Introduction to Coastal Dynamics and Shoreline Protection

Introduction to Coastal Dynamics and Shoreline Protection

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Introduction to Coastal Dynamics and Shoreline Protection G. Benassai University of Naples Parthenope, Italy

Introduction to Coastal Dynamics and Shoreline Protection G. Benassai University of Naples Parthenope, Italy

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 1-84564-054-3 No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2006 Printed in Great Britain by Lightning Source UK Ltd., Milton Keynes 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 Publisher.

Contents

Preface

xi

CHAPTER 1 Integrated approach to coastal dynamics 1. Coastal dynamics basic approach 2. Coastal erosion and remediation study 2.1 Data acquisition 2.2 Critical erosion evaluation 2.3 Planning analysis 3. Causes of coastal erosion 4. Space and time scales 5. Meteomarine factors 5.1 Wind 5.2 Waves 5.3 Currents 5.4 Sea level variations 6. Sediment transport and coastal structures 6.1 Modes of sediment movement and their appearance region 6.2 Coastal structures and sediment transport 7. Elements of coastal management

1 1 3 4 6 7 7 8 9 10 10 14 16 17 17 19 21

CHAPTER 2 Beach morphology and sediment analysis 1. Introduction 2. Beach classification 3. Beach morphology and sediment transport 4. Seasonal profiles, bars and berms 5. Equilibrium beach profile 6. Sediment analysis 6.1 Bathymetric and geophysical surveys 6.2 Physical and chemical analysis

27 27 27 30 33 33 35 35 37

6.3 Sediment size classification 7. Case study

39 42

CHAPTER 3 Linear wave analysis 1. Introduction to linear wave theory 1.1 Governing equations 1.2 Boundary conditions 1.3 Linearized boundary conditions 2. Results of the linear theory 2.1 Wave profile, length and celerity 2.2 Group celerity 2.3 Velocity components 2.4 Particle displacements 2.5 Wave pressure 3. Case study

45 45 45 49 50 51 51 54 56 58 60 62

CHAPTER 4 Sea level variability 1. Introduction 2. Astronomical tide 3. Long waves (tsunami and seiches) 4. Wave set-up and set-down 4.1 Radiation stress 4.2 Water level fluctuations due to radiation stress 5. Storm surge 6. Case study

67 67 67 71 73 74 76 77 85

CHAPTER 5 Random wave measurement and analysis 1. Wave measurements 1.1 Ultrasonic and pressure gauges 1.2 Wave buoys 1.3 Italian Sea Wave measurement Network 1.4 Satellite remote sensing 1.4.1 Radar altimeter 1.4.2 Synthetic Aperture Radars 2. Statistical properties of random waves 2.1 Data sampling 2.2 Data processing 2.2.1 Time domain analysis 2.2.2 Directional wave spectra 2.2.3 Pierson – Moskowitz and Jonswap spectrum 3. Statistical representation of wave climate 4. Case study

87 87 87 88 89 90 90 92 93 93 95 95 99 100 102 103

CHAPTER 6 Short term wave prediction 1. Introduction 2. Elements of wind measurement analysis 2.1 Wind information needed for wave hindcasting 2.2 Geostrophic and low-height winds 3. Wave prediction on deep water 3.1 Fetch and duration limited growth 3.2 Significant wave (SMB) model 3.2.1 Case study 3.3 Spectral wave models 3.3.1 First, second and third generation models 3.3.2 Third generation models 3.3.3 WaveWatch III 3.3.4 Case study (WWIII application for the Gulf of Naples)

107 107 108 109 110 111 112 114 116 117 118 119 122 124

CHAPTER 7 Long term wave statistics 1. Introduction 1.1 Wave data 1.2 Data selection 1.3 Extreme value probability distribution 2. Data fitting to the probability distribution 2.1 Normal Probability Distribution 2.2 Log-Normal Distribution 2.3 Gumbel distribution 2.4 Weibull distribution 3. Parameter calculation 3.1 Statistical tests of fit 3.2 Confidence intervals 3.3 Statistics of offshore extreme waves 3.4 Wave height persistence 3.5 Case study

127 127 128 129 129 129 130 131 131 132 133 133 134 135 137 138

CHAPTER 8 Wave transformation in the coastal zone 1. Wave energy and energy flux 1.1 Potential Energy 1.2 Kinetic Energy 1.3 Energy Flux 2. Refraction and shoaling 2.4 Discussion on Kr and Ks 3. Total reflection 4. Wave diffraction 5. Numerical models for wave propagation 5.1 Phase-averaged model 6. Finite depth spectral wave models 6.1 Other finite depth spectral wave models

143 143 143 144 145 146 150 152 155 156 158 159 161

6.2 Phase-resolving models 6.2.1 Boundary Integral Models 6.2.1.1 Mild Slope Equation Models 6.2.1.2 Boussinesq equation model 6.2.2 Lagrangian models – Ray method for wave transformation 6.2.3 Eulerian models 6.3 Grid models 7. Wave breaking

161 163 163 165 166 169 169 172

CHAPTER 7 Sediment transport 1. Introduction 2. Basic concepts of sediment transport 2.1 Critical bed shear stress 2.2 The Shields parameter and modified Shields diagram 2.3 Sediment fall velocity 2.4 Bed load and suspended load 2.4.1 Bed-load and shear stress 2.4.2 Steady bed load in sheet flow transport 2.4.3 Basics of suspended load transport formulation 2.5 The bottom boundary layer and the bed roughness 2.6 Bed load and suspended load: a simple parametrical model 2.7 Case study 3. Basic shore processes 3.1 Nearshore circulation 3.2 Wave run-up in the swash zone 3.3 Bar formation by cross-zone flow mechanisms

175 175 176 176 177 179 180 181 182 183 185 187 189 194 194 198 199

CHAPTER 10 Beach profile modeling 1. Cross-shore transport 2. Cross-shore sediment transport and equilibrium beach profile 3. Dean’s model for equilibrium beach profile 3.1 Equilibrium parameter A 4. Processes of accretion and erosion 4.1 Surf zone 4.2 Swash zone 5. Erosion/accretion parameters 5.1 Case study 6. Analytical profile modelling 6.1 Case study 7. Numerical beach profile modeling 7.1 Example of numerical model: SBEACH

201 201 203 204 207 208 208 208 209 213 214 219 220 222

CHAPTER 11 Shoreline modeling 1. Introduction

227 227

2. Longshore transport 2.1 Case study 3. Numerical shoreline modeling 3.1 GENESIS 3.1.1 Governing equations 3.1.2 Model parameters 3.1.3 Model implementation 3.1.4 Model calibration

228 230 231 234 234 236 239 241

CHAPTER 12 Comparison and choice among alternative protection systems 1. Introduction 2. Insertion of protection systems on the coastline 3. Shoreline protection systems 4. Hard measures 4.1 Detached emerged breakwaters 4.2 Detached submerged breakwaters 4.3 Emerged or semi-submerged groins 4.4 “T”-shaped emerged of semi-submerged groins 4.5 Adherent breakwaters 4.6 Seawalls 5. Soft measures 5.1 Artificial nourishment 5.2 Dune restoration 6. Schematic indications for the choice 7. Mechanisms of protection 7.1 Efficiency 7.2 Induced efforts

245 245 245 246 248 248 249 250 252 252 254 254 254 256 258 260 260 261

CHAPTER 13 Hydraulic design 1. Dimensional analysis 2. Wave run-up Ru and run-down Rd 3. Overtopping discharge 4. Transmission coefficient 5. Reflections 6. Case study

263 263 265 268 270 271 273

CHAPTER 14 Structural design 1. Introduction 2. Structural stability 2.1 Hudson formulation 2.2 Van der Meer Formulation 2.3 Comparison of Hudson and new formulae 3. Armour layers with concrete units

275 275 277 279 280 283 283

4. Low-crested structures 5. Reef breakwaters 6. Statically stable low-crested breakwater 7. Submerged breakwaters 8. Filter and core characteristics 9. Toe stability and protection 10. Breakwater head stability 11. Fundamentals of probabilistic design 12. Deterministic design – case study

285 286 286 287 287 288 289 289 291

CHAPTER 15 Beach fills 1. Introduction 2. Beach fill profile 3. Volume computation 4. Beach planform evolution 5. Longevity of beach fills 6. Effect of fill length and of wave climate 6.1 Case study 7. Compatibility of the borrow material 7.1 Case study 8. Sediment sources 9. Monitoring

293 293 296 298 298 301 303 305 306 308 309 310

References

313

Preface This book was developed from lecture notes for a course on Coastal Dynamics and Shoreline Protection addressed to students of Environmental Sciences. This is the reason why it is organized to introduce the reader to the fundamental principles of the topics treated in each chapter. It can be used as a training aid both for students and for practicing engineers, as almost every topic is developed with case studies. The book, which deals primarily with sandy coastlines, is divided into three parts. In the first part, which is limited to Chapter 1 – Integrated approach to coastal dynamics, the reader is introduced to the approach of a coastal erosion and remediation study and to coastal management. In the second part, the meteomarine factors cited in Chapter 1 are dealt with in some detail, together with the mechanisms of sediment transport. The topics addressed are, amongst others, linear and higher order waves, random waves and spectra, wave transformation in the coastal zone, water levels, short-term and longterm wave prediction, sediment transport, shoreline and beach profile modeling. The third part deals with the choice between various protection systems and tries to give the reader some basic elements of hydraulic and structural design for both rigid structures and beach fills. Acknowledgements have to be addressed to all the people who inspired me, in particular my father Edoardo for the first approach to the maritime structures, Giulio Scarsi and Laura Rebaudengo of Genoa University for their rigorous analytical approach to the maritime hydraulics. Many thanks have to be addressed to all the students to whom I had the privilege of teaching the concepts reported in this book for more than 14 years, and to the University Parthenope of Naples where I currently teach the course on coastal dynamics and shoreline protection. Additional thanks have to be addressed to the people who typed the manuscript for their patience and their contributions, especially Dr I. Ascione and Dr. E. Chianese. Finally, particular thanks is due to my wife Maria Pia and my sons Edoardo Maria and Rossella who have given me total support and encouragement, and to whom this book is dedicated. G. Benassai Naples, 2006.

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Chapter 1 Integrated approach to coastal dynamics 1 Coastal dynamics basic approach The littoral area is a system in unstable dynamic equilibrium: storms create short term erosional events, natural recovery after the storm and seasonal fluctuations may not be in balance to produce long term erosion. Shore protection projects moderate the long term average erosion rate and reduce damage caused by flooding and wave attack. The coastal environment constitutes a fragile, little known and complex ecosystem that is an important resource for most nations. The high economic interest relative to environmental resources and man’s extensive and growing use of the coastal zone drives the need for the optimization and the mitigation actions. In order to reach a satisfactory design, we must understand each element and its interactions, the inputs and the outputs, and how they affect the system and neighboring systems. In fact, any kind of protection scheme interfering with the littoral transport (groynes, breakwaters etc.) modifies this equilibrium. The relative consequences cannot be estimated without the analysis of the physical phenomena and the simultaneous consideration of the local process and their interaction with adjacent systems. In the past, mistakes were made by not considering the proper system boundaries or by not considering a super-system, when necessary. This restricted vision produced huge damages for the neighboring beaches, with catastrophic environmental, economic and social consequences. For this reason, an integrated approach to the coastal zone has to be done with a preliminary identification of all systems that influence the design and that are affected by it. The monitoring of the systems must be done on both a short and long time scale and has to be extended to the entire “physiographic unit”. This is characterized by the circumstance that the coastal sediments move inside it without external influences. The identification can be based on the study of the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

natural factors (wind, currents, human impact), or on the consequences of these factors: erosion and accretion. The identification requires the knowledge of the dynamic conditions and of the interaction with the littoral evolution. The limits of the area involved can be variable in time in relation to the events (natural or anthropic), which modify the coastal dynamics. The analysis of the wave conditions and in particular the interaction between the waves and the sediment plays a relevant role for understanding the physical processes. This analysis, then, has to be focused on the monitoring of the wave climate and on the statistical analysis of its spatial and temporal characteristics (wave height, direction and period). A statistically valid analysis can be obtained through the monitoring of the waves for a period of 10 years or more. When the waves approach the shoreline, they are modified by the seabed through processes such as refraction, shoaling, bottom friction and diffraction (if waves interact with structures). The waves, then, change in height and direction. This feature plays a relevant role in the littoral transport process. A numerical model is then required in order to consider the role of the bottom in changing the wave characteristics, due to lack of field local data regarding the littoral transport. This interaction between the waves and the bottom is due primarily to second order shallow water phenomena, such as the wave drift (which is responsible for mass transport) and the radiation stress, which is associated with the longshore current for obliquely incoming waves. The wave drift is directed normal to the shoreline, transporting the water particles near the surface in the direction of wave propagation. The shore-parallel component of the radiation stress generates the longshore current, which carries sediments along the shoreline (the so-called littoral drift). Other significant interactions can occur between waves and currents. Wavecurrent interaction may affect the development of rip currents. In fact the weak currents generated by a gentle alongshore variation in the wave field can cause significant refractive effect on the waves so as to change the structure of the forcing which drives the currents. This can cause the development of instabilities of the cellular circulation and can influence the rip channels spacing and depth. From the literature it is assumed that the longshore sediment transport is more relevant than the cross-shore one. The latter process, although significant during storms, gives little final influence on the sediment dispersion along the coastline. The littoral transport on a long time scale is directly connected with the longshore component of the energy transmitted during the breaking phenomena. This parameter can be computed on the basis of both the offshore wave climate (frequency of events classified by period, direction and wave height) and the propagation features of the waves. So a statistical analysis has to be carried out through all the possible combinations of wave height, period and direction, statistically computed from field measurements. The sum of these contributions, related to the breaking wave energy flux, allows to compute the

INTEGRATED APPROACH TO COASTAL DYNAMICS

3

longshore transport on several time scales. The applicability of the results depends on the analysis of the effective sediment transport, which means to put in the numerical model more information, such as grain size in relation to the depth and the breaking wave type (the most important parameters during the sediment suspension). The results obtained have to correspond with an independent assessment about the beach morphology and the influence of existing structures. The complexity of environmental phenomena makes the design process extremely complex. Even if the more sophisticated methods are used (backanalysis of a finite number of real cases, or the results of the close reproduction of phenomena), it is possible to make large errors in relation to the techniques and materials used and the boundary conditions imposed. This is the reason why the design is done by trial and error. This approach is obviously unacceptable in the prototype, but it is possible, however, in numerical models. These numerical models are used to verify the impact of structure in several environmental conditions, optimizing the localization, typology and the dimension of protection systems. In this phase a quantitative evaluation is carried out by numerical simulation of the coastline evolution in order to identify the potential environmental impact of the project. This prognostic step of the design process requires the definition of boundaries and the identification of the coastal dynamics through the analysis of available data. The impact assessment of the sediment budget supplied by the rivers, the analysis of available maps and aerial photographs and the study of the impact of existing structures define the boundaries of the physiographic unit. The numerical simulation in the final phase allows the quantification of the specific environmental impact, the time scale and the significant trend of the coastline modifications resulting from the interaction between the new structure and the mean wave climate.

2 Coastal erosion and remediation study The fundamental criteria of a coastal dynamics study are based on an accurate evaluation of the environmental, economic and social aspects of the area of interest on a spatial scale of the physiographic unit. In fact, the key concept in coastal design is the integration, that is the awareness that nothing should be done within a littoral cell without thinking about how it affects the rest of the cell, which by definition is part of the same system. The main goals of the coastal dynamics study are the assimilation and collection of a huge quantity of experiences, actually distributed among several private companies and government agencies, in order to reach a complete vision of the existing knowledge and in order to plan the coastal protection in the best way. In fact, a deep knowledge of the integrated physical processes can avoid unwise and dangerous actions and plays a substantial role in minimizing possible local negative consequences.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

With these premises, the main technical steps of a coastal dynamics study are the following: • • •

Data acquisition Critical erosion evaluation Planning analysis

A typical scheme of a coastal dynamics study is given in figure 1.1. Offshore Waves and Winds

Tides and Currents

Offshore Wave Climate

Hydrodynamics

DATA ACQUISITION

Coastal Morphology

PLANNING ANALYSIS

CRITICAL EROSION EVALUATION

Bathimetry

Coastal Dynamics Analysis

Inshore Wave Climate

Sediment Transport Analysis

Littoral Transport Analysis

Sediment Transport Evolution Satellite and Aerial Maps Bathimetry Dispersion of Sediments Effects of Structures

Numerical Model Shoreline Evolution

Wave-Current Interactions

Sediment Budget Physiographic Unit Significant Parameters

Trial-Error Design

Shoreline Simulations

Figure 1.1- The main steps of an integrated approach to coastal design. 2.1 Data acquisition The first phase of the study will involve the identification and acquisition of available data. Coastal data are usually acquired through field observation and may be divided into the following categories:

INTEGRATED APPROACH TO COASTAL DYNAMICS

5

• •

Field data (bathymetric and topographic surveys, sedimentology); Office data provided by: interpretation of historical maps, aerial photographs, etc; numerical simulations of the coastline, laboratory and office data analyses. Data are provided by various sources, such as: • Literature sources (University Departments, Journals and conference proceedings, reports of research projects); • Local sources, which provide detailed and sometimes unique data pertinent to the site (reports of local newspapers, interviews with fishermen, etc.); • Government agencies, which provide data archives and studies with relevant coastal information; • Private companies, such as oil, gas and construction companies. Some of these data are in the public domain and include also environmental impact reports containing extensive coastal process data; • Computerized literature databases containing information that may be acquired by key terms, subjects, titles and author names, available to major Universities and Government agency libraries. Data requested to characterize the significant coastal processes and to analyze the characteristics of severe storms include the following: Wave data, which include wave height, period, steepness and direction and breaker type. These data may be available through various techniques: recorded wave data, obtained using buoys and other recording instruments located offshore (piers or other coastal structures near the study site); hindcast data obtained from numerical prediction models and used to estimate wave statistics. Water level data are provided using tide gauges deployed near the study site. The water level is commonly measured through three types of instruments: pressure transducers, floating gauges and staff gauges. Typically, water level measurements are related to a given datum; daily records are usually published in reports, while predicted water levels and tidal current information for each day can be obtained from the annual Tide Tables. Geologic and sediment data include geologic maps and data collected during bathymetric surveys. These data are important to characterize the response of the coastline to severe storm events. Aerial photographs are useful for studies over long time scales to understand shoreline change assessments and to get information about coastal landforms and materials, behavior of engineering structures, location of rip currents, storm effects on the coastline, etc. Satellite data, like aerial photographs, are useful for assessing large-scale changes of the coastal zone. Finally, remote sensing allow to define spatial patterns of suspended sediments in shallow waters and freshwater discharges near estuaries.

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2.2 Critical erosion evaluation The most part of defense actions in the past were done in emergency conditions and economic constraints. Usually the actions were focused on the restoration of the previous conditions without paying attention to the causes of erosion. Besides, the actions were not focused on the efficiency, on the past experiences and, above all, on the need to reduce the environmental impact. For this reason, the first step of the critical erosion evaluation is the analysis of the existent structures, which means: • Impact analysis of the structures on the coastline evolution and on the local bathymetry; • Localization, time and costs evaluation for dredging operations and maintenance of ports and waterways; • Evaluation of the amount of volume for artificial nourishment; • Sustainability of the project location at all the technical and administrative levels. The basic critical shoreline changes can be summarized in three types: • Natural shoreline accretion • Shoreline erosion • Interactions between coastal structures and littoral processes The fundamental criteria to detect these critical situations are the following: • Physiographic unit • Littoral transport intensity • Coastline evolution • Sediment loss • Human impact Coastal erosion studies involve part of the following phenomena: • Erosion processes caused by natural or human impact • Sediment supply by rivers or dunes • Conflicting demands on the coastal area After the data acquisition and analysis, a hypothesis of critical area localization on provincial or regional scale has to be defined in order to focus the areas with the greater erosion problems. All the coastal area characterized by different erosion levels has to be detected and classified, in order to identify the urgency level and the correct strategy. The classification of the critical areas starts from the following identification of the damaged areas: • Areas with high erosion level: they need huge restoring activity of the littoral transport and/or beach nourishment • Areas with medium erosion level: they need a beach stabilization but specific restoring actions are not required. • Areas with low erosion level: they only need maintenance of existing equilibrium.

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2.3 Planning analysis The planning analysis requires: • • • • •

Analysis of the environmental local risks (erosion processes; loss of sediment; human impact, etc); Analysis of the existing negative responses; Final setting of physiographic unit limits; Classification of critical zones according to priority criteria; Checklist of possible alternatives to shore protection (retreat, do nothing);

In relation to the erosion level found out in the physiographic unit, the following options can be chosen: • • •

Passive hard structures: revetments, bulkheads, sea walls; Active hard structures: groins, detached breakwaters (emerged or submerged); Soft structures: artificial nourishment alone or protected with submerged structures, sand bypass.

The choice among alternative solutions is based on extensive coastal knowledge, which is the combination of theory and experience. Some of these solutions will be illustrated in chapter 12.

3 Causes of coastal erosion The main causes of coastal erosion can be divided into natural and human. Nevertheless, it has to be observed that the latter causes are indirectly related to the most part of the “natural” erosion processes. Natural causes The main natural cause of erosion is the imbalance between the loss of sediment due to offshore transport and the decrease of the longshore transport, which results in a negative sediment budget. The global climatic change emphasizes this problem and affect the sediment transport; the decrease of run off is also responsible for loss of littoral transport. The forces acting on littoral sediments are related to wave conditions. The strong dissipative processes associated to breaking events generate alongshore currents responsible for the littoral sediment transport between the breaker zone and the swash zone, which are responsible for spatial change. During severe meteomarine events, the sediment near the coastline is transported offshore and deposited in a bar far away from the shoreline by the steeper waves. During the following calm periods, the smaller waves tend to move the sand from the bar location toward the beach, even if part of the sediment is lost in the open sea. Other factors affecting the sediment transport are density currents and tidal currents which may change the wave circulation pattern. Another natural cause

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

of erosion is the mean sea level and the direct sediment transport by the wind, which results in the dunes formation. Human impact Human causes depend on the interaction between coastal structures and littoral transport, because they can modify the natural equilibrium through the imbalance of longshore sediment budget. Dams and other flood prevention structures reduces sedimentation in rivers and accelerates erosion processes. Besides, urbanization of coastal areas decreases the longshore sediment transport causing the erosion of some areas and the silting up of others. Localized structures like emerged and submerged breakwaters, groins and their combinations reduce the damages, although they need a focus study on sediment dynamics in the interested area and on the total extension of physiographic unit. This analysis has to be carried out to avoid additional damage both on the shoreline evolution and on the quality of the coastal environment.

4 Space and time scales Before the examination of the physical phenomena directly responsible for coastal processes, a first fundamental issue concerns the spatial and temporal scale on which processes take place. In this context the processes developing in relatively shallow water will be considered, that is between the coast line and the breaker zone where the interaction between the water forces and the bottom is particularly relevant. The usual approach when hydrodynamics (turbulence, waves, tides, stormsurges, currents, etc.) and morphodynamics (ripple formation, bar formation, cross-shore transport, longshore transport, etc.) are involved is the division of processes in relation to the spatial and temporal scale of observation (see figure 1.2). The processes involved can be distinguished in relation to the spatial/temporal scale of observation: • • •

Small scale (0.1mm – 10m, 0.1sec – 1 day) Intermediate scale (1m – 10km, 1sec – 1 year) Large scale (1 km – 100km, 1month – ten-years periods)

Different processes are connected to each other: in fact, the large scale processes modify boundary conditions for the processes on the intermediate scale, which, in turn, influence the processes on smaller scales. Some interesting questions about the influence of resolution on the correct description of the phenomena may be raised: the relationships between spatial resolution and predictability show that while increasing resolution provides more descriptive information about the patterns in data, it also increases the difficulty of accurate modeling.

INTEGRATED APPROACH TO COASTAL DYNAMICS

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Figure 1.2 - Space-time scales relationships, (modified from De Vriend, H.J., 1991). Processes on small scale Small scale processes involve interactions between bottom currents and sediments, which generate the sediments transport. Several bed forms are shaped by wave action (ripples and dunes). The bottom morphology and the bottom boundary layer dynamics are crucial at this scale to define the processes involved in the sediment transport. Processes on intermediate scale Intermediate scale processes involve the wave dynamics, the littoral wave circulation and the evolution of the bottom morphology due to sediment transport. Processes of wave dynamics include refraction and shoaling (caused by the depth decrease) diffraction, reflection (in presence of sheltering structures), breaking and dissipation. The circulation in the surf zone has to be evaluated in order to compute the sediment transport. The next step is to consider bathymetry evolution, which, in turn, modifies the circulation. The processes go on till an equilibrium is reached. Processes on large scale Large scale processes involve the behavior of wide coastal areas on long times, which is known in literature as LSCB (Large Scale Coastal Behavior). Large scale models are developed integrating the ones on small and intermediate scale. The starting input is a known morphology, then the forcing (waves, currents, river inlets, structures, etc) have to be imposed to find out a new configuration. The large scale observation is important from an economic and social point of view, and it represents an efficient tool for the correct coastal zone management.

5 Meteomarine factors Meteomarine factors are directly associated with climate, such as winds and atmospheric pressure variation, waves, currents, tides and storm surge, which affect coastal morphology.

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5.1 Wind Wind is caused by pressure gradients between adjacent areas. Pressure variations are generally the result of temperature differences due to disomogeneous insulation of adjacent zones reached by solar radiation (e.g. between lands and oceans). Wind fields vary in a wide range of spatial and temporal scales, from large-scale (permanent winds) to local (short winds). Hydrodynamics is strongly influenced by wind action, which is responsible for wave generation, wind setup, surge and surface currents. Wind forces directly modify coastal morphology through transport and deposition of sediments on the beach and the dunes, while they indirectly modify coastal morphology generating waves responsible for sediment transport. Wind action has a substantial role in some phenomena such as sea breezes, water level fluctuations and seiches. Sea breezes occur in a coastal area, especially in summer, and are generated by pressure gradients between water and coast: during the day, the warm air over the land becomes lighter and rises, thus forming an area of low pressure, resulting in a landward-directed breeze. During the night, the cooling of land is faster than water, which results in a seawarddirected breeze. Breezes don’t play a significant role in shaping the coastal morphology and give little contribution in sediment transport. Rapid changes in atmospheric pressure and directional shifts of strong winds on small seas and confined water bodies, cause periodic water level oscillations called seiches. 5.2 Waves Waves are the main factor in determining the shape and morphology of beaches and significantly influence the planning and design of harbours and coastal structures. The wave crest is the high water level reached by the free surface, whereas the wave trough is the low water level. The wave height (H) is defined as the vertical distance between the crest and the trough. The wave length (L) is the distance over which the wave pattern repeats itself. The wave speed (C) is the velocity of a single wave, while the wave period T is the time required for a wave to pass a particular location. The inverse of the period is the wave frequency f. When waves become large or travel toward shore into shallow water, higherorder wave theories are often required to describe wave phenomena. These theories represent nonlinear waves, while the linear theory is valid when waves are small and travel on deep water (but it still provides some insight for finiteamplitude periodic waves). However, the linear theory cannot account for the fact that wave crests are higher above the mean water line than the troughs are below the mean water line (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 2001). Waves are strongly related to wind action on surface water, which provides a source of energy for wave formation. Waves propagate energy supplied by the

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wind at air-sea interface, contribute to longshore transport and move bottom materials onshore and offshore. The mechanical characteristics of waves are important factors in planning and design of coastal structures because they interact with obstacles (breakwaters and coastal protection works), changing their characteristics until breaking occurs. Waves are divided in two main groups: short waves (period lower than 20s) and long waves, also called long-period oscillations (period between 30s and 40min). Water-level oscillations with a period longer than 1 hour (called astronomical currents and storms surge) are related to water-level variations. Short waves include wind waves and swell, whereas long waves can be divided into surf-beats, harbour resonance, seiche and tsunamis. Wave Classification A typical classification based on the frequency representation of all oceanic waves is given in Kinsman, B., 1965 (figure 1.3). A wave field is formed by many individual wave components, each characterized by a wave height, a wave period and a direction. Wave fields with many different wave periods and heights are called irregular, whereas wave fields with many wave directions are called directional. A wave field can be more or less irregular and more or less directional. Gravity waves are one of the most important factors acting on coastal morphology. Wave conditions vary extensively from one site to another, and depend on the wind climate and on the fetch extension, that is the water distance covered by wind without obstructions. Short waves are divided in two groups: •

•

Wind waves (also called sea waves), are generated by the local wind field. They are generally steep (high and short), irregular and directional waves. Their directionality makes difficult to recognize defined wave fronts, so they are also named short-crested waves. Swell are waves that traveled long distances away from their generation area and are more regular and unidirectional than wind waves. A portion of wave energy is lost during the propagation in deep water due to dissipation phenomena, which results in long period waves.

Wave Generation Wind waves are generated as a result of the action of the winds on the sea surface. Wave properties (height, period, direction) at a site depend on the following factors: • The wind field (speed, direction and duration); • The fetch of the wind field (meteorological fetch) or of the water area (geographical fetch); • The water depth in the wave generation area.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 1.3-Waves classification by frequency (after Kinsman, B.,1965). Wave height and period are closely related to wind conditions, from which they are estimated. The analysis of historical measured wind records is useful to reconstruct wave climate at a site. Such a computation is known as wave hindcasting. Forecast wind conditions are used to perform the wave forecasting. Wave transformation When waves approach a shoreline, they are affected by the seabed through refraction, shoaling, bottom friction and wave-breaking. Wave-breaking can also occur at deep water when the waves are getting too steep. The following transformations may occur: • Shoaling is the wave deformation, starting when water depth becomes less than about half the wavelength. This process causes a reduction in the wave propagation velocity and a shortening of the waves. • Refraction is caused by the fact that waves propagate more slowly in shallow than in deep water. When the wave front travels at an angle with the bathymetry, a change in direction of wave propagation occurs. An example of refraction phenomenon due to the bottom interaction is given in figure 1.4. • Diffraction is caused by sheltering structures such as breakwaters. This is the process by which the waves propagate into the lee zone behind the structures by energy transmittance laterally along the wave crests. • Bottom friction causes energy dissipation and thereby wave height reduction as the water depth becomes shallower. Friction is of special importance over large shallow water areas.

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13

Figure 1.4 – Waves usually do not approach the shoreline parallel to the coast, but interact with bottom, resulting in wave refraction (after Thurman, H.V., 1985). • Depth-induced wave breaking happens when the wave height becomes greater than a certain fraction of the water depth. The wave height of an individual wave at breaking is often said to be around 80% of the water depth, but this is an approximate number, depending on breaking type and beach slope. • Wave-current interactions occur in the presence of a large scale current field (like a river mouth) and in the presence of rip currents. They influence the associated sediment transport and the subsequent shoreline evolution. • Wave-wave interactions result from nonlinear coupling of wave components and result in energy transfer from some wave component to other ones. Statistical description of wave parameters A statistical description of waves is necessary in order to make a correct analysis of the wave climate and of the littoral transport. This statistical description is based on the directional distribution of the following wave characteristics: • The significant wave height Hs, which is defined as the mean of the highest third of the waves in a time-series of waves representing a certain sea state (Hs computed on the basis of a spectrum is referred to as Hm0). • The mean wave period Tm, which is defined as the mean of all wave periods in a time-series representing a certain sea state. • The peak wave period Tp, which is the wave period with the highest energy. The analysis of the distribution of the wave energy as a function of wave frequency for a time-series of individual waves is referred to as a spectral analysis. The peak wave period is extracted from the spectra. • The mean wave direction θm, which is defined as the mean of all the individual wave directions in a time-series representing a certain sea state.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

These parameters are often calculated from a time-series of the surface elevations once every three hours. The time-series is thereafter statistically analyzed to arrive at a synthetic description of the wave conditions in order to reach two main goals: •

•

Identify the occurrence probability of the extreme wave events for the design of maritime structures; this is often done through the percentage of exceedance of the wave height Hs; Identify the wave climate for the computation of coastal dynamics processes; this is often done through a directional distribution of the wave heights, often presented in the form of a wave rose or scatter diagram of Tp, versus Hs.

5.3 Currents Currents occur within the deep ocean, over the continental shelf and within the offshore and nearshore zones. They may be quasi-steady and persist for several hours to several weeks (ocean and shelf currents), or they may be oscillatory with periods of seconds (currents under waves). Currents may be limited to the surface or to the seabed, or they may extend over the full depth of water. Surface currents can have a different direction to those at the seabed. A partial currents classification based on the water depth divides the currents into the ocean, shelf and nearshore currents. The largest currents are those of the open ocean (ocean currents), which are driven by global scale interactions between the atmosphere and the sea. The continental shelf can extend between 1 and several kilometers from the coast. Continental shelf currents, are a complex mix of several components, among which there are internal waves, coastal trapped waves, tides and local wind induced currents. Shelf and ocean currents are generally of little significance within the shallower waters of the nearshore zone. This area is the preserve of wave induced currents which include: • • • •

oscillatory currents at the seabed prior to wavebreaking; mass transport of water shoreward as waves break; rip currents; longshore currents.

Currents are also an important aspect of many designs, particularly those involving the environment, water quality and habitat. Ocean currents are driven by the circulation of winds above the surface water, but coastal design needs the analysis of the wave-induced currents rather than the ocean currents. Wave-induced currents Nearshore currents are mainly caused by surface waves breaking on a beach. In fact, when surface waves break on a beach, wave energy is lost to turbulence generated in the process of breaking, and wave momentum is transferred into the

INTEGRATED APPROACH TO COASTAL DYNAMICS

15

water column generating nearshore currents. There are two current systems whose flow structures are predominantly horizontal, alongshore currents caused by obliquely incident waves and cell-like circulations, which can occur when waves are nearly at normal incidence. The scheme of a nearshore current system is indicated in figure 1.5. Cross shore currents It is known that the propagation of surface waves produces a mass transport in the direction of wave propagation, which is a second-order correction to the linear wave theory. This mass transport is called the wave drift and is directed toward the coast. When the waves are breaking, water is also transported in surface rollers towards the coast. These two contributions are concentrated near the surface. As the net flow must be zero, they are compensated by a return flow in the offshore direction, which is concentrated near the bed (undertow). Long shore currents The longshore current is generated by the shore-parallel component of the stress associated with the breaking process for obliquely incoming waves, the socalled radiation stresses. This current, which is parallel to the shoreline, carries the sediments alongshore and it is approximately proportional to the square root of the wave height and to sin(2αb), where 2αb is the wave incidence angle at breaking.

UNIT CELL

RIP HEAD

MASS TRANSPORT

WAVE BREAKING

RIP CURRENT

LONGSHORE CURRENT

SURF ZONE

SHORELINE

Figure 1.5 – Nearshore current system. Rip currents Rip currents have been defined as currents in the offshoreward direction which return the sea water transported shoreward by wave action. These currents,

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

which are concentrated within the surface layer, are part of the cellular circulations, depicted in figure 1.5, ‘‘fed’’ by the converging alongshore flows close to the shoreline. Rip currents cause a seaward transport of beach sand, hence have direct impacts on beach morphology. Recent studies (Yu, J. & D.N. Slinn, 2003) show that the offshore directed rip currents interact with the incident waves to produce a negative feedback on the wave forcing, reducing the strength and offshore extent of the currents. 5.4 Sea level variations Water-level variations are important because they lead to identification of the swash zone and the fluctuations of the coastline. The extreme water levels are associated to flooding events. Water-level variations can be divided in two types: • Regularly oscillating variations with periods from half day up to one year (astronomical tides); • Non regular variations with recurrence periods from days up to several years, mainly caused by meteorological conditions. Astronomical tide The astronomical tide consists of the periodic rise and fall of water level generated by gravitational interaction among the earth, moon and sun. These factors determine the tide at a given location, mainly generated in the deep oceans from which travels into the coastal waters. The tidal wave height in deep water is normally less than 0.5m, whereas in shallow water it is modified by shoaling and friction; at specific locations the tide can be up to 15m high. The tidal conditions at a specific location vary according to semi-diurnal and diurnal tidal constituents and are published in Tidal Tables. If the semi-diurnal constituents determine the tide at a given location, the tide is called semi-diurnal, and if the diurnal constituents dominate it is called diurnal tide. A semi-diurnal tide has two high waters and two low waters every day, whereas diurnal tides have only one of each every day. In addition to these diurnal and semidiurnal variations the contribution of the fortnightly variations must be added, that cause the tide to be higher than normal at full moon and at new moon (spring tide), and lower than the normal at the quarters (neap tide). Meteorological water-level variations The water level also varies as a function of the wind impact and atmospheric pressure variations on the water surface: • Wind action drives onshore and offshore the surface waters and is responsible for the water-level rise (wind set-up) in restricted areas subject to wind stress: when wind drives water offshore, deep waters move onshore, and vice versa. • Weather disturbances cause water level variations called barometric surge and storm surge. High onshore wind over the shallow water surface associated with low barometric pressure causes a temporary

INTEGRATED APPROACH TO COASTAL DYNAMICS

17

sea level rise, which results in flooding and coastal damage. The storm surge is the result of the combined impact of the wind stress on the water surface, and the atmospheric pressure reduction. The storm surge does not include the effect of the astronomical tide. The combined effect of astronomical and meteorological surges is often referred to as a tidal-wave. Barometric surges are often associated to storm surge and are generated by the inverse relationship between sea level and barometric pressure: water surface level rises as atmospheric pressure decreases of about 0.1 m for each kPa of pressure difference. In figure 1.6 an example of barometric surge superimposed on astronomical tide is given.

6 Sediment transport and coastal structures 6.1 Modes of sediment movement and their appearance region

Mean sea level (m)

The inshore sediment movement is usually divided into bed load, suspended load and sheet flow, as shown in the scheme of Figure 1.7. On deep water, waves have only a minor influence on sediment transport. As a wave enters shallow water, it begins to “feel” the bottom, its profile deviating from the linear one. In

Time (Days) Figure 1.6 - Example of barometric surge superimposed on astronomical tide (G. Benassai, 2002). fact, linear wave theory predicts that the crest and trough heights of the wave are equal (see Figure 1.8), that is, the wave is evenly distributed about the still water level. In contrast, finite amplitude wave theories predict waves with peaked crests and flat troughs. The crests are further above the still water level than the troughs are below this level (Young, I.R., 1999). When a resisting force of the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

sand particle on the bottom becomes smaller than a wave force on it, the sand begins to move. The depth of this point on the beach is called a critical depth for sediment movement and the critical velocity is defined as the water particle velocity at this depth. When a shearing force larger than the critical condition for sediment movement acts on the sea bed a sediment movement of traction mode (bed load transport) takes place. As the shear stress increases further, sand ripples are formed on the bed. Suspended sediment occurs by the advection of the trapped sediment in the vortex formed periodically around the sand ripple (Sawaragi, T., 1995). When the bottom shear stress increases beyond a certain limit, ripples disappear and the so-called sheet flow in which sediment in high concentration is transported within a thin layer appears. Because a great deal of the sand particles are transported in the thin layer in this range, fluid turbulence near the bottom is repressed and any systematic turbulence (such as vortex on the ripple), does not occur there. In the shallow water region where incident waves break, a great deal of sediment is brought into suspension by the turbulence caused by breaking waves. Breaking waves of the plunging type suspend more sediment than breaking waves of the spilling type. In fact in the plunging breakers the crest becomes faster than the trough, curls and violently collapses. swash zone

breaker zone

breaking point

Disapperance of ripples Initial motion Generation of ripples

Bed load transport

Figure 1.7- Sediment movement and mode of sediment transport on the beach. A considerable amount of energy is released into a downwardly directed mass of water and the turbulence reaches the area beyond the breaking zone. Sediment materials are suspended and transported offshore by undertow currents. A large trough is formed and becomes deeper until the energy is completely exhausted. Offshore a bar is formed, localized between the breaking point and the deepest zone reached by vorticity. In the swash zone, fluid motions become entirely different from that in the shallow water region and sand movement also differs from the other part of the

INTEGRATED APPROACH TO COASTAL DYNAMICS

19

beach. In fact, the swash zone is characterized by a fluctuation of coastline associated with sediment transport which occurs when the energy of the waves advancing up the beach is not completely exhausted. Swash dynamics depend on the wave frequency oscillations: at low frequencies, if the beach is permeable and the sand is not saturated, the water percolates through the substrate and the backrush decreases. An amount of sediment is accumulated on the beach in the swash zone and the beach slope increases. The result is a convex beach profile. At high frequencies (storm waves), if the substrate is saturated, the water cannot percolate through the sand and the backrush current cannot decrease. In this case the sediment transport associated with backrush is greater than uprush, which results in a concave beach profile. 6.2 Coastal structures and sediment transport Coastal structures interfering with littoral transport are the most common causes of coastline modification. The principal structures that may cause erosion are: • • • •

Groynes or similar shore normal structures Ports Inlet jetties at river mouths Detached breakwaters (a)

(b)

(c)

Figure 1.8 – Wave profiles: sinusoidal (a), cnoidal (b) and solitary (c). The accumulation and erosion pattern adjacent to coastal structures depend on the type of coastline, (the wave orientation of the shoreline) and on the extent and the shape of coastal structures. The typical impact of the coastal processes and the related erosion problems for different types of structures will be briefly discussed in the following.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Groynes Groynes are normally built perpendicular to the shoreline with the purpose of protecting a section of the shoreline by blocking (part) of the littoral transport, whereby sand is accumulated on the upstream side of the groyne. However, the trapping of the sand causes a deficit in the littoral drift budget, and the coastal protection is always associated with a corresponding erosion on the lee side of the structure. In other words, a groyne just shifts the erosion problem to the downstream area. This is the reason why groynes are often built in long series along the shoreline, a so-called groyne field. Ports The primary purpose of a port is to provide safe mooring and navigation for vessels but when built on the shoreline it interferes with littoral drift budget and the results are sedimentation and shoreline impact. Like a groyne, the port acts as a blockage of the littoral transport, whereby it causes trapping of the sand on the upstream side in the form of an accumulating sand filet, and the possible bypass causes sedimentation in the entrance. The sedimentation requires maintenance dredging and deposition of the dredged sand. The result is a deficit in the littoral drift budget which causes lee side erosion on the adjacent shoreline. A port must consequently minimize sedimentation and coastal impact. These requirements have not always been given adequate attention. The result is that many ports trap large amounts of sand and suffer from severe sedimentation. Inlet jetties at river mouths River mouths are often by nature shallow and variable in location, which makes them unsuitable for navigation. In order to improve navigation conditions (and to some extent flushing conditions), many river inlets have regulated mouths. The regulation may consist of jetties, possibly combined with maintenance dredging programs. For the above reasons, regulated inlets are normally obstructions to the littoral transport which means upstream sand accumulation along the upstream jetty, loss of sand due to sedimentation in the deepened channel and the associated maintenance dredging. Detached breakwaters Detached breakwaters are used as shore and coast protection measures. In general terms, a detached breakwater is a coast-parallel structure located inside or close to the surf-zone. Breakwater schemes have many variables, depending on the distance from shoreline and location relative to the surf-zone, length and orientation crest height, (emerged or submerged). The breakwater shelters the coast partly from the waves, however as the waves diffract into the sheltered area, a complete shelter cannot be obtained. Submerged breakwaters have a lower visual impact, but provide less shelter. The longshore current is partially blocked by the circulation currents which cause some of the longshore current to be diverted outside the breakwater. So the littoral transport in lee of the

INTEGRATED APPROACH TO COASTAL DYNAMICS

21

breakwater is decreased due the attenuated wave and longshore currents in the area sheltered by the breakwater. This causes trapping of sand behind the breakwater dependent on the length and shoreline distance, until a tombolo is eventually formed. For shorter or submerged breakwaters, only a salient in the shoreline will develop. In case of trapping of sand, a tombolo will be developed, which causes lee side erosion downstream of the breakwater very similar to what is developed for groynes.

7 Elements of coastal management The coastal system is comprised of a complex, dynamic web of interrelationships among human activities, societal demands, natural resources, and external natural and human inputs. The system is driven by human activities in terms of societal demands for use of the natural resources of the coastal area to produce desired products and services, e.g., seafood, marine transportation, recreation etc. Societal demands for outputs from a coastal area usually exceed the capacity of the area to meet all the demands simultaneously. Coastal resources, e.g., fish and recreations, are often “common property resources” with “open” or “free” access to users. Free access often leads to excessive use of the resource, e.g. degradation or exhaustion of the resource, coastal pollution and habitat degradation. The Integrated Coastal Management (ICM) is the process used to decide what mix of outputs will be produced. Integrated coastal management can be defined as “a continuous and dynamic process by which decisions are taken for the sustainable use, development, and protection of coastal and marine areas and resources. ICM acknowledges the interrelationships that exist among coastal and ocean uses and the environments potentially affected, and is designed to overcome the fragmentation inherent in the sectorial management approach. ICM is multi-purpose oriented, it analyzes and addresses implications of development, conflicting uses, and interrelationships between physical processes and human activities, and it promotes linkages and harmonization among sectorial coastal and ocean activities” (Cicin-Sain, B. & R.W. Knecht, 1998; Pernetta, J. & D. Elder, 1993). Pressures The coastal areas, which accommodate one of the more fragile and precious habitats of the world, are under pressure from economic causes, because recent large migrations have resulted in coastal stress and overloads (Kamphuis, J.W., 2000). The following pressures on the coastal zone are indicated: •

Population density: According to the 1994 distribution of population in relation to the distance from the nearest coastline, 20.6 percent of the world’s population lives within 30 km of the coast, and 37 percent within 100 km (Gommes R., J. Du Guerny, F. Nachtergaele & R. Brinkman, 1997). The coastal regions of

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

•

•

•

European countries are subject to constant pressures due to population density: half of European population lives in a narrow strip about 50 kilometres far away from the sea. The resources of the coastal zones produce a great part of the economic wealth. Population density has a substantial influence: on the equilibrium of natural ecosystems, as a result of human activities; on the quality and the amount of the natural resources (forests, grounds, waters, fishing zones, shores, etc.) as result of activity and population concentration. In turn, these factors increase the demand for use and exploitation of resources and require waste disposal; on the natural landscape, modified as a consequence of the human activities, the dimension and the scale of the relative structures and associated impact; Migration: as a result of migration to coastal areas, the coastal population is growing at a faster rate than the world population. Probably within the next 20 to 30 years, the coastal population will almost double; Tourism generates large fluxes of people who take vacations in far away coastal areas. The Mediterranean is one of the main tourist destinations of the world, giving accommodation to 30% of the international tourists: a third part of introits comes from the international tourism. Tourism usually has a seasonal frequency and increases in years. Probably the pressures on the littoral zones will continue to grow in future, with a doubling of tourism in the Mediterranean in the next 20 years. Tourism interacts with coastal environment in terms of land use and water resources consumption; Erosion caused by natural and human factors increase resources exhaustion, pollution and habitat degradation: these factors emphasize the fragility of coastal areas and push priority on protecting and upgrading the coastal system. Erosion of the habitat takes place mainly for the competitive use of the coastal zone. The analysis of erosion data (Table 1.1) shows that in the European coastal zones 1500 km are artificial coasts (Balearic islands, Gulf of the Lion, Sardinia, Adriatic, Ionian and Aegean), occupied by harbours (1250 km) (Corine, 1998). According to the data collected by Corine, approximately 25% of the Italian Adriatic coast and 7.4% of the Aegean coast shows an evolutionary tendency to erosion, whereas approximately 50% of the Euro-Mediterranean coastal zone is considered stable.

Conflicts The analysis of conflicts takes into account the interactions among natural resources and different economic, political and environmental factors. Conflicts among user groups and current social, economic and environmental conditions

INTEGRATED APPROACH TO COASTAL DYNAMICS

23

represent a problem that often requires an immediate solution. The challenge in this context is to combine the following interests: • •

Public interests: shore protection, resource preservation, development of infrastructure and public utilities, etc.; Private interests: development of projects and coastal protection, industrial development, navigation, etc.

There are often inherent conflicts of interests in such projects, both with respect to the objectives and with respect to costs sharing. Resolving these planning matters is often as difficult as it is to find a suitable technical solution. However, the dissemination of the planning concepts, of the principles of sustainable development and environmental protection as well as of the physical mechanisms in the coastal zone, are all important for the success of the entire planning and design process. Table 1.1 - Evolutionary trends of some coasts of the European part of the Mediterranean Sea for both rocky coasts and beaches as % of coasts (Corine, 1998). MARITIME REGION Balearic Islands Gulf of Lion Sardinia

Adriatic Sea Ionian Sea Aegean Sea

Total coastline (km)

Stability

Erosion

Sedimentation

No information

Not applicable

2861

68.8%

19.6%

2.4%

0.5%

8.7%

1366

46.0%

14.4%

7.8%

4.1%

27.8%

5521

57.0%

18.4%

3.6%

16.0%

5.0%

970

51.7%

25.6%

7.6%

3.9%

11.1%

3890

52.3%

22.5%

1.2%

19.7%

4.3%

3408

49.5%

7.4%

2.9%

37.5%

2.6%

Coastal management In the European coastal regions local administrations have the best knowledge of the real problems. They interact with citizens, private companies and non governing organizations. Regional agencies give the guidelines and principles to coordinate the local initiatives, whereas political and national programs facilitate the participations on regional and local level. In many cases the cooperation between States is also necessary: as an example, it would be better that countries

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

joining the same sea try to coordinate their own initiatives, rather than propose different and contrasting political actions. The strategy of coastal zone integrated management (figure 1.9) encourages this transnational approach in the countries joining the “regional seas” like the Mediterranean or the Baltic Sea. Another aim is to prevent political strategies that damage the coast. In the case of the agricultural pollution, the integrated coastal management will give to the PAC (Common Agricultural Policy) the measures to decrease the impact of fertilizers on coastal waters.

Figure 1.9 -Coastal management strategy (after Townend, I.H., 1994). Coastal management Objectives and Activities Chapter 17 of Agenda 21 so describes the objectives and processes of Integrated Coastal Management programs: Coastal States commit themselves to integrated management and sustainable development of coastal areas and the marine environment under their national jurisdiction. To this end, it is necessary to, inter alia: • • • •

Provide for an integrated policy and decision-making process, including all involved sectors, to promote compatibility and a balance of uses; Identify existing and projected uses of coastal areas and their interactions; Concentrate on well-defined issues concerning coastal management; Apply preventive and precautionary approaches in project planning and implementation, including prior assessment and systematic observation of the impacts of major projects;

INTEGRATED APPROACH TO COASTAL DYNAMICS

•

•

25

Promote the application of methods, such as environmental accounting, that reflect changes in value resulting from uses of coastal and marine areas, including pollution, marine erosion, loss of resources and habitat destruction; Provide access, as far as possible, for concerned individuals, groups and organizations to relevant information and opportunities for consultation and participation in planning and decision-making at appropriate levels.

The activities associated with the coastal management concern: •

•

•

Coordinating mechanisms (such as a high-level policy planning body) for integrated management and sustainable development of coastal and marine areas and their resources, at both the local and national levels. Such mechanisms should include consultation, as appropriate, with the academic and private sectors, nongovernmental organizations, resource user groups, and local communities; Data and Information: Coastal States, where necessary, should improve their capacity to collect, analyze, assess and use information for sustainable use of resources, including environmental impacts of activities affecting the coastal and marine areas. Information for management purposes should receive priority support in view of the intensity and magnitude of the changes occurring in the coastal and marine areas; International and regional cooperation and coordination: The role of international cooperation and coordination on a bilateral basis and, where applicable, within a sub-regional, interregional, regional or global framework, is to support and supplement national efforts of coastal States to promote integrated management and sustainable development of coastal and marine areas.

Planning and management of the coastal zones must necessarily be founded on the so-called “principle of precaution”: responsible subjects must try to preview the possible damages for the coastal zone and identify the more appropriate interventions. According to this principle, if they are not sure that some participation can take negative repercussions for a coastal zone, they must address their action with maximum caution. This approach assumes particular importance in vulnerable zones that could have negative consequences as a result of urbanization or tourism. In the past, many attempts to preserve the coastal regions failed, in spite of the good intentions, because they were addressed to sectorial aspects. As an example, the tourism in the coastal zones cannot be treated without paying attention to other concurrent factors, such as water supply, regional planning, impact of tourism on the existing natural habitats. The Integrated Coastal

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Management has the objective to facilitate contacts between local, regional and national administrations in order to allow the responsible subjects to obtain a precise picture of the real necessities involved. The interested local subjects and organisms must be involved in planning and management of the coastal zones: without the contribution of companies and citizens who live and work in the coastal zones, the ICM will not be able to work successfully.

Chapter 2 Beach morphology and sediment analysis 1 Introduction A beach is an accumulation of loose material around the limit of wave action. According to King (1972) the beach may be taken to extend from the extreme upper limit of wave action to the zone where the waves, approaching from deep water, first cause appreciable movement of beach material. According to the nature of the beach material, we can define gravely, sandy and silty beaches. Beaches have variable planar shapes (eg. tongues, tombolos, cuspids) and their classification depends on sediment characteristics (mineral and granular properties) obtained by topographic and bathymetric surveys, as described in the following sections. The beach profiles show a typical morphology extending between the low tide level and the first major change in topography (e.g. a dune), and covering also the submerged profile. The coastline is subject to cyclical changes on different temporal scales (a few seconds to geological ages). This variability provides great interest for researchers in the field of geology, sedimentology and coastal engineering.

2 Beach classification Geologists describe beach classification using different criteria: according to grain size criteria we distinguish gravely, sandy and silty beaches. Some criteria are based on lithology: marine processes modify chemical and physical properties of a rock (erosion and dissolution); strong temperature changes, crystallisation and weathering processes cause physical alteration of rocks (mechanical disintegration); chemical processes like oxidation, reduction and biochemical reactions cause decomposition of rocks. Coasts consisting of firm material are defined as consolidated coasts, whereas unconsolidated coasts are subject to vigorous erosive processes. Volcanic coasts are residues of calderas originated by volcanic explosion and are characterised by circular physiognomy (convex or concave) and deposits of volcanic material.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Early classifications, based on a genetic approach, distinguish coast affected by rising sea level (submergence), coasts affected by falling sea level (emergence), or compound coasts, affected by both phenomena (Davis, R.A., Jr., & Hayes, M. O. 1984; Johnson, D. 1919). Later classifications are based on onshore and shoreline morphology as those of Cotton (1952) and Shepard (1937). Inman and Nordstrom (1971) classification includes conditions of the offshore bottom. The most widely used coastal classification was introduced by Shepard in 1937, who distinguishes primary coasts - formed mostly by non-marine agents and secondary coasts – shaped primarily by marine processes or by marine organisms. As described in Shepard (1973), primary and secondary coasts include many groups and subgroups, so that Shepard’s classification is more detailed than others and includes all the existing world’s coasts. According to Shepard’s classification, primary coasts include: 1.

Land erosion coasts - shaped by subaerial erosion and partly drowned by postglacial rise of sea level (with or without crustal sinking) or inundated by melting of an ice mass from a coastal valley;

2.

Subaerial deposition coasts – for example, coasts originated by rivers, glacial and wind deposition;

3.

Volcanic coasts – originated by lava flows, fragmental volcanic products, volcanic collapse or explosion;

4.

Coasts shaped by diastrophic movements;

5.

Ice coasts – formed by various types of glaciers.

Primary coasts morphology reflects either tectonic or terrestrial or land-based processes. Generally they are coarse and irregular, because they have not been straightened out by currents and waves. Secondary coasts include: 1. 2. 3.

Wave erosion coasts – coasts eroded by wave action Marine deposition coasts - prograded by waves and currents Coasts built by organisms - formed by the growth of animals and/or plants.

Major features of secondary coasts are sea cliffs and wave-cut platforms; secondary coasts are affected by shoreline straightening over time. frequently they are characterised by the formation of beaches, sand spits, bars, and barrier islands. Longshore drift (the movement of sand along the shoreline in the direction of the waves) is one of the most important processes affecting secondary coasts. Energy-based classification Tides and waves are dominant forces driving littoral processes on open coasts because they sort the sediment and move it alongshore. Davies (1964) applied an energy-based classification to coastal morphology by dividing the world’s shores

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

29

according to tide range. Hayes (1979) expanded this classification, defining five tidal coastlines categories: 1. 2. 3. 4. 5.

microtidal <1m low-mesotidal, 1-2 m high-mesotidal, 2-3.5 m low macrotidal, 3.5-5 m Macrotidal >5 m

The Hayes (1979) classification (see Figure 2.1) was primarily based on shores with low to moderate wave power and was intended to be applied to trailing edge, depositional coasts. In the attempt to incorporate wave energy as a significant factor modifying shoreline morphology, five shoreline categories were identified based on the relative influence of tide range versus mean wave height (Hayes, M. O. 1979; Nummedal, D. & Fischer, I. A. 1978). 1. 2. 3. 4. 5.

Tide-dominated (high) Tide-dominated (low) Mixed-energy (tide-dominated) Mixed energy (wave-dominated) Wave-dominated.

The relative effects of these processes are significant, not the absolute values. Also, at the lower end of the energy scales, there is a delicate balance between the forces;

Mean tidal range

Tide dominated

Wave dominated Mean wave height Figure 2.1 - Energy-based classification of shorelines.

tide-dominated, wave-dominated, or mixed-energy morphologies may develop with very little difference in wave or tide parameters (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

3 Beach morphology and sediment transport In Italy the total development of shoreline amounts approximately to 7500 km, of which 55% are rocky coasts and 45% are beaches (most of them are sandy beaches). In northern Adriatic there are mild sloped sandy beaches, while steep shores are typical of Liguria, Calabria, Sicily and Sardinia, where little beaches (pocket beaches) are confined between adjacent capes. In Italy there are few silty beaches: actually they have an ecological importance because they are related to the so-called humid areas (areas covered by vegetation used by migratory birds). 32% of the Italian beach is subject to erosion, only 5% is in advance and 63% is stable; protection systems defend 15% (500 km) of beaches A general compendium (approximately 2000 km) of the morphology of Italian beaches is described in 60 maps (scale 1:100000) of CNR Italian Beach Atlas (Fierro G., AA. VV., 1999). Coastal zone definition is not simple because temporal and spatial variations modify boundaries and features. Besides, different definitions are used by researchers to describe elements and characteristics of a coast. Because of this, the following basic classification (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995) presents a general definition of coastal zone, (based on geological criteria) which defines four subzones: • • • •

Coast Shore Shoreface Continental Shelf

A coast is a transitional zone between land and water extending from the maximum reach of storm waves to the first major change in topography. The coastline is defined by the presence of cliff formed by wave erosion, dune or permanent vegetation. In some areas tide action obscures the landward limit of the coast (large deltas like Mississippi). The shore zone (Figure 2.2) is a sloping area where unconsolidated sediments are subject to wave action. The term ‘backshore’ is used for the zone above the limit of the swash of normal high spring tide, and is, therefore, only exceptionally under the direct influence of the waves. On a rocky coast it includes the cliffs, while a low coast may consist of sand dunes or mature salt marsh. The ‘foreshore’ zone is a sloping portion of the beach including all that part of the beach which is regularly covered and uncovered by the tide. On a tideless beach this zone will be narrow, only covering the distance between the limit of the swash and backwash of the larger waves. The foreshore zone includes the surf zone (a zone traversed by breaking waves extending between the wave breakpoint and the maximum run-up of the swash) and a swash zone, where wave swash (uprush of water) and backwash (back rush of water) foreshore.

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

31

The ‘berm’ is a terrace formed in the backshore zone above the limit of the swash at high tide to form a flat terrace, or a ridge with a reverse slope. Below low water the positive features on the sandy floor in the offshore zone are called ‘submarine bars’. The hollows found on the landward side of the submarine bars of sandy beaches are termed ‘troughs’. Another feature of coastal zones is the shoreface, a narrow sloping zone between the continental shelf and the low water limit, where unconsolidated sediments are subject to vigorous transport. Continental shelf is a slightly sloping zone of submerged continental margin extending from the offshore limit of shoreface to the slope of shelf break. Researchers divide continental shelf in a inner part, a middle part and a outer part.

Figure 2.2 Beach profile Sediment budget in the littoral zone is related to movements of materials between offshore and onshore zones. Sediments are mainly supplied by rivers, channels, beaches and dunes erosion. Secondary sources are artificial replenishment, industrial refuse materials, civil works, etc. In shallow water bottom materials are influenced by wave action, whereas in deep water currents action on bottom material is the main cause of sediment transport. In the littoral zone, sediment patterns and composition depend on the beach slope, characteristics of the area, external and internal sources of materials. Analysis of sediment characteristics, illustrated in the following subsection, plays a significant role in the study of littoral transport and sources of bottom materials. Offshore zone In deep water, waves have only a little influence on sediment transport. As a wave enters in shallow water, it begins to drag the bottom; wave length and speed increase, whereas the period remains constant. The wave energy increases with the wave height. In this phenomena, wave-bottom interactions produce some bed forms depending on the bottom contours and wave strength:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

• • • •

Moderate waves produce vortices of suspended particles, and generate some bed forms such as ripples. Strong wave action produces flat bottom and sediments are transported by sheet flows. When swell waves interact with bottom materials, ripples dimension depend on fluid particles excursion. In case of high wave speed, even if sheet flows occur, flat bottoms are not observed. Sea waves produce small ripples. Near the breaking zone, fluid motion becomes turbulent and many particles are suspended.

Surf zone and breaking zone: In this area breaking processes cause energy dissipation and produce vortices and turbulence phenomena. Breaking mechanisms are usually divided into two groups: spilling and plunging. •

•

Spilling occurs when steep waves advance up a gentle beach: when the wave breaks, the energy is gradually released over a considerable distance, and wave deformations are smaller than in plunging breakers. Suspended sediments are transported offshore by undertow currents or superficial currents associated with breaking. Breaking mechanisms continue their action with less intensity in the part of the surf zone nearest to the offshore zone, where the waves which generate the breaking process are recomposed. Plunging usually occurs on slightly steeper beaches: the crest becomes faster than the wave base, curls and violently descends into the wave trough. A considerable amount of energy is released into a downwardly directed mass of water and the turbulence reaches the area beyond the breaking zone. Bottom materials are suspended and transported offshore by undertow currents. A large bore is formed and becomes deeper and deeper until the energy is completely exausted. Ahead of the bore, a sediment accumulation is formed, localized between the breaking point and the deepest zone reached by vorticity.

Swash zone The swash is a fluctuation of coastline associated with sediment transport which occurs when the energy of the waves advancing up the beach is not completely exhausted. Swash dynamics depend on the frequency oscillations. At low frequencies, if the beach is permeable and the sand is not saturated, the water percolates through the substrate and the backrush decreases. An amount of sediment is accumulated on the beach in the swash zone and the beach slope increases. The result is a convex beach profile; At high frequencies (storm waves), if the substrate is saturated, the water cannot percolate through the sand and the backrush current does not decrease. In this case the sediment transport associated with backrush is greater than uprush, which results in a concave beach profile.

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

33

4 Seasonal profiles, bars and berms The beach slope is related to grain size: larger grain sizes generate steeper beaches as shown, for example in CERC (1984). During the winter storm waves move sand offshore, while the summer waves move sand onto the beach. Kamphuis (2000) shows that beach slopes are a function of the ratio (H/D) which represents the ratio of disturbing wave forces to restoring particle forces. On this basis, several qualitative accretion/erosion criteria were conceived (see Chapter 9). The interface where the sea surface meets the shoreline moves up and down the beach in a small area called swash zone. Even in case of low flow velocity, the small flow depth makes the shear stress extremely large. As a result, even coarse material can be mobilized by slow-moving swash. During the action of larger waves the beach material is redistributed offshore to become a longshore bar or sandbar, generally visible at low tide. The bars are an accumulation of material near the point of breaking waves and are considered offshore features of a beach (Kamphuis, J.W., 2000). Bars are formed near the breaking point (sometimes seaward of a trough) because the breaking waves set up a shoreward current with a compensating counter-current along the bottom. Sand carried by the offshore moving bottom current is deposited where the current reaches the wave break (Kamphuis, J.W., 2000). Berms form as a result of preferential shoreward transport within the swash. They are generally preserved (left behind) after a highwater event (stormy high tide, etc.). The berm is periodically overtopped during storms or extreme high tides. On sandy and shingle beaches berms build seaward through the multiple accretion of bars to the beach face. Vertical accretion to the berm is accomplished by swash, which is influenced by wave height. A beach may have more than one berm or none at all (e.g. an eroded beach backed by a seawall). Some beaches, particularly mesotidal gravel beaches, may exhibit multiple berms (e.g. LHT berm, HHT berm, summer berm, winter berm, storm berm, etc). High-water berms are formed during storm surges or spring tides (HHT berm).

5 Equilibrium beach profile The equilibrium beach profile results from steady wave forcing during the seasonal cycle. As described above, summer wave conditions move sand onto the beach, while winter storm waves move sand offshore. Beach profiles fluctuate with the seasonal cycle of wave energy: storms cause larger and more energetic waves in winter than in summer. As a result, the winter profile is characterized by the presence of a bar or by a berm on the foreshore (Figure 2.3). Dean (1983), conceived an equilibrium profile on the basis of constant energy dissipation.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Summer shoreline

Summer profile

Winter shoreline

berm

m.s.l.

cliff trough

bar Winter profile

bar

Figure 2.3 - Winter and summer beach profiles. The equilibrium profile is expressed in equation 2.1.

1 dF = De h dx

(2.1)

where: h is the water depth at distance x from the coast line F is the wave energy flux in shallow water; De is the dissipation coefficient of energy. Wave energy flux in shallow water is expressed as follows (see chapter 3): ρgH b2 ( gd )1 / 2 8 Hb is the breaking wave height, given by: F=

H b = d bγ

(2.2)

(2.3)

where γ is the breaker index (0.78 ↔1) and hb is the breaking depth. We define the equilibrium parameter A as: d = Ax 2 3

(2.4)

substituting this definition in (2.2) we obtain: F = (1 / 8) ρg 3 / 2γd 5 / 2 = (1 / 8) ρg 3 / 2γ 2 A5 / 2 x 5 / 3

(2.5)

With a little algebra we have: De =

1 dF 1 ρg 3 / 2γ 2 5 5 / 2 2 / 3 A x = d dx d 8 3

(2.6)

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

35

Finally, A can be expressed as a function of De according to the following expression:  24 1   A =  De 3/ 2 2  ρ g γ  5 

2/3

(2.7)

Dissipation of energy due to breaking waves destabilize sediment particles. When destabilizing forces are balanced by restoring forces, dynamical equilibrium occurs. The profile parameter A depends on grain size and fall velocity of particles. According to Moore (1982) and Dean (1983), it is possible to define relationships over ranges of grain sizes. For example: A = ( 1.04 + 0.086 ln( D )) 2

for 0.1×10 −3 ≤ D ≤ 1.0 ×10 −3 m

for 0.1×10−3 m ≤ D ≤ 0.2 ×10−3 m

A = 20 ⋅ D0.63

(2.8) (2.9)

Where D is the grain diameter. Dean (1983) also proposed a simple relationship between A and fall velocity: A = 0.50 ⋅w f 0.44

(2.10)

where wf is the fall velocity in m/s.

6 Sediment analysis 6.1 Bathymetric and geophysical surveys Temporal and spatial scales are important to define the zone of shoreline object of surveys as the time interval between two successive samplings. With regard to the spatial scale, it is necessary to survey the whole physiographic unit. From the engineer’s point of view, a physiographic unit covers the zone where any coastal change in plan or profile influences the adjacent coastline. In other words, it is a confined area where sediments move and the exchanges with the adjacent beaches are limited or negligible. The definition of a physiographic unit is important to define the limits of the area (generally variable) where the studies of the effects of coastal dynamics are necessary. With regard to the temporal scale, the study of the short term coastal dynamics requires a short time interval (analysis of the daily phenomena covering the observed period). In order to study long term phenomena, a long time interval is needed (seasonal phenomena require at least a time step of ten years). A preliminary characterization of physiographic units is carried out by analysis of coastal morphology. The first step is the comparison between historical charts and the definition of the area limits; the following step is the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

analysis of photographs and maps obtained by aero-photographic and topographic surveys. Then, the inspection of the coastline variations allows to examine the following characteristics: • • • •

the coastline cyclical fluctuations; evolution of river mouth as flèches (one-direction oriented flows) or cuspids (divergence of the flow); concave coastline (to which center sediment flow gradually converges); deposits and erosions of buildings near coastal structures.

Topographic surveys The topographic surveys are integrated with bathymetric surveys and coastal profiles. The analysis of the maps concurs to estimate the type of coastal morphology (emerged and submerged coast) and allows to characterize the elements of short-long term equilibrium. The topographic surveys are then related with the sediment characteristics, which pattern gives an alternate characterization of the physiographic unit. The topographic characteristics of the emerged and submerged beach are needed to verify the morphological evolution and they should be measured in the same period of the year, in order to avoid influence of the seasonal variability of the equilibrium profile. The ‘dry’ topographic survey is performed along transects (50 meters width or more) or from the coastline until buildings or any human infrastructure is reached; Bathymetric surveys are normally carried out using transects spaced by 50-100 meters or more until the 10-meters bathymetric line is reached. When steep zones are found, a more fine spacing and orthogonal sections are then needed. Bathymetric surveys Bathymetric surveys can be carried out until 1.5 meters depth, using an accurate echo sounder with an emission cone not wider than 10°. The depth measured has to be corrected in order to account for the tide excursions. The submerge profile for depths lower than 1.5 meters is generally measured with a graduated stick. Finally, the survey is plotted on scales 1:1000 or higher, interpolating the bathymetric lines every 0.5 or 1 meter. In transparent and shallow waters (10 meters depth) the surveys can also be integrated by aero photographic survey techniques. Sampling sediments The distribution of sediments and their selection can be studied through an accurate sampling strategy. The sampling site must be selected according to beach morphology and site variability: for example, trough, bar and foreshore regions are more variable than the nearshore and dune zones. Typically, grain size distributions are better sorted in the summer than in the winter. Finally, shoreline seasonal changes and engineering structures should be considered in selecting sampling points. Number of samples collected depends on research objectives:

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

• •

•

37

If the intent is to grossly characterize the sediments of a beach, usually only a few samples are needed; If the objective is to characterize the site as a whole, a set of samples needs to be collected. According to Hobson (1977), combining samples from different transects across the beach can reduce the high variability in spatial grain size distribution; If the intent of the study is to analyze the differences among the portions of the same beach, many more samples are needed.

In the third case, the first step of the survey is the development of a sampling project to define all the sampling locations. In the cross shore, according to Stauble and Hoel (1986), sampling points along the profile have to be located at all major changes in morphology: dune base, mid berm, mean high water, mid-tide, mid berm, mean high water and then at 3-m intervals until the depth of closure. In the longshore direction, sediment sampling should coincide with survey profile lines so that the samples can be spatially located and related to morphology and hydrodynamic zones (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995). 6.2 Physical and chemical analysis The most important sediment characteristic is the particle grain size (the measure of grain dimensions and their statistical distribution). Other interesting parameters are colour, texture, surface morphology (aspect and structure), shape and degree of rounding. The definition of sediment morphology and texture gives some indications of their origin and evolution; the analysis of petrography is useful to define the origin of sediment matrix. Grain size can be defined by direct measurement of particle diameters or, indirectly, by determination of their ‘hydraulic equivalents’ based on settling velocities of quartz spheres. The analysis is carried out using samples of 50 grams collected on the beach. The containers are opportunely labelled using a resistant plastic support where are signed place, date and hour of sampling. The oldest method of grain size determination uses a set of nested sieves with different mesh size. Each sieve is formed by woven stainless or brass steel wires. An amount of sediments pass through a set of nested sieves in which the size is gradually smaller down the stack. Grains are trapped on a sieve if their size is smaller than mesh openings. The sieves are agitated by hand (or mechanically) to make the selection more efficient (without forcing grains through the mesh). The weight of each size class is expressed as a percent of the total sample weight. The sieve analysis is a widely accepted method of grain-size determination because it is relatively easy and quick, but it can be a deadly process if the number of samples is too high. The amount of grains can affect the accuracy of the analysis: when the sample is too little, small errors can be more significant; if too much sand is used, one of the sieves in the stuck may become overloaded. Besides, grains can stick together or tend to get stuck in the mesh.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 2.4 - Sampling points along the beach profile (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995). Analysis of muddy sediments is commonly carried out by pipette analysis. The sample of sediments is put in a one litre graduated cylinder containing distilled water. An amount of dispersing agent is added to avoid particle flocculation. After agitation, 20 ml aliquots of mixture are taken using a pipette. The water in each aliquot is evaporated and the sediment amount is determined measuring weights of containers with and without sediments. In some cases, the presence of contaminants can have a significant influence on the accuracy of measurements. Pipette analysis is often a tedious process and requires great accuracy. Rapid Sediment Analyzer (RSA) is a method faster than the ones described above. RSA is a long cylinder containing distilled water, where an amount of sediment is introduced and allowed to settle. Particles are collected on a pan connected with a balance recording sediments weight. Some RSA have a computerized system to record weight data over time. The method is based on the principle that the falling velocity of grains in water varies with the diameter, shape and specific weight of particles. The system measures indirectly the grain size, on the base of settling and hydraulic behavior of particles. Measured velocities are compared with known settling rates and the distribution of particles is expressed as “equivalent diameters”. Some disadvantages are that RSA are calibrated with quartz spheres, whereas particles analyzed can have various shape and nature; finally, walls of the container can interfere with natural falling down of sediments.

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

39

6.3 Sediment size classification Sediments occur in a wide range of sizes, between micrometers and centimetres. The range of grain sizes of practical interest to coastal engineers is enormous, covering about seven orders of magnitude, from clay particles to large breakwater armour stone blocks. Diameter is the most important property; morphological parameters are defined by several geometric indices of rounding. Sediment size classification is usually performed with the assumption that particles are roughly circular and the grain size can be expressed as a projected cross section. Most common classifications are based on Wentworth scale (1922) and Krumbein scale (1936). Wentworth divides grains into four size classes based on particle diameter: mud (less than 0.06 mm) sand (between 0.05 and 2 mm) gravel (between 2 and 64 mm), cobble/boulder (larger than 64 mm). Statistical analysis of sediments demonstrates that size distribution of particles has a logarithmic distribution, so Krumbein (1936) proposes an alternate system called ‘phi scale’, in which grain diameter is expressed as:

ϕ = − log 2 ( D)

(2.11)

where d is the particle diameter in mm. Phi diameters are indicated by writing ϕ after the numerical value. For example, a 2.0 ϕ sand grain has a diameter of 0.25 mm. To convert from phi units to millimeters, the inverse equation is used: D = 2 −ϕ

(2.12)

The benefits of the phi unit include: 1) it has whole numbers at the limits of sediment classes in the Wentworth scale 2) it allows comparison of different size distributions because it is dimensionless. Disadvantages of this phi unit are: (a) As grain size get coarser, phi size gets smaller, which is both counterintuitive and ambiguous; (b) it is difficult to physically interpret size in phi units without considerable experience; (c) phi doesn’t represent a unit of length because it is a dimensionless unit (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995). After the sieve analysis, the weights of the sediment trapped on each sieve are tabulated and divided by the total weight of the sample in order to calculate weight percent. The cumulative frequency for each size is calculated summing up the weights for sediment trapped on that sieve and all those higher in the stack. Sediment character is described by a histogram that graphically shows the relative abundance of each size class. Poorly sorted samples can have multiple modes, whereas better-sorted samples will have a single mode (a single peak). Table 2.1 compares Wentworth and Krumbein scales. Method of moments Mathematical determination of grain size utilizes a statistical measure called the “method of moments”.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Table 2.1 - Wentworth and Krumbein scales

Gravel

Sand

Mud

Size categories

Millimetres

phi

Cobble/Boulder

>64

<-6.0

Pebble

64↔4

-6↔–2

Granule

4↔2

-2↔–1

Coarse

2↔0.5

-1↔+1

Medium

0.5↔0.25

+1↔+2

Fine

0.25↔0.06

+2↔+4

Silt

0.06↔0.002

+4↔+9

Clay

<0.002

>+9

The statistical moment is the moment of a given size class with respect to the arbitrary point (i.e., to the origin of the curve). The statistical moment of the distribution is the moment per unit frequency. It is determined by finding the moments of each size class (frequency percentage in each size class times the distance of each from the origin), adding them up, and dividing the sum by 100. The first moment is the mean calculated using the frequency (f) in percent for each size class and the midpoint (m) of each class in phi values. In the first moment the distance term (m) appears in the first power. Successively higher moments are defined by raising the distance term to progressively higher powers. In the higher moments the distance term is represented with respect to the mean, so the higher moments are moments about the mean, whereas the first moment is the moment with respect to an arbitrary point. The second moment is the square of the standard deviation. The third moment is the skewness, and the fourth moment is the kurtosis (see Table 2.2). Graphic statistics method The simplest method to calculate grain statistic distribution is based on few points from the plot representing grain size (on an arithmetic axis) versus cumulative frequency (on a probability ordinate). The resulting curve shows three important points: − − −

the 50th percentile ϕ50 (the sample median) the 16th percentile ϕ16 the 84th percentile ϕ84

Other points are the 5th and 95th percentiles (indicated as ϕ5 and ϕ95), used to calculate standard deviation. The graphic mean (Mϕ) is computed by the average

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

41

between the median (ϕ50) and the values one standard deviation unit to either side: Mϕ =

(ϕ16 + ϕ50 + ϕ84 )

(2.13)

3

Table 2.2 Formulas for calculating statistics using the method of moments. f= frequency in percent for each size class ; m= midpoint of each class in phi values;n=total number in sample. When f is expressed in percent, n=100 Mean (first moment)

∑fm 1 σ = f (m − x ) 100 ∑ 1 Sk = f (m − x ) 100σ ∑ xϕ =

Standard deviation (second moment)

1 n

2

ϕ

Skewness (third moment)

ϕ

3

ϕ

ϕ

3

ϕ

Kurtosis (fourth moment)

Kϕ =

1 100σ ϕ4

∑ f (m − x )

4

ϕ

Standard deviation can be computed using Folk (1974) formulation:

σϕ =

(ϕ84 − ϕ16 ) + (ϕ 95 − ϕ 5 ) 4

6

(2.14)

If the sample is distributed such that values for the ϕ5 and ϕ95 cannot be derived from the graph, then a less accurate determination (graphic standard deviation) can be made using:

σϕ =

(ϕ 84 − ϕ16 ) 2

(2.15)

The degree by which the distribution departs from symmetry is measured by the phi coefficient of skewness defined in Folk 1974 as:

αϕ =

(ϕ16 + ϕ 84 − 2ϕ 50 ) + (ϕ 5 + ϕ 95 − 2ϕ 50 ) 2(ϕ 84 − ϕ16 ) 2(ϕ 95 − ϕ 5 )

(2.16)

A perfectly symmetric distribution indicates that the skewness is zero. If the skewness is positive there is an excess of fine material and the curve has a tail to the right. Conversely, a negative value of skewness indicates excess of coarse material (a tail to the left). The phi coefficient of kurtosis βϕ is a measure of the peakedness of the distribution; that is, the proportion of the sediment in the middle of the distribution relative to the amount in both tails. Following Folk (1974), it is defined as:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

βϕ =

(ϕ 95 − ϕ 5 ) 2.44(ϕ 75 − ϕ 25 )

(2.17)

Frequency

Frequency

Mode Mean Mode Median

coarse

Particle size A

fine

Median Mean

coarse

Normal

Particle size B

fine

Positively skewed

Frequency

Mode Median Mean

Particle size

coarse C

fine

Negatively skewed

Figure 2.5 – Frequency curves illustrating the mode, median and the difference between normal frequency curves and asymmetrical (skewed) curves.

7 Case study Find the statistics of the sediment size distribution shown in Figure 2.6. Given the needed phi values are: ϕ05 = 2.224, ϕ16 = 2.498, ϕ25 = 2.727, ϕ50 = 3.506, ϕ75 = 4.211, ϕ84 = 4.506, ϕ95 = 4.718. In phi units, the median grain size is given as:

ϕ50 = MDϕ = 3.506 ϕ From equation (2.12) the median grain size in millimeters is found as: MD = 2-3.506 = 0.088mm The mean grain size is found in phi units as (equation 2.13): Mϕ = (ϕ16+ϕ50+ϕ84) / 3 =3.503ϕ From equation (2.12) this is converted to millimeters as: D = 2-3.503 = 0.088 mm

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

From equation (2.14) the standard deviation is found as:

σϕ = (4.506-2.498)/4 + (4.718-2.224)/6 = 0.918ϕ

Figure 2.6 - Sediment size distribution. From equation (2.16) the coefficient of skewness is found as:

αϕ =

2.498 + 4.506 − 2(3.506) 2.224 + 4.718 − 2(3.506) + = −0.016 2( 4.506 − 2.498) 2( 4.718 − 2.224)

From equation (2.17) the coefficient of kurtosis is found as:

βϕ =

4.718 − 2.224 = 0.689 2.44 ( 4.211 − 2.727 )

43

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

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Chapter 3 Linear wave analysis 1 Introduction to linear wave theory The problem of describing wave motion is generally approached through the hypotheses of a frictionless and incompressible fluid in irrotational motion. The equation of mass and momentum conservation written for this ideal fluid must be closed by appropriate boundary conditions. Two serious difficulties arise in the attempt to obtain an exact solution. The fact is that the free surface boundary conditions are nonlinear, and the second is that these conditions are prescribed at the free surface z=η which is initially unknown. The simplest and most fundamental approach is to seek a linear solution of the problem by taking the wave height H to be very much smaller than both the wave length L and the still water depth d: that is H<
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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

H/L H = d/L d

(3.1)

which is termed the relative wave height. Like the wave steepness, large values of the relative wave height indicate that the small-amplitude assumption may not be valid. A fourth dimensionless parameter often used to assess the relevance of various wave theories is the Ursell number, given by: 2

2 ⎛L⎞ H L H UR = ⎜ ⎟ = 3 d ⎝d ⎠ d

(3.2)

The value of the Ursell number is often used to select a wave theory to describe a wave with given L and H (or T and H) in a given water depth d. High values of UR indicate large, finite amplitude, long waves in shallow water that may necessitate the use of nonlinear wave theory. In fact, the Stokes expansion method is valid only if d/L >1/8 or kd>0.78 or UR<79 which restrict the applicability of Stokes theories only to deep waters. In shallow waters, the cnoidal wave theory is often used, for which reference is made to Sarpkaya & Isaacson, 1981. Le Méhauté (1976) presented a convenient plot showing the approximate limits of validity of various wave theories, as shown in Figure 3.1.

Figure 3.1 - The applicability of various wave theories. after Le Méhauté (Le Méhauté B., 1976).

LINEAR WAVE ANALYSIS

1.1

47

Governing equations

In developing the linear wave theory the following assumptions are made: • • • • • • • •

Fluid homogeneous; Density ρ constant; Surface tension neglected, so that pressure at free surface can be considered uniform and constant; Coriolis effect neglected ; Inviscid fluid (lack of internal friction); Irrotational flow (shearing forces negligible); Horizontal, fixed, impermeable bed, which provides the boundary condition that vertical velocity at the bed is null; Long-crested Waves (two-dimensional motion).

With reference to figure 3.2, we consider a right-hand Cartesian system of coordinates axes (x,y,z) with its origin on the still water level. Let x-axis be oriented in the positive direction of wave propagation, positive z-axis upward, yaxis orthogonal to x and z. C Direction of Propagation L

Crest

η(θ) 0

a

H

π/2

3π/2

π

SWL

-a

2π

θ=kx z=0

Trough w

u

Bottom z=-d

Figure 3.2 - Sinusoidal progressive wave With the assumptions of irrotational motion and incompressible fluid, a velocity potential exists which should satisfy the continuity equation: div v = 0

(3.3)

div(∇φ ) = 0

(3.4)

or: the divergence of a gradient leads to the Laplace equation:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

∂ 2Φ ∂ 2Φ ∂ 2Φ + + =0 ∂x 2 ∂y 2 ∂z 2

(3.5)

which states that the Laplacian of potential function Φ must be null. ∇ 2Φ = 0

(3.6)

∂ 2Φ ∂ 2Φ + =0 ∂x 2 ∂z 2

(3.7)

In a bidimensional case:

Equation (3.7) is the continuity equation for an ideal fluid (unviscid and uncompressible) in irrotational motion. The governing equations of motion in the fluid for the x-z plane are the Euler equations: ∂u ∂u ∂u 1 ∂p +u +w =− ∂t ∂x ∂z ρ ∂x ∂w ∂w ∂w 1 ∂p +u +w =− −g ∂t ∂x ∂z ρ ∂z

Substituting in the two-dimensional irrotationality condition ( ∇ × v = 0 ) ∂u ∂w = ∂z ∂x the equations can be rewritten as:

(3.8) (3.9)

(3.10)

∂u ∂ (u 2 / 2) ∂ ( w 2 / 2) 1 ∂p + + =− ∂t ∂x ∂x ρ ∂x

(3.11)

∂w ∂ (u 2 / 2) ∂ ( w 2 / 2) 1 ∂p + + =− −g ∂t ∂z ∂z ρ ∂z

(3.12)

Now, since a velocity potential exists for the fluid, we have u=−

∂φ ∂φ , w=− ∂x ∂z

(3.13)

Therefore, if we substitute these definitions into eqs. (3.9) and (3.12), we get ∂ ⎡ ∂φ 1 2 p⎤ + (u + w2 ) + ⎥ = 0 ⎢− ∂x ⎣ ∂t 2 ρ⎦

(3.14)

p⎤ ∂ ⎡ ∂φ 1 2 + (u + w 2 ) + ⎥ = − g − ⎢ ρ⎦ ∂z ⎣ ∂t 2

(3.15)

where it has been assumed that the density is uniform throughout the fluid. Integrating equation (3.14) with respect to x yields: p ∂Φ 1 2 (3.16) − + − ( u + w 2 ) = F1 ( z ,t ) ρ ∂t 2

LINEAR WAVE ANALYSIS

49

where the constant of integration varies only with z and time t. Integrating zequation yields: p ∂Φ 1 2 (3.17) − gz − + − (u + w 2 ) = F2 ( x, t ) ρ ∂t 2 Subtracting equation (3.16) from equation (3.17) gives gz = F2 ( x, t ) − F1 ( z , t )

(3.18)

Since g is not a function of x, it follows that F2 does not depend on x and is a function only of time and the same for F1=F2 − gz. Thus equations (3.16) and (3.17) can be reduced to the single equation − gz −

p

ρ

+

∂Φ 1 2 − (u + w 2 ) = F2 (t ) ∂t 2

(3.19)

The latter represents the “generalized Bernoulli theorem of Hydraulics” ∂Φ which must hold throughout the fluid. For steady flows, = 0 and ∂t F2=constant and equation assumes the expression: gz +

p

ρ

+

1 2 (u + w 2 ) = const 2

(3.20)

Which is the usual form of the steady-state Bernoulli equation.

1.2

Boundary conditions

In the hypothesis of irrotational and ideal fluid, the differential equations governing wave motion are Laplace equation and (Bernoulli) motion equation − gz −

p

ρ

+

∂Φ 1 2 − ( u + w 2 ) = const Motion equation (3.21) ∂t 2

∂2Φ ∂ x2

+

∂2Φ ∂ z2

=0

Continuity equation (3.22)

which must be valid throughout the fluid, that is in the region –d
50

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

(x,t) must remain the same during the motion and this implies that fluid parcel speed in the normal direction to free surface must match the velocity of free surface in the same direction: d [z − η ( x, t )] = 0 on the surface z = η dt

(3.23)

where η is the displacement of free surface about the horizontal plane z=0.

∂η dz ∂η − −u =0 ∂x dt ∂t

(3.24)

dz ∂Φ =w=− dt ∂z

(3.25)

Considering that

and that u=−

∂Φ ∂x

(3.26)

equation (3.25) can be written −

∂Φ ∂η ∂Φ ∂η − + ⋅ =0 ∂z ∂t ∂ìx ∂x

∂Φ ∂η ∂Φ ∂η + − ⋅ =0 ∂z ∂t ∂x ∂x

(3.27) (3.28)

Equation (3.28) kinematic boundary condition on the surface. On the bottom the non-penetration condition imposes that vertical component of velocity be null z = − d . The kinematic condition on the bottom becomes: w=−

∂Φ =0 ∂z

(3.29)

A further boundary condition is obtained by considering that fixed surfaces, such as the air-sea interface, cannot support variation of pressure across the interface and hence must respond in order to maintain the pressure as uniform. Therefore dynamic boundary condition on the surface requires that pressure be invariant along the wave form (i.e. z = η ) and be equal to atmospheric pressure. So, Bernoulli equation with pressure pη=constant=pa (usually taken as 0) is to be applied to free surface z=η(x,t)

∂Φ 1 ⎡⎛ ∂Φ ⎞ ⎛ ∂Φ ⎞ + ⎢⎜ ⎟ ⎟ +⎜ ∂t 2 ⎢⎣⎝ ∂x ⎠ ⎝ ∂z ⎠ 2

gη −

2

⎤ ⎥ = constant ⎥⎦

(3.30)

LINEAR WAVE ANALYSIS

51

1.3 Linearized boundary conditions The difficulty in obtaining a general analytical solution of continuity equation and motion equation arises from the fact that boundary conditions on free surface are non-linear, and that free surface is itself unknown in the problem. So a linear solution to the problem is looked for by hypothesizing that wave height H is much smaller than wave length L and than depth d:

H << L

H << d

(3.31)

So the following assumptions can be made:

• • •

free surface displacement η is very small, so that O(η)=0 wave motion is slow (i.e. u small) and hence the square of particle velocity is negligible: O(u2)<
A further assumption is made by treating Bernoulli constant as to be included into the velocity potential Φ(x,z,t). In this way non-linear terms, which involve products of terms with the order of H, are negligible with respect to linear ones having the same order as H; furthermore boundary conditions on the surface are applied in correspondence of still level (z=0). With these assumptions, the kinematic boundary condition on the surface (3.28) is linearized as:

∂η ∂Φ ∂η ⎡ ∂Φ ⎤ + =0→ = −⎢ ⎥ ∂z ∂t ∂t ⎣ ∂z ⎦ z = 0

(3.32)

Dynamic boundary condition on the surface becomes: gη −

1 ⎡ ∂Φ ⎤ ∂Φ = cost. = 0 → η = ⎢ ⎥ ∂t g ⎣ ∂t ⎦ z = 0

(3.33)

By deriving (3.33) with respect to t and substituting in (3.30), we have: 1 ∂ 2 Φ ∂Φ = g ∂t 2 ∂z

(3.34)

2 Results of the linear theory Integrating equation (3.34) through separation of variables, the solution yields potential velocity Φ(x, z, t), given by: Φ(x, z, t) =

ag cosh(d + z) ⋅ ⋅ cos(kx − σt) σ cosh(kd)

(3.35)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

2.1 Wave profile, length and celerity In linear theory, solution of equations are harmonic waves and vertical displacement η describing wave profile can be expressed through a regular progressive wave.

η=

1 g

⎡ ∂Φ ⎤ ⎢ ∂t ⎥ = a sin(kx − σt ) ⎣ ⎦ z =0

(3.36)

which is periodic in time and space.

η = a sin(kx − σt ) = a sin k ( x − ct )

(3.37)

where a=H/2 is the wave amplitude, k=2π/L is the wave number, σ=2π/T is the angular frequency, H is the wave height, L the wave length, T the wave period and C=σ/k=L/T is the wave celerity. The latter represents the speed at which a wave form propagates. ∂ 2Φ ∂Φ and we obtain the From equation (3.34) by explicating the terms ∂t 2 ∂z dispersion relation for the general case: σ 2 = gk ⋅ tgh(kd)

(3.38)

which can be otherwise written as: L=

gT 2 ⎛ 2 πd ⎞ ⋅ tgh⎜ ⎟ 2π ⎝ L ⎠

(3.39)

or through wave celerity

C=

gT ⎛ 2 πd ⎞ tgh⎜ ⎟ 2π ⎝ L ⎠

(3.40)

Equations (3.38), (3.39) and (3.40) state the relationship between wave lengths and wave periods; in particular equation (3.40) evidently shows that waves with different period travel with different speed. Equation (3.39) is noticeably implicit, and is to be solved numerically. In the resolution of practical problems it is usual to implement tabulated values of d/L0 and d/L in order to obtain corrisponding values of tanh (kd) and then to simplify the solution of equation (3.40) (see table 3.1). It may be convenient to distinguish the type of wave on the basis of the ratio between d and L called “relative depth” because the dispersion relation specializes according to the range which the ratio d/L lies in. d 1 > L 2

(deep water)

(3.41)

LINEAR WAVE ANALYSIS

53

⎛ 2 πd ⎞ tgh⎜ ⎟≅1 ⎝ L ⎠

(3.42)

gT 2 2π

(3.43)

gL0 L gT = 0 = 2π 2π T

(3.44)

L = L0 = C0 =

Because of the behavior of hyperbolic functions if relative depth d/L is less than 1/20, then the depth is small compared with wavelength and tgh(kd)≈kd. Equation (3.39) specializes into “shallow water” form (or “long wave”), d 1 < (shallow water) L 20

⎛ 2 πd ⎞ ⎛ 2 πd ⎞ tgh⎜ ⎟≅⎜ ⎟ ⎝ L ⎠ ⎝ L ⎠ L=

gT 2 2 πd ⋅ 2π L

L =T

gd

C = gd

(3.45) (3.46)

(3.47) (3.48) (3.49)

For a wave traveling in shallow water, wave celerity depends on the depth alone. If d/L is greater than 1/2 the waves are “short” and dispersion relation specializes into “deep water” form, because the hyperbolic fuction tanh(kd) approaches unity (see figure 3.3).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 3.3 – Relative depth and asymptotes to hyperbolic functions The case of 1/20
2.2 Group celerity When a group of waves propagates through a medium, the speed of the wave train (or of the wave group) is generally not the same as the speed with which individual waves within the group travel. Let us consider the superposition of two regular waves moving in the same direction and for simplicity let us suppose they have the same amplitude. Since the linear theory is applied, superposition of solutions is admissible and the two components can be summed.

η = η1 + η 2 = a cos(k1 x − σ 1t ) + a cos(k 2 x − σ 2 t )

(3.50)

LINEAR WAVE ANALYSIS

55

where, in the hypothesis of small ∆σ and ∆k in order that ki and σi satisfy dispersion relation k1 = k −

∆k 2

σ1 = σ −

∆σ 2

(3.51)

k2 = k +

∆k 2

σ2 =σ +

∆σ 2

(3.52)

In terms of wavelengths and periods, we can write

η=

⎛ 2πx 2πt ⎞ H ⎛ 2πx 2πt ⎞ H ⎟⎟ + cos⎜⎜ ⎟ − − cos⎜⎜ 2 T1 ⎠ 2 T2 ⎟⎠ ⎝ L1 ⎝ L2

(3.53)

Because wave lengths L1 and L2 are supposed different, for some values of x at a given time t, the two harmonics will be in phase enhancing wave amplitude to 2H, for some other x they will be out of phase and will “destroy” each other. Using trigonometric identities, the two sinusoidal waves may be combined: ⎡1 ⎣2

⎤

⎡1 ⎣2

⎤

η = H cos ⎢ [(k1 + k 2 ) x − (σ 1 + σ 2 )t ]⎥ cos ⎢ [(k1 − k 2 ) x − (σ 1 − σ 2 )t ]⎥ = ⎦

⎦

(3.54) ⎡1 ⎛ ∆σ ⎞ ⎤ = H cos(kx − σt ) cos ⎢ ∆k ⎜ x − t ⎟⎥ ∆k ⎠⎦ ⎣2 ⎝

The resulting equation represents a wave propagating with velocity C=σ/k which is modulated by a wave traveling with speed ∆σ /∆k. Figure 3.4 shows the characteristics of a group formed by sinusoidal waves. The last is an enveloping wave and the speed is called “group velocity”. Cg =

∆σ ∆k

(3.55)

The group velocity constitutes the speed at which wave energy propagates. In fact, by letting ∆k to tend to 0, we obtain the group velocity of a wave train of infinite wave length Lg=2π / ∆k. C g = lim

∆k → 0

∆σ dσ = dk ∆k

(3.56)

Using the dispersion relation, the derivative can be evaluated: σ 2 = gk tanh(kd) 2σ

dσ = g tanh( kd ) + gkd sech 2 ( kd ) dk

(3.57) (3.58)

Table 3.1 - Wave table (Kamphuis, 2000) d/L 0,0000 0,0179 0,0253 0,0311 0,0360 0,0403 0,0496 0,0576 0,0648 0,0713 0,0775 0,0833 0,0888 0,0942 0,0993 0,104 0,109 0,114 0,119 0,123 0,128 0,132 0,137 0,141 0,150 0,158 0,167 0,175 0,183 0,192 0,200 0,208 0,225 0,234

kd 0,000 0,112 0,159 0,195 0,226 0,253 0,312 0,362 0,407 0,448 0,487 0,523 0,558 0,592 0,624 0,655 0,686 0,716 0,745 0,774 0,803 0,831 0,858 0,886 0,940 0,994 1,05 1,10 1,15 1,20 1,26 1,31 1,41 1,47

sinh(kd) 0,000 0,112 0,160 0,196 0,228 0,256 0,317 0,370 0,418 0,463 0,506 0,547 0,587 0,627 0,665 0,703 0,741 0,779 0,816 0,854 0,892 0,930 0,967 1,007 1,085 1,166 1,254 1,336 1,421 1,509 1,621 1,718 1,926 2,060

cosh(kd) 1,00 1,01 1,01 1,02 1,03 1,03 1,05 1,07 1,08 1,10 1,12 1,14 1,16 1,18 1,20 1,22 1,24 1,27 1,29 1,31 1,34 1,37 1,39 1,42 1,48 1,54 1,60 1,67 1,74 1,81 1,90 1,99 2,17 2,29

d/L0 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,30 0,31 0,32 0,33 0,34 0,35 0,36 0,37 0,38 0,39 0,40 0,41 0,42 0,43 0,44 0,45 0,46 0,47 0,48 0,49 0,50 0,75 1,00

tanh(kd) 0,909 0,917 0,926 0,933 0,940 0,946 0,952 0,956 0,961 0,965 0,969 0,972 0,975 0,978 0,980 0,983 0,985 0,986 0,988 0,989 0,990 0,992 0,992 0,993 0,994 0,995 0,995 0,996 0,996 1,000 1,000

d/L 0,242 0,251 0,259 0,268 0,277 0,285 0,294 0,303 0,312 0,321 0,330 0,339 0,349 0,358 0,367 0,377 0,386 0,395 0,405 0,415 0,424 0,434 0,443 0,453 0,463 0,472 0,482 0,492 0,502 0,746 0,981

kd 1,52 1,57 1,63 1,68 1,74 1,79 1,85 1,90 1,96 2,02 2,08 2,13 2,19 2,25 2,31 2,37 2,43 2,48 2,54 2,60 2,66 2,73 2,79 2,85 2,91 2,97 3,03 3,09 3,15 4,69 6,16

sinh(kd) 2,18 2,30 2,45 2,59 2,76 2,91 3,10 3,27 3,48 3,70 3,94 4,15 4,41 4,69 4,99 5,30 5,64 5,93 6,30 6,69 7,11 7,63 8,11 8,61 9,15 9,72 10,3 11,0 11,6 54,4 236,7

cosh(kd) 2,40 2,51 2,65 2,78 2,94 3,08 3,26 3,42 3,62 3,84 4,06 4,27 4,52 4,80 5,09 5,40 5,72 6,01 6,38 6,77 7,18 7,70 8,17 8,67 9,21 9,77 10,4 11,0 11,7 54,4 236,7

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

tanh(kd) 0,000 0,112 0,158 0,193 0,222 0,248 0,302 0,347 0,386 0,420 0,452 0,480 0,506 0,531 0,554 0,575 0,595 0,614 0,632 0,649 0,666 0,681 0,695 0,709 0,735 0,759 0,782 0,800 0,818 0,834 0,851 0,864 0,887 0,900

56

d/L0 0,000 0,002 0,004 0,006 0,008 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0,045 0,050 0,055 0,060 0,065 0,070 0,075 0,080 0,085 0,090 0,095 0,10 0,11 0,12 0,13 0,14 0,15 0,16 0,17 0,18 0,20 0,21

LINEAR WAVE ANALYSIS

Cg =

dσ g tanh( kd ) + gkd sech 2 ( kd )σ = = dk 2gktanh ( kd )

57

(3.59)

2kd ⎞ C⎛ ⎟ = nC = ⎜⎜1 + 2⎝ sinh (2kd ) ⎟⎠

where the factor n=

1⎛ 2kd ⎞ ⎜⎜1 + ⎟ specializes for deep and shallow water as 2 ⎝ sinh (2kd ) ⎟⎠

follows:

•

•

Deep water:

n=1/2

and

n=1

and

Cg =

C 2

(3.60)

Shallow water: Cg = C

(3.61)

Cg = energy propagation speed. The energy travels at half the speed of the wave profile; in shallow water energy and wave profile move at the same velocity. 2.3

Velocity components

By the definition of the velocity potential, from equation (3.41) velocity components u,w can be obtained by deriving potential function Φ(x, z, t) with respect to x and z, respectively: u=−

w=

∂ Φ agk coshk ( d + z ) = ⋅ sin (kx − σt ) ∂x cosh ( kd ) σ

∂ Φ agk sinhk (d + z ) = ⋅ sin(kx − σt ) ∂z cosh(kd ) σ

(3.62)

(3.63)

The two velocity components u and w, both harmonic in space and time, are π/2 out of phase, and the comparison with equation (3.36) shows that horizontal component u is in phase with the profile, showing its maximum absolute value in correspondence for the crest and the trough, that is when: kx − σt =

π 2

+ nπ (n = 0,1,.........)

(3.64)

and is null in the nodes. kx − σt = nπ (n = 0,1,.........)

(3.65)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

η Cg 2α

η envelope

η=η1+ η2

C α 2H

-α

-2α

Lg

Figure 3.4 - Characteristic of a group formed by sinusoidal waves Vice versa the absolute value of vertical component w of velocity is maximum in the nodes and shows null value in correspondence of the crest. The horizontal component in crest is: uc =

agk coshk ⋅ (d + z) ⋅ σ cosh(kd)

(3.66)

Hence for z=0 and for z=–d (that is on the surface and on the bottom), the horizontal component is respectively: u c = u max =

agk

uc = u f =

σ

agk 1 ⋅ σ cosh(kd)

(3.67)

For what concerns w behavior in the node, we have: wn (z) = −

agk sinh k ⋅ (d + z) ⋅ σ cosh(kd)

(3.68)

Therefore for z=0 and z=–d, we have respectively: wn = wmax = −

agk

σ

tgh(kd )

wn = 0

(3.69)

The latter equation satisfies the cinematic boundary condition at the bottom.

LINEAR WAVE ANALYSIS

59

2.4 Particle displacements With reference to Figure 3.5 , let ζ be the horizontal displacement of surface parcels, and ξ be the vertical displacement. Velocity components are the time derivatives of the displacement components: dζ =u dt

dξ =w dt

(3.70)

Integrating with respect to time, we have:

ζ = ∫ udt = ∫

agk coshk (d + z ) ⋅ sin(kx − σt )dt = coshkd σ (3.71) agk coshk (d + z ) = 2 ⋅ ⋅ cos(kx − σt ) coshkd σ

ξ = ∫ wdt = ∫ −

agk sinh k(d + z ) cos(kx − σt )dt = ⋅ coshkd σ (3.72) agk sinhk(d + z) = 2 ⋅ ⋅ sin(kx − σt ) coshkd σ

The equations (3.71) and (3.72) can be simplified by using the dispersion relation (3.38). ζ=a

coshk (d + z ) ⋅ cos(kx-σx) coshkd

(3.73)

ξ=a

sinhk (d + z ) ⋅ sin (kx − σt ) coshkd

(3.74)

It is possible to demonstrate that particle paths are in general elliptical in shape. In fact equations (3.73) and (3.74) can be jointed (using the fundamental trigonometric relation): ζ

2

A2

+

ξ2 B2

=1

(3.75)

Being: coshk (d + z ) sinhk (d + z ) (3.76) B=a sinhkd sinhkd Equation (3.75) represents an ellipse with a major horizontal semi-axis A and a minor vertical semi-axis B. Equations (3.76) specialize depending on depth and trajectories assumes specific forms which can be easily determined by examining the values of A and B. A=a

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

u As

w

Bs

Ab

Figure 3.5 - Water particle orbits and relative displacement. shallow water: sinh k(d + z) ≅ sinh kd ≅ kd ; cosh k(d + z) ≅ cosh k d ≅ 1 A=

a kd

B=a

d+ z d

(3.77) (3.78)

Therefore the amplitude of horizontal semi-axis does not depend on vertical coordinate, whereas the amplitude of vertical semi-axis decreases with increasing depth until it becomes null on the bottom (z=–d), reducing the ellipse to a segment. sinhkd ≅ coshkd ≅

A = a⋅

k (d + z) e 2 kd e 2

kd e2

=

;

kz a⋅ e 2

sinhk (h + z ) ≅ coshk (d + z ) ≅

B = a⋅

k (d + z ) e 2 kd e2

kd e2

(3.79)

kz

= a⋅ e 2

(3.80)

Semi-axes being equal, the elliptical paths become circular, with the circumference radius exponentially decreasing with depth. For example in z=– L/2 we have A=B=0.042a. Below this depth, as already mentioned, wave perturbation is negligible. The described behavior of particle paths is depicted in Figure 3.6.

LINEAR WAVE ANALYSIS

d/L < 1/20

1/20 < d/L < 1/2

d/L > 1/2

Shallow water

Intermediate water

Deep water

61

Figure 3.6 - Water particle orbits for shallow, intermediate and deep water 2.5

Wave pressure

The generalized Bernoulli equation for an ideal fluid is: p

ρ

+ gz +

u 2 + w 2 ∂Φ − = f (t ) 2 ∂t

(3.81)

where the quadratic term is neglected (small amplitude waves). By matching in the same instant this relation to any depth z and to the surface z=η where the relative pressure is null, we have: ⎛p ∂Φ ⎞ ⎡ ∂Φ ⎤ ⎜⎜ + gz − ⎟⎟ = gη − ⎢ ⎥ ∂t ⎠ z ⎣ ∂t ⎦ z =η ≅ 0 ⎝ρ

(3.82)

The wave profile is obtained by equation:

η=

1 ⎡ ∂Φ ⎤ g ⎢⎣ ∂t ⎥⎦ z = 0

(3.83)

and hence: p

ρ Being:

⎡ ∂Φ ⎤ = − gz + ⎢ ⎥ ⎣ ∂t ⎦ z

(3.84)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

cosh k(d + z ) ⎡ ∂Φ ⎤ ⎢ ∂t ⎥ = a g ⋅ cosh kd sinh(kx− σt) ⎣ ⎦z

(3.85)

We have: p = − ρgz + ρga⋅

cosh k(d + z) sinh(kx − σt) = − ρgz + ρgη⋅ k p (z) cosh kd

(3.86)

The quantity k p (z) =

cosh k(d + z) cosh kd

(3.87)

is called the response factor and is always less than unit below the mean water level. In equation (3.86) the first term represents hydrostatic pressure, which would exist in absence of wave motion, the second term represents dynamic pressure due to wave motion. The dynamic pressure results from two contributions: hydrostatic pressure surcharge associated with free surface displacement and pressure (surcharge or discharge) produced by vertical accelerations accompanying the orbital motion. If the response factor Kp(z) is unitary, the pressure contribution from the free surface displacement is purely hydrostatic; indeed the response factor is equal to unity from the surface to the mean watel level (z=η≅0). Vertical acceleration is 180° out of phase with free surface displacement and its contribution modifies pure hydrostatic pressure by enhancing or decreasing its magnitude depending on wave phase. The behavior of pressure in correspondence of crest and trough is shown in figure 3.7. +η -η

η(1-Kp)

-η(1-Kp) -z

Actual pressure head

d

kx-σt

Hydrostatic p/γ=η-z

Hydrostatic p/γ=η-z

Figure 3.7 - Pressure diagram within the wave profile.

LINEAR WAVE ANALYSIS

63

At the crest, pressure is hydrostatic from z=η to z=0, below z=0 pressure is higher than hydrostatic pressure referred to mean water level, but lower than hydrostatic pressure relative to the crest level. At the trough, pressure is higher than hydrostatic pressure relative to the trough level, but lower than hydrostatic pressure corresponding to mean water level. Anyway, with increasing depth, pressure tends to the hydrostatic value corresponding to the mean water level.

3 Case study A calibrated pressure gauge located on the sea bed at a depth of z=13m, measures a value of maximum pressure ∆p=0.146 Kg/cm2 and a frequency f=0.11 Hz. Evaluate wave height and length and the relative wave celerity. Calculate the values of horizontal and vertical components of velocity for phase angles θ=kx-σt=90° e θ=180° at the surface, at half depth and on the bottom. Calculate also the value of semi-axes of the particle paths at the crest, at the surface, at half depth and on the sea bed. Given data: ∆p=0.146 Kg/cm2=1460 Kg/m2 z=13m f=0.11 Hz θ=kx-σt=90° θ=180°

•

Wave length, height and celerity

Consider the wave pressure equation:

p = − ρgz + ρgηK p ( z ) where Kp(z) is the pressure response factor given by Kp( z ) =

cosh(d + z) cosh(kd)

Because z=–d, the pressure equation can be written as: p = − ρg d + ρgη⋅

1 cosh(kd)

If the pressure gauge is calibrated, it gives only the dynamic pressure alone, and therefore we have: H ∆pmax ηmax = a = = ⋅ cosh(kd) 2 ρg Because k=2π/L the knowledge of wave length L is required for the calculus of wave number k and the determination of wave height. The value of L is determined from the period T and the depth d, with reference to the dispersion relation, which is implicit.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

f=0.11 Hz

T=1/f = 1/ 0.11 s=9.09 s

Deepwater wave length may be calculated from wave period: L0 =

gT 2 = 1.56 ⋅ T 2 = 1.56 ⋅ 9.09 2 m = 129 m 2π

Entering in the table 3.1 with the ratio

d 13 = = 0.1 , the value of tanh(kd) is L0 129

given as 0.709. From the formula d d = tanh(k d) L0 L

wave length corresponding to the depth of 13m can be determined as:

L = L0 tanh(kd) = 129 ⋅ 0.709 m = 91,46 m Now the wave number is given by: k=2π/L =2⋅ 3.14/91.46 m-1 =0.06869 m-1

•

Maximum elevation ηmax

∆p=0.146 Kg/cm2=1460 Kg/m2 ρg=γ=1020 Kg/m3

ρ=104 Kg s2/m4

g=9.8 m/s2

d=13m cosh(kd)= 1.4239 1460 ⋅ 1.4259m = 2.04m 1020 For the small amplitude waves, wave height is obtained as: H = 2η max = 2 ⋅ 2.04 = 4.08m Wave celerity is given by: L 91.46 = 10.1 m / s C= = 9.09 T

η max = a =

•

Horizontal and vertical components of velocity

Consider the equations (3.62) and (3.63) for velocity components: u=

agk coshk ( z + d ) sin (kx − σt ) σ cosh (kd )

w=−

agk sinhk ( z + d ) cos(kx − σt ) σ cosh (kd )

LINEAR WAVE ANALYSIS

65

Horizontal velocity u θ=kx-σt=90° and θ=180° z=0 , z=-d/2 , z=-d k=2π/L=0.06869 m-1 σ=2π/T=0.691 d=13 m When θ=180° (i.e. in the node) horizontal component is null. For θ=90°, that is on the crest, we have: agk

σ z=0 z=-d/2 z=-d

=

2.04 ⋅ 9.81 ⋅ 0.0686 m / s = 1.98m / s 0.691

cosh k(d + z) =1 cosh(kd) coshk(d + z) = 0.774 cosh(kd)

⇒ uc = ⇒ uc =

coshk(d + z ) 1 = = 0.703 cosh(kd) cosh(kd)

agk = 1.98m / s σ

agk ⋅ 0.774 = 1.53m / s σ

⇒ uc =

agk ⋅ 0.703 = 1.39m / s σ

Vertical velocity component w: θ=kx-σt=90° and θ=180° z=0 , z=-d/2 , z=-d k=2π/L=0.06869m-1 σ=2π/T=0.691 d=13 m When θ=90° (on the crest and in the trough) vertical component is null. In the nodes, θ=180° we have:

z=0 z=-d/2 z=-d

sinhk(d + z) = tanh(kd) = 0.711 cosh(kd) sinhk(d + z) = 0.323 cosh(kd) sinhk(d + z) =0 cosh(kd)

⇒ wn = −1.41m / s

⇒ wn = −0.64m / s ⇒ wn = 0

The last result is in agreement with the kinematic boundary condition on the bottom (vertical velocity component null).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

•

Semi-axes of particle paths

Major and minor semi-axes of particle paths are given respectively by: cosh k(d + z) sinh kd sinh k(d + z) B=a sinh k d A=a

Let us consider separately the behavior of the terms at the three depths: 1 ⎧ coshk(d + z) ⎪ sinh(kd) = tanh(kd) = 1.406 ⇒ A = a× 1.406 = 2.04 × 1.406 = 2.87 m z=0 ⎪⎨ ⎪ sinhk(d + z) = 1 ⇒ B = a = 2.04m ⎪⎩ sinh(kd)

z=-d/2

z=-d

⎧ cosh k(d + z) ⎪ sinh(kd) = 1.237 ⎪ ⎨ ⎪ sinh k(d + z) = 0.454 ⎪⎩ sinh(kd)

⇒ A = a× 1.237 = 2.52m ⇒ B = a× 0.454 = 0.93m

1 ⎧ cosh k(d + z) ⎪ sinh(kd) = sinh(kd) = 0.989 ⎪ ⎨ ⎪ sinh k(d + z) = 0 ⇒ B = 0 ⎪⎩ sinh(kd)

⇒ A = a× 0.989 = 2.02m

Horizontal semiaxis decreases with depth much more slowly than the vertical semiaxis, which becomes null for z=-d (on the bottom).

Chapter 4 Sea level variability 1 Introduction The mean sea level is referred to as the average level on a time interval of several minutes, in order to exclude both high frequencies, due to gravity surface waves, and sea level variations, which period is much longer than those of oceanographic interest. In fact coastal water levels are influenced by a variety of astronomical, meteo-oceanographic and tectonic factors, which often interact in a complex way to increase water levels significantly above normal tide level. Storms, which develop low atmospheric pressure, strong onshore winds and large waves, are the most common cause of abnormally high water levels. These increases of water levels are of concern because they intensify beach erosion and increase damage to coastal structures because they allow larger waves to cross offshore bars and break closer to the beach. Sea level fluctuations are classified according to time scales in: • astronomical tides (caused by the attraction of sun and moon on water masses); • long waves (tsunami and seiches); • wave setup (due to wave breaking); • storm surge (mean sea-level variations due to atmospheric causes). These categories of sea level fluctuations show periods ranging from a few minutes to some days.

2

Astronomical tide

The sum of forces caused by gravitational interaction among the Earth, Moon and Sun generates a periodic variation of big water masses. The rising and depression of water masses due to these forces is called astronomical tide, whereas the consequent horizontal motion is called tidal current. The most important tidal component is caused by Moon attraction and is called M2, with maximum values 12h and 25min intervals; consequently

68

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

maximum and minimum fluctuations of mean-sea-level are delayed 50min per day, in perfect agreement with the Moon day which lasts 24h and 50 min. The second component of the tide, called S2, is caused by Sun attraction, with a period of 12h, which in turn enhances or damps the semi-diurnal effects of the Moon. When the two attracting forces caused by Moon and Sun are in conjunction or in opposition of phase (full-Moon or new-Moon) the tide at a site shows the maximum level and is called spring tide; when the two forces are onequarter period out of phase (Moon in first or third half) the tide assumes the minimum value and is called neap tide. The tide fluctuations depend on the relative positions of Earth, Moon and Sun. The elements contributing to tidal forces are almost 100 on long time scales, 160 diurnal, 115 semi-diurnal and 14 one-third diurnal. But in practice only few components are accounted for and they are listed in Table 4.1. Examples of diurnal, semidiurnal and mixed tides are given in Figure 4.1. Table 4.1 - Tidal constituents and arguments (U.S. Army Corps of Engineers Coastal Engineering Research Center, 2002). TIDAL CONSTITUENTS AND ARGUMENTS Symbol M2 S2 N2 K1 M4 O1 M6 (MK)3 S4 (MN)4 ν2 S6 µ2 (2N)2 (OO)1 λ2 S1 M1 J1

Speed (deg/hr) 28.984 30.000 28.439 15.041 57.968 13.943 86.952 44.025 60.000 57.423 28.512 90.000 27.968 27.895 16.139 29.455 15.000 14.496 15.585

Period (hr) 12.421 12.000 12.659 23.935 6.2103 25.819 4.140 8.177 6.000 6.269 12.626 4.000 12.872 12.906 22.306 12.222 24.000 24.834 23.099

Symbol Mm Ssa Sa Msf Mf ρ1 Q1 T2 R2 (2Q)1 P1 (2SM)2 M3 L2 (2MK)3 K2 M8 (MS)4

Speed (deg/hr) 0.544 0.082 0.041 1.015 1.098 13.471 13.398 29.958 30.041 12.854 14.958 31.015 43.476 29.528 42.927 30.082 115.936 58.984

Period (hr) 661.765 4390.244 8780.488 354.680 327.869 26.724 26.870 12.017 11.984 28.007 24.067 11.607 8.280 12.192 8.386 11.967 3.105 6.103

To classify different sites of the Earth according to the tidal type, the following ratio is used: F = (K1 + O1)/(M2 + S2)

(4.1)

SEA LEVEL VARIABILITY

69

if: (1) F = 0÷0.25 gives a semi-diurnal tide: two daily maxima and two minima of almost the same height; ‘spring tide’ = 2(M2 + S2). (2) F =0.25 ÷ 1.5 gives a mainly semi-diurnal tide with irregular variation of amplitude and frequency between the minimum and maximum. (3) F =1.5 ÷ 3 gives a mainly diurnal tide, with one maximum a day; spring tide = 2(K1 + O1). (4) F > 3.0 gives a diurnal tide; spring tide like 3). Diurnal tide

Semi-Diurnal tide

Mixed Tide

M

6

12 First day

18

M

6

12 Second day

18

M

Figure 4.1 - Examples of diurnal, semidiurnal and mixed tides. Tides are waves several hundreds kilometres long. Being the dimensions of wave length often comparable with the geometric dimensions of the gulf or bay, it is important to study the tidal wave propagation in shallow water conditions, because rising of sea level in a site does not only depend on the magnitude of tidal forces, but also on resonance between tidal period and free oscillation period of the site. The shallow water motion equations for a constant depth h are: ∂U ∂U τ bx ∂U ∂ζ +U +V = fV − g − ∂t ∂x ∂y ∂x ρ (ζ + d )

(4.2)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

∂V ∂V τ by ∂V ∂ζ +U +V = − fU − g − ∂t ∂x ∂y ∂y ρ (ζ + d )

(4.3)

where: U,V = are respectively x and y components of mean velocity averaged over the water column. t = time f = 2ωsinϕ = Coriolis parameter, with ϕ latitude (in degree) ω = angular velocity of Earth (7.29 × 10-5 rad/s) g = gravity acceleration ζ = free surface elevation relative to the mean-sea-level τbx , τby = x and y components of bottom friction ρ = water density The mass conservation equation is: ∂ζ ∂ ∂ + [(ζ + d )U ] + [(ζ + d )V ] = 0 ∂t ∂x ∂y

(4.4)

The simplified equations can be obtained neglecting bottom friction, hypothesizing ζ + d ≅ d and considering only small acceleration terms, such that U ∂U ∂x and V ∂V ∂y can be neglected. The linearized equations obtained by (4.2), (4.3) e (4.4) are: ∂U ∂ζ = fV − g ∂t ∂x

(4.5)

∂V ∂ζ = − fU − g ∂t ∂y

(4.6)

 ∂U ∂V  ∂ζ  = 0 + d  + ∂t  ∂x ∂y 

(4.7)

In order to determine the tidal wave celerity and mean-sea level variation, let us consider a progressive tidal wave, travelling in a finite length channel (x direction). Equations become: ∂U ∂ζ = −g ∂t ∂x

(4.8)

∂ζ ∂y

(4.9)

fU = − g

∂ζ ∂U +d =0 ∂t ∂x

Elimination of U gives:

(4.10)

SEA LEVEL VARIABILITY

∂ 2ζ ∂t 2

= gd

∂ 2ζ ∂x 2

71

(4.11)

while the elimination of ζ gives a similar equation for U: ∂ 2U ∂t 2

= gd

∂ 2U ∂x 2

(4.12)

A simple harmonic solution of a progressive wave can be expressed in the form: ζ = Ae − my cos(kx − ωt )

(4.13)

U = U 1e − my cos(kx − ωt )

(4.14)

where: U1 =

g and m = f gd dA

where A is wave amplitude. Equations (4.13) and (4.14) represent a Kelvin wave propagating in x direction. Due to Coriolis force, wave amplitude exponentially decreases moving away from the right coast of the Northern hemisphere.

3

Long waves (tsunami and seiches)

Tsunamis are long-period water waves generated by undersea shallow focus earthquakes or by undersea crustal displacements (subduction of tectonic plates), landslides, or volcanic activity. Tsunamis can travel great distances, undetected in deep water, but shoaling rapidly in coastal waters and producing a series of large waves capable of destroying harbour facilities, shore protection structures, and upland buildings. There is no simple answer to how large an event needs to be to create a tsunami. Generally, it is thought that a quake (the most common cause of tsunami) needs to have a magnitude of 6.5 (Bryant, E., 1991) to 7.5 to produce a dangerous one. Tsunami are hard to detect in deep waters. Their wavelengths are very long (hundred of miles), but only a few feet high in the open ocean. Their speed is a function of the water depth: the deeper the water, the faster the tsunami. If they are created in the deeper parts of the Pacific, their speeds can be hundreds of miles per hour. Offshore tsunamis would strike the adjacent shorelines within minutes and also cross the ocean at speeds as great as 600 miles per hour to strike distant shores. As the ocean floor rises, the front of the wave slows, causing the rest of the wave to push up behind it, increasing the height.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Not all tsunamis break when they reach land. Some just rush ashore as a huge mass of water, like a sudden massive tide. Others break far from land and come ashore as a turbulent cascading mass called a bore. The size of the tsunami, its speed, as well as the coastal area’s form and depth are the factors that affect their shape. The power of a tsunami comes from the huge amount of water behind the leading edge of the wave. Normal waves have a small volume, so they dissipate quickly when they strike the shore. Tsunamis do not. Their huge volume pushes the water far inland. This phenomenon is called ‘run up’ and its size is what often determines a tsunami’s destructiveness (Myles, Douglas, 1985). Tsunamis rarely crash ashore in one huge wave. They are often preceded by coastal flooding, followed by a recession of the water, and then, finally, a series of waves. This effect is dangerous since many people assume the trouble is over after the first wave breaks. Unaware of the looming danger they venture too close to shore and are swept away by subsequent waves. The first wave to strike Crescent City, California, caused by the Alaska earthquake in Prince William Sound in 1964, was 9 feet above the tide level; the second, 29 minutes later, was 6 feet above tide; the third was about 11 feet above the tide level, and the fourth, most damaging wave, was more than 16 feet above the tide level. When an earthquake that might generate a Pacific Coast tsunami is detected, the Alaska Tsunami Warning Center calculates the danger to the northeast Pacific Coast and notifies the communities at risk. Those warnings may give people a few hours to prepare and evacuate (depending on the distance to the earthquake). If the earthquake occurs off our coast, however, there may be no time to send out hazard warnings. The first waves could arrive within minutes of the earthquake. The only tsunami warning might be the earthquake itself. Seiches are long waves with different causes. They develop when a enclosed body of water is shaken. The water literally sloshes around in its ‘container’ like water in a cup. Seiches are free oscillations with a natural frequency depending on the geometric dimension of the basin. Resonance effects can develop when the long wave period matches the frequency of oscillation of the body of water. Causes providing free surface oscillations, mainly in channels, are: • • • •

Pressure oscillations; Wind direction variations; River discharge in fluvial deltas and estuaries; Seismic oscillations during earthquakes.

The main cause in seiches generation has a meteorological feature. By supposing the force acting only at the initial process of generation of basin oscillations, seiches can be dealt with by using equations elaborated in the previous section. In particular, the example of a rectangular, narrow, long and constant-deep basin, similar to the one used in the previous section, but without open boundaries, will be re-considered here.

SEA LEVEL VARIABILITY

73

Solving the linearized shallow water equations, a solution for seiches of a steady wave is: ζ =

A0 cos kx cos ωt 2

(4.15)

with A0= amplitude of free surface oscillation Supposing that the cross-shore component of the velocity is null at the boundary of the basin; this condition implies that at the two ends of the basin, two anti-node are present, that is using equation (4.8) one obtains: senkx = 0

for x=0,l

(4.16)

This implies kl=nπ where n is the oscillation mode of the basin. Substituting k one obtains: L = 2l / n Figure 4.2 shows the basin wave length for three values of n. Basing on the characteristics of forces, seiches with several kinds of typical modes it is important to underline that energy is predominantly associated to the first modes of oscillation. Because: L = gd T the oscillation period of the basin is determined by: C=

T=

2l n gd

(4.17)

(4.18)

Wilson, 1972 has demonstrated that in the case of a rectangular open basin, equation (4.18) becomes: T=

4l n gd

(4.19)

that is: 1 n gd T 4 where n can only assume odd values. l=

4

(4.20)

Wave set-up and set-down

Wave set-up and Wave set-down are defined as the elevation and depression of the mean sea level due to breaking phenomenon. These variations of the mean sea level phenomena are associated with conversion of kinetic energy of breaking waves into quasi-steady potential energy (increased water level). When waves enter shallow water over a shoaling bottom, the characteristic vertical scale of wave motion is comparable with the local water depth. The

74

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

actual water depth in the arbitrary point of the coastal zone depends on many factors such as the geometric water depth, the mean water level influenced by the wave motion, tidal oscillations, seiching and storm surges. Waves approaching a sloping bottom are affected by some modifications of their parameters (wave height H, wave length L, crest angle α, depth d) resulting from shoaling, refraction, diffraction, reflection, and breaking processes (Massel, S.R., 1989).

Figure 4.2 – Seiches wave length for three values of n. The forward flow of the water particles in the breaking waves usually sets up longshore currents (see chapter 9) and “pumps” water across the breaking zone, increasing the water level there. The onshore momentum of the waves holds some of this water close to shore, causing an elevated water level near the shoreline (wave set-up). This phenomenon can be explained by the concept of radiation stress, introduced by Longuet-Higgins and Stewart (Longuet-Higgins, M. S., & R.W. Stewart, 1964) described in the following sub-section. 4.1 Radiation stress Let us first consider an off-shore steady condition, by assuming that a wave train beats against a coastline for a time long enough to establish an equilibrium condition. With reference to a system of Cartesian axes moving with the waves and oriented with x-axis in the direction of propagation, and y-axis orthogonal to it, the mean value of the total longitudinal momentum flux integrated over the vertical, deprived of the hydrostatic pressure per unit length, is equal to:

SEA LEVEL VARIABILITY ζ

S XX =

∫ (p + ρu )dz − ∫ 2

−d

0

p0 dz

75

(4.21)

−d

Where SXX is the normal component of radiation stress; d = bottom depth; ζ = mean-sea-level setup; p = pressure; p0 = hydrostatic pressure; u = horizontal wave velocity component; ρ = water density; The transversal component of radiation stress (SYY ) can be expressed as: ζ

SYY =

2 ∫ (p + ρv )dz −d

SYY =

∫

0

ρw2 dz +

−d

−

∫

0

p0 dz

(4.22)

−d

1 ρg ζ 2 2

(4.23)

The shearing stress component is denoted by SXY and results null because xcomponent of speed is zero as y-axis is orthogonal to the direction of wave propagation: S XY =

∫

ζ

ρuvdz

(4.24)

−d

In order to carry out an analytic treatment, radiation stress components are expressed in a system of coordinate with x-axis cross-shore direction, and y-axis long-shore direction. By transforming the old system of coordinates (X,Y) into the new system coordinates (x,y) one obtains: S yy =

1 (S XX − S YY ) − 1 (S XX − S YY ) cos 2α 2 2

(4.25)

S xx =

1 (S XX + S YY ) + 1 (S XX − S YY ) cos 2α 2 2

(4.26)

S xy =

1 (S XX − SYY )sen2α 2

(4.27)

where α is the angle between the wave fronts and the coast. Sxy is the shearing stress component with respect to long-shore direction caused by the difference between transversal and longitudinal momentum fluxes of an incident wave with an oblique front. Substituting orbital velocities u and w and surface elevation ζ and adopting linear wave theory, the following expressions are obtained: 1  S XX = E  2m −  2 

(4.28)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

1  SYY = E  m −  2 

(4.29)

and so: 1 1 3 S yy = E  m −  − Em cos 2α 2 2 2 S xy = Emsenα cos α

(4.31) (4.32)

where E is the total energy and m is the following ratio: m=

Cg C

=

1 2kd  1 +  2  senh2kd 

(4.33)

where Cg = group velocity and C = phase velocity. 4.2 Water level fluctuations due to radiation stress Waves approaching a sloping bottom are modified by shoaling, refraction, diffraction, reflection, and breaking processes. Therefore, the radiation stress components are modified and a new time-averaged equilibrium for the timeaveraged water level is estabilished. Longuet-Higgins and Stewart (1964) from the conservation of momentum flux in an incident wave train, derived the following momentum balance between the radiation stress variation and the changes in sea level: dS xx dζ + ρg (d + ζ ) =0 dx dx

(4.34)

If Sxx is known, the equation (4.34) can be integrated to yield a mean waterlevel set-down outside the breaker line and a mean water level set-up from the surf zone, until the shoreline. For spilling type breakers on dissipative beaches, the assumption commonly employed is that the breaker index γb (ratio of breaking wave height to mean depth at breaking) remains a fixed ratio throughout the entire surf zone :

γb =

Hb H ≈  (d + ζ )  d b

(4.35)

The breaking index can be calculated using various expressions, as described in chapter 8. If linear wave theory is applicable to compute the radiation stress component, Sxx=(3/2)E, where E is the total wave energy, so: 3 ρgγ b2 (d + ζ ) 2 (4.36) 16 Using the equation (4.36) in the momentum equation balance (4.34), yields: S xx =

SEA LEVEL VARIABILITY

 3γ b2  d ζ 3γ 2 dd  1 + =− b  8  dx 8 dx 

77

(4.37)

or:  8 dζ = −1 + 2  dx  3γ b

−1

  tan β  

(4.38)

Therefore, the mean water surface slope is proportional to the beach slope. The total rise of the mean water level in the surf zone can be calculated by integrating the equation (4.37):

ζ =−

3γ b2 8

−1

 3γ b2  1 +  d ( x) + C  8  

(4.39)

Evaluating the constant C at x=xb, the breaker line, where ζ = ζ , gives: b

ζ ( x) = ζ b +

3γ b2 8

−1

 3γ b2  1 +  [d b − d ( x)]  8  

(4.40)

Using shallow water theory, ζ becomes: b

ζb =−

1 γ bHb 16

(4.41)

For the maximum set-up at the shoreline, the equation (4.40) shows that: −1

2  2   3γ b2  1 + ζ max = ζ b + 3γ b 1 + 3γ b  d b  8  8  8  

(4.42)

or:

ζ max = ζ b +

3γ b2 db 8

(4.43)

Substituting (4.41) in (4.43) we get: ζ max =

5 γ bHb 16

(4.44)

Equation (4.44) shows that the mean water level at the shoreline is about 25% of breaker wave height due to wave set-up (Massel, S.R., 1989). Figure 4.3 gives a schematic representation of wave set-up and set-down.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

5

Storm surge

The term “storm surge” includes all the mean-sea-level fluctuations due to the passage of an atmospheric perturbation, which are due to both wind-wave interaction and sea response to pressure variations. Wind action gives a double contribution to sea-level variation: first, according to Miles’ theory, wind generates waves heights and so a wave setup is experienced; on the other hand, wind action on sea surface generates coastal currents, which in conditions of slightly sloping bottom cause an increase of mean-sea-level.

breaker line

Wave set-up

ζ

Wave set-down

ζs ζ max

ζb

d

Horizontal level ζ max = maximum wave set-up

ζ s = set-up at the still water level ζ b = wave set-down at the breaker line db

= breaking depth

Figure 4.3 – Wave set up and set-down. The last mechanism is called wind setup and depends on the magnitude, duration, and wind incidence angle with respect to the coast. Sea response to atmospheric fluctuations (storm surge) is not simultaneous to the transit of a cyclonic vortex, but requires the permanence of storm for a few hours. In the presence of a storm surge, equations describing coastal marine dynamics, that are mass conservation equation and motion equations, must account for the interaction between atmospheric effects and sea surface. In the following, the motion equations in the presence of a storm surge will be treated with some simplifications, that is they are vertically integrated and all terms are expressed with respect to current mean velocity, and to mean momentum flux over the water column. The following simplifications are established:

SEA LEVEL VARIABILITY

(1) (2) (3) (4)

79

vertical accelerations neglected; wave effects (wave setup) neglected; friction not considered; fixed and impermeable bottom.

A system of Cartesian axes is established with x and y oriented in horizontal plane with the origin on s.w.l., x axis cross-shore and y axis long-shore; θ is the anti-clockwise angle between wind direction and x-axis; W is wind speed. Motion equations are the linearized shallow water equations used at the beginning of this chapter: ∂U ∂U τ bx ∂ζ ∂U − = fV − g +V +U ∂x ρ (ζ + d ) ∂y ∂x ∂t

(4.45)

∂V ∂V τ by ∂ζ ∂V − = − fU − g +V +U ∂y ρ (ζ + d ) ∂y ∂x ∂t

(4.46)

and mass conservation equation is: ∂ ∂ ∂ζ + [(ζ + d )U ] + [(ζ + d )V ] = 0 ∂t ∂x ∂y

(4.47)

Symbols are also the same as those used at the beginning of this chapter. Solutions of this system of equations which describes marine dynamics can be obtained through numerical integration (e.g. finite difference method). The highest value of mean-sea-level variation during the transit of a storm, neglecting astronomical tide effects, is called maximum surge, while the absolute maximum over all values collected in all sites affected by the storm is called peak surge. In order to calculate mean-sea-level rise during a storm surge, a separate calculation of inverted barometer and wind setup is used. Sea-level variation caused by inverted barometer can be studied by considering sea-level fluctuations as a phenomenon of long wave propagation generated by atmospheric pressure fluctuation. Thus let us consider a negative pressure variation which propagates with speed U in x direction: p0 = f1 (Ut − x )

where x axis is cross-shore with entering shore direction. The linearized motion equations in x direction is: dζ d dp0 du = − gd − d dx ρ dx dt

(4.48)

(4.49)

Continuity equation can be expressed by considering a system of coordinates moving together with the long wave caused by pressure fluctuations, which makes the system stationary and the horizontal velocity equal to: u–U.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

By considering a constant transport Q, time and space-invariant, and by supposing that particle speed is proportional to free surface displacement: Q = (u − U )(d + ζ ) = −Ud

(4.50)

and thus linearizing: u =U

ζ

ζ

≅U

d +ζ

d

(4.51)

by assuming the form: ζ = f 2 (Ut − x )

(4.52)

mass conservation equation as a function of ζ dζ dζ = −U dx dt

(4.53)

)

(4.54)

Combining all equations: −

(

d dp0 dζ 2 U − gd = − ρ dx dx

which is an exact differential and can be integrated between a point of null pressure variation (and so of mean-sea-level) and a point of interest, obtaining:

ζ d

=

p0 ρ U − gd

(

2

)

(4.55)

In static conditions, that is in absence of propagation: ζS = −

p0

ρg

(4.56)

If the speed of translation U assumes a value close to gd = C , the denominator tends to null and mean-sea-level variation should become unlimited. In practice, friction effects damp this amplification. Figure 4.4 shows the absolute ratio between the sea-level variation caused by a moving depression and the same variation caused by inverted barometer in steady state. Also, the comparison between the cases with and without friction is given.

SEA LEVEL VARIABILITY

81

5 4.5 4

|ζ |

ζS

Frictionless case

3.5 3 2.5

A/(kdC)=0.5

2 1.5 1 0.5 0

0

1

2

3

4

5

U /C

Figure 4.4 - Dynamic response of translating pressure disturbance, with and without friction. For U << C, i.e. on deep bottoms, mean-sea-level variation is similar to the steady case, and assumes the value: ζ =−

p0

ρg

(4.57)

where ζ is mean-sea-level variation expressed in centimetres, p0 is atmospheric pressure variation expressed in millibar. Therefore to each negative variation of pressure of 1 millibar, sea responds with 1 centimeter of mean-level increase. In order to calculate storm surge due to wind setup the Cartesian plane previously discussed is adopted. X-component of wind stress is equal to τ θ = τ cosθ and let us suppose that the horizontal scale system is so small that Coriolis force effects are negligible. Linearized equations become: Sx

Sx

τ Sxθ τ bxθ ∂U dζ = −g + − ∂t dx ρ (d + ζ ) ρ (d + ζ )

(4.58)

after a long time period, supposing wind action remains constant, flux becomes stationary because of the presence of the coast, thus a dynamical equilibrium is established between wind stress which is balanced by pressure gradient coastward and by bottom friction. Bottom stress is not expressible in terms of flux U (being it null), so it is useful to introduce a scale parameter n defined as:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

nτ WX = τ Sxθ − τ bxθ

(4.59)

nτ WX = τ Sxθ − τ bxθ

(4.60)

that is:

τbxθ is negative as it is oriented in the opposite direction with respect to x, so -τbxθ is positive and n parameter is always greater that 1. Typical values of n vary between 1.15 and 1.30 (CERC, 1984); in the following it is set to the value 1.20. Equation (4.58) becomes: nτ wx dξ = dx ρg (d + ζ )

(4.61)

Wind stress is given by the following formula: τ S = CD ρ a Vw VW

(4.62)

Vw is wind speed measured in a fixed height (generally 10m);

ρa is air density

CD is the drag coefficient caused by wind action on free sea surface and it’s normally calculated at 10m above sea-mean-level (thus called C10); Equation (4.62) can be written as: τ S = C10 ρ a V10 V10

and the air density being ρa =0.001

(4.63)

kgs 2 m4

For a strong wind condition, the coefficient may be set to: nτ Wx = 1.2 × 0.001 × 2.5 × 10− 3 = 3 × 10− 6 C102

(4.64)

To calculate wind setup, let us consider a constant wind, being l continental shelf length and in this first case suppose a constant bottom depth d0

(d 0 + ζ ) dζ

dx

=

nτ Wx ρg

(4.65)

as d0 is not a function of x: 2 nτ 1 d (d 0 + ζ ) = Wx 2 dx ρg

(4.66)

by solving:

(d 0 + ζ )2 = 2nτ Wx x + C ρg

(4.67)

SEA LEVEL VARIABILITY

83

to determine integration constant C let us consider a null mean sea level variation in x=0 (at the beginning of continental shelf) located far enough from the coast.

(d 0 + ζ )2 = 2nτ Wx x + d 02

(4.68)

(d 0 + ζ )2 = 2nτ Wx x + d 02

(4.69)

ρg

and so:

ρg

ζ ( x) d0

 2nτ Wx xl   − 1 = 1 + 2 Ax − 1 = + 1 + 2  l ρgd 0 l  

(4.70)

with A parameter given by: A=

nτ Wx ρgd 02

(4.71)

With the purpose of determining wind setup on a variable bottom with constant slope, that is for linearly variable depth: d = d0 ( 1 −

x l

)

(4.72)

equation (4.61) can be written as

(d + ζ ) d (d + ζ ) − (d + ζ ) dd dx

where

dx

=

nτ WX ρg

(4.73)

d dd = − 0 = constant dx l

By dividing the two variables: −

(h + ζ )d (d + ζ ) = dx

d 02 l

  d +ζ  − A    d0

(4.74)

by solving:  d +ζ x + C = l  A − d0 

and by determining C as before:

   d +ζ  − A ln − A     d0

(4.75)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

  d +ζ x = l 1 −  d0  

 d +ζ  − A    d   − A ln 0   1 − A      

(4.76)

or in non-dimensional form  x  d + ζ = 1 − l  d0  

 d +ζ  − A    d   − A ln 0   1 A −     

(4.77)

Equations (4.70) and (4.77) solutions are reported in Figure 4.5 to show slope bottom effects on mean-sea-level variation. It is clear that for the same distance from the coast and same depth of continental shelf limit, a variable depth bottom produces a wind setup greater than the one pertaining to a constant depth. This characteristic may be explained by considering that mean-sea-level variation is increased approaching the coast and with bottom depth decreasing. Figure 4.6 shows the value of mean-sea-level variation for x/l=1 and h=0, that is on the coast, as a function of A parameter. This figure is used in this study because, sea-level rise is determined on the coast line by calculating parameter A. In order to obtain a value of free sea surface level in a site not coincident with the coast line, it is preferable to solve iteratively equation (4.77).

ζ0 d0

Figure 4.5 –Storm surge elevation versus shoreward distance for different wind shear stress and sloping bottoms.

SEA LEVEL VARIABILITY

85

(d + ζ ) d0

Figure 4.6 - Storm tide for x/l =1.0 for a sloping shelf. For the case of no Coriolis force, the ordinate is equal to ζ/d0, the storm surge at the coast, as d=0 at x/l=1.

6

Case study Example 1 Calculate wind setup caused by an atmospheric disturbance. Reference parameters values: c10 = 2.5.×10-3; ρair = 0.001; V10 = 29 m/s; l = 4000 m; =10m;

d0

The calculation of wind setup can be dealt with by using formula (4.71) which allows to calculate parameter A. With the help of Figure 4.5 and Figure 4.6 mean-sea-level variation can be estimated. By applying (4.71): A=

(3 × 10 × (29) × 4000) ≅ 0.01 −6

2

9.8 × (10 )

2

where A=0.01 and by supposing that the bottom between the coast and off-shore zone is smooth slope decreasing, from table in Figure 4.5, on coast line, that is x/l=1, one obtains that:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

ζ d0

= 0.03 ⇒ ζ = 0.03 × 10 ⇒ ζ = 0.3meters

mean-sea-level set-up in the in-shore, in calm conditions, is 30 cm. Example 2 Calculate wind setup across the surf zone (beach slope tanβ = 0.01) for a normally incident wave with breaker height Hb=2.7 m. The breaking index is γb=0.84. Reference parameters values: Acceleration due to gravity g = 9.81 m/s2 The breaking depth db is obtained by equation (4.35): db=Hb/ γb =2.7/0.84 = 3.2 m Set-down at the breaking is given by equation (4.41): ζb =−

1 γ b H b =1/16(0.842)·3.2= -0.14 m 16

The maximum set-up is given by equation (4.44): ζ max = ζ b +

3γ b2 3 ⋅ 0.84 2 d b = −0.14 + 3 .2 = 0 .7 m 8 8

Chapter 5 Random wave measurement and analysis 1

Wave measurements

Information on sea state severity can be obtained by wind hindcasting or forecasting, by visual estimation and by direct measurement of sea surface elevation through single point or remote sensing measurements. Generally wave measurement concerns only a few quantities such as height, period and direction, while other characteristics like wavelength and wave celerity are calculated by the previous data. In field wave experiments, three groups of instruments are used: instruments which need a reference height (wave staffs, acoustic and pressure sensors) and floating instruments, like wave buoys and remote sensing instruments. • Wave staffs can be resistive or capacitive, while acoustic sensors directly measure vertical displacement of free surface elevation. Pressure sensors indirectly measure the vertical displacement through a pressure measurement. Acoustic and pressure sensors usually adopt the bottom as the reference level, wave staffs generally refer to the height of a platform or pier. • Wave buoys are floating instruments which measure vertical acceleration by which vertical displacement is obtained through a double time integration. They are used in deeper water, where the supporting structures for wave staffs are not available and the water depth is too high to use a pressure gauge. • Conventional buoy systems are difficult to run and rather expensive in deep oceans, which comprise the vast majority of the globe. Satellite remote sensing systems have the capability of imaging wave conditions day and night through all weather conditions. 1.1 Ultrasonic and pressure gauges Wave gauges can be mounted on platforms offshore or on the sea floor in shallow water. Many different types of sensors are used to measure the surface elevation or wave pressure which is related to the former. Sound, infrared beams,

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and radio waves can be used to measure the distance from the sensor to the sea surface provided the sensor can be mounted on a stable platform that does not interfere with the waves. Bottom-mounted pressure gauges measure water level changes by sensing pressure variations with the passage of each wave. The gauges are either selfrecording or are connected to onshore recording device by cables. The pressure distribution below a gravity wave is given by: p = − ρgz + ρgηk p (z )

(5.1)

where: ρ is the sea water density; z is the vertical distance (positive upward) with the origin at the mean sea level; η is the surface elevation referred to the meansea level; kp(z) is the pressure response factor, given by: k p ( z) =

cosh(d + z ) ≤1 cosh(kd )

(5.2)

The first term in eqn. (5.1) ps=-ρgz represents the hydrostatic pressure which would occur in absence of wave motion, and the second term p=ρgηkp(z) is the dynamic pressure induced by wave motion. The pressure sensor, mounted on the bottom, is beforehand calibrated so as to record only dynamic pressure pd. In this case, vertical displacement is given by:

η=

pd ρgk p (− d )

(5.3)

Being: k p (− d ) =

1 cosh(kd )

(5.4)

kp(−d) is a function through k of the angular frequency σ by means of the dispersion relation. Short-period waves present very small values of kp(−d), whereas long waves show values of kp(−d) close to unity; this implies that very short waves are filtered by the pressure sensor. The depth range of pressure sensors goes from 10m to 20m. Lower depths may produce the breaking of the higher waves, which therefore are not correctly measured, while higher depths involve an attenuation of the response for short waves which becomes unacceptable. Pressure gauges can also be coupled to electromagnetic current meters, in order to obtain directional measurements of wave motion. 1.2 Wave buoys In deep water, where acoustic sensors and pressure gauges cannot be used, wave buoys are employed. They have generally a spherical or discus shape, with a

RANDOM WAVE MEASUREMENT AND ANALYSIS

89

diameter from 0.5 to 2.0 meters and weight from some tenth of kilos to some tons. The vertical acceleration is integrated twice with respect to time, to produce a measure of the instantaneous surface profile. In order to avoid measurement of unwanted accelerations due to roll and pitch of buoy, the sensitive axis of the accelerometer is mounted on a stabilized platform. Once the vertical acceleration measure is obtained, the buoy transmits the signal to a remote station for further analysis. The most common type of remote transmission is via radio; it is cheaper but requires, for a good reception, that the buoy is set within 30 miles; the second is via satellite but, due to the limitation of data exchange, it requires a microprocessor on board which allows the calculation of wave spectra and of selected parameters, so that the amount of transmitted data is smaller than that of recorded data. Thus the disadvantage of satellite transmission is that raw data are not generally available for further analysis. The transfer function between acceleration and displacement should be constant. For the most popular wave rider buoy manufactured by Datawell, the transfer function shows that the buoy response is essentially constant for wave periods between 5 and 10 s, while some attenuation is observed for periods between 10 and 25 s. A comprehensive comparison of various types of wave buoy is given by Allender et al. (1989). One of the most used buoys, the Datawell directional Waverider, has the following specifications: Hull Weight Range of wave height Resolution Accuracy Range of wave period Directional range Resolution Buoy heading accuracy Power supply

0.9m diameter 260kg ±20m 0.01m 3% FS 1.6-20s (when moored) 0-360° 1.5° 0.5-2° 25 VDC (internal batteries, 12 month capacity)

During the WADIC Project (Wave Direction Measurement Calibration Project) which was held at or in the vicinity of the Edda platform in the Ekofisk field in the North Sea during winter 1985-1986, several wave buoys, platforms and wave staffs were intercalibrated. The authors compared the performance of a particular buoy against the best estimated data set obtained from the platform based measurements. Although differences in accuracy have been found, comparisons for the most important engineering parameters, significant wave height, mean wave period and wave direction at the spectral peak are satisfactory. 1.3 Italian Sea WAve measurement Network In 1989 the Italian Ministry of Public Works started up the national Sea WAve measurement Network (SWAN). Since the beginning of 1996 SWAN is managed by the National Hydrological and Marine Survey (NHMS) which

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provides measurements and analysis of wave data systematically, water levels and other meteo-oceanographical parameters recorded in the Italian seas. SWAN was originally formed by 8 Datawell WAVEC buoys placed offshore La Spezia, Alghero, Ortona, Ponza, Monopoli, Crotone, Catania and Mazara del Vallo (see map in figure 5.1). Between 1999 and 2001 the RON was significantly upgraded with respect to instruments, management and data recording. The most significant change since March 1999 was the addition of two translation type WAVERIDER Datawell directional buoys at Cetraro and Ancona, in areas that were not covered by the first eight stations. In 2002 the network was completely restructured and upgraded. Four new stations were installed at Punta della Maestra near Chioggia, Civitavecchia, Capo Comino and Palermo, and the whole network was planned to work in real-time. Data were measured every thirty minutes and sent by radio link to local onshore stations, where they were processed and classified (APAT). Fourteen Triaxis (by Canadian Axis) translation type remote-controlled buoys are currently employed. Their main features are: • Spherical shape with diameter 0.9 m (weight = 200 Kg); • Solar panel and buffer battery power system; • Solid sensors; • Data sampling frequency of 4 Hz; • Onboard spectral and zero-crossing data analysis; • Radio link data transmission with assigned frequency of 44.8 MHz; • Onshore transmission of wave height and direction raw data time series, resampled as ∆t=0.78125 s; • Satellite tracing of buoy position by GPS and Inmarsat D+. These features allow more accurate analyses of wave parameters in Italian seas and minimize permanent data loss risks. The network operativity in real-time has enabled the realisation of automatic consultation services both on Internet, on RAI Televideo and transmission of measured data to the main meteorological centres through the World Meteorological Organization Global Telecommunication System. 1.4 Satellite remote sensing There are three standard active microwave instruments of principal interest for surface wave detection; namely, the altimeter, the Synthetic Aperture Radar (SAR) and scatterometer. These instruments can provide global, all weather, day and night data coverage. An overview of all satellites deploying instruments detecting the ocean surface is given by Komen et al., (1994). 1.4.1 Radar altimeter The most recent Radar altimeters missions include: • •

GEOSAT: November 1986 – January 1990 TOPEX: September 1992 – present

RANDOM WAVE MEASUREMENT AND ANALYSIS

• •

91

ERS1: August 1991 – present ERS2: May 1995 – present

GEOSAT transmitted 1000 pulses per second towards the Earth’s surface. The returned energy from these pulses was averaged to produce data values at a frequency of 1 per second. The ground speed of the satellite was approximately 6.5 km/sec and hence one data value was obtained every 6.5 km along the ground track. The pulses radiate away from the satellite altimeter antenna as a spherical shell. The pulse reflects from the water surface and the return pulse is sampled by the satellite. Rather than the return pulse being continuously sampled, it is measured in discrete intervals called “range gates”. Hence it is convenient to consider the pulse during each time increment corresponding to each range gate (Dobson and Monaldo, 1996). During the first range gate, the pulse illuminates a small circular region directly below the satellite. In subsequent time increments, the surface is illuminated by a series of expanding annular rings. The area of each of these annular rings is constant. The orbital characteristics depend on the use to which the data are to be put. For wave climate purposes, exact repeat orbits are the most valuable with the satellite returning to exactly the same ground track with time intervals of the order of 1 week to 1 month. During the exact repeat period the satellite’s ground track forms a web over the global oceans limited north and south by the satellite’s inclination. If the ocean surface was perfectly flat, the return pulse would exhibit a rapid rise, corresponding to the return from the inner circular region. This would be followed by a flat section resulting from each of the constant area annular rings. If the ocean surface is roughened by the presence of waves, the leading edge of the transmitted pulse will interact with the crests of the waves a small time before interacting with the troughs. As a result, the leading edge of the return will be broadened in comparison to the flat ocean case. As the wave height increases, this broadening of the leading edge of the return pulse increases. Therefore the slope of the leading edge of the return pulse can be used as a measure of the wave height.

Figure 5.1 - Location of the Italian directional wave buoy network (Corsini S. et al, 2002).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

3 1

2

Figure 5.2 - TRIAXYS (1) WAVEC (2) and WAVERIDER (3) buoys. 1.4.2 Synthetic Aperture Radars Synthetic aperture radar (SAR) is basically a method of ground mapping that uses computer processing to make radar work better. It first appeared in the early 1950s but did not reach a high state of development for almost 30 more years with the introduction of digital processing and other advances. SAR yields high-resolution, photograph-quality images because it combines radar images made many miles apart. SAR uses antennas positioned on moving bodies, such as satellites, and then mathematically combines the separate signals transmitted as the antenna moves, simulating or mimicking the transmission of radar from a source with a larger “aperture,” or a larger opening. For instance, a European Space Agency satellite with a 10-meter SAR antenna mimics the performance of a 4-kilometer-long stationary antenna. The motion of the satellite, combined with the wide beam of the radar, covers a swath along the ground, allowing a large area to be searched quickly typically 256 square kilometers. If a few dozen of these swaths are collected, an area of several thousand square kilometers will be covered. SAR is currently the only satellite-borne instrument which can measure directional characteristics of the ocean wave field independent of the weather conditions. However, from the wave analysis point of view, the actual image of the surface is not used directly, and a SAR wave analysis starts by a spectral analysis of the image. Although sometimes denoted the wave spectrum, the SAR image spectrum has turned out very far from being so. A rather complicated

RANDOM WAVE MEASUREMENT AND ANALYSIS

93

post-processing of the SAR image spectrum is therefore necessary for extracting quantitative wave information. The core of the methodology is the Hasselman nonlinear ocean-SAR spectral transform developed power the last few years (Hasselmann et al., 1991).

Figure 5.3 - SAR: example of ground tracks. Even if the methodology for extracting wave spectra from SAR data is available, there is still some way to go before this is becoming routine. The modulation mechanisms are incompletely understood and appear to be strongly dependent on centimeter waves generated by the local wind. The current resolution of space borne SAR also limits the observations to waves longer than about 40m. The SAR image spectrum is from the outset symmetric and thus suffers from a 180 degrees ambiguity in direction. This is the main reason for introducing the a priori spectrum in the inversion functional. However, in most cases, for example in coastal areas, the choice between the two possible wave directions is obvious.

2 Statistical properties of random waves 2.1 Data sampling The wave recorder will provide a continuous record of the surface elevation η(t) and should be used over a reasonably long period (a few years) so as to obtain data representative of the various conditions occurring. In order to avoid the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

accumulation of too much data, waves are recorded intermittently as indicated in figure 5.4. The recording interval is the time that lapses between successive starts of the recording instrument and could, for example, be 3 or 6 hours. The recording period is the duration of a single continuous recording and may be say 10 or 20 minutes. The sampling interval between two digitized ∆t is included between 1/10 and 1/20 of the significant period, generally goes from 0.25 to 1.0 second, more often equal to 0.39 or 0.5 seconds. Once ∆t is chosen, the highest frequency at which the spectrum can be estimated is automatically determined by: fc =

1 2∆t

(5.5)

denoted as Nyquist frequency. Wave component with this frequency is sampled with two point per wave, and the energy contained in frequencies higher than fc is added to the constant energy in the range of 0
η

recorder on

recorder on

t recording period

recording period

recording interval

Figure 5.4 - Recording interval and recording period for a random signal η(t).

RANDOM WAVE MEASUREMENT AND ANALYSIS

2.2

95

Data processing

Data processing is preceded by a preliminary filtering in order to identify and eliminate instrumental errors; the most frequent is the comparison of one or more peaks (spikes) occurring due to interference of radio signal. Starting from the vertical displacement data, the characteristics of the significant wave are determined through two different ways: the first (time domain) passes through the zero crossing method (up- or down-crossing) in order to obtain wave heights and periods of random waves, and then the characteristic waves (e.g. significant wave); the second (frequency domain) requires the determination of spectral energy density of vertical displacements; from which spectral moments are obtained which allow the estimation of a spectral significant wave height and a significant wave period (see figure 5.5).

Automatic vertical displacements

Quality control

Zero up-crossing analysis Random waves

{H i } {Ti }

Spectral analysis

Energy Spectrum

S(f) Spectral moments

Characteristic waves

{H1 / c }

{T } H 1/ c

∞

m0 = ∫ S ( f )df 0

Figure 5.5 - Diagram for the estimation of a spectral significant wave height and significant wave period. 2.2.1 Time domain analysis The random wave is defined as that part of registration included between two subsequent crossings of the zero line (i.e. MSL) with a positive derivative of surface elevation η(t) (zero up-crossing method) or negative derivative (zero down-crossing method). The single wave is characterized by a waveheight

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

intended as the difference of the maximum and minimum values recorded, and a period defined as the difference of temporal coordinates of two zero crossing. Zero up-crossing height of random waves is defined as the vertical distance between the two points of maximum and minimum height (that) surface elevation assumes between two subsequent SML crosses. H u = η max − η min

(5.6)

whereas zero up-crossing wave period Tu is the time interval between the mentioned subsequent crossings. To obtain the significant wave, which by definition is a regular wave derived from a third of highest waves, heights of n random waves are sorted in increasing order yielding a continuous succession {Hi} such that i=1,n (where i=1 represents the minimum wave, i=n the highest wave). Significant wave height Hs comprises the n/3 highest waves Hj where j=(1-1/3)+1. Height and period of this significant wave are respectively given by:

Hs =

3 n ∑ Hi n i= j

Ts =

3 n ∑TH n i= j i

(5.7)

Random waves are arranged ascending order with relative periods so that the first wave is the highest of the wave record. On the total of m waves of a record, the ratio c=m/n and the mean height of the c highest waves H1/n, and relative periods T1/n, is calculated; usual values of n are 3,10,20 and in particular H1/3 and T1/3 are known as significant height and period. In the hypothesis that random waves are characterized by a narrow band spectrum, Longuet-Higgins (1952) proposed the Rayleigh distribution for the significant wave height Hs, which probability distribution and cumulative probability are respectively expressed by: p( H ) =

 − H 2  exp  2  4σ η2  8σ η  H

 H  P( H ) = 1 − exp − 2   8σ η 

(5.8)

(5.9)

~ Introducing the non dimensional variable H = H / σ η and reminding the relation ~ ~ p ( H ) dH = p ( H ) dH , equations (5.8) and (5.9) become: ~ ~  H2 H ~ p( H ) = exp−  4  8  ~  H2 ~ P ( H ) = 1 − exp −   8 

(5.10) (5.11)

RANDOM WAVE MEASUREMENT AND ANALYSIS

97

The Forristall (1978) distribution, is given by: ~ ~α   H ~ H (α −1)  p( H ) = α exp− (5.12)  β  β  ~α ~  H  p ( H ) = 1 − exp − (5.13)   β  where α=2.126 and β=8.42. The following quantities can be obtained through the time domain analysis:

• • • • •

Hm = mean height [m] obtained from the average of n zero up crossing waves Hs = significant wave height [m] Ts = significant wave period [m] Hmax = maximum wave height [m] obtained as maximum value of height of a record, after a crescent sorting of wave heights. Tmax = period of the maximum wave [s]

Having significant example, H1/ 3 = H S

adopted a theoretical distribution, it is possible to estimate the wave height for the surface displacement characteristics. For using Rayleigh distribution, Longuet–Higgins obtained = 4σ . H 1 / 10 = 2.03H

H 1 / 10 = 1.27 H 1 / 3

H 1 / 3 = 1.60 H

(5.14)

Values obtained by field data respectively give the following results: H 1 / 10 = 1.26 H 1 / 3

H 1 / 10 = 2.00 H

H 1 / 3 = 1.58 H

(5.15)

in perfect agreement with (5.8). Analogous to heights, characteristic wave periods are mutually correlated. Goda (1985) suggests the following ranges of variability. TH max = (0.6 ÷ 1.3)TH 1 / 3

(5.16a)

TH1 / 10 = (0.9 ÷ 1.1)TH1 / 3

(5.16b)

TH1 / 3 = (0.9 ÷ 1.4)TH1 / 1

(5.16c)

The average value of these variations may be given as: TH max ≅ TH 1 / 10 ≅ TH 1 / 3 ≅ (1.1 − 1.3)TH

(5.17)

from the analysis of University Parthenope field data we obtained: TH1 / 10 = 1.02 TH1 / 3 = 1.18 TH1 / 1

(5.18)

For what concerns the maximum wave height recorded, it is a random variable, because different records which show the same value of H1/3 may show different

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values of Hmax. Nevertheless it is possible to define a probability density for the ratio Hmax/H1/3 which depends on the number N of wave records. The distribution mode, that is the most probable value, is given by: ( H max / H1/ 3 ) mode ≅ 0.706 ln N

(5.19)

Thus the prediction of the maximum wave height in a wave record falls approximately in the range: H max = (1.6 ÷ 2.0) H 1/3

(5.20)

For example, Goda assumes Hmax=1.8 H1/3 as design wave height. In the hypothesis that the temporal series η(t) is stationary, information on the characteristics of sea state may be obtained in frequency domain by a spectral analysis, that is by applying Fourier transform to the autocorrelation function. The auto-correlation function of vertical displacement correlates the value of η at time t to its value at time t+τ and therefore gives an information of the correlation of the signal with itself for different temporal shifts τ. 1 R (τ ) =< η (t ) ⋅η (t + τ ) >= lim  T →∞  T 

T /2



−T / 2



∫ η (t ) ⋅η (t + τ )dt 

(5.21)

In practice the period T used to obtain the temporal mean is sufficiently large to ensure that its value does not influence the estimation of R(τ). The autocorrelation function R(τ) expressed in nondimensional form yields to the autocorrelation coefficient Ψ(τ) given by: Ψ (τ ) =

R(τ )

σ η2

=

< η (t ) ⋅ η (t + τ ) >

σ η2

(5.22)

Obviously the signal is perfectly correlated with itself for a null temporal shift, whereas it is completely uncorrelated with an infinite time shift. τ = 0 ⇒ Ψ (τ ) = 1 τ = ∞ ⇒ Ψ (τ ) = 0

(5.23)

The autocorrelation function is directly connected to the spectral energy density because R(τ) and S(f) are given by the Fourier transformation, that is ∞

S ( f ) = ∫ 4 R(τ ) cos(2πfτ )dτ

(5.24)

0

∞

R (τ ) = ∫ S ( f ) cos(2πfτ )df

(5.25)

0

Spectral moments are given by the following relation: ∞

m n = ( 2π ) n ∫ f n S ( f )df 0

(n=0,1,2,…)

(5.26)

RANDOM WAVE MEASUREMENT AND ANALYSIS

99

In particular, moment of zero order, equal to the variance of the recorded data, is given by: ∞

m0 = ∫ S ( f )df

(5.27)

0

For each spectral moment, a corresponding frequency exists: fn =

1 2π

 mn  m  0

   

(n=0,1,2,….)

(5.28)

In particular f2=(m2/m0)1/2 is the zero up-crossing frequency of the stationary time series {η(t)}, and the relative zero up-crossing period (mean period) is given by Tz=T0,2=1/f2. Considering a Rayleigh distribution for random wave heights, the significant wave height obtained by a spectral analysis is equal to Hm0=4m01/2, while the root mean square wave height Hrms results equal to H rms = 2 2 m0 . From spectral analysis, also peak frequency fp may be obtained, that is the frequency corresponding to the maximum value of energy spectral density, which inverse is the peak period Tp. Through a spectral analysis the following quantities are obtained: • • • • •

Hrms Hm0 Tz Tp Dm

= root mean square height [m] = significant height [m] = mean period [s] = peak period [s] = mean direction [°N]

The theoretical spectrum most used in practical applications is the JONSWAP spectrum (Hasselmann et al., 1973), given by (see § 2.2.2):

S J = αg 2 2π −4 f

−5

{

}

exp − 1.25( f / f p−4 ) γ

[

exp 0.5( f / f p −1) 2 / ω 2

]

(5.29)

For γ=1, The spectrum is that of Pierson-Moskowitz, from which JONSWAP spectrum is derived, the mean value of γ obtained in North Sea is 3.3, the mean value obtained in Italian coasts is about 2.0. 2.2.2 Directional wave spectra

In time domain directional spectra are treated using the same method as non directional ones while for the definition of spectral forms generally used is the auxiliary function G(j∆F,θ) which models the spectral shape defined above: S ( j∆F ,θ ) = S ( j∆F )G ( j∆F ,θ )

(5.30)

where G satisfies the relation: π

∫ G( j∆F , θ )dθ = 1

−π

(5.31)

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allows to make the present definition of wave spectrum, coherent with the nondirectional ones and to develop a directional analysis for each frequency value. The function G(j∆F,θ) is given as discrete directions in some cases (Endeco) and as mean value, directional spreading, asymmetry coefficients and exceed parameters in other (Datawell). In the time domain directional spectra may be obtained considering that the quantity S(f,θ)dfdθ represents the contribution to the variance ση2 due to waves with frequency between f and f+df and direction between θ and θ+dθ, that is: π

∞

σ η2 = ∫

∫ S ( f , θ )dfdθ

(5.32)

0 −π

Directional spectra are often proposed in the form of a product of a monodimensional spectrum S(f) and a function of directional dispersion called “directional spreading function” D(f,θ): S ( f , θ ) = S ( f ) D( f , θ )

(5.33)

Several semi empirical expressions have been proposed for D(f,θ). The first exemplification is that to consider D(f,θ)=D(θ) independent of frequency. Longuet-Higgins, Cartwright (1961) proposed a power cosine form: 1  D(θ ) = C ( s ) cos 2 s  (θ − θ ) 2  

(5.34)

being θ the propagation direction of the whole wave motion sea state, and: C ( s) =

Γ( s + 1) Γ 2 π ( s + 0.5) 1

(5.35)

The value of s describes the degree of dispersion around the direction θ , a high value of s corresponds to a more concentrated spectrum around θ . In the particular case of random waves propagating only in direction θ , D(θ) tends to the Dirac function. 2.2.3 Pierson – Moskowitz and Jonswap spectrum

The Pierson – Moskowitz (1974) is given by: 2

S ( f ) = αg (2π )

−4

f

−5

−4   f   5   exp −   4  fp      

(5.36)

where α=8.1×10-3 (Phillips constant). This spectrum depends only on the wind speed U and so refers to fully developed conditions. The JONSWAP spectrum constitutes a modification of the Pierson - Moskowitz spectrum to account for the effect of fetch restrictions and provides a much more sharply peaked spectrum. It is given by:

RANDOM WAVE MEASUREMENT AND ANALYSIS

S( f ) =



−4 

5  f   a αg 2 γ exp −  5 4  4  fp   f (2π )   

(

σ a = 0.07 σ = σ b = 0.09

(5.37)



)2 / 2σ 2 f p2 ]

a = exp[− f − f p

101

for

f ≤ fp

for

f > fp

(5.38) (5.39)

Here fp is the peak frequency at which S(f) is a maximum and is found to be related to the fetch parameter by fp=2.84(gF/U2)-0.33; σa and σb relate respectively to the widths of the left and right sides of the spectral peak; α is equivalent to the Phillips’ constant but is now taken to depend on the fetch parameter α=0.066(gF/U2)-0.22; and finally γ is the ratio of the maximum spectral density to that of the corresponding Pierson Moskowitz spectrum. 2.00 fp

α=(0.65 f p )

∫ (2π )

4

g 2 f 5 S ( f ) df

(5.40)

1.35 fp

The peak enhancement parameter γ can be calculated as the ratio between the maximum spectral energy and the maximum energy of the Pierson-Moskowitz spectrum with the same value of α:

γ = S(fp)(2π)4 fp -5exp(5/4)(αg2)-1

(5.41)

where α is obtained from equation (5.50). The nondimensional energy, peak frequency and fetch are expressed by the following equations:

ε = σ2g2/ Ua4

(5.42)

ν = S(fp) Ua/g

(5.43)

χ = g x/ Ua

(5.44)

2

are related with each other the following regression lines (Hasselmann et al.,1973):

ε =7.13×10 –6 ν -3.33 ν = 3.5χ -0.33

(5.45) (5.46)

A relevant number of regression lines was obtained by other studies, all of them valid for fetch limited conditions: Kahma (1981) obtained a dependence of ν with χ comparable to the JONSWAP one but with growth rates for ε greater than JONSWAP of a factor about 2. So Kahma achieved the following growth curves:

ε = 3.60×10 -7χ

(5.47)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

ν = 3.18χ - 0.33

(5.48)

Benassai (1994) obtained the following relationships between the spectral parameters:

ε =2.71×10 -6 ν -3.76 α = 0.027ν

0.51

ν = 2.6χ -0.29 ε = 4.1×10 χ -8

(5.49) (5.70) (5.71)

1.16

α = 0.044χ - 0.15

(5.72) (5.73)

3 Statistical representation of wave climate Wave climate indicates the statistical characteristics of synthetic wave parameters recorded for a number of years averaged on a yearly basis (but also on a monthly or seasonal basis). Reference is normally made to their occurrence with reference to a single or a couple of wave parameters. Below, the attention is focused on two different probabilistic distribution: • •

joint-occurrence frequency of wave height/direction mean exceedance or persistence of significant wave height above assigned thresholds.

The first information is very useful in the analysis of coastal morphodynamics and in coastal protection as well as in harbour design, while the second one gives useful indications for maritime and navigation purposes. Occurrence frequencies can easily be calculated by dividing the value interval of all measurements taken into different classes. The frequency, for each class, is the ratio between the number of events falling within the considered class and the total number of data. Similarly, exceedance frequencies can easily be calculated by dividing the interval of all measured values into different classes. The frequency, for each class, is the ratio between the number of events above the minimum threshold and the total number of measured events. In order to calculate the joint-occurrence frequency of wave height/direction, sea states with significant wave heights below 0.50 m may be defined as calm and may not considered further. The interval of measured wave heights is divided into classes, while each direction sector covers an angle of 10°, 22.5° or 30°. The information is graphically represented with polar diagrams, similar to those also used for winds. They enable an immediate visual identification of the most frequent wave direction (prevailing seas) and the direction of sea states with the most frequent highest waves (dominant sea). Their eventual coincidence marks a predominant sea. In order to calculate the wave height over threshold persistence, the interval of measured values can be divided into classes. The information is graphically

RANDOM WAVE MEASUREMENT AND ANALYSIS

103

represented with histograms which allow an immediate calculation of the mean annual duration of sea states above a given wave height. Finally, the joint-occurrence of wave/direction can be calculated also on a quarterly/seasonal basis as follows: Winter = January, February, March; Spring = April, May, June; Summer = July, August, September; Autumn = October, November, December. This information, represented with frequency polar diagrams, may be useful to design temporary installation or sea activities (e.g. summer navigation lines, seasonal moorings, etc.). The main elements of wave climate are parameters relative to wave height, period and wave direction. Generally one refers to significant wave and significant period obtained by the analysis in time domain and to the peak period obtained by spectral analysis, and mean direction. Data relative to wave climate must cover a time period long enough, one year at least in order to have a sufficiently reliable description, ten years minimum if it is necessary to abstract information about extreme waves. In fact Goda showed that mean annual value of significant wave height obtained in a single year may deviate by 15% with respect to the value obtained on long time intervals. In any case, the representation of wave climate in a determined site may be performed in different ways, by reporting for example: • Mean annual, seasonal or monthly value, of significant wave height, complete or not with standard deviation and with the maximum value assumed in the considered time interval. • Polar diagram of appearance frequency of significant wave height that is histogram of significant wave height which fall in assigned provenience directions. • Frequency tables of the joint distribution significant wave height- mean period (or significant period) distinguished or not basing on a assigned directional class (the first hypothesis is preferable). The results of the analysis for the Italian Seas have outlined the existence of two different groups of directional wave climate, which can be related to different geographical areas of the Italian Peninsula. Namely, the Western group (Alghero, La Spezia, Ponza and Mazara) and the Eastern group (Pescara, Monopoli, Crotone and Catania). The first cluster has typically an unimodal annual direction climates, characterized by high waves mainly originating from the West, while the second one, instead, often exhibits a bimodal wave climate tending towards a North-South axis. Furthermore, the frequencies of waves coming from the western sectors are low in the Western group and High in the Eastern one. Mean seasonal climate shows that stations in the Western group have a lower seasonal variability with respect to stations in the Eastern ones. Finally, the persistence of significant wave heights over threshold shows that Western seas are more stormy than the Eastern ones.

4 Case study Calculate H1/3, H1/10, T1/3 and T1/10 from table 5.1 , the distribution mode and compare the values reported in eqn. (5.20a) and (5.20b) with empirical values of Hmax / H1/3 and H1/10 / H1/3

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Figure 5.6 –Sea level variation recordings. The values of Hu and Tu from figure 5.6, are reported in table 5.1. The following values of H1/3, H1/10, T1/3 and T1/10 can be obtained from table 5.1: H1/3=(1.94+1.92+1.71+1.56+1.55+1.52+1.50+1.49+1.40+1.31)/10=1.59 m H1/10=(1.94+1.92+1.71)/3=1.86 m T1/3-=(5.990+8.955+6.240+7.633+7.727+7.054+8.190+8.975+6.408+7.987)/10 =7.515 s T 1/10 =(5.990+8.955+6.240)/3=7.0617 s Hmax / H1/3 = 1.2

H1/10 / H1/3 = 1.16

According to the theoretical expressions (5.20) Hmax/H1/3 = 1.6÷2.0 m According the equation (5.19) the distribution mode is given by: ( H max / H1 / 3 ) mode ≅ 0.706 ln N =0.706× [ln(31)]1/2=1.308

RANDOM WAVE MEASUREMENT AND ANALYSIS

Table 5.1 Hu

Tu

1.920 1.550 0.690 1.400 0.650 1.060 1.310 1.710 1.940 0.970 0.240 0.360 0.650 0.910 0.810 1.520 0.760 0.100 0.340 0.550 0.510 1.560 0.340 0.180 1.160 0.920 1.490 1.500 0.850 0.720 0.240

8.955 7.727 6.310 6.408 4.425 3.276 7.987 6.240 5.990 4.856 2.284 2.750 4.088 3.882 4.476 7.054 3.511 1.011 3.747 3.006 2.867 7.633 3.042 4.657 6.157 7.126 8.975 8.190 7.702 4.143 2.795

Wave Number 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 31

Order Number 30 27 12 23 11 20 22 29 31 19 4 7 10 17 15 26 14 1 5 9 8 28 6 2 21 18 24 25 16 13 3

105

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Chapter 6 Short term wave prediction 1

Introduction

The study of ocean wave forecasting dates back centuries. During the second world war, Sverdrup and Munk (1947) introduced a forecasting method to predict the so called ‘significant wave’, defined as ‘the wave which describes the sea state’ for the invasion of Normandy in 1945, which was later published. They proposed a parametrical description of the sea state and used empirical wind and swell laws. The method was later improved by Bretschneider, 1958 and was called SMB (1958). An important advance in the sea state prediction was in the fifties, with the introduction of the wave energy spectrum, based on the assumption that the sea surface can be represented as a Fourier series of superimposed waves with different wave lengths and with statistically random phases. Each of the Fourier modes was associated to the mean wave energy of the spectrum. The first dynamical equation describing the evolution of spectrum was given by Gelci et al. (1957) with the formulation of the spectral balance equation. The lack of adequate theories forced Gelci to use empirical expressions to define the spectral evolution. The Phillips (1957) and Miles (1957) studies introduced a deeper insight on wave generation. An experimental milestone was the JONSWAP field study in the North Sea (published in 1973) in which the spectral evolution was formulated as a function of fetch length, through a parametrical approach, giving rise to the JONSWAP spectrum, which is still widely used. Another important concept was the introduction of the self-similarity shape, which was later developed for the spectral transfer on shallow water. After Hasselmann et al., (1985) who derived the source function for the nonlinear interactions it was possible to write down a general expression for the source function (Hasselmann et al., 1985), consisting of three terms representing the input from the wind, the nonlinear transfer and the dissipation due to whitecapping. The spectral wave energy balance in its complete form is actually the basis for all numerical wave prediction models.

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The first models, developed in the early 1960s, only took into account the wave energy growth and dissipation. Later these models became known as “first generation” wave models. From their results it became clear that interactions between waves of different frequencies were important in determining the wave spectral evolution with the fetch length. These non-linear interactions were computationally expensive for the time (1970’s), so parameterizations were developed to account for this effect, the most popular of which was the JONSWAP one. Wave prediction models using a parameterization of the nonlinear interactions are known as “second generation”. Further development led to an approximation for the non-linear transfers, which considers the four most important interacting waves at each frequency out of the infinite number of interactions theoretically possible. This approximation is more expensive to compute than the parameterizations of a second-generation model, but it improves significantly the model, because the wave spectrum is not forced to take a particular form for growing wind-sea, which is free to evolve according to the physical equations. Such a wave model is called “third generation”, and now is routinely run by several meteorological offices

2

Elements of wind measurement and analysis

The wind is an air flow caused by pressure gradients between adjacent areas. Horizontal pressure gradients arise in the atmosphere primarily because of density differences, which in turn are generated primarily by temperature differences. Wind results from nature’s effort to eliminate the pressure gradient, but is modified by many other factors. The altitude divides the boundary layer into three sub-layers (Figure 6.1): in the geostrophic layer (up to 1000 m height) the wind has a constant speed and experiences the Coriolis forces to the right (in the northern hemisphere). The pressure gradient is balanced by the Coriolis force and is perpendicular to the wind direction. Below the geostrophic region, the boundary layer may be divided into two sections: a constant stress layer (<100 m) called hydrodynamic layer and above which an Ekman layer (100 to 1000 meters in height). A detailed description of the boundary layer mechanics is given in Resio and Vincent (1977). The pressure gradient is nearly always in approximate equilibrium with the acceleration produced by the rotation of the earth. The geostrophic wind (defined by assuming that exact equilibrium exists) varies proportionally with the pressure gradient and is a function of the Coriolis parameter (f=2Ωsinϕ; Ω=7.29⋅10-5 rad/s); its velocity depends on the air densityρa (Kg/m3), the distance between two adjacent isobars ∆n (in meters) the pressure gradient ∆p (Pa) through the equation: U =

1 ∆p

ρ a f ∆n

(6.1)

SHORT TERM WAVE PREDICTION

109

The equation (6.1) can be explicitated in terms of the latitude ϕ (degrees) as: U =

Z ≅ 1000 m

4.82 ∆p sin(ϕ ) ∆ϕ

(6.2)

Geostrophic layer

Ekman layer

Z ≅ 100 m Constant shear layer

Figure 6.1 - Atmospheric boundary layer over waves. The geostrophic wind blows parallel to the isobars leaving the low pressure to the left, when looking in the direction toward which the wind is blowing, in the Northern hemisphere. Geostrophic wind is usually the best estimate of the true wind in free atmosphere. When the trajectories of air particles are curved, equilibrium wind speed is called gradient wind. Gradient wind is stronger than geostrophic wind for flow around a high pressure area, and weaker than the geostrophic wind for flow around low pressure. The magnitude of the difference between geostrophic and gradient winds is determined by the curvature of the trajectories. If the pressure patterns do not change with time and friction is neglected, trajectories are parallel with the isobars. 2.1 Wind information needed for wave hindcasting Wave hindcasting from meteorological data must be preceeded by the following activities: a. Estimate the mean surface wind speed and direction; b. Delineate a fetch distance over which the wind is reasonably constant in speed and direction, and measure the fetch length; c. Estimate wind duration over the fetch. These calculations can be made in different ways depending on the location and the type of meteorological data available. For restricted bodies of water, like

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

the Mediterranean Sea, the fetch length is the distance from the forecasting point to the opposite shore measured along the wind direction. The most common form of meteorological data used are depicted on synoptic surface charts, which report lines of equal atmospheric pressure called isobars. In Figure 6.3 the distance between adjacent isobars is 3 mb. (1 bar=105 Pa=105 N/m2; 1 mb=105/1000 Pa=105/1000 N/m2). H and L indicate high and low pressure areas, respectively. Triangular markers identify the presence of an atmospheric front. The distance between the flex points of two adjacent isobars is the wind fetch. The wind speed and direction are represented by the symbol : each mark value is 5 m/s and a half mark is 2.5 m/s, so the wind speed is 22.5 m/s at 35°N 130°E. Winds can be divided into three groups: high frequency winds, high speed winds and winds with both characteristics, called “prevailing winds”. Wind frequency, speed and direction are usually plotted on polar diagrams, in which different wind speeds are grouped into classes (e.g. 0-5 m/s, 5-10 m/s, 10-15 m/s) represented by segments of different width departing from the centre of the diagram. The segments length is proportional to the wind frequency. In Figure 6.2 winds are divided into six speed classes: the first (<1 m/s), the second (1-2 m/s), and so on. It is evident that the prevailing winds come from Eastern directions. Wind speed (m/s)

Figure 6.3 - Polar diagram showing directional occurrence of wind speed. 2.2 Geostrophic and low – height winds The wind speed in the geostrophic region is used to calculate the wind speed at 10 meters height : 1/ 7

 10  U (10) = U ( z )   z 

(6.3)

z is the height above the mean water level in the geostrophic layer (z=1000 m), e.g. if U(z)=22.8 m/s ⇒U(10)=11.8 m/s.

SHORT TERM WAVE PREDICTION

111

In the aerodynamic layer, it is possible to write an equation for the wind speed profile: U ( z) =

U*   z   z  ln  − Ψ   0.4   z0   L 

(6.4)

U*= friction velocity (the shear stress is given by ρU ) k= Von Karman constant z0= surface roughness Ψ= function which accounts for the effects of stability of the air column on the wind velocity L =a length scale associated with the mixing process, dependent upon the airsea temperature difference. Wind speed in the aerodynamic layer is measured by anemometers. The measured wind speed is corrected to obtain the corresponding value at 10 m height above the mean sea level. In order to take account of the different resistance encountered by wind on land and on sea, another correction factor is needed : 2

*

(6.5) U(10)= RL ⋅U’(10) RL is a function of wind speed at 10 m height above the sea level, as shown in Figure 6.4. We assume that RL=1 when the anemometer is at 10 m above the mean water level. The Figure 6.4 shows that RL >1 at low wind speed because the shear stress on land is higher on sea. At high wind speed has a higher friction sea surface with waves than land surfaces (RL <1).

3

Wave prediction on deep water

Short term wave prediction (at time scales of 1 hour) is carried out in the usual practice by the calculation of the characteristics of the significant wave. This is a regular wave which describes the sea state, and is characterized by two parameters: the significant wave height (Hs) and the corresponding wave period (Ts). The significant wave height (Hs) is defined as the average height of the waves which form the highest 33% of waves in a given sample time interval (typically 20 to 30 minutes). The significant wave period (Ts) is the average of the periods of the highest one-third of wave heights. In order to characterize the wave characteristics, two methods are frequently used: - Significant wave model (SMB) - Spectral wave models SMB model, introduced by Sverdrup and Munk (1947) and refined by Bretschneider (1958), is the simplest and the most frequently used method. The model is based on empirical formulations of wave height and period as functions of wind speed (U), fetch (x) and time duration (t) of wind events. Spectral models are based on the analytical formulation of spectral wave action and take

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 6.3 -Surface synoptic chart of Pacific Ocean – 27 October 1950 – ( Shore Protection Manual, 1984).

account of the interaction between different frequency components. The spectral models solve the spectral energy balance as a function of a 2-D spectrum, which frequency and direction are elements of a 2D matrix.

3.1 Fetch and duration limited growth The development of a wave field on deep water is governed by wind speed duration and direction variations, shape and position of the coastline. The combination of these variables leads to a vast number of situations which cannot be simply characterized. Two idealized cases are, however, commonly examined: fetch limited growth and duration limited growth. Although highly idealized, these cases provide a valuable insight into many of the physical processes responsible for wind wave evolution and provide estimates of wave conditions which could be expected at a site. Fetch limited growth occurs when a wind of constant magnitude and direction blows perpendicular to a long and straight coastline. The water is assumed deep and the wind blows for a

SHORT TERM WAVE PREDICTION

113

sufficiently long time that the wave field reaches a steady state (independent of time). Hence, for the given wind speed, the wave field becomes a function of the only fetch, x. 2.0

1.5 RL

1.0

0

0.5

10

U’(10) (m/s)

20

30

Figure 6.4 – RL is a function of wind speed at 10 m height above the sea level. A related problem to fetch limited growth is duration limited growth. This case considers the development of the wave field from an initially calm sea. All land boundaries are assumed sufficiently distant that there is no fetch limitation to growth. The wind field is of constant speed and direction and spatially homogeneous. The water is also assumed to be deep. For a given wind speed, if it blows for a limited time, the resulting wave field will be spatially homogeneous and only a function of the duration or time that the wind has been blowing, t. Sverdrup and Munk (1947), and Kitaigorodskii (1962, 1970) considered the variables which may be active in fetch and duration limited growth. They assumed that the most significant quantities were the variance of the surface elevation, the wind speed at a reference height, the fetch, the gravity acceleration, the wind duration, the frequency of the spectral peak. Dimensional analysis indicates that there should be four non-dimensional groups of parameters:

ε= ν=

σ 2 g 2 the non dimensional energy

(6.6)

f pU a

(6.7)

U a4 g

the non dimensional peak frequency

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

χ=

gx the non dimensional fetch U a2

(6.8)

ς=

gt the non dimensional duration Ua

(6.9)

Therefore for fetch limited conditions:

ε = f1 ( χ ) ; ν = f 2 ( χ ) (6.10) and for duration limited conditions:

ε = f 3 (ς );

ν = f 4 (ς )

(6.11)

where f1, f2, f3 and f4 are functions to be determined. 3.2 Significant wave (SMB) model The wave energy balance in the generation area without currents is given by:

∂E + ∇ ⋅ (C g E ) = Qin + Q ds ∂t

(6.12)

E is the wave energy, Cg is the group speed (the energy propagation speed). The right side of the equation (6.12) is the net source of energy per unit time in the generation area obtained summing up the wind energy (Qin) and the energy dissipated by whitecapping (Qds). A first part of the source term is a supply of wave energy and a second part is transferred away from the generating area. The wave energy is a function of the sea density (ρ), gravity (g), time (t), space (x) and wind speed (U) as shown by: (6.13) E = f ( ρ , g , t , x, U ) The equation (6.13) can be rewritten in a nondimensional expression:

gE  gt gx  = f , 2  4 ρU U U 

(6.14)

where gt/U is the time duration parameter and gx/U2 is the fetch parameter. The equation (6.12) can be applied to many wave generation events: if the wind does not blow for enough time to reach the stationary case, the wave energy is not transferred out of the local generating area (transient case), so the wave energy increases locally:

∂E = Qin + Q ds ∂t

(6.15)

SHORT TERM WAVE PREDICTION

gE  gt  = f  4 ρU U 

115

(6.16)

If the wind blows for enough time, the local generating area is saturated, so wave energy can be transferred to adjacent areas (stationary case). The wave energy is only a function of the fetch parameter:

∇ ⋅ (C g E ) = Qin + Qds

(6.17)

 gx  (6.18) = f 2  U  In this case, the duration parameter is greater than a threshold value tm. If both the fetch parameter and the duration parameter are higher than a minimum value the maximum energy input is reached with the associated wind input, so that any further energy increase is dissipated (fully developed sea), this means that the wave energy content is constant for the given wind speed: gE

ρU 4

Qin + Qds = const

gE

ρU 4

= const

(6.19) (6.20)

The SMB model calculates the significant wave height Hs and period Ts on deep water as functions of wind speed (U), fetch extension (x), gravity (g), and time duration (t):

H S = f 1 (U , x, t , g ) ,

TS = f 2 (U , x, t , g )

(6.21)

At a given wind speed, the fetch and the time duration are given by:

gH s  gt gx  gTs  gt gx  (6.22) = f1  , 2  ; = f2  , 2  2 ρU  U U  ρU U U  In the transient case, the significant wave height and period depend on the time duration parameter gt/U (duration limited sea): gH s  gt  gTs  gt  (6.23) = f1   ; = f2   2 ρU  U  ρU U  In the stationary case the significant wave height and period are expressed as functions of the fetch parameter (fetch limited sea): gH s

ρU 2

 gF  gTs  gF  = f1  2  ; = f2  2  U ρ U   U 

(6.24)

The fully-developed sea occurs when the fetch no longer controls the development of the waves and there is no more net transport of energy from the wind to the waves. In this case gx/U2> gxm/U2 and the significant wave height and period are constant and only functions of the wind speed:

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gTs gH s = const = const ; 2 ρU ρU

(6.25)

According to Bretschneider (1958, 1973) the dependence of the significant wave height and period by the fetch and duration parameters can be formulated as: 0.42  gH s  gx   = ⋅ 0 . 283 tanh 0 . 0125     2 ρU 2  U   

(6.26)

0.25  gTs  gx   = 1.2 ⋅ tanh 0.077 2   ρU  U   

(6.27)

The equations (6.26) and (6.27) are valid only for the stationary case, (t>tm) with tm given by:

{[

]

0.5 gt m = 6.5882 ⋅ exp 0.0161λ2 − 0.3692λ + 2.2024 + 0.8798λ ρU

}

(6.28)

where λ = ln( gx / U 2 ) . If gx/U2 exceeds a critical value, the fully developed sea occurs:

gH s = 0.283 ρU 2 gTs

ρU

= 1. 2

(6.29) (6.30)

Equations from (6.26) to (6.30) are depicted in figure 6.5. The lines plotted in Figure 6.5 are used to calculate Hs and Ts for given values of U, t and x. The wave condition is identified by the intersection between the values of U and x. The dotted line t=tm is the minimum time duration needed for the stationary case. If t
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Figure 6.5 - Significant wave height and period forecasting curves (Shore protection manual 1977 )

(a) In figure 6.5 the intersection between U=35 Kn and x=60 n.m. gives:

6htm ⇒ stationary case (fetch limited sea). The intersection between U=35 Kn and x=60 n.m. gives Hs=11 ft ≅3.3 m and Ts =7.3 s. (b) In figure 6.5 the intersection between U=35 Kn and x=200 n.m. gives tm=18 h⇒ t
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of terms considered, models are divided into three groups: first, second and third generation models. A key concept in the description of random waves is the two dimensional spectral distribution function of surface elevation variance. It contains the information required for a dynamic and probabilistic description of the local wavefield. To lowest order of wave steepness, surface elevation variance and wave energy density are equivalent, apart from a constant factor. Energy divided by the intrinsic frequency is called wave action, and presents a very useful concept because it is conserved in case of wave-current interactions (in absence of energy sources or sinks). The spectral distribution function of variance, energy or wave action is the principal prognostic quantity in the phase-averaged modelling of random waves. Its slow evolution in time and space can be calculated on the basis of an energy or action balance. This approach was initiated by Gelci et al. (1956, 1957) (using energy) and has been followed ever since, with numerous and continuing improvements and extensions (given by Komen et al. (1994), Battjes(1994). Phase averaging models predict average or integral properties of the wave field. These may be quantities such as significant wave height and peak period. Phase averaged models solve a single equation, the energy (or action) balance equation:

∂E ∂ [(C g ⋅ ∇θ ) E ] = S + ∇(C c E ) − ∂t ∂θ

(6.31)

where E(x, t, f, θ) is the directional wave spectrum, x is the position in space, Cg is the group velocity. S is a source function which includes a description of the physical processes representing wave growth, interaction and decay. This equation allows swell to be represented in addition to local wind generated waves. (6.32) Stot = Sin + S nl + S ds where Sin is the atmospheric input, Snl represents quadruplet non linear interactions between spectral components and Sds is the dissipation by whitecapping breaking. 3.3.1 First, second and third generation models The first generation models, developed in the early 1960s, took only account of wave energy growth and dissipation (Q=Qin+Qds). The dissipation source Qds was obtained assuming that the wave components suddenly stopped growing as soon as they reached a universal saturation level (Phillips, 1957). The saturation spectrum, represented by Phillips’ one dimensional f−5 frequency spectrum and empirical equilibrium directional distribution, was prescribed.

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The source energy transferred by the wind was expressed as the sum of the Phillips (1957) increase during the initial wave motion development and the Miles (1957) exponential growth of energy. The relative importance of the nonlinear transfer Qnl on the spectral evolution become more evident after extensive wave growth experiments (Hasselman et al., 1973) and direct measurements of the wind input to the waves (Snyder et al., 1981). Non-linear interactions between different wave components are the main cause of spectral growth with fetch extension; low frequency wave components causing the spectrum to quickly increase. This led to the development of second generation wave models, which attempted to simulate the nonlinear interaction in the source balance. A major impediment to the inclusion of Qnl was the significant computational time required for its evaluation. In order to overcome this problem a class of wave models which utilized approximations to this term were developed (Barnett, 1968; Ewing, 1971; Sobey and Young, 1986; Young, 1988). The significant advancement with this class of models was the inclusion of a parametric representation for the nonlinear interaction term Qnl . In almost all cases these approximate forms represented the model spectrum in terms of a small number (3 to 5) parameters. The JONSWAP spectral parameters were a common choice (Barnett, 1968; Ewing, 1971; Young 1988). Of course, problems occurred when the model spectrum did not conform to any of those for which Qnl had been precomputed. A further sub-classes of models, termed hybrid models, took this process one step further and predicted the evolution of the parameters rather than the discrete values of the two dimensional directional spectrum. The Sea Wave Modeling Project, SWAMP (SWAMP Group, 1985) demonstrated that it was necessary to overcome the limitations in the parameterization of Qnl of the second generation wave models. Some of the SWAMP (1985) participants developed a third generation wave model (WAM, 1988) which applieed no restrictions on the spectral shape, gave an exact Qnl parametrization (Hasselmann et al., 1985) and calculated Qds from field measurements of Snyder et al., (1981). The WAM model is a third generation wave model which solves the wave transport equation explicitly without any presumptions on the spectral shape. Third generation models utilize the discrete interaction approximation (DIA) to the Qnl term (Hasselmann et al., 1985). Although the DIA is still an approximation, it is very different in its formulation to the simple parametric forms used in second generation models. The DIA retains the basis physics of the nonlinear interaction process, but considers a very small sub-set of all the possible interactions. In contrast to second generation parameterization for Qnl, the DIA has many degrees of freedom as the values in discretely specified directional spectrum.

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3.3.2 Third generation models Third generation models solve the energy balance equation using a finite difference method based on a gridded representation of the bathymetry allowing refraction and shoaling to be modelled. The spectral balance equation can take many different forms, dependent on: • • • • • • •

Time variation (stationary or nonstationary) geometry of propagation space (plane or spherical) space coordinates (Cartesian, polar, spherical) spectral domain (wavenumber vector, frequency/direction) spectral density (energy or action) medium (deep water; finite depth, constant or variable; currents) source functions (processes and representation)

Wave Spectral Density Sη(f)

Envelope of spectra

Frequency

Figure 6.9 - Evolution of one-dimensional frequency spectrum. To illustrate the basic structure, a formulation in terms of action density in wave number space is summarized in the following. Dispersion equation 1

σ = ( gk tanh kd ) 2 = ω − kα U α

(α = 1,2)

(6.33)

G σ =intrinsic frequency; ω = absolute frequency; k = wave number; k = k

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121

Propagation equation

xα = c gα + U α =

∂σ + Uα ∂kα

∂U β ∂k ∂k ∂σ ∂d − kβ kα = α + x β α = − ∂t ∂x β ∂d ∂t ∂xα ω =

Action density

∂U α ∂ω ∂σ ∂d ∂ω + kα = + xα ∂t ∂xα ∂d ∂t ∂t

G G G G N (k ; x , t ) = E (k ; x , t ) / σ

(6.34) (6.35)

(6.36)

(6.37)

in which E is energy density Action balance equation

∂N ∂N  ∂N + xα + kα = ∑ S Qi ∂t ∂xα ∂kα i

(6.38)

Qi is the source term (i=1,2, …) The propagation formulation applies to each of the three domains considered here (oceans, shelf sea and nearshore) but distinguishing the source terms. Ocean models: The spectral evolution of wind-driven ocean waves, in the absence of currents, is mainly determined by wind input, whitecapping and nonlinear resonant quadruplet interactions; a suitable parameterisation for these source functions has allowed the development of the so-called third generation wave model, in which the spectrum is calculated without a priori constraints. Shelf sea models: the additional effects of restricted depth and relatively stronger currents have to be taken into account. The major change in energy budget is increased dissipation due to bottom friction (Weber, 1988) and, under extreme conditions, depth-induced breaking. Refraction due to varying depth and currents plays a minor role. Consider the individual source terms: • •

Wind input and whitecapping are taken to be the same as in deep water, with the finite depth influence on wavenumber and phase speed taken into account (WAMDI, 1988). The deep-water parameterisation of the quadruplet interactions is scaled up with a multiplying factor dependent on k d , where k is the spectrally-averaged wavenumber. This approximation is said to be valid for k d > 1 (WAMDI, 1988).

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•

•

In shelf seas, bed friction is important. Significant advances have been made in the understanding of the hydrodynamics and the modelling of the oscillatory near-bottom boundary layer. However, prediction remains problematic, particularly because the development of flow– dependent roughness on sedimentary bottoms and the role of wavecurrent interactions in the bottom boundary layer are insufficiently known. Depth dependent breaking does not need to be taken into account in shelf sea models under normal conditions. However a number of numerical studies (e.g. extreme waves in Northern Sea using Tolman’s model by Holthuijsen et al. 1994) have shown that inclusion of this process is necessary.

Near-shore models: If shelf sea models are made to include even depthinduced breaking among the finite-depth effects, one can wonder whether there is a need for so-called near-shore models and why such shelf sea models are not simply used all the way to the shore. The reason is that the very small depth and the more rapid depth variation typical for coastal areas would require a high spatial resolution, which for time dependent models also translates into small time increments of the numerical integration. This makes application of shelf sea models in near-shore regions unattractive and causes the need to use separate models to cover a rather restricted area from the coastline out to seaward boundary which is usually taken at some chosen depth contour, say 10m or 20m below MSL. 3.3.3

WaveWatch III

WaveWatch III is a third generation wave model developed at NOAA/NCEP after the WAM wave model, as a further development of WaveWatch I, (Delft University of Technology) and WaveWatch II (NASA, Goddard Space Flight Center). The governing physical equations, the physical parameterizations and the numerical methods reflect some modifications of previous models. The solution of the governing equations is based on a first and a third order accurate numerical scheme. The breaking waves physics are not modeled, hence the applicability of this model is outside of the surf zone and on large scale. Outputs from the model include significant wave height gridded fields with the associated wave directions and periods, spectral information about wave energy at the different wavelengths. The governing equations simulate variations in time and space of wave growth and decay produced by the surface wind, dissipation (e.g. due to whitecapping), and the bottom friction effects. For irregular wind waves, the random variance of the sea surface is described using variance density spectra (usually denoted as energy spectra). The variance spectrum F depends on all independent phase parameters, i.e., F(k,σ,ω), and furthermore varies in space x

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123

and time t, e.g., F(k,σ,ω,x,t), where k,σ and ω are the wave number vector, the intrinsic frequency and the absolute frequency respectively.

σ 2 = gk tanh kd

(6.39)

ω = σ + k ⋅U

(6.40)

where U is the current speed averaged in time and space and d is the water depth. If the individual spectral components satisfy the linear wave theory (locally), the dispersion relation and Doppler type equation interrelate the phase parameters; only two independent phase parameters exist, and the local and instantaneous spectrum becomes two-dimensional. Within WWATCH the basic spectrum is the wavenumber-direction spectrum F(k,θ), which has been selected because of its invariance characteristics with respect to physics of wave growth and decay for variable water depths. The output of WWATCH, however, consists of the more traditional frequency-direction spectrum F(fr,θ). The different spectra can be calculated from F(k,θ) using straightforward Jacobian transformations. Without currents, the variance (energy) of a wave packet is a conserved quantity. With the addition of currents the energy or variance of a spectral component is no longer conserved, due to the work done by current on the mean momentum transfer of waves (Longuet-Higgins et al., 1961). In a general sense, however, wave action A≡E/σ is conserved (Whitham,1965; Bretherthon and Garrett, 1968). This makes the wave action density spectrum N(k, θ)≡F(k, θ)/σ the spectrum of choice within the model. Wave propagation then is described by:

DN Q = Dt σ

(6.41)

where D/Dt represents the total derivative (moving with a wave component) and S represents the net effect of sources and sinks for the spectrum F. In a numerical model, a Eulerian form of the balance equation (6.41) is needed. The balance equation for the spectrum N (k,θ,x,t) in a spherical grid as used in WWATCH is given as (for convenience of notation, the spectrum is henceforth denoted simply as N):

∂  ∂  Q ∂N kN + + ∇ x ⋅ xN + θN = ∂k ∂θ σ ∂t

(6.42)

x = c g + U

(6.43)

∂σ ∂d ∂U k = − −k⋅ ∂s ∂d ∂s

(6.44)

∂U  1  ∂σ ∂d −k⋅ θ = − − k  ∂d ∂m ∂m 

(6.45)

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where cg is the group celerity and θ is the wave direction, s is a coordinate in the direction θ and m is a coordinate perpendicular to s. The above equations are valid for a Cartesian Grid. The net source term is generally given by summing up a wind-wave interaction term Qin, a nonlinear wave-wave interactions term Qnl and a dissipation (whitecapping) term Qd .In shallow water additional processes have to be considered, most notably wave-bottom interactions Qbot (e.g., Shemdin et al., 1978). The general source terms used in WWATCH as:

Q = Qin + Qnl + Qds + Qbot

(6.46)

Nonlinear interactions are optionally modelled using the Discrete Interaction approximation (DIA, Hasselmann et al., 1985), or the Webb-Resio-Tracy method (WRT). The model includes two source term options: the first one is based on WAM model cycles 1 through 3 (WAMDIG 1988); the second one is based on Tolman and Chalikov (1996) formulation. Input to WWIII can consist of wind, current, water level, temperature and ice concentration fields on the spatial wave model grid. WWIII model gives various types of output, such as fields of mean wave parameters on the spatial grid and input fields driving the model (wave height, maximum wave height, primary and secondary wave direction, primary and secondary wave period, sea height, swell height, sea period, swell period, sea direction, swell direction, and whitecap probability). 3.3.4

Case study (WWIII application for the Gulf of Naples)

The spectral third-generation ocean wind-wave model WAVEWATCH III (WW3), operational since January 2005 at the Department of Applied Sciences of the University of Naples “Parthenope” (Italy), was adopted for simulating wave propagation in the Gulf of Naples. The model was coupled with PSU/NCAR mesoscale model (MM5), which gives wind forcing at 1-h intervals. The model was implemented using a four-nested grid configuration covering the Mediterranean Sea until the Gulf of Naples, the inner mesh with higher resolution (1 km x 1 km). The simulated directional spectral waves were compared with APAT storm wave data recorded in year 2000 offshore the Gulf of Naples and with wind and wave data collected by Servizio Idrografico e Mareografico offshore the mouth of river Sele in the Gulf of Salerno. The MM5/WW3 model domains cover four areas (from regional ocean to small scale): • DOMAIN 1 (Mediterranean sea) • DOMAIN 2 (Seas around Italy) • DOMAIN 3 (Tyrrhenian sea) • DOMAIN 4 (Gulf of Naples) Information about the spatial dimension of four domains is summarized in table 6.1.

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Table 6.1 – Spatial information about the four domains of WWIII Latitude range

Longitude range

Latitude Increments

Longitude Increments

DOMAIN 1

30.02 47.84

-5.53 41.83.

0.24

0.24

DOMAIN 2 DOMAIN 3 DOMAIN 4

36.11 48.31 39.80 41.67 40.41 41.08

3.76 22.41 12.50 16.47 13.72 14.69

0.08 0.03 0.01

0.08 0.03 0.01

Domain 3

Domain 4

Fi

Figure 6.10 – Example of significant wave height, domains 3 and 4. An example of nested grids is in figure 6.10 (domains 3 and 4). The comparison between simulated and observed waves is given for two locations close to the Gulf of Naples during the November 2000 storm: the first one for deep water (Ponza 40°52’00’’N e 12°57’00’’E) and the second one for intermediate depth (Sele mouth 40°29’06”N, 14°55’30”E). Time histories of simulated and recorded wave storms gives more insight into the physical aspects of the simulation: the wave simulations November 2000 storm (characterized directions spread from 200°N to 230°N) are in good agreement with the data (fig. 6.11 and 6.12). Figure 6.14 gives map of WWIII 2004 storm in the Gulf of Naples.

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

DDw [°N]

126

Hs [m

Figure 6.11 – Measured and simulated wave directions- Ponza and Sele mouth stations - November 2000. M=measured; S=simulated.

Figure 6.12 – Measured and simulated significant wave height Ponza and Sele mouth stations - November 2000. M=measured; S=simulated

Figure 6.14 – WWIII results for the 2004 storm in the Gulf of Naples

Chapter 7 Long term wave statistics 1 Introduction The characterization of wave climate in a coastal zone for a long period deals with the estimation, through an extrapolation from data directly collected or hindcasted, of the wave parameters besides the period of observation, in order to obtain an extreme wave, also called “design wave”, which by definition is the highest wave which may occur in a given return period. The procedure for the evaluation of a design wave is based on the analysis of recorded or estimated data and passes through the following steps: • • • •

Wave height data selection relative to a period of observation; Choice of a known probability distribution of extreme values; Fitting of wave height data to the chosen distribution; Data extrapolation through the probability distribution chosen to identify the extreme wave height in a given return period.

The sample independence is somewhat guaranteed by fixing a threshold value for the time interval between two subsequent sea storms. The size of this interval can be calculated on the basis of the auto-correlation function of the observed time series. By imposing a weak sample independence, a long enough time lag implies that the two subsequent sea storms are independent, which makes the auto-correlation function value sufficiently small (the value of 0.4 is normally recommended in literature). Generally, this time interval is equal to 48 hours. In order to assure the independence of two consecutive events, the combination of different criteria is used (e.g. a time interval of 48 h and a deviation of 60° by the mean wave direction). The choice of threshold wave height is also a delicate operation, because it has a considerable influence on wave height estimates with a given return period. In fact, if the threshold value is too low, it produces too many sample elements, most of which do not really represent extreme conditions. These circumstances will determine stable but over-estimated predictions of the wave height with a

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given return period. On the contrary, if the threshold value is too high, it produces very few sample elements, even if they do not represent extreme conditions. These circumstances determine strongly variable wave height predictions according to the number of samples collected. The analysis of SWAN (Sea Wave Italian Network) data shows that H0=2.0 m is a value acceptable for all the Italian sites, with a number of storms variable between 10 and 30. 1.1 Wave data Long term wave prediction can be performed on the basis of three categories of data: 1. 2. 3.

Visual data; Data obtained by hindcasting model based on wind data; Instrumental data collected in situ or by remote sensing.

Visual data, reported by ships during their normal activity, are less reliable than those obtained by hindcasting. In the past visual observations of waves and wind were the only basic source of ocean wave statistics. Although other types of wave data are now available, visual observations are still the source which covers most of the ocean areas. In the early stages of observations, the sea state was represented using the Beaufort scale, which gave the gradation of wind speed associated with some description of the sea state. However, there was not a straightforward relationship between wind speed and wave height. Only in 1947 did the World Meteorological Organization introduce the international code for the observation of wind and waves. There are two main sources of wave visual data, i.e. data from merchant ships and observations by weather ships (Ocean Weather Stations — OWS). Merchant ships are generally expected to avoid bad weather, thus being subjected to less severe wave conditions than weather ships. The stations cover most of the ship routes between Europe and North America. The Pacific Ocean is not so well documented and only a few stations operate in the northern part of the Pacific. Major sources of visual data are the compilations made by Hogben and Humb (1967) and Hogben (1988). Wave hindcasting are used at locations where no information on the wave climate is directly available. Several wave hindcasting models starting from the wind field have been used to generate large data set, eventually calibrated with instrumental data. Instrumental data collected in situ by wave gauges or through remote sensing are the most reliable source of data. Nevertheless their use is reduced due to the limited availability (few subsequent years of observation) and because of the high cost of acquisition. A wave recorder at a site will provide a continuous record of the surface elevation η(t) and should be used on a reasonably long period (ten years or longer) so as to obtain a data archive representative of the various conditions occurring.

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129

1.2 Data selection The first step to calculate the design wave height is the selection of the set of wave heights and periods in the site of interest. The time duration of the sample is limited by the data availability. In the case of measured data, the sample length may be five or ten years, in the case of hindcasting the length may be longer (from 20 to 50 years). Selection of wave height values is done in different ways, for example the maximum significant wave height occurring during a storm can be considered, or the maximum annual significant wave height (if the length of data sample is a few tens of years). Data used for the extreme wave analysis should be taken only from significant events in the recorded time history and each value should be from a different event to ensure stochastic independence. The preferred approach to data selection is to take the maximum value from each wave storm to create a series of extreme values. Thus the extreme analysis can focus on a smaller series representing truly significant events. A set of declustered events can be identified by local maxima with a sufficient separation in time such that independence is a reasonable assumption, as already stated before. 1.3 Extreme value probability distribution There is no theoretical basis for selecting any particular probability distribution to represent the observed data. Therefore generally a best-fitting distribution function is chosen among the candidates (log-normal, Gumbel, Frechèt, Weibull). All these distributions have a theoretical base, but in long term statistics they are used as empirical data fitting. To describe the different distributions, it is useful to introduce the following parameters: • α - Shape parameter: determines the basic shape of a particular distribution • θ - Scale parameter: controls the degree of spread along the abscissa • ε - Location parameter: locates the position of the density function along the abscissa.

2 Data fitting to the probability distribution The probability that a wave height H ' is less than a specified wave height H is defined as: P = P( H ' < H )

(7.1)

The probability that H ' is greater than a specified wave height H can be defined as: Q = Q( H ' > H ) = 1 − P

(7.2)

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The first is called Cumulative Distribution Function, the second Probability of Exceedence. In the following some probability functions are listed: they are all possible candidates as Extreme Value Probability Distributions. 2.1 Normal Probability Distribution We begin with this common distribution, because it forms the basis for the LogNormal distribution described below. The equation for the CDF, based on a Normal distribution with sample mean H and sample standard deviation σH is: P=

1

σ H 2π

H

∫e

1 H −H −  2 σ H

   

2

−∞

H −H = Φ  σH

  = Φ ( Z ) 

(7.3)

where H is wave height, H mean waveheight, σH standard deviation of waveheight and Z the standard normal variate H −H Z =   σH

 H 1  = H− σ σ H H 

(7.4)

Standard Normal Probability Tables are used to relate P to Z. However because we are interested in extreme values, the common version of Normal Probability tables do not cover a large enough range. Table (7.1) presents an extended range, in which only negative values of Z are shown (only the left hand of the normal distribution is represented here), If Z is known, such a table will yield P=Φ(z) as defined by equation (7.4). Since the standard normal probability distribution is symmetrical, the positive counterpart of the probability distribution (i.e. the values P for positive values of Z), is obtained as Ppos =1-Pneg. The tables can also be used in reverse: Z can be obtained from P. We define that inverse operation symbolically as:

Z = Φ −1 ( P)

(7.5)

Table 7.1 - Probability table: P=Φ(z) Z 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

0 0.5000 0.4602 0.4207 0.3821 0.3446 0.3085 0.2743 0.2420 0.2119 0.1841

-1 0.1587 0.1357 0.1151 0.0968 0.0808 0.0668 0.0548 0.0446 0.0359 0.0287

-2 0.0228 0.0179 0.0139 0.0107 0.0082 0.0062 0.0047 0.0035 0.0026 0.0019

-3 1.350E-03 9.677E-04 6.872 E-04 4.835 E-04 3.370 E-04 2.327 E-04 1.591 E-04 1.078 E-04 7.237 E-05 4.812 E-05

-4 3.169 E-05 2.067 E-05 1.335 E-05 8.456 E-06 5.417 E-06 3.401 E-06 2.115 E-06 1.302 E-06 7.944 E-07 4.799 E-07

-5 2.871 E-07 1.701 E-07 9.983 E-08 5.802 E-08 3.340 E-08 1.904 E-08 1.075 E-08 6.008 E-09 3.326 E-09 1.824 E-09

Equation (7.4) shows that Z is actually a linear function of H and therefore, the appropriate transformation of axes for a normally distributed CDF would be:

LONG TERM WAVE STATISTICS

Y = Z = Φ −1 ( P );

131 (7.6)

X =H

The slope and the intercept of the straight line, according to eq. (7.4), would be 1 ;

A=

B=−

σH

H

(7.7)

σH

This transformation can be tested if a series of points is normally distributed. The test for normality is: Are the points on a straight line? The Y-axis transformation comes from eqn. (7.6). When reduced variate is plotted against H, points are much closer to a straight line than they would be if the probability value were used. 2.2 Log-Normal Distribution Using variables Y=Φ-1(P) and lnH as X, Y and X should produce almost a straight line. The CDF of P versus lnH is given by: Y = Z = Φ −1 ( P ) =

ln H − ln H

σ ln H

=

1

σ ln H

ln H −

ln H

σ ln H

(7.8)

or:

A=

Y = Φ −1 ( P )

(7.9)

X = ln H

(7.10)

1 ;

σ ln H

B=−

ln H

σ ln H

(7.11)

The individual points do not lie exactly on a straight line, but their relationship may be approximated by the straight line. The equation of the straight line of best fit is obtained by linear regression analysis. 2.3 Gumbel distribution In addition to log-normal probability distribution, it is possible to use distributions developed specifically for analysis of extreme values. These models were originally derived for a limited number of “ordered statistics” such as a set of maximum annual floods sorted in descending order. The Gumbel distribution is given by:   H −γ P = exp − exp − β  

   

(7.12)

This equation can be linearized by taking the logs by both sides.  H −γ ln P = − exp − β  and taking logs again

  

(7.13)

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ln(− ln( P)) = −

H −γ

(7.14)

β

or 1 γ  1  H −γ − ln  ln  = = H− β β β  P

(7.15)

The reduced variate Y will be called G. The resulting transformation is: 1  1 Y = − ln ln  = G ; X = H ; A = ; β  P

B=−

γ β

(7.16)

2.4 Weibull distribution The above distributions all have two parameters. A more versatile extreme value distribution is the three-parameter Weibull distribution: α

  H −γ P = 1 − exp −    β 

   

  

(7.17)

which may also be expressed as:   H −γ Q = exp −    β 

α

  

   

(7.18)

Linear transformation may be accomplished by taking the logs of both sides:  H −γ − ln Q =   β

α

  

(7.19)

H −γ

(7.20)

which results in: 1/α

 1  ln   Q

=

β

1/α

 1 Y =  ln   Q

= W; X = H; A =

1

β

;B=−

γ β

(7.21)

The Weibull distribution has three parameters (α, β and γ). Linear regression provides only two constants (A and B) and if we want to continue to use linear regression analysis, the determination of the third coefficient (α) will require some trial and error. Assuming a different value of α the curvature of the points will change. The parameter γ in the Weibull and Gumbel distributions has physical meaning. It is the lowest limit of H (when H=γ, Q=1 or P=0). Thus γ is

LONG TERM WAVE STATISTICS

133

theoretically equal to the threshold value in a Peak over Threshold data set. This can be used as a check.

3 Parameter calculation Once the probability distribution has been chosen, the selected wave heights are reduced to a set of points to be plotted on a probabilistic paper. With this aim, data are sorted in descending order with reference to an index m such that m=1 corresponds to the highest wave, and m=N to the smallest one and a value of exceedence (or non-exceedence) probability must be assigned to each extreme data value. A simple estimation of the exceedence Q(Hm) and of the total probability P is given respectively by equations: Q( H m ) =

m ; N +1

P ( H m ) = 1 − PS ( H m )

(7.22)

So it is possible to individuate an interpolating line which represents the best fitting of data to the chosen distribution. Generally for this purpose two different approaches may be used: graphical or computational. Graphical approach: any set of measured data is plotted such that the selected distribution lies on a straight line; in such a way the goodness of fit and the extrapolation procedure are visually evaluated. Thus, once the particular probability paper is selected, abscissa and ordinate scales must be constructed in order to make the distribution appear as a straight line. The linear ordinate scale y is related to the cumulative probability P, and the linear abscissa x is related to the variate H according to the relationship relative to the chosen probability distribution. Computational approach: the Method of Moments is often used, but also Least Squares Method and Maximum Likelihood Method may be employed. Method of Moments consists in equating the first two or three moments of the distribution to the estimated moments calculated by data set. Regardless to the method used, it is preferable to plot the computed distribution and data together to ensure that the fit is consistent with good engineering judgement. 3.1 Statistical tests of fit When several candidate distribution functions are under consideration, one is selected as a best fit to the data. The selection criteria can range from visual inspection of plotted results and simple statistics such as the correlation between data and model. A possible method for testing the goodness of probability distribution is the chi-square test. The general procedure involves the use of a statistic with an appropriate chisquare distribution as a measure of discrepancy between an observed probability density function and the theoretical probability density function which is

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expected for the theory in exam. A hypothesis of equivalence is then tested by studying the sampling distribution of this statistic. To be more specific, consider a sample of N independent observation from a random variable x with a probability density function p(x). Let the N observations be grouped into K intervals, called class interval, which together form a frequency histogram. The number of observations falling within the i-th class interval is called observed frequency in the i-th class and will be denoted by fi. The number of observation that would be expected to fall within the class interval if the true probability density function of x were p0(x) is called expected frequency in the i-th class interval and will be denoted by Fi. Now, the discrepancy between the observed frequency and the expected frequency within each class interval is given by fi−Fi. To measure the total discrepancy for all class interval, the squares of the discrepancies in each interval are normalized by the associated expected frequencies and summed to obtain the sample statistic K

( f i − Fi )2

i =1

fi

χ2 =∑

(7.23)

The goodness of the chosen probability distribution is related to the tendency of chi-square to approach to zero. In the ideal case that the chosen probability distribution is coincident with the observed one, chi-square matches zero. For each value i, the χ2-test calculates the squared weighted mean of differences between analytical and empirical frequencies. For χ2=0 the sample and the distribution agree; the difference between the two distributions increases with χ2. The ratio (fi−Fi)2/fi represents the relative difference between the analytical and empirical frequency, for a given value n. The χ2-test is carried out thought the following steps: • • • •

Divide the sample into k classes with the same amplitude; Select a confidence threshold P; Individuate the value χ2 by the table 7.2, for a given number of degrees of freedom (ν=k−1). Compare the empirical and the analytical values χ2 .

3.2 Confidence intervals Confidence intervals associated with the chosen distribution function should also be estimated, to appraise the closeness of fit of the data points to the fitted distribution on either sides of the fitted line. The scatter of data is best described in terms of confidence limits individuated by a pair of curves drawn on either sides of the best-fit line; they provide the confidence bands within which data are expected to lie with corresponding probability, namely a 50% or 90% confidence bands border a zone within which data are expected to lie with a 50% or 90% probability.

LONG TERM WAVE STATISTICS

135

For given sample size N and chosen confidence probability level, a pair of height limits for the particular statistics is provided. This method assumes that the residual are normally distributed and is only valid for the range of data used in the fit; thus it cannot be extrapolated beyond the observed data and therefore cannot be used for the prediction. An alternative method of confidence bands determination involves the Monte Carlo simulation to generate random data sets deriving from the best-fit distribution that have been obtained. The spread of simulated data can be used to estimate the confidence or the uncertainty attached to any chosen value. Differently from the previous, this approach is advantageous as it can be used on either observed and predicted values of wave height. Once data have been fitted to the probability distribution, the latter must be extrapolated in order to evaluate design wave height corresponding to a fixed return period. Return period is defined as the average time interval running between two successive events of the design wave height being equalled or exceeded, and is correlated to the exceedance probability QS by the relation: TR 1 1 = = r Q S ( H TR ) 1 − Q ( H TR )

(7.24)

where r is the interval of record relative to each wave height measurement. From equation (7.24) the value of cumulative probability associated to a fixed return period is derived and the corresponding wave height may be determined from the best-fit line that has been plotted. This design value is defined as the individual data points used in the analysis (significant height, maximum height, etc. depending on adopted data) and should be equalled or exceeded on average once over the duration of any return period. In the project contest the encounter probability E may also be used, which is the probability that the design wave height is once equalled or exceeded during the structure lifetime L. The relation between the encounter probability and the return period is the following:  r E = 1 −  TR

  

L/r

(7.25)

3.3 Statistics of offshore extreme waves For the determination of the so called “design wave” needed for the functional and structural design it is necessary to perform a statistical elaboration of the highest wave heights, recorded during the peak of independent sea storms relative to the time series deduced by wave data. When wave height data are available, the statistical analysis of incomplete data series is applied. In this method, the significant wave heights for different directional sectors are selected; each height selected is greater than a known threshold value and identifies an independent storm peak. The incomplete series method (POT) is generally preferred to the annual maxima method (because of

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the limited number of data) and the complete data series method (because some wave events may be not independent). The POT method defines the temporal characteristics of storms by the selection of the events associated to wave heights greater than a threshold wave height value H0. In the simplest case of the local maxima method the random and independent variables are equally distributed and governed by a random counter. The random counter is the expected number of storms exceeding the threshold wave height H0 during a year. If the storms with height H>H0 have a Poisson distribution, the following equation is valid: P ( Na = n) = λn exp − λ / n!

(7.26)

Table 7.2 - χ2 table for given values of degrees of freedom ν.

Where λ = λˆ = n / M is the mean number of storms with H>H0 during a year, n is the total number of storms with H>H0. The implicit relationship between Hs,T and T is:

LONG TERM WAVE STATISTICS

F ( H s ,T ≥ H 0 ) = 1 − 1 / λˆT

137

(7.27)

for a given value T, Hs is determined through the equation (7.27). The equation (7.27) is independent by the process and is generally valid if the storms with H>H0 have a statistic distribution and if λˆ represents how many times the threshold value H0 is exceeded per time unit. Let us introduce the exponential distribution F: F ( H s | H s ≥ H 0 ) = 1 − exp[( H s − H 0 ) / β ]

(7.28)

we obtain: H s = H 0 + βˆ ln λˆ + β ln T

(7.29)

The eqn. (7.29) relates the wave height and the return period, and βˆ , λˆ are empirical parameter. For a given H0, β is the estimation of the maximum probability and can be calculated also through the method of moments. The variance associated to the estimation of Hs,T, is given by: Var ( Hs, T ) =

β2 N

[1 + (ln λˆ + ln T ) ] 2

(7.30)

Through the equation (7.30) the standard error σ = Var ( H s,T ) and the confidence interval can be obtained. If the probability is 95%, the “true” value of Hs,T is included between Hs±2σ, where Hs,T is given by the equation (7.29) and σ is given by equation (7.30). The data fitting is carried out by selecting the wave height values, plotted on a probability map. Data are taken in decreasing order and are represented by the following expression: C H m = {H m }mN =1

(7.31)

where the m=1 and m=N are the highest and the smallest height indexes, respectively. The coefficients of the fitting curve are identified through the method of moments, which is based on the identity between the analytically determined mean µH , and variance σ2H and the statistical mean µHm and variance σ2Hm . After the data fitting, the wave height associated with the return period can be determined. The return period Tr is the mean time interval between two consecutive wave events with H≥H0. Tr depends on exceeding the probability as follows: Tr 1 1 (7.32) = = r 1 − P ( H Tr ) Q ( H Tr ) where r is the mean time interval at which the statistic data are referred.

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3.4 Wave height persistence Wave persistence traditionally refers to duration of conditions in which wave heights are above or below a certain threshold value. This information is needed in connection with the operations of many marine structures (for example, most pipeline dredges cannot work in wave heights greater than 1m). A simple parametrical model is described in the Italian Wave Atlas (APAT, 2004) to evaluate the main persistence. Another model commonly used is the one of Kuwashima & Hogben (1986), which is based on the mean persistence τ ( H ) obtained by the Weibull distribution which parameters can be determined by the Least Squares Method: τ = A[ − ln( Q )] − β

(7.33)

where Q is the exceedence probability of the threshold H ' given by the Weibull distribution:

{

Q( H ' ) = exp − [(H '−ε ) / ϑ ]α

}

(7.34)

and the exceedence probability of the nondimensional duration x (τ normalized with the mean duration τ ) is given by:

Q ( x ) = exp − ( x / ϑ s ) α s

(7.35)

where αs and ϑs are given by the lower-limited Weibull distribution:

α s = 0.267γ ( H ' / H ) 0.4

(7.36)

ϑ s = 1 /[ Γ( 1 + 1 / α s )] α s

(7.37)

being H’ the threshold value height and H the mean value given by: H = ϑ Γ( 1 + 1 / α ) + ε

(7.38)

γ = α + 1.8ε /( H − ε )

(7.39)

and:

ε being the lower limit of the Weibull distribution. 3.5 Case study

In order to obtain long term statistics of wave data, we refer to significant wave heights measured by the APAT wave buoy located offshore the coastline of Monopoli, in the Adriatic Sea, which refers to a period of observation r=9 years (1989-1997). The values of 29 peak-over-threshold significant wave heights for the direction DD=350°N have been reported in table 7.3 together with their exceedence probability Q(Hm). The mean and variance of data sample are given by:

LONG TERM WAVE STATISTICS

µ= σ2 =

1 N

N

1 N

139

∑ Hm

(7.40)

∑ (H m − µ) 2

(7.41)

m =1

N

m =1

The Probability Distribution Function is given by:   H −γ P = exp − exp − β  

    

where γ and β are obtained through the method of moments: µˆ = γ + kβ = γ + 0.577 β the variance σ2 is given by:

σ = 2

π

2

6 So the method of moments gives:

µˆ =

β ≅ 1.645β

1 N ∑ H m =2.2690 m N m =1

2

and

2

σˆ 2 =

1 N

N

2

∑ ( H m − µˆ ) 2 =1.1429 m

m =1

the above values give:

σˆ = 1.645β ⇒ β = 2

σˆ

2

2

1.645

=0.8335 m-1

µˆ = γ + 0.577 β ⇒ γ = µˆ − 0.577 β =1.7880 m The PDF is P(H) =exp(−exp[−1.1997 (H−1.7880)]) The transformed variable is : W=-ln{-ln[P(H)]}= 1.1997 (H-1.7880). The significant wave heights are represented in figure 7.1. On the y-axis the transformed variable W and the PDF P is reported. The application of the χ2 test requires the definition of the following four significant wave height classes: 1) Hs<1.5 m 2) 1.5 ≤Hs<2.5 m 3) 2.5 ≤ Hs <3.5 m 4) 3.5≤ Hs <5.0 m The significant wave heights are obtained as: H=1.7880−{ln[−lnP(H)]}/1.1997 2

In order to perform χ test, in table 7.4 the analytical and empirical significant wave heights are given, with a confidence level P=95%. The analytical value of χ2, reported in table 7.2 with reference to the confidence level of 95% and the number of degrees of freedom, is greater than

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the empirical value. The hypothesis that the sample follows the Gumbel distribution is so verified with a confidence value P=95%. The mean time interval between the peak values of the significant wave height in the storms is r=9years/29=0.30 years. The degrees of freedom are given by ν=k-1 where k are the four classes. The number of linearly independent conditions is 3; one of these conditions identifies the fourth class, the other two conditions are associated with the parameters calculated from the sample. The empirical value of χ2 is calculated as: χ2=(9-8)2/9+(10-10)2/10+(5-5)2/5+(5-6)2/5=0.31 Tr [years] 6.00

W

0.9975

5.00

P(Hs) 0.9932

4.00

0.9818

3.00

0.9514

2.00

0.8734

1.00

0.6922

0.00

0.3678

-1.00

0.0659

-2.00

0.0006 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Tr=100 yr Tr=50 yr Tr=20 yr Tr=10 yr

Hs [m]

Figure 7.1 – Gumbel distribution fitted with data. The cumulate probability can be calculated from the definition of return period, and the “design” wave height can be obtained by knowing the empirical parameters above calculated: TR 1 = r 1 − P ( H TR )

P(HTR)=1−r/TR =1−0.30/50 = 0.994 W=5.113=1.1997 (H-1.7880)

LONG TERM WAVE STATISTICS

141

HTR =(5.113/1.1997) + 1.7880 =6.05m The wave period can be given by (Boccotti, 1997): 1/ 2 TH TR = 0.95 ⋅ 9π {H Tr / 4 g } =10.55s

The calculated value of the design wave height is 6.05m (TR=50 years) and its wave period is 10.55s. Table 7.3 - Ordered peak significant wave heights and associated Q(Hm).

m

Hm

Q(Hm)

1

4.4

0.9808

2

3.9

0.9464

3

3.8

0.9121

4

3.7

0.8777

5

3.6

0.8434

6

3.5

0.8091

7

3.4

0.7747

8

3.1

0.7404

9

3

0.7060

10

2.8

0.6717

11

2.7

0.6374

12

2.4

0.6030

13

2.3

0.5687

14

2.2

0.5343

15

2.1

0.5000

16

2

0.4657

17

1.9

0.4313

18

1.8

0.3970

19

1.7

0.3626

20

1.6

0.3283

21

1.5

0.2940

22

1.4

0.2596

23

1.3

0.2253

24

1.2

0.1909

25

1.1

0.1566

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

26

1

0.1223

27

0.9

0.0879

28

0.8

0.0536

29

0.7

0.0192

Table 7.4 - Analytical and empirical significant wave heights Hm[m]

class

Hmt[m]

class

4.4

4 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1

4.8

4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1

3.9 3.8 3.7 3.6 3.5 3.4 3.1 3 2.8 2.7 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7

3.9 3.5 3.5 3.5 2.8 2.6 2.6 2.5 2.5 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.6 1.5 1.5 1.3 1.2 1.2 1.1 1.0 0.9 0.7 0.6 0.3

Chapter 8 Wave transformation in the coastal zone 1 Wave energy and energy flux When a wave train propagates through a fluid, part of its of energy is transmitted throughout the fluid; the rate of energy transfer matches the group celerity Cg which is the propagation velocity of the wave packet obtained by the envelope of the group of random waves. Energy consists of two contributions: potential energy resulting from the displacement of the free surface and kinetic energy due to the motion of water particles throughout the fluid. The total energy and its transmission is of extreme importance in determining how wave properties change when a wave approaches to a shore or/and interacts with a structure. 1.1 Potential Energy