Introduction to nearshore hydrodynamics

INTRODUCTION TO
NEARSHORE
HYDRODYNAMICS


ADVANCED SERIES ON OCEAN ENGINEERING
Series Editor-in-Chief
Philip L- F Liu (Cornell University)

Vol. 9

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Vol. 17

Offshore Structure Modeling
by Subrata K. Chakrabarti (Chicago Bridge & Iron Technical
Services Co., USA)
Water Waves Generated by Underwater Explosion
by Bernard Le Mehaute and Shen Wang (Univ. Miami)
Ocean Surface Waves; Their Physics and Prediction
by Sfanislaw R Massel (Australian Inst. of Marine Sci)
Hydrodynamics Around Cylindrical Structures
by B Muflu Sumer and Jnrgen Fredsne (Tech. Univ. of Denmark)

Water Wave Propagation Over Uneven Bottoms
Part I - Linear Wave Propagation
by Maarten W Dingemans (Delft Hydraulics)
Part II - Non-linear Wave Propagation
by Maarten W Dingemans (Delft Hydraulics)
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Random Seas and Design of Maritime Structures (2nd Edition)
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Vol. 18 Beach Nourishment: Theory and Practice
by Robert G. Dean (Univ. Florida)
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by Subrata K. Chakrabarti (Offshore Structure Analysis, Inc., Illinois, USA)
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by Robert 7: Hudspefh (Oregon State Univ., USA)
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by Edward M. Lewandowski (Computer Sciences Corporation, USA)
Vol. 23 Theory and Applications of Ocean Surface Waves
Part 1: Linear Aspects
Part 2: Nonlinear Aspects
by Chiang C. Mei (Massachusetts Inst. of Technology, USA),
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Dick K. P. Yue (Massachusetts Inst. of Technology, USA)

Vol. 24 Introduction to Nearshore Hydrodynamics
by Ib A. Svendsen (Univ. of Delaware, USA)


Advanced Series on Ocean Engineering -Volume 24

INTRODUCTION TO
NEARSHORE
HYDRODYNAMICS

Ib A. Svendsen
University of Delaware, USA

Scientific
1: World
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N E W JERSEY * LONDON * SINGAPORE

BElJlNG * SHANGHAI

HONG KONG

TAIPEI * CHENNAI


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To my wife
Karin


Prologue

My husband, distinguished professor emeritus, Dr. Ib A. Svendsen, died

on Sunday, December 19, 2004.
He had been working intensely to finish this book for the past several
months following his retirement from University of Delaware on September
1, 2004. It seemed to be both a rewarding, but on the same time highly
frustrating job, as anybody who has given birth to a book will probably
recognize.
As he mentions in his preface, one of the problems he was facing was
deciding what to include in the book. He knew that some topics might
have been included or covered in more details, and he was considering the
possibility of an additional book exploring these subjects, and embodying
the response from this edition.
On December 10 Ib sent the manuscript to World Scientific Publisher.
On Tuesday December 14 he made the last organizational changes to his
files on the book, and inquired of the publisher how much longer he would
have for changes and additions. He was looking forward to discussions with
colleagues and students about the contents of the book.
But early on December 15 he collapsed with cardiac arrest at the fitness
center at University of Delaware. He died without regaining consciousness.
It is my hope that this book will become the means of learning and inspiration for future graduate students and others within coastal engineering
as was Ib’s sincere wish.
The royalties from this and Ib’s other publications will be used to
finance a memorial fund in his honour: Ib A. Svendsen Endowment,
c/o Department of Civil Engineering, University of Delaware, Newark,
DE 19716. This fund will benefit University of Delaware civil engineering students in their international studies.
Karin Orngreen-Svendsen
Landenberg, March 21, 2005


Preface

The objective of this book is to provide an introduction for graduate
students and other newcomers to the field of nearshore hydrodynamics that
describes the basics and helps de-mystify some of the many research results
only found in journals, reports and conference proceedings.
When I decided to write this book I thought this would be a fairly easy
task. From many years of teaching and research in the field of nearshore
hydrodynamics I had extensive notes about the major topics and I thought
it would be a straight forward exercise to expand the notes into a text that
would meet that objective.
Not so. From being a task of considering how to expand the notes which I found enjoyable - the work rapidly turned into the more stressful
task of deciding what to omit from the book and how to cut. I had completely underestimated the number of relevant topics in modern nearshore
hydrodynamics, the amount of important research results produced over
the last decades, and the complexity of many of those results.
In the end I came up with a compromise that became this book. I have
considered some topics are so fundamental that they have to be covered in
substantial detail. Otherwise one could not claim this to be a textbook.
On the other hand, for reasons of space, sections describing further developments have been written in a less detailed, almost review style and
supplemented with a selection of references to the literature. The list of
references is not exhaustive but rather meant to give the author's modest suggestions for what may be the most helpful introductory reading
for a newcomer to the field. This unfortunately means that many excellent papers are not included which in no way should be taken as an indication of lesser quality. The transition between the two styles may be
gradual within each subject. It is hoped that the detailed coverage of the

vii


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Introduction to nearshore hydrodynamics

fundamental topics such as linear wave theory, the basics of nonlinear Stokes
and Boussinesq wave theory, the nearshore circulation equations, etc., will
bring the reader’s insight and understanding to a point where he/she is able
to benefit from the sections that discuss the latest developments, is able to
read the current literature, and perhaps to start their own research.
Though no computational models are described in detail and the presentation focuses on the hydrodynamical aspects of the nearshore the choice of
topics and the presentation is oriented toward including the hydrodynamical basis in a wider sense for some of the most common model equations.
The principles that form the basis of good modelling can perhaps be simplified as the following:

If you want to model nature you rrmst copy nature
It you want to copy nature you must understand nature
This has been the motto behind the writing of this book.
The purpose of hydrodynamics is the mathematical description of what
is happening in nature, and the basic equations such as the Navier-Stokes
equations are as close to an exact copy of nature as we can come. Therefore misrepresentation of nature only comes in through the simplifications
and approximations that we introduce to be able to solve the particular
problem we consider. No model/equation is more accurate than the underlying assumptions or approximations. An important task in providing
the background for responsible applications of the equations of nearshore
hydrodynamics is therefore to carefully monitor and discuss the physical
implications of the assumptions and approximations we introduce. I have
tried to do just that throughout the text.
Todays models are becoming more and more sophisticated and complex. Usually this also means more and more accurate and the use of them
is becoming part of everyday life. Mostly this also means they become
more and more demanding of computer time and of man power to use and
interpret them. So in many applications there will be a decision about
which accuracy is needed. Is linear wave theory good enough? Are we outside the range of validity of a particular Boussinesq model? Nobody can
prevent users from deciding to use model equations/theories for situations
where they are insufficient or do not properly apply. Sometimes the results
are acceptable sometimes they are misleading. One parameter, such as for

example the wave height, may be accurately predicted for the conditions
considered while another, say the particle velocity, is not. It is generally


Preface

ix

outside the scope of this book to provide estimates of errors for particular
theories.
It may be easy to run a computer model. One has to remember, however, that all that comes out of that is numbers. An enormous amount of
numbers. They may be interpreted and plotted in diagrams to look like
nearshore flow properties. But knowing/understanding the powers and the
limitations of models requires understanding the basis for the equations.
Which features are represented in the equations, which not, why this or
that effect is important, and when, etc., is a first condition for generating
confidence in the results. It is the hope that the content of the book will
help serving that purpose and thereby promote the prudent and constructive use of models.
For reasons of space many important topics and aspects of nearshore
hydrodynamics have been left out.
One such is the testing of the theories using laboratory measurements.
A major reason is of course lack of space, but there are some important
concerns too. In a moment of outrageous provocation and frustration I once
wrote about laboratory experiments: “If there is a discrepancy between
the theory and the measurements it is likely to be due to errors in the
experiments”. The reason is that, while it is fairly easy to create good
theories, it is so difficult to conduct good experiments, in particular with
waves. Anybody who has tried can testify to all the many unwanted - and
often unanticipated - side effects and disturbances that occur even in a
simple wave experiment in a wave flume. And often those are the major

reasons for the deviations between the theory and the experiment designed
to test it. Therefore we have to be careful before we use an experimental
result to deem a well documented theory inaccurate or poor as long as we
are within the range of validity of the assumptions. This is also why I
prefer to replace the commonly used term L‘~erifi~ation”
of a model against
experimental data with the term “testing”. So though comparisons with
measurements can be found many places a systematic testing of theories
against laboratory measurements has not been one of the main objectives
of the book. In fact comparison of the simpler theories to more advanced
and accurate ones is often more revealing.
In a different role experimental results have been quoted extensively to
gain physical insight into areas where theoretical understanding is lacking.
This particularly applies to the hydrodynamics of waves in the surfzone.
Extensive field experiments have been conducted in particular over the
last two decades. The comprehensive and careful data analysis of those


X

Introduction t o nearshore hydrodynamics

experiments has provided insights and ideas for further study. However,
those results are described in the book only to the extent it is needed
to understand the hydrodynamical phenomena and the theories covering
them. It should be noted, though, that the way nearshore modelling is
developing today the direct comparison of model results with the complex
conditions on natural beaches will be one of the promising research areas in
the coming years. Unfortunately we can only just touch upon this subject
in an introductory book like this.

Wind wave spectra is an area that is more related to data analysis than
to hydrodynamics. Only the concept of energy spectra is explained as an
example of wave superposition and a brief overview of the ideas is given.
This also applies to nonlinear spectral Boussinesq models. The reader is
referred to the relevant literature.
Again for space reasons, evolution equations and concepts for time and
space varying waves based on Stokes’ wave theory have not been explored
at all. This applies to topics such as the side band instability, which mainly
occurs in deeper water, to the theory of slowly varying Stokes waves and
the nonlinear Schrodinger equation. An important reason for this choice is
that the Stokes’ wave theory has an uncanny habit of not working well in
the shallow water regions nearshore. Instead the Boussinesq wave theory,
which leads to nonlinear evolution equations for waves in shallow water,
has been covered in great detail. This theory has over the last one or
two decades been developed into an extremely useful and accurate tool for
nearshore applications. In fact it has even been extended to depths that
approach the deep water limit of the nearshore region which further adds
to its relevance.

Acknowledgements
A book like this is really influenced by a great number of contributions
over many years, often from people who do not even realize they have
contributed. I cannot here mention them all but I do want to thank my
colleague through many years Ivar G. Jonsson for numerous discussions
that helped develop my insight into the topics described in this book. He
was also co-author on an earlier book on The Hydrodynamics of Coastal
Regions, which has been the starting point for the description of linear
waves in this book.
Also a special thank to Howell Peregrine. Our extensive scientific discussions have been ongoing for decades and he more than anybody helped
open my eyes to the fascinations of fluid mechanics.

Preface

xi

Over the years I have also received many comments from graduate students t o the notes I have used in my courses. Those notes have formed the
initial basis for the book.
More focused on the present book has been valuable comments and
suggestions from Mick Haller and Ap Van Dongeren who reviewed early
versions of the first chapters and helped improving their content and form.
Also thanks to Jack Puleo for input t o the chapter on swash, to Kevin Haas
for his many suggestion for the chapter on breaking waves and surfzone
dynamics, and to Francis Ting for generously providing his unpublished
data shown in that chapter.
Also sincere thanks to Qun Zhao who throughout the work has been
assisting in many ways, including in preparing the many drawings and
patiently responded to my many requests for changes. And to my secretary
Rosalie Kirlan for taking care of scanning figures from papers and reports.
I also want express my gratitude t o Per Madsen for numerous discussions
and extensive help with and insight into many subjects, particularly on the
more advanced topics on linear and nonlinear waves.
And special thanks to Jurjen Battjes who undertook the task to read a
late version of the entire manuscript. His meticulous comments and suggestions have been invaluable as they helped not only to remove typos but
t o improve many unclear or ambiguous passages in the manuscript.
More than anybody, however, I a m indebted to my wife Karin. It is
an understatement to say that without her patient and caring support this
book would not have been finished.
However, inspite of all efforts t o avoid it, it is inevitable that a book like
this will have misprints and errors and, even worse, reflect my misunderstandings of other scientists work. I apologize to the authors for any such

mistakes and hope that they will have time and patience to point them out
t o me.
The work on this book has been partially funded by the National
Oceanographic Partnerships Program (NOPP) under the ONR grant
N0014-99-1-1051, which has lead to the development of the open-source
Nearshore Community Model (NearCoM) briefly described in the last chapter. It is hoped that this book will help more students, engineers and coming scientists to understand the basic theories of nearshore hydrodynamics
which such models are based on and thereby be able to use them wisely.
IAS
Landenberg, PA, December 2004


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Contents

Preface

vii

1. Introduction

1

1.1 A brief historical overview . . . . . . . . . . . . . . . . . .
1.2 Summary of content . . . . . . . . . . . . . . . . . . . . .
1.3 References .Chapter 1 . . . . . . . . . . . . . . . . . . . .

2 . Hydrodynamic Background

1
2
9
11

Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
Kinematics of fluid flow . . . . . . . . . . . . . . . . . . .
2.2.1
Eulerian versus Lagrangian description . . . . . .
2.2.2
Streamlines, pathlines. streaklines . . . . . . . . .
2.2.3 Vorticity w i and deformation tensor eij . . . . . .
2.2.4
Gauss’ theorem, Green’s theorems . . . . . . . . .
2.2.5 The kinematic transport theorem, Leibniz rule . .
2.3 Dynamics of fluid flow . . . . . . . . . . . . . . . . . . . .
2.3.1
Conservation of mass . . . . . . . . . . . . . . . .
2.3.2
Conservation of momentum . . . . . . . . . . . .
2.3.3 Stokes’ viscosity law, the Navier-Stokes equations
2.3.4
The boundary layer approximation . . . . . . . .
2.3.5
Energy dissipation in viscous flow . . . . . . . . .
2.3.6
The Euler equations, irrotational flow . . . . . . .
2.4 Conditions a t fixed and moving boundaries . . . . . . . .
2.4.1 Kinematic conditions . . . . . . . . . . . . . . . .
2.4.2 Dynamic conditions . . . . . . . . . . . . . . . . .

2.5 Basic ideas for turbulent flow . . . . . . . . . . . . . . . .
2.1
2.2

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11
12
12
13
14
15
16
17
17
18
20
22
23
23
24
25
27
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Introduction t o nearshore hydrodynamics

2.6
2.7
2.8

2.5.1 Reynolds’ decomposition of physical quantities . .
2.5.2 Determination of the turbulent mean flow . . . .
2.5.3 The Reynolds equations . . . . . . . . . . . . . .
2.5.4 Modelling of turbulent stresses . . . . . . . . . . .
Energy flux in a flow . . . . . . . . . . . . . . . . . . . . .
Appendix: Tensor notation . . . . . . . . . . . . . . . . .
References - Chapter 2 . . . . . . . . . . . . . . . . . . . .

3 . Linear Waves
3.1 Assumptions and the simplified equations . . . . . . . . .
3.2 Basic solution for linear waves . . . . . . . . . . . . . . .
3.2.1 Solution for q5 and r] . . . . . . . . . . . . . . . .
3.2.2 Evaluation of linear waves . . . . . . . . . . . . .
3.2.3 Particle motion . . . . . . . . . . . . . . . . . . .
3.2.4 The pressure variation . . . . . . . . . . . . . . .
3.2.5 Deep water and shallow water approximations . .
3.3 Time averaged properties of linear waves in one
horizontal dimension (1DH) . . . . . . . . . . . . . . . . .
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.3.2 Mass and volume flux . . . . . . . . . . . . . . . .
3.3.3 Momentum flux-radiation stress . . . . . . . . .
3.3.4 Energy density . . . . . . . . . . . . . . . . . . .
3.3.5 Energy flux . . . . . . . . . . . . . . . . . . . . .
3.3.6 Dimensionless functions for wave averaged
quantities . . . . . . . . . . . . . . . . . . . . . .
3.4 Superposition of linear waves . . . . . . . . . . . . . . . .

3.4.1 Standing waves . . . . . . . . . . . . . . . . . . .
3.4.2 Wave groups . . . . . . . . . . . . . . . . . . . . .
3.4.3 Wave spectra . . . . . . . . . . . . . . . . . . . .
3.5 Linear wave propagationover unevenbottom . . . . . . .
3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.5.2 Shoaling and refraction . . . . . . . . . . . . . . .
3.5.2.1 Simple shoaling . . . . . . . . . . . . . .
3.5.2.2 Determination of the refraction pattern
3.5.3 Refraction by ray tracing . . . . . . . . . . . . .
3.5.4 The geometrical optics approximation . . . . . . .
3.5.5 Kinematic wave theory . . . . . . . . . . . . . . .
3.6 Wave modification by currents . . . . . . . . . . . . . . .
3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . .

29
30
34
39
43
45

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49
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55
66

72
77

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100
102
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108
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154


Contents

Waves on a steady. locally uniform current . . . .
Vertically varying currents . . . . . . . . . . . . .
The kinematics and dynamics of wave propagation
on current fields . . . . . . . . . . . . . . . . . . .

Combined refraction-diffraction . . . . . . . . . . . . . . .
3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.7.2 The wave equation for linear long waves . . . . .
3.7.3 The mild slope equation . . . . . . . . . . . . . .
3.7.4 Further developments of the MSE . . . . . . . .
3.7.5 The parabolic approximation . . . . . . . . . . .
References - Chapter 3 . . . . . . . . . . . . . . . . . . . .

3.6.2
3.6.3
3.6.4
3.7

3.8

4. Energy Balance in the Nearshore Region
4.1
4.2
4.3

4.4
4.5
4.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
The energy equation . . . . . . . . . . . . . . . . . . . . .
The energy balance for periodic waves . . . . . . . . . . .
4.3.1 Introduction of dimensionless parameters for l-D
wave motion . . . . . . . . . . . . . . . . . . . . .
4.3.2 A closed form solution of the energy equation . .

4.3.3 The energy equation for steady irregular waves .
The general energy equation: Unsteady wave-current
motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The wave action equation . . . . . . . . . . . . . . . . . .
References - Chapter 4 . . . . . . . . . . . . . . . . . . . .

5. Properties of Breaking Waves
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The highest possible wave on constant depth . . . . . . .
5.3 Qualitative description of wave breaking . . . . . . . . . .
5.3.1 Analysis of the momentum variation in a
transition (Or: Why do the waves break?) . . . .
5.4 Wave characteristics at the breakpoint . . . . . . . . . . .
5.5 Experimental results for surfzone waves . . . . . . . . . .
5.5.1 Qualitative surfzone characteristics . . . . . . . .
5.5.2 The phase velocity c . . . . . . . . . . . . . . . .
5.5.3 Surface profiles v(t) . . . . . . . . . . . . . . . . .
5.5.4 The surface shape parameter Bo . . . . . . . . . .
5.5.5 The crest elevation q c / H . . . . . . . . . . . . . .

xv

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165
166
166
168
171
179

189
199
207
207
207
213
213
215
221
222
227
228

229
229
230
232
238
242
246
247
248
250
252
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Introduction to nearshore hydrodynamics

xvi

5.5.6 The roller area . . . . . . . . . . . . . . . . . . . .
5.5.7 Measurements of particle velocities . . . . . . . .
5.5.8 Turbulence intensities . . . . . . . . . . . . . . . .
5.5.9 The values of P,,, B, and D . . . . . . . . . . .
5.5.10 The wave generated shear stress u,W, . . . . . .
5.6 Surfzone wave modelling . . . . . . . . . . . . . . . . . . .
5.6.1 Surfzone assumptions . . . . . . . . . . . . . . . .
5.6.2 Energy flux E f for surfzone waves . . . . . . . . .
5.6.3 Radiation stress in surfzone waves . . . . . . . . .
5.6.4 Volume flux in surfzone waves . . . . . . . . . . .
5.6.5 The phase velocities for quasi-steady breaking
waves . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.6 The energy dissipation in quasi steady surfzone
waves . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Further analysis of the energy dissipation . . . . . . . . .
5.7.1 Energy dissipation for random waves . . . . . . .
5.7.2 Energy dissipation with a threshold . . . . . . . .
5.7.3 A model for roller energy decay . . . . . . . . . .
5.7.4 Advanced computational methods for surzone
waves . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Swash . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 References - Chapter 5 . . . . . . . . . . . . . . . . . . . .
6 . Wave Models Based on Linear Wave Theory
6.1
6.2
6.3

6.4
6.5

6.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
1DH shoaling-breaking model . . . . . . . . . . . . . . . .
2DH refraction models . . . . . . . . . . . . . . . . . . . .
6.3.1 The wave propagation pattern . . . . . . . . . . .
6.3.2 Determination of the wave amplitude variation .
Wave action models . . . . . . . . . . . . . . . . . . . . .
Models based on the mild slope equation and the parabolic
approximation . . . . . . . . . . . . . . . . . . . . . . . .
References - Chapter 6 . . . . . . . . . . . . . . . . . . . .

7 . Nonlinear Waves: Analysis of Parameters

255
258
262
263
269
271
271
274
279
281
282
285
287
287
291
292

293
295
303
311
311
313
313
313
317
318

319
320
323

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
323
7.2 The equations for the classical nonlinear wave theories . . 325
7.3 The system of dimensionless variables used . . . . . . . . 328


Contents

7.4
7.5

7.6
7.7

Stokes waves . . . . . . . . . . . . . . . . . . . . . . . . .

Longwaves . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 The Stokes or Ursell parameter . . . . . . . . . .
7.5.2 Long waves of moderate amplitude . . . . . . . .
7.5.3 Long waves of small amplitude . . . . . . . . . . .
7.5.4 Long waves of large amplitude . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
References - Chapter 7 . . . . . . . . . . . . . . . . . . . .

8. Stokes Wave Theory
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Second order Stokes waves . . . . . . . . . . . . . . . . . .
8.2.1 Development of the perturbation expansion . . .
8.2.2 First order approximation . . . . . . . . . . . . .
8.2.3 Second order approximation . . . . . . . . . . . .
8.2.4 The solution for 4 2 . . . . . . . . . . . . . . . . .
8.2.5 The surface elevation . . . . . . . . . . . . . . .
8.2.6 The pressure p . . . . . . . . . . . . . . . . . . . .
8.2.7 The volume flux and determination of K . . . . .
8.2.8 Stokes’ two definitions of the phase velocity . . .
8.2.9 The particle motion . . . . . . . . . . . . . . . . .
8.2.10 Convergence and accuracy . . . . . . . . . . . . .
8.3 Higher order Stokes waves . . . . . . . . . . . . . . . . .
8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
8.3.2 Stokes third order theory . . . . . . . . . . . . . .
8.3.3 Waves with currents . . . . . . . . . . . . . . . .
8.3.4 Stokes fifth order theory . . . . . . . . . . . . . .
8.3.5 Very high order Stokes waves . . . . . . . . . . .
8.4 The stream function method . . . . . . . . . . . . . . . .
8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
8.4.2 Description of the stream function method . . .

8.4.3 Comparison of stream function results with a
Stokes 5th order solution . . . . . . . . . . . . . .
8.5 References - Chapter 8 . . . . . . . . . . . . . . . . . . . .
9. Long Wave Theory
9.1
9.2

Introduction . . . . . . . . . . . . . . . . . . . . . .
Solution for the Laplace equation . . . . . . . . . . .

xvii

331
333
335
335
336
337
338
338
341
341
342
342
345
346
349
351
355
358

362
364
366
369
369
370
371
372
373
374
374
374
378
379
381
381
383


xviii

Introduction to nearshore hydrodynamics

The Boussinesq equations . . . . . . . . . . . . . . . . . .
Boussinesq equations in one variable . . . . . . . . . . . .
9.4.1 The fourth order Boussinesq equation . . . . . . .
9.4.2 The third order Korteweg-deVries (KdV) equation
9.5 Cnoidal waves - Solitary waves . . . . . . . . . . . . . . .
9.5.1 The periodic case: Cnoidal waves . . . . . . . . .
9.5.2 Final cnoidal wave expressions . . . . . . . . . . .

9.5.3 Infinitely long waves: Solitary waves . . . . . . .
9.6 Analysis of cnoidal waves for practical applications . . . .
9.6.1 Specification of the wave motion . . . . . . . . . .
9.6.2 Velocities and pressures . . . . . . . . . . . . . . .
9.6.3 Wave averaged properties of cnoidal waves . . . .
9.6.4 Limitations for cnoidal waves . . . . . . . . . . .
9.7 Alternative forms of the Boussinesq equations - The linear
dispersion relation . . . . . . . . . . . . . . . . . . . . . .
9.7.1 Equations in terms of the velocity uo at the
bottom . . . . . . . . . . . . . . . . . . . . . . . .
9.7.2 Equations in terms of the velocity us at the MWS
9.7.3 The equations in terms of the depth averaged
velocity ii . . . . . . . . . . . . . . . . . . . . . .
9.7.4 The equations in terms of Q . . . . . . . . . . . .
9.7.5 The linear dispersion relation . . . . . . . . . . .
9.8 Equations for 2DH and varying depth . . . . . . . . . . .
9.9 Equations with enhanced deep water properties . . . . . .
9.9.1 Introduction . . . . . . . . . . . . . . . . . . . . .
9.9.2 Improvement of the linear dispersion properties .
9.9.3 Improvement of other properties . . . . . . . . . .
9.10 Further developments of Boussinesq modelling . . . . . .
9.10.1 Fully nonlinear models . . . . . . . . . . . . . . .
9.10.2 Extension of equations to O ( p 4 )accuracy . . . .
9.10.3 Waves with currents . . . . . . . . . . . . . . . .
9.10.4 Models of high order . . . . . . . . . . . . . . . .
9.10.5 Robust numerical methods . . . . . . . . . . . . .
9.10.6 Frequency domain methods for solving the
equations . . . . . . . . . . . . . . . . . . . . . . .
9.11 Boussinesq models for breaking waves . . . . . . . . . . .
9.11.1 Eddy viscosity models . . . . . . . . . . . . . . .

9.11.2 Models with roller enhancement . . . . . . . . . .
9.11.3 Vorticity models . . . . . . . . . . . . . . . . . . .

9.3
9.4

387
389
389
392
396
399
402
406
410
411
415
420
423
425
426
426
427
429
430
435
437
437
437
442

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445
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448
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Contents

xix

9.11.4 Wave breaking modelled by the nonlinear shallow
water equations . . . . . . . . . . . . . . . . . . . 451
9.12 Large amplitude long waves U >> 1: The nonlinear shallow
water equations (NSW) . . . . . . . . . . . . . . . . . . . 451
454
9.13 References - Chapter 9 . . . . . . . . . . . . . . . . . . . .
10. Boundary Layers
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 The boundary layer equations . Formulation of the
problem . . . . . . . . . . . . . . . . . . . . . . .
10.1.2 Perturbation expansion for u . . . . . . . . . . . .
10.1.3 The 1st order solution . . . . . . . . . . . . . . .
10.1.4 The 2nd order solution . . . . . . . . . . . . . . .

10.1.5 The steady streaming us in wave boundary layers
10.1.6 Results for u, . . . . . . . . . . . . . . . . . . . .
10.2 Energy dissipation in a linear wave boundary layer . . . .
10.3 Turbulent wave boundary layers . . . . . . . . . . . . . .
10.3.1 Rough turbulent flow . . . . . . . . . . . . . . . .
10.3.2 Energy dissipation in turbulent wave boundary
layers . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Bottom shear stress in 3D wave-current boundary
layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
10.4.2 Formulation of the problem . . . . . . . . . . . .
10.4.3 The mean shear stress . . . . . . . . . . . . . . .
10.4.4 Special cases . . . . . . . . . . . . . . . . . . . . .
10.5 References - Chapter 10 . . . . . . . . . . . . . . . . . . .
11. Nearshore Circulation

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Depth integrated conservation of mass . . . . . . . . . . .
11.2.1 Separation of waves and currents . . . . . . . . .
11.3 Conditions at fixed and moving boundaries, I1 . . . . . .
11.3.1 Kinematic conditions . . . . . . . . . . . . . . . .
11.3.2 Dynamic conditions . . . . . . . . . . . . . . . . .
11.4 Depth integrated momentum equation . . . . . . . . . . .
11.4.1 Integration of horizontal equations . . . . . . . .
11.4.2 Integration of the vertical momentum equation .

469
469
470
473

474
480
483
486
490
493
497
500
502
502
502
504
509
513
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520
523
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xx

Introduction t o nearshore hydrodynamics

11.5 The nearshore circulation equations . . . . . . . . . . . .
11.5.1 The time averaged momentum equation . . . . .
11.5.2 The equations for depth uniform currents . . . .
11.6 Analysis of the radiation stress in two horizontal
dimensions, 2DH . . . . . . . . . . . . . . . . . . . . . . .
11.6.1 Sap expressed in terms of S, and S, . . . . . . .
11.6.2 Radiation stress for linear waves in two horizontal
dimensions (2DH) . . . . . . . . . . . . . . . . . .
11.7 Examples on a long straight beach . . . . . . . . . . . . .
11.7.1 The momentum balance . . . . . . . . . . . . . .
11.7.2 The cross-shore momentum balance: Setdown and
setup . . . . . . . . . . . . . . . . . . . . . . . . .
11.7.3 Longshore currents . . . . . . . . . . . . . . . . .
11.7.4 Longshore current solution for a plane beach . . .
11.7.5 Discussion of the examples . . . . . . . . . . . . .
11.8 Wave drivers . . . . . . . . . . . . . . . . . . . . . . . . .
11.9 Conditions along open boundaries . . . . . . . . . . . . .
11.9.1 Introduction about open boundaries . . . . . . .
11.9.2 Absorbing-generating boundary conditions . . . .
11.9.3 Boundary conditions along cross-shore boundaries
11.10 References - Chapter 11 . . . . . . . . . . . . . . . . . .

12. Cross-Shore Circulation and Undertow
12.1 The vertical variation of currents . . . . . . . . . . . . . .
12.1.1 Introduction . . . . . . . . . . . . . . . . . . . . .
12.1.2 The governing equations for the variation over
depth of the 3D currents . . . . . . . . . . . . . .
12.2 The cross-shore circulation, undertow . . . . . . . . . . .
12.2.1 Formulation of the 2-D problem and general
solution . . . . . . . . . . . . . . . . . . . . . . .

12.2.2 Boundary conditions . . . . . . . . . . . . . . . .
12.2.3 Solution for the undertow profiles with depth
uniform vt and 011 . . . . . . . . . . . . . . . . . .
12.2.4 Discussion of results and comparison with
measurements . . . . . . . . . . . . . . . . . . . .
12.2.5 Solutions including the effect of the boundary
layer . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.6 Undertow outside the surfzone . . . . . . . . . . .
12.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . .
12.3 References - Chapter 12 . . . . . . . . . . . . . . . . . . .

540
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557
557
559
566
573
579
581
587
587
589
598
600
603

603
603
605
608
608
610
613
615
618
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xxi

Contents

13. Quasi-3D Nearshore Circulation Models
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Governing equations . . . . . . . . . . . . . . . . . . . . .
13.2.1 Time-averaged depth-integrated equations . . . .
13.2.2 Choices for splitting the current . . . . . . . . . .
13.3 Solution for the vertical velocity profiles . . . . . . . . .
13.4 Calculation of the integral terms: The dispersive
mixing coefficients . . . . . . . . . . . . . . . . . . . . . .
13.4.1 Final form of the basic equations . . . . . . . . .
13.5 Example: Longshore currents on a long straight coast . .
13.6 Applications and further developments of quasi-3D
modelling . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.6.1 The start-up of a longshore current . . . . . . . .
13.6.2 Rip currents . . . . . . . . . . . . . . . . . . . .
13.6.3 Curvilinear version of the SHORECIRC model .
13.6.4 The nearshore community model, NearCoM . . .
13.7 References - Chapter 13 . . . . . . . . . . . . . . . . . . .

14. Other Nearshore Flow Phenomena

627
627
631
631
632
633
637
640
64 1
645
645
647
649
654
655
661

14.1 Infragravity waves . . . . . . . . . . . . . . . . . . . . . .
14.1.1 Introduction . . . . . . . . . . . . . . . . . . . . .
14.1.2 Basic equations for infragravity waves . . . . . .
14.1.3 Homogeneous solutions Free edge waves . . . .
14.1.4 IG wave generation . . . . . . . . . . . . . . . .

14.2 Shear instabilities of longshore currents . . . . . . . . . .
14.2.1 Introduction . . . . . . . . . . . . . . . . . . . . .
14.2.2 The discovery of shear waves . . . . . . . . . . . .
14.2.3 Derivation of the basic equations . . . . . . . . .
14.2.4 Stability analysis of the equations . . . . . . . . .
14.2.5 Further analyses of the initial instability . . . . .
14.2.6 Numerical analysis of fully developed shear waves
14.3 References - Chapter 14 . . . . . . . . . . . . . . . . . . .
~

661
661
664
666
673
681
681
681
684
688
692
696
701

Author Index

705

Subject Index

711


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Chapter 1

Introduction

1.1

A brief historical overview

The nearshore coastal region is the region between the shoreline and a
fictive offshore limit which usually is defined as the limit where the depth
becomes so large that it no longer influences the waves. This depth depends
on the wave motion itself and in simple terms it can be identified as a depth
of approximately half the wave length. Thus in storms with larger and
longer waves the offshore limit moves further out to sea. This definition
is practical because the influence of the bottom on the waves is one of the
most important mechanisms in nearshore hydrodynamics.
Nearshore hydrodynamics could probably be said to have been founded
by G. G. Stokes, who in 1847 developed the first linear and nonlinear wave
theory. Today this theory is often referred to as Stokes waves (see also
Stokes, 1880). Over the following century various wave phenomena were
analysed and a great number of results, remarkable from a mathematical point of view, were obtained. Of particular importance from todays
perspective was the development by Boussinesq (1872) of the consistent
approximation for nonlinear waves in shallow water, a situation for which
Stokes himself recognized that his theory was failing. Korteweg and DeVries

(1895) added to this result by finding analytical solutions to the Boussinesq equations. These solutions are known as cnoidal and solitary waves.
Interestingly the infinitely long solitary waves had already been observed
in real channels by Russell (1844). Finally even this ultra brief historical
review would be incomplete without mentioning the pioneering discovery
of the wave radiation stress by Longuet-Higgins and Stewart (1962). This
established the insight that forms an essential element in all later research
related to currents and long wave generation in the nearshore.

1


2

Introduction t o nearshore hydrodynamics

The advent of computers has radically changed the perspective of what
is relevant hydrodynamics in todays world. Equations or theories that,
when developed before the computer age, were merely of theoretical interest
have become central to modern engineering applications while many of
the remarkable mathematical results that helped the understanding of how
waves behave have become mainly of academical interest. The content of
this book partially reflects that in the choice of which subjects and results
are pursued in detail.

1.2

Summary of content

As an introduction Fig. 1.2.1 from Svendsen and Jonsson (1976) shows
a schematic of most of the major wave phenomena that occur in the

nearshore. These, and some more that are not visible in such a picture,
are the phenomena that are analysed further in the following chapters.
Chapter 2
The first chapter (Chapter 2 ) is meant as a reference chapter that essentially presents the main hydrodynamical results used later in the book.
For most sections there are no derivations in this chapter. If the reader
needs further explanation reference is made to the textbooks quoted in the
list of references at the end of the chapter. Exceptions are the sections on
boundary conditions, turbulence and energy flux which contain material
not so easily found in standard books.
Chapter 3
Remaining central to the understanding of nearshore wave and current
motion is the Stokes theory, which in its simplest linear form represents
the most important theoretical background for nearshore hydrodynamics.
Chapter 3 therefore gives a thorough analysis, not only of the linear wave
theory itself but also of the most important of the results that have been
derived on the basis of that theory.
The main objective of the linear theory is to establish a first approximation for all the flow details of small amplitude waves on a constant depth.
This is done in Section 3.2.
The characteristic surface profile of such waves is described by the sine
function, whence they are also called sinusoidal waves. It turns out that
even though the average over a wave period of such wave profiles is zero