COHSTHL AND OCEAN ENGINEERING Philip L.-F. Liu, Editor
Volume 7
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World Scientific
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,
ADVANCES IN COASTAL AND OCEAN ENGINEERING
CORSE HND OCEHN ENGINEERING Volume 7
Editor
Philip L.-F. Liu Cornell University
V f e World Scientific wb
Singapore Sinqapore • New Jersey • London L • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
ADVANCES IN COASTAL AND OCEAN ENGINEERING Volume 7 Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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PREFACE TO T H E R E V I E W SERIES
The rapid flow of new literature has confronted scientists and engineers of all branches with a very acute dilemma: How to keep up with new knowledge without becoming too narrowly specialized. Collections of review articles covering broad sectors of science and engineering are still the best way of sifting new knowledge critically. Comprehensive review articles written by discerning scientists and engineers not only separate lasting knowledge from the ephemeral, but also serve as guides to the literature and as stimuli to thought and to future research. The aim of this review series is to present critical commentaries of the stateof-the-art knowledge in the field of coastal and ocean engineering. Each article will review and illuminate the development of scientific understanding of a specific engineering topic. Our plans for this series include articles on sediment transport, ocean waves, coastal and offshore structures, air-sea interactions, engineering materials, and seafloor dynamics. Critical reviews on engineering designs and practices in different countries will also be included. P. L.-F. Liu
PREFACE TO THE SEVENTH V O L U M E
This volume consists of five papers covering a wide range of topics in coastal oceanographic engineering. Drs. Maarten Dingemans and Ashwini Otta prepare the first paper on the subject of "Nonlinear Modulation of Water Waves". This comprehensive review article starts with several illustrative sections to guide readers to the nonlinear wave processes in deep and intermediate water. The Nonlinear cubic Schrodinger (NLS) equations are then presented and discussed for both deepwater and varying water depth with or without an ambient current. Discussions are extended to higher-order modulation equations, such as Dysthe's equation and the Zakharov's equation. Derived from the Hamiltonian principle, the Zakharov's equation provides a broader basis for higher-order equations. Drs. Dingemans and Otta point out the importance of including the formulations of dissipation due to breaking in the modulation equations. Several experiments have suggested that the frequency downshift could be a rather sudden process associated with wave breaking. The second paper is entitled "Bubble Measurement Techniques and Bubble Dynamics in Coastal Shallow Water". Both authors, Drs. Ming-Yang Su and Joel Wesson, are experts in field measurements and instrumentations. In particular, they have been involved in studying bubble dynamics in deepwater waves and in coastal shallow water waves for more than fifteen years. In this paper, they first give a comprehensive review of various sensors for measuring physical parameters of the bubble field generated by wave breaking and the corresponding deployment methods for some of these sensors in the coastal water. These sensors are based on the optical, acoustical, and electromagnetic principles. Several field experiments are used to illustrate the functionality of these sensors. Based on their experience in the field experiments, Drs. Su and Wesson give an insightful account of dynamical and statistical features of wave breaking and bubble field in the nearshore environment. Drs. Panchang and Demirbilek present the third review paper, entitled "Simulation of Waves in Harbors Using Two-Dimensional Elliptic Equation Models". They provide a comprehensive review of mathematical modeling procedures developed in recent years in the area of elliptic wave equations, which vii
viii
Preface to Volume 7
are suitable for simulating wave agitations and resonance in ports and harbors. Modeling techniques and extensions of the linear mild-slope equation to include steep slope, realistic boundary conditions, and dissipative mechanisms such as wave breaking and bottom friction; wave-wave and wave-current interactions are discussed. Several practical applications are demonstrated. The fourth paper is written by Dr. Losada and is entitled "Recent Advances in the Modeling of Wave and Permeable Structure Interaction". This paper focuses on the theoretical development of various mathematical models for wave and structure interactions. The structure could be impermeable and permeable. In the case of permeable structures, the determination of empirical coefficients characterizing the porous materials is discussed. The paper also reviews the state of arts wave models based on the Reynolds Averaged Navier Stokes equations. The volume of fluid method is used in the model to trace the free surface location so that wave-breaking process can be simulated. In the last paper Drs. Harry Yeh and Kiyoshi Wada report their laboratory observations on lock exchange flows. The Laser-Induced Fluorescent-dye technique is used to examine the qualitative characteristics and behaviors of gravity currents and internal bores. The similarity and dissimilarity between gravity flows and internal bores are discussed based on vortex dynamics. Philip L.-F. Liu, 2001
CONTRIBUTORS
Zeki Demirblek US Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MS 39180 USA Maarten Dingemans Delft Hydraulics P. O. Box 152 8300 ad Emmeloord The Netherlands Inigo J. Losada Ocean Sz Coastal Research Group, Universidad de Cantabria, E.T.S.I, de Caminos, Canales y Puertos, Avda. de los Castros s/n, 39005 Santander Spain Ashwini Kumar Otta Splash Hydrodynamics Kerkstraat 20a 8011 RV Zwolle The Netherlands Vijay Panchang School of Marine Sciences, University of Maine, Orono, ME 04469 USA (Temporarily at NOAA Sea Grant Office, 1315 East-West Highway, Silver Spring, MD 20910, USA)
IX
x
Contributors
Ming-Yang Su Naval Research Laboratory, Oceanography Division, Stennis Space Center, MS 39529 USA Kiyoshi Wada Department of Civil Engineering Gifu National College of Technology Sinsei-cho, Gifu-ken, 50104 Japan Joel C. Wesson Neptune Sciences, Inc. Slidell, LA 70458 USA Harry Yeh Department of Civil and Environmental Engineering Box 352700 University of Washington Seattle, WA 38195-2700 USA
CONTENTS
Preface to the review series
v
Preface to the seventh volume
vii
Contributors
ix
Nonlinear Modulation of Water Waves Maarten Dingemans and Ashwini Otta Bubble Measurement Techniques and Bubble Dynamics in Coastal Shallow Water Ming-Yang Su and Joel C. Wesson
1
77
Simulation of Waves in Harbors Using Two-Dimensional Elliptic Equation Models Vijay Panchang and Z. Demirbilek
125
Recent Advances in the Modeling of Wave and Permeable Structure Interaction Inigo J. Losada
163
Descriptive Hydrodynamics of Lock-Exchange Flows Harry Yeh and Kiyoshi Wada
203
NONLINEAR MODULATION OF WATER WAVES
MAARTEN W. DINGEMANS and ASHWINI K. OTTA
Contents 1. Introduction
2
2. Basic Insight into Modulational Processes 2.1. A simple example of instability 2.2. Basic ideas of the Benjamin-Feir instability mechanism
4 5 7
3. Nonlinear Schrodinger-type Equations: Horizontal Bottom 3.1. A heuristic derivation of the NLS equation 3.2. The scaling in the NLS equation 3.3. A sketch of the derivation in two horizontal dimensions 3.3.1. An alternative 2DH set of equations 3.4. Conservation laws 3.5. Special cases of NLS-type equations 3.6. Effects of surface tension
8 8 10 11 15 16 18 19
4. Nonlinear Schrodinger-type Equations: Uneven Bottom 4.1. Propagation in one dimension 4.2. Propagation in two horizontal dimensions 4.3. Shallow-water limit 4.4. Effect of an ambient current on ID propagation
22 22 25 28 28
5. Some Solutions of the NLS-type Equations 5.1. Decaying solutions 5.2. Soliton-type solutions
31 31 31
6. Higher-Order Modulation Equations 6.1. The Dysthe equation 6.2. Modification due to an ambient current 6.3. The Zakharov equation 6.4. Reduction of Zakharov equation to NLS-type equation 6.4.1. Narrow-band approximation in both dispersion and nonlinearity 6.4.2. Modification due to surface tension 6.4.3. Narrow-band approximation in nonlinearity only 6.5. Extensions of the Zakharov equation
35 35 37 38 43 43 45 47 51
1
2
M. W. Dingemans
& A. K. Otta
7. Generation of Free Long Waves 7.1. Formulation of the equations 7.2. ID situation, no ambient currents
52 53 56
8. Observations of Wave Modulations 8.1. Theoretical aspects of modulational instability 8.2. Laboratory observations 8.3. Deep-water modulation: initial stage and demodulation 8.4. Deep-water modulation: modulation leading to breaking 8.5. Spectral evolution
59 59 63 64 66 67
8.6. Comparison between theory and experiment
68
9. Summary
69
References
71
1. Introduction Customarily, in water-wave propagation problems, distinction is made between regular and random waves. Regular waves typically consist of a few components (not necessarily harmonics of each other) while random waves consist of many components in which the phases are distributed randomly, usually uniformly. Transformation of waves take place as they propagate due to interactions between components, variation of the bottom and current and forcing conditions such as wind. In wave motion, effect of shear is usually confined to a thin boundary layer. Experiences show that the important features of nonlinear wave interactions, refraction and shoaling due to bottom and current variation may be represented satisfactorily using a simpler mathematical description under the assumption of the fluid being inviscid and irrotational. This allows introduction of a wave potential Q(x, z, t) such that the velocity field (it, w)T is given by (V$, d$/dz)T where u = (u,v)T = (ui,ii2) T is the horizontal velocity vector and w is the vertical component. We shall use in this text a coordinate system such that x = (xi,X2) T = (x,y)T is directed horizontally while z is directed upwards (opposite to the gravity acceleration vector). The still-water level and the sea bed are defined respectively by z = 0 and z = —h{x). The governing equations for wave motion are then given by a field equation, the Laplace equation for $, a kinematic and a dynamic condition at the free surface z — ((x,t) and a kinematic condition at the bottom, V
2
$+|^=0;
-h(x,t)
< z < C(x,t),
(la)
Nonlinear Modulation
of Water Waves
3
and the three boundary conditions, (V$) 2
^ and
9< = 0
V*.V< = -
+
9$ ~dt
V $ - V / i = 0;
at
z =
with
z =
((x,t),
T[(] = 7V7V .
(lb)
(lc)
£(x,t),
z — —h(x),
where P = Pa - T[C]
at
(Id)
C f[(l + Vivci 1 ) / ] 2 1 2
(le)
5
7 being the surface tension. Both 7 and the atmospheric pressure pa are usually taken to be constant. Because the pressure in the water is reckoned with respect to the atmospheric pressure, pa is taken to be zero for simplicity. Taking pa = 0 and 7 = 0, we can eliminate ( from the two free-surface conditions to get, «92$ dt2
d 1„^ „ 15$ d h-V$•VH 9t 2 2dzdz
5$ dz
+9
£)+'-'2 at
= 0 2 = C - (2)
For later use, we write the free-surface conditions of Eqs. (lc) and (Id) in terms of the free-surface wave potential ip(x,t) = $>{x,((x,t),t} and the s free-surface vertical velocity w — (o is used. If U2(k)cjQ(k) < 0, the characteristic velocities of Eq. (8) are complex and the system in Eqs. (7a) and (7b) is elliptic. That means small sub-harmonic modulations will grow with time and that the wave system is unstable in this sense. For waves on deep water, the dispersion relation reads, 2
LO
gk(l + (ak)2 + • • • ) ,
'gk[l+l-(ak)2
+ --- ) .
(9a)
(9b)
Nonlinear Modulation
We thus have LU0 = \fgk and LO2 = gl^2kb/2/2 for all k for waves on deep water. 2.2. Basic ideas of the Benjamin-Feir
of Water Waves
and we have
instability
LO2LOQ
7
= -gk/8
it)],
a lower side band
a-i exp[i(k2X — u>2t)],
(12a)
and (12b)
with ai, C2 -C a. (4) Nonlinear interaction between the second harmonic and these side-band perturbations produces, a2ai exp[i(2fc — k\)x — i(2to — to{)t\,
(13a)
and a2a,2 exp[i(2fc — k2)x — i(2u> — W2)t],
(13b)
and also the sum-interaction terms. (5) Suppose now that: kx + k2=2k,
LJ1+U2
= 2LO.
(14)
Then a2a,2 exp[i(fcia; — cuit)] ~ upper side band,
(15a)
and a2a\ exp[i(k2X — u^i)] ~ lower side band.
(15b)
8
M. W. Dingemans
& A. K. Otta
We see that the simultaneous presence of the upper and lower side bands in association with the second harmonic results in a mutual reinforcement of resonance. The instability mechanism for Stokes waves is essentially the exponential growth in time resulting from this synchronous resonance. Mathematical features of modulational instability will be discussed later in section 8.1. 3. Nonlinear Schrodinger-type Equations: Horizontal B o t t o m The nonlinear Schrodinger (NLS) equation is the simplest example of an evolution equation for weakly nonlinear waves with strong frequency dispersion. The NLS equation describes the nonlinear evolution of a wave group with carrier wave number k and frequency LO. We first consider the properties of a wave group (see also Chu and Mei, 1971). Let the group consist of a superposition of two sinusoidal waves with amplitude oo and different (u>i, ki) and (u>2, k2). The resulting wave is written as a cos x with: / 5k a(x, t) = 2oo cos ( —x X{x,t)
\ —t I ,
5LJ
(16)
= -(k1+k2)x--(to1+uj2)t,
(17)
with 5k = k\—ki and 5u> = u>i — <JJ2- We now introduce the carrier wave number and frequency by k = dx/dx = (ki + k2)/2 and u> = —dx/dt = (u>i + u>2)/2. It is seen now that when the individual waves satisfy the dispersion relation u>j = fl(kj), it is not true for o> = fi(fc) except for nondispersive waves. Indeed, one has w = (£l(ki) + Q,(k2))/2 which becomes Taylor expansion for small 5k/k,
u=
3.1. A heuristic
o (i) + M w
derivation
8
dk2
+
M
of the NLS
384
dk4
+
....
(18) v
'
equation
A heuristic derivation of the NLS equation has been given by a number of authors, amongst which are Karpman and Krushkal' (1969), Kadomtsev and Karpman (1971), Karpman (1975, section 27), Jeffrey and Kawahara (1982, p. 59), Yuen and Lake (1982, p. 75), and Dingemans (1997, section 8.3.2). The derivation starts with a harmonic wave with basic frequency and wave number
Nonlinear Modulation
given as
of Water Waves
9
(u0,k0),
C(x,t) = R e ^ O r , * ) ^ 0 * - ^ ^ ^ } = R e ^ O r , ^ 7 ^ } = Re{A(x,t)e-^x'^}.
(19)
The dispersion relation is written in the general form, ft(fc,a2).
w=
(20)
As we consider modulated waves, we now focus on small changes in the carrier wave frequency and wave number U>Q and ko: u> = OJQ + SLO and k = ko + Sk with SUJ/LOQ = 0(/3) )|i.>+.aJV). Pi)
with ()o denoting that the quantity is evaluated for the basic state (ko and a = 0). We introduce the abbreviations, a;0 = ft(fco,0), c
f l
=(^)
and
< = (|j^)
(22)
.
so that Eq. (21) may be written as: - ( w - w„) + cgSk + ^{Skf
+ (^\
a2 + • • • = 0 .
(23)
We now introduce operator correspondence for which we have Sk —>• idx and w — u>o = Su> —> —idx with X is the slow spatial coordinate X = j3x and T the slow time T = /3t. Operating on A* results in:
i£r+°-&y-?*!m
+ (m)wA-
=
°-
(24)
Introduction of the moving coordinate £ and the time variable r by (see also next subsection): £ = X -cgT and r = sT, (25)
10
M. W. Dingemans
& A. K. Otta
and changing to A leads to the NLS equation in its usual form, d2A , k) and (u>',k'). If the differences Q, = u>' — u> and K = k' — k are small, then, the resulting wave will have a long-wave envelope with wave number K and frequency Q,. The dispersion relation is: _,
1 d2ui r ^ 2
/(,-, T,
n = c
"()*+2^*
1 d3uj _~ +
6WK
+
'
(31)
where cfl(°) = dw/dk is the group velocity of the carrier wave (u, k). Then, the linear phase velocity of the wave envelope, V = u/K is: y
= c (9o
)
++
I^+i^2 + 2dk2 Qdk3
(32)
Nonlinear Modulation
of Water Waves
11
On the other hand, considering the long wave with (Cl,K) to be a nonlinear long wave (i.e., without frequency dispersion), its propagation velocity may be described by the characteristic velocity dx/dt which expanded in powers of e, can be written as:
g = c g (°)+ £Cs W+ £ 2 c g ( 2 ) + .-..
(33)
The coupling between the modulation and the nonlinearity is strongest when both effects are of equal order of magnitude. Thus, K = 0(s). Because we already had K — 0((3), this means that /3 ~ e. Notice that this is similar to the case of stationary fairly long waves where the linear dispersive long wave velocity is c = ^fgh{\ — (kh)2/3 + • • • } and the nonlinear nondispersive long wave velocity is c = \/gh{l + a/(2h) + • • • } . For permanency, it was required that (kh)2 ~ a/h. Because K = C(/3) is measured with respect to the carrier wave number k, the scale on which the wave group has to be considered is A = A//3, and so X = j3x where x is scaled with A = 2ir/k. In order to remain near the centre of the wave group, a moving coordinate system should be applied. The coordinate along the characteristic curve is then £ = (3(x — cg(°H). Because one has x = (3~li + cg^H, it follows that dx/dt = p-ld£,/dt + cg(°\ From Eq. (33), one obtains dx/dt = ecg^' + cg(°>. In order that these expressions are the same, it is necessary that: 9
pedt
'
Introducing the slow time scale r = £/3t, one obtains d£/dt = cg^(^,r). produces, £ = (3{x - cg(°H), r = e(3t. 3.3. A sketch of the derivation
in two horizontal
This (34)
dimensions
We will give here a sketch of the derivation of the NLS-type of equations in two horizontal dimensions for the case of a horizontal bottom. This derivation follows Davey and Stewartson (1974). At time t ~ 0, the free-surface elevation is given as: C(x, 0) = ie-a(Pxu
px2)eikxi
+ CC ,
(35)
12
M. W. Dingemans
& A. K. Otta
which thus represents a progressive wave in the xx direction with a slowlyvarying amplitude. The amplitude a is measured here in m 2 s _ 1 , i.e., a is a measure for the amplitude of the velocity potential $(x,z,t). The governing equations are the Laplace equation for $ in —h < z < ((x,t), the kinematic conditions at z = —h and z = (, and the dynamic condition at z = (. A solution of these equations is taken to be of the following form,
((X,t)=
J2 tmE™, $(x,Z,t) = J2
77171
(36a)
with C-m = Cm .
4>*-m = m(x,t) are expanded as:
cm = E £ ^ ( n , m ) (^ 7 ?' T )-
(37a)
n—m oo
= J2en^m>(Z,T,,z,T), £ = e(xi - cgt),
rj = ex2 ,
T
= e2t,
(37b) (e = j3) .
(37c)
where cg is the linear group velocity and no distinction between the two scales e and j3 is made anymore. The zeroth-order, zeroth-harmonic terms £(°>°) and 0. The reader would find it worthwhile to refer to Ablowitz and Segur (1979, 1981) and Djordjevic and Redekopp (1977) for an elaborate discussion on the behaviour of the solutions in the separate regions. For water waves for which surface tension effect is negligible, one is primarily interested along the ordinate (7 = 0). Some more properties of the system along this ordinate are discussed in section 5. 4. Nonlinear Schrodinger-type Equations: Uneven B o t t o m 4.1. Propagation
in one
dimension
In the same way as a weakly-dispersive long-wave equation such as the KdV equation for water of constant depth can be extended to a KdV-like equation for the case of varying depth, h = h(x), an inhomogeneous NLS equation can be derived for the propagation of wave packets on an uneven bottom. In both cases, the reflection has to be neglected because both the NLS and the KdV equations describe waves propagating in one direction only. Djordjevic and Redekopp (1978) gave a derivation of an inhomogeneous NLS equation in a way which is similar to that in which the Davey and Stewartson equations
Nonlinear Modulation
of Water Waves
23
(48) are derived. The depth is slowly varying, h = h{(32x) and /? is supposed to be proportional to e where e is the wave slope which conforms with the common assumption in the NLS scaling. Because j3 = A/A is the modulation parameter and the modulation of the carrier wave gives rise to a wave group, A may be seen as a measure for the horizontal extent of the wave group. Because the group velocity is a function of the depth h and therefore also a function of e2x, the following multiple scales are introduced now, dx
{[•
Z = e2
t
(70)
Note that the role of r and £ is reversed compared to the constant-depth case. It is supposed that u> = constant, i.e., no temporal variation of the medium is considered. It is supposed that the group velocity cg, the phase velocity c and the wave number k can locally be defined as a function of the local depth h(£) and therefore the variation of cg, c and k is with £: c s (£), c(£) and fc(£). The free surface is expanded as: oo
COM)
(
n
£ < E c(n,m)(£,T)En
(71a)
n—1 \m=
with E
exp
/ k(£)dx - wt I
(71b)
and C ( n '" m ) = (C ( n , m ) )*,
(71c)
with ()* denoting the complex conjugate. This expansion thus is similar with the expansion used before, the difference being an adoption to the nonuniform depth which necessitates the adoption of a coordinate moving with a nonuniform velocity so as to remain near the centre of the wave group. Proceeding in the usual way, we find from the e3E° terms an equation for ^M1'0) and from the e3El terms an equation for the amplitude B ( £ , T ) of the solution for t^ 1 ' 1 ) emerges. Introducing the quantity Q ( £ , T ) by: QQ&T)
dcj>^°) k2cg H ; ?r dr gh - cg
{2^ + l-a2}|S| "i ^
(72)
the equation for (1,0) becomes simply, dQ dr
0,
with the solution
Q = Qo(£) :
(73)
24
M. W. Dingemans
& A. K. Otta
and the equation for B{^,T)
becomes,
dB d2B i-^r + Ai -dr^2 + ifnB = vi \B\2B + u2QQB ,
(74a)
where the coefficients are given by d : (1 - cr2)(l - kha) Mi Ai
v\
a+
d{kh)
2
kh(l-a )
1
a(l - kha) a + kh(l-a2)
m) by: ^(n,m)
=
£(n,m)
and similarly for <jy-r
&xp 71=1
gives the expansions,
n
m=-\-n
c = Y^£ Yl c n=l
(n,m)
£n
Yl for <j>^1'0') and A for A for convenience, dA : ,_,„,!,„ i ffeor,/9cg ko xA 8t • c , . V A + - ( V . c s M - - | ^ V ^ ^ . V
\k0
J
+ ikluQK\A\2A+l-vA
ko
\ko k0
= Q,
)
k0
)
(82a)
Nonlinear Modulation
of Water Waves
27
and d2
-V-(ghVcf>)=V-
m2
\A?
2 ~9 \A? 2w
(82b)
4 dt Vsinh 2 g
0
where K=
2-(cosh4q + 8 - 2 t a n h 2 g ) , kx = — , 16sinh q g a = cothg and q = koh ,
(83a) (83b)
and the coefficients fj, and v represent functions of the derivatives of depth and wave number of which the expressions have been given in Liu and Dingemans (1989, Eqs. (B.l) and (B.2)). Both /J. and v are zero in case of a horizontal bottom. Notice that V = (dx,dy)T is the horizontal gradient operator. For a horizontal bottom, the evolution equation, Eq. (82a), simplifies considerably. Taking the main wave direction in the rc-direction (and thus fco is directed along the a>axis so that fco • V = kod/dx), the resulting evolution equation reads for horizontal bottom, dA dt
dA dx
Cg 9^Z--A
i (dcqd2A c„d2A\ ,2 , 4l2 „ t „ , 2 2 \+^k2u0K\A\2A-iAG
0 (which is always the case in shallow water), the governing equations for the envelope-hole solution reduce to the inhomogeneous KdV equation with variable coefficients. That Eq. (78a) reduces to this generalised KdV equation can be shown in the following way. Write B = i?(^,r)exp[i J #(£, fdf)] with R and 9 be the real functions. R and 6 are then expanded in a power series to a small parameter 6: R = Ro + SRi + 52R2 + • • • and 9 = 56\ + 0). The following further coordinate stretching is introduced,
'»"{?£)-*}• *="" 0 and thus kh > 1.363. For the case that period conditions on the amplitude b are imposed, more possible stationary solutions are found. The conditions imposed are now: db/dX —» 0 for b —» ao as X —» X(. With the notation r = b2, the differential equation is: dr
~ox
2—r(r—ro)(r
— r3)
with
ro = a^
and
r$ = 2
Ai
r0 .
(104)
V\
In this case three cases for viable solutions have to be considered: (1) v\ > 0 and r$ > 0 leading to the dn-solution, (2) v\ > 0 and r3 < 0 leading to the cnsolution, and (3) v\ < 0 and r3 > 0, giving the sn-solution. These solutions are: (1) i/i > 0 a n d r 3 > 0 1/2
A(£,r) = &3dn
X
" 2Ai 1
,„
£— i • exp i—— _,ir.3(2_m) 2Ai
V2
x +
__
(105a)
Nonlinear Modulation of Water Waves 33
with
la
T3 -T-p
m
»"3
2r0
(105b)
< 1.
u
v\
(2) i ^ > 0 a n d r3 < 0 2 A(£,T)
= a0cn
\ 1/2 X m
2TOAJ V
• •exp
y
l 1V TO
.1 2 2
2Ai
with TO = ( 2 —
(106a) 4Aii/iag
2aA:
(106b)
(3) i^i < 0 a n d r3 > 0 /
A ( £ , T ) = a0sn
• exp
\ 1/2
Kir) * v
•
r
-1 2
m =
2s ^OQVI
2Ai
with
m V" 4Ai^ia,
M , TO 2
\ -i
4Aii/ 1 a 2 ,y
(107a)
an rz — ' 'J TO
V 4A1J7!
(107b)
Because for m —» 1, we have d n —• sech, sn —>• t a n h a n d en —> sech, we see t h a t t h e limiting values for m —> 1 of t h e solutions (105)-(107) are: 1/2 .A(£, i) = 6 3 sech
with
X
vi\i
r 3 = 63 = 2 — > 0
W" iST
exp
and
(108)
vx > 0 ,
2\ I/2 A(£, t) = ao sech with
X
" 2Ai
exp ^
M
^
+ SAT (109)
a < 0, 1/2
A(£,,t) = ao t a n h with
-
i^i < 0
X and
exp
.
a = cos2 •& - 2 sin2i9 .
(112)
For tp > 0 (tan 2 •& < 1/2), solutions for the group envelope in terms of the elliptic functions dn and en always exist, i.e., groups of permanent waves and of infinite extent exist, which also vary periodically in space and time. The common limit (771 —• 1) is the sech profile. For the case that ip < 0, the situation is much more complicated. In the critical direction i?c such that tan 2 i? c = 1/2 (or V = 0), only constant amplitude plane waves are possible. For the case %jj < 0, we refer to Hui and Hamilton (1979). 6. Higher-Order Modulation Equations The equations discussed in the previous sections govern modulation of gravity waves valid up to 0(e3). These equations have been found to be capable of producing several broad features of nonlinear modulation. However, comparisons with experiments have also revealed features like deviation of the predicted growth rate of unstable modes for steeper waves (e > 0.15) and asymmetric growth which lie beyond the NLS approach. These limitations of the NLS equation have drawn attention to the necessity of higher-order modulation. In this section, we will first discuss a higher-order modulation due to Dysthe (1979) which is valid only in deep water. Modification of this set of equations due to an ambient current will be treated following the recent work of Stocker and Peregrine (1999). Finally, the section will be closed by a description of "the Zakharov equation". Zakharov's set has two distinct features of being derived from an alternative principle and being more general, encompassing Dysthe's equation as a special case. 6.1. The Dysthe
equation
To express the potential and free surface elevation, we use the form (Dysthe's form has been modified slightly for consistency), C = C + \[AJ* + A2ei2i> + ... + CC],
(113a)
36
M. W. Dingemans
& A. K. Otta
-[Bekz
$ =
D 2fcz i2tf i>2e e
A'd
+ --- + CC],
(113b)
with $ and ( denoting the potential and elevation respectively of the slowly varying mean flow and where $ = k • x — tot and k = \k\. The governing equations for the modulation of B corresponding to Eqs. (2.19), (2.20) and (2.10) of Dysthe (1979) are given in dimensional variables by: dB_ ~dt
2k~dx % to
"l6fc3 A-3 4w'
5$
~di
2 u> d B 8k2 dx2
LO_0B
•gC = 0
dt
d3B dxdy2
d2B 4fc2 dy2
d3B dx3
dB ' dx
2UJ'
3k3 -BIB
\B\2B
dB* dx
4ui
B*
as dx (114a)
ox
at
(114b)
0,
k-V(\B\2)
at
z = 0.
(114c)
Equation (114a) incorporates the correction of a misprint in the original Eq. (2.19) of Dysthe (1979) as pointed out by Janssen (1983). Another misprint appearing in Eq. (2.17) of the same article is the factor 3 of the second term of r which should be 8 and has been noted by Brinch-Nielsen and Jonsson (1986). The terms on the left-hand side of the evolution equation (114a) are all of 0{ka)3 while the terms on the right-hand side are all of fourth order. In other words, the usual NLS equation for deep water is retrieved if the higher-order correction terms contained in the right-hand side are set equal to zero. As ( is of third order, the term d(/dt in Eq. (114c) may be neglected. This simplifies the substitution oidQ/dz in Eq. (114a). In that case, Eq. (114a) reduces to: dB ~dt
+
w dB ~5k~dx~
16 k
3
d3B dxdy2
uj
d2B
8k2 dx2 d3B dx3
d2B 4fc2 dy2 k3
\B\2B 2OJ'
.dB*
+ iT-B B dx
6B*
dB_ dx
kB
dx
z=0
(115)
Nonlinear Modulation
of Water Waves
37
which is also identical to Eq. (10) of Trulsen and Dysthe (1996). As also discussed in Lo and Mei (1985), these equations are derived under the condition that kh = 0{(fca) - 1 } (k)
41 =
1
1/2
25
[6(M) + &*(-M)],
(132a)
1/2
without changing the value of the integral in Eq. (135). The interaction coefficients can be symmetrised as noted by Stiassnie and Shemer (1984). The Zakharov equation has been reconsidered by Krasitskii (1994) who showed that previously used forms did not give a truly Hamiltonian system of equations; this had to do with the definition of the interaction coefficients. It appears that in the older form of the Zakharov equation, the coefficients were not sufficiently symmetric. For an extensive discussion of these matters is referred to Krasitskii (1994) and Badulin et al. (1995). As put forward by Krasitskii (1994), the symmetry conditions are not clear without considering the Hamiltonian formulation. Notice that Rasmussen (1998, Eq. (2.61)) writes the Zakharov equation (135) in the form, 8B_, -{k,t) = -i ~dt
/ / / dkidk2dk3X^l2t3C'3.2
(137a)
,
with for C" the expression, 'n-l
CL=
n-1
l[b*(km,t))
l[b(km,t))-S[k+J2km-
m=\
x exp
Y2um~ y^Wm )t
J2k" m=n
(137b)
Nonlinear Modulation
of Water Waves
43
Taking T^ = -X^2\ the same equation as given in Eq. (135) is obtained. For X Q I 2 3 = —T0 ! 2 3' Rasmussen (1998, Eq. (2.60)) gives the expression, ^M),1,2,3 = Q^ 0,1,2,3 + ^0,1,3,2) i
(138)
whenever both k + ki — k2 - k3 = 0 and \u + co(ki) — w{k2) — w(fc3)| < 0(e2); otherwise, X^ = 0. The coefficient YQ^2 3 n a s been given in Rasmussen (1998, Eq. (A.18)). Taking the symmetric form of Krasitskii (1994) (in his notation, T^> is called V^), the underlying system is a Hamiltonian system and we have the property that (see also Badulin et al., 1995): T< 2 )(fc,k u k 2 ,k 3 ) = T^(kuk,k2,fc3)
= T^(kuk,
fc3,
k2)
= T(-2\k2,k3,k,k1). 6.4. Reduction
of Zakharov
equation
(139)
to NLS-type
equation
6.4.1. Narrow-band approximation in both dispersion and nonlinearity Stiassnie (1984) showed that the NLS equation can be derived from the Zakharov equation by restricting the waves to have narrow spectra only. To this end, it is supposed that the energy is concentrated around the wave number k = ko = (fco,0)T which is in accordance with the usual assumption for NLS equations that the waves have one predominant direction, taken here as the x = X\ direction. We then write, k = k0 + K,
K=(KI,K2)T
with
|K|/A;O
= o(l).
(140)
To facilitate the expansion for narrow spectral width, a new amplitude variable B is introduced by: B(K, t) = B(K, t) exp{-i[w(fc) - w(fe0)]} .
(141)
The Zakharov equation (135) becomes in terms of B, dB i — (K,t) - Hfc) - u>(ko)]B(K,t) ///
dKidK2dK3TQjl23(k0
x B*{K1)B{K2)B(K3)6(K
+ K,k0 + Ki,k0 + K2,k0 + K3)
+ K1-K2-
K3) .
(142)
44
M. W. Dingemans
& A. K. Otta
Substitution of Eq. (141) in Eq. (136) yields the following expression for (,
C(x,t)
J,[kax-u(k0)t]
dn
2TT
Lo(k0 + K)
1/2
[B(K,t)iKX
29
+ CC].
(143)
We also write ( as: C(x,t) = Re{a(a:,£)ei[fcoa!-a'(fc°)t]} ,
(144)
Expansion of y/u(ko + K) to first order in K for the case of deep-water waves (the case considered by Stiassnie, 1984) permits us to approximate the complex amplitude a to:
In the Zakharov equation (142), we now expand the term ui(k) — w(fco), u>(\k0 + K\)
-u(k0)
- Pt
L_ _)
K
2 V k0
4k0
±_
2k0
o
8k2
(146)
k3
The Zakharov equation (142) has to be expressed in terms of the complex amplitude a instead of A. Therefore (Stiassnie, 1984), Eq. (142) is multiplied by yj2uj{ko)lg • (1 + K/(4/CO)) and subsequently the inverse Fourier transform is taken. This results in (Stiassnie, 1984): .da l ~di
+
1 2~V k0
1 d2a 4k0dx2
da dx
/2u;(k0)\l/2
1
{
2n
9
J
+
1 d2a 2k2 dy2
d,Kid,K2dK3
.(2)
'^0,1,2,3(^0 + K2 + K3 - Ki,k0
1
K-2 + K3 -
Ki
4fo>
+ Ki, fc0 + K 2 , fco + K 3 )
•^*(/c1)e(K2)^(K3)ei(K2+K3-Kl)'x . In deep water, one has u = \J g\ko + K\ = \/gko(l first order, co(k0 + K) = w(fc 0 )(l + K\/(2k0)).
8k3
d3a dxdy2
(147)
+ 2K\/ko + |/c| 2 /fcg) 1 / 4 and thus, to
Nonlinear Modulation
of Water Waves
45
For the case of deep-water waves, it is possible to show that a first-order Taylor expansion of the interaction coefficient T0 12 3 m the spectral width becomes, T
CII2
3( f c 0 + K 2 + « 3 - K l , ^0 + K l , fco + « 2 , ^ 0 + K3)
A- 3
3
/ 2^2
4TT 2
(«1 - K2) 2 •«3)
-
(«1 - K3)
2A;0|KI-K2|
2k0\Ki
- K3
\K\*
+0
(148)
k2
Using Eq. (148) in Eq. (147), the following equation is found, .da dt
1/a 2 V fo
2uj(k0)
1/2
,9a ' 9a;
1 d2a Ak0 dx2
ik$ 2 da* K\a\'a--^-a dx
1 9a 2k0 dy
d3a 8k . dx3
3i 5 3 a Ak\ dxdy2
2
, o, ,o da
3i*g|a|»-
kfial 1^2"'
(149)
where the sign of the second term in square brackets in the right-hand side of the equation is negative instead of positive as in Stiassnie's equation (10). Notice that this correction is the same as the correction of Janssen (1983) of Dysthe's equation. See also Hogan (1985, p. 371). In Eq. (149), I is an integral which can be related to the derivative of the wave potential $ at z = 0 (Stiassnie, 1984) as: 4gn2 d$ 1 = 2 (150) cu (ko) dx z = 0 Substituting Eq. (150) in Eq. (149) and rewriting results in: . /da \dx
2k0 da\ wQ dt)
ki\a\2a 01 '
1 d2a 4k0dx2
1 d2 2k0 dy2
- a 2 —- -3ik20\a\2~ dx 2 dx
d3a 8k2 dx3 2k0 (feo) dz
3i d3a Ak$ dxdy2 (151) z=0
6.4.2. Modification due to surface tension Hogan (1985) extended the analysis of Dysthe (1979) by also taking surfacetension effects into account and derived a fourth-order equation valid for deepwater waves with surface tension effects included. The dispersion relation used is u)2{k) = (1 + s)gk and s = 7/c2/(pg) with 7 the surface tension which is given as force per unit length (N/m). He starts with the Zakharov equation
46
M. W. Dingemans
& A. K. Otta
with surface-tension effects included and then follows Stiassnie's (1984) method to obtain the fourth-order evolution equation. Writing, ((x, t) = Re{a(x, t)e i(kox-uJot) }.
(152)
new scaled (primed) variables are introduced by t' = u>ot, x' = kox, a' = koa, $ ' = (2fco/w0)$ and cg' = {kQ/u}0)cg. The dimensionless higher-order NLS equation then is, dropping the primes, da
2i
~di -is
da
+ C9
+ p
dlc'
d3a dxdy2
d2a ^2 dx
+q
da
Xl —~ W1 dy
d3a dx3
,da* dx
,da dx
o d2a
UJQ
3 .UJQ da 8 &o dxdy2
1
,
1 .u0 d3a
l2
16 &Q dx3
1 4
2
da* dx
3., . l2da , 9$ -i/c 0 w 0 |a| — + k0a—-
(156) z=0
Furthermore, as in Dysthe (1979), the same condition for 3>x follows from the kinematic condition, 1 d(a2) 2 W °" dx
a7
(157)
0.
at
6.4.3. Narrow-band approximation in nonlinearity only The limited bandwidth for which the NLS and the modified equation (mNLS) as given by Dysthe (1979) are derived hampers the application to real waterwave problems as noted amongst others by Trulsen and Dysthe (1996). These authors enhanced the extent of the bandwidth. Both the NLS and the mNLS equations are derived under the conditions that: lAfcl k
0(s)
and
kh = 0(s~l)
with
e = ka .
(158)
The resulting equation, valid up to 0(e4) has been given in Eq. (115). We rewrite that equation in the following form, S(B) = C2{B) +
N2{B),
(159a)
where S{B) = 0 stands for the NLS equation with S = C\ + N\, C\B and Af\B being the linear part and the nonlinear part of the nonlinear Schrodinger equation respectively. Furthermore, we still have Eqs. (114b) and (114c). The extensions of the NLS equation needed for the mNLS equation are given by: _,„, .fdB UJ dB\ u (\ d2 °) and / for r^ 1 - 0 ), we have the following relation between C and 4>, see Kirby (1983),
or Eq. (184) for C-
Nonlinear Modulation
of Water Waves
55
In absence of an ambient current field and reverting back to the unstretched variables x and t, Eq. (184) simplifies to: d2C
w
LoA2
4 sinh2 kh
H * £ i .
and in terms of <j>, we then have, W-V.W»-£v.(kM)-->i-?l*£.. yy YJ dt2 2 \ tor J 2sinh2A;h X\. Suppose that we have a permanent wave group with accompanying bound long wave in the region x < XQ. Upon progressing of this wave group in the region with variable depth, the carrier waves forming the wave group experience shoaling and the group itself changes and is not permanent anymore. This has as a consequence that the long wave also changes; during this change also free long waves are formed. This situation is not very much different from
56
M. W. Dingemans
& A. K. Otta
which is encountered by propagation of solitary and cnoidal waves in regions with uneven bottoms. Also in these situations long waves are formed, see the discussion and the references mentioned in Dingemans (1997, section 6.6.3). 7.2. ID situation,
no ambient
currents
We consider the long-wave equation (185b) in this section. Because A is a function of £ = x — cgt, as becomes clear from the amplitude equation (181) a particular solution i to Eq. (185b) is also a function of x — cgt. As done in Dingemans et al. (1991), we can integrate the long wave equation once and we obtain, for the case of a horizontal bottom, 6 d
( c /2
L\ & -gh)—
1 g2k 2 u
gkcg \A\\ 2sinh2fc/i
(186)
where cj>£ = 0 is chosen for A = 0. For the amplitude of the wave envelope, we now use the NLS equation which reads (Dingemans et al. (1991, Eq. (8)), ( dA
+c
dA\
('*- °irx)
ldcQd2A + +A
, -
2-d^ ^-
,, u
, „ l2 A „, „ A nA
* *M = '
, „„ , (187a)
where the operators Q and H are given by:
o
