep cut of depth 260 m extending half way down the eastern side of the basin. Although the southerly flow in the Norwegian Trench, computed by Davies and Heaps (1980), occurred in. the steady state and with a uniform southerly wind field, their results explain the southerly flow which occurred on April 3, 1976. During April 3, 1976 the depression to the north-west of Scotland moved eastward, and the region of southerly winds spread over the North Sea. Although the wind direction over the North Sea changed during this period from southerly to westerly (see Fig. 8.2a and b), the southerly winds were dominant over the North Sea for part of the time, and gave rise to the southerly flow in the Norwegian Trench. Furnes (1980) has also shown analytically that a southerly flow in the Norwegian Trench occurs when the wind field over the North Sea is from the south.
368
SURFACE 106OOh 3/4/76)
. . . . .....
. . . ., .. .. .. .. /-:..... ,.I. . :.. ...., \....
!!% .:fn, (b)
*
...A
.... ,,.... .b.. . .. .. .. ....... ..... ........ .,,,, /,,---.... . . . . ... . . .... .. ,/',.,:J.l ... ... . I / / , ' / - - ........ .. .. .. .. ... . ... ...... 7: : : : :. :. :. .... ... ..... .... ...........-.. . ... ........... ....,, .... ... ....,' .... ... ... ....
)-"? f ,
(1..
,,I.
. l ', ll il f . .
..,
. 1 :;ZL.!:
...' ./
\i;
-
-,
.. .... .. . .I. . . ......
.'I..
- - - - , . , a
.-rLr,rr.
.
a ,
I
i . ,
I.,.
>,
,.
2:? ...I..
,-,,, . ...... ............... .. ............ .. ........... ..... ::r . . ............... ................. . . . . . . . . . . . . . . . ....... ... ................ I , , , .
L . d \ / . / - / I ,
. . . . .. -_-... .... . . .... .... ..-........ ........ ... ..... . . . ... .... . . .... .. .. .. ......... ......... .... ............... .... ............ .... . . . . ..... ...,,, ....... - ,,,-.. _ ..... ... .... . . . ...
&
JJ
1 .
DEPTH MEAN 10600h 3/4/76)
Fig. 8.3. Computed current vectors (surface and mid-depth) at 0600h 3/4/76
369 I
April 1976
Fig. 8.4.Comparison of observed transport (solid line) over the area between meridian 3 " W E and the Norwegian coast according to Furnes and Saelen (1977) and computed transport for the same section (dotted line). Computed transport in the Norwegian Trench, defined as the area between meridian 3"30'E and the coast is given by (broken line).
The observed transport was computed by taking 25-h means to remove the tidal current. The computed transport, however, was obtained by subtracting hourly values of the tidal flow from hourly values of the total flow involving tide and surge. Despite these differences both model and observations show a southerly transport in the Norwegian Trench during April 3, 1976. During the twenty-four hour period from 0600h April 3 to 0600h April 4, 1976, the depression to the south of Iceland moved to the north-east and south-westerly winds predominated over the North Sea (Fig. 8.2b). Surface currents of between 20 and 60 cm s - ' were evident in the central and northern North Sea at 0600h April 4, 1976 (Fig. 8.5a), although in the Southern Bight and English Channel surface currents were less than 5 cm s-'. Depth mean currents (Fig. 8.5b) show a transport of water from the area to the north-west of Scotland, eastward across the north of Scotland, and then southward along the English coast. This flow subsequently turns eastward in the central North Sea, and then flows northward. This flow does not appear to enter the Skagerrak, but flows northward along the Norwegian coast. This northward transport of water in the Norwegian Trench, during April 4,is shown in Fig. 8.4. The northward transport in the Norwegian Trench is consistent with the transports computed by Davies and Heaps (1980) and Furnes (1980). Davies and Heaps (1980) demonstrated (see fig. 11 of their paper) that a south-westerly wind over the North Sea produces a northerly flow in the Norwegian Trench. They also show that the magnitude of the northerly flow in the Norwegian Trench increases if the wind direction changes from south-west to west. An increase in northerly transport within
'9L/
40090 Ie (usam qidap puz a3qms) sio13a~iuaunr, paindm03 '5'8 %d ( 9 L / b / / p 40090) NV3W Hld3a
(SL/WP 1100901 33v3ms
.....-...,.-,,,
OLE
371
MID- DEPTH ( 1800h 5/4/76)
Fig. 8.6. Computed current vectors (surface and mid-depth) at 1800h 5/4/76.
312
SURFACE ( 0 6 0 0 h 6/4/76)
MID-DEPTH
(0600h 6/4/76)
373
............... .. .. ... ... ... ... ... ... ... BOTTOM (0600h 6/4/76)
D E P T H MEAN (0600h 6/4/76)
Fig. 8.7. Computed current vectors (surface, mid-depth, bottom and depth mean) at 0600h 6/4/76.
314
the Trench during April 4 is evident in Fig. 8.4, associated with a change in the wind from south-westerly to westerly winds of the order of 10 knots, associated with the north-east movement of the depression which was situated to the south of Iceland. During the morning of April 5 , 1976 the magnitude of these westerly winds increased and by 1800h April 5 , 1976 strong westerly winds of between 20 and 30 knots were blowing over the northern and central North Sea (Fig. 8 . 2 ~ ) .Surface currents exceeding 40 cm s - ' in the northern North Sea were induced by these winds, and in the shallower central North Sea, surface currents exceeded 120 cm s- I (Fig. 8.6a). Currents at mid-depth (Fig. 8.6b) were appreciably smaller than the surface currents, with velocities of the order of 10 to 20 cm s - ' although with higher velocities of up to 60 cm s-I off the Danish coast. The direction of the current at mid-depth is significantly different from that of the surface current, especially in the German Bight and along the west coast of Denmark. During this period sea levels in the North Sea rose, as water flowed into the North Sea along the east coast of England. As the depression progressed eastward, the north-westerly winds moved over the North Sea. By 0600h April 6, 1976 (Fig. 8.2d), the low-pressure region had moved over Scandinavia. Strong north-westerly winds of up to 35 knots still persisted over the northern North Sea and 20 knot onshore winds were present in the German Bight. Strong southerly surface currents of between 40 and 100 cm s - ' occurred over the North Sea at this time (Fig. 8.7a), although in the English Channel surface currents were only of the order of 5 cm s-I. The onshore winds in the German Bight induced a westerly flow at the sea surface of between 40-80 cm s- I . It is apparent from the plot of depth mean currents given in Fig. 8.7d that during this period there was a significant south-easterly transport of water across the North Sea, under the direct influence of the north-westerly winds. This transport of water caused residual elevations in the German Bight to rise to one metre during April 6 (Fig. 8.8), and this increase in elevation produced a north-south gradient over the North Sea. This gradient induced a northward flow of water along the west coast of Denmark (Fig. 8.7d). This flow, at the sea surface, was opposed by the strong (of order 25 knots) north-westerly winds. However, the northerly flow did occur below the wind-driven surface layer at mid-depth (Fig. 8.7b) and near the seabed (Fig. 8 . 7 ~ )Over . the Norwegian Trench the wind effect is weaker than over the North Sea, due to its greater depth, and the direct wind-driven layer is restricted to the near surface water. Consequently, the Norwegian Trench acts as a channel, through which water can be transported northward, at depth, out of the North Sea. The northward transport within the Norwegian Trench (which can be readily seen in Fig. 8.7d), must be balanced by an inflow of water at its southern end, which is at the eastern end of the Skagerrak. This flow into the Norwegian Trench, within the Skagerrak, can be readily seen in Fig. 8.7d. Some of the water comes from the northward flow along the west coast of Denmark, which subsequently turns eastward into the Skagerrak, and then into the Norwegian Trench. However, part of the flow into the Norwegian Trench is produced by water that flows eastward across the northern North Sea, driven by the north-westerly winds (see Fig. 8.7d). The increased transport of water into the North Sea, along the east coast of
375
England, produced by the north-westerly winds, and its subsequent northward flow out of the North Sea, within the Norwegian Trench, explains the rise in northerly transport within the Trench, which occurred on April 6. The influx of water into the North Sea, caused elevations to rise along the east coast of England, and along the continental coast, during April 6, 1976 (Fig. 8.8). There was little southerly flow through the Strait of Dover at this time. Presumably such a flow was opposed by the westerly winds blowing over the English Channel. Between 0600h and 1800h April 6, 1976, the magnitude of the north-westerly wind field over the North Sea decreased, and by 1800h, a moderate northerly wind field of the order of 5 knots covered the North Sea (Fig. 8.2e). However, off the west coast of Denmark, strong north-westerly winds of order 35 knots persisted for several hours. This spatial distribution of the wind field is reflected in the surface currents computed with the model (Fig. 8.9a). Depth mean currents (Fig. 8.9b) show a high degree of spatial variability over the North Sea. This variability has been found by Riepma (1978) in an analysis of the observations taken during JONSDAP '76. Comparing Fig. 8.9b with Fig. 8.10 (derived from the spatial distribution of currents given by Riepma, 1978), it is evident that the model reproduces the major features of the wind-induced circulation in the North Sea. In particular, the model shows the flow of water into the North Sea, along the north-east coast of Scotland. This flow splits, at a position in the North Sea to the east of Aberdeen, with part of the flow going northward, subsequently turning south-eastward and then flowing along the edge of the Norwegian Trench into the Skagerrak. The second branch of the flow, transports water to the south-east, across the central North Sea and into the German Bight. To the west of the German Bight, this flow bifurcates with part of the water mass moving northward along the Danish coast, and some water going southward. Part of this southerly flow, returns northward along the east coast of England, with some flow moving westward into the German Bight. A portion of this southerly flow, however, appears to pass through the English Channel. I t is evident from a comparison of Fig. 8.9b and Fig. 8.10 that the flow pattern computed with the model is in excellent agreement with the observational data, and the flow pattern inferred by Riepma (1978) from these observations. A north-south gradient of sea surface elevation over the North Sea, maintained by the northerly winds is evident at this time (Fig. 8.1 1). Over the next twelve hours, light northerly winds persisted over the North Sea. The gradient of sea-surface elevations over the North Sea could not be maintained by this reduced wind stress, and a northward hydrostatically forced flow of water out of the German Bight occurred (Fig. 8.12d) accompanied by a reduction in residual elevations in this area. The northerly wind stress opposed any northward flow at the sea surface, and a southerly transport of water is evident (Fig. 8.12a). However, a northerly flow did occur at mid-depth and at the seabed (Fig. 8.12b and c). An analysis of model results revealed that on average the surface layer of southerly flowing water was between 5-10 m thick, and that in this layer the magnitude of the current diminished very rapidly with depth below the free surface. Below this surface layer the current magnitude changed very little with depth, up to within a few metres of the seabed.
316
As residual elevations in the German Bight fell, the north-south gradient of elevation over the North Sea diminished. Since the gradient was primarily responsible for forcing water northward within the Norwegian Trench, as its magnitude diminished during April 7, so the northward transport in the Trench was reduced. Comparing Fig. 8.12d with Fig. 8.13 (derived from the spatial distribution of currents given by Riepma, 1978), it is apparent that the model reproduces the major features of the flow. The observed current gyre off the east coast of Scotland between Wick and Aberdeen is reproduced by the model, as is the north-westerly flow of water out of the German Bight, postulated by Riepma (1978). The northerly flow along the east coast of England is also reproduced by the model. The southerly flow along the western edge of the Norwegian Trench is in good agreement with the observations, as is the northerly transport in the Trench. This
HOEK Y R N H n i L R N C
I JnUI DEN
TERSCHELLING
HELGOLRND
CUXHRVEN
ESBJERG
1
1
-
+ *
i
PlANDRL
S T R V ANGER
i RERGEN
c
I
.
.
.
I
.
.
.
L
.
.
.
I
.
.
.
I
I
311
1 .6 1.2
0.4
0.0
..
. . . . . . ............... ... .;--+--+ ........... ....................... 3 .... WICK
.... I M M I NGHRM
LOWESTOFT
WRLTON
SOUTHEND
....
%..*
DOVER
Fig. 8.8. Comparison of computed residual elevations ( 0 0 0 0 ) and observed residuals (AAA).
northerly transport reached a maximum value of 2.6 Sv on April 7, 1979, which agrees well with the observed value of 2.5 Sv (Fig. 8.4). Although current meters were deployed across the Norwegian Trench at latitude 59"20'N, the rigs which were positioned in the Trench to the west of 4"E were lost (Furnes and Saelen, 1977). Consequently, the observed transport shown in Fig. 8.4 does not cover the total cross-sectional area of the Trench (defined here as the area between meridian 3'30'E and the Norwegian coast). Comparing observed and computed transports over the area between meridian 3"50'E and the coast (Fig. 8.4) shows that the model, on average, underestimates the observed transport. This difference between observed and computed transports may be due to the inflow of Atlantic water into the North Sea, along the western edge of the Norwegian Trench; this water subsequently flows into the Skagerrak and then northward within the Norwegian Trench (Dooley, 1974). Although the model does not include the Atlantic, flow to and from the Atlantic across the open boundary of the model is permitted by using the radiation condition 25. However, in eq. 25 the current qMdue to meteorological forcing over the Atlantic is set to zero, and consequently the flow
E
a
0
m I D z
I
V A
m 0
9
0
-m-m
0
2
c a v)
W W I .
379
Fig. 8.10. Observed current vectors, and postulated currents (dotted dashes) for 6/4/76
from Riepma
( 1978).
of Atlantic water into the North Sea, and the subsequent outflow of this water in the Norwegian Trench may be underestimated by the model. Although the current meters positioned in the Trench to the west of 4"E were lost, it is possible to compute the northward flow across the total area of the Norwegian Trench by using the model (dashed line in Fig. 8.4). On average, this flow exceeds the flow through the area between meridian 3"50'E and the coast
380
ELEVATIONS k m ) (1800h 6/4/76)
Fig. 8.11. Computed residual elevations (cm) at 1800h 6/4/76.
(dotted line in Fig. 8.4) although both flows show the same time variations. It is evident from Fig. 8.4 that the model reproduces the major features of the flow in the trench, however, there are shorter period oscillations of the order of 12 h in the computed flow which do not appear in the observed flow. One possible explanation for this is that the meteorological data used in the sea model was derived from an atmospheric model which is updated with observational data at 12-h intervals. The meteorological situation from April 8-10, 1976 was predominated by westerly winds. Offshore winds associated with the passage of a front across England during the late evening of April 8, 1976 and early morning of April 9, 1976 produced the small (of order 20 cm) negative surge at ports along the east coast of England (Fig. 8.8). Depth mean currents over the North Sea during this period were, on average, less than 10 cm s-'. It is apparent from Fig. 8.8 that the numerical model accurately computes the magnitude of the positive surge which occurred at English ports on April 6, 1976. However, the negative surge, which followed, was not reproduced by the model particularly at Walton and Southend. The reason for this is not clear. The three-dimensional model at present does not contain the advective terms, which are known to be important in shallow water areas. The computed three-dimensional meteorologically induced circulation of the
38 1
SURFACE (0600h 7/4/76)
MID-DEPTH (0600h 7/4/76)
6a
7
0
0 0, 0
-
2 D
5
I
4I
m U
. . . . . . . . . . . . . ..............I.>
I:.::
1':"
. A , .
f
4'
5;
f;;
. __. ,.,-- \ r.. . . . . . . . .
-.a.,
3
0
0)
0 0
E
0
......
.......... ..........., .............
w
N 00
383
O0
10"
Fig. 8.13. Observed current vectors, and postulated currents (dotted dashes) for 7/4/76
from Riepma
( 1978).
North Sea, however, appears to be in good agreement with observations. The current measurements, however, were made in the deeper offshore waters of the North Sea, where the contribution of the advective terms would be small, and hence the omission of these terms in the numerical model would have had a negligible effect upon the computed currents.
3x4
5. SURFACE CURRENT TO WIND-SPEED RATIO
The ratio between surface current velocity and the wind speed ten metres above the sea surface is particularly important in oil pollution problems. In this connection several experiments [Hughes (1956), in the ocean, Tomczak (1964) in the North Sea] have been made to determine this ratio by measuring the drift of plastic envelopes floating close to the sea surface. Results from these experiments suggest that the surface current is of order 2% (ocean) to 4.2% (North Sea) of the surface-wind velocity. Measurements of current profiles just below the water's surface (Shemdin, 1972) show that the current magnitude decreases rapidly with depth. Hence it is necessary to have a high resolution within the surface layer if the surface current is to be computed accurately. The present model, unlike the more conventional grid box or layered models, uses an expansion through the vertical in terms of continuous functions, and hence the current throughout the surface layer can be determined. Surface current to wind-velocity ratios in the North Sea were computed using the numerical model for the period considered here. At 0600 GMT April 3, 1976 wind velocities over the central North Sea were on average less than 3 m s - ' and surface currents were typically between 1-5 cm s - ' (Fig. 8.3a). Surface current to wind ratios during this period were spatially very variable, ranging from about 0.1 to 2%. Higher ratios of between 3 to 7% were computed at 0600h April 4, 1976, the higher value was associated with the area in the North Sea where the currents were the order of 80 cm s - ' (Fig. 8.5a). The magnitude of the surface wind over the North Sea at this time varied considerably, reaching a maximum value of about 10 m s - ' in the region of highest surface current. A ratio of the order of 15% was computed in an area to the west of the Danish coast at 1800h April 6, 1976, associated with surface currents of the order of 150 cm s- (Fig. 8.9a) and spatially varying wind speeds of between 10-12 m s-'. The region over which the ratio was 15% was very limited and in adjacent sea areas where the surface currents were the order of 100 cm s-I, this ratio was between 4 and 8%. Over the majority of the North Sea at this time the ratio was less than 3%. Near coastal or sheltered areas a significantly lower ratio was obtained than in the open sea. These results suggest that the ratio of surface current to surface wind depends upon the magnitude of the surface current and the position of the sea area in relation to the coast. On average a ratio of between 3-7% was computed over the North Sea, although this value dropped to near zero during calm periods. The ratio of 15% computed at 1800h April 6, 1976 does not agree with the observed value of 4.2% in the North Sea (Tomczak, 1964). However, since the value of 4.2% was obtained by tracking floating cards over the North Sea for a period in excess of 10 days, it represents a space and time-averaged value, whereas the computed value of 15% applies only over a small area and for a few hours. Comparison with the average computed value of between 3-776, however, appears reasonable. Recent analysis of surface currents and winds from a data buoy moored off Lowestoft (Collar and Vassie, 1978), gave a surface current to wind ratio of 0.9%.
'
385
This low ratio may be due to the fact that the buoy was moored in a sheltered location close to the coast, and also because the current measurements were taken at a depth 3 m below the sea surface. Davies (1977) has shown that the computed surface current is particularly sensitive to the value of surface eddy viscosity; decreasing as the surface eddy viscosity increases. Also the value of surface eddy viscosity increases with increasing wind speed (Munk and Anderson, 1948). Consequently, as the wind speed increases, the surface eddy viscosity increases, and this tends to reduce the magnitude of the surface current. In the numerical model used here, the eddy viscosity is proportional to the square of the depth mean current, and hence it increases as the current velocity rises with increasing wind speed. The high ratio (15%) of surface current to wind velocity, found during periods when the wind velocity was the order of 12 m s I , suggests that perhaps the parameterisation of eddy viscosity used in the model, underestimates the magnitude of the viscosity at the higher wind speeds, and hence the computed surface currents are too high. However, without observations of surface current and wind velocity in the open sea during a stormy period, a detailed parameterisation of eddy viscosity at high wind speeds is difficult. ~
6 . CONCLUDING REMARKS
A three-dimensional model, previously proved by computing the M, tidal distribution of elevations and currents over the shelf (Davies and Furnes, 1980) has been used to compute the wind-induced circulation on the shelf for the period April 1-9, 1976. The computed wind-induced circulation during and after the storm surge of April 6, 1976 has been compared with spatial distributions of observed currents given by Riepma (1978) and found to be in good agreement. A time-averaged value of surface current to surface-wind ratio of between 3-7% has been computed with the model. This range of values is in good agreement with observed values in the North Sea of 4.2%.
ACKNOWLEDGEMENTS
The author is indebted to Dr. H.W. Riepma for providing him with copies of the observed residual circulation of the North Sea on April 6 and April 7, 1976 (ref. Figs. 8.10 and 8.13). The many useful comments and suggestions made by Dr. N.S. Heaps during the course of this work are very much appreciated. The care and effort taken by the Meteorological Office in extracting the meteorological data from their atmospheric model is much appreciated, as is the work performed by Mr. R.A. Smith in preparing the diagrams presented in this paper. The work described in this paper was funded by a Consortium consisting of the Natural Environment Research Council, the Ministry of Agriculture and Fisheries, and the Departments of Energy and Industry.
386 REFERENCES Benwell, G.R.R., Gadd, A.J., Keers, J.F., Timpson, M.S. and White, P.W., 1971. The Bushby-Timpson 10 level model on a fine mesh. Sci. Pap. Meteorol. Office, London, 32: 1-23. Cartwright, D.E., 1976. Shelf-boundary tidal measurements between Ireland and Norway. Mem. SOC.R. Sci. Liege, Ser. 6, 10: 133-139. Collar, P.G. and Vassie, J.M., 1978. Near surface current measurement from a surface following data buoy (DB 1)-11. An harmonic and residual analysis of current meter records. Ocean Eng.. 5 : 291-308. Davies, A.M., 1977. Three-dimensional model with depth-varying eddy viscosity. Proc. 8th Liege Colloquium on Ocean Hydrodynamics. In: J.C.J. Nihoul (Editor), Bottom Turbulence. Elsevier, Amsterdam, pp. 27-48. Davies, A.M., 1979. Application of numerical models to the computation of the wind induced circulation of the North Sea during JONSDAP '76. "Meteor" Forschungsergeb. Reihe A, 22: 53-68. Davies, A.M., 1980. On formulating a three-dimensional sea model with an arbitrary variation of vertical eddy viscosity. Comput. Methods Appl. Mech. Eng., 22: 187-21 1. Davies, A.M. and Flather, R.A., 1978. Application of numerical models of the north west European Continental shelf and the North Sea to the computation of the storm surges of November-December 1973. Dtsch. Hydrogr. Z. Eng.-H.A., 14: 1-72. Davies, A.M. and Furnes, G.K., 1980. Observed and computed M, tidal currents in the North Sea. J. Phys. Oceanogr., 10: 237-257. Davies, A.M. and Heaps, N.S., 1980. Influence of the Norwegian Trench on the wind-driven circulation of the North Sea. Tellus, 32: 164-175. Davies, A.M. and Owen, A., 1980. Three-dimensional numerical sea model using the Galerkin method with a polynomial basis set. Appl. Mathem. Modell., 3: 421-428. Dooley, H.D., 1974. Hypothesis concerning the circulation of the northern North Sea. J. Cons. Int. 1'Explor. Mer, 36: 54-61. Flather, R.A., 1976. A tidal model of the north-west European continental shelf. Mem. Soc. R. Sci. Liege Ser. 6, 10: 141-164. Furnes, G.K., 1980. Wind effects in the North Sea. J. Phys. Oceanogr., 10: 978-984. Furnes, G.K. and Saelen, O.H., 1977. Currents and Hydrography in the Norwegian coastal current off Utsira during JONSDAP-76. Norwegian Coastal Current Project, Rep. 2/77, University of Bergen, Bergen. Hasse, L. and Wagner, V., 1971. On the relationship between geostrophic and surface wind at sea. Monthly Weather Rev., 99: 255-260. Hughes, P., 1956. A determination of the relation between wind and sea-surface drift. Q. J. R. Metall. Soc., 82: 494-502. Munk, W.H. and Anderson, E.R., 1948. Notes on a theory of the thermocline. J. Mar. Res., 7: 276-295. Riepma, H.W., 1978. Residual currents in the North Sea during IN/OUT phase of JONSDAP '76 (First results extended). ICES Pap. CM 1978/C: 42, Hydrography Committee. Shemdin, O.H., 1972. Wind generated current and phase speed of wind waves. J. Phys. Oceanogr., 2: 41 1-419. Tomczak, G., 1964. Investigations with drift cards to determine the influence of the wind on surface currents. In: Studies on Oceanography. Tokyo University, Tokyo, pp. 129- 139.
387
CHAPTER 9
DEVELOPMENT OF A THREE-LAYERED SPECTRAL MODEL FOR THE MOTION OF A STRATIFIED SHELF SEA. I. BASIC EQUATIONS N.S. HEAPS
ABSTRACT Dynamical equations are formulated for a three-layered model designed to represent the motion of a stratified shelf sea. The motion in each layer is considered to be three-dimensional with density and vertical eddy viscosity prescribed in each. Continuity of current and shear stress is satisfied across the internal interfaces. A spectral method is developed for solving the equations in which horizontal components of current are expanded through the oertiral within each layer in terms of a set of eigenfunctions. The time-varying horizonral distributions of surface elevation, interfacial elevation and current may be determined from a two-dimensional numerical time-stepping procedure- based on the dynamical equations transformed by vertical integration.
INTRODUCTION
A spectral method has been developed in recent years for the numerical solution of the three-dimensional hydrodynamic equations for tides and storm surges (Heaps, 1972, 1973, 1974, 1976, 1981; Heaps and Jones, 1975, 1977, 1981). In all this work the water was considered to be vertically homogeneous and solutions were therefore obtained applicable to motion in a shallow well-mixed sea. In this chapter, the method is extended to take account of movements due to wind, atmospheric pressure and tide in a continental shelf sea with vertical density stratification. A three-layered model is formulated with density taken as constant and uniform in each layer but differing between layers. The system may represent surface, thermocline and bottom layers in reality so that thermocline displacements and the associated currents may be predicted. Application to the Celtic Sea is envisaged: to yield currents, elevations of the sea surface and elevations of the internal interfaces between the layers, during the summer season of thermal stratification. Eddy viscosity is prescribed in each layer and the three-dimensional structure of horizontal current is determined in each. Continuity of current and stress across an interface is satisfied. The present design marks an advance on that of earlier models (O’Brien and Hurlburt, 1972; Simons, 1972) in that the latter do not resolve the vertical structure of current in each layer. Also in those models, the interfacial stresses require formulation in terms of the depth-averaged currents in the respective layers. The ensuing study (Part I) establishes the basic mathematical equations of the new model. In each layer, honzonta!. components of current are expanded through the vertical in terms of a set of eigenfunctions. Coefficients in these expansions may
388
be evaluated using a two-dimensional numerical time-stepping procedure based on the equations of motion and continuity transformed by vertical integration. The transformation of the equations and the development of the eigenfunction expansions are the main concerns in the analysis which follows.
BASIC EQUATIONS
The equations of motion of the sea are taken in the form:
where: F= -pN-
aU
az ’
G = -pN-
av aZ
(4)
and :
The solution of these equations is supposed to extend over a sea region of sufficiently small area for the curvature of the Earth to be neglected. The notation is: time Cartesian coordinates, forming a left-handed set, in which x and y are measured in the horizontal plane of the undisturbed sea surface and z is depth below that surface components of current at depth z in the directions of increasing x, y , respectively components of internal stress in the x, y directions by which the water above depth z acts on the water below that depth the equilibrium tide components of the tide-generating forces in the x, y directions pressure in the water density of the water coefficient of vertical eddy viscosity geostrophic coefficient, assumed constant acceleration of the Earth’s gravity The three-layered model is illustrated in Fig. 9.1. The surface, intermediate and bottom layers have undisturbed depths h , , h,, h,, respectively, where h , and h , are constants and h , is a function of x and y , defining the seabed topography. In equilibrium the sea surface is horizontal at z = 0 and the two internal interfaces
389
b
z = h,+h2+h3
Fig. 9.1. Three-layered model.
horizontal at z = h , and z = h , + h,. During the motion the surface is elevated by {,, to z = - {,, the upper interface by l2to z = h , - 12, and the lower interface by l3to z = h , + h , - 13. Using suffices 1, 2 and 3 generally to denote values in the surface, intermediate and bottom layers we have: for thesurfacelayer(-{, < z < h , - 5 , ) : p = p , , u= ui, v = v , , N = N , , p = p ,
(6)
for the intermediate layer ( h , - l2< z
< h , + h , - {,):
p = p , , u = u 2 , v = v 2 , N = N 2 , p= p 2
and for the bottom layer ( h i+ h ,
-
(7)
l3< z < h ) :
p=p3,u= u3,~=v3,N=N3,p=p3
(8)
where h = h , + h , + h , denotes the total mean depth. Integrating eq. 3 through each layer, assuming homogeneity in each with p , , p2 and p 3 taken as prescribed constants, it follows that: PI
=Pa
+ g p , ( z + (1
P2=Pa+gPl(hl + L - l 2 ) + g P 2 ( z - h ,
+l2>
(9)
P3 = P a + g p , ( h , + 2, - 3 2 + gp,( h2 + l 2 - l 3 ) + g P 3 b - h , - h2 + 2 3 ) where pa denotes atmospheric pressure on the sea surface. Then the substitution of
3 90
eqs. 6-9 into the equation of motion 1 yields:
a
a? 1 ap, +g----+z ax
p3 ax
while eq. 2 similarly yields:
Small motion is investigated so that eqs. 10 and 13 of the surface layer apply for 0 < z < h , , eqs. 11 and 14 of the intermediate layer for h , < z < h , + h,, and eqs. 12 and 15 of the bottom layer for h , + h , < z < h. Correspondingly, evaluating surface and interfacial conditions on the undisturbed levels we have:
atz=h,: u1=u2,
(17)
u1=u2
at z = h , + h,: u 2 = u 3 ,
v2 = v3
p2N2---=p au2
N -au3 ,
at
3
3
az
(19) p N1=p av
22az
N - 8%
33az
Here, eq. 16 introduces components of wind stress F,, G, and eq. 21 components of bottom stress FB G,. Equations 17, 18, 19 and 20 express the continuity of current and stress across the interfaces. Completing the mathematical formulation, equations of continuity for small motion, corresponding to the three layers, may be written:
39 1
Fulfilling eqs. 16-21, solutions of eqs. 10-15 and 22-24 are sought for u,, u5, l5 (s = 1, 2, 3). Initial values of these variables are prescribed over the sea region
considered; appropriate dynamical boundary conditions along the periphery of the region (associated with coast or open sea) are satisfied for all time. External forcing in the equations comes from wind stress ( F s , Gs), atmospheric pressure gradient ( a p , / d x , d p , / + ) and tide-potential gradient (ds/ax, as/+). Other tidal and meteorologically generated disturbances may enter the sea region across its open boundaries and such disturbances are specified in the peripheral boundary conditions.
VERTICAL INTEGRATION OF THE EQUATIONS OF MOTION
The equations of motion are now integrated vertically using a kernal function f ( x , y, z, t) defined through the depth as follows:
f=f,
O
Consider eqs. 10, 11 and 12. Multiply eq. 10 by p , f , and integrate with respect to z from z = 0 to z = h,; add to this the result of multiplying eq. 11 by p2f , and integrating from z = h , to z = h , h,; add further the result of multiplying eq. 12 by p3f 3 and integrating from z = h, h, to z = h; then divide the outcome by p l h . It follows that :
+ +
where:
ti =
1( l h l f , u ,d z + p. /"""f2u2 h
PI
dz + -
h,
6 = ~ ( ~ h ' fd ,z ~+ -lp2 j h 1 + " f 2 u 2 d z PI h ,
I ( j h l f ld z + j h ' + h 2 f 2d z "I=% -
h
0
+jh hI+h2
hl
( J h 1 + h 2 f 2d z + j h f 3 d z ) hl
+-
hI+h2
f3dz)
392
f,d z +,/,I+,%
PI
dz + %Jh PI
h,
f3 h,+h,
dz)
(33) P2
dz +2
jh1+h2u2&
J x = ~ ( i h l ud zl +~-PI
h,
at
PI
jh
Jf3
( 34)
h,+h,u at d z )
Integrating by parts twice in each integral of eq. 33 gives: r=h
) + j g h 1 p I ~ , $ (N , $ )
dz +i:lih2p2u2$(
Examining the form of eq. 35 leads us to take:
(;
N2%)
=
-Xfl,
O
=
-Af2,
h,
< z < h , + h,
h,
+h 2 < z
where X is independent of z, and also: atz=O:
fl=l,
atz=h,:
= h:
az
dfl
fl = f 2 ,
atz=h,+h2: and at z
afI - - 0
f2=f3,
-df3= o az
af2
P I N I ~ = P ~ N ~ , af2
p N-=P 2
2az
3
df3 N 3-a z
N2$)
dz
z=hl +h,
(35)
393
Then, with conditions eqs. 16-21, eq. 35 simplifies to:
where f 3 ( h ) denotes the value of
a , JPa PI ax
+
f3
at z
4 -f,(h)FEJ
=
h . Consequently, eq. 26 may be written:
+Jx
Plh
Similarly, working from eqs. 13, 14 and 15 it may be shown that:
where Jy is given by eq. 34 with u replaced by v . Equations 42 and 43 are vertically integrated forms of the equations of motion 10-15. The evaluation of FB, G , , J, and JY is discussed later. Next, consideration is given to the determination of A , f l , f2 and f3.
EIGENVALUES AND EIGENFUNCTIONS
Solving eq. 36 with respect to conditions 37-40 yields, formally, a set of ascending eigenvalues:
X=A,
(44)
and corresponding eigenfunctions: fi
=fi,r,
fi = f i . r ,
f3 = f , r
(45)
( r = 1, 2,. . . , 00). In general, eq. 36 is solved with respect to the independent depth coordinate z for each time t and each position (x,y ) . The eddy viscosities N, (s = 1, 2, 3) are regarded as known functions of x, y , z and t , their definition being the subject of hypothesis and possibly being related to properties of the motion. Easy trigonometric solutions of eq. 36 are obtainable when viscosity is assumed to be uniform through the depth in each layer (but still allowed to vary with x, y and t ) and this case will be considered on p. 397. Hence, taking account of eqs. 44 and 45, from eqs. 42 and 43 we get a doubly infinite set of vertically integrated equations:
NOI.L.VVIX0rISNW.L ZSX3ANI 2H.L
The current components us and us (s = 1, 2, 3 ) are now expressed in terms of their transforms ti,, 6,.This is the inverse operation to that carried out in eqs. 27 and 28. It enables the equations of continuity 22, 23 and 24, and J,,,,J,,,from eq. 54 to be written in terms of ti,, Cr. Also, expansions for depth-mean current involving these transforms may be derived. From eq. 36:
395
that integrating over the respective layers, factoring the integrals by p l , p2, p3 and then adding them together, gives:
SO
where A,, Br are independent of z . Multiplying ul by p l f l , s , u 2 by p2fZ,cand u3 by p3f3,s,integrating through each layer and adding the results; then applying eq. 58 and taking note of eq. 48; it follows that:
A, =G s k
(60)
Similar operations on v l , v2 and v3 in eq. 59 give:
B, = Gs4 Here:
Therefore, combining eqs. 59, 60 and 61 we get: W
uI=
C +rGrfI,r>
r= 1
m
C Gr6rfl.r
r= 1
( O G Z G ~ ~ )
396
Each current component is thus expressed in terms of a series of current modes through the vertical corresponding to r = 1, 2, 3 , . .. . The r th mode is defined by fl,r(z), 0 G z Q h , ; f 2 , r ( z ) ,h , Q z Q h , + h,; f3,r(z), h , + h , Q z Q h . Substituting the forms of eq. 63 into the equations of continuity 22, 23 and 24, and taking account of eqs. 50, 51 and 52 yields:
Similarly, from eq. 54: m
W
Jx.r =
C +sfisbr,s? s= 1
~ y ,= r
C +scsbr,s s= 1
where:
Using eq. 63, the x-directed components of depth-mean current in the respective layers may be expanded as follows:
Correspondingly, averaging through the entire depth of water gives the mean current u = (h,U,
+ h , u-, + h , E , ) / h
W
=
C +rfirul,r
(70)
r= 1
The associated y-directed components of depth-mean current, derived similarly, are given by eqs. 69 and 70 with u replaced by v .
397
GENERAL PROCEDURE
In applying the preceding theory it is supposed, as in the earlier work (Heaps, 1974), that infinite series may be truncated to M terms. Computations of elevation and current may then be based on the following operations. (a) Use eqs. 64, 65, 66, 46 and 47, written in finite-difference form, to increment the set of 2 M + 3 variables l,, l,, ti, ( r = 1, 2,. . . , M ) , 6,( r = 1 , 2,. . . , M ) from values at time t to values at the later time t A t . Repeating t h s procedure, determine the spatial distributions of these variables through time, starting from known initial values at t = 0 and making successive evaluations at t = mat ( m = 1 , 2, 3 , . . . ). An explicit updating scheme may be employed similar to that given by Heaps (1 974). (b) For inclusion in the above computations, determine X r , f l , , , f 2 , , , f 3 , , from eqs. 36-40 as described on p. 393. Thence find a 2 , , ,a , ,, a4,, from eqs. 50-53, +, from eq. 62 and br,sfrom eq. 68. (c) In eqs. 46 and 47 evaluate .I, Jy,, ., from , eq. 67 and, assuming say a square law of bottom friction, take:
c2,
F,
= kp3u3(h ) ( u:(
h ) + o:( h))”’
G,
= kp3v,( h ) ( u:(
h ) + v:( h ) ) ’ I 2
+
where k is a prescribed friction coefficient and u,(h), q ( h ) denote the components of horizontal current at z = h. (d) On the basis of eq. 63 determine u 3 ( h ) and v , ( h ) in eq. 71 from:
r= 1
M
(e) Alongside the evaluations of ti,, 6, through time, derive horizontal current components u s , us (s = 1 , 2, 3) from eq. 63 and depth-averaged currents U, V, Us,U, (s = 1, 2, 3) from eqs. 69 and 70. THE CASE OF EDDY VISCOSITY UNIFORM THROUGH THE DEPTH IN EACH LAYER
Suppose that N , , N2, N, are each assumed to be independent of the depth z . In these circumstances simple analytic solutions of eq. 36, subject to conditions 37-40, may be derived and the theory becomes readily applicable. At the outset it is convenient to define the following non-dimensional parameters:
398
together with fractional depths for the entire water column:
5=z/h,
O<[
5, = h ,/h, 5,
+ h,)/h,
= ( hI
(74)
lower interface
and for the individual layers: 17 = 5/61
7
p= ( ~ - ~ 1 ) v=((-[,)/(l
0 < 17 < 1, 1,
intermediatelayer
O < v < 1,
bottomlayer
/ ( ~ 2 - ~ 1 ) ,O < p <
-(,),
surface layer
Then, writing
A =Nla2/h2 it is easily shown that solutions of eq. 36 subject to eqs. 37, 38 and 39 are:
0<17< 1 06p6 1 0
f,= cos( a a v ) f2 = b, cos( b a p ) - c2 sin( b a p ) f3= b, cos( cav)- c, sin( c w ) where: b,
= cos
c2 = d sin ua
aa,
b, = b, cos ba
-
c3 = eb, sin ha
c, sin ba
+ ec, cos ba
Satisfying eq. 40 leads to the following equation for a :
( q + e + d + 1) s i n [ ( a + b + c ) a ]
+ (4-
e - d + 1) sin[(a - b
+(q+e-d-
1) s i n [ ( u + b - c ) a ]
+ c ) a ]+ ( 4 - e + d -
1) sin[(a
-
h - c ) a ] = 0 (79)
Denoting the non-negative roots of this equation, arranged in ascending order of magnitude, by: a=ar(r=
1 , 2 , ..., a)
(80)
it follows from eqs. 76, 77 and 78 that:
A,
=
N,a:/h2
(81)
and: f1.r
= cos(uar?)
f,,,= b2,rcos( harp) - c , , ~sin( harp) f3,r
= b3,rcos( c a r v ) - c , , ~ sin( c a r v )
where: b,,,=cos(aa,),
c,,,=dsin(aa,)
b3,r=b2,rcOs(bar)-c,,rsin(bar) c ~ =,eb,,, ~ sin( b a r )
+ ec,., cos( b a r )
0
399
Expressions required for the computation of the three-dimensional motion, following the procedure outlined on p. 397, have been derived above. Examining these expressions and the parameters on whch they depend, it is apparent that generally a,, A,, [5.r, as.r, vary with horizontal position (x, y ) and time t ; the fs,T also, of course, vary with depth z. Equation 79 has to be solved to find a , for each x, y , t . However, special cases introducing simplification may be recognised, described below. (a) If N , , N2, AT3 are assumed to be independent of time, varying only with position (x,y ) , then a r , Ar,fs.r, us+rr are also time-independent (but still positionally dependent) and may therefore be evaluated, once and for all, before the time-stepping calculations begin. Equation 79 has to be solved for each (x,y ) . (b) More restrictively, if N , , N,, N3 are assumed to be constants then a r ,A,, j,,r, a s , rGr , (again time-independent) have a positional variation depending on the depth topography only. (c) If it be further assumed that the sea is of uniform depth, then a r , A,, +r are constants andf,,, functions of the depth coordinate only. Under these conditions, eq. 79 has to be solved only once. Such a simplified situation is considered in Part 11 (Ch. 10) of this work for the study of motion in a rectangular basin. When N , , N2, N3 are independent of time as in (a), (b) and (c) above, since[\,, is time-independent it is evident from eqs. 67 and 68 that br$s= Jx,r = J,,r = 0.
+,
+,
400 REFERENCES Heaps, N.S., 1972. On the numerical solution of the three-dimensional equations for tides and storm surges. MCm. SOC.R. Sci. Libge, Ser. 6, 2: 143-180. Heaps, N.S., 1973. Three-dimensional numerical model of the Irish Sea. Geophys. J. R. Astron. SOC.,35: 99- 120. Heaps, N.S., 1974. Development of a three-dimensional numerical model of the Irish Sea. Rapp. P. V. Rtun. Cons. Int. Explor. Mer, 167: 147-162. Heaps, N.S., 1976. On formulating a non-linear numerical model in three dimensions for tides and storm surges. In: R. Glowinski and J.L. Lions (Editors), Computing Methods in Applied Sciences, Springer, Berlin, pp. 368-387. Heaps, N.S., 1981. Three-dimensional model for tides and surges with vertical eddy viscosity prescribed in two layers-I. Mathematical formulation. Geophys. J. R. Astron. Soc.,64: 291-302. Heaps, N.S. and Jones, J.E., 1975. Storm surge computations for the Irish Sea using a three-dimensional numerical model. Mem. Soc. R. Sci. Libge, Ser. 6, 7: 289-333. Heaps, N.S. and Jones, J.E., 1977. Density currents in the Irish Sea. Geophys. J. R. Astron. SOC.,51: 393-429. Heaps, N.S. and Jones, J.E., 1981. Three-dimensional model for tides and surges with vertical eddy viscosity prescribed in two layers-11. Irish Sea with bed friction layer. Geophys. J. R. Astron. SOC.,64: 303-320. O’Brien, J.J. and Hurlburt, H.E., 1972. A numerical model of coastal upwelling. J. Phys. Oceanogr., 2: 14-26. Simons, T.J., 1972. Development of numerical models of Lake Ontario: Part 2. In: Proc. 15th Conf. Great Lakes Res., Int. Assoc. Great Lakes Res., pp. 655-672.
40 1
CHAPTER 10
DEVELOPMENT OF A THREE-LAYERED SPECTRAL MODEL FOR THE MOTION OF A STRATIFIED SEA. 11. EXPERIMENTS WITH A RECTANGULAR BASIN REPRESENTING THE CELTIC SEA N.S. HEAPS and J.E. JONES
ABSTRACT
The three-layered model of Part I (Ch. 9) is applied to determine the dynamic response of a stratified rectangular sea area (representing the northern part of the Celtic Sea) to a uniform wind-stress pulse. Inertial oscillations are generated by the wind action and their properties are evaluated in a series of numerical experiments -varying boundary conditions, frictional coefficients, also wind-stress magnitude and duration. On the basis of the results thus obtained, a first interpretation is given of observed inertial currents in the Celtic Sea. These can be comparable in magnitude to the tidal currents there, particularly near the sea surface. In the numerical computations of the water motion, based on a finite-difference formulation of the hydrodynamical equations, the barotropic component is split in both space and time from the baroclinic. The model calculations, which would otherwise be prohibitively long and expensive, then become feasible. Radiation conditions are employed along the edges of the rectangular area and enable a range of open-boundary conditions to be simulated.
INTRODUCTION
The three-layered model, formulated in Part I, is now applied to determine wind-induced motion in a rectangular basin-representing the northern part of the Celtic Sea during the summer season of thermal stratification. The dynamic response of the water to a wind pulse is investigated. Numerical experiments with the model are carried out to determine the influence on that response of grid resolution, open-boundary radiation, eddy viscosity, wind strength and wind duration. Thereby the model is subjected to a first series of behavioural tests and, at the same time, information relating to currents in the Celtic Sea is obtained. The hydrodynamic equations derived in Part I are solved numerically employing a time-stepping finite-difference scheme on uniform horizontal grids. To render the computations tractable, the barotropic motion is split in both space and time from the baroclinic. Thus, in the space domain, the barotropic motion is evaluated on a coarse grid, adequate to resolve the comparatively long horizontal length scales of that motion. On the other hand, the baroclinic motion is evaluated on a fine grid, necessary to resolve the shorter horizontal length scales of the internal waves. In the
402
time domain a separation is effected between barotropic and baroclinic wave propagation. Then, a longer computational time-step may be used in the generation of the slower baroclinic component than in the generation of the faster barotropic component. These procedures introduce necessary economies into the computations. An important element in the work is the application of radiation boundary conditions along the edges of the rectangular model. This permits flow to take place across the edges, governed by progressive-wave relationships. While such conditions cannot be associated directly with known observational states in the Celtic Sea, nevertheless the effects on the motion of differing degrees of outward radiation may be studied. Thereby the general influence of open boundaries may be assessed. When the boundaries are closed, it is shown that internal wave forms develop corresponding to Poincare waves throughout the interior and Kelvin waves against the boundaries. Such features have been observed in the Great Lakes (e.g. see Mortimer, 1977, 1980). When the boundaries are opened the Kelvin waves pass o u t of the model. Inertial oscillations of both current and elevation are shown to be the predominant response to the wind and the model is used to investigate their structure, propagation and varying distribution. Here, inertial oscillations refer to an entire group of waves with periods both at and near to the inertial period. In this sense, Poincark waves are inertial in character. The results obtained show that while inertial oscillations of current are generated directly by wind acting over the area of the model sea, other inertial oscillations of current (and internal elevation) spread slowly inwards from the boundaries. These are baroclinic waves generated at the boundaries by the radiation conditions imposed there. Increasing the radiation constants, the boundary-generated inertial waves may be progressively weakened to leave wind-generated inertial currents such as would be produced in an open-ocean situation with no boundary influence. There is a considerable literature on inertial waves in oceans, seas and lakes. During recent years for example, observations and theory have been presented by Pollard (1970) and Pollard and Millard (1970) for the open ocean, by Kundu (1976) for a region off the Oregon coast, by Mortimer (1977, 1980) for Lake Ontario and Lake Michigan, by Tang (1979) for the Gulf of St. Lawrence, by Krauss (1981) for the Baltic Sea, by Millot and Crepon (1981) for the Gulf of Lions in the Mediterranean Sea, and by Thomson and Huggett (1981) for Queen Charlotte Sound and Hecate Strait off the west coast of Canada. Most of the theory presented in past work has been analytic in character; there appears to have been little or no effort made to reproduce inertial currents within the framework of a numerical model of a stratified sea as in this chapter. Such a model offers the prospect of obtaining more realistic solutions to the hydrodynamic equations than can be obtained by analytic means. Results of the numerical model experiments carried out in this work are described in a series of diagrams, with comments. Physical explanations of the computed motions of the water are given-insofar as the variations obtained can reasonably be identified with known wave forms and modes of oscillation. The experiments seek to elucidate the model’s performance for a range of input parameters determining open-boundary conditions, frictional levels and wind-stress characteristics.
403
As a conclusion to the work, inertial currents observed in the Celtic Sea are discussed briefly. A first interpretation of them is given in terms of results obtained from the numerical experiments. The practical importance of these currents (especially near the sea surface) in attaining magnitudes which are comparable to those of the tidal currents, is stressed. Characteristically, a wind pulse produces an initial current surge followed by damped inertial motion. Particularly relevant to the reproduction of observed inertial currents is the simulation of their damping by friction and their reduction (or enhancement) by processes of wave cancellation (or reinforcement). These features are investigated in the experiments. The work started here needs to be continued with further model development and model verification against observational data. In this respect, possible avenues of future research are suggested. Continued use is made of the notation introduced in Chapter 9. RECTANGULAR SEA MODEL
Consideration is given to motion generated by wind within a rectangular area of the Celtic Sea, ABCD, delineated in Fig. 10.1. The area, 380 km long and 180 km ! 55"N
50"N
Fig. 10.1. Celtic Sea showing area of interest.
404
S A L I N I T Y ( %o 1
T E M P E RAT U R E ("C)
OOJ
I
100
1
I
Fig. 10.2. Mean vertical distributions of temperature and salinity for the month of August at Station MU in the Celtic Sea. (5 1 OWN, SOOO'W)
wide, runs parallel to the south coast of Ireland from the continental slope to the entrances of the Irish Sea and the Bristol Channel. Heaps (1965) showed it to be an important region for storm-surge development. During the summer the region is stratified into essentially three layers, typified by the vertical distributions of temperature and salinity shown in Fig. 10.2. These distributions, quoted by Bowden (1955), are observed means for the month of August at Station MU (Fig. 10.1). They yield a vertical density profile which may be resolved into a surface layer of depth 25 m with a, = 25.8, a thermocline layer of depth 15 m with a, = 26.5, and a bottom layer of depth 60 m with u, = 27.2 (Fig. 10.3). The present theory is applied assuming this three-layered structure everywhere within the rectangular area. Total depth within the rectangle is taken to be uniform at 100 m, a reasonably good first-order approximation to the real topography. Thus: h , = 25 m,
p , = 1.0258 g ~
h,
pz = 1.0265 g cm-3
=
15 m,
h , = 60 m, h = h , + h , + h 3 = 100m
r n - ~
p3 = 1.0272 g cm-3
The region has a central latitude (p = 50O40'. Therefore, with the angular speed of the Earth's rotation given by w = 7.292 X l o p 5 s-', the geostrophic coefficient (assumed constant) is taken as: y = 2 x 7.292 X l o p 5 X sin(50'40') = 1.128 X 10-4s-1.
(2)
405
2
10
20
30 Depth (metres)
J.
40
50
60
70
80
90
100
Fig. 10.3. Mean vertical distribution of density for the month of August at Station MU, derived from the temperature and salinity profiles of Fig. 10.2. Resolution into surface, intermediate and bottom layers is shown by the dashed lines.
Assuming eddy viscosity to be uniform and constant within each layer, the following values are regarded as generally representative of summer conditions: N , = 300,
N,
=
10,
N3 = 100 cm2 s - ’
(3)
The value of the upper-layer viscosity N , is an order-of-magnitude estimate corresponding to a surface wind speed of 8.7 m s - ’ (Defant, 1961, p. 423; Kullenberg, 1976; Svensson, 1979) and therefore a wind stress of 1 dyn cm-’ (Heaps, 1965). The comparatively low value of N, is chosen to reflect a situation of high vertical stability in the thermocline, while the value for N3 is chosen to be considerably larger than N2 (on the supposition that vertical stability in the bottom layer is comparatively weak) but smaller than N , (due to the separation of the bottom layer from the wind forcing by the intermediate layer). However, tidal mixing in the bottom layer-arising from shearing of the tidal current against the seabed-might enlarge N 3 , even to the point
406
of its value being greater than that of N , . Clearly the values of viscosity in eq. 3 are tentative and deviations from them need to be considered and the resulting effects on the motion determined. In defining friction at the seabed the coefficient in the quadratic law is taken as: k = 0.0026
(4)
Omitting the tide-generating forces and atmospheric pressure gradients over the sea surface, accounting for the uniform depth and the uniform constant eddy viscosities, and truncating infinite series to include only the first M terms, the equations for numerical solution are:
where:
These equations are derived from eqs. 64-66, 46, 47, 71 and 72 of Part I (henceforth ( s = 1, 2, 3) and the current called I). Solutions are sought for the elevations 1% transforms ti,, 6, ( r = 1, 2,.. ., M ) over the area ABCD. The A,, us,,, +,, & ( h ) are given by eqs. 81-86 of I and depend on basic parameters a , b, c, d, e, q, d, and d, (from eq. 73-1) along with a , , Y = 1, 2,. . . ,M (the first M non-negative roots of eq. 79-1). With constant values of depth and eddy viscosity the a , b , c, d , e , q , d 2 ,d, are all and +,. Table 10.1 lists the values of constants and hence so also are the a,, A,, a , , as,r (s = 1, 2, 3) and for r = 1(1)30 corresponding to the values of depth, density and viscosity given by eqs. 1 and 3. The first ten current modes through the vertical, associated respectively with r = 1(1)10, given by eq. 82-1, are plotted against fractional depth ( in Figs. 10.4 and 10.5. The r t h mode f , ( ( ) is such that f,=f,,r (0 < ( < 0.25, surface layer), f,=f2,, (0.25 < 6 < 0.40, intermediate layer), f,=f,,, (0.40 < 6 < 1.00, bottom layer). Each mode has a slope discontinuity at ( = 0.25 and at ( = 0.40, i.e. at the two interfacial levels. The first mode ( r = 1) is barotropic and the higher modes ( r > 2) baroclinic. The values of u1,, in Table 10.1 distinguish
+,
407
TABLE 10.1 Model parameters corresponding to the values of depth, density and viscosity in eqs. 1 and 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.00000 0.99039 2.83826 4.12799 5.98675 7.56906 9.12960 10.87843 12.10537 13.00125 14.77785 15.87084 17.94728 19.26002 21.10660 22.64834 24.16323 25.12440 26.68724 27.69243 29.86512 30.98653 33.07026 34.37635 36.17944 37.24516 38.61673 39.59863 41.7 1055 42.78835
1.00000000 0.00037301 0.00007280 0.00000077 0.00013532 0.00017 100 0.00006776 - 0.00000 187 0.00000595 0.000005 11 - 0.00004332 -0.00006210 - 0.0000 1718 - 0.00000036 - 0.000032 14 -0.00003558 - 0.00000620 0.00000479 0.00000032 0.00000158 0.00003772 0.00004272 0.00000905 - 0.00000005 0.00000932 0.00000591 - 0.00000879 - 0.00001383 - 0.00000105 - 0.00000217
0.75000
0.60000
- 0.24708
- 0.29954
- 0.22947
0.12279 0.20680 - 0.03 185 - 0.12541 - 0.01641 0.04039 0.00078 - 0.01584 0.02799 0.04474 - 0.0291 1 -0.05112 0.00709 0.02648 - 0.00084 - 0.007 10 0.01374 0.01 924 - 0.02432 - 0.03056 0.0 1442 0.02155 - 0.00342 - 0.00563 0.00701 0.00873 -0,01868 - 0.01916
- 0.20793
- 0.16644 -0.12518 - 0.08288 - 0.03765 - 0.00949 0.00835 0.03549 0.04627 0.05429 0.05 165 0.04000 0.02566 0.00993 0.00009 - 0.01420 - 0.02 I 56 - 0.03096 - 0.03204 - 0.02768 -0,02148 - 0.01024 - 0.00303 0.00588 0.01153 0.02021 0.02234
0.99908 2.27996 0.24219 0.38599 0.09076 0.33685 0.15258 1.32205 1.55396 3.52300 0.47426 0.43788 0.11612 0.29545 0.12374 0.79337 0.8901 1 4.13528 1.03090 0.48298 0.18072 0.25236 0.12868 0.50766 0.53429 3.52997 2.13649 0.55485 0.30327 0.20100
quantitatively between the barotropic mode ( = 1) and the baroclinic modes (ai,, 0, r >, 2). As r increases it is apparent that the modal sh’ape becomes more oscillatory, particularly in the thermocline region. Figure 10.6 shows a finite-difference grid for the numerical solution of eqs. 5-7 over the rectangular sea area. Grid lines are x- and y-directed, parallel to the sides of the rectangle. Grid points are of three types: 0 a {-point at which {,, 12, {, are evaluated, + a u-point at which zi, ( r = 1, 2,. . . , M ) are evaluated and X a u-point at which 6,( r = 1, 2,. ..,M ) are evaluated. Points of each type form a rectangular array of I rows and n columns, and are numbered consecutively along successive rows as follows: i = ( j - 1)n + 1, ( j - 1)n + 2,. . . ,jn f o r j = 1, 2,. . . ,1 where n = 20 and I = 10 (Fig. 10.7). Mesh lengths between points of the same type are A x in the x direction and Ay in they direction. The network has square elements with a 20 km
-
408
Fig. 10.4. Current modes 1-5 through the vertical, plotted against fractional depth (for N , = 300, N , N3 = 100 cm2s-').
=
10,
side, so that Ax
= A y = 20
km
(9)
With the sides of the rectangle taken through u and v-points as indicated in Fig. 10.6, the sea region has 19 mesh elements in the x direction making up the length 19 X 20 = 380 km and 9 mesh elements in the y direction making up the width 9 X 20 = 180 km. Current transforms, currents, elevations and wind-stress components are defined on the finite-difference grid such that at u-point i :
and at r-point i :
where s = 1, 2, 3. With these definitions, using an explicit difference scheme similar
409
Fig. 10.5. Current modes 6-10 through the vertical, plotted against fractional depth (for N , = 300, N, = 10, N3 = 100 cm2s-').
to the one employed by Heaps (1974), the differential eqs. 5-7 are approximated by:
[ls,i(t
+ A t ) - ~ s , i < t )/At l
r= 1
(13)
410 I
2
3
4
5
6
7
8
9
10
I1
12
13
14
15
16
17
18
19
x
x
::
x
Y
::
x
::
::
x
x
x
x
::
::
z
x
20
I
, A x
I
0
2
0 - - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - -
3
0 - - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - ~
4
0 - - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - -
5
0 - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - -
6
x x x x x x x x x x x x x x x x x x x x o ~ - o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o - x x x x x x x x x x x x x x x x x x x x
7
0 - - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - -
8
0 - - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 -
9
0 - ~ 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - -
10
0 - - 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 - -
x
x
x
x
x
x
x
x
x
x
D
x x
x
x
x
x
x
x
x
'
x
x
"
-
x x x
"
x x x
x x x
x x x
x x x
x x x
x x x
,.
x x x
x x x n
x x x
x x x n
x x x
x x x n
x x x
x x x n
x x x
x x x n
x x x
x x x n
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
-inn
C
Fig. 10.6. Finite-differencegrid covering the rectangular area of the Celtic Sea shown in figure 1 ; l. u and 0,+ and X, respectively; straight lines mark out the sides of the rectangular area.
v points are denoted by
41 1
I
l
l
1
Fig. 10.7. General plan of the finite-differencegrid showing: (a) the overall numbering system for the grid points, and (b) the localised notation for grid-point positions.
Equations 13-15 are applied in turn at all the interior l, u and v-points of the rectangular sea to derive the values of 12,i13,i,z2r,i and at time t from their values at time t + At. This procedure is repeated for t = 0, At, 2At, 3At, .. . and the variables are developed through time over the sea area starting from an initial state
412
For a closed rectangular basin, coastal boundary conditions of zero normal flow have to be satisfied, namely: f i r , i = 0 at each u-point i on they-directed land boundaries =0
at each v-point i on the x-directed land boundaries
(18)
for all t > 0 The time-step At is limited by the stability criterion At < A ~ / ( 2 g h ) ' / ~ which relates to the speed of the surface gravity waves. For the network and basin of Fig. 10.6, numerical stability is achieved by taking: = 360 s (19) Accompanying the computation of the fields [z,i, [3,i, a , , and fir,i over the rectangular basin, through time, goes the evaluation of horizontal currents at u and v-points. Thus, from eq. 63-1 using eq. 82-1, the three-dimensional current structure (in terms of the first M current modes) is given by:
At
M
us.i=
C +rcr,iL,r,
M
vs,i=
r= 1
C Gr6r.iL.r
~=1,2,3
r= 1
and, from eqs. 69-1 and 70-1, the depth-mean currents by:
Horizontal current components at each interior [-point are determined by taking simple averages across the point of the components evaluated (using the above expressions) st the neighbouring u and v-points. Thus:
The averages yielding water column.
usqi,K,i are taken for each E (0 < 6 < 1) through the vertical
413
RADIATION BOUNDARY CONDITIONS
As an alternative to the closed boundary conditions given by eq. 18, radiation open-boundary conditions are now formulated based on one-dimensional wave propagation in an inviscid three-layered system. Considering propagation in the x direction, from eqs. 10-12 and 22-24 of I the relevant equations are:
The assumption of zero friction means that u l , u2 and u3 here are independent of the depth I. Progressive wave solutions are sought of the form:
s,=
Z,F(t - x/c0), u,= U,F(t - x/c0),
S=
1,2, 3
(29)
where co is the speed of wave propagation and F a general function of t - x/co; Z,, Us are unknown coefficients. Substitution of eq. 29 into 23-28 gives:
where: H = c,'/g Since 1 - p 1 / p 2 , 1 - p 1 / p 3 , 1
(32) -
p 2 / p 3 << 1, eq. 31 may be written in the approxi-
414
mate form:
( H- h)( H 2 -poH + 90)= 0
(33)
in which: PO = (h,h2/h)(l
40
- Pi/P2)
= (h,h2h3/h)(1
+ (hih3/h)(l
-PI/P2)(’
- P i / P 3 ) + (h2h3/h)(l
-P2/P3)
- P2/P3)
(34) (35)
Therefore, from eq. 33:
H = h , H ( 2 ) ,H ( 3 )
(36)
where:
H‘2)=[ p o + ( p : - 4 q o ) ’ / 2 ] / 2 , H ~ 3 ) =p[ O - (p:-4q,)1/2]/2
(37)
From eqs. 30 and 32, approximating as before, it follows that corresponding to
H=h: c0 = c p ,
h, z , , z --z, h3 z, = h2 + h 3 h ~
C p u,= u2= u3 ---z, h
corresponding to H
=
If(*):
and, similarly, corresponding to H
= H(3):
where:
and:
Equation 38 defines a barotropic mode (mode 1) propagating with speed cb” and eqs. 39 and 40 baroclinic modes (modes 2 and 3 ) propagating with speeds ci2), .A3). In general, wave progression in the x direction consists of contributions from each
415
mode yielding:
5; = -11 h 3 (1) + p q p +
p(3yJ3)
h
uI=
p) c(3) “ p Lg’0( 3 ) ,(2)
-
h
h,
C p
C p
u2=-
h u3 =
h {{I’
= Fl (
l2
c(3) - p ( 2 ) ) p + 0 ( 1
h*
-0 p c(l)
h,
+ -CpY )
(43) - P(3)){43)
h2
l2 +
( 2 ) (2)
c(3) -p(3){43’ 0
h3
h3
t - x/c~”)
{12) = F, ( t - x / c ~ ) ) { : 3 ’ = 4 ( t - X / C p )
The F , , F2, F3 are functions representing wave propagation in the respective modes. Superscripts denote mode number. Modifying eq. 43, reflected waves travelling in the negative x direction, with reflection coefficients rl, r,, r3 for modes 1, 2, 3, respectively, may be written: H(2)
{ l = r pr 2-T p 1
H(3)
- r3-{$3’
h,
416
Summing incident and reflected waves from eqs. 43 and 44 at x 1 - rs
E,
=
1 + rs
s=1,2,3
= 0,
writing: (45)
we get:
+ r l) Fl(t ) {i2)= (1 + r2)F2(t ) 5i3)= (1 + r3)F3(t ) {,‘I)=
(1
rl,
The first three equations of 46 are used to solve for [[I), 14’) and li3) in terms of 12, 13.Then substituting these expressions into the following three equations for u , , u 2 , u j yields: 3
UJ =
c
s= I
UJ,&
j = 1,2, 3
(47)
in which: c(l) c pco( 3 ) 0 u I ,s = El -6 s - E 2 -Ps - E -4s
h
hl
hl
417
(49)
where, in turn
Equation 47 is taken as a radiation boundary condition with the coefficients E , , E , determining the degree to which modes 1, 2 , 3 are radiated in the x direction. Particular cases governing the transmission of mode s in the x direction are: ( r , = 1):
E,=O
1 (r,=o): ( r , = - 1) :
&,=
E , + 00
total reflection, zero flow no reflection total reflection, zero elevation
(51)
Introducing eq. 47 into 48-1 enables the radiation condition to be expressed in the usable form: 3
‘r
=
c
- uZ.r)Ul.s
+ (P2/Pl)(aZ,r
- a3.r)U2,s
+ (P3/Pi)‘3,rU3.s]
ls
(52)
s= 1
This equation is applied along the sides BC, AD of the rectangular region ABCD (Figs. 10.1 and 10.6) and, with zir replaced by 6,,along the sides AB, DC of the region. In this way the equation governs x-directed propagation across BC, AD and y-directed propagation across AB, DC. Within the computational scheme based on eqs. 13-15, condition 52 determines zi, ( t A t ) or 6,( t + A t ) at each boundary u or u-point from 5, ( t + A t ) , 5, ( t + A t ) , 5, ( t + A t ) at the nearest interior [-point. Evaluation of zi, ( t + A t ) on the boundaries BC, AD may follow directly after evaluation of zi, ( t + A t ) in the interior using eq. 14. Similarly, evaluation of 6, ( t + A t ) on the boundaries AB, DC may follow directly after evaluation of 6, ( t A t ) in the interior using eq. 15. Thereby, elevations and current transforms at time t + A t are deduced from their values at time t both on and within the boundaries of the rectangular region. Specifically, for untrammelled outward radiation of all three modes along the four sides of the region, eq. 52 is employed with:
+
+
1
E,
=
E,
= -1
E,
=
E,
= -1
(s=
1
onsideBC on side AD on side AB (with zi, replaced by 6,) on side DC (with ti, replaced by 6,)
(53)
1,2,3)
Setting E , = e2 = E , = 0 all round produces closed boundaries and a return is made to the conditions of eq. 18.
418
MODEL WITH FINE GRID: TIME A N D SPACE SPLITTING
To improve the horizontal resolution of the internal motion the finite-difference grid is refined by 1/3 to yield a network with square elements of side 20/3 km: Ax = Ay = 20/3 km (Fig. 10.8b). With the present finite-difference scheme, numerical stability on the new grid requires a corresponding 1/3-reduction in the time-step: A t = 360/3 = 120 s. As a result, computation running times increase by 3 x 3 X 3 = 27 causing solution development to become prohibitively long and expensive. To reduce the size of the computations, and hence make further progress possible, the updating procedure for elevations and current transforms is split between {,; ti,, 6, (barotropic) and S;, 5;; ti,, 6,:r = 2, 3,. . . , M (baroclinic). Specifically: (i) c,, ti,, 6 , are updated on the original coarse grid A AX, 3Ay) every time-step At. Here we may take A t = 360 s as before. The updating is based on the appropriate equations of continuity and motion from eqs. 13-15 with sz, 13, ti,, 6, ( r = 2, 3,. ..,M ) omitted except in the friction term. A degelierate form of the radiation boundary condition of eq. 52 is employed relating li,, 6,to {,. (ii) J2, 13,ti,, 6,( r = 2, 3,. . . ,M ) are updated on the fine grid (Ax, A y ) every q' time-steps, i.e. at intervals of At' = q'At. That updating is based, as appropriate, on the full equations of continuity, motion and boundary radiation given by eqs. 13- 15 and 52. The l,,ti,, 6,occurring in the equations are evaluated on the fine grid by linear interpolation between their values on the coarse grid determined in (i). Note that {,, ti,, 6 , are evaluated independently of 12,13, ti,, 6,( r = 2, 3,. . . , M ) apart from the link through friction. A separation is thereby achieved between barotropic wave propagation in (i) and baroclinic wave propagation in (ii). Therefore, since the internal waves have lower propagation speeds, a longer time-step Ar' may be taken in (ii) without disturbing the computational stability in (i). An economy is thus made in the computations. Advancing I,, ti,, 6, through time on the coarse grid is justified by the longer horizontal space scales assumed for the barotropic motion. The calculation of these variables on the coarse grid, and their subsequent interpolation on to the fine grid, avoids having to compute them on the fine grid directly-with the heavy computational penalty imposed by smaller space and time increments. The complete set of dynamical variables l,, 12, 13, ti,, 6,: r = 1, 2,. . . , M is calculated every q' time-steps when eqs. i and ii are performed together in succession. Then, along with the surface and interfacial elevations, the three-dimensional horizontal current structure may be determined-using eqs. 20, 21 and 22. The difference equations governing the revised computation scheme described above are now formulated. Operations on both the coarse grid and the fine grid are expressed using a grid-point numbering system based on the fine grid with n = 58 and I = 28 (Fig. 10.7). Thus, working from eqs. 13-15, equations for updating {,, ti,, 6,on the coarse grid at t = 0, At, 2At, 3At,. .. are:
(a) GRID G,
419
(b) GRID G3
P
'I
L__c
u3
X
-
( c ) GRID G5
yt
X
Fig. 10.8. Rectangular basin mapped out on grids G , , G, and G,. The grid lines pass through u and o-points, forming square elements with { evaiuated at the mid-point of each. The grid-points of G , (20-krl mesh) are shown in Fig. 10.6. Refinement of G, gives G, (20/3-km mesh) and G, (4-km mesh). Th( central elevation point C,, and the edge elevation points P, P,, Pa, Ps,Q , Q , , Q4, Q, are marked.
420
Here, ~ 2 , , ~ ( t * 6r,i(t*) ), denote the latest updated values of Gr,i, 6r,i( r = 2, 3,. . . ,M ) evaluated at an earlier time t = t * . Equations 54-56 are applied successively (in that order) to determine, at interior grid points of the coarse network, values of 11,ti,, 6, at t + A t from their values at t . The boundary values of ti,, 6,at + A t are deduced from the radiation condition of eq. 52 with only mode 1 of eq. 46 included. That condition then takes the form: +At)=[(al,l
-',,I)+
(P2/PI)('2,1
( ~ 3 / ~ ~ ~ ' 3 , 1E ]I C h ' ) { l ( t + A t ) / h
(59)
where (on the coarse grid) l1 corresponds to the interior {-point nearest to the u-point on the boundary at which fi, is being evaluated. Evaluation of the boundary 2,(t + At), using eq. 59, may follow directly after evaluation of the interior ti,( t + A t ) using eq. 55; while evaluation of the boundary 6 , ( t + A t ) , using eq. 59 with ti, replaced by a,, may follow directly after evaluation of the interior 6 , ( t A t ) using eq. 56.
+
42 1
From eqs. 13-15, equations for the updating of thefine grid for t = 0, At‘, 2 A t ’ , 3At’, ... are:
12,5;, ti,,
8, ( r = 2, .3,.. ., M) on
l s J t + At’) = S S . I ( t ) M
-
C has,r+r{[tir.l(t) - ~ r , i - l ( t ) ] ~ t ’ / ~+x[ c r , , - n ( t ) - 4 . i ( t > l ~ t ’ / ~ y } r=2
- ha,,l+l { [ til.l( t * * ) - til,l- ( t * * ) ] Ar’/Ax s=2,3
+ [ dl.l- ( t * * ) -
(60)
( t**)] A t ’ / A y )
422
In the above we take: t**
=t
+ At’ - At
as a representative lower time level. Equations 60-62 are applied successively to determine, at interior grid points of the fine network, values of 13,fir, 13, ( r = 2, 3,. ..,M ) at t + At’ from their values at t . The boundary values of these ir,, 6,at t At‘ are deduced from the radiation condition of eq. 52 employing updated values of elevation from eq. 60 and updated values of elevation from eq. 54 interpolated on to the fine grid. Adopting a mixture of time levels in eqs. 57, 60 and 63 facilitates the ongoing progression of the time-stepping calculations. The acceptance of such heterogeneity requires that the longer time increment At‘ should adequately resolve the barotropic as well as the baroclinic motion, through time. Moreover, the influence of {,, ir,, 6 , in the time-stepping procedure based on eqs. 60-62 is then adequately represented.
sZ,
+
GRID SCHEMES
Numerical experiments were carried out to evaluate the dynamic response of the rectangular basin to a steady, uniform, x-directed wind stress field of 1 dyn cm-2 (denoted by W,), created over the entire water surface at t = 0, maintained constant for 10 h, and then removed. The motion of the water was computed over periods of 200 h or more following the application of the wind at t = 0. Initial tests assumed outward radiation across all four sides of the basin with:
1, 1, 1 onsideBC -1, - 1 , -1onsideAD = 1, 1, 1 on side AB = - 1, - 1, - 1 on side DC These coefficients corresponding to the case of no reflection in eq. 51, form a “radjatjon” matrix:
q ,E ~ , E , =
=
1 -1
-
(66)
Three separate grid systems were employed, shown in Fig. 10.8. They are: grid G , (20-km mesh), grid G, (20/3-km mesh) and grid G, (20/5 = 4-km mesh). The grids G, and G, represent successive refinements, by factors of 1/3 and 1/5 respectively, of grid G I .The network of grid points associated with G I is shown in Fig. 10.6: the grid-poiilt arrays associated with G, and G, follow the same pattern (Fig. 10.7) where n = 20 and 1 = 10 for grid G I ,n = 58 and 1 = 28 for grid G,, n = 96 and 1 = 46 for grid G,. Computations of the motion in the basin based on eqs. 13-15 and 52 were carried out on grid G,. Parallel computations on grids G, and G,, respectively, were also performed but had to be limited in length due to their considerable size and consequent high computing cost.
423
However, time and space splitting between the barotropic and baroclinic motions (see p. 418) reduced the computations on grids G, and G, to manageable proportions without significantly affecting the accuracy of the results. Thus, using grid G, as described on p. 418, with the barotropic motion generated on the 20-km network and the baroclinic motion on the 20/3-km network, time splitting between those motions was found to be possible without any significant loss in accuracy taking q’ = 5 ( A t = 360 s, A t ‘ = 1800 s). The effectiveness of this approach as a means of achieving computational economy is seen from the following CPU times (on the CRAY-IS computer) required to simulate the first 42 h of motion: 160 s (no time or space splitting, the combined barotropic and baroclinic motions calculated on the 20/3-km network), 61 s (space splitting only: q’ = 1, At‘ = A t = 360 s) 37 s (time and space splitting: q’ = 2, A t ’ = 720 s) 29 s (time and space splitting: q’ = 3, A t f = 1080 s) 25 s (time and space splitting: q’ = 4, A t f = 1440 s) 23 s (time and space splitting: q f = 5, A t f = 1800 s) These times were obtained with M = 10 and are quoted to indicate generally the order of time saving which is possible. Doubling the number of vertical modes, taking M = 20, led to run times longer by a factor of about 1.5. Grid G, was used similarly, with the baroclinic motion generated on the finer 4-km network. Again, taking q’ = 5 incurred no significant loss in accuracy in the results. Run times were found to be about six times longer, correspondingly, than those with grid G,: the price paid for increased resolution of the internal motion. The most severe restriction was imposed by limited computer memory and, for this reason, the computations with grid G, were performed with no more than ten modes ( M = 10). Using grid G,, both the barotropic and baroclinic motions were evaluated on the 20-km network. Introducing time-splitting between those motions with q’ = 5 reduced run times to about two-thirds their former length without significantly impairing accuracy. Run times were then about one quarter the length of corresponding runs with grid G,. However, the results obtained with grid G , suffered seriously from the poor spatial resolution of the internal motion.
COMPUTATIONAL RESULTS
Diagrams are now presented describing the computed response of the rectangular basin to the wind-stress field W, (longitudinally x-directed, of magnitude 1 dyn cm-2, applied uniformly over the water surface from t = O to t = 10 h). The computations were based on the values of depth, density, rotation, eddy viscosity and friction-coefficient given by eqs. 1-4. The first ten current modes were employed ( M = 10). To determine the effects of increased horizontal resolution in the computations,
424
Fig. 10.9. Mesh element of G,: subdivided by the continuous lines into elements of G, and by the dashed lines into elements of G,. Positions P , P,. P., and Ps are marked.
comparable model runs were carried out on all three grids G I , G, and G, (Fig. 10.8); time and space splitting with q' = 5 was used as described on p. 418 and discussed further on p. 423. 0
A
(a)
yI D
Q. x
C
Fig. 10.10. (a) Wind-stress pulse W, acting longitudinally over the rectangular basin; (b) the time variation of that stress.
425
Figure 10.8 indicates the central elevation point C, and the edge elevation points: P, Q on grid G I ; P, P,, Q, Q, on grid G,; P, P4, P,, Q, Q4, Q, on grid G,. Figure 10.9 compares the relative positions of P, P4,P3 and P,, respectively situated 10, 6, 3.3 and 2 km from the mid-point of the side BC of the rectangular basin; the positions of Q, Q4, Q, and Q, are similarly distributed with respect to the mid-point of the side CD. Time variations of current and internal elevation at these points are examined. Under the influence of the wind (Fig. 10.10) and the Earth's rotation, surface water moves towards P and Q causing the internal interfaces to be displaced vertically downwards at those positions (12, {, < 0). Gradients of surface level, sloping upwards towards P and Q, develop at the same time. As described below, inertial oscillations of current are subsequently induced in all three layers of the basin, together with associated vertical oscillations of the internal interfaces. Currents are symmetric and elevations of the surface and the interfaces are antisymmetric about the central position C,. Time variations of current and internal eievation
Adopting radiation boundary conditions defined by the matrix (eq. 66) corresponding to no reflection of outgoing waves, computations on grid G, over 200 h yielded variations in depth-mean current at C , shown in fig. 10.11. In each layer the currents exhibit damped inertial oscillations, the inertial period 27r/y being 15.5 h. exceeding 4 cm The current components GI, 6, in the surface layer are largest, s - I . , those fi,, 6,in the bottom layer are smaller by about half and sensibly out of phase by 180". The surface-layer oscillations are mainly produced by the direct action of the wind, while the bottom-layer oscillations arise primarily from the gradients of surface elevation, set up by the wind, which produce changes in the pressure field through the entire water column (Krauss, 1979). With open-boundary radiation as prescribed, surface gradients build up while the wind acts, but dissipate rapidly after the wind ceases. The current components G 2 , C2 in the intermediate layer are relatively small, amplitudes of oscillation there being less than 1 cm s-I. Currents fi, t? averaged through the entire depth show a non-oscillatory flow pulse during the first ten hours of motion, which may be associated directly with the wind forcing during that time. The surface-layer and bottom-layer currents of Fig. 10.11 are displayed in polar form in Fig. 10.12. Thus, changes in the magnitude and direction of the surface-layer current are illustrated by plotting (el,C1) at hourly intervals in the ( u , u ) plane. This diagram clearly portrays an initial current surge produced by the x-directed windstress pulse, followed by a clockwise rotation of the current vector in damped inertial motion. The bottom-layer current vector (Us,U 3 ) behaves similarly, rotating in the same sense but lagging behind the surface vector by essentially half the inertial period. That 180' phase lag becomes established after about 20 h following the commencement of the motion. Manifestly, the initial current surge in the bottom layer tends to be y-directed and associated with forcing by gradients of sea level diagonally oriented across the basin set up by the wind. Repeating the computations on grid G, gave depth-mean currents at C, in close
426
U
4
V
~ " ' " " 50 " " " " 1'00 " '
150
200 h
Fig. 10.1 1. Depth-mean currents at C , , computed on grid G,.
agreement with those shown in Fig. 10.11 and 10.12. However, the same computations on grid G I gave significantly different currents at C,, as illustrated in Fig. 10.13 where U, from G I and G, are compared. The El from G , is seen to vary in an
421
( a1
t4
4
(b 1
3 2
I
-2
-3
-4
-5
Fig. 10.12. Depth-mean currents at C, plotted at hourly intervals in polar form, for (a) the surface layer showing(:,, D , ) from 1 = 0 to t = 200 h and (b) the bottom layer showing(ii3, D 3 ) from r = 0 to f = 100 h. Units are in cm s - I .
irregular way, a behaviour interpreted as being due to the failure of the 20-km mesh to properly resolve the internal oscillations. This failure might be expected, since the internal Rossby radius of deformation for a simple two-layer system representing the density profile of Fig. 10.3 (with an upper layer of depth 30 m and density 1.0258 g cm-3 and a lower layer of depth 70 m and density 1.0272 g ~ r n - ~ rotating ), with
I
-
Fig. 10.13. Depth-mean current ii, at C,, computed on grids G I and G,.
428 = 1.128 x l o p 4 s-’, is approximately 5 km. To achieve a horizontal resolution of this order the networks G, and G, are certainly required; G, is too coarse. Even finer resolution than that provided by G, and G, might ultimately be necessary to give a more accurate and comprehensive description of the internal wave field. Time variations of l2 (the elevation of the upper interface) are plotted in Fig. 10.14. The variations at points P and Q are shown, obtained from the computations on G, and G,. Damped inertial oscillations are again evident following initial depressions of the interface produced by the wind-driven movement of water in the surface layer. The oscillations are reproduced in closely similar farm by G , and G, respectively. However, on G,, the oscillations are somewhat smaller in amplitude and are preceded by higher initial depressions of level. Manifestly, mean interfacial levels at P and Q are depressed by the wind action, an effect dependent on the degree of vertical stratification. The return of mean level to zero, after the wind has ceased, is slow under the influence of the density forces.
y
cm
30 20 10
0
- 10 -20
-30
30 20
10
0 -10
- 20 -30 -40
Fig. 10.14. Elevation of the upper interface t2 at points P and Q; computed on grids G, and G,, respectively.
429
Figure 10.15 compares the variations of 5; at P and P,, on grid G,, and at P, P4 and P,, on grid G,. These P-positions are located near the mid-point of the edge BC of the basin (Figs. 10.8 and 10.9). Clearly the initial depression of the upper interface becomes larger quite rapidly as the edge is approached. However, after 30 h or so, the oscillations at the P-positions on each grid come into close agreement. The variations at P, and P, are approximately the same-as one should expect of results for neighbouring points; nevertheless, the agreement does indicate a consistency between the output from G, and G,. Figure 10.16 shows a similar set of curves giving l2at Q, Q, on grid G, and at Q, Q4, Q, on grid G,. These Q-positions are near to the mid-point of the edge CD of the basin. Note the much larger initial response here, due to a hgh cross-basin surface flow rapidly created after the onset of the wind.
crn.
80
60 40
20 O h
0
- 20 t(hours)-
-40
-60
- 80
80
60
(G,)
40
20
O h 0
- 20
-80
Fig. 10.15. Elevation l2at P and P3 computed on grid G, and at P, P4 and Pscomputed on grid G,.
430
Figure 10.17 compares the vertical displacements 12, l3of the upper and lower interfaces at the edge positions P3 and Q3, computed on grid G,. It is evident that the oscillations of the respective interfaces are damped in unison and take approximately the same amplitudes. However, during the first 20 h of motion, the upper interface is depressed further than the lower one and the lower one then rises further than the upper one. This behaviour, involving the closing together of the interfaces, affects horizontal flow in the intermediate layer (see, later, Fig. 10.22).
40
20 O h
0
- 20 t
-120
--
(hours)
-
Q3
-140 --
- 160 --
Fig. 10.16. Elevation
zz at Q and Q 3 computed on grid G , and at Q, Q4 and Qs computed on grid G,.
43 1
:iIVVV -20
-40
-120
--
-140 --
-160
3
--
Fig. 10.17. Comparison of
l2 and l, at P,
and Q 3 : from computations on grid G,.
The changing spatial distributions of current and internal elevation, as determined from the computations on grid G,, are now described. The inertial oscillations of TZ (exemplified in Fig. 10.14) are concentrated around the edges of the basin during the early stages of the motion. Correspondingly, the rotating currents in the various layers (exemplified in Fig. 10.11) are essentially uniform in the horizontal and only depart from that uniformity around the edges. As time advances, the T2 oscillations spread slowly into the interior and the spatial uniformity of the currents begins to
432 CO
co
x-
. . . . . . . . . . . . . . . . . . . . . . . . . . .
1::
: :
I
-Y : : : : : : :
!
:
:
!
+
1
10
10
0
0
-10
- 10 I I
\
-20 I
1.40 h
-20
i
t, (crn)
- 30
-30
- 40
-40
t=8h
-50
I
I
-50
-60
-60
-70
-70
- 80
-80
I
Fig. 10.18. Distributions of computations on grid G,.
lZ
along the central sections COP, and CoQ, at various times, from
break down from the edges inwards. Both the elevations and the currents are damped continuously by outward radiation and frictional dissipation. These changes through space and time are illustrated in Figs. 10.18 and 10.19 where l2and U , are plotted along the central sections COP, and CoQ, (Fig. 10.8) for various times. Exemplifying the spatial distribution of internal (interfacial) elevation, Fig. 10.20 displays contours of 5; at t = 136 h. The internal wave pattern shown in the figure is basically cellular in structure. Longitudinal crests and troughs predominate as a result of cross-basin spreading of the oscillatory motion from the longer sides. The effects of spreading from the ends of the basin are also evident. An explanation of the above changes appears to depend on postulating the presence of Poincare internal standing waves of various orders, rotating clockwise within the rectangular basin. Cellular wave motions, depicted by Mortimer (1977, 1980) for a two-layer channel model, are envisaged. In support of this hypothesis, it is apparent from Fig. 10.11 and 10.14 that waves at Q lag by one quarter of an
433
4.0
4.0
3-5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0
0
1=4h
,
1
-U, ( c m r ' )
1-68h
Fig. 10.19. Distributions of U , along the central sections COP, and C,Q, at various times, from computations on grid G,.
inertial period behind those at P while the bottom currents at C, rotate such that peaks of ii3,V3 may be associated in time with troughs of l2 at Q and P, respectively. Thus, an assemblage of internal Poincare modes is prescribed where generally a mode consists of no x I , rectangular cells of rotating wave motion ( n o along the basin in the x-direction and I , across the basin in the y-direction); n o , I , separately take the odd-integer values 1, 3, 5,. ... Initially it is supposed that the modes are superimposed to produce the early conditions in the basin (described above) involving uniform motion in the interior and departures from that uniformity around the edges. It is then argued that, through time, those modes (differing in period and rotational speed) separate out and produce the cellular wave patterns seen in the computed results. In such circumstances, multiple nodalities are caused by the presence of the hgher-order wave forms corresponding to n o , I , > 1. Higher cross-basin nodalities, I,, are particularly noticeable. The ultimate resolution provided by the finite-difference grid, in this case G,, clearly places an upper limit on the order of a mode which can be satisfactorily represented in the computations. Explanations given for the occurrence of inertial oscillations in Lake Michigan and Lake Ontario (Mortimer, 1980) and on the continental shelf of the Gulf of
3
435
Lions (Millot and Crepon, 1981) also apply here. Two distinct generating mechanisms were suggested. The first involves the local response of the sea to wind stress, appropriate to open-ocean conditions (Pollard and Millard, 1970). Thereby, an approximation to pure inertial motion is produced in a region distanced from boundary constraints such as the interior of the rectangular basin considered in the present work. The second mechanism involves the creation of long barotropic and baroclinic gravity waves at boundaries. These motions are required, in combination with the local response to the wind, for the satisfaction of the boundary conditions. In the present problem, as already described, the baroclinic waves appear to take the form of near-inertial Poincare oscillations of internal elevation and current which spread into the rectangular basin from its edges. On the other hand, the barotropic waves produce comparatively rapid changes in surface gradient over the basin which excite inertial oscillations manifested at lower levels in the water column, 180" out of phase with the inertial oscillations directly excited by the wind near the sea surface. Figure 10.21 shows power spectra of ii, at C,, l2at P and l2 at Q. Spectra are given corresponding to values of ii, and l2 computed on grids G I , G, and G,, respectively; effects of mesh refinement are thus portrayed. Each diagram was derived from hourly values in the range 0-200 h, using a Fast Fourier Transform with a 10% cosine taper. Smoothing was kept to a minimum to enhance the resolution of the spectra. A single major peak just below 0.067 cph (period 14.9 h) is indicated by all the spectra derived from values computed on grids G, and G,. This near-inertial signal might be regarded as confirming the presence of internal Poincare modes-with periods lying between 14.9 and 15.5 h (the inertial period). Calculations of the kind carried out by Mortimer (1977), based on the simple two-layer model already cited, show that there are two such modes for no = 1: one of period 15.41 h corresponding to I, = 1 and another of period 15.01 h corresponding to I, = 3. Also there are higher-order modes, with periods lying within the same range, corresponding to no = 2(1)6, I , = 1 and no = 2(1)3, I, = 3. Taking the coarseness of the spectral bandwidth into account, the major peak allows for the possible occurrence of pure inertial oscillations (period 15.5 h). In Fig. 10.21, large sub-inertial peaks appear in the G I spectra, indicating the presence of longer-period oscillations judged to be spurious to the proper solution for the motion in the rectangular basin. The G, and G, spectra agree closely and are considered to reflect the true dynamics. However, a small broad peak at 0.137 cph (period 7.3 h) shows up in the G, spectra of S; but not in the corresponding G, spectra. Initially this secondary peak was thought to be the first harmonic of the major inertial peak and present as a result of nonlinearity introduced through the quadratic law of bottom friction. However, the secondary peak still appeared when a linear law of bottom friction was employed. Therefore the peak is now attributed to the weak presence of the basin's slowest mode of barotropic free oscillation, more conspicuous when the basin is completely closed (see, later, Fig. 10.38). That the peak occurs with G, and not with G, would appear to indicate that the outward radiation of the slowest mode is more strongly accomplished on the coarser grid. The influence of the barotropic oscillation (period 6.8 h), in giving rise to the secondary peak, is mainly exerted during the first few hours of the basin's motion. Combination of the barotropic and baroclinic oscillations at P and Q would then appear to
436
_!__ 0.06
0.1
0.16
0.2
5, ( P
* Grid G ,
Grid G ,
Grid G5
Fig. 10.21. Effect of grid refinement on spectra of ii, at C,, Sz at P and sz at Q. The spectra are derived from hourly values (0-200 h) in the computed response to wind-stress pulse W,. Units: frequency in cph (abscissae); ii, in (cm s-’)’ cph-’ and in cmz cph-’ (ordinates).
s2
431 8h
6h -6
10 h
-5
-4
-3 -2
0
-3
-p
I
12 h
2
3
I4 h -4
I6 h
-I
-I
0
I
2
20 h
18 h
Fig. 10.22. Vertical current profiles of u and u at C, at r = 6(2)20 h. From computations on grid G, taking M = 10. The dotted lines indicate interfacial levels. Current components u and u are in cm s-’.
c2
delay the initial downward displacements in at those positions, as shown in Fig. 10.14. Interestingly, the spectral analysis of currents measured in Lake Ontario and in the Baltic Sea during summer stratification has yielded secondary inertial peaks (Schwab, 1977; Krauss, 1981). Whether the cause of their occurrence in those
438
situations is the same as here, is not known. However, non-linearity (of one form or another) might well be the mechanism which is generally responsible. The development of the vertical structure of current at the central position C , is shown in Fig. 10.22. Components u and v are each plotted through the depth at two-hourly intervals from r = 6 h to t = 20 h. The results come from the computations on grid G,. Under the action of the x-directed wind, relatively large u and v currents develop near the surface during the first six hours of motion. The vertical profiles at t = 6 h show that u diminishes with increasing depth through the surface layer to small values below, while v also diminishes with depth, changing direction in the intermediate layer to become a small uniformly distributed opposite current in the bottom layer. Subsequently, until the wind ceases at t = 10 h, the u-profile develops to give a vertical distribution of current which reverses direction twice in the water column, indicating upper and lower wind-driven circulations. During the same time, the character of the v-profile remains unaltered. After the wind ceases, u
t = 8h.
-6
-5
-4
-3
-2
-I I
M 10
_ _ ~ - 20 ....... .... 30 50
Fig. 10.23a. Caption on p. 439.
0
I cms? 1
2 I
3
/’,”’
I
4 I
439
-2
-I
0.4
0.5
M - X I -
10
30
0-6
0.7
0.9
Fig. 10.23. Vertical profiles of the current components u , u at C, for (a) t computations on grid G , taking M = 10, 20, 30, 50.
=8
and (b) t
=
20 h. From
and v profiles develop in which flow is essentially uniform but oppositely directed in the surface and bottom layers. Current shear is then concentrated in the intermediate layer. The profiles at t = 12, 14, 16, 18 and 20 h exhibit these features which are manifestly associated with inertial oscillations. Following through the sequence of changes in profile mapped out in Fig. 10.22, a distinct alteration in the current regime at t = 10 h, from one in which wind forcing predominates to one in which free oscillatory motion predominates, may be recognised. The characteristic profiles of u and v associated with the inertial motion develop rapidly between t = 10 h and t = 12 h following the zeroing of the wind. The skewness of those profiles, with respect to the vertical axis, indicates the presence of both barotropic and baroclinic modes. Clearly, the highest current shears occur in the intermediate layer while the wind is acting and it is at this stage therefore that shear instability might occur in
440
reality, accompanied by erosion of the thermocline (Krauss, 1981) and an increase in the eddy viscosity N2. Figure 10.23 shows the vertical profiles of u and v at C , for: (a) r = 8 h; and (b) r = 20 h, determined from separate computations on grid G I with M = 10, 20, 30, 50, respectively. The use of G I (rather than G, or G,) achieved necessary economies in computer storage and running times for the higher values of M taken. Employing more than the first ten vertical modes (illustrated in Figs. 10.4 and 10.5) scarcely improves the accuracy of the profiles which develop after the wind has ceased, as exemplified in Fig. 10.23b. The character of the profiles is then basically represented by modes 1 and 2 ( M = 2). However, to achieve good accuracy in the u-profile while the x-directed wind acts, the first twenty or more modes are required ( M >, 20) as demonstrated in Fig. 10.23a. The additional modes are needed to define u in the surface layer (particularly near the sea surface) and in the thermocline, where convergence with respect to increasing M is comparatively slow. During the wind forcing, the v-profile is again well reproduced by ten modes. Therefore, by taking
c = 0.00
c= 0.25
Iv
Iv
€ = 1.00
1
1
Tv 1
Fig. 10.24. Current ( u , w ) at C,, plotted in polar form at hourly intervals from t = 0 to r = 100 h, at five levels through the vertical given by 5 = 0.0 (surface), 0.25, 0.40, 0.70, 1.00 (bottom). Each division along the axes represents 1 cm s-'. These results correspond to the vertical profiles of current in Fig. 10.22.
44 1
M = 10 in the main computations of the present investigation, an essentially accurate representation of the changing vertical distribution of current is obtained under all circumstances. However, more vertical modes are needed to improve the estimation of currents in the surface layer when the wind is acting. A further description of the horizontal currents at C,, as determined from the computations on grid G,, is given in Fig. 10.24 where ( u , v ) is plotted at hourly intervals from t = 0 to t = 100 h at five levels through the vertical: 6 = 0 (surface), 6 = 0.25 (upper interface), 6 = 0.40 (lower interface), 6 = 0.70 (mid-depth of bottom layer) and 6 = 1.00 (bottom). Manifestly, the patterns of rotating current at 6 = 0 and 5 = 0.25 approximate the pattern of rotation of the depth-mean current in the surface layer shown in Fig. 10.12a. Similarly, the patterns of rotation at 6 = 0.40, 0.70 and 1.00 approximate the pattern of rotation of the depth-mean current in the bottom layer shown in Fig. 10.12b. These features reflect the essential two-layer structure of the inertial motion. It is interesting to note that the inertial circuits at the specific depths in Fig. 10.24 have a slight ellipticity not exhibited by the inertial circuits of the depth-averaged motions in Fig. 10.12. Passing downwards through the
U
4
U
d
Fig. 10.25. Surface current ( u , 0 ) at the edge positions P, P3, Q and Q 3 :plotted at hourly intervals from = 0 to t = 100 h. The results correspond to those in Fig. 10.24. Units: cm s-I.
t
442
intermediate layer (0.25 < 6 < 0.40), the magnitude of the horizontal current vector becomes small and then increases again. Correspondingly, the phase of rotation changes by 180°. Thereby, the surface- and bottom-layer regimes of current are linked in the vertical water column. Figure 10.25 shows the variations of surface current at the edge points P, P3, Q, Q 3 .It can be seen that the surface-current rotations at P and Q are similar to those at the central point C, (Fig. 10.24) but are more heavily damped. At P3 and Q 3 the currents are smaller than at P and Q even though the initial current surge at Q 3 is enhanced. Proximity to the boundaries evidently constrains the magnitudes of the currents and increases the damping due to energy loss by outward radiation. Effects of open-boundaiy radiation
The computations on grid G , were repeated with differing degrees of openboundary radiation. Taking: el,E
~
e3, = =
= =
1,
e0 on side BC
c0,
-1,
-to,-
1,
E
-1,
~
- E ~ ,
,
~~onsideAD e0 on side AB
-e0onsideDC
implying a radiation matrix
K =
-1
1
-&o &o
-&o to
the motion in the basin was evaluated for e 0 = 0 , 10, 50, covering a range of conditions governing the outward propagation of the baroclinic waves (eq. 5 1). The motion previously considered was for e0 = 1 (eq. 66). Keeping q’ = 5 , the time step A t in the computations had to be reduced to 90 s with E~ = 10 and to 45 s with E~ = 50 in order to maintain numerical stability. cm 40
20
t (hours)h
0
- 20 -40
-60
- 80 Fig. 10.26. Elevation l2 at P, computed on grid G,, employing radiation matrix R with E~
= 0,
1, 10, 50.
443 cm. T
Fig. 10.27. Elevation
c2 at P3, computed on grid G,, employing radiation matrix R with
E,,
= 0, 1, 10, 50.
Results given in Figs. 10.26-10.29 show that by increasing E~ from 1 through 10 to 50, 5; at the edge points P, P,, Q, Q, is reduced to near zero. The internal elevations 5;, S3 are suppressed in this way all round the edges of the basin. Consequently, the inward spreading of inertial boundary waves (Figs. 10.18 and 10.19) is weak and internal elevation remains small everywhere. Correspondingly, current ( u , v) is, for the most part, horizontally uniform-edge effects being small.
Fig. 10.28. Elevation c2 at Q, computed on grid G,, employing radiation matrix R with E,,
= 0, 1, 10, 50.
444 cm 50 t (hours)
-
0
h
-50
- 100
-150..
-200..
-250 ..
-300-.
-350- -
Fig. 10.29. Elevation S2 at Q,, computed on grid G,, employing radiation matrix R with E~
= 0, 1, 10, 50.
Thus, the patterns of current at the central position C, (Fig. 10.24) apply with good approximation to the edge positions P, P3, Q, Q3. Hence, increasing E , progressively removes the influence of the edges on the inertial motion. For the case E, = 0 (corresponding to total reflection of the internal waves at the boundaries) the variations of l2in Figs. 10.26- 10.29 display inertial oscillations superimposed on substantial longer-term changes. The latter can be seen more completely in Figs. 10.30 and 10.31 where S; is plotted over an extended period of time. At P and Q (located 10 km from the sides BC and DC, respectively), l2has a slowly varying part taking the form - at each position - of a lowering of level followed by a gradual return to zero. The depression at Q bottoms out some 150 h earlier than that at P, indicating an anticlockwise wave progression within the basin. Corresponding variations at the neighbouring positions P3 and Q3 (located 3.3 km from BC and DC, respectively) are significantly different. At Q3, initially there is a large steep downward plunge in level followed, 100 to 120 h later, by a rapid rise. At
445 Crn
50
T
Fig. 10.30. Elevation l2 at P and P3 computed on grid G,, from t = 0 to t = 400 h, employing radiation matrix R with e0 = 0.
P3 there is a downward plunge in level between t = 50 h and t = 70 h, which may be identified as the effect of the initial downward displacement at Q3 after its propagation around the edge of the basin from Q3 to P3. Similarly, the rise at Q 3 between t = 100 h and t = 120 h may be identified as the propagated effect of an initial steep rise in level at Q3, the position opposite to Q 3 on the other side of the basin. Again, an anticlockwise transmission around the edge of the basin is involved: from Q; to Q 3 . The evidence points therefore to the presence of internal Kelvin waves propagating in an anticlockwise sense around the basin, concentrated withn a band of thickness 3.3-10 km adjacent to the edges. Calculations with the two-layer model give internal Kelvin-wave modes (formed by Kelvin waves propagating along
em
50 Q3
100
400 h
n
f
t (hours)
Q -150
Fig. 10.31. Elevation matrix R with e0 = 0.
-
i at Q and Q 3 computed on grid G , , from t = 0 to t = 400 h, employing radiation
446
the basin in opposite directions, as described by Mortimer, 1977) with estimated periods (398/m) in hours, where m = 1, 2, 3,. . . ,25. Here, m denotes the number of nodal points along the length, with an upper limit of 25 to keep frequencies sub-inertial and thereby meet the condition for Kelvin-wave reflection at the ends of the basin. The shapes of the responses of Sz at P3 and Q3 suggest that a number of those Kelvin-wave modes are excited by the wind. With m = 2, a period of approximately 200 h is obtained which corresponds well with the propagation characteristics identified above. Specific confirmation of the occurrence of that periodicity comes from the spectral analysis of S2 at P3 and Q3, which yields a major peak at 200 h. For e0 = 0, the variations of surface current at P, P3, Q, Q3 are shown in Fig. 10.32. These variations, which may be compared with those for eo = 1 in Fig. 10.25, are now discussed. At P and Q, inertial currents occur with patterns of rotation
U
A
U +
L
I t t
t
I Fig. 10.32. Surface current ( u , u ) at the edge positions P, P3, Q and Q3: plotted at hourly intervals from t = 0 to f = 100 h. From computations on grid G, with a radiation matrix R ( E =~ 0). Compare with ( u , u ) given by Fig. 10.25 corresponding to R ( E=~1). Units: cm s-I.
447
similar to those obtained with e0 = 1. However, as a new feature, the centres of rotation are displaced from the origin, indicating a small steady u-current at Q and a small steady v-current at P. These currents are evidently associated with the long-term changes in internal level at P and Q shown in Figs. 10.30 and 10.31. At P3 and Q3 the Kelvin waves have a dramatic effect on the currents. Thus, at Q3, there is an initial current surge (corresponding to the initial downward plunge of l2shown in Fig. 10.31) which diminishes through a series of damped inertial oscillations (associated with similar oscillations of l2 in Fig. 10.31). The current pattern displayed terminates at t = 100 h and, therefore, does not exhibit a further surge associated with the rapid rise in lz between t = 100 h and t = 120 h shown in Fig. 10.31. At P3, inertial currents develop up to about t = 50 h (parallelled by similar oscillations of lz in Fig. 10.30) but then a current surge occurs (corresponding to the downward plunge of in Fig. 10.30) which takes the current vector out of orbit and extends it along the v-axis. Subsequently, there is a slow oscillatory diminution of that current, associated with a similar behaviour of l2in Fig. 10.30. The inertial currents at P3 and Q3 are smaller than those at P and Q, presumably due to the closer proximity of the former to the edge of the basin. A constraining effect of the closed boundary condition on the rotations of current is thus demonstrated. Also for E ~ =0, Fig. 10.33 shows the variations of both the surface and the
c2
(t=0)
-
U
d
Fig. 10.33. Current ( u , u), at the surface (5 = 0) and the bottom ( 5 = I), at positions P and Q : plotted at hourly intervals from f = 0 to f = 100 h. From computations on grid G , with a radiation matrix R ( E =~ 0). Units: cm s - ' .
448 CO x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F
- Y
CO
c:: : : :
!
: : : :
QJ !
!
~
!
0
>
- 10
- 20
I
-30
-40
-50
-60 I
, I
-70
- 80
Fig. 10.34. Distributions of l2 along the central sections COP, and CoQ3 at various times, from computations on grid G, with a radiation matrix R ( E=~0). Compare with l2 given in Fig. 10.18 corresponding to R ( e 0 = 1).
bottom currents at P and Q. While rotating inertial currents predominate, displacements of the centres of rotation indicate the additional presence of small steady currents: u-directed at Q (positive at the surface and negative at the bottom) and u-directed at P (again positive at the surface and negative at the bottom). These currents, considered along with the slow variations of at P and Q shown in Figs. 10.30 and 10.31, suggest the presence of the basin's slowest free mode of internal oscillation-essentially represented by the internal Kelvin-wave mode corresponding to rn = 1, with period 398 h and a node at the central point (Rao, 1977). However, the non-oscillatory form of at P and Q would appear to indicate that the mode is critically damped by friction. Overall, therefore, for the case E~ = 0, inertial motion predominates in the interior of the basin and approximates that obtained for the case E~ = 1. However, closure around the edges means that the wind also sets up internal Kelvin waves (first and second-order modes have been recognised). These induce large slowly varying
c2
c2
;
F
x-
CO I
:
:
:
:
,
*
:
:
:
:
:
:
:
40
I
co
: : : ; : : : ; : ; : : : : :
- Y
449
c
4 0
35
35
30
30
1=4h
25
25
i
u,
I
20
20
15
15
10
10
05
05
0
0
-0 5
-0 5
-1 0
-10
(crnr')
Fig. 10.35. Distributions of Ti, along the central sections COP, and CoQ, at various times, from computations on grid G, with a radiation matrix R ( e O= 0). Compare with ii, given in Fig. 10.19 corresponding to R(E,,= 1).
internal elevations near the edges and significantly affect currents in that boundary region. The Kelvin-wave influence, predominating around the edges, may be seen in Figs. 10.34 and 10.35 where S; and U, are plotted along the central sections COP3and CoQ3 for various times. The influence is clearly portrayed in Fig. 10.36 where ~ 0) may be comcontours of l2at t = 136 h are shown. Figures 10.34-10.36 ( E = pared directly with Figs. 10.18- 10.20 (E~ = 1). The motion in the basin, with e0 = 50, was recomputed with the radiation coefficient for the barotropic waves increased in magnitude from 1 to 30. Accordingly, the radiation matrix employed was:
P
WO
>O,
,
ci 2.5
< 0.6
I
I
Fig. 10.36. Contours of l2 at t = 136 h, from computations on gnd G, with a radiation matrix R ( E=~0). Compare with sz given in Fig. 10.20 corresponding to R ( E =~ 1). Units in centimetres.
45 1
I"
T"
t
Fig. 10.37. Current ( u , u) at the surface $. = 0 and the bottom $. = 1 at C,: plotted at hourly intervals from to t = 100 h. From computations on grid G, with a radiation matrix R' in which q, = 50 and (a) E; = 1, (b) E; = 30. Units: cm s-'. 1=0
where e0 = 50 and E; = 30. Numerical stability in the computations was achieved by taking At = 20 s. Considering the results obtained, the effects of the change in radiation coefficient on the surface and bottom currents, at the central position C,, are shown in Fig. 10.37. Clearly, the inertial currents at the surface are increased while those at the bottom are decreased. These changes may be attributed to a reduction in the horizontal surface gradients (and the associated horizontal pressure gradients) set up by the wind. Thus, with the larger radiation coefficient, smaller surface elevations develop around the edges of the basin, leading to smaller elevation gradients everywhere over the water surface. Since the horizontal pressure gradients oppose the wind stress, near the surface, their reduction leads to a greater impulsive force acting on the water in the surface layer; consequently, larger inertial currents are excited there. On the other hand, the horizontal pressure gradients form the main driving force in the bottom layer and their reduction therefore leads to smaller inertial currents at those depths. Figure 10.37 also shows that increasing eb from 1 to 30 removes much of the ellipticity of the inertial current orbits. Evidently, therefore,
45 2
that ellipticity is due to the influence of the surface gradients, which diminishes as E; increases. Since those gradients arise from the edge boundary conditions, it follows that the ellipticity reflects a constraining effect imposed by the edges. If E , and EL together became progressively larger, it may be inferred that the surface and internal interfaces would become increasingly flat and the motion in the basin would tend to become pure inertial, applicable to open-ocean conditions, with no boundary influence. In those circumstances, the inertial currents would increasingly be concentrated in the upper part of the vertical water column since, without appreciable horizontal pressure gradients, wind stress would be the only effective force acting. However, in the present system of computation, larger values of E~ and EL require a smaller time-step At for the maintenance of numerical stability. In practice this places an upper limit on the values of E, and E; which can be used. Finally, the motion was recomputed assuming a completely closed basin, taking a radiation matrix R' (eq. 68) with E, = 0 and E; = 0, i.e. R' = Rb where: 0 RL=I0
0 0
0
o0 ]
0 0 0 Note: a previous computation determined the motion with the edges of the basin closed to the baroclinic waves but open to the barotropic waves, taking R' with e0 = 0 and EL = 1, i.e. R' = R ; where: /
l o o \ -1 0 0 R { = l 1 0 01 -1 0 0 Figure 10.38 compares corresponding results from these two cases. Time plots of surface elevation l , at P and Q show that closing the edges of the basin to the barotropic waves introduces a 6.8-h oscillation corresponding to the basin's slowest mode of barotropic free oscillation (Rao, 1966). The waves of that mode rotate anticlockwise, evident by comparing the phases of l, at P and Q. Manifestly, the barotropic oscillation significantly affects the u-component of current throughout the depth at the central position C,. Thus, the barotropic u-currents at C, are seen to be less than the baroclinic at the surface 5 = 0 and at the bottom E = 1 , but the former currents predominate over the latter at 5 = 0.35 in the thermocline layer.
Effects of changes in eddy viscosity
The response of the basin to the wind-stress field W,, for different values of eddy viscosity, is illustrated in Figs. 10.39-10.41. In each figure the computed x-directed current u at the central position C,, at the surface 5 = 0 and the bottom 5 = 1, is plotted against time. The results were obtained applying the radiation conditions defined by eq. 66. Figure 10.39 demonstrates the effect of increasing N2 from 10 through 50 to 100, keeping N , fixed at 300 and N3 at 100 (all values in cm2 s-'). Clearly, raising N 2 , the viscosity of the intermediate layer, produces significantly heavier damping of the
453
0
t (hours)+
too I
200 '
.
'
'
.
.
'
'
'
2 cm
I
0 -I
Fig. 10.38. Surface elevation {, at Q and P; current u at C , (at three depths $. = 0, 0.35, 1.0). For: (a) a completely closed basin, radiation matrix Rb; and (b) a basin closed to the baroclinic waves but open to the barotropic waves, radiation matrix R ; .
454 cms-'
5r
50
Fig. 10.39. Current u at C,, at the surface .$ = 0 and at the bottom .$ = 1, showing the effects of changing N2: taking N , = 300; N2 = 10, 50, 100; N3 = 100 cm2s-I.
inertial currents both at the surface and at the bottom. With N2 = 50 or 100 there is little left of the oscillatory motion after 100 h, so that the motion is essentially extinguished in about four days following its excitation. A recent analysis of wind-driven inertial currents observed within a broad coastal sea on the west coast of Canada (Thomson and Huggett, 1981) has indicated that bursts of inertial oscillation there have durations of this order of magnitude. Observed inertial oscillations are intermittent with a clear signal lasting for no more than a few periods (Tang, 1979). There is, therefore, evidence to suggest that taking N , = 50 is probably more realistic than taking N2 = 10 as in the experiments described so far in t h s chapter. In Fig. 10.40, N , is increased from 100 through 300 to 1000, with N , = 50 and N3 = 100. Manifestly the main effect of this enlargement of the surface-layer viscosity is to substantially reduce the first (and highest) peak of the surface current; subsequent peaks of that current remain relatively unaffected. The changes in N , have little influence on the bottom current. Thus, N , conditions the magnitude of the surface current while the wind is acting. Finally, in Fig. 10.41, N3 is increased from 50 through 100 to 300, keeping N , fixed at 300 and N, at 50. As expected, raising the value of eddy viscosity in the bottom layer produces a stronger damping of the bottom current than the surface current.
455
5 4
3
2 I
0
0 t (hours)-
-I
-2 -3
2 I
0
-I
-2
Fig. 10.40. Current
u
at C,, at the surface 5 = 0 and at the bottom
5 = 1,
showing the effects of changing
N,:taking N , = 100, 300, 1000; N, = 50; N3 = 100 ern's-'.
While an increase in each viscosity yields a more highly damped system, the damping appears to be most sensitively controlled by the viscosity of the intermediate layer, N,. Hence, in reality, the occurrence of shear instability in the thermocline region, leading to an enlargement of N,, might significantly add to the damping of the inertial oscillations. Increasing the coefficient of bottom friction k from 0.0026 to 0.0050 (for the case N , = 300, N2 = 50, N3 = 100) was found to have only a small effect on the currents. This is to be expected, since the bottom stress here might anyhow be considered as unrealistically low and ineffective due to the absence of tidal currents in the quadratic law. Following up the last point, the quadratic law was replaced by a linear law:
FB = k’p,u,( h ) , GB = k’p,t~,( h ) (71) where the frictional constant k’ may be adjusted to represent the presence of
456 ems-' 5 4
3
2 I
0 -I
-2
I
0 -I
-2
Fig. 10.41. Current u at C,, at the surface 5 = 0 and at the bottom 6 = 1, showing the effects of changing N3: taking N , = 300; N2= 50; N3 = 50, 100, 300 cm2s-I.
Fig. 10.42. Current u at C,, at the surface 6 = 0 and at the bottom $. = 1, as determined using (a) the quadratic law of bottom friction with k = 0.0026, and (b) the linear law of bottom friction with k ' = 0.2 cm s-I: taking N , = 300, N2 = 50, N3 = 100 cm2s-'.
451
background tidal currents (Heaps, 1974). For the strong tidal regimes on the British continental shelf, experience has indicated that k’ = 0.2 cm s- is an appropriate value. Figure 10.42 shows the computed surface and bottom u-currents at C , obtained using eq. 71 with this value of k’, conditions being otherwise the same as before with N , = 300, N2 = 50, N3 = 100 cm2 s-I. Comparison with the corresponding currents obtained using the quadratic law (k = 0.0026) demonstrates, see Fig. 10.42, that accounting for the tidal effect in the bottom friction introduces a further significant amount of damping. Manifestly this causes the inertial currents at the surface to be reduced to near zero in about 70 h, and those at the bottom to be reduced quite drastically over the entire range of time. The reduction in current near the seabed produces there a bending of the vertical profile of current (compare with the fairly straight profiles of Fig. 10.22, obtained without the tidal effect in the bottom friction). Basin with one coastal boundary; effects of changes in the duration and magnitude of the wind pulse
In some further experiments, carried out on grid G, as before, the side AB of the rectangular basin was closed to represent the south coast of Ireland (Fig. 10.1). At the same time, the other three sides BC, AD and DC were opened to represent sections of the Celtic Sea. Radiation conditions given by the matrix:
were employed, this being a modified form of: 1
1
1
1\
given originally by eq. 66 for the basin with open-boundary radiation on all four sides. The zeros in the third row of R , allow for the closure of AB. Assuming a closed AB implies that the stratified region reaches the southern Irish coast and is bounded by it. If, however, a coherent coastal front exists there (Holligan, 1981), then the edge of the stratified region falls short of the land and the application of an open-boundary condition along AB might be more appropriate. In the respective circumstances described by eqs. 72 and 73, the motion produced by the wind-stress field W , was computed talung N , = 300, N, = 50, N3 = 100 cm2 s I . The value of N2 was increased from 10 to 50 cm2 s-’ to achieve a more realistic damping of the inertial oscillations, as already discussed. The quadratic law of bottom friction, with k = 0.0026, was employed. Results obtained using R , when compared with those obtained using R , indicate the influence of the land boundary AB. On examination, that influence may be recognised as arising from the presence of internal Kelvin waves, set up against the boundary and propagating from B to A, their main effects being concentrated within a few kilometres off the coast. Thus,
458
currents obtained with R , , corresponding to a completely open basin, take forms such as are illustrated in Figs. 10.24 and 10.25 (however, the inertial oscillations are now damped more heavily due to the larger N 2 ) . In contrast, currents obtained with R , , corresponding to the land boundary AB in position, take forms as for Q and Q, in Fig. 10.32 at Q' and Q; (opposite points to Q and Q3 on the other side of the basin-close to the mid-point of AB). Away from the boundary the currents differ only slightly from those for the open basin. Also, near the end points of the boundary, A and B, the currents are basically like their open-basin counterparts but with some elongation of the inertial circuits reflecting the presence of the boundary. Naturally, the internal elevation l2also exhibits the Kelvin-wave influence. Thus, at Q' and Q;, variations of l2occur which are similar but opposite in sign to those at Q and Q, shown in Fig. 10.31. The Kelvin-wave influence on l2is negligible away from the boundary and is small at the end points A and B. Considering further the basin with coast A B (radiation matrix R , ) , a series of experiments was carried out on grid G, to investigate the effects of varying the
L
(a) 7.5- hour wind pulse
I
(b) 15-0-hour wind pulse
Fig. 10.43. Current ( u , u ) at C, plotted at hourly intervals from t = 0 to t = 100 h at the sea surface, 5 = 0, and at the seabed, = 1. Results are shown for: (a) a 7.5-h wind pulse, and (b) a 15.0-h wind pulse, each of magnitude 1 dyn ern-'. From computations on grid G,. Units: cm s-'.
459
duration and magnitude of the wind-stress pulse. Firstly, keeping the wind stress fixed at 1 dyn cmP2(Fig. lO.lO), the motion in the basin was computed for pulses of duration T = 2.5, 5.0, 7.5,..., 22.5 h. Figure 10.43 shows the time variations of horizontal current at the central point C,, at the sea surface ( E = 0) and at the seabed ( 5 = l), for the two cases: T = 7.5 h and T = 15.0 h. For T = 7.5 h the inertial currents are comparatively large due to the reinforcement of disturbances emanating from the rise in wind stress at t = 0 by disturbances emanating from the fall in wind stress at t = 7.5 h (approximately half an inertial period later). On the other hand, for T = 15.0 h, the inertial currents become comparatively small due to the cancellation of disturbances originating from the wind-stress changes at t = 0 and t = 15.0 h, respectively. Overall, the results show that, following the initial current surge, the magnitude of the inertial currents increases as T increases from 0 to 7.5 h, decreases as T increases from 7.5 to 15.0 h, and increases again as T increases from 15.0 to 22.5 h. Thus, the size of the inertial currents which develop depends importantly on how closely the duration of the wind pulse approximates to either an odd or an even multiple of half the inertial period. Notwithstanding this behaviour, it should be pointed out that the maximum inertial current, attained during the initial surge, increases with T up to about T = 5.0 h at the sea surface and up to about T = 7.5 h at the seabed, but thereafter, for higher values of T, remains constant (Fig. 10.44). Secondly, keeping the duration of the wind pulse fixed at 10 h, the motion in the basin was computed for wind-stress magnitudes of 0.5, 1.0, 3.0, 5.0 and 7.0 dyn cm-2. The surface-layer eddy viscosity N , was assumed to increase proportionally with the applied stress (Csanady, 1976; Svensson, 1979), correspondingly taking values 150, 300,900, 1500 and 2100 cm2s-'. The curves marked a in Fig. 10.45 show maximum inertial current (derived from the computations) plotted against wind stress and the associated wind speed. At the sea surface the maximum is seen to rise
Duration (hours)
Fig. 10.44. Maximum inertial current, at the sea surface ,$ = 0 and at the seabed ,$ = 1, produced by a wind-stress pulse of 1 dyn cm-' of various durations.
460 cmi'
,
o! (
1
1
2
3
4
5
6
dyn cm-*
7
10
5
dyn cm-Z
0 0
1
2
3
4
5
6
7
4
d ,
I
I
2 6
8
10
ms-' 12
14
16
Fig. 10.45. Maximum inertial current, at the sea surface E = 0 and at the seabed 5 = 1, produced by a 10-h wind pulse taking various magnitudes. Those magnitudes are given in terms of wind stress (dyn cm-*) and the associated wind speed (m s-I). The striped areas delineate currents associated with a range of possible open-boundaryconditions.
to 27.5 cm s-' and, at the seabed, to 10.5 cm s-l-as the wind stress (speed) increases to 7 dyn cmP2( - 16 m s-I). For comparison, the curves marked b in Fig. 10.45 give maximum inertial current computed assuming open-boundary conditions defined by the radiation matrix R' with E~ = 50 and E& = 30 (see eq. 68). Employing such high radiation coefficients on all four sides of the basin, an approach is made to horizontally uniform conditions: involving relatively small surface and internal elevations, enhanced inertial currents at the sea surface and reduced inertial currents at the seabed (see Fig. 10.37). Inertial current maxima in the Celtic Sea might well lie between values given by curves a and b since those curves could be considered as limits associated with a representative range of open-boundary conditions. The area between the curves, for sea surface and seabed respectively, is therefore striped: indicating best estimates, on the basis of the present calculations, of maximum inertial currents induced by a wind pulse of duration 10 h (or longer). Including the effects of tidal currents in the bottom friction (eq. 71), the maximum inertial
46 1
currents at the seabed given by Fig. 10.45 would of course be significantly reduced-as indicated in Fig. 10.42. Inertial currents in the Celtic Sea
Generally, the Celtic Sea receives a complex series of wind pulses through time and any inertial currents respectively excited by these will combine in a process of reinforcement and cancellation to produce a resultant field of currents. Wind veering (in the northern hemisphere) with a rotational speed near to the inertial frequency is clearly the most effective condition for the generation of inertial currents, implying a resonant forcing, and the extent to whch that forcing mechanism occurs will depend on the track and speed of storm depressions and the movement of associated meteorological fronts (Thomson and Huggett, 1981). Also the frequency of storms will influence the progressive development of inertial currents: successive storms occurring at intervals equal to low multiples of the inertial period will tend to build up inertial current activity as the currents generated by one are boosted by those generated by the next, etc. The Celtic Sea is well exposed to the winds of Atlantic depressions which pass from west to east over the British Isles, and is particularly affected by secondary depressions which move north-eastwards over Ireland bringing strong veering winds to bear on the West Coast sea areas (Lennon, 1963; Heaps, 1965). Inertial currents are therefore to be expected in the Celtic Sea and, indeed, have been observed in moored current-meter records obtained from there during the summer and autumn seasons of thermal stratification. Such records, covering the period June 3-July 4, 1973 (Howarth and Loch, 1976), have been analysed by M.J. Howarth (pers. commun., 1982) to show the presence of inertial currents in the surface layer at A' (51"24'N, 7'40'W), C' (51"03'N, 7"OO'W) and D' (50°45'N, 6'27'W). These positions lie within the area of the present model (Fig. lO.l), roughly on a line from Cork in Ireland to St. Ives in Cornwall. Inertial current magnitudes reached maxima of 18, 16 and 8 cm s-' at A', C' and D', respectively. Winds during the period, as recorded at Roches Point on the south coast of Ireland, fluctuated in Strength between 0 and 12 m s- (some of the strongest being in the south-westerly quadrant) and therefore on the basis of Fig. 10.45 inertial currents possibly reaching 15 cm s-' are predicted in the surface layer. While an order-of-magnitude agreement with the observations is thus obtained, a proper estimation of the currents requires a detailed numerical simulation of the observed motion, taking account of the continuously changing magnitude and direction of the wind field and determining the ongoing response of the three-layered model to it. The response to variations in the atmospheric pressure field may also be sufficiently important to be calculated and included. Moored current-meter records from the Celtic Sea, obtained during September and October 1980 (Heaps and Howarth, 1981) also show clear evidence of inertial current activity. Thus, for a position K (50"30'N, 9"49'W), located near the shelf edge and lying withn the area of the present model, spectral analyses of observed current residuals in the surface, intermediate and bottom layers indicate a predominant inertial peak. A significant peak at around 11 h also shows up, which may correspond to a free mode of oscillation of the entire Celtic Sea shelf (Flather, 1976).
'
462
Winds at Roches Point during this period reached a maximum of 17 m s - ' from W2OoS, and inertial currents up to about 25 cm s - l were recorded in the surface layer at K. These observational values provide a further order-of-magnitude check on the model results shown in Fig. 10.45. Next, going beyond the scope of this chapter, the model needs to be tested more rigorously against observational data, as described above. The amplitudes of the surface tidal streams in the main body of the northern Celtic Sea, covered by the present model, vary between about 20 and 40 cm s - ' for the mean tide M,. Those streams are increased by about 40% at springs. (Howarth and Loch, 1976; Davies and Furnes, 1980; Pingree, 1980). Clearly, therefore, inertial currents in the surface layer of the Celtic Sea can be comparable in magnitude to the tidal currents there. As such they make an important contribution to the surface flow, either enhancing it or diminishing it according to when they occur in relation to the phase of the tide. On the basis of both observation and theory, therefore, inertial currents are seen to occur in the Celtic Sea during the season of stratification. They appear to be second only to the tidal currents in the magnitudes they can attain during periods of strong winds. The mechanics of their generation in the Celtic Sea requires further study, including model simulations of observed events (the largest inertial currents are likely to occur in September and October covering the equinoctial gales). For such purposes, the present three-layered model is being improved by the inclusion of a more realistic depth topography. Also, its area is being extended southwards and westwards to take in other areas of the Celtic Sea and the neighbouring shelf edge. In the longer term, the processes of turbulent mixing in the stratified state need to be modelled more realistically in accordance with up-to-date observational evidence. Moreover, the representation of fronts in the Celtic Sea (these being part of the overall hydrographic scenario there, separating homogeneous from stratified water) constitutes an important future modelling problem (Pingree, 1980; Holligan, 1981). Tidal motions, not explicitly involved in this chapter, need to be included in due course to achieve a more complete dynamical description; the generation of internal tides in the area is a problem of considerable interest in its own right (James, 1980; Baines, 1982).
CONCLUDING SUMMARY
The first stages in the development of a three-dimensional numerical model for the motion of a stratified sea have been described. A spectral method for the solution of the hydrodynamical equations (in three layers) is formulated in Part I. The method is then applied in Part I1 (this chapter) to determine wind-induced motion in a rectangular basin representing the northern part of the Celtic Sea during the summer season of thermal stratification. The numerical experiments of Part I1 test the general behaviour of the model. Computational problems centre on providing a fine enough horizontal grid ( A x , Ay) and a sufficient number of vertical modes M to properly resolve the internal three-dimensional current field. It seems clear that some form of computational split
463
between the barotropic and the baroclinic motions is necessary to limit the size of the calculations and hence make them practically possible. The radiation conditions, applied along the edges of the rectangular area, cover a range of open-boundary conditions (see eq. 51). At one extreme, the boundaries may ~ 0). At the other extreme, they may be opened with the be completely closed ( E = surface and internal elevations suppressed all round ( E , + 0 0 ) : the field of currents is then nearly uniform in the horizontal at any time, showing only weak boundary influence, and might be considered as resulting from the action of a uniformly distributed wind acting over and beyond the model area. Between these two extremes is the case in which disturbances propagate outwards across the boundaries without reflection ( E , = 1). This has been chosen as the basic condition in the present investigation and is formulated in eq. 66. Under such circumstances, the motion might be regarded as resulting from the action of wind mostly confined to the rectangular area. Results of the numerical experiments show how inertial oscillations, involving both barotropic and baroclinic modes, are excited by a wind-stress pulse. Sea-level gradients, set up against the boundaries, introduce horizontal pressures in the water which constitute part of the forcing. Those pressures introduce an ellipticity into the mainly circular inertial-current orbits. Baroclinic inertial waves are generated at the boundaries and spread slowly into the interior of the rectangular basin, producing an internal wave pattern in which longitudinal crests and troughs predominate. Thus, it is demonstrated that inertial currents come into existence through boundary influence, supplementing those which arise from the direct action of the wind over the sea. Long-period internal Kelvin waves are shown to develop when the boundaries are closed. The slowest mode of barotropic free oscillation of the rectangular basin is then manifestly present. It is evident that the attenuation of the inertial oscillations is importantly dependent on the frictional parameters N , , N2, N3, k'. The damping action associated with the viscosity N 2 , located in the thermocline region, appears to be critical. Also, the bottom friction coefficient k', associated here with the background tidal currents, has a strong influence near the seabed. Cancellation between successivelygenerated inertial wave trains can reduce the inertial currents very effectively. By the same token, the currents can be reinforced if the trains combine in unison. As a practical result of the modelling, order-of-magnitude estimates are made of maximum inertial current in the Celtic Sea produced by a 10-h wind pulse taking various magnitudes (Fig. 10.45). More detailed investigations of the mechanisms of generation of inertial currents in the Celtic Sea are required. Further steps of model development lie ahead whlch, in the shorter term, involve the consideration of non-uniform depth topography. Some important problems for the longer-term future are concerned with the more accurate modelling of turbulent processes in the stratified vertical water column, the representation of frontal boundaries, and the inclusion of nonlinear processes. The present work might therefore be regarded as the starting point for a sequence of investigations using stratified-sea models.
464 ACKNOWLEDGEMENT
This work (both Parts I and 11) was sponsored by the Department of Energy as part of a commission to the Natural Environment Research Council.
REFERENCES Baines, P.G., 1982. On internal tide generation models. Deep-sea Res., 29: 307-338. Bowden, K.F., 1955. Physical oceanography of the Irish Sea. Fish. Invest., Ser. 2, 18(8), 67 pp. Csanady, G.T., 1976. Mean circulation in shallow seas. J. Geophys. Res., 81: 5389-5399. Davies, A.M. and Fumes, G.K., 1980. Observed and computed M, tidal currents in the North Sea. J. Phys. Oceanogr., 10: 237-257. Defant, A., 1961. Physical Oceanography, Vol. I. Pergamon Press, New York, N.Y., 729 pp. Flather, R.A., 1976. A tidal model of the North-West European continental shelf. Mem. SOC.R. Sci. Liege, Ser. 6, 10: 141-164. Heaps, N.S., 1965. Storm surges on a continental shelf. Philos. Trans. R. Soc. London, Ser. A, 257: 351-383. Herps, N.S., 1974. Development of a three-hmensional numerical model of the Irish Sea. Rapp. P. V. RBun. Cons. Int. Explor. Mer, 167: 147-162. Heaps, N.S. and Howarth, M.J., 1981. RRS John Murray Cruise 9/80: 27 August-9 September 1980, RRS Challenger Cruise 16/80: 17-30 October 1980; Celtic Sea. Institute of Oceanographic Sciences, Cruise Report No. 114, 50 pp. Holligan, P.M., 1981. Biological implications of fronts on the Northwest European continental shelf. Philos. Trans. R. Soc. London, Ser. A, 302: 547-562. Howarth, M.J. and Loch, S.G., 1976. Moored current meter records, Celtic Sea 2 June-7 July 1973, 10s Bidston moorings 30-36. Institute of Oceanographic Sciences, Data Report No. 6, 101 pp. James, I.D., 1980. Thermocline formation in the Celtic Sea. Estuarine Coastal Mar. Sci., 10: 597-607. Krauss, W., 1979. Inertial waves in an infinite channel of rectangular cross-section. Dtsch. Hydrogr. Z., 32: 248-266. Krauss, W., 1981. The erosion of a thermocline. J. Phys. Oceanogr., 11: 415-433. Kullenberg, G.E.B., 1976. On vertical mixing and the energy transfer from the wind to the water. Tellus, 28: 159-165. Kundu, P.K., 1976. An analysis of inertial oscillations observed near Oregon coast. J. Phys. Oceanogr., 6: 879-893. Lennon, G.W., 1963. The identification of weather conditions associated with the geqeration of major storm surges along the west coast of the British Isles. Q. J. R. Meteorol. SOC., 89: 381-394. Millot, C. and Crepon, M., 1981. Inertial oscillations on the continental shelf of the Gulf of Lions-observations and theory. J. Phys. Oceanogr., 11: 639-657. Mortimer, C.H., 1952. Water movements in lakes during summer stratification; evidence from the distribution of temperature in Windermere. Philos. Trans. R. SOC.London, Ser. B, 236: 355-404. Mortimer, C.H., 1977. Internal waves observed in Lake Ontario during the International Field Year for the Great Lakes (IFYGL) 1972 :I. Descriptive survey and preliminary interpretation of near-inertial oscillations in terms of linear channel-wave models. Spec. Rep. No. 32, Center for Great Lakes Studies, The University of Wisconsin-Milwaukee, Milwaukee, Wisc., 122 pp. Mortimer, C.H., 1980. Inertial motion and related internal waves in Lake Michigan and Lake Ontario as responses to impulsive wind stresses. I. Introduction, descriptive narrative, and graphical archive of IFYGL data. Spec. Rep. No. 37, Center for Great Lakes Studies, The University of Wisconsin-Milwaukee, Milwaukee, Wisc., 192 pp. Pingree, R.D., 1980. Physical oceanography of the Celtic Sea and the Bristol Channel. In: F.T. Banner, M.B. Collins and K.S. Massie (Editors), The North-West European Shelf Seas: the Sea Bed and the Sea in Motion. 11. Physical and Chemical Oceanography and Physical Resources. Elsevier, Amsterdam, pp. 415-465.
465 Pollard, R.T., 1970. On the generation by winds of inertial waves in the ocean. Deep-sea Res., 17: 795-8 12. Pollard, R.T. and Millard, R.C., 1970. Comparison between observed and simulated wind-generated inertial oscillations. Deep-sea Res., 17: 813-821. Rao, D.B., 1966. Free gravitational oscillations in rotating rectangular basins. J. Fluid. Mech., 25: 523-555. Rao, D.S., 1977. Free internal oscillations in a narrow, rotating rectangular basin. Canada Marine Sciences Directorate, Manuscript Rep. Ser., 43: 391-398. Schwab, D.J., 1977. Internal free oscillations in Lake Ontario. Limnol. Oceanogr., 22: 700-708. Svensson, U., 1979. The structure of the turbulent Ekman layer. Tellus, 31: 340-350. Tang, C., 1979. Inertial waves in the Gulf of St. Lawrence: a study of geostrophic adjustment. Amos.-Ocean, 17: 135-156. Thomson, R.E. and Huggett, W.S., 1981. Wind-driven inertial oscillations within Queen Charlotte Sound and Hecate Strait, May-September 1977. Canada Institute of Ocean Sciences, Pacific Marine Science Rep. 81-20, 90 pp.
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467
SUBJECT INDEX
Advective terms --, contribution of in the generation of higher harmonics, 183 --, importance of in shallow water, 380 Amphidrome -, displacement of, 179 -, relation of position of to friction, 179 Amphidromic points, 138 Atmospheric model, 358 Averaging time, I90 Baroclinic mode, 406, 414 Barotropic mode, 406, 414 Bars, longshore, 71 Beach -profile, 67 -slope, 73, 120 -, topographic effects, 71 Bed -, effective level of, 57 -, erosion of, 28 -layer, 195-197 -load sediment trap, 91 -load transport on beaches, 101, 104 -, natural profiles of, 16 -profiles, representation of by a series of harmonics, 18 -roughness, 39 -, sedimentation of, 28 -shear stress, 2, 7, 190 -shear-stress distribution, 282 Bores, 120 Bottom -motion, 1 -percolation, 1, 45 -ripples, 10, 46 -sediment, 1 -stress, 127, 183 -stress, dependence on roughness length, 118 -stress, mean value of, 120, 183 -stress under waves, 39 Bottom boundary layer, 189 --- , subdivision of, 192 ---, thickness of, 189 Bottom boundary stress, 111, 118 Bottom friction, 1, 183, 359
--- coefficient, 360
___ ,
contribution of to generation of higher harmonics, 183 Boundary conditions --, lateral, 314 --, radiation, 270, 315, 358, 413-417 Boundary conditions (radiation), effect of, 442-452 Boundary layer (depth-limited), 232-240 -_-_ , current structure in, 239 _-_- , distribution of thckness of around British Isles, 232 -_--, turbulence structure in, 240 Boundary layer (leading edge flow), 240 ---__ , current structure in, 240 -_---, turbulence structure in, 241 Boundary layer over topography, 242, 243 -_-_ , behaviour of bottom stress in, 244 ___- , turbulence in, 244, 245 Boundary layer (stratified), 227-232 _-_ , current structure in, 227 Cell circulation, 67, 8 1 Charts (co-tidal), 170 --, for North-West European Seas, 173-178 --, relation of currents to, 171 Charts (tidal), 128 Circulation (residual), 139 --, around sandbanks, 325-327 --, resulting from bottom friction, 351 --, resulting from bottom topography, 351-353 Circulation (wind-induced), 357, 364-383 Coastal winds, 82 --, effect of on longshore currents, 82 Constant stress layer, 114 -_-, departure from, 288 --- hypothesis, 131 --- profile, 114, 279 Current (inertial), 402 --, in Celtic Sea, 162, 461, 462 Current (longshore), 70-83 --, profile of, 75, 79, 80 --, variation with depth, 81 --, unsteadiness in, 81 Current (meteorologically induced), 358 Current (residual), 321, 357
468 Current (surface), 384 Current (tidal), 1, 135 measurement of, 155-157 --, vertical structure of, 184 Current, variation of with time, 425-442 Current meters, deployment of in North Sea,358
--.
Depth-averaged model, 269, 272, 289, 313-317
--_ , use of in the determination of residual flow, 331-335 Dissipation -, length scale, 113 -of energy, non-linear, 180 -of turbulence energy, 113 -of wave energy, 1, 39,45 Dissipation of wave energy by breaking, 9 Drag, form, 2 Drag coefficient, 70, 73, 191, 201 --, beneath waves, 44, 47 geostrophic, 210
--.
Eddies (residual)
--, associated with headlands, 323-325 --, effect of lateral viscosity on, 327 --, generation of by friction, 324 --, in basins and embayments, 327 Eddy viscosity, 48, 191, 357, 385 --, dependence on current, 8 --, effect of changes in, 452-457 --, enhancement of by tidal mixing, 405 --, lateral, 327 Eigenfunctions, 393 Eigenvalues, 393 Elevation (internal), variation of with time, 425-442 Energy equation (turbulent), 49 Energy density in waves, 68 -, flux factor, 83 -(turbulent), 112, 121, 190 --, distribution of beneath waves, 125 --, redistribution of, 113 Exchange coefficient, 269, 270 --, horizontal, 122, 123
Finite difference --;equations, 272, 273, 408-412 --, grid, 358, 406 Free-stream --, elliptic flow, 225 --, rectilinear flow, 223 --,velocity, 7 Friction coefficient, 119, 128 --, dependence of on roughness length, 120, 129, 20 1 --, dependence of on wave amplitude, 120
--, dependence of on wave period, 120, 130 --, optimised value of, 119, 128 Friction -, factor, 44-46 -, law (quadratic), 118, 127, 19', 269, 405 Friction velocity, 127, 190 --, comparison of with depth-averaged velocity, 127 Fronts, 228, 229
Galerkin method, 360 Gauges (tidal) --, characteristics of, 152-155 --, deep-sea, 152 Grain diameter, 195 Grid schemes, 422 Harmonic constituents, 166 --, variability of analysed parameters, 166 Harmonic constituents --, variations resulting from ocean variability, 168 --, variations resulting from tide-surge interaction, 169 Hydrostatic approximation, 274 Inertial oscillations, 402 Interaction of waves with bed, 2, 67-69, 87 Kelvin wave, 402
--, relation of to tides, 178-182 Laplace's equation, 1 1 Length scale --, attenuation of, 279, 282 --, viscous, 196 Littoral drift, 67, 83-93 --, evaluation of, 84 --, measurement of hy sand tracers, 90 --, measurement of by sediment traps, 91, 102 --, measurement of by tracer dilution, 90 --, problems associated with jetties and breakwaters, 85 Littoral sand transport, 93 Local equilibrium conditions, 113 Logarithmic layer, 43, 193, 197-200 --, constant stress hypothesis, 131, 132 Logarithmic velocity profile, 43,48, 114, 131, 27 1 MeteorologiFal data, 363 Mixing length theory, 113 Mixing scale --, horizontal, 123 --, vertical, 113, 270 Motion induced by waves, 1
469 Nearshore currents, wave generated, 68-83 Non-linear processes in tides, 139 North Sea, circulation in, 364-383 Oscillatory flow, shear-stress distribution in, 226 Pneumatic coastal tide measuring systems, 139, 149-151 Poincare waves, 402 Potential flow --, layer of, 12 --, over rippled bed, 10-16 Radiation stress, 68 --, relation of rate of sand-transport to, 93 Residual flow --, around sand banks, 325-327 --, difference between circulations and through flows, 322 --, formal definition of, 328-331 --, forming closed streamlines, 322 --, measurement of, 322 --, use of depth-averaged equations, 332 --, use of vorticity concept, 322 Residual transport, 1 Response analysis, 162- 166 --, application of to shelf currents, 165 --, computation from of equivalent harmonic constants, 166 --, refinement of for use in shallow water, 165 Response to wind stress field, 423 Reynolds-averaged velocity, 268 Reynolds number --, roughness, 195 --, wave, 40 Reynolds stress, 113, 190 --, relation to bursting phenomena, 251-255 Resonance, 180 -, tidal amplification by, 180
Rough -, hydrodynamically, 40 -, turbulent, 40 Rough turbulent flow, 197 Roughness -, bed, 39 -, effect on bursting phenomena, 256 -, equivalent, 40 -, length, 114, 200-203, 270 -, of mobile sand bed, 42 -, regime, 195 -, relative, 40
Sand-ripples, 2 --, perturbing effect of on near-bed flow, 3 Sand transport, 67, 83
--, distribution of in breaker zone, 97 --, threshold of motion, 4-8 Sea model, 403-413 Sediment - 3 f e c t of thickness on evaluation of littoral drift, 90 -, entrainment of, 3 -, estimation of suspended concentration by opacity measurements, 103 -, stratification by suspensions of, 230 -, thickness of movement, 88 -, threshold of motion, 4-7, 21 Sediment transport, 2, 7, 20, 50 --, on beaches, 67, 83 Shallow sea, well-mixed, 387 Shallow water, generation of higher tidal species in, 159, 162 Shallow-water flow over topography, 267 Shallow-water models --- with one horizontal dimension, 269 --- with two horizontal dimensions, 313 Shear-stress profile, 208-21 1 -_-, from numerical model, 278 --_ over topography, 283-289 Shelf model, three-dmensional, 357, 359-363 Shelf tides, dynamics of, 178-184 Shingle and gravel movement along beaches, 9 1 Similarity hypothesis, 123, 270 Skin friction, 2 Smooth -, hydrodynamically, 40 -turbulent flow, 196 Spectral model, 387, 401 --, three-layered, 387-399, 401 Spectral technique, pseudo, 116 Stability -, datum, 139 -, dependence on sediment concentration profiles, 230 -parameter, 227 -, wave flow, 9 Stilling wells, 139- 148 --, design of, 148 --, theoretical aspects of, 141 Stratification -, effect of on turbulence, 232 -, thermal, 230 Stratified sea, 387, 401 Streaming velocity, 116 --, dependence on roughness length, 116 Surface current to wind-speed speed ratio, 384 Surface waves, 1 --, energy flux in, 1, 27 --, generation of longshore currents by, 68 --, induced potential flow over rippled surface, 3
470 --, propagation over a rippled bed, 21-29 Suspended-sediment transport --_ , o n beaches, 101-104 --- , relative importance of bed-load transport, 101-104 Thermocline, 387 Tidal -analysis, 157- 170 -constituents, principal energy containing, 159 -currents, 136 -energy budget, 181 -, forcing at boundary, 358 Tidally driven residuals, 321 Tides in Atlantic, 172 Time and space splitting procedure, 418-422 Transfer -coefficient, vertical, 112, 113 -law, gradient, 112, 122, 127 -term, vertical, 112 Transport, rate of for sand grains of different sizes, 100 Troughs, longshore, 7 1 Turbulence, 189 -, bursting phenomena, 25 1-258 -, closure, 112, 270 -, dissipation range, 25 1 -energy budget, 125 -, generation of by breaking waves, 120 -, in planetary layer, 248 -, in stratified layer, 248 -, in surface layer, 246 -, in wave flows, 9 inertial subrange of, 251 -, spectra of, 245-251 -, spectra of in tidal flows, 249 -, spectra of over topography, 25 I -, in surf zone, 74 Turbulent -boundary layer, 40-42 -oscillatory boundary layer, 48-50
-.
Variance, 190 Velocity -, depth-averaged, 127, 269, 331 -, effect of bed-ripples on, 26 -, near-bed, 8 Velocity profile from numerical model, 278, 283-289 ---__ , comparison with Coles’ Wake Law, 279 --_-- , comparison with experiment, 281 --_-- , comparison with field observations, 310 ---_-, over topography, 283 Viscous sub-layer, thickness of, 196, 281 Von Karman’s constant, 113, 198, 203, 278 Vortex formation and shedding, 2, 47, 48, 52 Vorticity -, application of concept to tidal circulation, 335-339 -, application of concept to residual circulation, 322, 335, 338-342 -, Eulerian solution for residual, 346-351 -, generationof by topographical features, 342 -, quasi-LagranDan solution for residual, 343-346 Wave boundary layer, 39-50
--_ , thickness of, 39, 42, 43 Wave energy, reflection of by bed ripples, 21, 27, 29-39 Wave height, longshore variations in, 81 Wave-induced turbulent flow, 111 Wave -refraction, 2 -set-up, 69 Waves -, effect of on nearshore sand transport, 93 -, inertial, 402 Wind pulse, effect of change in duration and magnitude of, 457-461
