advances in coastal and ocean engineering volume 7

COHSTHL AND OCEAN ENGINEERING Volume 7 Philip L.-F. Liu, Editor

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World Scientific

ADVANCES IN COASTAL AND OCEAN ENGINEERING

C O R S E HND OCEHN ENGINEERING

Volume 7

Editor

Philip L.-F. Liu Cornell University

V f e World Scientific w b Sinqapore • New Jersey L Singapore • New Jersey • London • Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

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ADVANCES IN COASTAL AND OCEAN ENGINEERING Volume 7

Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd.

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PREFACE TO THE REVIEW 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 state-of-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 VOLUME

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

C O N T E N T S

Preface to the review series v

Preface to the seventh volume vii

Contributors ix

Nonlinear Modulation of Water Waves 1 Maarten Dingemans and Ashwini Otta

Bubble Measurement Techniques and Bubble Dynamics in Coastal Shallow Water 77

Ming-Yang Su and Joel C. Wesson

Simulation of Waves in Harbors Using Two-Dimensional Elliptic Equation Models 125

Vijay Panchang and Z. Demirbilek

Recent Advances in the Modeling of Wave and Permeable Structure Interaction 163

Inigo J. Losada

Descriptive Hydrodynamics of Lock-Exchange Flows 203 Harry Yeh and Kiyoshi Wada

NONLINEAR MODULATION OF WATER WAVES

MAARTEN W. DINGEMANS and ASHWINI K. OTTA

Contents

1. Introduction 2

2. Basic Insight into Modulational Processes 4 2.1. A simple example of instability 5 2.2. Basic ideas of the Benjamin-Feir instability mechanism 7

3. Nonlinear Schrodinger-type Equations: Horizontal Bottom 8 3.1. A heuristic derivation of the NLS equation 8 3.2. The scaling in the NLS equation 10 3.3. A sketch of the derivation in two horizontal dimensions 11

3.3.1. An alternative 2DH set of equations 15 3.4. Conservation laws 16 3.5. Special cases of NLS-type equations 18 3.6. Effects of surface tension 19

4. Nonlinear Schrodinger-type Equations: Uneven Bottom 22 4.1. Propagation in one dimension 22 4.2. Propagation in two horizontal dimensions 25 4.3. Shallow-water limit 28 4.4. Effect of an ambient current on ID propagation 28

5. Some Solutions of the NLS-type Equations 31 5.1. Decaying solutions 31 5.2. Soliton-type solutions 31

6. Higher-Order Modulation Equations 35 6.1. The Dysthe equation 35 6.2. Modification due to an ambient current 37 6.3. The Zakharov equation 38 6.4. Reduction of Zakharov equation to NLS-type equation 43

6.4.1. Narrow-band approximation in both dispersion and nonlinearity 43 6.4.2. Modification due to surface tension 45 6.4.3. Narrow-band approximation in nonlinearity only 47

6.5. Extensions of the Zakharov equation 51

1

2 M. W. Dingemans & A. K. Otta

7. Generation of Free Long Waves 52 7.1. Formulation of the equations 53 7.2. ID situation, no ambient currents 56

8. Observations of Wave Modulations 59 8.1. Theoretical aspects of modulational instability 59 8.2. Laboratory observations 63 8.3. Deep-water modulation: initial stage and demodulation 64 8.4. Deep-water modulation: modulation leading to breaking 66 8.5. Spectral evolution 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

^ + V * . V < = - at

and

where

9$ ~dt

V $ - V / i = 0;

9< = 0 at z = ((x,t),

z = £(x,t),

z — —h(x),

7 . V f VC 1

[(l + ivci2)1/2] 5

(lb)

(lc)

(Id)

(le) P = Pa - T[C] with T[(] = 7V

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$ 5$ dt2 +9dz

d 1 „ ^ „ 1 5 $ d h - V $ • V H

9t 2 2dzdz £) + ' - ' 2 = 0

at 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 free-surface vertical velocity ws — (<9<&/<9z)|z=f, e.g., see Dingemans (1997, Eqs. (64)),

^ + i ( V ^ ) 2 - i K ) 2 [ l + (VC)2] + ^ + K = 0:

dt + v^-vc-«/[i + (vc)2 0.

(3a)

(3b)

While the framework described above is complete within the assumption of potential theory, tractable solutions of the exactly nonlinear equations are at the least computationally too heavy to permit computations over large area and gain insight into the physical processes. It is therefore essential to simplify the analytical setups with which the "relevant" physical processes


may be quantified. One of the fundamental processes in wave propagation is "resonant" or "near-resonant" interaction between wave components. While for capillary waves (wave length A less than 2 cm) on one end of the wave spectrum and shallow water waves (A > 7h, h being the water depth) on the other, the significant interaction takes place between three waves (quadratic), the same occurs between four waves (cubic) over deep and intermediate water for gravity waves. A key assumption used in the simplification of this process is the narrow-bandedness of the spectrum. This allows one to represent the entire spectrum or components within a narrow-band by a single component with a fast-varying phase function (varying at a basic central frequency u>o and a corresponding wave number ko) and a slow-modulation of the amplitude. Nonlinear transformations taking effect over a length scale much larger compared to 2ir/kQ are accounted for in the slow-modulation of the amplitude function. Mathematical definitions and elaboration of the narrow-bandedness and the slow-modulation will be presented in the following section. We will first provide an insight to the nonlinear modulation followed by a derivation of the nonlinear (cubic) modulation equation, well-known as the nonlinear Schrodinger equation (NLS). This equation is the simplest form to study nonlinear modulation over deep and intermediate water. We will then discuss how the assumption of narrowness may be relaxed and higher-order description may be used. A summary of the experimental observations is compiled to discuss the "relevant" features during interactions and the need to go beyond NLS, the simplest form of the modulation equation. Of importance to coastal engineers over intermediate depth, we have paid attention to propagation over a varying bottom and in the presence of an ambient current. An example of the application of the nonlinear Schrodinger equation to the computation of the velocity field under waves has been given by Trulsen et al. (2001).

This article is by no means an exhaustive review of modulation phenomena. Capillary waves, except for a short discussion on their effects, are beyond the purview of this review. Similarly, shallow water waves are kept outside the present scope. For interested readers, additional references of previous reviews of modulation of water waves are by Yuen and Lake (1980, 1982), Hammack and Henderson (1993), and Dias and Kharif (1999).

2. Basic Insight into Modulational Processes

One of the earliest motivations for study of nonlinear modulation came from the observations of instabilities of water waves in wave flumes (Benjamin and


Feir, 1967). Although instability is a special feature of nonlinear modulation, we shall start with instability as an introduction. The essence of modulational instability can be treated in a number of factually equivalent ways. First, we consider an example based on the action equation and the equation of conservation of wave crests. Next, we consider the principal features of the Benjamin and Feir instability. Subsequently, we consider the nonlinear Schrodinger (NLS) equation of which we give an heuristic derivation. The Benjamin and Feir instability also results from a perturbation of the nonlinear Schrodinger equation.

A succinct physical explanation for the onset of instability of weakly nonlinear waves has been given by Lighthill (1965, 1967). We here follow his analysis,

(1) Consider a pulse of weakly nonlinear waves in deep water which initially contains waves of uniform lengths.

(2) Since the nonlinearity causes the crests of the waves with the larger amplitudes to travel more quickly {to2 = gk(l + k2a2)), wave numbers tend to increase in the front of the pulse and decrease at the end of the pulse.

(3) Because dcg/dk < 0, the shorter waves in front of the pulse and the longer waves behind the pulse cause energy to approach the centre of the pulse, resulting in an increase of the amplitude in the centre of the pulse. This accelerates the instability.

For the case of finite-amplitude waves in water of finite depth, the spatial variation of the waves induces a mean flow and a change in mean water level. These mean flow and mean water level variations have a stabilizing effect because they cause wave numbers to increase behind the pulse and decrease in front of the pulse.

2 .1 . A simple example of instability

We first notice that, in absence of ambient currents, the modulation of a wave train can be described by the equation for wave action conservation and the equation for conservation of wave crests,

da2 d / 8LU 2 \ n ^+^Urr0' (4a)


An essential property of nonlinear behaviour is the dependence of the frequency w on the amplitude a of the wave. For near-linear waves, u> may be written as:

UJ = uj0(k) + L02(k)a2-\ . (5)

Simply using this expression for u in Eq. (4b) results in:

dki fdu>o 9u>2 2\ ®kj . .da2

-a •w2(k)—=0. (6) dt \ dkj dkj J dxi dxi

The important term in Eq. (6) is u>2da2/dxi. This term leads to a correction of 0(a) in the characteristic velocities. The other extra term merely gives an 0(a2) correction of an existing term in dkj/dxi. Substitution of Eq. (5) into the wave action equation (4a) only results in extra terms of 0{a2). Neglecting this term and considering wave propagation in one spatial direction only, results into:

da2 ,,,^da2 ,,.,. 0dk ._ , — +.'0(k)^ + < ( f c ) a » - = 0, (7a)

dk , .,. dk ,,. da2 .„ N

-5E+-'o(k)^+^k)—=0, (7b)

where a prime denotes differentiation to k. The characteristics (or wave rays) of this set are given by:

^=#)TyS#)a. (8)

If u>2(k)ujQ (k) > 0, we have two different characteristic velocities. This is clearly a nonlinear effect because for vanishing a (the linear approximation), we have dx/dt = oj'0(k), as follows directly from Eq. (4b) when w = u>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,

LO 2 gk(l + (ak)2 + • • • ) , (9a)

'gk[l+l-(ak)2 + --- ) . (9b)


We thus have LU0 = \fgk and LO2 = gl^2kb/2/2 and we have LO2LOQ = -gk/8 < 0 for all k for waves on deep water.

2.2. Basic ideas of the Benjamin-Feir instability mechanism

A Stokes wave can be unstable due to side-band perturbations as shown by Benjamin and Feir (1967). The essential steps are as follows (see also Stuart and DiPrima, 1978),

(1) Consider the small-amplitude wave,

aexp[i(kx — cot)}. (10)

(2) Nonlinear interactions force the generation of harmonics of this mode in particular the second harmonic which is proportional to:

a2 exp[2i(kx - ut)] . (11)

(3) Suppose that two modal perturbations arise,

An upper side band a\ exp[i(kix — u>it)], (12a)

and a lower side band a-i exp[i(k2X — u>2t)], (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)


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 Bot tom

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 5LJ \ a(x, t) = 2oo cos ( —x —t I , (16)

X{x,t) = -(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 = o ( i ) + M + M + .... (18) w 8 dk2 384 dk4 v '

3.1. A heuristic derivation of the NLS 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


given as (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,

w = ft(fc,a2). (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) <C 1 and 5k/ko = C(/3). We now consider small changes of this basic state, i.e., we expand around the basic state consisting of (wo,ko) an-d zero amplitude (thus around the linear approximation). As amplitude measure, we use the small parameter e which stands for either ak or a/h whichever parameter is the most restrictive. Then, we have,

u = n(*,.o)+(£)o«+i (*g)o(«,.+(g>)|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 = ( ^ ) a n d < = ( | 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)


and changing to A leads to the NLS equation in its usual form,

dA d2A , „ l 2 „

<9£2

where the abbreviations,

have been used and where e = j3 has been set. For deep water we have,

1 + -k2a2 gk 1 + - f c V ^ 0 ,

and we thus obtain for deep water the NLS equation,

dA lu)0d2A 1 , 2 | „,,,„

(26)

(27)

(28)

(29)

3.2. The scaling in the NLS equation

The NLS equation reads,

dA , d2A . „,2 „

where

(30a)

£ =/3(z - csi) and r = e/3t. (30b)

In order to see why r and £ are scaled as Eq. (30b), we follow simple arguments of Asano (1974). Consider two waves with frequency and wave numbers (u>, 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:

_, /(,-, T, 1 d2ui r^2 1 d3uj _~ n = c " ( ) * + 2 ^ * +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 ( o ) + I^+i^2 9 + 2dk2 + Qdk3 (32)


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=cg(°)+£CsW+£2cg(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:

pedt 9 '

Introducing the slow time scale r = £/3t, one obtains d£/dt = cg^(^,r). This produces,

£ = (3{x - cg(°H), r = e(3t. (34)

3.3. A sketch of the derivation in two horizontal 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)


which thus represents a progressive wave in the xx direction with a slowly-varying 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 = <Pm , E = exp[i(A;a;i - Lot)}.

Subsequently, the functions (m(x,t) and <j>m(x,t) are expanded as:

(36b)

cm = E £ ^ ( n , m ) (^ 7 ?' T ) -n—m oo

= J2en</>^m>(Z,T,,z,T),

(37a)

(37b)

£ = e(xi - cgt), rj = ex2 , T = e2t, (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 <^°'°) are taken to be zero, £(0,°) = </>(°'°) = 0. Upon substitution of the representation for <J? in the Laplace equation, we get,

E E ^ ^ ( n ' m ) * ^ V ( - ) + 2 f e f c m - ^ ( B , m )

dz2 d$

9 2j , (n ,m) Q2±(n,m)

«9£2 dr]2 0. (38)

Using the bottom condition, it can be seen that <^1,0) and c/>(2'0) are independent of z. <f>(3'°} is the first zeroth-harmonic term which depends on z,

dz [ + > [d? + dV2 r •


For <^1,:L) and <fi(2'2\ the following expressions are found,

and for c^2,1) is found,

.dB (z + h) sinhk(z + h) — htanhkhcoshk(z + h) <9£ cosh k h

(39b)

Subsequently, the expressions for C, and <£> are substituted in the kinematic and dynamic free-surface conditions. The coefficients of enEm are equated to zero for n = 1, 2,3 and m = 0,1, 2 and then it is possible to calculate the quantities £(n,m) for n = i i 2 a n d m = 0,1, 2. For n = 3, the component d^'^/dz of the vertical velocity gives a contribution in the kinematic free surface condition. Eliminating C^2'°\ a differential equation for </>(1,0) is found,

2 , a V l i 0 ) L0V (1 ,O) ,2r , 2 N I S | S | 2

where c is the phase velocity, c = w/fc and a — tanh /c/i. Also, an equation for 0(3>°) follows, but, that one is not needed here.

The coefficients of e3E1 in the kinematic and dynamic free-surface conditions yield two equations for </>(3,1) and £(3,1) at z = 0. Elimination of one of these from the two equations shows that the equations are only compatible if the following condition is fulfilled,

dB d2ud2B d2B 1 , 4 / , , 4 M „ , 9 r , 2 i ^ + "dkiW + CC9W = 2k{9a ~ u + 13a ~ 2 a m B

+ k2(2c + cg(l-a2))B^^. (41)

It is noted that Eqs. (40) and (41) describe the evolution of the progressive wave to first order in e. i^1'0) is the first-order mean current and B is the amplitude of i^1'1) evaluated at z = 0. Equation (40) is a Poisson-type of equation for the mean flow with forcing and Eq. (41) is a NLS equation, coupled to the mean flow.


For later use, these equations are written in the following form,

.dB x d2B d2B , l 2 n „90 ( 1 , o ) , ,

l^ + X l W + >*1W = Xl1 l X2 dT' (42a)

a ^ + g h ^ = ^ ( i l B P h ( 4 2 b )

with the coefficients given by:

xf = ^ ( 9 ( 7 - 2 - 1 2 + 1 3 a 2 - 2 a 4 ) , (43a)

X2 = ^[2c + cg(l-a2)], (43b)

a = gh- cg2 , (43c)

4 = fc2[2c + c 0 ( l - o - 2 ) ] , (43d)

A i = ^ - [ - c s2 + gh(l - cr2)(l - kha)}, (43e)

« = ^ | > 0 . (430

Because B is the envelope of the wave potential, we write xf a n d 4 to remind one of the fact that these coefficients depend on the definition of B being the envelope of the wave potential or the free-surface elevation; in the latter case, we write Xi a n d 4- We drop the superscript in those situations in which the value of the coefficients is not immediately needed. We have,

X\=^x\ and 4 = -4. (44)

The solutions for the £(n>m) follow from the free-surface conditions for the various orders of approximation using the solutions of Eq. (39). We obtain (Davey and Stewartson, 1974),

v 2

g(^=iuB, gCW)=cga-^--k2{l-o2)\B\2, (45a)

g C ^ =iu,D + c g ^ , flC(a'2) = k2B2 ( ^ ) , (45b)

LOF = 3ik2B2


1 - a 4 ' 4CT2

3.3.1. An alternative 2DH set of equations

A quantity Q(^,rj,T) is introduced by:

Q{Z,V,T) = d^ifi) l

k2 d£ gh-cg2 9

The mean surface elevation £(2'0) then is given by:

£(2,0)

cg2(l-o-2)\B\

k2 „ k[a + 2kh(l-a2)} 2

g gh- cg2 B\2.

Equations (41) and (40) can then be written in the form,

i ™ + \ ^ + ^ = vt\B\2B + v2QB, dr 0^ + ^dr,2

{gh-cg2)'~- +gh

92Q , ghd2Q r,d

2\B\2 ^ <9£2 dr]2 l dr)2

with

>1 = i „ » ( t ) < o , w = ^ ) s | > „ ,

k4

Acoa2 9 - 10a2 + 9CT4

2a2

V2

gh-cg

fc4

:{4c2 + 4ccs(l - a2) + gh{\ - a2)2}

2U)Cr

K\ = ghc,

[2c + c f f ( l - a 2 ) ] > 0 ,

2c + c 3 ( l - c r 2 ) gh - cg

2

(45c)

(46)

(47)

(48a)

(48b)

(49a)

(49b)

(49c)

(49d)

Equations (48) are sometimes referred to as the Davey and Stewartson equations (D&S Eqs.). In fact, it would be more appropriate to call these and similar equations the Benney and Newell equations because they were first derived for


water waves in the paper of Benney and Newell (1967). It is noticed that, in absence of capillarity, one has gh — cg

2 > 0 for waves of finite length and therefore, Q satisfies an equation of Poisson type (i.e., an elliptic equation). We also note that Eqs. (48) are derived under the conditions that e = ka, P = A/A = Ak/k and AA^/Afci are all small parameters of the same order where k = (k2 + k^)1/2.

3.4. Conservation laws

It is well known that the NLS has a infinite set of conservation laws. Writing the NLS equation as:

da d2q , ,n „ {~b\ + W^+(Jl^ 1 = ° m - o o < x < o o , t>0, (50)

Zakharov and Shabat (1972) (see also Lamb, 1980, p. I l l ) gave the conserved quantities in the form,

{2i)nCT /

CO

fn(x)dx, n > l , (51a)

-oo

with

/n+i = 9i(^) + . £ fjfk> * = |"ii* (51b) j+k=n

and /o = 0. The first two conservation laws are readily obtained as follows (e.g., see Debnath, 1994, p. 349). The first conservation law is obtained by multiplying Eq. (50) by qx and the complex conjugate equation by qx and subtracting these equations. An integration to x then leads to d/dt J_oo \q\2dx = 0.

The second conserved quantity fc is obtained as follows. First, we multiply Eq. (50) with qx and its complex conjugate with qx and add these equations to give,

«(<Mx - qtqx) + {qxxqx + qxXqx) + vi\q\2(qqx + qqx) = o. (52)

Secondly, Eq. (50) and its complex conjugate are differentiated to x and we multiply the resulting equations with q and q respectively. Adding these equations leads to:

i{qqxt - qqxt) + {qxXXq + qxxxq) + o\ [q{\q\2q)x + q{\q?q)x] = o. (53)


Subtraction of Eq. (53) from Eq. (52) and integration to x then leads to:

/

oo

i(qqx - qqx)dx = 0 . (54)

•oo

d_ rco

dt

Usually, real conserved quantities are used with different numerical factors, see, e.g., Watanabe et al. (1979), also given in Dingemans (1997, p. 928). The first two are:

/

OO -I /-OO

dx\q(x,t)\2dx and I2 = — / (q*qx - q*xq)dx. (55) •oo • " J -oo

Watanabe et al. (1979) gave also a different conserved quantity IQ as:

7° = I - " I*'9'2 " ^iiq*qx ~ m*x)} ^ ' (56)

which upon differentiation to t leads to the relation,

d r°° dt /

OO

x\q(x,t)\2dx = I2, (57) -oo

which can be used to define the velocity of the centre of gravity of q(x, y) because the left side of Eq. (57) is independent of time as Ii is a conserved quantity and thus time-independent. With the centre of gravity XE of q(x,t) defined as:

S^°oo x\q\2dx dxE li %E = roo3 | ,9 , , it follows that ——- = — , (58)

J_oo \q?dx dt h

and thus XE is also a conserved quantity. Ablowitz and Segur (1979) gave the following conserved quantities belong

ing to the D&S equations (42),

h= JJ\B\2d^drj, (59a)

h = IJ(BW~A'w)d^dn' (59b)

W/H^-<)^' (59c)


SI [{-dB 2

+ Mi dA drj

4 , aX2 fd(f>

d£ (xD3 (d±

drj d^drj. (59d)

These conservation laws can be useful as a check on numerical computations

of NLS-type equations.

3.5 . Special cases of NLS-type equations

In the reduction to the ID case, the dependence on the lateral coordinate 77

is ignored. Equation (42b) can then be integrated once so tha t the resulting

system is:

8B x d2B

30(1.0)

vi\B\2B,

dt

where the condition d<f>^1^ /d£

given by:

v\ = Xi

(60a)

- « 2 | S | 2 , (60b)

0 has been used and v\ is

-X2 • (60c)

0 whenever \B\

«2

The long-wave potential (T>(1,0) can be determined after the complex amplitude

has been determined from the NLS equation (60a).

In the limit for deep water, we have kh —> 00 and a —> 1. Keeping now

ekh <C 1 (and, therefore, in fact, Skh <C 1, or the length of the wave group

is much larger than the depth) , Eq. (40) reduces to V ^ 1 ' 0 ) = 0 and Eq. (41)

reduces to:

which is the two-dimensional nonlinear Schrodinger equation for deep water.

When the envelope B is independent of the lateral coordinate 77, the equation

reduces further to the one-dimensional NLS equation for deep water. Here,

the envelope B is the amplitude of the potential. Notice tha t Eq. (61) is in

fact uniformly valid for kh —> 00 (the condition ekh <C 1 need not be imposed)

because the troublesome terms cancel in the next approximation, see Longuet -

Higgins (1976).


Davey and Stewartson (1974) gave the following set of equations as the shallow water limit {kh —• 0 and cg -» c) of Eqs. (40) and (41),

2ik8B_{khfd^+8^ = ^ ^B3J^^ ( 6 2 a )

JghdT v ' d£2 drj2 2gh3* ' yfgh dt,

The validity of these equations has been checked by Freeman and Davey (1975) who started from the original equations of motion and introduced the appropriate long-wave scaling. They showed that the double limit e —>• 0, kh —> 0 in the asymptotic expansions for $ and £ was uniform when the condition e/{kh)2 <§; 1 was fulfilled. Because e = ka, this condition can be written as (a/h)/(kh). This condition does not necessarily imply that the Stokes parameter which is given by (a/h)/{kh)2 is a small quantity.a

Under the condition that e/{kh)2 = 0(1), Freeman and Davey obtained the Kadomtsev-Petviashvillii equation (the two-dimensional generalisation of the Korteweg de Vries equation).

When B and i^1'0) are independent of 77, i.e. B and fit1'0' do not vary in the direction perpendicular to the wave motion, the set of shallow-water equations reduces to a NLS equation for B,

2ik OB fl,,nd2B 9k2 , „ l 2 „

w*-{kh)w = -w*™B' (63)

here use has been made of d<j>W/d£ = 0 for \B\ = 0.

3.6. Effects of surface tension

Djordjevic and Redekopp (1977) extended the analysis of Davey and Stewartson (1974) in that they also included the effect of surface tension. The linear dispersion relation and the phase and group velocities then are given by:

T/fc2

uT = (1 + >y)gk tanh kh, with 7 = -— , (64a)

aNotice that Djordjevic and Redekopp (1978) defined e/(kh)2 to be the Stokes parameter while e measured the wave slope ka.


^(( l+7) tanhfc / i .k

1 -c 2

2

1

l + kh1 a + 2 ^ (7 l + 7 _

(64b)

(64c)

It is found that the second-harmonic term £(2>2) becomes singular when 7 = cr2/(3 — cr2). In that case, one has second-harmonic resonance which is possible for capillary waves. Assuming that the wave numbers are not too close to the ones for which 7 = er2/(3 — a2), the third-order terms may be considered. Instead of Eq. (40), now the following equation is obtained for the leading-order mean flow </>(1,0),

2 ^ ( 1 , 0 ) t 3 V ( 1 , 0 ) , 2 f ,, 2 2 1 d\Bf (65)

Introducing a quantity Q byb:

8*(1'°> , «. f„. 7 , . „ ,,

the set of Eqs. (48) is recovered now with the coefficients0 modified by the surface tension effect, see Djordjevic and Redekopp (1977), Ablowitz and Segur (1979, 1981, p. 320) or Dingemans (1997, p. 903).

A few notable differences between the situation with and without surface tension are listed below,

• With surface tension included, it becomes possible that cg > gh. This is easiest seen by considering a series expansion for small kh of cg from Eq. (64c) which yields,

cg = gh(l+j) (l + ^ ) = 0{{khf). (67)

As 7 > 0, we have cg > \fgh in the shallow-water approximation.

bNotice that in Eq. (2.14) of Djordjevic and Redekopp (1977), a printing error occurs: the first term between curly brackets has 2c instead of 2c-y in the numerator. cNotice that a misprint in Djordjevic and Redekopp (1978, Eq. (2.17)) is present: the third term between square brackets in the expression for v\ should have the numerical factor one and not four, see also Eqs. (48).


The coefficient v\ of the term \B\2B is singular when cg = y/gh and when 7 = cr2/(3 — <72). For cg = y/gh, one has long-wave/short-wave resonance in which the group velocity of the short (capillary) waves equals the phase velocity of the long (gravity) waves. Near cg

2 = gh, another scaling of the independent variables has to be used and the asymptotic expansion of $ and ( is also different (Djordjevic and Redekopp, 1977),

$ = £2/3<f>0 + e4>i + ei/3h + ••-,

with ( = e2/i(xi -cgt), r) = e2/ix2, T 2/3 0 =-4/3 t.

In this case, the ID version becomes,

FB, dB , d2B

dr di2

where

and

nt Si

9 F -« i^( l^ l 2 ) .

(68a)

(68b)

(68c)

(69)

i,o)

d£

Si l + g ( l -a 2 ) ( l+ 7 ) and Qi = -5ik2{l-a2) > 0.

A major aid in discussing the general behaviour of the equation system including surface tension is the figure for the parameter space of the coefficients with respect to the surface tension parameter 7 = "~fk2/(pg) and kh (Fig. 1, taken from Ablowitz and Segur, 1981). A similar picture is also given by Djordjevic and Redekopp (1977). The corresponding set of equations is Eq. (42) where, as noted before, the coefficients are modified due to the effect of surface tension. An explicit definition of these coefficients are found in Ablowitz and Segur (1981, p. 320) with v = v\ = \i ~ ^2X2/0: and A = Ai,

X = Xi-The character of the solution is determined by the sign of the coefficient.

Therefore, each region of this figure has specific characteristics regarding the dynamical behaviour of waves. Along the two lines bounding the region F, singularities of v occurs. The other three lines represent a simple zero of a coefficient. The regions C, D and F where \v < 0 admit solitons as solutions for


0 .25 .5 .75 1.0 1.25 1.5

Fig. 1. Parameter space of the coefficients inclusive surface tension, showing where they change sign as functions of surface tension 7 (abscissa) and kh (ordinate); from Ablowitz and Segur (1981).

the amplitude envelope. No soliton solutions are possible in the regions A, E and B where Xu > 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 Bottom

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


(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

t Z = e2 (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 £: cs(£), c(£) and fc(£). The free surface is expanded as:

with

and

COM)

E

oo ( n

£ < E c(n,m)(£,T)En

n—1 \m=

exp / k(£)dx - wt I

(71a)

(71b)

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:

dcj>^°) k2cg H ; ?r "i ^ QQ&T)

dr gh - cg

the equation for </>(1,0) becomes simply,

dQ

{2^ + l - a 2 } |S | (72)

dr 0 , with the solution Q = Qo(£): (73)


and the equation for B{^,T) becomes,

dB d2B i-^r + Ai - ^ + ifnB = vi \B\2B + u2QQB , dr2

where the coefficients are given byd:

(1 - cr2)(l - kha) d{kh) Mi

Ai

v\

a + kh(l-a2) <%

1 {l-Bh.(l-kha){l-a2)\ C9 ) 2UJC

a(l - kha) k' a + kh(l-a2) ' ~k

1 d2u 2cg

3 Ok2

(74a)

(74b)

(74c)

4toa2c 9 - 10a2 + 9cr4 2azc, 2„ 2

hfl

+ 4-(l-<r2) + 4 ( 1 - f f 2 ) 2

C(7 CQ

yi *1 1 2a c„

\-a' > 0 .

(74d)

(74e)

It is noted that Eq. (74a) can be written in a simpler form upon application of the transformation,

B (£ , r )=B(£ ,T )exp 'J V2(OQo {£)<% (75)

The resulting inhomogeneous NLS equation then reads in terms of B,

-iUxB. (76) .dB . d2B ...2-

It is obvious that the term V2Q0B gives only a phase shift. The essential difference between the NLS equation (26) and (76) is the term —ifiiB in the right-hand side of Eq. (76). Another difference is that the coefficients Ai and Ui in Eq. (76) are functions of £.

Equation (76) describes the evolution of wave packets propagating over an uneven bottom under the condition that reflection can be neglected and

^Note that (1 — a2)2 occurs in the last term between curly brackets in the expression for V\ and not (1 — a)2 as given by Djordjevic and Redekopp (1978, Eq. (2.17)).


consequently the depth varies very slowly, h = h(/32x). Only for constant depth, we have Hi = 0 and Aj, v\ are constants.

Equation (76) can be transformed to a homogeneous equation with independent coefficients by introducing the transformation,

By carrying out the transformation and choosing a~1da/d£ obtains,

3D , 82D , „ „ Xi^r^- -h\D\2D = 0, dt dr2

with

B = aD , u\ = a.v\ and a(£) = exp - / »i(i)d£

(77)

-fii, o n e

(78a)

(78b)

4.2. Propagation in two horizontal dimensions

The starting point for the derivation is the usual set of equations for inviscid, irrotational fluid motion with a free surface z = ((x,t) and velocity potential $>(x, z, t) on water of varying depth with the bottom given by z = —h(x).

For the derivation of the evolution equations for complex amplitude of the wave group and the accompanying long wave motion is referred to Liu and Dingemans (1989) and the references referred to there. Basic to the perturbation approach is the introduction of two small parameters viz. a modulation parameter S and a nonlinearity parameter e. The nonlinearity parameter is related to the slope of the carrier wave ka with a and k the amplitude and wave number of the carrier wave. The modulation parameter is related to both the inhomogeneity of the medium, i.e., the bottom slope and the variation of the incoming wave field in both time and space. Use is made of slow scales x\ = Sx — [Sx, Sy] and t\ = St so that Vi = d/dx\ and the most general set of equations is obtained when e and S are of equal order.

The free surface elevation C(x,i) and velocity potential <&(x,z,t) are expanded in terms of the nonlinearity parameter s and, as we are interested in the propagation of harmonic waves, also in terms of harmonics,

=+n C = ^ £ " ] T ^n<m)Em, $ = ^2en

m—-\-n i)Er,

with E = exp ':X{xi,h) (79)


and Qn>~m) is the complex conjugates of Qn<m) and similarly for </An'-m). Expansion of the phase function x as x = S n = o enXn(xi, t\) and introducing new functions £(">m) by:

^(n,m) = £(n,m) &xp

gives the expansions,

7 1 = 1

and similarly for <jy-r

m=-\-n m=-j-n

c = Y^£n Yl c ( n , m )£™, $ = J2£n Yl <r4(7i'm)-Bo1, n = l m=—n n=l m——n

where £ 0 — exp[ixo(a:i,*i)/£], (80)

and u>o = —dxo/dti is the constant carrier wave frequency and ko = ViXo is related to WQ through the linear dispersion relationship LOQ = gko tanh koh with ko = \ko\- For the several orders in efn) and the harmonics (m) equations for the £(ra-m) and </>(n'm) are obtained. For the first-order problem is obtained,

^ = C ( i , o ) = 0 | ^ ^ M c o ^ k o i h + z)) C(1,1) = 1 A ; ( 8 1 )

oz 2wocosh(fco/i) 2

where A is an unknown complex amplitude. To ensure the nonsecularity of the higher-order solutions, solvability con

ditions have to be imposed, see Chu and Mei (1970) and Liu and Dingemans (1989). The final result is an evolution equation for the complex amplitude A and a wave equation with forcing for the wave induced current i^'1'0) which is a real function (see Eqs. (5.12) and (5.14) of Liu and Dingemans, 1989). These equations are, without the fast varying part hi of the bottom, and in physical variables, while writing <j> for <j>^1'0') and A for A for convenience,

dA : , _ , „ , ! , „ x A i ffeor,/9cg ko 8t • c , .VA + - (V.c s M-- |^V^^ .V

\k0 J ko \ko k0 ) k0 )

+ ikluQK\A\2A+l-vA = Q, (82a)


and d2

m2 -V-(ghVcf>)=V- ~92\A? 2w0

\A? 4 dt Vsinh2g

(82b)

where

K= 2-(cosh4q + 8 - 2 t a n h 2 g ) , kx = — , (83a) 16sinh q g

a = cothg and q = koh , (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 con

siderably. 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 dA i (dcqd2A c„d2A\ , 2 , 4l2 „ t „ , , ,

C9^Z--A ^ " 5 1 7 + T-^2 \+^k2u0K\A\2A-iAG<P = 0, (84a) dt g dx 2 \ dk0 dx2 k0 dy2

with

The corresponding wave equation for a horizontal bottom with fc0 directed along the a>axis is:

^ - f l / l V 0 - ^ o " ^ ^ - 4 s i n h 2 , dt • ( 8 5 )

The reduction of Eq. (82) to ID is readily obtained by supposing fc0 is directed along tree a;-axis and ignoring all y-dependence,

dA dA\ ldcgd2A „ ^ J , 2 IAI2A

lK+C^) + 2dt^+AG*-k0"0KlAlA =

1 f d fdcg\ } dA fidcq 1 \ „ , N

Mtek£)-'i\te-{2i£-2v)A> (86a)

and

d24> d ( d<j>\ d ( fc0 2. „ l 2 \ w2 d ( u p . , , ,

W - d~x {9h£) = dx {^An - Tat {^tq) ' (86b)


4.3. Shallow-water limit

In the shallow-water limit, kh —> 0, we obtain, under the condition that the Stokes number is small, e/(kh)2 <C 1, the following expressions for the coefficients fix, Ai, v\ and f2,

«~\M"*- "Aw**"- (87b)

where a prime denotes differentiation to the argument. For \\V\ > 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\ + <52#2 + • • • where S is a measure of the slope of the modulation of a wave train about a uniform finite-amplitude state. The discussion here is restricted to any small-amplitude long-wave perturbation of a wave which is modulationally stable (i.e., Aii^i > 0). The following further coordinate stretching is introduced,

'»"{?£)-*}• *=""<• c(X)

Substitution of the expansions for R and 6 yields expressions for RQ(X) and c(X) and a relation between 9\{X,T) and R\(X,T). The secularity condition for R2 and #2 yields the generalised KdV equation which can be written with i?i =h7/2H(X,T) as:

where RQ = roh~~l/A.

4.4. Effect of an ambient current on ID propagation

We proceed with the assumption of kh = 0(1) and a current U such that U/s/gK < 1. In the absence of waves, the current variation may be determined


by the nonlinear shallow water equation, i.e.,

d(c dU(h + Cc)

dt dx 0,

dU_ dU_ dCc

dt dx dx

(90)

(91)

Further, the variation of depth and current is assumed to be an order of magnitude higher than the wave nonlinearity parameter e = ka, i.e.,

1 dh kh dx

1 dU kU dx

1 dU toU dt

= 0{e2). (92)

Using a perturbation analysis with respect to the underlying current (U, (c), Turpin et al. (1983) showed that the amplitude A of the waves satisfies the equation (note that the form of the equation mentioned below is a result of multiplying icg to their original Eq. (2.22) and converting the pressure amplitude to surface amplitude),

IAT +icgA^ + X\ATT - vi\A\2A + ijcA = 0, (93)

where T = e2t, £ = e2x and r = e(J(dx/cg) — t). In Eq. (93), A is the complex amplitude of the first order elevation, identical to c^1,1) in the expansion series of Eq. (71). The coefficients vary slowly as a function of depth and current and are given by:

Ai(£,T) 1 {ca?

2uv (cg

1

uy d\

l-^(l-a2)-(l-akd)

"i(t,T) =

2{cg + U)2 dk2 '

g2kA \ Aula2 1

(94a)

IOCT2 + 9CT4 3f± gd-c2

41-1 c„

•4 1 - ) ( l - a J ) gd 2 \ 2 ( i - O

1 dtor

2u~r~&T 7c(?, T) — +

ujr d l c„

Tdl; with a — tanh(fcd) and d = h + (c.

U

(94b)

(94c)


The effect of current in the first instance is reflected in the linear dispersion relation, i.e.,

UJ = ur + kU with J^ = gk tanh kd, (95a)

c = — , vg = Cg + U, and (95b)

c° = d^ = ir^+kh(l-^> <9 5 c> where vg is the absolute group velocity, taken with respect to a fixed reference frame and d represents the mean water level including the set-down due to current and tur the apparent frequency for an observer moving with the current U.

Equation (93) with the coefficients defined by Eq. (94)e is an extension of the equation derived by Djordjevic and Redekopp (1978) and represents a general one-dimensional modulation equation for narrow-banded short waves on an ambient current or a long wave (long enough compared to the short waves to validate the scales of Eq. (92). For an opposing current, waves are prevented from propagating upstream as the group velocity cg becomes zero. The equation fails near such points and its validity is limited to milder opposing current such that the blocking condition is not met.

Additional properties of the modulation equation, Eq. (93), can be derived from a simplified form made possible through a transformation (Djordjevic and Redekopp, 1978; Turpin et al., 1983). An important parameter that emerges is:

K=-T* > Wr / i / U r

(96)

where s may be recognised as the shoaling factor for infinitesimal waves. For a given variation of depth and current, broad features of the evolution of a wavepacket may be determined from the parameter K. Formation of a soliton is expected with increase of K to a positive value. In both cases of with and without current, K becomes zero when k(h + (c) = 1.36 where k is the local wave number taking into account the ambient current, if present.

eA in Bq. (93) is the complex amplitude of surface elevation unlike in the expression ((2.23c), p. 5) in Turpin et al. (1983) where A corresponds to the pressure amplitude.


5. Some Solutions of the NLS-type Equations

We consider the Nonlinear Schrodinger equation in its standard form,

.dA , d2A . .., „ n

dr (97)

where A is the envelope of the free surface elevation ((x, t), Ai = co"(k)/2 < 0 and v\ > 0 for kh > 1.363. For v\ > 0, so-called soliton solutions are possible, while for v\ < 0, decaying solution can be found.

5.1. Decaying solutions

A specific decaying solution with an oscillatory tail was given by Benney and Newell (1967), see also Segur and Ablowitz (1976). This decaying solution of Eq. (97) reads,

^ , r ) = ( - A 1 r ) - * ( ^ - ) Aexp 4Air

|A|2log(-A1r) + (98)

where A and <j> are constants. Segur and Ablowitz take this form of solution as starting point for the case of the NLS equation for uneven bottom and they let the constants A and (j> then be slowly varying in £ and r .

In 2D, a decaying solution which satisfies Eqs. (42) reads,

B(£,n,T) = - e x p

db 4>{m (£, V, r) = -X2 -^i + C(T)T, + d{r)

(99a)

(99b)

For scaled versions of Eqs. (97) and (42), Ablowitz and Segur (1979) gave similar solutions.

5.2. Soliton-type solutions

We now consider the case v\ > 0 so that soliton-type solutions are possible. The free-surface elevation is given by:

COM) = -Aei<-koX-u'0^ +CC. (100)


Since we look for stationary solutions, we put,

A($, T) = &(X)e i^-ST) where X (101)

is a moving frame with respect to the frame (£, r) and v and £ are constants. Substituting Eq. (101) in the NLS equation (97) leads to an ordinarily differential equation in b(X),

Ai d2b dX2 + ab- i/ib3 = 0 with

4AT' (102)

and where has been substituted £ = v/{2\{) in order to obtain real solutions for the amplitude b.

Imposing the condition that b(X) and db(X)/dx —> 0 for X —»• ^oo, the following solution is obtained (for details, see Dingemans, 1997, p. 919),

^4(£;T) = ao sech , V2

V ^ exp Z 2A7^ ' 2Uia° + 4AT (103)

where v and the amplitude ao are still two free parameters. The parameter v is usually taken to be zero. Notice that this solution is valid only under the condition that i^iXi < 0, or, otherwise stated, v\ > 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 r0 . (104)

Ai 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 cn-solution, and (3) v\ < 0 and r3 > 0, giving the sn-solution. These solutions are:

(1) i/i > 0 and r 3 > 0

A(£,r) = &3dn

• exp

" 2Ai

1/2

i—— £ — i 2Ai

X

1 ,„ x V2

_ , i r . 3 ( 2 _ m ) + _ _ (105a)


with m

(2) i ^ > 0 a n d r3 < 0

A(£,T) = a 0cn

•exp

T3 -T-p

»"3

2 \ 1/2

la 2r0

v\ u

< 1. (105b)

2TOAJ X m

1 • V y .1 2 l -2Ai 2 VTO 4Aii/iag

with TO = ( 2 — 2aA :

(106a)

(106b)

(3) iî < 0 and r3 > 0

A ( £ , T ) = a 0sn / \ 1/2

Kir) * m

• exp • v r - 1 2 M , 2Ai

V"

with m = 2s

TO 4Aiîa,

2 \ - i an V

rz —

(107a)

(107b) ÔQVI 4Aii/1a2,y ' ' J TO 4A1J7!

Because for m —» 1, we have dn —• sech, sn —>• tanh and en —> sech, we see

tha t the limiting values for m —> 1 of the solutions (105)-(107) are:

1/2 .A(£, i) = 63 sech

vi\i X exp W" iST

with r 3 = 63 = 2 — > 0 and vx > 0 ,

A(£, t) = ao sech 2 \ I /2

" 2Ai

(108)

X exp ^ - M ^ + SAT

with a < 0 , (109)

A(£,,t) = ao tanh 1/2

X exp . v + • / 2 v 1

with i i < 0 and a < 0 . (110)


Some examples of these solutions have been considered in Dingemans (1997, p. 925). We replot solutions for the dn, en and sn-solutions given there.

In two dimensions, let us consider the deep-water case Eq. (61) because of its simplicity. In nondimensional quantities such that to = 1 and k = (£, m)T = (1,0)T and g = 1, we then have,

.dA 1 (d2A d2A\ . AI2A , , „ ,

We note that for NLS-type of equations for water waves, the coordinate along the propagation direction £ and the lateral one r/ are not interchangeable.

Fig. 2. A dn-solution.

Fig. 3. A cn-solution.

Fig. 4. A sn-solution.


Several special solutions of Eq. ( I l l ) have been given by Hui and Hamilton (1979). Denoting the angle between the direction of the carrier wave and the direction for which solutions are sought for by •&, the £, r\ plane is split into regions according to the sign of the quantity ip defined by:

i> = cos2 •& - 2 sin2i9 . (112)

For tp > 0 (tan2 •& < 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 tan2i?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)


$ = <£> -[Be kz A'd D 2fcz i2tf i > 2 e e + --- + 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

LO_0B

2k~dx u> d

2B d2B

5$

~di

% to

"l6fc3

A-3

4w' ' dx

•gC = 0 at

8k2 dx2

d3B d3B dxdy2

dB

dx3

4fc2 dy2

3k3

\B\2B

ox

0,

4ui -BIB

2UJ'

dB*

dx B* as

dx

dt k-V(\B\2) at z = 0.

(114a)

(114b)

(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 w dB ~dt + ~5k~dx~

uj d2B d2B

16 k3

8k2 dx2

d3B d3B

dxdy2 dx3

4fc2 dy2

k3

+ iT-B B

2OJ'

.dB* dx

\B\2B

6B* dB_ dx

kB dx z=0

(115)


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} <C 1. Lo and Mei argue that the equations also remain valid for kh = 0{{ka)~1} when also the condition d^/dz = 0 at z = —h is added.

6.2. Modification due to an ambient current

Modification to Dysthe's equation in the presence of an ambient current is considered by Stocker and Peregrine (1999). In addition to the wave-induced mean flow, denoted by $>(x,z,t) and (, we now introduce $ c (x , i ) and (c to represent the potential and elevation due to an ambient current. Thus, the expressions for potential and elevation are:

1, C = C + Cc -{Ae itf A2e

i2» + • • • + CC], (116a)

$ = $ + $ c + I [BekzeM + B2e IkzAld + • • • + CC]. (116b)

Assuming further that the ambient current U has a slow variation, the current field may be expressed as a perturbation about a constant (both spatial and temporal) mean —V, i.e., U = — V + (U\(x,t), Vi(x,t))T.

In a frame of reference moving with the mean velocity —V, the current modified higher-order equations read,

d 3 „ „ 1 „ (gko)1/2

dt+Cgodx

kA

KQ

v-v, B 8fc0

2 -{Bx

2( f lfeo)V2

. ( g f c o ) 1 / 2

1 16fcj}

B\B\2 - k0$CxB = i k0

4(fffc0)1/2

(Bxxx - GByyx) + k0B$x

2Byy)

B{BB*X - 6BXB*)

hPi, (117) i = 0

where the term Pi contains the higher-order contribution due to the current and is given by:

-Pi = ik0 V - V $ c z L B - i V $ - V B . (118) v2(fffc0)

1/2 [di+C90~dx~

To complete the system, the current potential $ c and the wave-induced mean potential 3> need to be defined. The current potential is determined by:

V 2 $ c = 0; z<0, (119)


dt

$C 2 -

V-V $c +

[l"H Cc 0; 0,

while the wave-induced mean flow valid up to 0{e4) is given by:

d 2 $

9 $ ~dz~

d_

dt '

' ~2

V2$ +

v - v

0; z< 0. dz2

<£ + 5C = 0 ; z = 0,

1/2

5a: 0; 0.

(120)

(121)

(122)

(123)

(124)

As seen from Eq. (117) B, the amplitude of the velocity potential, depends explicitly on the external current. However, the surface elevation terms are related to B in an identical manner as that in Brinch-Nielsen (1988) and Brinch-Nielsen and Jonsson (1986) without any explicit dependence on the current U.

To evaluate the performance of the 0(£4)-modulation equation, Stocker and Peregrine (1999) have undertaken numerical solution for a specific case of waves being modulated by a sinusoidally (of much longer wave length) varying current. Computed profiles from both a lower-order (C(e3)) and the higher-order models are compared with those from an exactly nonlinear boundary integral model. Comparison shows that starting from the identical initial condition, the evolution predicted from the lower-order NLS equation kept deviating with time. Significant improvement was achieved by adopting the C(e4)-nonlinear equation. Agreement with the fully nonlinear solution was good till about breaking (breaking was said to occur if sharp curvature appeared on the computed surface in the exactly nonlinear model) was initiated.

6.3. The Zakharov equation

That the NLS equation is a special case of the Zakharov equation has been proven by Stiassnie (1984) for the case of deep water. Originally, Zakharov (1968) derived a deep-water evolution equation for the amplitude of a wave field. A little later, the equations for restricted depth were given by Zakharov and Kharitanov (1970), still for horizontal bottom.


We first sketch the steps along which the Zakharov equation can be obtained. Here, we follow Stiassnie and Shemer (1984), see also Shemer and Stiassnie (1991), where the bottom has been assumed to be horizontal. A detailed account can also be found in Rasmussen (1998).

The kinematic and dynamic free-surface conditions are written in terms of the free-surface potential ip(x,t) = $>{x,z = £(x, £),£}, the vertical velocity ivs = (d$>/dz)\z=£ and have been given in Eqs. (3). Together with the Laplace equation V 2 $ = 0 and the kinematic bottom condition d^/dz = 0 at z = — h, these equations constitute the description of the physical problem. We stress the fact that in this derivation, the bottom is taken to be horizontal. The derivation of the Zakharov equation proceeds in the following steps,

(1) The horizontal Fourier transform of Eqs. (3) yields two integro-differential equations for the Fourier transforms ( and tp where the Fourier transform of a function f(x) is defined by:

f(k) = — dxf(x)e-ik*, (125a) 2 7 r i -oo

and the delta function is defined as:

1 f°°

5(k) = j^y]JxJkx- (125b) In addition to the variables £ and (p, ws also features in the transformed form of Eqs. (3). An expression for ws has to be found. This is achieved in the following steps.

(2) Taking the horizontal Fourier transform of the Laplace equation and satisfying subsequently the bottom condition yields a separation of the vertical structure in the following way,

$(fc, z, t) = 4>(k, t) cosh[|fc|(z + h)]; (126)

this makes it possible to express the free-surface variables tp and ws in terms of <fi(k,t) and ((x,t) in the following way,

1 f00

<p(x,t) = — / dfc[cosh[|fc|/i] cosh[|fc|£(x,£)] 2TT 7-oo

+ sinh[|fc|/i] smh[\k\((x,t)]]eikx , (127a)


ws(x,t) 2?r

dfc[|fc|0cosh[|fc|/i] sinh[|fc|C(a:,t)]

+ sinh[|fc|/i] cosh[|fc|C(a3, t)]]e k a (127b)

(3) The next step is to express ws in terms of ( and (p. It is here tha t the first

approximations have to be made. The expressions sinh(|fc|£) and cosh(|fc|£)

are replaced by their Taylor expansions up to 0{( | fc |£) 3}; ( is expressed

by its Fourier transform (. Finally the Fourier transform of Eq. (127) is

considered. This yields two equations in which <p and ws are expressed in

terms of <j> and £. An iterative solution of the equation for ip is applied to

obtain <j> as a function of ip and the subsequent use of 4>{(p) in the equation

for ii!s yields an expression for ws in terms of ( and (p. This expression

for ws is now used in the Fourier transform of the free-surface equations

(3). Multiplying the equation for C, by y/g/(2uj(k)) and multiplying the

equation for <p by y/u(k)/(2g) and adding these two equations together,

the result is an evolution equation for the complex variable,

b(k,t)

1/2

C(k,t) + i u(k)

_2u>(k)

where the dispersion relation is:

w = [g\k\tanh(\k\h)}^2

The evolution equation for b{k,t) then is:

1/2

ip(k,t), (128)

(129)

db (k,t) + iu(k)b(k,t) + i^2 / / dkldk2V^\k,kl,k2)C2n

n = 1 J J - o o

+ iJ2 dk1dk2dk3W^\k,k1,k2,k3)C3n

n = l

5

S / / / / dkidk2dk3dk4Xin)(k, fci, fe2, * 3 , k^C^ = 0 , (130)

n = l

with the C(n given by:

/ n - l \

Cln=[ l[b*(km,t))[ H b(km,t)\ •6[k+J2krn-J2k™)> ^ = 1 m = l


with * denoting the complex conjugate and $^m=n(-) = 0 an<^ llm=n(') = 1 whenever £ < n. Notice that C, and (p can be expressed in terms of b as:

u>(k) 1 / 2

C(M) = 25

<£(M) = -*

[6(M) + &*(-M)],

1/2

[6(fe,t)-6*(-fe,t)]. 2w(fc)

(4) We now use the transformation,

B{k,t) = b{k,t)e~iu^t

(132a)

(132b)

(133)

The term iiob then disappears from Eq. (130) while for b can be read B in Eq. (130). The next assumption now is that we suppose that B is composed of a (in time) slowly-varying part B and faster varying parts B', B" and B'",

B(k, t) = eB(k, t2, t3) + e2B'(k, t, t2, t3) + e3B"(k, t, t2, t3)

+ e4B'"(k,t,t2,h), (134)

where tj = eJt, j = 2, 3. The slow time ti is omitted because no exact resonance between three waves is possible. When surface tension is included, exact resonance is possible for three-wave interaction, but, these waves are very short and not of interest for us here. It is assumed also that most of the wave energy is contained in B. The representation of Eq. (134) is substituted into the evolution equation (130); separating terms of equal power in e leads to evolution equations for the B and B', B" and B'". For e1, no information is obtained. Terms with e2 yield an evolution equation for dB' jdt which depends on B and therefore can be integrated to t while keeping t2 and £3 fixed.

At 0(e3) is obtained an equation for idt2B + idtB" in which the right-hand member depends on terms with B where both slow and fast-varying terms are present. Separating this equation into a slow and fast-varying part and also writing B = eB, we obtain the evolution equation,

•£-///: dk1dk2dk3T^l2t3BlB2B3S(k • fei - k2 - fe3)

x exp[i(w + UJI- UJ2- wz)t], (135)


where Bj stands for B(kj,t). For B", an evolution equation is obtained in which the right-hand side does not depend on the slow time-scale so that an integration to t is possible.

Equation (135) is the so-called Zakharov equation which is also valid for restricted depth when the dispersion relation u>(k) and the definition of T2 are adapted for finite depth.

Once B has been determined, the free-surface elevation ((x,t) follows by:

((x,t)= J dk(^-\ [B{k,t)ei{-k-x-ut) + CC]. (136)

A problem with the Zakharov equation is that many different forms for the interaction coefficient T^2' exist. The reason for this is that there is some freedom in the definition of T^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_,

~dt -{k,t) = -i / / / dkidk2dk3X^l2t3C'3.2 ,

with for C" the expression,

(137a)

'n-l n - 1

CL= l[b*(km,t)) l[b(km,t))-S[k+J2km- J2k" m=\ m=n

x exp Y2um~ y^Wm )t (137b)


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,kuk2 ,k3) = T^(kuk,k2,fc3) = T^(kuk, fc3, k2)

= T(-2\k2,k3,k,k1). (139)

6.4. Reduction of Zakharov equation 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 + K,k0 + Ki,k0 + K2,k0 + K3)

x B*{K1)B{K2)B(K3)6(K + K1-K2- K3) . (142)


Substitution of Eq. (141) in Eq. (136) yields the following expression for (,

C(x,t) J,[kax-u(k0)t] 2TT

We also write ( as:

dn Lo(k0 + K)

29

1/2

[B(K,t)iKX + CC]. (143)

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 2 V k0

8k2

K L_ _) ±_

4k0 2k0

o k3 (146)

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 1 l~di + 2~V k0

da 1 d2a 1 d2a +

d3a

/2u;(k0)\l/2 1

{ 9 J 2n

dx 4k0dx2 2k2 dy2 8k3 dxdy2

K-2 + K3 - Ki d,Kid,K2dK3 1

4fo>

.(2) '^0,1,2,3(^0 + K2 + K3 - Ki,k0 + Ki, fc0 + K2, fco + K3)

• * ( / c 1 ) e ( K 2 ) ^ ( K 3 ) e i ( K 2 + K 3 - K l ) ' x . (147)

In deep water, one has u = \J g\ko + K\ = \/gko(l + 2K\/ko + | /c|2/fcg)1/4 and thus, to

first order, co(k0 + K) = w(fc0)(l + K\/(2k0)).


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,

TCII2 3( fc0 + K2 + « 3 - K l , ^0 + K l , fco + « 2 , ^0 + K3)

A-3

4TT2

3 / 2 ^ 2 • « 3 ) -

(«1 - K2)2 («1 - K3)

2 A ; 0 | K I - K 2 | 2k0\Ki - K3 + 0

Using Eq. (148) in Eq. (147), the following equation is found,

\K\*

k2 (148)

.da 1 / a

dt 2 V fo

1/2 ,9a 1 d2a 1 9a d3a 3i 53a ' 9a; Ak0 dx2 2k0 dy 8k2. dx3 Ak\ dxdy2

2uj(k0) K\a\'a--^-a

ik$ 2 da* dx

, o, ,o da 3 i * 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 =

cu2(ko) dx (150)

z = 0

Substituting Eq. (150) in Eq. (149) and rewriting results in:

. /da 2k0 da\ 1 d2a 1 d2 d3a 3i d3a \dx wQ dt) 4k0dx2 2k0 dy2 8k2 dx3 Ak$ dxdy2

<9$ ki\a\2a - a 2 —-

01 ' 2 dx -3ik2

0\a\2~ 2k0

dx a>(feo) dz (151)

z = 0

6.4.2. Modification due to surface tension

Hogan (1985) extended the analysis of Dysthe (1979) by also taking surface-tension effects into account and derived a fourth-order equation valid for deep-water 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


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,

2i da ~di

da d2a dx2 + C9dlc' + p ^ + q—~XlW1

da dy

d3a d3a ,da* ,da <9$ -is dxdy2 dx3 dx dx dx 2 = 0

where the coefficients are given by:

3S2 + 6s - 1 P 4(1 + s2) '

( l - 5 ) ( l + 6 s + g2) 8(1 + 1 ) 3

( l - s ) ( 8 + s + 2s2) 16 (1 -2s ) ( l + I ) 2 '

3(4s4 + 4s3 - 9s2 + s - 8) 8(l + s ) 2 ( l - 2 S ) 2

q =

Xi =

c„ =

l + 3s 2(1 + s) '

3 + 25 + 3s2

4(1+ s)2 '

8 + s + 2s2

8 ( l - 2 s ) ( l + s )

UQ 1 + 3S 9 2k0 1 + s '

with

99

(153)

(154a)

(154b)

(154c)

(154d)

(154e)

In variables with dimension, one obtains from Eq. (153),

'da da\ UJQ d2a tuo d2a 2 2 • wo d3a 2l{M+Cad-x)+^d^2+qe0w~Xl^koHa = -lsK'^

UJ d3a . , 2da* . , . l2da n , d$ —ir—^^r — lukoujoa — \-tvkouJo\a\ ^—h zfcoa—

/CQ 9a;3 dx "' ' dx

where the coefficients of Eq. (154) have been used.

dx (155)

2 = 0


To be able to compare this equation with other higher-order equations, we notice that in absence of surface tension (5 = 0), Hogan's result reduces to:

UJQ d2a u>o d2a 1 , l2 1 .u0 d3a da da\ ~dt+Cg&c) 8/cg dx2

16 &Q dx3

3 .UJQ da 1 2 da* 8 &o dxdy2 4 dx

3 . , . l2da , 9 $ -i/c0w0|a| — + k0a—- (156)

z = 0

Furthermore, as in Dysthe (1979), the same condition for 3>x follows from the kinematic condition,

a7 1 d(a2) 2W°" dx

at 0. (157)

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 water-wave 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 <92 \ Jfe4,„ l2„ , , , SW = ' 1 7 + ^ - 7 1 2 U « - « B " ^.\B?B > ( 1 5 9 b )

£2(B)

Ms (5,$)

VJJ d3

16k3 \ dxdy2 d^)B-

IK B

dB* dx - £ ) * • • " > % •

dx

(159c)

(159d) z = 0


Trulsen and Dysthe (1996) now assumed a wider bandwidth to exist,

lAfel k

0(e^2) and kh = 0{e~1'2). (160)

With e1'2 being the new expansion parameter, the new slow spatial and time variables are e1'2x, el'2t and also the vertical coordinate is somewhat faster than before: e1'2z. With a similar expansion procedure as before, now up to fifth order in the expansion parameter e1'2, the resulting equation is, in variables with dimension,

S(B) = C2(B)+N2(B,$)+£3(B), (161a)

the difference being only in the linear dispersive terms. For the nonlinearity, the narrow-band assumption is retained. We have,

u> ( 5 <94 <94 94 \ A ( J B ) = 32fc^ ( " 4 5 ^ + 15^W + 3d?) B

li (\ <95 _ <95 3d5

64 V 4 9a;5 dx3dy2 2 dxdy4 B. (161b)

While Eq. (161) is derived for wider bandwidth, it is still an approximation for a somewhat narrow bandwidth, albeit wider than before. The effect is seen in a further shrinking of the instability region for Stokes waves. Further progress is given in the paper of Trulsen et al. (2000) where no conditions on the bandwidth are set. Using pseudo-differential operators which capture the full dispersive behaviour, further progress is achieved.

Trulsen et al. (2000) start by expressing the surface displacement as:

1 r°° C(x,t) = - a{k)ei{-k*-u{-Wdk + CC, (162a)

2 J-oo

with the dispersion relation given by the deep-water form u = y/g\k\. Supposing that the free-surface elevation may be described by a modulation of a carrier wave with wave number fco = (ho, 0)T and frequency U)Q — w(fco), C c a n

also be expressed as:

C(a:, t) = \A{x, t)el(-koX-UJ°t^ + CC . (162b)


We now use k = ko + K as the modulation vector. Equating the two expressions for C in Eqs. (162), one obtains,

/

oo

A(K,t)eiKXdK, (163a)

-oo

with A{K, t) = a(k0 + K)e<["—M*o+">-«(feo»] . (i63b)

Differentiation of Eq. (163b) with respect to t shows that A satisfies,

dA — + i[u)(k0 + K) - Wo}A = 0 . (164)

To return to physical space, we use operator correspondence where we have to account for the fact that K is the change of wave number relative to fco. Writing K = (KJ, K2)T, we thus have,

> idx and > idv . (165) fco fco

In physical space, we then have,

dA — +L(dx,dy)A = 0, (166a)

with

L(dx,dy) = »{[(1 - idx)2 - d2

v}l,i - 1} , (166b)

and thus,

—- + —~ dKi{uj{k0 + K) - tuo] / dx'eiK<x-x^A{x',t)=0. (166c) at 4-7T2 J_00 J_00

By expanding Eq. (166b) in the derivatives, linear evolution equations to all orders can be obtained. Expansion up to fifth-order derivatives yields the linear part of Eq. (159). Trulsen et al. (2000) use the exact linear dispersive equation (166b) for the linear part of the higher-order NLS-type equation instead of some power-series approximation such as Eq. (159). Keeping the usual nonlinear term, the equation considered then is:

3 A 1 — + L(dx,dy)A + -iuj0k

20\A\2A = 0 . (167)

50 M. W. Dingemans & A. K. Otto.

Trulsen et al. (2000) also consider the equation with the nonlinear terms as introduced by Dysthe (1979) and then get the system,

dA 1 3 dA — + L(dx,dy)A + -iuj0kl\A\2A + -LO0k0\A\2 —

with

" v J- , y ' ' 2 u 0I '

1 , A2dA* ., A6i$ -cj0k0A

A— \-ik0A—

5 $ 9^

, 2 - u ^ , - , d

= 0, Z=0

1 d 1 Al2

V 2 $ = 0 for - o o < z < 0,

<9$ —— = 0 for z —• -oz

-oo.

X

(168a)

(168b)

(168c)

(168d)

Equation (167) can also be derived from the Zakharov equation (142) with ( given by Eqs. (143) and (144) in essentially the same way as done by Stiassnie (1984). Expansion of representation in Eq. (143) to leading order in the spectral width yields (cf. Trulsen et al, 2000) expression (144) with the amplitude a given by:

a(x,t)=( — ') f dKB{K,t)eiK-x . (169)

The narrow-band approximation of the kernel T0 i 2 3 yields,

T^l2,3(ko,ko,k0,k0) = ^ . (170)

Multiplying Eq. (142) by (2u0/g)1^2, taking its inverse Fourier transform and applying the narrow-band approximation, Eq. (170), for the nonlinear term, yields the following NLS-type equation with fully dispersive behaviour and narrow-band nonlinearity,

da ~dt ' 4TT2

1 f°° 1 — / dx'dKi[uj(ko + K,) - iv0]a(x',t) +-iio0kQ\a\2a = 0 . (171) ^ J-oo 2


6.5. Extensions of the Zakharov equation

By allowing for spatial variations, Rasmussen (1998) derives for the deep-water case a so called Zakharov-Nonlinear Schrodinger equation (his equation (3.50)) which reads,

8B_ ig_ (k2x-2k2

yd2B k2

y - 2k2x d

2B 6kxky d2B \ — +c9-^B+8ujk y k2 dx2 + k2 dy2 + k2 dxdyJ

= -i fff dkxdkzdk-sC1^ , (172)

where C% 2 is already defined in Eq. (137b). As Rasmussen noticed, the left-hand side is similar to the linear part of the nonlinear Schrodinger equation, apart from the mixed derivative term which can be transformed away by changing the x-direction into the main propagation direction. We notice that the extra terms (compared to the original Zakharov equation) are due to the partial variation of the wave amplitudes. Because the case of deep water is considered and no current is present, the modulation is solely due to the frequency modulation, introduced from outside the computational domain. In this respect, the situation is similar as in Stiassnie (1984) in which a NLS equation is derived from the Zakharov equation for deep water.

It can be shown (see Rasmussen, 1998) that for the case that the wave field consists of only one dominant wave component, B{k, t) = B(k', t)6(k — k') and also choosing k' = (fc, 0)T, the deep-water NLS equation results,

dB w dB ig (dB dB\ ik3 , „ , , „ , N

m+2-kd-x + s5k{^-2^) = - ^ \ ^ - (173)

Here is used the fact that:

A 0 , l ,2 ,3 - ^ . (174)

as can be found in Stiassnie (1984, p. 432). Neglecting the variation on the slow spatial scales, Eq. (172) reduces to:

— + cg • VB = -i fff dfeidfeadfcaC^a , (175)

which is called the inhomogeneous Zakharov equation by Rasmussen. By considering a Stokes wave with constant amplitude,

B{k,t) = BeiatS(k~k'), (176)


Rasmussen poses the question how much k' may deviate from its exact value

without changing the nature of the solution. Pu t in another form: what is the

appropriate value of the bandwidth for application to random waves. To this

end, the solution is multiplied by a small variation due to a small per turbat ion

wave number K,

B(k,t) = Be<at+KXh{k - k'). (177)

By substi tut ing this expression in Eqs. (172) and (175), Rasmussen finds

tha t the solution for the free-surface elevation C(a;i*) c a n D e wri t ten in both

cases as:

C(x, t) = a cos[K • x - (1 + (Ka)2)uj(K)t}, (178)

where K = k + n and Eqs. (172) and (175) are fulfilled when n/k ~ O(s) and

0{e2) respectively.

7. G e n e r a t i o n of Free Long Waves

In coastal areas, long waves with typical periods of minutes can be generated

due to several physical phenomena. One of the generation mechanisms is due

to the nonlinear effect on modulated wave trains. Generally two types of long

waves exist: (1) locked (forced) long waves and (2) free long waves which prop

agate with their own celerities according to the linear dispersion relation. In

coastal areas and for a narrow-banded wave group, the celerities become \fgh

where h{x) is the depth.

For horizontal bot toms, usually only the locked waves are generated. The

effect of an uneven bot tom is tha t free long waves are generated and,

moreover, tha t part of the locked wave energy transforms to free waves. Mei

and Benmoussa (1984) [see also Liu, 1989] have shown tha t the free long

waves could propagate in a direction different from the wave group and the

carrier waves. For shear-current regions, similar phenomenae occur, see Liu,

Dingemans and Kostense (1990). Other mechanisms of long wave generation

have been discussed in Holman and Bowen (1982).

The importance of knowing the locked and free long waves in coastal areas is

because they influence the sediment t ransport rates and especially the amount

of free long waves is important for harbour oscillation problems. For harbours

with berths for large vessels, often the effect of the long waves exceeds tha t of

the short wind waves not only due to possible resonance of the harbour itself

but also due to the mooring systems which do have resonance peaks at much

lower frequency closer to the free long waves. So, the knowledge of the amount


of free long waves at the harbour mouth is essential for a good harbour design. The first paper discussing the importance of free and locked waves for harbour design was the one of Bowers (1977). Bowers showed that free long waves were generated because of the difference in the strength of the bound waves across the junction between the two channels. Another cause of the generation of free waves with which we are concerned here is due to the refraction of wave groups, either due to an uneven bottom or due to shear currents, see e.g., Mei and Benmoussa (1984), Liu (1989), Liu et al. (1990), Liu et al. (1992) and Dingemans et al. (1991).

In this section, we primarily consider the second-order formulation for the wave amplitudes. In third-order, the NLS-equation formulation is obtained. For that case, we refer to Dingemans et al. (1991).

7 .1 . Formulation of the equations

We consider a train of modulated linear waves propagating over a slowly-varying bottom. In first instance, we also include an ambient current field in the considerations. Because the length scales of the resulting wave groups are much larger than those of the carrier waves, slow variables are introduced by:

X = j3x and T = j3t, / 3 « 1 . (179)

When an ambient current field U{X,T) is also considered, the first-order displacements can be written in the following form (see Liu et al., 1990),

C(x, t; X, T) = l-A(X, T) exp[iX(x, t)] + CC , (180a)

*(x,z,t;X,z,T) = - i g A £ T ) C O S ^ f c ; Z)] exp[tX(x,t)] + CC, (180b)

with

x(x,t)= dx'k(x')-cjQt, (180c)

where the carrier wave frequency u is determined by:

w = u)r + k • U and J^ = gk tanh kh, (180d)

and fc is given by: k = Vx{x,t). (180e)


In the same way as derived in Liu and Dingemans (1989) for the case without an ambient current, an equation for the complex amplitude with current becomes,

dT \ujr V x - (Cg + U)

L0r o, (181)

where V x = {d/dX1,d/dX2)T.

Writing ( for ((2>°) and </> for r^1-0), we have the following relation between C and 4>, see Kirby (1983),

<; = - l E ± w g DT 4g sinh2 kh '

where D/DT = d/dT + U • Vx and

dT x (U + hVx$+^k 2ojr

= 0.

(182a)

(182b)

As noted by Liu et al. (1990), taking the horizontal gradient of Eq. (182a) and eliminating ( from the resulting equation and Eq. (182b) yields the long-wave equation for the potential </>,

g|+(Vx.t/)§|-9Vx.(ftV„

£!vx 2 X

k^ gk d\A\2

2sinh2fc/i dT Vx gk\A\*

2sinh2fc/i U (183)

which was also derived by Kirby (1983) by using an averaged Lagrangian approach.

Alternatively, we can write the long-wave equation in terms of the second-order free-surface elevation ( by taking the total derivative of Eq. (182b) and substituting D<p/DT from Eq. (182a), yielding,

j^L - VX • (ghVxC) + ~{C^x • U)

'x • hV x JA2

4 sinlr kh DT X V 2to (184)

In the case of an ambient current and an uneven bottom, the governing equations are the amplitude equation (181) with either Eq. (183) for <j> or Eq. (184) for C-


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*£.. (185b) dt2 yy YJ 2 \ tor J 2sinh2A;h <9i v y

It is to be noted that both the amplitude and the long-wave equations are of second order and the validity of the formulations is restricted to spatial scale |X | = \/3x\ = 0(1) and similarly for the time. Notice that these longwave equations have already been encountered in the derivation of NLS-type equations, see, e.g., Eq. (82b). The main attention is directed in this section towards the forced long-wave equation. The forcing itself is described by the amplitude equation. When proceeding to third order, a NLS-type equation is obtained for the amplitude. Using this NLS-type equation instead of the second-order equation (181) means that the forcing in the right-hand side of the long-wave equation can be determined more accurately than is possible with the second-order amplitude equation. In the present section, the accuracy of the forcing itself is not so important but the fact that a forcing is present matters.

Notice that the coupling between the amplitude equation (181) and the long-wave equation (183) occurs in one way only. No coupling from long wave to amplitude equation exists. In the case of the NLS equation for restricted depth, two-way coupling exists. Therefore, we will only consider the case of a NLS equation with its companion long-wave equation.

That bound waves become free when the wave group is progressing into a region with variable depth seen as follows. Suppose we have a ID situation with shelf-like region consisting of a horizontal part with h = h0 for x < Xo, a variable part h = h(x) for XQ < x < Xi and again a horizontal part for x > 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


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 <f>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

2 L\d& ( c / -gh)— 1 g2k gkcg

2 u 2sinh2fc/i \A\\ (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 dA\ ldcQd2A , - , , , „l2 A „, „ , „„ ,

( ' * - + c ° i rx ) + 2-d^+A^- *u*MA=nA' (187a)

where the operators Q and H are given by:

o*=(^c-"I-*!)*• <187b)

and

dx\dk ^ dx V 2 dk 2

cosh4fc/i + 8 - 2tanh2 kh , to2 , , , , ,„„„,.. K= -. , Koo = — and cr = cothfc/i, (187d)

16sinh4fc/i 9 with fi and ^ coefficients depending amongst others on the bottom slope which are given in Eqs. (Bl) and (B2) of Liu and Dingemans (1989).

In order to find an initial condition on the horizontal-bottom part of the shelf-like geometry, Dingemans et al. (1991) substituted the locked wave solution <f>e in the expression for Q§ in Eq. (187a). This leads to the NLS equation

gNotice that, because gk/sinh2kh — u>2/4sinh2 kh, the long-wave equation as given in Dingemans et al., 1991, Eq. (9), is equal to the formulation (185b).


for horizontal bottom,

3A

with

vi

dA ' dx

bJfcooC-g

ldcgd2A 2

kg2

+ k 2" +A«n£kh +k2UK.

(188a)

(188b) Jig sinh^ kh ' ) cg

2 - gh

As shown in section 5, an initial condition for solution of the wave envelope

follows from the horizontal-bottom equation (188a) as:

A(x, t) = a sech / -V2

dcg/dk a {x — cgt) (189)

The corresponding bound long-wave is given by the differential equation (186). The solution of <fi and A is achieved by a simultaneously numerical solution

of Eqs. (187a) and (185b). The solution for </> is the combination of free and locked long waves. The omission of the reflection in the derivation of the evolution equation for the wave envelope (see Liu and Dingemans, 1989) leads to the consequence that there is no reflected component of the locked-wave. For the free wave, we have both a forward (</>t) and a backward (<^7) propagating component. We have,

4> = 4>t + 4>+ + 4>j . (190)

The solution of the long-wave equation is considered now. Because Dingemans et al. (1991) wanted a solution for 4> itself and not only for its derivatives, the following method was used. Divergence-like and curl-like differential operators T> and 1Z were introduced by:

Vu = T> dui du2

— + — and TZ(p)

( *£ \ dx

V dp

"at J

(191)

It can be proven that any twice differentiable vector field u with T>u = 0 has the property u = TZ{p) for some scalar function p determined uniquely up to a constant, in the same way as the usual stream function for incompressible flow is determined.11 The long-wave equation (185b) can also be written in the

The proof follows simply by substitution of 11 = TZ,{p) into T>U = 0.


form of T>u. With the special choice p = —ghip, with ip an auxiliary function, Eq. (185b) then is equivalent to the system,

2

f + ! < » " * > - y y ^ <"»•) <9?/> <9<?f> c/fco

dt dx 2tooh (192b)

The set (192) now replaces the long-wave equation (185b). It has to be stressed that the variable ip is only an auxiliarly variable, it has no physical significance and ip even does not fulfill the wave equation (185b).

The computations are performed in the interval 0 < x < L. At both the inflow and the outflow boundaries, we take the bottom to be horizontal in order to facilitate the application of radiating boundary conditions. For the envelope A, we give the value at x = 0 and at x = L, we give the weakly reflection condition (dt + cgdx)A = 0. For the boundary condition of the longwave potential </>, distinction has to be made between the three long-wave components: the locked-wave component §\ and the free waves 4>~t and <j>J. For a horizontal bottom, these components satisfy,

(dt + cgdx)4>+ = 0 , (dt + y/gkdx)$$ = 0 , (dt- yfghdx)4>~f = 0 . (193)

The inflow condition for <j> is found as the sum of these equations. Using also the solution of Eq. (186), we obtain as weakly reflective condition at the inflow boundary,

g2k 9 K / W \ z

^ - v V ^ = - ^ ^ " " " " " \A\* - 2^gh-^ . (194a) dt ox ^gh - cg ox

The inflow condition for the auxiliary function ip can be obtained from the system of Eq. (192) with </> being substituted into the boundary condition, Eq. (194a), resulting in:

gk / to \2

dA _ ^ ^ L = Cg^tl^nhfcfej_|A|2 _ M (194b)

at ox y/gh - cg ox

Similar conditions can be derived at the outflow side, see Dingemans et al. (1991). It has to be remarked that the weak point in the analysis of Dingemans


et al. (1991) in this respect is that for the locked wave, the assumption of a single wave group has been used. However, after progressing over an uneven bottom part, usually the wave group is split in two or more groups with their own velocity. At the inflow boundary, one can make sure that a single wave group is entering the computational domain. This restriction can also be rephrased in the following way: the coupling between the wave envelope and the long wave should be weak or the coupling term Q4>A should be weak.

From the computed results, the mean free-surface elevation £ follows from Eq. (182a).

The determination of the free and bound components can be performed only for a horizontal bottom. Because both d4>/dt and d(f>/dx are available from the numerical computation, we also have,

at _ ~df + ~W + ~~m and Tx - ~dx~ + ~d^ + ~dx~ ' (195j

The sets (193) and (195) yield five equations for six unknowns. Together with Eq. (186), the system is closed and can be solved. The splitting in free and bound wave components is thus achieved for a horizontal bottom only. This procedure is usually also used for uneven bottom with the understanding that the procedure then is not very accurate but with the hope of sufficient accuracy. However, this is still to be proven. The solution of these six equations leads to:

dx4% = dx$ - d - C\A\2, djj = Cu dx4>+ = C\A\2 , (196a)

with

C ,2-gh

and

92k l u) y 2to 9\2smhkh)

(196b)

Ci = ^[dt$ + cdj - (c - cg)C\A\2}, (196c)

and c = y/gh. The solutions for dtftt, dtfyl a n d dt4>\ follow immediately from Eqs. (193). An example is given in Dingemans et al. (1991).

8. Observations of Wave Modulations

8.1. Theoretical aspects of modulational instability

Basic ideas of Benjamin-Feir instability mechanism have been presented in the introductory section 2. The first mathematical treatment of instability of water


waves was presented by Benjamin and Feir (1967). In an alternative approach, instability of water waves can also be analysed on the basis of modulation equations. While instability analysis based on the classical NLS equation leads to identical results as derived by Benjamin and Feir, use of higher-order modulation equation leads to improved results. We skip the details of the stability analysis based on the NLS equation which may be found in several books (e.g., Dingemans, 1997, p. 9311) and present some results for one-dimensional wave propagation. In short, proceeding with the equation,

i^+\^-»M\2A = 0, (197)

where the coefficients Ai and v\ are defined according to the Davey-Stewartson equation (section 3), one may seek solution to the amplitude envelope A(£,T)

in the form,

A(Z,T)=a(Z,T)exp[iX{t,T)]; a = a0+b(Z,T). (198)

In the assumed form, ao is the amplitude of the uniform Stokes solution and b is the perturbation, the nature of which needs to be investigated. The main conclusions are that for a perturbation of the form &(£, r) = 6exp[i(/c£ — fit)],

(1) fl2 = X1K2{\1K

2 + 2u1al). (2) Instability occurs for fl2 < 0, i.e., X2K2 + 2\iVia2 < 0. For real solutions

of K, this condition is met only if A^i < 0, equivalently, v\ > 0 because Ai < 0 always. This is equivalent to the condition that kh > 1.363.

(3) With kh > 1.363, the waves are unstable to a perturbation n such that:

1 2 ^ vi 2 2« < - ^ o ,

in deep water (kh —>• oo), this condition translates to (n/k) < 2y/2kao. (4) Under the condition of instability, the growth rate Im(fi) is:

/ 1 \ 1 / 2

Im(fi) = K I |Aii/i \a20 - -\\K2 j . (199)

As may be seen from Eq. (199), the growth rate has a maximum and is therefore responsible for selective magnification of perturbation modes. For deep water

'An error has crept into the analysis presented in that section. The origin of the error is the coefficient of the term aTT in Eq. (8.227a) on p. 931 which should have been 1/2.


waves, perturbations K/k < 2\Z2ka0 are unstable and the maximum growth rate occurs for K/k = 2kao at which the growth rate Q,m is uj(kao)2/2.

Exact analysis of Longuet-Higgins (1978) for deep water reveals marked differences from the preceding results based on the NLS equation. This difference is reduced by using higher-order modulation equation. An illustrative example of this is provided by Dysthe (1979). From Dysthe's analyis, valid only for deep water, the perturbation modes for instability and the maximum growth rate are:

(200a)

Im(fi)|max = ^(ka0)2[l - 2(ka0)]. (200b)

The normalised maximum growth rates (Im(fi)|max/o;) as predicted by the NLS and Dysthe's theory are shown for deep water case in Fig. 5.

The presence of a varying current makes the medium inhomogeneous for waves. Equivalently, this is reflected in the coefficients of the Schrodinger equation having both temporal and spatial variation. While the essence of the stability analysis remains the same as before, a general analysis of the side-band instability due to the inhomogeneity introduced by a current becomes difficult.

0.09

0.06

0.03

o 0 0.2 0.4

ka,,

Fig. 5. The normalised maximum growth rate ( Im(n) |m ax/w) as a function of steepness kao from NLS and Dysthe's theory.

^ < 2V2ka0\/l K

k


Here, we discuss some results, due to Gerber (1987), which are valid only for deep-water. For waves with a constant dominant wave vector k = (k, 0)T, the current-modified Schrodinger equation used in the analysis is:

dA . 8 A ^7 + (c9 + U) dt

1 dcn

dx ' \2 dx

1 UJr 82A 1 1 LUr 8

2A 8 k2 dx2 4 k2 dy2 2

- — 4 dx

u>rk2\A\2A = 0, (201)

where U is a slowly-varying current and wr is the linear relative frequency equal to \fgk. To discuss the effect of an ambient current collinear to the wave direction, we consider the one-dimensional version of Eq. (201) in which case, the derivatives with respect to y are omitted.

Unlike in case of waves alone, the Stokes solution in the presence of a varying current is no longer of uniform amplitude. The solution to Stokes amplitude ao(x) is given by:

ao (x) = a,Q exp Jxo y

A 3 dU 4 dx

u (202)

where oo is the amplitude at x = XQ. The relationship between Q, and the perturbation mode K is found to be,

n2 UJ2rK

2

8k2 k alexp

Thus, instability occurs for,

0 < - < 2V2ka0 exp k

(203)

(204)

Equation (204) is equivalent to the case of waves only if the current gradient is zero. Secondly, utilising Eq. (202), the stability condition

0 < ~ < 2V2ka0 k

(205)

becomes identical to the Benjamin-Feir condition for waves in terms of the local steepness ka,Q. Thus, under the assumption that the group length scale is much smaller than the scale of current variation, the stability criterion remains


unchanged in terms of local wave steepness. The local steepness is of course modified by the variation of the current from its original value.

For the specific case of initial current being zero (U = 0), explicit use of the deep water relationship in Eq. (204) leads to:

0 < ^ < 64y/2k aQ

[i + 4f]i(i + [i + 4f]i)5 (206)

where all the variables with the superscript "tilde" denote their initial values. It is evident from Eq. (206) that an opposing current (U/c < 0) increases the region of instability whereas a following current has the reverse action. It follows further from Eq. (203) that the maximum growth rate occurs for,

K

k 2(kdo) exp

Jx0

dx 1& 3 dU 4 dx

+ u (207)

at which the growth rate 17 is given by:

Im(fi) = -(kaQ) exp dx + M

• u (208)

8.2. Laboratory observations

Following the observations of instability of deep water waves by Feir as reported in Benjamin (1967), several experimental investigations of the modulational behaviour have been undertaken to date. Almost all of the reported experiments over this pertain to deep water waves (see Table 1 in Tulin and Waseda, 1999) except perhaps a singular case (Shemer et al, 1998). More experiments over finite depth may therefore be suggested to cover a broader range.

Limited length of the wave flumes and energy damping due to side-wall boundary layers are two factors which make interpretations of experimental results difficult. The other factor is the background noise or specific seeding of the input signal in the experiments. Experiments without seeding show that background noise is enough to lead to modulational instability. One of the sources of the background noise in the experiments is the transient front (Melville, 1982; Tulin and Waseda, 1999). Seeded experiments on the other hand make it easier to study growth in a controlled way with systematic variation of the amplitudes and frequencies of the disturbance modes. Modulation


and consequent increase in local steepness may lead to breaking depending on the initial wave steepness and disturbance parameters. We describe first the initial stage of modulation and demodulation for nonbreaking waves followed by modulation leading to breaking.

8.3. Deep-water modulation: initial stage and demodulation

A pioneering set of experiments in this regard is due to Yuen and Lake (1975) and Lake et al. (1977). Their experiments were carried out in a 0.9 x 0.9 x 12 m tank. By using modulated input signals of prescribed side-bands, they were able to record measurable growth of the side-band amplitude along the length of the tank. For the purpose of laboratory experiments, spatial growth rate is more relevant than the temporal growth rate for which theoretical expressions have been presented in the preceding section. The equivalent spatial growth rate fiw of the side-band amplitude based on the linear stability analysis of Benjamin and Feir (1967) is given by:

n(x) = A[2{ka0)2 - A2]H , 5=—. (209)

to

Their measurements supported both the theoretical prediction (209) for the growth rate fl^ = (d(logS)/dx) (5 being the side-band amplitude) and the earlier observations of Benjamin (1967). They observed, however, that the wave steepness measured near (but further away for evanescent modes) the wave-maker needs to be corrected before being used as the initial steepness of the carrier waves in the theoretical expression. It was construed (Lake and Yuen, 1977) that this correction was necessary to account for the apparent nonlinear-ity of waves, generated by a sinusoidally moving wavemaker, being smaller than the theoretical Stokes waves (with its bound superharmonics). From analysis of data of the side-band width A of the most unstable mode and the measured steepness, this correction was specified to be (fca)o = 0.78(fca)measured-Later work (Longuet-Higgins, 1978) has shown that the analysis of Benjamin and Feir shows significant deviation from the exact values (potential flow) for (fca)o > 0.1. Thus, while the difference between Stokes waves and that generated by a sinusoidal wavemaker is a fact, the mismatch between the measured growth rate and that from Eq. (209) is also likely due to the small steepness limitation of the latter.

The exposition in Lake et al. (1977) went beyond the initial growth of instability. They gave evidence of demodulation, i.e., the waves tend to return


to their initial state existing before modulation. This observation that modu-lational instability does not necessarily lead to a disintegration of water waves renewed support to the recurrence of modulation exhibited by the NLS equation. However, there remained many questions hidden behind this evidence of near-recurrence and their data did not cover sufficient ground to clarify them. Many of the issues involved have been re-examined in the recent experiment by Tulin and Waseda (1999).

One of the cases investigated by Tulin and Waseda (1999) in a wave tank of 4.2 x 2.1 x 50 m, corresponds to an initial steepness of 0.1 and a "seeded" modulation of 5LU/U> — 0.0894. No breaking was observed and the length of the tank allowed the completion of one modulation cycle. This case is perhaps the best evidence of recurrence among reported laboratory studies. At the peak of the modulation near 25 m from the wavemaker, the amplitude at the carrier frequency had nearly vanished. Beyond this peak modulation, however, the waves evolved back nearly to the original three modes (carrier, upper and lower side-bands) before the end of the tank. This is a strong corroboration of the near recurrence referred to by Lake et al. (1977). As described later, for values of steepness and modulation band within a certain range, waves break during modulation. The long-term behaviour in those cases is markedly different than the near-recurrence of the wave modulation.

In the aforementioned recurrence of nonbreaking modulation, we have used the qualifier "near" indicating that the wave system does not exactly reset back to the initial three waves at the end of the modulation. One key difference is that the evolved spectrum shows a noticeable (on a log-linear scale) discretised high-frequency energy spread. Although the energy content of the spread is low, this feature may be responsible for the deviation from a perfect recurrence. Secondly, while Lake et al. (1977) found that the carrier waves experience a down-shift of the frequency, this down-shift was absent in Tulin and Waseda (1999) in the absence of breaking. This feature, if true, is another deviation from a perfect recurrence. In spite of these deviations being small, they are important in the sense that the tank length has limited such studies to a maximum of one modulational scale. Multi-cycle behaviour of these deviations for nonbreaking waves remains an interesting study.

In summary, it is important to recognise that side-band instability does not lead to breaking for waves of all steepness and all modulations. A rough guide to determine the region of breaking on the parameter space of e = (fca)o and 6u)/u> is given by a diagram presented by Tulin and Waseda (1999) (their Fig. 17, p. 220). This diagram is based on experiments and fully nonlinear


computations over a range of e and Su>/ui. Roughly, waves of initial steepness (ka)o smaller than 0.1 seem to be stable to undergo a modulation cycle without breaking. For larger steepness, the region of breaking is centered around the locus of maximum growth rate; i.e., Au/w « \/kao-

8.4. Deep-water modulation: modulation leading to breaking

For waves of initial steepness greater than 0.1 and modulational disturbances within a certain band, breaking occurs during evolution. For a systematic exposure on the observation of modulational behaviour during and after breaking, we turn mostly to the studies due to Melville (1982, 1983) and Tulin and Waseda (1999). Important contributions to this field have also been put forward by Ramamonjiarisoa and Mollo-Christensen (1979) and Su and Green (1985).

Increase in local steepness during evolution leads to breaking. In the pre-breaking stage, the measured growth rate agrees well with the theory. The local steepness, expressed byj kHm/2, seems to lie in a range from 0.25 to 0.4. This value is lower than the theoretical value of 0.44 for maximum steepness of stable waves. One immediate consequence of wave-breaking is the loss of near-recurrence described in the previous section. Secondly, the energy distribution over the wave components during and after breaking is drastically modified. Aside from the energy loss due to breaking, a significant difference between modulation with breaking and without lies in the relative growth of the sidebands. This point is best illustrated in Fig. 6 (reproduction of Fig. 18 of Tulin and Waseda, 1999).

The amplitude of the lower side-band remains high after breaking while for cases of no-breaking the amplitude subsides significantly almost to its original state. In contrast, the upper side-band drops back from its peak towards its pre-growth value. Although the carrier wave recovers somewhat it stays lower than its initial value. Thus, two processes seem to be taking place: discriminatory energy loss from the carrier and upper side-band modes, and an irreversible energy transfer to the lower side-band. The latter conclusion is more difficult to justify due to the length limitation of the wave tanks in the two experiments (Melville, 1982; Tulin and Waseda, 1999). The end result is that the lower sideband remains most energetic after peak modulation and breaking resulting in a permanent down-shift of the spectral peak within the length of the tank.

JI t is not perfectly clear whether k is the wave number of the carrier wave or the one of the modulated wave. The issue seems not to be very important since a range is denned.


1.0

0.8

a

0.4

0.2

(a)

— ^

- \

\> 1 .

V ' / V -Breaking

1 1 20 40 60

kxlln 20 40

kxlln 20 40

for/271

Fig. 6. Spatial modulation of carrier, sub and super side-band modes with and without breaking. Reproduced from Tulin and Waseda (1999) (Fig. 18, p. 221).

8.5 . Spectral evolution

In the two preceding sections, evolution of only the side-bands and carrier waves

have been discussed. Although an important par t of the evolution process, be

haviour of the side-band modes alone does not give the complete picture of

the spectral transformation. Occurrence of discretised high-frequency spread

of energy (seen clearly in log-linear plots) is another feature of importance

with respect to the long-term evolution of the wave field. Melville (1982) was

the first to draw at tention to this aspect from his experimental results. He

noted tha t aside from the spikes at the side-band frequencies, spikes appear at

other frequencies. These spikes are first prominent at the subharmonic interac

tion frequency of Ao; and super harmonic modes na>o with n as an integer. W i t h

evolution downstream from the wavemaker, prominent spikes become more nu

merous in the discrete spectrum appearing for example at frequencies WQ+2ALJ,

n(u>o ± Aw). At locations near and after breaking, there seems to be a t rend

towards a broad-band continuous spectrum around the primary frequency UJQ.

The initial steepness of waves used in Melville's experiments was 0.232 and

0.292 respectively. Tulin and Waseda (1999) observed tha t the generation of

a continuous spectrum depended not only on the steepness but also on the

bandwidth of the disturbance. From the known experimental cases, it does not

seem feasible to conclusively specify the conditions which lead to a continuous

spectrum. The results only show tha t under certain conditions, wave-breaking


during modulation is also associated with a scattering of energy from discrete spikes to a continuous spectrum apart from the energy dissipation.

8.6. Comparison between theory and experiment

Numerical investigations have been presented by several authors based on several forms of nonlinear equations. However, comparisons between experimental measurements and numerical solutions are somewhat less exhaustive and should be extended. The cubic NLS equation is the basic theoretical model for studying nonlinear modulation. Side-band instability which is a special class of nonlinear modulation is admitted by this theory. Both the unstable modes and the initial growth rate may also be extracted from this theory. More importantly, the theory predicts recurrence of modulation, a phenomenon which is clearly beyond the linear instability analysis. As mentioned in the previous section, for waves of initial steepness less than 0.1, laboratory observations show near recurrence. However, in spite of this success of the NLS equation in reproducing the broad features of the growth of side-band modes, peak modulation and subsequent demodulation and several other features remain unanswered attracting renewed attention for more advanced analysis.

In an inter-comparison study, Landrini et al. (1998) have compared the evolution behaviour predicted by NLS, mNLS, Zakharov's equation, Krasitskii's equation and the fully nonlinear potential flow computations. In consistence with the theoretical expectation, the computed results from an approximate equation system became closer to that from the fully nonlinear potential flow computation with increasing order of approximation. Fully nonlinear potential flow computations have been found to reproduce the experimental measurement of wave modulation up to the point of breaking (Tulin and Waseda, 1999). From whatever limited number of cases, fully nonlinear computations have been carried out seem to predict the breaking at the right location. Among the approximate nonlinear equations, NLS which is the simplest equation capable of modelling modulation does not reproduce any asymmetry. Comparisons by Shemer et al. (2000) between numerical computations and experimental observation show that asymmetry during the evolution of a group may be satisfactorily produced by mNLS and Zakharov's equation, the latter doing better. In an subsequent work, Shemer et al. (2001) have shown in a more elaborate way that the Zakharov equation could be used for an accurate description of the modulation and the skewed shape of the amplitude envelope of a wave field (non-breaking) during its evolution along a tank.


A major source of difficulty is the role of dissipation. Dissipation arises from the wall layer of the experimental flume and due to naturally occuring processes of surface wave-breaking. Many of laboratory and field observations are inseparably influenced by such dissipative mechanisms. Analysis of experimental measurements (Melville, 1982, 1983; Tulin and Waseda, 1999) indicate that energy transfer between wave components in the incipient stage of (and during) breaking happens rather rapidly in a complex way. Further understanding of this process is necessary to be able to enter the next stage of modelling in which post-breaking modulation may be calculated.

9. Summary

Wave transformation in the field involves wide-ranging processes including interactions among components and with varying depth and current. Modulation equations present a simpler framework with which both analytical and numerical insight and quantification of the processes may be achieved. An essential assumption in the derivation of the modulation equations is that length scale of variation of the amplitude envelope is much longer than that of the carrier waves. Thus, validity of such equations become limited when the wave properties (amplitude envelope, wave number) exhibit fast variation. With an ambient current, the current is assumed to be of large-scale compared to the waves. We also like to draw attention to the fact that ID coherent wave groups (soliton-type solutions) are likely to become unstable in two dimensions. This is due to the instability of groups to oblique perturbations (e.g., Ablowitz and Segur, 1979). Because of this, coherent wave groups are more readily observed in wave flumes than in the field.

This review begins with some illustrative sections to guide uninitiated readers to the nonlinear processes in deep and intermediate water. Modulation equations involving cubic interactions have been presented for deep water followed by the more general cases of varying depth with or without an ambient current. Many theoretical descriptions deal with the classical case of deep water modulation. In engineering practices, the role of a finite and varying depth is important. The simplest form of equations on finite depth is the set of the so-called Davey-Stewartson equations [Eqs. (48)]. The cubic evolution equation on finite depth differs from the deep-water version in two ways. First, the coefficients of the equation depend on the depth. More importantly, the depth-averaged wave-induced current (long-wave) manifests itself in the amplitude modulation equation resulting in a coupling between the two. In 2D


propagation on water of finite depth, the evolution equation for the complex amplitude and the wave equation for the long-wave potential need to be solved simultaneously. In ID, this is simplified since the long-wave potential can be obtained after the evaluation of the amplitude [Eqs. (60)].

Conservation laws and the special solutions of the NLS equation may be used to serve as good guidelines for the validation of the numerical models. With waves propagating over a varying depth, an associated process is the generation of free long waves. Long waves are of importance in regards to harbour oscillation and coastal sediment transport. Nonlinear modulation is not an essential mechanism behind the magnitude and generation of free long waves but adds to the features. Generation of long waves, driven by the wind waves, is discussed in some length in the section "generation of free long waves". Effect of surface tension, although touched briefly in this review, is considered outside the present scope.

The motivations behind modulation equations of higher-order are more than only the natural theoretical extension. Both quantitative and qualitative changes are introduced through higher-order modulation equations. These changes agree with experimental measurements in a superior way than the cubic Schrodinger equations. For example, the side-band growth rate predicted by NLS equation is in satisfactory agreement with experimental measurement and exact computations only over a low range of initial steepness (ka)o < 0.1. This range is extended significantly by using the higher-order basis of Dysthe's equation [Eqs. (115)]. Further, asymmetric growth rates of the upper and lower side bands could be at least qualitatively reproduced by this set. This observed feature of asymmetric growth rates is beyond the scope of the NLS equation. Dysthe's equation (1979) is the first among the widely-known higher-order extensions of the NLS-type equation. Both Dysthe's equation and the current-modified form given by Stocker and Peregrine (1999) are formulated for deep water only. Derived from the Hamiltonian principle, the Zakharov equation provides a broader basis for higher-order modulation equation. The equation proposed by Dysthe (1979) has been shown to be a special case of the Zakharov's equation under the assumption of narrowness of the spectrum. Use of the Zakharov equation or a reduced form of it (where the reduction is based on the assumption of narrow-band approximation only in regards to nonlinearity) is recommended while considering a broad-band spectrum [Trulsen et al., 2000 and Eqs. (168)]. Inclusion of an ambient current, allowing perhaps a depth-varying profile, is a desired future development for the higher-order formulations.


Experimental measurements have on one hand lent justification to the basis

of nonlinear modulat ion equation. On the other hand, they have identified

several gaps which motivate further developments. One area where theoretical

developments are lacking is the modelling of dissipation. Apart from a few iso

lated a t tempts (e.g., Lake et al, 1977; Trulsen and Dysthe, 1990), formulations

of dissipation due to breaking and surface turbulence are largely absent in cur

rently used modulation equations. This is in spite of the fact tha t dissipation is

necessary in being able to model multi-cycle modulational evolution in many

practical computations. It is also a generally held notion tha t dissipation is an

essential mechanism behind frequency down-shift observed in uni-directional

propagation. Notwithstanding the possibility tha t nonlinear interaction alone

may cause a permanent down-shift in three-dimensional wave trains (Trulsen

and Dysthe, 1997), breaking related dissipation remains an important aspect in

future development of understanding the behaviour of evolution of steep waves

of practical interest. Experiments by Melville (1982) and Tulin and Waseda

(1999) indicate tha t the down-shift may be a rather sudden process associated

with breaking instead of being gradual. Clearly, there is a strong need for a

bet ter understanding of such process both from experimental and theoretical

investigations.

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Rasmussen, J. H. (1998). Deterministic and Stochastic Modelling of Surface Gravity Waves in Finite Depth. Ph.D. Thesis, ISVA, Denmark, Series Paper 68, 245pp.

Segur, H. and M. J. Ablowitz (1976). Asymptotic solutions and conservation laws for the nonlinear Schrodinger equation, I. J. Math. Phys. 17(5): 710-713.

Shemer, L., H. Jiao, E. Kit and Y. Agnon (2001). Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation. J. Fluid Mech. 427: 107-129.

Shemer, L., E. Kit, H. Jiao and O. Eitan (1998). Experiments on nonlinear wave groups in intermediate water depth. J. Waterway, Port, Coastal, and Ocean Eng. 124(6): 320-327.

Shemer, L. and M. Stiassnie (1991). The Zakharov equation and modified Zakharov equations and their applications. In: Nonlinear Topics in Ocean Physics, ed. A. R. Osborne, Proc. Int. School of Physics "Enrico Fermi", North-Holland, Amsterdam, pp. 581-620.

Shemer, L., H. Jiao and E. Kit (2000). Nonlinear wave group evolution in deep and intermediate-depth water: experiments and numerical simulations, Proc. 15th Int. Workshop on Water Waves and Floating Bodies, eds. T. Miloh and G. Zilman, pp. 166-169.

Stiassnie, M. (1984). Note on the modified nonlinear Schrodinger equation for deep water waves. Wave Motion 6(4): 431-433.

Stiassnie, M. and L. Shemer (1984). On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143: 47-67.

Stocker, J. R. and D. H. Peregrine (1999). The current-modified nonlinear Schrodinger equation. J. Fluid Mech. 399: 335-353.

Stuart, J. T. and R. C. DiPrima (1978). The Eckhaus and Benjamin-Feir resonance mechanisms. Proc. Roy. Soc. London A 362: 27-41.

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Trulsen, K. and K. B. Dysthe (1990). Frequency down-shift through self modulation and breaking. Water wave Kinematics, eds. A. T0rum and O. T. Gudmestad, pp. 561-572.

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Trulsen, K. and K. B. Dysthe (1997). Frequency down-shift in three-dimensional wave trains in a deep basin. J. Fluid Mech. 352: 359-373.

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BUBBLE MEASUREMENT TECHNIQUES A N D BUBBLE DYNAMICS IN COASTAL SHALLOW WATER

MING-YANG SU and JOEL C. WESSON

This review consists of two parts. The first part deals with the state of the art for various sensors for measuring physical parameters of the bubble field generated by wave breaking, and the corresponding deployment methods for some of these bubble sensors in the littoral zone. These bubble sensors make use of different kinds of optical, acoustical, and electromagnetic principles to effect their intended purposes. The second part covers the description of dynamical and statistical features of wave breaking and bubble fields in the oceanic shallow-water environment particularly near the surf zone where the special characteristics of shallow water depth on breaking waves and bubble generation are most prominent. The main physical parameters considered herein are the temporal and spatial variation of air void fraction and bubble size density.

1. Introduction

Surface wave breaking occurs in deep water and shallow water zones of lakes, seas and oceans. In deep water, surface gravity waves in general break only when their slope reaches beyond certain values under strong wind stress. On the other hand, wave breaking occurs in shallow water near shore, especially near the beach, even under no-wind situations, when the slopes of incoming swells from remote areas increase to high values due to water depth decreasing toward the shore (surf zone). Thus, one sees more wave breaking events in shallow water than in deep water. In this review, we shall restrict our attention to the shallow water zones.

Bubbles of different sizes are generated by wave breaking. The bubble generation and subsequent evolution are substantially different in saltwater (seas and oceans) than in freshwater (rivers and lakes). These differences will be explained more fully in the next section. The important role of bubbles in heat and gas exchange (primarily CO2 and O2) in the air-water interface of freshwater rivers and lakes are well recognized, but, these are out of

77

78 M.-Y. Su & J. C. Wesson

our review areas. Thus, this review shall further be restricted to saltwater coastal zones.

Breaking waves play a very important role in nearshore dynamics especially in the surf zone and its immediate neighboring area. Their principal effects are loss of energy and momentum from breaking waves to the water column that create rip currents, longshore currents, turbulence, and sediment transport. Another important effect of breaking waves is air entrainment into the water in the form of bubbles ranging in radius from about 10 fi to several centimeters. These bubbles affect the heat and mass transfer across the sea surface, and may cause strong effects on underwater acoustics and optical scattering and propagation (Inman et al., 1971; Svendsen and Putrevu, 1994; Melville, 1996). As bubbles escape from the surface and burst into smaller droplets into the air above, small salt particles are formed. These salt particles are the main components of near surface aerosols that may absorb strongly in the infrared light spectrum (de Leeuw and Kunz, 1992).

Unlike moderate slope surface waves, whose dynamics and statistics have been studied for the past one hundred years, wave breaking is an extremely nonlinear process (Longuet-Higgins, 1988; Peregrine, 1983, 1991; Mei and Liu, 1993), and the bubbles generated by breaking have been investigated even less (Thorpe, 1982, 1992). This is not due to the fact that wave breaking is a less important aspect of wave dynamics in general (Melville, 1996), but, rather due to the fact that wave breaking is so nonlinear, transient, and complicated in its dynamics and difficult in its measurement (both in the laboratory and in the field), that many wave investigators try to stay away from the study of breaking waves, if they can. Current practical necessity for both civil and military needs are such that breaking waves and their effects shall become one of the most important subjects in nonlinear wave dynamics investigations in the next decades.

The main focus of this review is twofold: (1) Bubble measurement techniques and (2) bubble dynamics/statistics in shallow saltwater coastal zones. The review is divided into the following six areas for convenience sake:

(1) Primary mechanisms of shallow water wave breaking and bubble generation in Sec. 2.

(2) Various types of bubble sensors for measuring physical parameters and their characteristics and suitability in the shallow water environment in Sec. 3.

Bubble Measurement Techniques and Bubble Dynamics 79

(3) Various types of sensor deployment methods in shallow water conditions in Sec. 4.

(4) Descriptions of several field experiments conducted over the past 15 years, in which bubble measurements occupied an integral part. Analysis of data from these experiments allow better understanding of the wave generated bubble field in Sec. 5.

(5) Large air cavities and large bubbles are formed immediately after the breaking wave crest enters the water surface. The percentage of air per unit water volume is commonly called void fraction (VF). Large temporal and spatial variations of void fraction are direct consequences of the passing and evolution of each breaking wave, and over a wider area under the tidal effect in shallow water. These variations will be reviewed in Sec. 6.

(6) For many important applications, not only the void fraction, but also how the air content is distributed among the various sizes of bubbles, is the deciding factor. Thus, the statistics of bubble size density (in terms of number of bubbles of particular radii within a unit volume of water) is a significant subject of this review in Sec. 7.

After the reviews of the above six aspects, some general remarks on bubble dynamics in shallow water are in order (Sec. 8), in which some major shortcomings in the current state of the art in each aspect shall also be pointed out, in the hope that more research will be addressed to these issues in the future. In a nutshell, the investigations of breaking wave and bubble dynamics in shallow water are still in the early stages of their full development in all aspects: laboratory experiments, field observations, and theoretical/numerical modeling.

2. Primary Mechanisms of Wave Breaking and Bubble Generation

In order to discuss the sensors and their deployment techniques for bubble measurements, we first need to know some general characteristics of the bubbles; why and how surface waves should break, and how bubbles are generated by wave breaking. We shall not be able to deal with all these complicated issues, but only describe briefly their primary dynamical mechanisms to aid our descriptions in the next several sections (Banner and Peregrine, 1993; Longuet-Higgins, 1994; Longuet-Higgins and Cokelet, 1976, 1978). We shall start with


(a) wave breaking mechanisms and then describe (b) bubble generation mechanisms, as follows.

(a) Breaking wave mechanisms

Both deep water and shallow water surface gravity waves break before reaching their highest theoretical possible form, i.e. the so-called Stokes limit (Lamb, 1945). Roughly, the effective slope S of the Stokes limiting wave o m is defined as

2 u i Sm = H/(l/2L) = — ^ - , (1)

where H is the wave height and L is the wavelength. This is due to the fact that the wave becomes unstable when its S reaches about 25% to 50% of Sm. These intrinsic instabilities of surface waves push the initial high slope waves even higher until the point of their breaking by highly nonlinear wave-wave interaction mechanisms. Major growth of surface waves from shorter wavelengths to longer wavelengths is also due to their instability, prompted by other nonlinear wave-wave interaction mechanisms. Moreover, these two types of nonlinear wave instability are fundamentally different in nature (so-called: 3-wave, 4-wave, and 5-wave interactions and their coupling effects) (McLean, 1982a; Su, 1982; Su et al, 1982a; Su and Green, 1984; Lin and Su, 2000; and others).

When the mean water depth (D) on which waves travel becomes a small fraction of the wavelength, another physical parameter, the depth ratio D/L enters into the picture of wave instabilities (McLean, 1992b; Su et al., 1982b). Even with a low initial D/L, a wave will grow quickly in height as D/L becomes smaller as a gently sloped beach is approached.

Many swells break before they actually reach the beach itself, resulting from the submerged sandbar in front of the beach head. The commonly called "surf zone" is the nearshore area where most wave breaking occurs. The location of the surf zone is critically affected by the water depth (D), which is in turn controlled by the local semi-diurnal tidal variations. There are various types of beaches: those with no bar, a single bar, or multiple bars. When two or three submerged bars exist, then, one sees two or three parallel surf zones.

Finally, most waves break again on the beach even though they have lost much of their energy from earlier breaking over the submerged bar. Beaches are often curved in shape, creating small points and bays along the shoreline. This curved bathymetry acts to focus or de-focus the waves along the


beach to further affect the location of strongest wave breaking (Battjes, 1988; Peregrine, 1983).

(b) Bubble generation and evolution mechanisms

The immediate outcomes of wave breaking are (i) loss of wave energy/momentum to the water below, (ii) entrainment of some air into the water below in the form of bubbles with various radii, and (iii) generation of sprays from the breakup of the sharp crest and/or the impact of the crest on the surface. The generation of bubbles can be observed in the process of boiling water, opening a pressurized beer can, or as a swimmer jumps into a pool. As a matter of fact, they are observed everywhere and quite often. However, the detailed dynamical mechanisms of bubble generation are not fully understood. As for our current interest in bubble generation by wave breaking (plunging, creeping, and/or swash type), the significant parameters controlling their generation are the air/water density ratio, the air/water temperature difference, gravity, surface tension of the water, presence of surfactants, and of course, the shape and velocity of the crest portion of the breaking wave itself. Once the crest portion curves inward and enters into the water surface, some air is enclosed to form an air cavity which quickly breaks up as soon as the wave travels past the breaking site. Another part of the air is entrapped into smaller air pockets which in turn break up into even smaller bubbles of various shapes. Only small bubbles (with radii <C 3 mm) are more or less spherical in shape while larger ones can have more irregular shapes. Because of the combined effect of surface tension and the dissolution of air into the surrounding water, air bubbles with radii smaller than a few fi are few in seawater. For most practical applications, we may set the smallest bubble radius at 1 \x and the largest bubble radius at 5 mm. Thus, there is almost a four decade range in bubble radius, from 10~6

to 5 x 1 0 - 3 m. As a comparison, we note that the ocean surface wavelengths range from about 10~2 m (capillary waves) to about 400 m (for long gravity waves with period of about 15 seconds) which is about four decades too. The whole bubble-wave spectrum covers about an eight-decade range.

Once bubbles are generated, their dynamics are more complicated than wave dynamics in that bubble motions are driven by two separate forces: the upward gravitational force due to buoyancy and the surrounding turbulent water movement. For very small bubbles (with radii < 100 fi), they move like solid particles and are carried passively by the water motion, without


significant effects of upward buoyancy. For large bubbles with radii > 5 mm, the buoyancy force is dominant. For bubbles with their radii in between (100 x 10 - 6 m to 5 x 10~3 m), they are pulled back and forth, up and down by these two separate external forces besides the action of surface tension in the air-water interface. In short, strong turbulence of the velocity field of the surf zone is another crucial factor in the formulation of any dynamical model for investigating the bubble field in and near the surf zone (Lin and Liu, 1998).

Two endgames of bubbles are (i) moving up to the water surface and bursting into a series of smaller droplets and (ii) dissolution by the surrounding water. Most larger bubbles follow the first route while most smaller bubbles follow the second route. Between the generation and the endgames, some bubbles may collide with others and coalesce into somewhat larger bubbles. One factor that influences most the coalescing effect is the salinity (salt content) in the water. For freshwater (with zero salt content), the coalescing effect is energetically favorable and thus very common while for saltwater such as the average seawater with a salt content of 34°/oo, the coalescing effect is less energetically favored and occurs less often. In fact, in saltwater with salinity 20°/ooand above, the coalescing effect is already similar to that of average sea-water (Detsch, 1990; Scott, 1975). This effect significantly affects the average lifetime of bubbles in a container after they are generated by the exact same mechanisms say for example by an air pump (like those used in an aquarium). Under the same bubble generation process, we see more smaller bubbles present in the saltwater tank than in the freshwater tank while there are fewer large bubbles in the saltwater tank than in the freshwater tank (Scott, 1975). A controlled experiment in the large scale wave tank (12 x 15 x 450 ft) at Oregon State University (OSU) was conducted in 1989 using the computer-controlled wave paddle as the breaking wave (and bubble) generator. The main finding of the experiment is that the bubble size density (for 30 < r < 800 fi) is up to 10 times higher in the saltwater (20%o) than in the freshwater (Cartmill and Su, 1993). The salinity effect on bubble generation, evolution, and lifetime may need to be taken into consideration in coastal areas where significant amounts of fresh water input from large rivers are present. The exact physical-chemical mechanism for this effect of salinity on bubble coalescence is still unknown.

Two additional complicating factors are particularly important for the area close to the sand beach. The first is solid particles (sand or other material) suspended in the water and the second is the surfactants produced by biological organisms and/or oil discharge carried from far away. Both of these factors may interact with the surface of bubbles. Small bubbles may attach to the


surface of these particles and vice versa. Surfactants may produce a coating on the bubble surfaces to retard the gas dissolution process, thus prolonging the bubble lifetime. Significant coating which increases the possibility of small particle attachment will affect the optical and acoustical response of bubbles to external excitations. Finally, the relative temperature between air and seawater also has some effects on both generation and evolution of bubbles.

3. The Bubble Field Description and Bubble Sensors

We shall first consider what kinds of physical information one often needs from bubbles before we can sensibly discuss the types of bubble sensors available today and their advantages, limitations, and suitability for shallow water coastal environments.

3.1. Physical parameters of the bubble field

According to the increasing order of information content, the physical parameters for describing bubbles may be divided into four categories:

(a) Total air volume in a unit volume of water. This is the so-called void fraction expressed in percentage.

(b) Bubble size density in a unit volume of water which gives the statistical distribution of bubble sizes.

(c) Position vector and radius of each individual bubble in a unit volume of water.

(d) Position vector, velocity vector, and radius of each individual bubble in a unit volume of water.

Category (a) obviously contains the least information while category (d) contains the most detailed information to describe the bubble field. In fact, if the last two categories [(c) and (d)] of information are available, then, the first two categories [(a) and (b)] can be determined from them by integration and averaging. It goes without saying that the measurement of category (d) will be much more difficult than those of the first three [(a), (b), and (c)].

One needs to further consider under what situations these physical parameters of the bubble field are to be measured: either in laboratory controlled conditions or in the field where severe weather and sea states may be encountered. Obviously, laboratory conditions are much easier to deal with than those in the nearshore under heavy wind and wave pounding.


3.2. Physical principles for bubble measurements

There are three main kinds of physical principles used for measuring these physical properties (parameters) of the bubble field:

(a) Optical principles. (b) Acoustical principles. (c) Electromagnetic principles.

Some general features of these principles will be explained further below.

3.2.1. Optical principles

Let r denote the radius of a bubble. In clear water, the air bubble has a smooth interface with the surrounding water. This interface is a very good reflector. Since the spherical interface has a different degree of reflectance toward a fixed parallel light beam, some portion of light will be reflected while other parts will be transmitted from the water into the gas (air) within the bubble and finally come through the air-water interface again after several internal reflections. The complete and complicated details of the reflectance and transmission of light rays and their intensity for clean bubbles have been worked out under Mie theory (Marston, 1980).

In principle, any part of the Mie theory that is related to the reflectance at given angles can be used for designing a sensor to measure bubble radius. But in reality, only a small set of features from the Mie theory are easier to utilize (Marston, 1980; Agrawal et al., 1991; Sequoia Application Note, 1997). The nature of the external light for illuminating bubbles is also important: normal incandescent light and laser light produce significant differences in the response from the bubbles.

When the water is "dirty", i.e. not clear any more, the light response from bubbles are affected too. This "dirtyness" can be caused by a smooth coating of surfactant or an irregular shape of the interface due to attachment of many microscopic particles on the bubble surface. The effects of "dirtyness" are particularly serious in nearshore (near surf zone) water. The presence of a high density of particulate matter in the water medium outside of and separate from bubbles may further complicate the light transmission into and out of a bubble.

3.2.2. Acoustical principles

Both the air (gas) inside and the water outside of a bubble are compressible fluids of different density p(pWater ~ 1000 x pair). Under an external excitation


of pressure from some acoustic source, the combination of a bubble and its surrounding water behave like a vibrating system. As in any such system, the air/water bubble system has its special resonance frequency (JR) near which the bubble oscillates violently (changing its radius or undulating its spherical shape circumferentially) (Medwin and Clay, 1997; Leighton, 1994). For a bubble with 10 \i < r < 1000 fj,, this resonance frequency is related to r and given in Medwin and Clay (1997). The smaller the bubble radius, the higher the resonance frequency.

Within a bubbly flow with many bubbles more or less uniformly distributed throughout a volume of water, the speed of sound is different for both pure water (« 1500 m/s) and air (« 330 m/s). However, the sound speed in a bubbly flow, a mixture of air and water, does not vary monotonically from 1500 m/s to 330 m/s based on increasing air content in water, but, rather follows the Wood formula (Wood, 1941). The sound speed falls down to a minimum at about 20 m/s for an air void fraction range from 0.1% to 1%. This peculiar feature shows the significant impact of the presence of bubble clouds on underwater sound propagation. A more precise relationship of the sound speed (Vs) as a function of the bubble size density B(r) is given in Terrill and Melville (1998).

3.2.3. Electromagnetic principles

Assuming that the electrical conductivity of air is almost zero or negligibly small compared with the electrical conductivity of saltwater, the overall resistance RB and capacitance CB of a volume of such bubbly fluid are different from those for a bubble-free fluid. The degree of difference in resistance and capacitance is related to the bubble size density (B(r)) and its total void fraction (VF). A precise, accurate formula for the resistance of a bubbly fluid (with B(r) not too high) has been inferred from the work of Sir James Maxwell over 100 years ago (Maxwell, 1891). More details are given in the next section.

Thus, we may see that physical principles of optics, acoustics, and electro-magnetism are available for us to design bubble sensors for various purposes. In the next section, we shall examine some of the existing bubble sensors and point out specifically their advantages and limitations.

3.3. Types of existing bubble sensors

Based on various optical, acoustical and electromagnetic principles as described above, some sensors have been designed, tested and utilized in laboratory


and/or field measurements. We shall next introduce these sensors in three classes, as follows:

3.3.1. Optical bubble sensors

To our knowledge, there have been eight kinds of optical bubble devices, with some slight variations among them, developed as follows:

(a) Tri-camera system (Johnson and Cooke, 1979; Su et al., 1994). (b) CCD video imaging system (Bowyer, 2000). (c) Light blocking system (Hwang et al., 1990). (d) Linear reflectance system (Ling and Pao, 1988; Su et al., 1988). (e) Laser holography (California Inst, of Tech., circa 1970). (f) Laser scattering/transmissometry system (Agrawal et al., 1991; Sequoia

Application Note, 1997). (g) Particle imaging velocimetry (PIV or DPIV) system (Dabiri and Gharif,

1997). (h) Wide angle surface video imaging (Holman et al., 1991; Lippmann and

Holman, 1990).

A brief introduction for each of these follows.

(a) Tri-camera system

This bubble system consists of three underwater strobes arranged 120° apart and pointing to a common focus area at which a single camera is used to take a macro image of a water volume approximately 10 x 10 x 5 cm3 in size. The simultaneous flashing of these three strobes generates three bright spots on film from each bubble within the focus volume. By means of direct measurement of each tri-spot's separation on the film, under a microscope, the radius of the bubble can be determined. This data processing can be automated. Its advantage is that it is a direct and relatively accurate method, but, limited to bubble radii > 50 ji. Too many particulates other than bubbles present will also cause confusion and mistaken recognition (Johnson and Cooke, 1979). It is most useful in the laboratory for calibrating other types of bubble sensors (Su et al, 1994).

(b) CCD video imaging system

This bubble sensor uses a single CCD video camera focusing on a small volume about the same size as the above system. The method of data analysis is also


similar too. It has an advantage to use consecutive images for determining the velocity of bubble movement. This latter advantage is just beginning to be utilized. However, its resolution in bubble size is poorer than the above system due to the CCD resolution (Bowyer, 2000).

(c) Light blocking system

This bubble system consists of one light source and one light receiver arranged in line (Hwang et al, 1992). Any bubble within the optical path between these two blocks away some amount of light energy within its cross section, which gives a means to determine the bubble radius. Obviously, too high a bubble density may cause more than one bubble to be present in this path at any given time, so as to cause error in the bubble size determination. Too many particles, other than bubbles, present another problem. The resolution power of this system is about the same as (a) and (b).

(d) Linear reflectance system

This bubble sensor consists of one white light source and one light detector with their optical axes angles crossed at 125°, at which the strongest surface reflection occurs. The intersecting optical field is quite small, having dimensions 5 x 6 x 0.4 mm3. The reflected light intensity from a bubble varies linearly with the square of the bubble radius. Through another calibrating device that makes use of the relationship between the bubble size and the vertical terminal velocity for r < 1 mm, the bubble radius can be determined from the time series of the reflectance signal (Ling and Pao, 1988). This bubble sensor can be used for 10 < r < 200 fi, but its small imaging volume and the requirement of only a single bubble present at any time, make it more useful for laboratory use or long-time averaging in the field. It cannot be used for investigating the transient bubble phenomena immediately after one wave breaks.

(e) Laser holography

This bubble sensor is a regular laser holograph system submerged in water with its focus volume centered around any small volume of a bubbly flow under consideration. It has been used to determine small bubbles down to about 1 /i in radius (California Inst, of Tech, circa 1970). Its complexity in hardware and post processing analysis does not provide much advantage above the current advanced DPIV systems.


(f) Laser in-situ scattering and transmissometry system

This system is designed for determining the solid particle sizes by means of the small angle diffraction of coherent laser light (Agrawal et al., 1991). Under the restriction of small angle diffraction, both small particles and bubbles with radius from 1 to 100 \i behave alike. Thus, it was arranged to test in the field for small bubble determination. It was found that this laser system is not suitable for use in nearshore situations due to very high density of particles present, often more than ten times higher than the bubble density.

(g) Particle imaging velocimetry (PIV or DPIV) system

This system is based on the holograph principle and light slicing with a sophisticated digital image recording and data analysis system. In principle, this system may provide the most detailed information on the bubble field, including position vector, velocity vector, and bubble radius simultaneously. So far, its main use is to provide the bubble velocity vector. Its resolution of bubble size is as good as the other types of sensors [(a) and (b)] discussed here.

(h) Wide angle surface video imaging

Bubble plumes generated by wave breaking also are manifested on the water surface as whitecaps which are the sunlight reflectors from the combined surface bubbles and sprays. To detect the overall spatial distribution and temporal motion of many bubble plumes, a remote imaging system by a video camera from a tower at the beach or from a mast on a ship will be useful. During daytime, the natural sunlight is available, while at night, an artificial white and/or IR light source can be used to illuminate the breaking wave field. Clearly, quantitative calibration of the surface reflectance is problematic due to varying light sources and weather conditions, but, the system is useful for overall imaging of movement and distributions of breaking waves and bubble plumes in the surf zone.

Another attempt by a commercial company with some support from NRL to develop a bubble sensor was based on the Mie scattering theory, but, this was ended after two years of effort due to difficulties in designing a very accurate mechanical arrangement for the required multiple-angle measurement.


3.3.2. Acoustical bubble sensors

There are basically four kinds of acoustic bubble sensors tha t have been

constructed and tested as follows:

(a) Sidescan sonar (Thorpe, 1982; Vagle and Farmer, 1992).

(b) Acoustic resonator (Breitz and Medwin, 1989; Su et al, 1994; Commander

and McDonald, 1991; Commander and Moritz, 1988; Farmer and Vagle,

1997, 1998).

(c) Sound velocity meter (Melville et al, 1995; Terrill and Melville, 1999;

Caruthers et al, 1999a).

(d) Two frequency resonator (Newhouse and Shankar, 1984).

A brief description of each of these is given below:

(a) Sidescan sonar

This bubble sensor is really a regular sidescan sonar, except it is installed in

such a way tha t the acoustic beam is pointed upward to the sea surface. It has

been operated at a single frequency (Thorpe, 1982, 1992) or simultaneously at

four different frequencies (Vagle and Farmer, 1992). The system is t ime-gated

so tha t the re turn signals, reflected and scattered by bubbles from various

depths of water below the surface, are received by the detector and correlated

with the sound source. The bubble field normally has a maximum density

near the sea surface. This technique is really a remote sensing technique for

measuring the bubble field from the surface to 20-30 m below. Its advantage

is tha t the sensor system is below the wavy surface so tha t it encounters fewer

impacts in comparison with other bubble systems tha t must be present inside

the breaking waves and the bubble field. The price to pay for this advan

tage is tha t the acoustic signal needs to travel a much longer pa th from the

source to the sea surface and re turn to the detector located about 20 to 30 m

below the sea surface. When the bubble density is high, such as those close

to a large breaking wave, the acoustic signal will be scattered and absorbed to

such a degree tha t the range gating is no longer accurate and the noise to signal

ratio is so high that any bubble density estimations will be erroneous. Sev

eral publications on the bubble size density using the multifrequency side scan

sonar were later found to be in error by their own authors, who switched to

using acoustic resonators instead. This system, however, is useful for detect

ing the motion of low density bubble plumes, for example under Langmuir


circulations (Thorpe, 1992), or those bubbles generated on the beach and

t ransported offshore by rip currents (Caruthers et al., 1999b).

(b) Acoustic resonator

This system consists of two acoustic transducers about 25 cm in diameter and

separated by about 20 cm, facing each other to make an open parallel plate

resonator (Breitz and Medwin, 1989; Su et al., 1994; Farmer et al, 1998;

Su and Wesson, 1999). Bubbly flow can freely pass between the two parallel

transducer plates. One transducer produces continuous broadband (more or

less white) noise from about 10 Hz to 200 kHz. This acoustic signal will tra

verse the space between the two transducer plates back and forth many times

and some of the acoustic energy will escape from the system. However, the

two parallel plates actually produce a geometric resonance such tha t the power

spectrum at the receiver transducer will show regular spikes at approximately

even frequency spacing. Each of these resonant peaks in the spectrum cor

responds to a harmonic of the geometrical resonance. Furthermore, bubbles

of particular size with their resonance, associated with each of these harmon

ics will undergo intrinsic bubble resonance which results in acoustic scattering

and absorption. Thus, two distinct kinds of acoustic resonances are involved

in this bubble sensor system.

The first advantage of this system is its in-situ operation, thus avoiding

the weak point in the sidescan sonar (a). The second advantage is tha t the

single pair of transducers is actually performing the work of multiple pairs

simultaneously. The fundamental frequency for the above system is near 3 kHz,

so for a 200 kHz system, there are 200/3 « 66 harmonics. In turn , the bubble

density at 66 different bubble sizes can be determined simultaneously by one

pair of two transducers alone. Its limitation is also its in-situ operation, as it

requires a series of such acoustic resonators located at every depth of interest

below the sea surface.

It was found from laboratory tests and field experiments tha t although some

degree of solid particles (such as sedimentary particles) exist in the surf zone,

they do not affect the operation of the resonator appreciably. This is due to the

fact tha t the solid particulates do not resonate as bubbles do under acoustic

excitations. Since the acoustic scattering volume has dimensions of about

20 x 20 x 25 cm3 , it responds much quicker to the transient rapid variation of

bubble size density, immediately after any wave breaking. Its operating upper

limit of V F is about 0.01% to 0 .1%.


There are several data processing techniques for transforming the acoustic resonator spectra to bubble density which are reviewed in Su and Wesson (1999) and Vagle and Farmer (1998). The estimation of the bubble density distribution from the power spectrum of the acoustic resonator is a mathematical inversion problem from the measured acoustical attenuation to the bubble size density. The attenuation is determined from the difference between the spectra obtained with bubbly water and bubble-free water. Some off-resonance attenuation, particularly of larger bubbles at high frequencies, complicates the inversion problem. The first inversion method by Breitz and Medwin (1989) was based on an iterative technique. This method begins by initially approximating the measured attenuation at a given frequency as due to only those bubbles that are resonant at that frequency. Thus, this initial estimate is considered pointwise and local. This initial solution is then iteratively corrected to account for the off-resonance attenuation of the entire bubble population. Improvements to this method have been developed subsequently by Commander and McDonald (1991), Farmer, Vagle, and Booth (1998), Duraiswami, Prab-hukumar, and Chahine (1998), and Su and Wesson (1999). Commander and McDonald formulated the inversion problem by approximating the solution as a sum of B-splines. This was done to cast the entire problem as a single matrix inversion. Thus, their solution is global rather than pointwise. However, the matrix thus obtained is often ill-conditioned, so, it is inverted using regularized singular value decomposition (SVD) in order to obtain a realistic bubble size density. This is equivalent to requiring only smooth estimations for the bubble density. Its drawback is that it may reduce actual curvature in the bubble density especially at small bubble radius. Farmer, Vagle, and Booth (1998) extend the analysis of Commander and McDonald by examining both the real and imaginary parts of the complex sound speed and details of the resonator physics in order to produce a measure of the quality of the estimated bubble density. Duraiswami, Prabhukumar, and Chahine (1998) use a constrained optimization estimation rather than regularized SVD to compute the bubble density using inverse methods, although their apparatus is a source and receiver rather than a resonator system. Su and Wesson (1999) have made further modifications of these methods. Their method uses an optimization rather than SVD and a modification of the B-spline approximation by using variable (rather than equal) spacing of the bubble radii at which the solution is discretized. These improvements were motivated by the need to resolve the bubble density at small radius and also extend its range to larger


radius. The optimization technique is also found to be more resistant to noise in the attenuation data.

(c) Sound velocity meter

This bubble system consists of a sound source and a detector separated by about 10 to 20 cm that measure the sound speed variation through the bubbly flow between the pair of transducers. As explained above, the sound speed is affected by the presence of bubble plumes. Thus, the measured sound speed can be used to infer the bubble density at several bubble radii if the frequency of the sound source is varied over the range of interest (Terrill and Melville, 1999). This system is operated in-situ as for (b), so, it needs to have a series of source-detector pairs located at every depth of interest below the sea surface. In terms of hardware, this bubble sensor is simpler in construction than that for (a) and (b), but, it needs to have more sophisticated electronics and data processing hardware in order to accurately determine the speed of sound in the bubbly medium.

(d) Two frequency resonator

This bubble system consists of two transducers whose primary sound fields cross each other more or less at 90°. The crossing area defines the volume of interest for bubble measurement. The first transducer transmits the resonant frequency of the particular size of bubbles (this frequency is varied sequentially to cover a range of different bubble sizes) while the second transducer with a much higher frequency and more focused sound field acts as both the sound source and receiver in turn. The amplitude variation of the nonlinear mixing of the two frequencies between the bubble resonant frequency and the high frequency is used to determine bubble density. Thus, this system is a hybrid of the above (a) and (b), i.e. it makes use of the bubble resonance and it also operates remotely. The price to pay for this advantage is the much larger spatial dimensions of the total system, suitable perhaps for deeper water depth but difficult to arrange in shallow water areas such as the surf zone. It has been tested in the laboratory (Newhouse and Shankar, 1984), but, so far it has not been developed far enough for field measurements.

3.3.3. Electromagnetic bubble sensors

There are three kinds of electromagnetic bubble sensor available as below. All three are for measuring only void fraction of the bubble field.


(a) Resistance void fraction sensor (Su and Cartmill, 1994a; Su et al., 1999a; Lamarre and Melville, 1992).

(b) Capacitance coil loop sensor. (c) Ring shape capacitance sensor (Falmouth Scientific).

A brief description for each of these follows:

(a) Resistance void fraction sensor

This bubble sensor consists of two electrodes (either plates or wires) with their separation depending on the type of applications under consideration. Normally, the separation is about 2 to 4 cm for use in the nearshore field. The presence of bubbles between the two electrodes changes the electrical conductivity of the water between them. The exact calculation based on two-dimensional infinite plates separated by a layer of dieletric medium with spherical impurities with high resistance has been derived over 100 years ago by J. C. Maxwell (Maxwell, 1891). His theory can be approximately utilized for the special case of bubbly flow, assuming air bubbles with zero electrical conductivity (Su and Cartmill, 1994a). The ratio of the two electrical resistances between the case with and without bubbles present is the quantity that is related to void fraction (VF) as follows:

y p _ {Rw/Rwo) - 1-0 ,„.. {RV1/RWO)+0.5' { ]

where Rwo is resistance without bubbles and Rw is resistance with bubbles. From other direct calibrations, it is found that for VF < 30%, the above equation gives fairly good estimates of VF, but for VF > 30-80%, the above equation overestimates by about 10 to 20%. In theory, the VF could reach the range of 80-100% under certain conditions for a portion of the bubbly flow under wave breaking. Such high VF portion has a very short lifetime and is beyond the current state of the art to measure with any accuracy. In practice, the current VF is limited to about 80% at the highest.

(b) Capacitance coil loop sensor

This bubble sensor consists of a coil loop of insulated wire that is excited by a high frequency AC signal. The dielectric medium around the coil will affect the capacitance of the coil. Thus, the varying density of bubbles in the surrounding water will change the capacitance. This dependence is the physical basis for


measuring VF. It has been tested, but, its response and accuracy is found not as good as the resistance sensor (No published references available).

(c) Ring shape capacitance sensor

This sensor has a shape of a ring with the radius of the inner and outer ring circumference about 1 cm and 3 cm respectively. Its operating principle is the same as the above coil sensor (b) and it is commercially available (Falmouth Scientific). To the knowledge of the authors, no publications on its field application in the surf zone are available.

3.4. A brief summary of bubble sensors for deploying in nearshore zones

Because of the frequent presence of a high density of sand and other solid/ biological particles in the nearshore zone especially close to the surf zone, all the underwater optical bubble sensors described above are deemed to be unsuitable for field measurements, except the wide-angle remote imaging of the overall surf zone by a video camera from a platform above the sea surface. For measuring bubble size density, the acoustic bubble sensors, especially the acoustic resonator and the sound speed sensor, are the most popular and have been tested extensively (as exhibited by the references cited above). For measuring the void fraction, the resistance sensor has been used most frequently.

4. Shallow Water Deployment Techniques

Once certain types of bubble sensors have been selected for the measurement of certain bubble parameters in the nearshore shallow water zone, the next important question is how to deploy them. The shallow water zone is a very difficult location to place most types of sensors and especially for bubble sensors. We shall discuss this problem next.

The main difficulties arise from the following three factors:

(a) The nearshore undulating wave surface with or without wind forcing has a vertical variability due to passing of large swells often of the same order as the local mean water depth.

(b) The tidal effect on the mean water depth nearshore is large particularly near the surf zone and beach. The incoming and receding tide may over a six-hour period change the beach line by 50 to 100 m for a gentle-slope beach.


(c) The bubble field, either the VF or the bubble size density, normally has its maximum value close to the wavy sea surface with a rapid decrease downward to the sea bed.

Most conventional deployment methods for sensors (or arrays of sensors) are bottom mounting. The location of each and every sensor on this bottom mount is fixed at certain heights above the bottom and certain fixed distances away from some reference points on the beach. As the tide changes, the mean surface level moves with a longer period (on the order of 6 hours) and individual passing waves move with a much shorter period (on the order of 10 s). Thus, these conventionally mounted sensors cannot be located in the most desirable and ideal locations all the time and some may even be exposed temporarily above the sea surface. Certain types of surface following arrays have been of course deployed for many years in the deep ocean or even nearshore in water depths from 10 to 20 m, but not in areas with a mean water depth of only a few meters and with the significant wave height of the same magnitude.

The experience by the senior author (MYS) in his first participation in a field experiment for measuring breaking waves across the surf zone at Duck 94 (1994) brought home the shortcoming of this conventional sensor deployment problem. It was a direct response to solve the shallow water deployment problem that a new approach was invented. The new technique consists of a swinging bar that is neutrally buoyant with one end attached to a small surface following buoy and the other end attached to a universal joint anchored to the sea bed. The length of the bar is about 1.5 times the mean water depth for its optimum operation during mid-tide but it could become 5 times the mean depth (or more) at the low tide period and about 1.2 times the mean depth at the high tide period. Beyond these two extreme water depth ranges, the swinging bar loses its utility. Any number of sensors, such as bubble sensors, temperature, tilt, and compass may be mounted along the length of the bar. The length of the bar (from the bottom to the surface) and its tilt angle which is recorded with the sensor data provide the time series of sensor locations relative to the sea surface continuously. Some of the bubble sensors can be mounted quite near the top end of the bar in order to measure the maximum intensity of the bubble density. All of the sensors on the bar are always beneath the sea surface and cover the entire vertical column while responding to the undulating wavy sea surface. Following the tidal change, the whole array of sensors automatically adjust their locations. Because of inertia and flow


resistance of the swinging bar with its mounted sensors, the top buoy does not exactly follow the sea surface undulation, particularly near the sharper wave crest. Nevertheless, this swinging bar deployment technique does provide a big advance in deploying bubble sensors and instruments in the surf zone for measuring the rapid changes in void fraction produced by swell breaking.

At a mean water depth of 5 m or more, with the corresponding significant wave height less than 30% of the mean water depth, the conventional surface following buoy deployment can be used for bubble sensor mounting. This buoy will be anchored to the bottom by one or more cables. Two or three cables attached to the buoy from different angles may prevent the buoy from excessive rotation under the forcing of the tidal current ellipse.

Some short time deployments with an accompanying ship may even use the free-floating buoy without any bottom anchored cables. This type of deployment is used more often in deeper water but would be almost impossible near the surf zone under heavy weather conditions.

When a suitable long research pier is available, some bubble measurements may take advantage of it. However, the pier with many pilings underneath it will often generate substantial bubbles that may cause serious bias in the measurement. One interesting candidate is the Sensor Insertion System (SIS) of the US Army Corps of Engineers, Field Research Facility at Duck, NC. On its nearly 600 m long pier, the SIS is movable on a track along the entire pier. A controlled arm on the SIS can be used for mounting sensors that will be almost 20 m away from the pier from the surface to the sand bottom. Under appropriate wind and wave conditions, the effect of the pier-generated breaking waves and bubble field will be much less than those caused by the surf zone itself. Some useful bubble (and other) measurements have been obtained using this system in 1998 (Su et al, 2000).

Some of the off-shore platforms (either commercial gas and oil platforms or research ones), for example the North Sea Tower in the German Bight, have been used for deploying bubble sensors. Again, caution should be exercised in order to minimize the platform effects on the bubble measurements.

Finally, we should mention FLIP of the Scripps Institute of Oceanography. Its long arm (more than 10 m beyond the vertical buoy) has been used for suspending some bubble sensors. Normally, FLIP needs at least 30 m or deeper water depth, so its range of operation is closer to offshore fixed platforms than the deployment methods near the surf zone mentioned above.


5. Large Scale Shallow Water Field Experiments (1985-1999)

Compared with very extensive measurements of oceanic waves or even with the less extensive measurements of oceanic breaking waves in the last 60 years, the amount of measurements of bubbles generated by wave breaking in both deep and nearshore has been very limited, even though white capping is such an obvious and ubiquitous surface phenomenon to notice in windy seas and along the beach. The importance of such bubble fields on both civilian and military operations has been recognized as early as the Second World War. The major reason for such a scarcity of field bubble measurements is the lack of appropriate bubble sensors. Here, we are referring to accurate measurements of bubble size density and large void fraction close to the wave breaking event. On the other hand, the effects of bubbles on acoustic and optical scattering and transmission have been observed and measured many times.

Johnson and Cooke (1979) were among the earliest reported measurements of the bubble size density in coastal water. Their device was the tri-camera system (originally proposed by H. Medwin) with the minimum detected bubble radius limited to about 50 \i. The maximum bubble radius observed was about 500 \±. During the early 1980's, S. A. Thorpe and colleagues in England started to use a side-scan sonar at a fixed single frequency near 250 kHz to explore the bubble clouds in lakes and nearshore. They discovered many interesting near surface phenomena resulting from surface wave breaking (Thorpe, 1982, 1992). Having used only a single frequency of 250 kHz (corresponding to the resonant frequency of bubbles with 13 /i radius), this sonar could neither determine the bubble size density nor large void fraction conditions.

About the same period, S. C. Ling (Ling and Pao, 1988) developed an in-situ optical bubble sensor under ONR sponsorship. The bubble sensor (described in Sec. 3) could measure bubble radius ranging from about 10 to 200 fi. Its first extensive field deployment was done on the North Sea Tower at a mean water depth of 30 m. This tower was located about 10 km offshore in the German Bight. The bubble sensor was lowered from the tip of a 20 m boom beyond the research platform. Extensive measurements (about 120 hours in total) were obtained in heavy weather during a six-week period in Nov.-Dec, 1985 (Su and Cartmill, 1988; Ling and Pao, 1988). The main findings from these bubble size density measurements are (a) a very prominent peak occurs near 20-30 /j, in radius, (b) the bubble size density for bubbles with radius 10 < r < 200 n remains almost constant even down to 15 m depth, (c) for r > 80 /i, the density follows a r - 5 to r~6 power law with the increasing


radius, and (d) for r < 20 (i, the density follows a r3 power law. Between the period 1986-1993, this optical device has been used again several times for bubble measurements in both deep and coastal water. The essential feature of the bubble size density (having a peak value near 20-30 fi) was found to be similar.

In this period, the single frequency side scan sonar was expanded to a four frequency system covering frequencies from about 40 to 400 kHz by D. Farmer and his colleagues. Using this device and the assumption of the bubble size density peak at 25 /x (based on Su et al., 1988), the first bubble size density measurements by the side scan sonar were published (Farmer and Vagle, 1989). Several other publications based on this multiple frequency sidescan sonar appeared later. Around 1994, researchers finally realized that the accuracy of the bubble size density obtained by this method is questionable due to the problems mentioned in Sec. 3, particularly in cases near breaking waves with very high bubble densities near the surface.

Also during this period, several other attempts to develop new bubble sensors were under way. One of the most successful designs is the acoustic resonator (Breitz and Medwin, 1989). The prototype device was further refined and improved at NRL-SSC (Su and Cartmill, 1994b).

The resistance sensor for void fraction measurements was first demonstrated by NRL-SSC in recognition of the need for measuring the rapid change in bubble VF near breaking wave events and was used in NRL-SSC laboratory tests since 1990 and in field experiments since 1992. This design has been followed by several other research groups with some minor variations (Lamarre and Melville, 1993).

During the Duck 94 Experiment at Duck, NC, a vertical fixed array (2 m height) with 8 resistance sensors provided by NRL-SSC was incorporated onto the instrumented surf zone measurement sled operated by the Naval Postgraduate School. It was found that these resistance sensors when in air above the wave trough have such high electronic impact on those still under the surface that all the sensors in the array were severely affected so that the measured void fraction is fragmentary and difficult to assess its accuracy. As a result, all those void fraction data have been discarded and the urgent need for an improved deployment technique to measure bubbles very near the surface was fully recognized for the first time.

During 1995-1997, under ONR sponsorship, a comprehensive coastal bubble system was designed, constructed, and tested by NRL-SSC (Su et al., 1999a,

Bubble Measurement Techniques and Bubble Dynamics _ 99

1999b). The main parts of the system are as follows:

(a) Three swinging bar systems, each with six resistance sensors for void fraction measurement. These systems can be deployed in mean water depths from 3 m to 6 m, covering the surf zone.

(b) Five surface following and bottom anchored buoys (each of 4 m vertical length), each having 4 void fraction resistance sensors, 4 acoustic resonators, and vertical accelerometers (for wave height measurements). These five buoys can be anchored to the sea bed by three cables on each buoy. The mean water depth is from 8 m to 15 m. The offshore distances were about 1 km to 2.3 km.

All the sensors on each system have on-board data collection systems which are further connected to shore computers by marine cables for power, control, and data transfer on demand. Without question, the whole system is the largest coastal bubble system ever constructed.

This whole system from NRL-SSC was first field tested at the Scripps Acoustics and Bubbles Experiment, March, 1997 near the Scripps Pier, as one of seven research teams involved in the joint experiment. Other bubble sensors including a sound speed sensor array, acoustic resonators, and inverted echosounders were also deployed at the same time by various teams. Unfortunately, the wind was so calm (both near and offshore) during the two weeks of pre-scheduled experiment that the only bubbles generated were from light swell breaking on the beach and carried outward by rip currents near the pier (Caruthers et al., 1999b).

This NRL-SSC coastal bubble system was an integral part of a much larger field experiment for nearshore wave/current/sedimentary studies (Sandy Duck 97) in September and October, 1997, at Duck, NC (Birkemeier, 1997a, 1997b). It was an unfortunate fact of life that the coastal weather should have been so calm again during this two-month period for the main experiment of Sandy Duck 97 that not many interesting events occured to measure breaking waves and bubbles.

In two weeks in February, 1998 (during the Duck 98 Experiment), the NRL-SSC bubble sensors, temperature sensors, wave gauges, and current meters were mounted near the tip (about 3 m from the end) of the extended arm of the FRF Sensor Insertion System (SIS). This was a joint experiment between NRL-SSC and FRF. This time, there were several days of strong wind and heavy seas that caused many interesting wave breaking events and


produced many bubble clouds within the reach of the instrumented arm. A string of four resistance void fraction sensors, each mounted on a separate small surface-following buoy, was laid away from the a rm t ip by a cable. The string of bubble sensors provided a sequential measurement of passing breaking wave events. Fixed on the submerged portion of the arm tip were four acoustic resonators for measuring bubble size densities. At the same time, a video camera was mounted at about 10 m above the surface on the vertical support of the SIS, recording the surface whitecapping above the area where the submerged sensors are located. Another video camera was mounted at the pier level to record these same phenomena. All of these sensors did provide, for the first time, the quantitative description of void fraction and bubble size density over the surf zone from about 200 m to 400 m off the beach where the submerged bar was located.

In view of the failure to have sufficient suitable wind and wave conditions for the bubble measurements over the entire littoral zone from the beach to about 2 km offshore during Sandy Duck 97, the NRL-SSC team went back to Duck, NC again in February and March, 1999 to deploy three swinging bar arrays of void fraction sensors and four bot tom fixed arrays. Each fixed array had two acoustic resonators mounted and a string of four resistance sensors floating at the surface in a line toward shore. These seven arrays were placed in a line 200 m north of the Duck pier. Again, we were lucky this time to have three storms passing by during the two weeks of the measurement period. Since all the wind conditions measured were from the nor th or northeast , we thus believed that there was no undesirable bias from the pier itself. During the daytime, a video camera mounted on the 140 ft (34 m) high tower located on the beach was operated to cover the entire surf zone area where the sensor arrays were located. We have thus a very comprehensive set of bubble da ta in the surf zone obtained during the Duck 99 bubble experiment. During this experiment, several other research teams from NRL-DC, NRAD (San Diego), T N O (Netherlands), NASA and several universities from the US and UK were also involved in this coastal aerosol investigation over the same coastal area. It was based on the fact tha t spray droplets from bursting bubbles on the surface are the major source for marine aerosol (salt particles) generation, that such a comprehensive joint experiment was planned and executed.

6. B u b b l e Void Fract ion Variat ions N e a r Surf Zones

As described earlier, a single wave breaking event produces two integral parts

of air entrainment:


(a) An initial large void fraction composed of cavities and large bubbles. These large air pockets escape to the surface very quickly, generally in less than a tenth of the wave period.

(b) A bubble plume with smaller bubbles that stay in the water longer than a wave period.

In this section, we shall describe the characteristics of some field data collected during the field experiments described in Sec. 5 by means of the bubble sensors and deployment techniques in Sees. 3 and 4, respectively.

Surface wave breaking in deep water and in shallow water do share some commonality as well as some differences. In order to appreciate more clearly the characteristics of shallow water breaking and void fraction, we shall mention, from time to time as needed, their corresponding counterparts in deep water. So far as the generation of void fraction by wave breaking is concerned, the major difference between the situation in deep and shallow water is their intensity and duration relative to the wave period. In deep water, almost every breaking wave is short crested, i.e. the spanwise length of breaking portion is much shorter than its wavelength and its breaking duration is much shorter than its wave period as well. In contrast, in shallow water, particularly in the surf zone and close to the beach, the process of wave breaking is often continuous in space and time, sometimes even longer than both its wavelength and wave period respectively. The dynamical reason for this difference is that the shoaling process (in shallow water) continuously causes the incoming waves to become steeper and steeper and induces breaking as a consequence. Figure 1(a) shows the time series (over 10 min) of two surface elevation and three void fraction time series collected by the resistance void fraction meters mounted on a swinging bar deployment located near the submerged bar at the beach at Duck, NC (Su et al., 1999a). Time series (a) is the surface elevation determined from the tilt angle of the swinging bar while series (b) is the surface elevation obtained by a pressure gauge located in the instrument housing on the sea bed. The correlation between these two surface elevations is 0.86, typical of all the cases under actively breaking waves. This correlation will reach 0.95 and higher for the situation of nonbreaking swells. The three void fraction sensors — VFl, VF2, and VF3 — are mounted from near the surface, ranging downward toward the bottom. Their locations change with time but can be precisely computed from the tilt sensor data. Comparing these three (c), (d), and (e) time series simultaneously, one sees the corresponding increase/decrease in their void fractions.


Duck 99 Bubble and Aerosols Experiment (Feb 19, 1999)

VFSystem#2 Sensors lf2,3, PR, Tilt •(641)« 0:79m1 T -'5.79a ' W -10.89^/1 »(p*)4 l.lBm HB(tilt)- 1.07m,

Time (min) 15:30:00-15:40:00

(a)

Duck 99 Bubble and Aerosols Experiment (Feb 19, 1999)

VFSystemtt2 Sensors 1,2,3, PR, Tilt [•(641)- O.Blm T - 5.62m W »10.84m/» It(pr)> 1.17m H«(tilt)- 0.99m

Havti From Tilt D(tilt) •< 1.84m

Correletion(pr:tilt)> 0.84 Delay* O.90s Havaa from PR Gaug* D(pr) - 1.70m

Time (tec) 15:53:00-15:54:00

(b) Fig. 1. Time series of void fraction and surface wave height, (a) Time series of surface wave height (upper two curves) and void fraction (lower 3 curves) over 10 min, collected by the resistance void fraction meters and pressure gauge mounted on a swing bar array located near the submerged bar at the beach at Duck, NC (Su et al, 1999a). The time series (a) is the surface elevation determined from the tilt angle of the swinging bar, while the series (b) is the surface elevation obtained by the pressure gauge, (b) The same format as (a) with an expanded time scale (by a factor of ten).


Figure 1(b) expands the time scale ten times in order to exhibit more details of these variations. The correlations between the surface elevations (a) and (b) together, and the void fractions (c), (d), and (e) together, is not obvious from this figure and many others as well. This observation seems to imply that the surface elevation alone is not a sufficient criterion for detecting breaking waves in the surf zone.

Figure 2 shows an even more enlarged time series of vertical variations of void fraction in the surf zone at six levels, from near the surface (VF1) to close to the bottom (VF6), over a 30 s period (Su et al., 1998). Since the swinging bar is installed with its floating buoy leaning toward the beach, the bubble plume (with its void fraction) generated by the breaking wave actually reaches the lowest bubble sensor earlier than the top sensor as shown through the relative delay to reach their maxima for these six time series. Regarding the passing of a wave breaker in the surf zone as a single event, we can perform ensemble averages of many of these events with conditional sampling. We select the maximum value of void fraction variation from each event as the central reference time and consider only the ±2 s with respect to it to form the event. Such a conditionally sampled collection of breaking wave events over a 3-hour period is shown in Fig. 3(a). Its maximum VF value is 58%. The mean and its standard deviation of the void fraction at each time difference is shown

o 20 Tims (sec at 4.404 hrs , 10/17/97)

Fig. 2. Time series of void fraction from six void fraction meters from void fraction staff V F # 3 over a 30 s period near low tide.


Duck 98 Bubble and Aerosols Experiment (Feb 24, 1998) SIS VF String Chan; 5

a •So.6

Tina 10>38:09-14|00|00 Waves: s tdev=0.12, kur t= 0.392

N_JPul6es= 928 T h r e s h o l d s . 10 minimum=0 .03

(a)

Duck 98 Bubble and Aerosols Experiment (Feb 24, 1998) SIS VF S t r i n g Chan: 5

_T:

•0

'go.4 !>

"T~ ~r 10:38:09-14:00:00

Waves: s tdev=0.12 , kur t= 0.392 N Pul363= 928 Throshold=0 .10 miniinum=0. 03

Time ( s )

(b)

Fig. 3. Conditional sampling of void fraction plume time series, (a) Conditional sampling of void fraction plumes above 10%. The time origin is set to the time of peak void fraction for each pulse, (b) Composite void fraction time variation of plumes, with one standard deviation limits shown.


Sandy Duck 97 Bubbles Experiment (1997) 100

Mean void Fraction

Tides High - 0836/2100 Low - 0217/1506

50

40£

c o •H

30« XI

- 20§

t

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Void Fraction

Fig. 4. The relative (+) and cumulative (x) contributions of void fraction (over 1 hour) for observations on Oct. 17. 1997.

in Fig. 3(b). The shape of the mean VF shows slight skew toward the wave front, consistent with the geometric description of breaking waves by Bonmarin (1989). The maximum value of the VF mean is 27% with one standard deviation nearly 12% VF. Figure 4 shows another type of statistics for void fraction as the relative (+) and its cumulative (x) contribution to void fraction over a one-hour period. Clearly, there are more frequent occurences of small void fraction for VF < 25% and its cumulative contribution reaches 0.9 from the contributions less than VF = 17%. Certainly this statistic is highly location dependent, i.e. dependent on the positon of the bubble sensor array with respect to the submerged bar in the surf zone. Figure 5 provides another statistic of the probability density function for the peak void fraction observations from the same data in Fig. 3. It shows that the most probable void fraction to encounter at that location is about 7% with about a 25% probability while the chance of a VF « 20% is only about 2%. Figure 6 shows the temporal variations of void fraction events observed over a 24-hour period with relatively active breaking waves inside the surf zone. Two high and two low mean water levels (marked in the figure) due to the diurnal tide caused the corresponding variations in breaking wave frequency and intensity as expressed by their void fraction. Each dot in Fig. 6 is a data point. An hourly running mean of this void fraction data is also shown. This figure shows dramatically the deciding factor of the tide on the nearshore dynamics on a barred beach, by means of


Sandy Duck 97 Bubbles Experiment (1997)

0.4 Max VF

N_plumea- 2733

Threshold' 0.05

Haan> 0.107965

STD DEV- 0.0680896.

Fig. 5. Probability density function for peak void fraction observations from the same period (Oct. 17, 1997) for all pulses.

0 . 5 0

0 . 4 0

•rt O 0 . 3 0

' 0 . 2 0


Low Tide 0217

io itli^ftJiJinL^

Void F r a c t i o n #3-1

High Tide 0836

Low Tide 1506

High Tide 2100

iiiiiA, 6 8 10 12 14 16 18 20 22 24

Hours (October 17, 1997)

Fig. 6. Temporal variation of void fraction over a 24-hour period (October 17, 1997) showing the tidal effects on breaking waves. Low water is at 0217 and 1506 local time. High water is at 0838 and 2100.


its influence on both the temporal variation and spatial location of the breaking waves, energy/momentum loss to the current field, and the generation/distribution of bubble fields. The overall spatial variation (across the entire range of the surf zone to the beach) of a continuously breaking wave field with its air entrainment is certainly of much interest. The field experiments at Duck, NC in 1997-1999 did conduct some of these measurements. But, even for a fairly uniform and gentle beach like Duck, NC, there is still much local variation in the bathymetry caused by the breaking waves themselves. At the same time, the incoming swells vary all the time in their steepness and periods such that it is difficult to give any general statement (assessment) about their spatial variations normal to the entire beach.

One useful approach to assess this variation is to analyze the video imagery of a section of surf zone based on its surface reflectance (indicating the presence of sprays/bubbles/foams on the surface) such as being done for several years by R. Holman and his colleagues from OSU at Duck and other beaches (Lippmann and Holman, 1990; Holman et al., 1991). The drawback for such a remote sensing approach alone is the lack of quantitative estimates of associated void fraction below the surface. In 1998 and 1999, NRL experiments at Duck did remedy this situation by placing both the bubble sensors inside the surf zone and a video camera mounted at the top of the FRF 34 m tower at the same time. The objective is to establish possible correlations using the reflectance from the video images and from the direct in-situ measurement of void fraction. Analyses of such correlations have not been advanced far enough for making any definitive assessment of their quantitative correlation at this writing.

7. Bubble Size Distributions in Littoral Zones

We now discuss the distribution of bubble size density, B(r), which is defined as the number of bubbles within a unit volume (m3) per micron increment in bubble radius (#/m3 / /z) . In discussing this bubble size density and its distribution both horizontally with respect to the beach and vertically with respect to the water depth, several main questions of concern are as follows:

(a) B(r)'s dependence on prevailing wind speed, duration and associated wave field.

(b) _B(r)'s dependence on the time lag with respect to a single wave breaking event.


(c) The radius range rm of bubbles that have the maximum (peak value) of B(r).

(d) The functional dependence of B(r) for r > rm. (e) The functional dependence of B(r) with depth. (f) Effects of local current (longshore or rip current) on B(r). (g) Effects of local water qualities such as salinity and surfactants on B(r).

These concerns are not totally independent of each other and there have simply not been enough field measurements conducted to answer all of these with certainty in sufficient quantitative detail. Our knowledge on the last two items, (f) and (g), are particularly lacking. In the shallow water environment, there are additional complications of unknown (and varying) bathymetry near the surf zone acted upon by the changing tide. The nearshore bathymetry is actually formed by breaking waves themselves. During each large storm, the beach bathymetry changes substantially. With all the above disclaimers in mind, we shall next make some comments on the bubble size density in littoral zones.

One of the earliest bubble size density measurements of considerable duration in the coastal zone was carried out at the North Sea Tower using an optical bubble sensor in the winter of 1985 (Ling and Pao, 1988; Su and Cartmill, 1988). The results of this six weeks of measurements provides some answers for the first four factors ((a), (b), (c), and (d)).

The dependence of B(r) on wind speed W in a more or less equivalent sea-state after a long period (> 5 or 6 hours) of wind forcing is approximately:

, •200^

/ B{r) dr oc W4-3 . (3)

This dependence changes dramatically at the initial stages of wind-wave growth.

As breaking waves pass by the bubble sensor, B(r) will experience a sudden increase and then decrease more gradually. The change shows more clearly for r > rm as shown in Fig. 7, following the sequence from (a) to (b). The last is the mean B(r) over a long period. In other words, there are more large bubbles present immediately after a breaking wave event. These large bubbles escape quickly by their buoyancy to the sea surface and disappear. This process leads to a balanced mixing of various sizes of bubbles as shown in curve (e). The reason for the slower change in bubble density for r < rm is due to a different phyical mechanism: the dissolution of gas (air) from inside the bubble into the surrounding water. This dissolution process is enhanced


b

c

d

Fig. 7. Schematic diagram of bubble density variation at different phases of a wind event. (a) is at the beginning of a storm where large bubbles are more frequent and (e) is after equilibrium has been reached.

by the inverse of bubble radius due to increasing the surface to the volume ratio as a bubble becomes smaller. In other words, there is a continuous disappearance of small bubbles within the water medium. This dissolution process is affected considerably by the presence of surfactants which form a coating at the air-water interface. The higher the amount of surfactants, the slower the dissolution rate. Thus, in dirty coastal waters, the density of smaller bubbles are often higher than those in cleaner deep water.

The bubble size density also shows a clear peak in the radius range r m « 20-30 fx. The physical reason for the existence of this peak can be argued reasonably but a detailed derivation of its exact value is still not available. Essentially, the peak is maintained by a balance between the continuous supply of bubbles by wave breaking and the disappearance of bubbles by the combined effects of buoyant escape and the dissolution process.

For the functional dependence of B(r) for r > rm, the measurements show a relation as follows:

B(r) oc r~n for 200 fj, > r > 50 fi with n = 5-6. (4)

Interestingly enough, a recent measurement by an acoustic bubble sensor (Terrill and Melville, 1999) yield a relation B(r) oc r~5-7 which falls within


the above bounds that were obtained in 1985 by a totally different optical sensor. In this recent acoustic measurement, the peak of B(r) appears only in one of four curves. The index (n) is known to be sensitive to the presence of surfactants (as well as the wave ages and close proximity to other breaking waves, among other factors) but their exact relationship is also unknown at the present time.

In many active breaking wave environments, the index appears often to be between 3 and 4. A recent bubble measurement using video imaging analyses in coastal waters outside the surf zone (Bowyer, 2000) for bubble radii from 70 to 3000 fx shows that the value of n varies between 1 and 2 for B(r) near surface breaking waves, between 3.5 and 5 away from the breaking wave surface, and for the overall average of n is between 2 to 3.

From all the above past measurements and others, we should be very careful in stating the functional dependence (n) by giving detailed conditions under which such B{r) is measured.

The bubble size density B{r) can be further scaled to the air volume distribution with respect to the bubble radius as: r3B(r)dr. This distribution normally has a peak near r « 100 /x (Terrill and Melville, 1999; Farmer, Vagle and Li, 1999), implying that the most significant contribution to air volume in bubble plumes is due to bubbles with radius around 100 p..

Several formulas to model the variation of B(r) with water depth have been proposed over the years, but none of them has had a sound physical basis. For this reason, this review shall not put an emphasis in this respect. The interested reader may refer to the last several cited references.

During the Scripps Bubble and Acoustics Experiment, March, 1997, NRL-SSC Acoustics Division (Caruthers et al, 1999b) made high frequency attenuation measurements near the beach at a depth of « 6 m when some low density bubble clouds were carried offshore by rip currents from the beach. Using an iterative approach (Caruthers et al., 1999a), some bubble size densities were obtained (Caruthers et al, 1999b) for 10 p < r < 100 /x. Most of these distributions show a peak near r = 20 to 40 /j,, which is consistent with the observation in the North Sea experiment and other bubble experiments.

Finally, we shall now examine the significantly different responses of the bubble size density B(r) with respect to a storm passing at two nearshore locations; one located 2.3 km offshore (mean water depth of about 14 m) and one within the surf zone (mean water depth about 4 m) (Su et al, 1999). The bubble sensors used are acoustic resonators. The hourly wind speed, wind


direction, significant wave height, wave period and wave slope are shown in Fig. 8. The wind and wave conditions show that before the wind speed picked up near midnight of September 20, the waves consisted of swells with about a period of 13 s and with a significant wave height of only 0.25-0.30 m. The weather had been calm in the preceeding week; the wind speed ranged from 6 to 7 m/s in the longshore direction. As the front passed through near midnight of September 20, the wind direction shifted to an onshore direction, the wind speed reached its maximum of about 15 m/s in just one hour and it remained near that speed for about six hours. It then decreased steadily to about 10 m/s at noon on September 21. The significant wave height increased steadily from midnight to its maximum of about 2.3 m around 0700 on September 21 and then decreased steadily. The time lag between the two maxima of wind speed and significant wave height is about five hours. This time lag of wave height growth with respect to the wind forcing is due to its fetch. Based on the significant wave height, Hs, and the mean wave period, Tm, an effective wave slope S is computed as follows:

5 = A f = L 6 ^ % ' ( 5 )

where Am is the mean wavelength calculated from linear deep-water gravity wave theory. Note that the effective slope follows closely the wind speed change (Fig. 8). During the period of initial rapid wind speed increase, the wind speed is much higher than the mean phase speed of locally generated waves. As such, the mean wave slope will be high under the storm wind forcing and frequent wave breaking of small to mid-size waves occurs. These breaking waves entrain air below the surface and produce bubble plumes which are further dispersed by the wave orbital motion and turbulence generated by wave breaking. Significant changes in the bubble densities were observed by acoustic resonators from near the surface to deeper than 2 m.

The characteristic shape of the bubble density depends critically on the stages of wave growth and decay. The bubble density over the two hours of storm growth using 10 minute intervals (in Fig. 9) shows the characteristics of evolution of bubble size density during a storm cycle. The initial bubble density is very low at all bubble radii. Then, the bubble density in the radius range from 30 to 300 /J, is especially elevated during the most rapid growth of the storm. An hour later, bubble density is still high but no longer dominated by the bubble density in the 30 to 300 fi range.


Sandy Duck B u b b l e s E x p e r i m e n t ( 1 9 9 7 )

00 2 4 6 Time / ( h r )

Fig. 8. (Top) Wind speed and direction at FRF, September 20-21, 1997, during the Sandy Duck Experiment. (Middle) The significant wave height and mean wave period for the same time interval. (Bottom) The effective wave slope for the wave field calculated from the wave data above. The high tide for 9/20 is 2306 while the low tide on 9/21 occurs at 0512 and high tide at 1136 (Data provided courtesy of CERC FRF).

We now switch our attention to the bubble measurement inside the surf zone. The bubble size density measured on the bottom mounted array (AR#5) is presented for the period during the same storm (9/20-9/21/97). Results from the upper acoustic resonator on the array (at 2.5 m depth) are shown in Fig. 10. The bubble size density over radii from 25 /J, to 500 fi inside the surf zone at the time period 0500-0600 is about 5-7 times higher than the bubble density at AR#1 at 2.3 km offshore at the same time. The contrast in bubble densities in the surf zone before and after the storm is large. These significant differences in the bubble densities inside and outside the surf zone, and the associated time lag with respect to the arrival time of the storm, can be attributed to the different dynamical mechanisms responsible


S a n d y Duck B u b b l e s E x p e r i m e n t ( 1 9 9 7 )

a o n o

I 10* h 6

• •o i o 2

a

10°

1 1 1

\ i;-' \ '*' \

0 1 : 5 0 - 0 2 : 0 0 N

-

•

1 t 1

1 ' '

AR# 1 Chan 1 Sep 21 _ 10 m i n u t e i n t e r v a l s

r 2 m i n u t e p r o f i l e s ^ N (*) f i r s t p r o f i l e

^H^ \ ( + ) l a s t p r o f i l e

"'**'•• '"• \ f t W \

' • ' • ' A \ \ W I ~~

**"-'•• ^ V »

>-.';•. ' \ \ X K

V\ %& $&* %. l ^ V j J l : 0 0 - 0 1 : 1 0

\L " ''•Y' a* H V\, '1

T^< "'\ *'-U

0 0 : 0 0 - 0 0 : 1 0 ^ . E v

1 1 • 1 . 1 r

100 Radius / micron

1000

Fig. 9. A summary of the bubble densities for three 10 min intervals between 0000 and 0200 on September 21, overlaid in a single figure.

for production of bubbles in these two characteristically different locations in the littoral zone. In the location of AR#1, 2.3 km offshore at a mean water depth of 14 m, the bubbles are produced mainly by local wind-wave breaking (onshore wind). At AR#5, only about 200 m off the beach with a mean water depth of 4 m, bubbles are produced primarily by the swell breaking over a submerged sandbar together with secondary breaking on the beach.

Figure 8 shows that the significant wave height increases only from 0.3 m to about 1.0 m during the two hours, 0000-0200. This period was midway between the high tide at 2306 and low tide at 0512. For the next two hours, approaching the low tide, the significant wave height increases to about 2 m,


Sandy Duck 97 Bubbles Experiment (1997) H T~ 1 1 — I — I I I ! ^~l 1 1 1 — I — I I I

10 100 1000 Radius / micron

Fig. 10. The hourly averaged bubble densities from an acoustic resonator array in the surf zone, before (00:00-01:00) and after (05:00-06:00) the effects of the storm are felt in the surf zone.

hence, more frequent and larger breaking waves are expected which produce more bubbles. The significant wave height increases to a maximum of 2.3 m during the next two hours until 0600 and remains relatively high at 1.8 m until 0900. These rather steady wave conditions result in similarly constant intensity and frequency of wave breaking inside the surf zone. The bubble density variation observed in the surf zone followed the breaking wave height and intensity.

Therefore, the variation of the bubble densities inside and outside the surf zone, respectively, are controlled by different aspects of breaking wave dynamics even under the same storm forcing. This leads to considerable differences


in the bubble density and in the time lag with respect to the prevailing storm cycle and tidal condition.

The void fraction and associated low frequency sound speed due to bubble plumes are two integral characteristics of the bubble field. The average bubble void fraction is obtained by integrating the bubble density over the range of bubble radii from 17 fi to 1200 fi. Such average void fraction is in fact only a fraction of the total air entrainment by wave breaking, since the current NRL-SSC acoustic resonator is limited to measure bubbles with radii r < 1200 /x. We shall refer to this integrated value as the averaged void fraction over any chosen period.

The hourly average void fraction (VF) from the acoustic resonators at both AR#1 and AR#5 are shown in Fig. 11. The averaged VF outside the surf zone (at AR#1) follows the increase of the wind speed closely to its maximum, then, decreases even faster than the effective wave slope as the wind speed decreases. This feature of the response of the averaged VF is characteristically similar to the associated bubble size densities. On the other


to"4

tiio"5 (0 u fa

•o •H

•0 - 6 £ 1 0 0) d u o

5

18 20 22 00 02 04 06 08 10 12 Time / Hours (Sept 20-21, 1997)

Fig. 11. The hourly averaged void fraction (obtained by integrating the hourly averaged bubble size density) both inside (AR#5) and outside (AR#1) the surf zone for the period from 1800 on September 20 to 1200 on September 21. The associated low frequency sound speed (c) for several void fraction values are indicated. High and low tides during the period are also indicated on the time scale.

n I I I I i i i i r

AR#5 Channel 1 Inside Surf Zone,

n i i i r

1237 m/s

High Tide 23:06

_ l I I I I I I L

Low T i d e H i g h Tid< 0 5 : 1 2 1 1 : 3 6


hand, the hourly averaged VF inside the surf zone (at AR#5) follows closely the growth of the significant wave height to its maximum at the same period (0500-0600). In addition, the maximum value of the hourly average VF inside the surf zone is about 10 times higher than that outside the surf zone. Their associated low-frequency sound speeds, computed based on Wood's formula (Wood, 1941), are 1220 m/s and 1440 m/s, respectively, with the nominal sound speed for bubble-free water being 1498 m/s. Hence, the deficit of the low-frequency sound speed inside the surf zone can be large enough to affect significantly sound speed dependent phenomena, such as shock wave propagation from an underwater explosion inside the surf zone.

8. General Remarks on Bubble Dynamics in Shallow Water

Both wind wave and swell breaking inside the surf zone can produce large amounts of bubbles. The peak value of the void fraction (VF) in wave breaking reaches 80% or higher within about 10-20% of a wave period. But, in the bubble plume, bubbles substantially larger than about 3000 fj, (3 mm) in radii quickly escape to the surface and disappear. Smaller bubbles with radii less than 1000 y or so have smaller buoyancy and are controlled by the dispersion of wave orbital motion and turbulence, and remain much longer in the water column. The averaged void fraction due to the larger bubbles may be measured by the conductivity-type void fraction meter. The hourly average void fraction can range from 5 x 10~3 to 1.5 x 10~2 near the water surface from these large bubbles. However, the averaged VF for the smaller bubbles with radii less than 1000 fi is limited within 10_4% to 10_ 2% which is two orders of magnitude lower than the corresponding VF from the larger bubbles (> 3000 y). Since these two types of averaged VFs have different effects and applications in littoral zones on optical, acoustical, air-sea interactions, and aerosol production, we shall refer to the average VF from the larger bubbles as "macro VF" and the average VF from the smaller bubbles as "micro VF". For example, to determine marine aerosol levels and the low-frequency sound speed, one needs to use "micro VF" while for air-sea heat/mass transfer and energy /momentum loss by wave breaking, one should use "macro VF".

The evolution of bubble size density during a storm nearshore and in the surf zone is controlled by different wave dynamics. Data from the offshore location show that the growth of bubble populations follows the increase of the local wind speed and wave slope closely rather than the growth in significant wave height. This is in clear contrast to the corresponding growth of bubble


densities in the surf zone, which are closely correlated with local significant wave height. In normal coastal situations, with wind speed around 30 knots (15 m/s), the growth of the significant wave height may lag the onset of wind speed increase by four to six hours.

Differences also exist between the growth rates of bubble densities inside and outside the surf zone with respect to the same wind forcing. The growth rate outside of the surf zone is due to wave breaking at all scales cumulatively with bubble dispersion by wave orbital motions and turbulence while the growth rate inside the surf zone is due to the direct plunging breakers which produce large bubble plumes within a small fraction of the wave period. Since the bubble size density and the bubble void fraction have different effects on acoustical and optical applications, their different responses with respect to the wind speed, wave growth, and decay are important considerations for these applications.

During the rapid wind speed increase with increasing wave breaking, the growth rate of bubble densities for various bubble radii are different. Bubbles with radii from 30 to 300 // are observed to have higher growth rates than those bubbles with radius either larger or smaller than this range. For larger bubbles (r > 300 fi), the rapid loss of the larger bubbles is visible by observation both in the field (above and below the surface) and in the laboratory, because these larger bubbles with corresponding larger buoyancy rise quickly to the surface and disappear. For smaller bubbles (r < 30 /x), the physical explanation why their growth is slower than for slightly larger bubbles is more complicated. However, in laboratory experiments using a small jet (with diameter < 1 mm) from a syringe directed onto the surface of a salt-water tank with the jet angle larger than the critical angle of about 30°, most bubbles generated and visible to the eye are observed to be larger than 50 /x (about 1/20 of the jet diameter). Based on these observations from laboratory experiments, the surface tension of the salt water surface appears to cause a straining force against the impinging jets of breaking waves and consequently causes air entrainment into the surface in such a way that only bubbles with radii > 40 (J, to 50 \i are produced at the moment of wave breaking. The high surface tension may prevent the formation of smaller bubbles. After bubbles are generated, their air content will be lost by dissolution into the surrounding salt water and they will thus become smaller and smaller in time prior to reaching the surface by buoyancy and bursting into micro-droplets in the air. This suggests that larger bubbles (r > 50 fi) are generated by direct air entrainment of wave breaking while smaller bubbles (r < 30 /x) are produced by subsequent dissolution of those larger bubbles.


Some fraction of small bubbles will disappear entirely by the dissolution process. However, as the salt water medium is saturated with atmospheric gases (O2, N2, and CO2), the bubble dissolution process may be suppressed to such a low rate that bubbles with radii < 30 fi to 40 /i cannot be reduced further and will remain in the water for a long time. Surfactants, if present, may further reduce the gas dissolution rate across air-water interfaces. In coastal regions where sufficient surfactants are present (due to biological activities, oil spills, and/or land-borne sources), the bubble dissolution process may be completely stopped. Bubbles with radii about 10 /x or less may remain suspended for long times as passive particles.

Inside the surf zone, bubbles are generated by breaking of both swells (more or less constantly) and locally generated seas (whenever storms are present). Thus, the bubble size density in the surf zone does not respond as quickly to the wind speed increase as does the bubble size density offshore, where the contribution from surf zone wave breaking is slight. On the other hand, the major feature of bubble density in the surf zone is its higher intensity due to breaking of ubiquitous swells both from distant (stormy) sources as well as nearshore wave growth.

Bubble size densities were found to have a maximum near radii 20 /i to 30 fi by an optical technique in a North Sea tower in the 1985 winter season and in other investigations (Su et al., 1988; Terrill and Melville, 1999). Some other investigation did not find this maximum (Breitz and Medwin, 1989; Su and Cartmill, 1994b; and others). If bubble sensors are limited in their measurement to radii 40 \i or larger, the bubble size density may be inferred to have a maximum near 40 /J to 50 \i. Near the ocean surface, bubbles with radii smaller than 15 /x which is the lower bound of current acoustic resonator systems operating to 200 kHz are expected. Based on the bubble density data presented earlier, the bubble size densities at radius 10 fi are likely to be an order of magnitude higher than at 17 /i. This contradicts earlier estimates (Su et al, 1988; Farmer and Vagle, 1989; Phelps and Leighton, 1998; and others) that the bubble size density has a peak near 30 /z-50 /j, radius. These apparent differences are not caused by the resolution of the acoustic resonators but rather are related to different wave growth and decay conditions under which different bubble densities are measured.

Other factors which must be considered, include the relative influence of swells versus local seas, stages of local wind and wave growth, and the local bathymetry that controls the location and intensity of wave shoaling and wave


breaking. Effects from all these environmental factors are often difficult to sep

arate at the current stage of our measurement capability and our understanding

of wave breaking dynamics and bubble dynamics in the nearshore region.

In the past 20 years, we have developed a bet ter understanding of wave

breaking dynamics and bubble dynamics in both deep and shallow coastal wa

ters through many advances in sensor development and several large-scale field

experiments as well as theoretical/numerical modeling. Because of small-scale

variability, a large number of sensors and dense sensor array configurations

are needed in the littoral zone. For instance, given the somewhat irregular

bathymetry even for the relatively straight coastline near the F R F location

at Duck, NC, several hundred bubble sensors would be needed to address the

detailed spatial changes in the wave breaking and bubble fields. Hence, there

is still a long way to go before the degree of understanding of bubble dynamics

reaches tha t of surface wave dynamics.

A c k n o w l e d g m e n t s

The senior author (MYS) would like to acknowledge the generous support by

ONR/NRL-SSC in the past 15 years for breaking wave and bubble investiga

tions. Much of what has been learned in the field and reviewed in this article

was obtained through long time continuous support . The senior author would

also like to thank Mr. Ray Burge of NRL-SSC for his very capable assistance

in bubble sensor construction, testing, and field experiments over the past

10 years.

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SIMULATION OF WAVES IN HARBORS USING TWO-DIMENSIONAL ELLIPTIC EQUATION MODELS

VIJAY PANCHANG and Z. DEMIRBILEK

Ports and harbors are the center of social, cultural, economic, governmental, military activities and are closely tied to the national economy of the continental United States. Many local and state economies are dependent on waterborne commerce as a major source of transport and enhanced port capacity is vital to Nation's economy, trade and commerce. Several major U.S. ports/harbors currently have renovation plans in response to the expansion of ocean-borne world commerce and coastal engineering projects (dealing with wave agitation in harbors caused by expansion (landfills), channel deepening and widening, flaring channels for improved navigability, channel maintenance and other infrastructure modifications) generally require a detailed knowledge of the wave field in the project areas. Physical and numerical model studies are often conducted concurrently for these projects to evaluate technical feasibility and to optimize design alternatives. This paper provides a comprehensive review of mathematical modeling procedures developed in recent years in the area of elliptic wave equations suitable for simulating waves in ports and harbors. Modeling techniques and extensions of the well-accepted mild-slope wave equation to include steep-slopes, realistic boundary conditions, dissipative mechanisms like friction and breaking, wave-wave and wave-current interactions are discussed. Assumptions such as constant water depths outside the modeling area and fully reflecting exterior coastlines, which plagued earlier elliptic models, have been eliminated in recent treatments of the open boundary. The development of several improved boundary conditions that can be used along coastlines, islands and structures (jetties, breakwaters, etc.) in the modeling domain is discussed. These improvements in the boundary conditions, along with the inclusion of dissipative effects like breaking, allow for more accurate treatment of the scattered and reflected waves throughout the model domain and lead to a realistic representation of waves in complex regions of highly varying bathymetry and boundary types. We describe advances pertaining to the computing efficiency of elliptic wave models which has until recently been a major drawback for these models. Using advanced parallelization schemes, we have been able to reduce the computational time for prototype-scale applications to the order of a few seconds for monochromatic waves and to less than one hour for the simulation of multi-directional irregular sea states. We present a computational framework for including wave-wave and wave-current interactions in elliptic wave modeling. Application of elliptic modeling methods to wave transformation in the Los Angeles/Long Beach Harbor complex and in Barber's Point Harbor are described. Some research areas have been identified.

125

126 V. Panchang & Z. Demirbilek

1. Introduction

Ports and harbors play a vital role in the growth and well-being of many nations. They are important hubs for commercial, military and recreational activities. For instance, about $600 billion in foreign trade passed through ports in the United States in 1997; this trade is projected to triple by 2020 (YOTO, 1998). Engineers must provide the infrastructure that can handle this growth. Harbor facilities must accommodate ever larger ships ("megaships") with increasingly demanding schedules and complex environmental regulations. It is therefore critical that these facilities be designed in a manner that enhances efficiency and safety of harbors operations like cargo loading/unloading, etc.

One of the physical features that can have an adverse impact on harbor operations is the wave climate. For example, some waves (not necessarily big waves) lead to undesirable vessel motions (resulting in operational difficulties such as broken mooring lines, downtime for cargo handling, etc.) or undesirable sediment movement (resulting in more frequent dredging). Reliable estimation of wave conditions in and around a harbor is vital to the success of harbor operations. This estimation must often be accomplished through mathematical modeling techniques. However, most harbors confront the modeler with numerous complexities. Geometrically, as may be seen in Fig. 1, the domain to be modeled may include completely arbitrary coastline shapes and bathymetric features as well as man-made structures like piers, jetties, breakwaters, etc. These features induce wave refraction, diffraction, reflection and dissipation by friction and breaking to varying degrees. The incident waves of interest may cover a wide spectrum, from very short waves to extremely long period waves that cause resonance and may approach the harbor from any direction. For short waves, the number of grids needed to discretize the domain can be extremely large, making the modeling difficult. Longer waves may need fewer grids, but require a better specification of boundary reflectivities since they are more susceptible to reflections in all directions from structures, coastlines, and bathymetric slopes. In addition to these complexities, the modeler may also have to account for the effects of the interaction between various wave components and of tidal or other currents which can magnify or diminish the wave climate in different parts of the domain.

In this paper, we describe a methodology that has become well-accepted in recent years for modeling the situation described above (See Panchang et al, 1999 for a review of recent coastal wave modeling methods). In its basic form, the methodology is based on solving the following two-dimensional

Simulation of Waves in Harbors 127

Fig. 1. Los Angeles/Long Beach Harbor area. Numbers show gage locations for hydraulic model study (after Seabergh and Thomas, 1995).

elliptic equation, v-(ccgV(f>) + k2ccg4> = o, (i)

where,

<f>(x, y) = complex surface elevation function (= <pi + ifa)

i = y/^1

a = wave frequency under consideration

C(x, y) = phase velocity = a/k

Cg{x,y) = group velocity = dcr/dk

k(x, y) = wavenumber (= 2ir/L) related to the local depth d(x, y)

through the dispersion relation,

a2 = gk tanh (kd). (2)


The wave height H can be obtained from complex surface elevation function 4> as follows,

2a tf = y V ^ + ^ ) - (3)

Essentially Eq. (1) represents an integration over the water column of three-dimensional Laplace equation used in potential wave theory. The integration, originally described by Berkhoff (1976) and Smith and Sprinks (1975), is necessary because the solution of the three-dimensional problem is computationally difficult for harbors with a characteristic length that is several times of the wavelength. The integration is based on the assumption that the vertical variation of the wave potential is largely the same as that for a horizontal bottom, i.e.,

rtW)«<^dfc+*) #*,»)• (4)

This approximation is obviously valid for a "mild slope" characterized by \Vd\/kd « l , a criterion that is usually met in practice (extensions to steep slopes are described later). Being elliptic, Eq. (1) represents a boundary value problem which can accommodate internal nonhomogeneities. It hence forms a widely-used basis for performing wave simulations in regions with arbitrarily-shaped (manmade or natural) boundaries and arbitrary depth variations. Unlike "approximate" mild slope wave models (e.g., REFDIF and RCPWAVE described by Dalrymple et al, 1984; Kirby, 1986; Ebersole, 1985), there are no intrinsic limitations on the shape of the domain, the angle of wave incidence, or the degree and direction of wave reflection and scattering that can be modeled with Eq. (1). In essence, Eq. (1) represents the complete two-dimensional wave scattering problem for the nonhomogeneous Helmholtz equation as demonstrated by Radder (1979). While it is valid for a monochromatic (single incident frequency-direction) wave condition, irregular wave conditions may be simulated using Eq. (1) by superposition of monochromatic simulations.

For further development in this paper, we may consider the following extended form of Eq. (1),

V-(CCgV(/>) + {k2CCg + iCgaW)(t> = 0, (5)

in which a dissipation term W has been included. By separating the real and imaginary parts of Eq. (5), Booij (1981) has shown that Eq. (5) satisfies the energy balance equation in the presence of dissipation. The term W may represent breaking and/or friction and is described later.


Several computational models based on Eq. (1) or Eq. (5) have been developed in recent years. These models differ in the choice of the numerical method used (e.g., finite-difference method, boundary element method, finite element method), in the choice of boundary conditions, in the method used to solve the linear system of equations that results from discretizing the elliptical governing equation, and in the inclusion of additional mechanisms. In this paper, we provide a review of the modeling techniques pertaining to the application of Eq. (1) to harbors. The layout of this paper is as follows. Sections 2 and 3 describe the various kinds of boundary conditions and numerical solution methods that have been developed in recent years. Section 4 describes the incorporation of additional mechanisms in Eq. (5). The application of a comprehensive finite-element model to simulate waves in the Los Angeles/Long Beach harbor complex (Fig. 1) and in Barber's Point Harbor is described in Sec. 5.

2. Boundary Conditions

Domains on which the elliptic equation Eq. (5) is solved are enclosed by closed boundaries (represented by coastlines and surface-penetrating structures like pier walls or pier legs, breakwaters, seawalls, etc.) and open boundaries (which represent an artificial boundary between the area being modeled and the sea region outside). A separation between the model domain and an outer water area from where no waves enter the model domain (e.g., a creek or tributary at the backbay or down wave end of the domain) may be considered to be a fully-absorbing closed boundary. An open boundary is considered to be the one where an incident wave is specified (and may contain other radiated waves). Along all boundaries, appropriate conditions must be specified to solve Eq. (1); however, even in the best of circumstances, only approximate boundary conditions can be developed (e.g., see Dingemaans, 1997).

2.1. Closed boundary conditions

Along coastline and surface-protruding structures, the following boundary condition has traditionally been used (e.g., Berkhoff, 1976; Tsay and Liu, 1983; Tsay et al., 1989; Oliveira and Anastasiou, 1998; Li, 1994a),

d<j> (6)


where n is the outward normal to the boundary and a is related to a user-specified reflection coefficient as follows,

Kr varies between 0 and 1 and specific values for different types of reflecting surfaces have been compiled by Thompson et al. (1996).

It may be verified that Eq. (6) is strictly valid only for fully-reflecting boundaries (i.e., Kr — 1). For partially reflecting boundaries, it is valid only if waves approach the boundary normally. For other conditions, Eq. (6) is approximate and may produce distortions in the model solutions. These limitations may be eliminated by describing the solution at the boundary more fully as a sum of incident and reflected waves,

4> = A{exp[ik(n cos 9 + ssin9)} + Krexp[ik(—ncos9 + ssin9 + /?)]} , (8)

where A is the amplitude of the approaching waves, 9 is the direction at which they intersect the boundary (9 = 0 for normally incident waves), s is the coordinate along the tangent to the boundary, and (3 is a phase shift between the incident and the reflected wave. Equation (8) leads to the following boundary condition,

^ = ikcosel-feM;1:%. (9) dn l + Krexp{ik/3y y '

Unfortunately, 9 and (3 are not known a priori inside the model domain and must be estimated by approximation. For fully absorbing boundaries (Kr = 0), Li and Anastasiou (1992) and Li et al. (1993) have used Eq. (9) after estimating 9 from Snell's Law and the deep-water incident wave angle. Alternatively, Isaacson and Qu (1990) estimated 9 as follows,

9 = avctan{(dx/ds)/{dX/dn)} , (10)

where x ls the argument of the complex quantity </> (i.e., the phase of (f>). For implementation, they first used Eq. (6) as a boundary condition, obtained x from the results, determined 9 from Eq. (10), used Eq. (9) as a boundary condition to perform a second iteration of the model, recalculated x a n d 9, performed a third model iteration using Eq. (9) and so on. Like Pos (1985), they assumed (3 = 0 while using Eq. (9), based on limited numerical tests that showed little sensitivity to f3. Clearly, like the Snell's Law approach,


Eq. (10) is valid only for Kr = 0 (although problems with nonzero Kr were also considered). To include the effect of the reflected waves (i.e., the second term in the right hand side of Eq. (8)), Isaacson et al. (1993) suggested estimating 9 as follows,

9 = (1/fc) arcsin {dX/ds} . (11)

Again, an iterative method with repeated model calculations were needed. Steward and Panchang (2000) analyzed these methods and noted difficulties with convergence of the above iterative methods and with the quality of the solutions obtained with Eqs. (10) and (11). They were able to eliminate these difficulties by estimating 9 from the following expression,

n dx/fdx 2Krk(cos(k/3) + Kr) ...\ tan0=-^/ - ^ + —±—K-j-r7r, ~ cos(9) ) . (12)

dsl \dn l + 2Krcos(kf3) + K? K>) y '

Equation (12) is a generalization of Eq. (10) that allows nonzero Kr and /3. For a detailed comparison of results, see Steward and Panchang (2000). As an example, Fig. 2 shows a simulation of waves propagating into a rectangular harbor area obtained with Eq. (12). Unlike the results demonstrated by Steward and Panchang (2000) and by Beltrami et al. (2000) using other boundary conditions, Fig. 2 contains no spurious oscillations or noise.

Despite the increasing sophistication seen progressively in Eqs. (6) and (9) and in the various ways of estimating 9, some fundamental problems remain. The most important one is inherent in Eq. (8), i.e., the assumption that the total wave field near the boundary can be represented either by one set of plane waves (in the case of Eq. (10) or the Snell's Law approach of Li et al. (1993)) or by two sets of plane waves (in the case of Eqs. (11) and (12)) propagating in constant depth. In domains of complex shapes (as in Fig. 1) with arbitrary bathymetry and boundaries with varying reflectivities, a complex pattern of waves can result; simple wave trains are not easily discernible and, as noted by Isaacson and Qu (1990), the definition of a single 9 (and (3) can become meaningless. Further, even when there is a well-defined train of waves near the boundary (justifying the use of the above methods), precise estimation of Kr and /3 is still problematic. Values of KT provided by Thompson et al. (1996) certainly do not cover full range of reflecting surfaces that the modeler encounters, nor do they cover the dependence of these parameters on the incident wave frequency. Efforts to incorporate the work of Dickson et al. (1995) and Sutherland and O'Donoghue (1998) pertaining to (3 in models such as the one described here are lacking. In some ways, it may be best to


Fig. 2. Modeled waves in a rectangular harbor, using Eqs. (9) and (12). Phase diagram (shows cosine of phase angle). Semicircle represents open boundary. Kr = 0 on closed boundaries.

recognize these difficulties at the outset and use the simplest expression of Eq. (6) by combining all the uncertainties noted above into a single parameter a which may be regarded as a tuning parameter.

2.2. Open boundary conditions

Along the open boundary, an incident wave fa must be specified. Along this boundary, however, waves backscattered from within the domain will also exist and their magnitude is generally not known. In the context of simple rectangular domain models with one side (aligned, say, in the y direction) constituting the open boundary, Panchang et al. (1988, 1991), Li (1994a, 1994b), and Oliveira and Anastasiou (1998) have used the following condition,

^ = ik(2<f>i - 4>) • (13)


Equation (13) is obtained by assuming that the incident and backscattered components along this boundary can be described by fa = Ai exp(ikx) and fa = B exp(—ikx) respectively (where Ai is the (specified) amplitude of the incoming wave and B is an unknown), adding the two components and differentiating. Obviously, this is valid only if the incident and backscattered waves near the boundary are plane waves propagating in the +/— x direction.

For more complex domains involving multidirectional scattering, Eq. (13) is inappropriate. Harbor applications generally use model domains such as that described in Fig. 3 where the semicircle is used to separate the model area from the open sea. In the exterior domain f2', the potential <j> is comprised of three components,

4> = fa + fa + fa, (14)

where fa = the incident wave that must be specified to force the model, fa = a reflected wave that would exist in the absence of the harbor and

Transect 2 Incident Wave P 2 e;

Transect 1 Pi

Fig. 3. Harbor wave model domain; definition sketch.


4>s = a scattered wave that emanates as a consequence of the harbor and must satisfy the Sommerfeld radiation condition. With appropriate descriptions for these components, a boundary condition can be developed along the semicircle.

In traditional harbor models (Mei, 1983; Tsay and Liu, 1983; Thompson et al., 1996; Chen and Houston, 1987; Xu and Panchang, 1993; Demirbilek and Panchang, 1998), the exterior wave conditions are described as follows,

4>i = Ai exp[ikr cos(8 — #*)], which is the specified input, (15)

4>r = At exp[ifcr cos(6> + 0*)], (16)

OO

<t>„ = ^2 Hn(kr)(An cosnO + Bn sinnO), (17) n=0

where (r, 9) denotes the location of a point in polar coordinates, Hn is the Hankel function of the first kind and order n, and An and Bn are unknown coefficients.

For the specified incident wave field given by Eq. (15), Eqs. (16) and (17) result from the solution of the relevant eigenvalue problem in the traditional method. As demonstrated by Xu et al. (1996), however, this eigenvalue problem in which 4>s and <j)r are coupled, may be solved only under the following conditions,

(i) the exterior region must have a constant depth, (ii) the exterior coastlines Q\R\ and Q2R2 must be fully reflecting and

collinear.

These requirements usually cannot be met in practice where the exterior geometry varies arbitrarily and the unrealistic bathymetric representation used by the modeler invariably has an adverse influence on the solution. In field applications, the exterior bathymetry is irregular and the depth generally increases in the offshore-direction. Condition (i) is thus violated, causing two problems as demonstrated by Panchang et al. (2000). First, the modeler must arbitrarily select a representative "constant" depth and test the sensitivity of the solutions to these depths. This can be extremely time-consuming. Second, the effect of reflections from the sloping exterior bathymetry is ignored. These effects are often significant especially for long periods that are of interest in harbor resonance studies. Condition (ii) is also problematic. Exterior coastlines are not always fully reflecting for all wave conditions and imposing full reflection in such cases yields extremely large amplification factors and


rapid variations in the wave pattern in the outer regions of the domain. (See examples in Xu et al. (1996), Demirbilek et al. (1996) and in Thompson et al. (1996)). One may of course enlarge the interior region in the hope that these effects do not contaminate the results in the area of interest, however, there is no guarantee that these effects are confined to specific regions. In addition, the extra memory requirements and grid-generation for a larger domain are usually exceedingly demanding. Thus, while Eqs. (16) and (17) constitute rigorous solutions of the eigenvalue problem, their use renders the application of harbor wave models problematic in practice. (One consequence of the above is that many of the models in this category cannot correctly simulate fairly simple phenomena like waves approaching a sloping beach. Investing confidence in model results when applied to field situations is therefore difficult).

To overcome these difficulties, Panchang et al. (1993) have described a procedure that requires the exterior domain to be suitably divided into a finite number of regions of constant depths. A boundary integral equation is then developed for each of these exterior regions using the appropriate Green's function. The boundary element formulations for these regions are then matched with each other along the interfaces and with a finite-element network in the model interior to obtain the solution. It was found, however, that this type of model is extremely cumbersome to code and construct for general implementation. Other difficulties may also be expected if mechanisms such as dissipation, wave-current interaction, etc., are to be introduced into the governing equation. Chen (1990) also has suggested discretizing the exterior domain into a finite number of radial "infinite elements" with a prespecified shape function in each element. However, this shape function is entirely dependent on the farfield approximation for the Hankel functions, suggesting that a fairly large computational domain is still needed.

An effective alternative is to use a "parabolic approximation" to describe

where,

d<f>s _ dn

1 2r

i 8kQr2 ' q ~

(18)

2 , (18a) 2k0r

where r and 9 represent the polar coordinates of a point on the open boundary and fco is a representative wavenumber for the open boundary (p and q are not unique and alternative forms, each obtained with an appropriate rationale, have been investigated by Givoli (1991), Xu et al. (1996), Panchang


et al. (2000)). The parabolic approximation, Eq. (18), allows the scattered waves to exit only through a limited aperture around the radial direction. Unlike Eq. (17), it does not rigorously satisfy the Sommerfeld radiation condition. However, using this formulation decouples <f)s from the other components. These components (fa and <j>r) may be obtained by making a compromise between a detailed exterior bathymetric representation (which as noted earlier, is difficult) and the constant depth representation (which is unrealistic). A one-dimensional representation where the depths vary in the cross-shore direction only (Figs. 3 and 4) may be selected. This is reasonable since, in general, this is often the direction in which the depths vary the most. If natural variations do not permit the representation of the exterior depths by only one section, a second one-dimensional section shown as transect 2 in Figs. 3 and 4 may be constructed. For transects 1 and 2 with varying depths, no simple analytical expression (such as Eq. (16)) can be found for the reflected wave (since 4>i a n d 4>r are coupled). However, the quantity,

Pa = <Pi + ' (19)

may be obtained by the solution of the one-dimensional version of Eq. (5) since the depths along these transects vary in one direction only. This

Fig. 4. Two ID transects representing the exterior bathymetry (do not have to be identical).


one-dimensional equation is (Schaffer and Jonsson, 1992; Panchang et al. 2000),

T{CC'T ax \ ax

where, for one-dimensional geometry.

— [CCg-j^] + kCCg(k cos2 0 + iW)ijj = 0, (20)

4>o = ip{x) exp(ifcy sin 9). (21)

Equation (21) is an elliptic ordinary differential equation requiring two boundary conditions. It may easily be solved via a simple finite-difference scheme (for the present, the dissipation factor W is considered to be prespecified). Assuming that transect 1 extends out to a region of constant depth (or deep water), a condition at Pi may be obtained by combining a specified incident wave,

4>i{Pi) = Ai exp(ikxcos6i + ikys'mOi), (22)

(where A^ = a given input wave amplitude) and an unknown reflected wave,

4>riP\) — -Bexp(—ikxcosdi + ikysmOi). (23)

Without loss of generality, the point Pi may be located at x = 0 which allows elimination of B to yield,

- ^ = ik cos 0i (2Ai - V) • (24)

At the coastal boundary point Q\, the partial reflection boundary condition of Eq. (9) may be used in the following form,

dtp _ iy/k2-k2sm2S(l-Kr)

fa - YTK; ^' ( }

where Kr is the reflection coefficient for the exterior coastline (i.e., near Qi), and k sin 6 is constant for one-dimensional problem.

The solution of Eq. (20) using boundary conditions, Eqs. (24) and (25), along with Eq. (21) produces <j>0 along transects 1 and 2. These solutions are denoted by c/>oi and 4>02 • The desired </>o along the semicircle may be obtained by laterally translating </>oi and ^02 via interpolation between transects 1 and 2 as follows,

<j>o = (1 — m)<j>oi exp(—ifc(j— y)sin#) + m0o2exp(ifc(r + y)sin0), (26)


Fig. 5. Modeled wave refraction on a sloping beach; angle of incidence 60°. Phase diagram.

where we have set y = 0 at the center of semicircle, the interpolation function

m = (r — y)/2r, r is the radius of the semicircle, y is the lateral coordinate of

the open boundary node relative to the origin of semicircle (Fig. 3).

The boundary condition for (/> along the semicircle T may be obtained by

using the continuity of the potential (Eqs. (14) and (19)) and its derivative

along with Eqs. (18) and (26),

d</> d4>o . . . . d2<f> (27)

Thus, the solution of Eq. (20) provides </>o along the one-dimensional transects.

These values can be translated laterally and subst i tuted into Eq. (27) to obtain

the open boundary condition for the two-dimensional equation of Eq. (1). Zhao

et al. (2000) and Panchang et al. (2000) have demonstrated tha t this procedure

provides extremely satisfactory solutions for a large number of test cases. An

example of wave refraction along a sloping beach is shown in Fig. 5. The

expected bending of the crests can be observed with no spurious effects.

3 . N u m e r i c a l S o l u t i o n

Equat ion (5) is generally solved using the boundary element method, the finite-

difference method, or the finite element method. In general, finite-difference

discretizations are not well-suited to represent the complex domain shapes de

scribed, for example, in Fig. 1. Not only are the boundaries distorted, but

the number of uniformly spaced grids may also be excessively large (adequate

resolution, typically 10 points per wavelength, demands tha t the spacing be de

termined from the smallest wavelength). Most studies with the finite-difference


method have been limited to largely rectangular domains (e.g., Li, 1994a, 1994b; Panchang et al, 1991; Li and Anastasiou, 1992). Boundary element models can handle arbitrary shapes and require minimal storage since only the boundaries are discretized; however, they are limited to subdomains with constant depths only (e.g., Isaacson and Qu, 1990; Lee and Raichlen, 1972; Lennon et al, 1982). Finite element models, on the other hand, allow the construction of grids with variable sizes (based on the local wavelength) and give a good reproduction of the boundary shapes. Most finite element models (e.g., Tsay and Liu, 1983; Tsay et al, 1989; Kostense et al, 1988; Demirbilek and Panchang, 1998; Panchang et al., 2000) have used triangular elements, and modern graphical grid generating software permits efficient and accurate representation of harbors with complex shapes. For example, the Surface Water Modeling System described by Zundell et al. (1998) and Jones and Richards (1992) can be used to conveniently generate as many as 500,000 elements of varying size, based on the desired (user-specified) resolution, and to specify the desired reflection coefficients on various segments of the closed boundary. The solution of Eq. (1) by the finite element method is described in detail by Mei (1983) and by Demirbilek and Panchang (1998) when different types of open boundary conditions are used.

Whether one uses finite differences or finite elements for discretization, the numerical treatment of Eq. (1) with appropriately chosen boundary conditions leads to system of linear equations,

[AP] = [B], (28)

where [</>] represents the vector of all the unknown potentials. For solving Eq. (5), a similar system results as long as W is prespecified. The matrix [A] is usually extremely large. In earlier models (e.g., Tsay and Liu, 1983; Tsay et al., 1989; Chen, 1990; Chen and Houston, 1987), the solution of Eq. (28) was accomplished by Gaussian Elimination which requires enormous memory and is prohibitive when the number of wavelengths in the domain is large (i.e., short waves or a large domain). Pos and Kilner (1987) were able to alleviate this difficulty somewhat by using the frontal solution method of Irons (1970).

In recent years, the solution of Eq. (28) has been obtained with minimal storage requirements for [A]. This is due to the development by Panchang et al. (1991) and Li (1994a) of iterative techniques especially suited for Eq. (1). These techniques, based on the conjugate gradient method, guarantee convergence and have been found to be extremely robust in a wide variety


of applications involving both finite differences and finite elements for several kinds of boundary conditions. Hurdle et al. (1989) have used the biconjugate gradient algorithm. This variation is efficient when it works but (as noted by Kostense et al., 1997) it does not guarantee convergence for this type of governing equation. More recently, Oliveira and Anastasiou (1998) explored the use of the Generalized Minimum Residual method and the Stabilized Biconjugate Gradient method and reported greater efficiency with finite-difference models based on Eq. (1). With a finite-element formulation, however, Zhao et al. (2000) found that the GMRES method of Oliveira and Anastasiou (1998) failed to converge whereas their latter method yielded erratic efficiency.

Alternative solution techniques have also been explored by Li and Anastasiou (1992) who first express <p as exp(^x) and obtain a new equation for /J,. Since /x is sometimes a less rapidly varying function than 4>, Li and Anastasiou (1992) suggest that as few as 2 or 3 grid points per wavelength will suffice. However, as noted by Li and Anastasiou (1992) and by Radder (1992), the presence of rapidly varying topography or of reflections in various directions will necessitate much finer resolution (say 10 points per wavelength) and the solution obtained by using the log of the potential may lead to excessive smoothing. For mildly varying bathymetry with low reflections, it may in any case be more efficient to solve instead of the "parabolic approximation" of Eq. (1) which is intended for such applications. Li and Anastasiou (1992) have used the multigrid method to minimize storage problems. When higher resolution is required, though, this method does not offer any significant advantage over the conjugate gradient schemes described by Li (1994a) and Panchang et al. (1991) since at least one grid with the desired resolution must be constructed. Further, the multigrid method is best suited to rectangular finite-difference discretizations.

Another method, proposed by Li (1994b), involves solving the following parabolic equation,

a^ = V • (CCgV<f>) + k2CCg<P, (29)

where a is a constant. Equation (29) is an approximation of the time-dependent hyperbolic wave equation associated with Eq. (1). It is solved by marching forward in time until steady state is reached. Equation (29) is similar to the heat equation and standard techniques (e.g., the ADI method) for solving such equations are used by Li (1994b). It must be noted, though, that the elimination


of equations. For conjugate gradient solvers, 90% of the CPU time is spent on matrix-vector products and inner product kernels. Therefore, OpenMP (OARB 1997) may be used to parallelize the kernels. Two-level parallelization schemes can use OpenMP to accelerate the solution for each component and MPI to simultaneously obtain solutions to multiple incident wave components. More details regarding parallelization schemes for harbor wave models may be found in Bova et al. (2000) who report a reduction in run times by a factor of 250-580 compared with serial codes for an application in Ponce de Leon Inlet (Florida). A problem with nearly 300 input spectral components was solved on a 25 square km domain containing 235,000 nodes in 72 hours. An example from Bova et al. (2000), shown in Fig. 6, suggests that the simulation produces, qualitatively, a sea-surface that looks realistic. Model results for this site are discussed in greater detail by Zhao et al. (2000).

4. Incorporation of Additional Mechanisms

As noted earlier, Eq. (1) incorporates the effects of refraction, diffraction and reflection induced by any nonhomogeneity in the model domain. Equation (5) is an extension of Eq. (1) that includes, in addition, the effects of friction and wave breaking. Similar extensions are possible to include the effects of wave-current interaction, wave-wave interaction and of steep slopes. The modeling of these mechanisms in the context of the elliptic equation, Eq. (1), is described in this section.

4 .1 . Dissipation

In Eq. (5), W represents the combined effects of friction and breaking which may be separated as follows,

W = w/Cg+j, (30)

where w is the friction coefficient defined by Dalrymple et al. (1984) and 7 is a breaking factor. These coefficients are empirical and parametrizations for these have been described by Dalrymple et al. (1984), Tsay et al. (1989) and Chen (1986) for friction, and by Battjes and Janssen (1978), Dally et al. (1985), Massel (1992), Chawla et al. (1998) and Isobe (1999) for breaking. Some of these parameterizations have been extensively validated against field data (e.g., Larson, 1995; Kamphuis, 1994). We do not repeat the parameterizations here; rather, we note that they are all dependent on the wave amplitude.


Published studies demonstrating the effects of friction in harbor models (e.g., Chen, 1986; Tsay et al, 1989; Demirbilek and Panchang, 1998; Kostense et al, 1986) have estimated w on the basis of the incident wave amplitude. It is then easy to pre-specify w while solving Eq. (5). These studies appear to show that friction can change the magnitude of resonant peaks in harbor models quite substantially; at other frequencies, the effect seems to be minimal. Jeong et al. (1996) and Moffatt & Nichol Engineers (1999) have attempted to utilize W to include harbor entrance losses, however, no details regarding the modeling technique are presented.

In general, however, since both w and 7 are functions of the wave amplitude which is unknown a priori inside the domain, their inclusion makes the problem nonlinear and requires iteration. For the first iteration, W is set equal to 0 and Eq. (1) is solved (e.g., nonbreaking solutions are obtained). The resulting wave heights are used to estimate W via the parameterizations for w and 7 and Eq. (5) is solved. The process is repeated until convergence is obtained.

Since dissipation (especially breaking) occurs outside the computational domain also, open boundary conditions like Eqs. (13), (15) and (16) may not be appropriate. Inclusion of breaking inside the domain and its exclusion in the exterior descriptions create artificial discontinuities along the open boundary, especially in shallow areas, and consequently, spurious effects would propagate into the model domain. In this event, Eq. (20) is a more appropriate description of the exterior and may be used to develop the necessary boundary conditions. A digitized bathymetry file is used to obtain the depths d(x) along transect 1. These depths are interpolated onto uniformly spaced nodes and the wave properties C, Cg and k are calculated. Equation (20) may be easily solved by finite-differences using boundary conditions, Eqs. (24) and (25). Again, iterations are required because W is not known initially. When the solutions converge, the procedure is repeated for transect 2. These converged solutions of Eq. (20) along transects 1 and 2 include, albeit in a one-dimensional sense, the effects of dissipation and hence constitute more appropriate forcing functions than Eqs. (13), (15) and (16) do. 0o along the semicircular open boundary is obtained via Eq. (26).

Performing nonlinear iterations within the model domain as W varies from iteration to iteration can be time-intensive. We have explored the possibility of combining the iterative conjugate gradient methods for the linear system with the iterations required for the nonlinear modeling, i.e., the conjugate gradient iterates obtained while solving Eq. (28) were perturbed by upgrading W


periodically. Unfortunately, this maneuvering destroys the robust convergence properties of the conjugate gradient solvers for Eq. (28). At present, each linear system, for a specified W, must be completely solved until convergence is obtained and the whole procedure repeated with a new W. More effective methods to accelerate the solution need to be developed.

Zhao et al. (2000) developed a finite-element model using Eqs. (6), (20) and (27) to formulate the boundary conditions and applied it to several tests involving breaking. These tests involved a sloping beach, a bar-trough bottom configuration, shore-connected and shore-parallel breakwaters on a sloping beach, and two field cases in the North Sea and Ponce de Leon Inlet (Florida). Five breaking formulations, given by Battjes and Janssen (1978), Dally et al. (1985), Massel (1992), Chawla et al. (1998) and Isobe (1999), were examined. They found that the Isobe (1999) criterion was difficult to use within the context of the elliptic model and that the absence of a lower breaking limit generally contributed to excessive dissipation (compared with data) in the Chawla et al. (1998) and Massel (1992) formulations. In general, the formulations of Battjes and Janssen (1978) and Dally et al. (1985) were found to be the most robust from the point of view of incorporation into an elliptic model based on Eq. (5) and to provide excellent results compared to data.

For simulations involving several spectral components, Zhao et al. (2000) examined two approaches. In the first approach, complete simulations were made one at a time for all monochromatic components where the amplitude of each component was used in the relevant breaking formula. The results were subsequently assembled using linear superposition. They found that this approach led to some overestimation compared with data; this was attributed to the individual component amplitudes being too small to induce breaking in the model. In view of this overestimation, a second approach was considered where the breaking factor was calculated on the basis of the significant wave height instead of the component wave height. This approach eliminates the independence of individual component simulations, thereby changing the overall model numerics. With the second approach, one round of nonlinear iterations for all components must be performed, the significant wave height is calculated at each grid point and this larger wave height is used to estimate the breaking factor (Chawla et al, 1998). This approach led to the initiation of breaking occurring further offshore in the case of their simulations of wave transformation around Ponce de Leon Inlet. An example of the model simulations near the US Army Field Research Facility at Duck, North Carolina, is shown in Fig. 7 for an input wave condition given by a significant wave height of 2.3 meters.


Fig. 7. Modeled wave amplitudes (m) at FRF Duck. Top, no breaking; Bottom, breaking based on significant wave height.

A complex pattern of waves is created in the middle of the domain due to the complicated bathymetry. Clearly, breaking plays an important role (the dots aligned in the shore-perpendicular direction in the middle of these figures represent circular piles which were assigned full reflection). These simulations were performed with 208 spectral components for a storm in 1996. The


simulations took 265 hours CPU time using 41 processors and ran nonstop for about 3 days on the US Army Corps of Engineers super-computer. Comparison to field data and other details are provided elsewhere.

4.2. Wave-current interaction

Many coastal regions experience high background currents. Wave propagation is influenced by these currents (e.g., waves opposing the currents become larger and vice-versa). Based on the derivation by Kirby (1984), one may incorporate currents in Eq. (5) as follows,

V • (CCgV<t>) - V • (U(U • V $ ) + (^-a2 + iCgarW + a2 - a2 + iaV • U

<f> + 2ia\J • V(j> = 0 , (31)

where \J(x, y) = current vector (provided by a flow model), oy = a — k • U = relative frequency, {a2 = gk tanh (kd); C — ar/k). Several sophisticated hydrodynamic models are available nowadays for obtaining the desired flowfield information TJ(x,y). When hydrodynamic models provide three-dimensional flowfields, the vertical dependence may be removed for use in Eq. (31) via the "equivalent uniform current" defined by Hedges and Lee (1992); this quantity is obtained by vertically averaging the current over a depth eL over the water column where eL = (1/k) tanh (fed).

The generalized mild-slope wave equation, Eq. (31), is still elliptic and may be solved by the techniques noted previously. For a prespecified U(x, y), a linear system of equations like Eq. (28) results. However, to compute the Doppler shift in the wave frequency (oy = a — k • U), the wave vector k is needed. While the magnitude of k is known a priori from the dispersion relation, Eq. (2), its direction is not. This problem may be resolved by first solving Eq. (31) without the effects of wave-current interaction, obtaining an estimate of the local wave direction, computing the relative frequency ar = a — k • U, solving Eq. (31) again, revising the wave direction, and repeating until the model runs converge. Such an approach has been taken by Kostense et al. (1988); Li and Anastasiou (1992), however, prefer not to calculate the direction of k in view of the computational burden. While their results for one test-case (pertaining to waves approaching a rip current on a sloping beach) are reasonable, iterations are indeed necessary for complex flowfields.

Further difficulties arise in the specification of open boundary conditions. Published studies using the elliptic model of Eq. (31) have assumed that


currents are absent on this boundary. Research is needed to develop open boundary conditions that include the effects of currents (an extension of Eq. (20) is possible for the case of currents varying only in the cross-shore direction outside the computational domain).

4 .3 . Wave-wave interaction

By including most of the nonlinear terms in the vertical integration of the three-dimensional Laplace equation, Kaihatu and Kirby (1995) obtained an extension of Eq. (1) that incorporates wave-wave interactions. Expressing the potential in terms of harmonics as:

N

where,

<p(x,y,z,t) - ^2fn(kn,d,z)(j)n(kn,ujn,x,y,t). 71 = 1

_ coshkn(d + z) In —

(32)

cosh knd

and performing an integration over the vertical modifies Eq. (5) as follows,

Vfc • [(CCg)nV<l>n\ + kl(CCg)n<pn+i(CgaW)n<t>n

" r a - l

(33)

J2 2(TnV<f>i • Vcf> n— 1 "i" Gl<j>l^ $11—1 "i" &n— l^n — 1 ^ n

1=1

<Tj<Tn_i(Tn 2 2

5 (<7( + c r ; c r n - l + <Tn_i)<pl<Pn-l

N-n

^ 2cr„V<?!>,* • Vcj>n+i + an+i4>n+iV24>* - (Ti(j>*V24>n+i

L i=i

2 (°"/ ~ <Tl<Tn+l + <Tn+im <Pn+l

complex conjugate value of <

(34)

in which q Similar equations were also derived by Tang and Ouellet (1997) who have

further demonstrated that this type of extension provides the governing equation, Eq. (1), with the same level of nonlinearity as that in Boussinesq wave models. This is particularly noteworthy since models based on Eq. (1) or


/ / / / f / / 0.45 ( 0.35 ( 0.25 ( 0.15

\ \ V \ \ \. \.

A = 0.98 cm

T = 3s

t 0.46 m

0.15 m

±_

(a)

Model • Data

0.02

0.015

•S 0.01

0.005

0

0.02

0.015

§ 0.01

0.005

0

Second Harmonic

•

Model • Data

20 25

- Third Harmonic

=»£***

— Model • Data

10 15 x(m)

20 25

(b) Fig. 8. (a) Bathymetry (m) of Whalin (1971) used for wave-wave interaction study, (b) wave height comparison for wave-wave interaction.


Eq. (5) simultaneously offer the computational stability and the advantages of finite-element gridding in harbors and complex coastal areas (which Boussi-nesq models sometimes lack). From the perspective of the solution technique, the coupling of harmonics represented by the right hand-side of Eq. (34), if prespecified, leads to a linear system of equations like Eq. (28). However, for a given component <j)n, the other components constituting the right hand side are not known a priori. Again, an iterative technique must be used where the values from the previous round can be used to calculate the right hand side. Figure 8 shows a finite element model simulation (based on Eq. (34)) of wave propagation and interaction over the "tilted cylinder" bathymetry of Whalin (1971). The results match the laboratory data of Whalin (1971) very well and shows that higher harmonics can build up from zero to a magnitude similar to the linear solution and can hence contribute much to the overall solution (hitherto the coupling represented by the right hand side was included only in simple (parabolic approximation) models; e.g., Tang and Ouellet, 1997 and Kaihatu and Kirby, 1995).

4.4. Combined nonlinear mechanisms

Equation (34) contains the effects of two of the additional mechanisms described so far, i.e., wave-wave interactions and dissipation. By rederiving Eq. (34) on a moving frame of reference, the equation may be extended to simultaneously include the effects of wave-current interaction also (Kaihatu and Kirby, 1995). As demonstrated above, the modeling procedures for each of these mechanisms are individually nonlinear and require numerical iterations. However, combining all the nonlinear effects in numerical simulations has as yet been unexplored. An efficient model must juxtapose iterations and also assure convergence. Further, appropriate tests for the enhanced model are not readily available (especially for the combination of wave-wave and wave-current interactions). For systematic model verification, data isolating and combining these mechanisms are needed. In that regard, the numerical advancements appear to be preceding data availability.

4.5. Steep-slope effects

Unlike the inclusion of the nonlinear mechanisms described above, overcoming the "mild slope" requirement discussed in Sec. 1 is relatively easy. Mas-sel (1993), Porter and Staziker (1995), Chamberlain and Porter (1995), and


Chandrasekera and Cheung (1997) developed extensions of Eq. (1) to include steep-slope effects. Their extensions may be described by the following equation,

V • (CCgV(f>) + {k2CCg + dx (V/i)2 + d2V2h)(j> = 0, (35)

where d\ and d2 are functions of local depths. Reference may be made to these publications for the various definitions of d\ and d2; in general, though, differences in the proposed definitions of these functions impact model results to a very small extent. The steep-slope terms are fairly straightforward to include in the model because they are linear. Further, they have the advantage of being "automatic", i.e., they have little contribution for mild slopes, do not change the solution technique and the additional computational demand is negligible. However, steep slopes lead to breaking and model performance in the vicinity of steep slopes will involve iterations (an analytical model has been developed by Massel and Gourlay (2000) to include breaking and steep-slope effects near coral reefs).

5. Application to Harbors

So far, we have described various developments made in recent years to construct more reliable models based on Eq. (1) for use in domains with arbitrary shape and bathymetry. In this section, we describe application of one such model to the practical problem of simulating harbor resonance in the Los Angeles/Long Beach Harbor complex (Fig. 1) and in Barber's Point Harbor (Hawaii). Both harbors are undergoing considerable renovation to accommodate increased shipping. A finite-element model called CGWAVE was developed to solve Eq. (5) using Eqs. (6), (18), (20) and (27) to formulate the boundary conditions. The Surface Water Modeling System (Zundell et al., 1998) was used for grid generation.

5.1. Simulations in the Los Angeles/Long Beach Harbor complex

The Los Angeles/Long Beach Harbor complex (Fig. 1) is one of the largest harbors in the world; therefore, the model domain is quite large covering an area of approximately 120 square km. Bathymetric input was obtained by digitizing NOAA chart number 18749. For numerical modeling, a grid containing 285,205 triangular finite elements was developed. It was based on a resolution


of 10 points per wavelength for a 30-second wave. The two one-dimensional transects in the exterior were extended in the offshore direction to a distance of 9.2 km beyond which the depth was assumed to be constant. At this location, the input wave was specified.

For initial quality control simulations, the coastal reflectivity was initially set equal to zero (i.e., fully absorbing) since this case is easier to examine qualitatively than the case when a large number of reflections are present. Figure 9 shows the phase diagram for a 50 second wave. The results appear to be quite satisfactory. A reduction in the wavelength in the onshore direction is evident. No spurious boundary effects are seen. Penetration through the breakwater gaps is precisely as one would expect. Bending of the crests as they approach from onshore also indicates a correct reproduction of refractive effects.

For further simulations, the coastal boundary was assumed to be fully reflecting (both inside the model and for the one-dimensional transects). Also, the geometry of the offshore breakwaters was changed. These breakwaters are known to be permeable to waves (e.g., Chiang, 1987) and it is hence not appropriate to consider them as closed boundaries. Permeable structures cannot be easily handled within the context of an elliptic boundary value problem.

Fig. 9. Modeled phase diagram for Los Angeles/Long Beach Harbor complex; 50 second obliquely incident wave.


One approach may be to treat the breakwater as a water area and ascribe an appropriate dissipation factor in that region. We used an alternative approach whereby the breakwater was divided into several segments so that energy could propagate through gaps in the breakwater. 50% of the breakwater length was opened up by means of numerous gaps interspersed among several solid segments.

Seabergh and Thomas (1995) conducted hydraulic model simulations for this complex at the US Army Waterway Experiment Station in Vicksburg, Mississippi. They collected data at several gages, shown in Fig. 1, for various harbor plans. The bathymetric data used for numerical modeling obtained from the more recent NOAA chart was a reasonable approximation of the harbor geometry described as "Stage II" by Seabergh and Thomas (1995). However, neither the bathymetry data nor the boundary geometries used in the two studies were identical.

Seabergh and Thomas (1995) performed their hydraulic model experiments for a large number of input frequency components varying from 30 seconds to 512 seconds. At each gage location, the amplification factor was measured for several frequencies and a resonance curve was developed. These curves were found to be extremely noisy, i.e., the response varied quite rapidly with frequency at the gages (see example in Fig. 10). For convenience of analysis, therefore, they partitioned the data into three groups: short period waves (30 s

Gage 56 Resonance Curve

16

14

J! 12

a. e <

10

8

6

4

2

0

10 100

Wave Period (s)

1000

Fig. 10. Resonance curve at Gage 56.


to 42 s), medium period waves (42 s to 205 s) and long period waves (205 s to 512 s). For each gage, the amplification factors within each group were averaged over the respective frequencies.

Numerical simulations were performed for three incident angles, for normal incidence and for 30° on either side of it, to account for the effects of the wave maker. The results of the three directional inputs were averaged for each frequency. The exact location of each gage was not known, so results in the general vicinity of the gage as determined from Fig. 1 were extracted and averaged over the frequency bands stated earlier. In all, simulations were made for 10, 30 and 17 frequency components in the three bands. These components are irregularly spaced and correspond approximately to the discrete frequency components used by Seabergh and Thomas (1995) in their hydraulic model simulations.

An example of the modeled resonance curve is shown in Fig. 10. At T = 45 s, the lab data show a remarkably high amplification that the model underpredicts; conversely, for T between 300 s and 400 s, the model value is greater than the hydraulic model data. The overall results for all gages, using the averaging described above, are compared against the hydraulic model data in Fig. 11. In general, the numerical simulation predicts the response at the gages as well as the hydraulic model data. The agreement is quite good for the short and medium period waves. Greater discrepancy is seen for the long waves which also exhibit greater gage-to-gage variability. For the long waves, there seems to be systematic overprediction near certain gages. These discrepancies could be attributed to several factors. First, the two bathymetry sets are not identical and the high variability implies that small differences in the geometry can result in large differences in the response. The location of the input wave is also different in the hydraulic and numerical models. Further, the exterior sea is bounded in the hydraulic model, thus, possibly preventing radiation out to the open sea. Finally, reflection coefficients and the degree of permeability of the breakwaters are not sufficiently well-known.

It is of course possible to introduce dissipation and/or adjust reflection coefficients or breakwater closure to tune the model better so that a calibrated model for the Los Angeles/Long Beach complex would be available for future use. However, there is no assurance that the hydraulic model is the true benchmark. The high level of agreement between the hydraulic model and numerical model results for the short and medium period waves and the moderate agreement for the long period waves indicates that the performance of the numerical and hydraulic models are certainly compatible, although not identical.


Wave Periods from 30s to 42s

— Model -Q-Data, Upper Bound -<>-Data, Lower Bound

5 10 15 20 25 30 35 40 45 50 55


5 10 15 20 25 30 35 40 45 50 55


10 15 20 25 30 35 40 45 50 Gage Number

55

Fig. 11. Wave height comparison for Los Angeles/Long Beach Harbor complex.


5.2. Simulations in Barber's Point Harbor

Seiches in Barber's Point Harbor (Hawaii) have been studied by Okihiro et al. (1993) and Okihiro and Guza (1996). Okihiro et al. (1993) also used a numerical model (Chen and Houston, 1987) based on Eq. (5) with Eqs. (15)-(17) describing the open boundary conditions. Such a model, as noted earlier, confronts the modeler with having to select a constant exterior depth and to assume that the exterior coastline is fully reflecting. Further, this model solves Eq. (28) by Gausian elimination. As noted earlier, this creates storage problems and hence allows only coarse resolution for some frequencies. To overcome these limitations, Eqs. (18), (20) and (27) were used to formulate the open boundary conditions.

Bathymetric data used by Okihiro et al. (1993) were used to develop a new grid containing about 65,065 elements. The model was run for 136 frequency components. Full reflection was used on all closed boundaries. Field data were available at four locations inside the harbor (denoted by East, West, North and South gages); see Fig. 12. Data were also available at a gage outside the harbor (denoted by "offshore" gage in Fig. 12); these were used to normalize the amplification factors inside the harbor. Model results are

Fig. 12. Barber's Point Harbor, bathymetry and gage locations.


south

+ model - observations

0.005 0.01 0.015 0.02 north

0.025 0.03

0.005 0.01 0.015 0.02 west

0.025

0.005 0.01 0.015 0.02 frequency (Hz)

0.025

Fig. 13. Wave height comparison for Barbers Point Harbor.

0.03

0.03

compared against field data in Fig. 13. There is fairly good agreement between the model calculations and the measurements especially for the long periods. For the short periods, there appears to be some overprediction by the model. This is attributed to the fact that shorter waves experience less reflection. The


simulations with a lower reflection coefficient for these waves and more detailed results will be presented elsewhere. However, the results at all four locations inside the harbor are a fairly reasonable reproduction of the field data.

6. Concluding Remarks

In this paper, we have provided a review of recent developments in simulating ocean waves with models based on the elliptic refraction-diffraction equation. In general, finite element models appear to be best suited for practical applications covering the full spectrum of waves to which a harbor may be exposed. (Some practical applications may be found, for example, in Tang et al., 1999; Pos et al, 1989; Mattioli, 1996; Kostense et al., 1988). Advances in the treatment of boundary conditions and of matrix systems associated with the dis-cretized equations have made it possible to eliminate many of the difficulties that led to inferior solutions. They have also eliminated the need for approximations of the elliptic model. Further, the advances reduce the burden on the modeler who does not have to test the sensitivity of model results to unrealistic assumptions (such as constant depths in the exterior). Applications to the Los Angeles/Long Beach Harbor region and to Barber's Point Harbor presented here demonstrate that finite element modeling with the techniques described in this paper produces results that are at least as reliable as those obtained by other methods. Inclusion of additional mechanisms like dissipation, wave-wave interactions, wave-current interactions and steep slope effects can enhance the usefulness of these models. However, how a model will behave when these effects are combined is not yet clear. Further research in modeling methods as well as data where some of these effects can be combined and isolated are desirable.

Acknowledgments

Partial support for this work was provided by the Office of Naval Research, the Maine Sea Grant Program and the National Sea Grant Office. Many of the developments described here were made with the assistance of five graduate students at the University of Maine (Karl Schlenker, Wei Chen, Luizhi Zhao, Doncheng Li and Khalid Zubier) and Dr. Michele Okihiro of the Scripps Institution of Oceanography. Their contributions are gratefully acknowledged. Permission to publish this paper was granted by the Chief, Corps of Engineers, to publish this paper.


References

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RECENT ADVANCES IN THE MODELING OF WAVE A N D PERMEABLE STRUCTURE INTERACTION

INIGO J. LOS ADA

Artificial and natural porous structures are of great interest in coastal and harbor engineering. The modeling of wave interaction with permeable structures is therefore a key issue to determine the functionality and stability of this kind of structures. In most circumstances, an averaging process is introduced in the analysis of the flow in terms of a seepage or discharge velocity and some coefficients depending on the flow. In order to solve the wave and structure interaction, the porous flow model is matched with a flow model for the fluid region. In this paper, it will be shown that several new equations including the resistance forces in the porous medium have been derived. Newly developed models based on Boussinesq-type equations or direct resolution of the Navier—Stokes equations using VOF techniques have opened a new range of possible applications. However, these models still highly depend on porous flow coefficients. Predictive formulae for these constants under oscillatory flow conditions require further research especially if these models are considered to be an alternative to physical modeling in the design of coastal structures.

1. Introduction

A porous medium is a two-phase material in which the solid matrix, usually assumed to be rigid, constitutes one phase and the interconnected voids or pores constitutes the other. One of the main characteristics of porous media is the irregular shape and size of its pores, randomly distributed, conferring the flow through this heterogeneous formation considered a very complex nature. Our interest will be to determine the flow through the porous formation with typical length scales much larger than the characteristic pore size.

Artificial porous structures such as rubble-mound breakwaters, submerged structures, outfall protections, artificial fishing reefs or armor layers for the protection of seawalls or vertical structures are of great interest in coastal and harbor engineering since they provide one of the best means to induce incident wave dissipation by friction inside the structures. Therefore, the knowledge

163

164 /. J. Losada

of the flow motion in and around the porous media and the corresponding

pressure and force fields will be a key issue to determine the functionality and

stability of these coastal structures.

Furthermore, sand and shingle beaches may be considered as the natura l

porous media and therefore the determination of wave dissipation on permeable

layers, the beach water table evaluation, or of how beach permeability may

affect the fundamental surf zone hydrodynamic processes are also important

issues to be addressed.

The complex internal geometry of a porous medium, artificial or natural ,

is difficult if not impossible to determine. Furthermore, in general, in the

coastal and harbor engineering field, there is a lack of interest in knowing

the internal details of the structure or the microscopic flow. In fact, in most

circumstances, our interest will be in determining the characteristics of the

flow in large portions of the porous s tructure considered and introducing an

averaging process in the analysis of the flow. The averaging process has a

smoothing effect, filtering out small scale variations associated with the media

heterogeneity and pore irregularities.

In several other fields, the transition to average macroscopic variables is

based on a statistical approach, especially in hydrology when analyzing flow

and t ransport in porous formations and aquifers. However, to date, the

stochastic approach is not considered yet in the field of coastal engineering

since several additional complications have to be addressed.

The modeling of wave and permeable natural or artificial s t ructure interac

tion is based on the coupling of two models, one tha t describes the flow acting

on the s tructure and one that describes the flow through the porous structures.

The accuracy of the modeling will be limited by the hypotheses and simpli

fications formulated for the flow in the outer fluid region and by the validity

and hypotheses of the porous flow model, usually relying on some constants

depending on the flow and finally on the matching conditions imposed.

This paper summarizes some of the most recent work available in the lit

erature on wave interaction with porous structures. It will be shown tha t the

study of wave interaction with permeable structures has evolved in parallel

with wave theories in fluids. In the last few years, special a t tent ion has been

paid to the development of equations and numerical models to analyze the

interaction with porous structures with very promising results. However, the

application of these models to prototype scale has to be carried out with care.

This paper is organized as follows. First, a review of the existing porous

flow models is presented. An emphasis is made on the difference between

Recent Advances in the Modeling of Wave 165

stationary and nonstationary flows. In Sec. 3, the general governing equations for flow in porous media are derived. In order to formulate simple solutions, the linearized problem is shown in Sec. 4. Considering potential flow inside and outside the porous medium, solutions in terms of eigenfunction expansions are formulated to analyze wave interaction with vertical permeable structures. Furthermore, the derivation of an extended mild-slope equation for wave propagation on permeable layers is presented. The equation is applied to wave interaction with submerged permeable breakwaters including wave breaking. In Sec. 5, the most recent developments for shallow water equations are indicated. Wave diffraction and transmission by a permeable vertical breakwater is modeled using Boussinesq-type equations. The time and depth-averaged equations for waves in permeable media are derived in Sec. 6. Some preliminary applications are shown to model mean water level variations in permeable submerged breakwaters. Finally, in Sec. 7, a model based on the Navier-Stokes equation is presented. This model, called COBRAS is able to simulate wave interaction with permeable structures including wave breaking and turbulence.

2. Porous Flow Models

2.1. Stationary flow

The success of the theoretical formulation of the wave and porous structure interaction largely depends on the accuracy of the empirical formulas and coefficients used to describe the frictional forces exerted by the porous media.

The study of the flow in porous media can be traced back to 1856 when Darcy found empirically, using a vertical permeameter, that the ID steady flow in sand or other fine granular material can be described by the following formula (Bear, 1972),

I = K-lud = apud , (1)

where i" = (—dpo/dy)/pg is the hydraulic gradient, p the fluid density, g the gravitational acceleration, po = p + pgho the effective pressure, ho the vertical distance from the selected datum, ud the discharge velocity and K = 1/a.p (m/s) the permeability coefficient and ap an empirical coefficient.

The Darcy law is often written in the form I = (v/gKp)ud where Kp (m2)

is the intrinsic permeability. This expression, which later has been referred to as the Darcy law, is esentially valid for laminar flows, breaking down when the flow velocity becomes sufficiently large or when the characteristic length scale of the porous material is large. A certain critical Reynolds number

166 /. J. Losada

Re = ucDc/v above which Darcy law becomes invalid can be defined in terms of a characteristic discharge velocity in the porous media uC: a characteristic length scale of porous media Dc and the molecular viscosity v. Above this critical number, the turbulence effect becomes apparent and the flow resistance appears to increase.

Forchheimer (see Bear, 1972), at the beginning of the century extended Darcy's law to include a quadratic term accounting for the frictional force induced by turbulence such that:

I = apud + bp\u

d\ud, (2)

where bp is another empirical coefficient with dimension (s2/m2). In order to describe the coefficients ap and bp, several expressions were

proposed. Some of them were obtained based on various analogies. Considering nonconvective laminar flow in a number of circular capillaries with diameter dc, Darcy's law compares to the Hagen-Poiseuille equation yielding to the following relation,

7 = 3 2 ^ . (3)

Applying the pipe analogy, the combined efforts of Kozeny (1927) and Carman (1937) lead to:

/ = 3 6 « < ^ - ^ « ' , (4) n% gdz

where ne is the porosity defined as the fluid volume divided by the total volume d is the diameter of spheres with equivalent geometry to the pores and K is a coefficient which is taken to be 0.5.

Ergun (1952) extended the work by Kozeny and Carman for the linear flow resistance applying the pipe analogy to the Forchheimer flow regime and introducing a new constant for the quadratic term arriving at the following expression,

/ = 150 { i z £ > ! » u ' + 1.75 i i Z ^ l V I - (5) n% gd1 n% gd

In general, the following expressions for the ap and bp coefficients can be used,

( l - n e ) 2 v ap=iap^^g^!> ( 6 )


6P = / 3 P ^ 4 r - (7) n% gDc

A summary of the different values of the nondimensional coefficients ap, j3p and the characteristic length-scale for the porous media Dc proposed by several authors can be found in van Gent (1993).

2.2. Nonstationary flow

Wave action on structures induces a nonstationary flow. Polubarinova-Ko-china (1962) added a time-dependent term to the Forchheimer equation as Eq. (2). The resulting equation including the inertia term accounting for the acceleration is referred to as the extended Forchheimer equation,

dud

I = apud + bpu

d\ud\ + cp— , (8)

where cp is a dimensional coefficient (s2/m). This formula can be derived from the Navier-Stokes equation (van Gent, 1991).

Gu and Wang (1991) and van Gent (1991) found an expression for cp after a theoretical derivation,

cP = - ^ ~ , (9) neg

where 7 is a nondimensional coefficient that accounts for the added mass. The concept of added mass is associated with the fact that in order to accelerate a certain volume of water, a certain amount of momentum is needed. To accelerate the same volume of water in a porous medium, an additional amount of momentum is needed. This is called added mass since the extra amount of momentum suggests that a larger volume of fluid has to be accelerated.

Please note that Eq. (8) does not include a possible resistance force due to the presence of a convective term. Such a resistance term, probably important for flow through porous media with considerable large-scale convective transport, could be incorporated into the bp term because it would be quadratic in the velocity (van Gent, 1991).

2.2.1. Relative importance of the resistance forces

Gu and Wang (1991), van Gent (1993) and Losada et al. (1995) discussed the relative importance of the three contributions to the total resistance force for nonstationary flow in porous media represented in Eq. (8), namely the

168 /. J. Losada

resistance due to laminar flow f^, resistance due to turbulent flow /jy, and the inertial resistance / / .

Following Gu and Wang (1991), the importance of the resistance forces can be analyzed in terms of two different Reynolds-type numbers,

D ucDc Kf = ,

a-^-,3. <10) Kl~ v ~l K C '

where uc is a characteristic velocity, Dc is the characteristic particle size of the porous material, v is the kinematic viscosity and KC is the Keulegan-Carpenter number and to is the angular wave frequency. Both Reynolds numbers give the relative importance of the inertial forces to the viscous forces; however, in Rf, the inertia is of convective nature and the resistance is due to velocity changes in space, whereas in Ri, the resistance force arises locally due to the change of velocity at a specific location of the porous structure.

In Fig. 1, regions with different dominant resistance components are shown. Regions / , L, and N are dominated by one resistance force only, that is, the dominant force is at least one order of magnitude larger than the other two.

1E6

1E5

IE4

1E3

1E2

1 El

Qi~ 1E0

1 El

1E-2

1E-3

1E-4

1E-5

1E-6

IE-6 1E-S 1E-4 1E-3 IE-2 1 El 1E0 1 El 1E2 1E3 1E4 1E5 1E6

Fig. 1. Relative importance of the resistance forces (after Gu and Wang, 1991).

fN > 10fL

fn > 10f, N Region

h > JN

k > I Of, h > 10fH

L Region

Smith (1991) • - - vanGent(1994)

Losada et al. (1995) A./,

f,>10fL

f, > 10fH

/Region


Table 1. Dominant force components under coastal wave actions (Gu and Wang, 1991).

Material description D(m) uc (m/s) Rf Rt

Dominant force

Coarse sand or finer

Pebble or small gravel

Large gravel Crushed stone

Boulders Crushed stone

Artifical blocks Large rocks

< 0.002

0.01

0.10

0.3-1.0

> 1.0

< O(10~3)

O(10" 2 )

O(10 _ 1 )

O(10°)

> O(10°)

< O(10°)

O(102)

O(104)

O(106)

> O(106)

< O(10°)

O(102)

O(104)

O(105)

> O(io6)

Laminar

Laminar Turbulence Inertia

Turbulence Inertia

Turbulence Inertia

Turbulence Inertia

In one region, the three resistance forces are of equal importance while there are three intermediate regions where two out of the three forces could be important.

Considering several characteristic parameters under coastal wave conditions, Gu and Wang (1991) gives an illustration of dominant force components for bottom material of various sizes. This table is only orientative but it is a general guideline of practical interest. Furthermore, the results in Fig. 1 and Table 1 are important for analyzing possible scale effects in physical models.

2.2.2. Determination of the parameters for nonstationary flow

The friction coefficients ap and bp from the Forchheimer equations were measured in tests with stationary flow. Until the work by Smith (1991), Hall et al. (1995) and van Gent (1993, 1995), no available data for the determination of the friction coefficients under oscillatory flow were reported.

Smith (1991) provided a set of friction coefficients obtained experimentally in an oscillating water tunnel through different arrangements of prepared packing spheres. One sample of rock material was tested.

van Gent (1993, 1995) carried out permeability measurements in a U-tube tunnel to study flow through five samples with various types of stones with D50 = 0.0202 — 0.0610 m. The differences between stationary and oscillatory flow were studied and the contributions of laminar, turbulence and inertia

170 /. J. Losada

terms were determined. Figure 1 shows the range where the experimental work by Smith (1991), van Gent (1993) and Losada et al. (1995) were carried out.

Based on the experimental results, new expressions for nonstationary porous flow friction coefficients were formulated. According to van Gent (1995), the friction coefficients for the extended Forchheimer equation should be expressed as:

(l-ne)2 v

gDl ' ' 5 0

Ml + i ^ ^ - k - where KG ^ v \ T K C J n\ gD50 neD e-^50

1 + 7 ^ , „ „ 0.015 — where 7 = 0.85 — where

neg Ac uc 0.015

>

(11)

negT j ^ + 0.85 '

where uc is a characteristic velocity of the flow, T is the wave period and D$Q is the median grain size diameter. It is recommended to take the maximum discharge velocity as representative of the flow.

Although the coefficients ap and j3p may still depend on parameters like grading, shape, aspect ratio or orientation of the stones, the following values are recommended, ap = 1000 and j3p = 1.1.

Further work on unsteady flow equations can be found in Burcharth and Andersen (1995).

Equation (11) represent a significant contribution to the modeling of wave interaction with permeable structures. However, the discrepancies appeared in the application of the formulae by other authors (see, Liu et al., 1999; Lynett et al., 2000); the limited range of existing experimental data and the importance of an accurate prediction of the coefficients on the modeling of wave and structure interaction seem to be important reasons to do further research on this topic in the near future especially if the modeling is to be applied to prototypes.

3. General Governing Equations and Matching Conditions

3.1. Governing equations

The flow in porous media can be described by the general Navier-Stokes equations,


du* * ^ i _ _ l ^ a 2 < s

where v is the molecular viscosity, u* is the ith component of the instantaneous velocity in the pores and p* the instantaneous effective pressure. As already stated using the macroscopic approach, based on averaging over a small but finite volume with a representative length scale larger than the typical pore size but smaller than the characteristic length scale of the problem, perturbation in the field due to the presence of individual particles and pore irregularities can be ignored.

To replace the actual velocity with the seepage velocity, Sollitt and Cross (1972) resolved the local instantaneous velocity field u* into three components,

u* = Ui + u\ + u\, (14)

where Ui is the seepage velocity, that is, "the average velocity within small but finite and uniformly distributed void spaces; u\ is the spatial perturbation accounting for local velocity components due to pore irregularities or boundary layers, and u\ is the time perturbation accounting for local transient fluctuations within the pores" (Sollitt and Cross, 1972). Likewise, the pressure field may be split up into analogous components.

The effect of the transient or turbulent and spatial perturbations on the mean flow in the pore can be determined by substituting these definitions in the Navier-Stokes equations. Expanding the total derivative in the Navier-Stokes equations, substituting Eq. (14) and the analogous expression for the pressure field in this equation, and performing the time averaging for a period much smaller than the time scale of the macroscopic unsteadiness yields,

— (Ui + u\) + {ut + u\) • V(Ui + u\) + vfiv\

= — <j> + p"+iz)+vV2(ui + u\), (15) P

where (A) denotes time average in a period of time smaller than the wave period.

Proceeding in the same way in the continuity equations leads to:

V - ( U i + u ? ) = 0. (16)

172 I. J. Losada

Integrating the equations of motion over a small but finite volume, the effect of spatial fluctuations within the pore may be isolated.

Finally, the following set of Reynolds Averaged Navier-Stokes equations (RANS) are obtained,

- ^ + UiVui + uyVut + < • VM| = — V(p + 7-z) + vV2Ui (17a)

and

V«i = 0 , (17b)

where (A) denotes spatial average. Due to the nonlinearity related to the convective terms, the terms associated to the spatial and turbulent fluctuations uf • Vuf and u\ • Vu* remain in the equations after the averaging process. In analogy to turbulence analysis, these two terms may be interpreted as stresses with respect to the mean motion.

In order to solve the equations, it is necessary to find closure equations. Based on the work by Ward (1964) for steady and nonconvective flow conditions, Sollitt and Cross (1972) established the following equivalency,

vV2Ui - [ u p V u ^ + u f - V u * ] = -

where Kp is the intrinsic permeability, Cf a dimensionless turbulent coefficient and uf the discharge velocity which is related to the seepage velocity by the following relationship uf = neUi, where ne is the porosity.

The following equation is obtained,

dui 1 .

where s = necpg is introduced as a co-factor in the local acceleration term to account for the added mass and the following additional assumption has been made,

UjVuj < usj • Vut + uyVul. (20)

Sollitt and Cross (1972) claimed this assumption to be valid for problems of practical importance, for which wave length is much greater than pore diameter. Please note that the viscous terms in Eq. (17a) have been included in

vul u-t\uf\ (18)

vneUi Crrr (19)


the equivalency. One could have assumed this term to be negligible compared to the fluctuations.

Using a vertical-lattice type porous medium made of rectangular wooden sticks nailed together producing uniform pores, Losada et al. (1995) performed a very comprehensive set of measurements inside and outside the structure including free surface and water particle velocities using Laser Doppler Velocimetry and pressure records.

Instantaneous measurements at 9 points at each pore were processed in order to calculate the instantaneous and spatial fluctuations, the time and space-averaged velocity, and the convective terms associated with each of the components.

The analysis points out two important results: (1) spatial fluctuations are always more important than temporal fluctuations since temporal fluctuations are confined by the pore size, and (2) even if under certain circumstances, convective terms associated with fluctuations are more important than those associated with seepage flow, the latter should not be neglected.

There is no unique way to decompose the instantaneous velocity. Liu et al. (1999, 2000) supported by the experimental results by Losada et al. (1995) considered dividing the fluid variables into two parts only, a spatially-averaged component and a spatially-fluctuating component, assuming the temporal fluctuations to be negligible. The resulting spatially-averaged Navier-Stokes equations are:

1 + cA duf ufduf_ I dp v d2uf 1 duiuj

ne dt nl dxj p dxt ne dxjdxj n% dxj ' (21)

where CA = 7((1 — ne)/ne) is the added mass coefficient. The correlation of spatial velocity fluctuations, the last term on the right-

hand side of Eq. (21) is modeled by Liu et al. (1999) by a combination of linear and nonlinear frictional forces as follows,

I duluSj 2 TT~ = ~gaPUi - gbpucUi, (22)

in which uc = y/uiul so that the first term on the right side represents the linear, viscous force, while the second term represents the nonlinear turbulent force. Both ap and bp are empirical coefficients which are functions of Reynolds number and the geometric characteristics of the porous media.

174 /. J. Losada

Please note that Eqs. (22) and (18) are similar. However, the second (viscous) term on the right-hand side in Eq. (17), generally much smaller than the third term for problems of engineering interest, is retained in Liu et al. (1999) since this term is responsible for transferring shear force and may become increasingly important near the interface between porous media and outside flow for smaller scale problems.

3.2. Matching conditions

Matching conditions are necessary to guarantee the continuity of the solution for the interface between the fluid and porous regions. In general, continuity of mass flux and pressure are the matching conditions considered.

For long wave models, continuity of free surface, velocity and their derivatives are usually enforced.

However, when the models in the fluid region or inside the porous structure include the modeling of turbulence or boundary layers, the matching at the interface has to be carried out with caution.

4. Wave Interaction with Structures. Linear Solutions

4.1. Linearized problem

Assuming a simple harmonic wave of frequency to, Eq. (19) may be linearized on the basis of Lorentz's hypothesis of equivalent work (Sollitt and Cross, 1972; Madsen, 1974), replacing the nonlinear terms in Eq. (19) by an equivalent linear term fuJUi where / is a dimensionless friction coefficient. This yields a linearized form of the equation,

istoui — —V • I —h gz ) — tofui. (23) \P J

Taking the curl of this equation shows that the flow in the porous medium is irrotational and therefore can be described by a potential $ that satisfies,

m = V $ . (24)

Substituting Eq. (24) into Eq. (23) results in a Bernoulli-type equation for unsteady flow within the porous medium,

s^- + -+gz + fuj<S> = 0. (25)


Finally, substituting Eq. (24) into the continuity equation in Eq. (17b) yields Laplace's equation,

V 2 $ = 0. (26)

At the free surface 77, the Bernoulli equation in Eq. (25) can be combined with the linear kinematic free surface boundary condition,

to yield,

?l-u>2(s-if)-=0. (28) dz g

Furthermore, the following linear complex dispersion relationship has to be satisfied by the waves propagating inside the porous medium,

to2(s-if)= gTtanhTh, (29)

where T is a complex wavenumber. Solutions to these equations depend on the values of the porous material

parameters, s, ne, Kp, and Cf, known for a given material and the linearized friction coefficient / . Therefore, an additional condition is required to evaluate this coefficient. Following Sollitt and Cross (1972) and Madsen (1974), / is evaluated from the following equation,

1 Jo Jv ' l^l2 + ^ l"il3 ^ dtdV

f = - ^ " r — " > 3 0

w Jo JvnelutfdtdV

for a porous structure of volume V under a wave-cycle of period T. Please note that Uj is taken to be the real part of the seepage velocity and therefore, an interative procedure is needed to evaluate / .

4.2. Solutions based on eigenfunction expansions

Based on this irrotational and linear approximation, Sollitt and Cross (1972) presented a model to analyze wave interaction with vertically sided porous structures. This work was later extended by Dalrymple et al. (1991) to include oblique incident waves. Dalrymple et al. (1991) considers the interaction of a gravity wave train with a single homogeneous, isotropic, porous structure of width b between two semi-infinite fluid regions of constant depth h. The wave

176 /. J. Losada

field outside the structure can be specified by velocity potentials $ ! in the seaward region and $3 in the leeward region of the breakwater by specifying the well-known linear boundary-value problem for water waves in constant water depth. In the rigid porous medium, region 2, a boundary value problem can be defined using Eqs. (24), (28), and (29) and adding the kinematic bottom boundary condition 8<&2l'dz = 0 in z = —h.

Each of the boundary value problems is formulated in terms of linear homogeneous equations. Separation of variables leads to Sturm-Liouville problems where the potentials may be expressed in terms of an eigenfunction expansion. Since the solution in adjacent regions must be continuous at each interface, continuity of mass flux and pressure at x = 0 (interface between regions 1 and 2) and at x = b (interface between regions 2 and 3) is required. These conditions may be expressed as:

$ix = ne$2x , $1 = (s - i / ) $ 2 at x = 0 , (31a)

and

®ix = ne$2x , $1 = (s - i / ) $ 2 at x = b, (31b)

where the continuity of pressure is derived from the Bernoulli equation Eq. (25). Substituting the potentials into the matching conditions in Eqs. (31a) and

(31b), a system of equations is obtained. Unknowns are the complex amplitudes of the progressive and evanescent modes in the potentials $1, $2, and $3. Applying the orthogonality of the eigenfunctions over the water depth in the 3 regions results in a simpler system. Once the system of equations is solved together with Eq. (30) and the corresponding dispersion relationships, the potential and therefore, the flow is completely defined.

Dalrymple et al. (1991) presents the variation of reflection and transmission coefficients for several rectangular geometries, finite, semi-infinite and infinite breakwater with an impermeable wall considering several relative water depths and analyzing the influence of wave incidence. They found that a minimum reflection coefficient occurs for different angles of incidence depending on the / value. It has to be pointed out that results in this work have been obtained assuming a given constant / .

The eigenfunction approach has also been applied to crowned breakwaters (Losada et al, 1993) or to submerged porous steps (Losada et al., 1997a).

The complex nature of the dispersion equation as Eq. (29) leads to two particular difficulties when Sollitt and Cross (1972) model is used in conjunction


with eigenfunction expansions technique. First, it is difficult to locate the complex roots of the dispersion relation by standard numerical methods. Second, the vertical eigenfunction problem is not self-adjoint and standard expansions theorems do not apply. These problems have been discussed by several authors (f.i. Dalrymple et al., 1991). Mclver (1998) presents a method that allows the explicit calculation of the roots of the complex dispersion relation and uses the theory of non-self-adjoint differential operators to show how the formal construction of eigenfunction expansions can be carried out for the interaction of water waves with porous structures.

In order to consider different geometries generalizing the theory of Sollitt and Cross (1972), Sulisz (1985) developed a boundary element method to investigate wave transmission and reflection from a multilayered breakwater with arbitrary shape.

Based on a linearized long wave theory, Massel and Mei (1977) and Massel and Butowski (1980) were the first to consider random wave interaction with permeable structures. Following Dalrymple et al. (1991), Losada et al. (1997b) considered the interaction of directional random waves with vertical permeable structures. Using an eigenfunction expansion, Losada et al. (1997b) simulated the transformation of a given incident spectrum in the vicinity of the partially reflecting structures. The influence on the results of the structure's geometry, permeable material characteristics and incident wave spectrum is analyzed.

Further information regarding linear solutions for monochromatic waves based on a linearized version of Sollitt and Cross (1972) equations can be found in Chwang and Chan (1998).

4 .3 . Mild-slope equation

Within the framework of linear wave theory, Berkhoff (1972) proposed a two-dimensional theory which can deal with large regions of refraction and diffraction. This new equation, the mild-slope equation, has been extensively used for wave propagation modeling.

In order to consider wave propagation on porous slopes and wave interaction with trapezoidal permeable submerged breakwaters, Rojanakamthorn et al. (1989), Losada et al. (1996a) and Mendez et al. (2000) present extended versions of the mild-slope equation.

The extension of the mild-slope equation for permeable layers is derived by multiplying the Laplace equation by its correspondent vertical eigenfunctions,

178 /. J. Losada

Fig. 2. Schematic description of the submerged permeable structure geometry.

MQ{Z) and Po(z), expressed in terms of the propagating mode only, neglecting evanescent modes and integrating over depth.

Following Losada et al. (1996a), the new governing equation is:

/ M0(z) (v2h§2 +

J-h+ah \ dz2 dz + ne(s — if)

r-h+ah p0(z)(V£$4 + ^ | d z = 0,

where

(32)

(33) $ 2 = (p(x,y)M0(z) and $ 4 = ip(x,y)PQ{z),

and $2 is the velocity potential in the fluid region above the permeable layer or submerged breakwater, $4 the potential inside the permeable region, y> is the complex amplitude of the water surface and V/, = (d/dx, d/dy).

The boundary and matching conditions for variable depth are:

• Combined free surface boundary condition,

r-ll $ 2 = 0 at 2 = 0. dz g

• Bottom boundary condition,

<9$4

dz

Continuity of mass flux,

oz

Vh • W - $ 4 = 0 at z = -h.

(34)

(35)

<9$4 + Vh-hVh-$>4 at z = - / i + a/i. (36)


• Continuity of pressure,

$ 2 (s - i / )$4 at z = -h + ah . (37)

Integrating Eq. (32) using the boundary and matching conditions and finally the mild-slope assumption, the following equation is obtained,

V h • (XVfc • <p) + (rgx " *>xfD)>P = 0,

where

X (£) [£ M$(z)dz + ne(s-if) -h-\-a.h

r J-h

h-\-cth

P*(z)dz

(38)

(39)

and where the term iu>fifD<p has been added to account for wave breaking, fo being an energy dissipation function, g the acceleration of gravity, and i the imaginary unit. MQ(Z) and Po(z) are the vertical eigenfunctions. The expressions of the vertical eigenfunctions, dispersion equations and complete potentials can be found in Losada et al. (1996a) and Mendez et al. (2000).

The energy dissipation function due to wave breaking on a submerged permeable breakwater takes into account the processes of wave decay and recovery and is expressed as (Rojanakamthorn et al, 1990):

SD = (40)

where Cg is the group velocity, TR is the real part of To, tan£ is the equivalent bottom slope at the breaking point which is defined as a mean slope in the distance 5heftb offshore the breaking point and,

v hef

vs =0.4(0.57 +5 .3 tanC),

M (41)

vr = QA-, h. ef

where the subscript b means the value at the breaking point. hef is the effective water depth over the porous layer as defined in Losada et al. (1997a).

To solve the problem, a finite difference scheme and proper boundary conditions at the domain boundaries are used. The resulting system of equations

180 /. J. Losada

including Eq. (30) can be simultaneously solved. Assuming a breaking criteria, fo is calculated using the expression in Rojanakamthorn et al. (1990). In order to arrive to a solution, an iterative procedure has to be carried out.

Losada et al. (1996b) extends the model to consider the interaction of directional random waves with submerged breakwaters. The influence of structure geometry, porous material properties and wave characteristics including oblique incidence, on the kinematics and dynamics over and inside the breakwater is considered in Losada et al. (1996a, 1996b) and Mendez et al. (2000). Models are validated against experimental data.

Figure 3 shows root-mean-square wave height Hrms evolution in a submerged permeable breakwater. The numerical model results obtained in Mendez et al. (2000) are compared with experimental data (Rivero et al., 1998). The submerged breakwater with 1:1.5 slopes on both sides has a crown width of 0.61 m is constructed on an impermeable core and an armor layer of quarrystones with a mean weight of 25 kg. The structure is placed on a 1:15 rigid slope bottom. The incident irregular wave characteristics are given by

rmsi 0.396 m and Tp = 4 s. Applying Eq. (11) and with ne = 0.4, the rest of the permeable material characteristics are: Kp = 2.5 x 10~5 m2, Cj = 0.3, s = 1. The resulting linearized friction coefficient is / = 2.49.

Results show very good agreement even under breaking conditions. In front of the structure, wave reflection induces a modulation of the wave height. Wave

o.io

' 1 ' 1 ' 1 ' 1

* \ /• i——*

H.„.,= 0.396 m T„=4.0s _ rms.1 p

a =1.5 |R 1=0.02 *=45° - K^ 2.5E-05 m2 Cf= 0.3 _ f=2.49 s=l

1 1 ' 1 •

• \

1 1 ' 1 '

_ -

-

Fig. 3. Wave transformation by a permeable submerged breakwater. Comparison of experimental and numerical results.


breaking takes place on the crest. A modulation of the wave height is also visible leewards the submerged breakwater due to the reflection induced by the bottom slope.

Mase and Takeba (1992) extends the mild slope equation, deriving time-dependent and time-independent wave equations for waves propagating over porous rippled beds. By using the time-independent equation, the Bragg scattering is examined in a one-dimensional case showing that energy dissipation in the porous layer contributes to smaller reflected and transmitted coefficients than those in the case of an impermeable rigid rippled bed.

5. Shallow Water Models

5.1. Introduction

There are numerous situations where accurate computations of the wave field on permeable beaches or around permeable structures are not possible using the mathematical models presented in previous sections.

Nonlinearity is an important feature in the process on wave interaction with most coastal structures usually located between intermediate and shallow water depths. For example, in order to model the generation of higher harmonics on regions of abrupt depth variations such as crowns of permeable submerged structures (Losada et al., 1997a), the mild-slope equation is limited by the use of linear theory. A review on shallow water models can be found in van Gent (1995).

5.2. Recent developments

Using a perturbation method, Cruz et al. (1992) derived a set of time-dependent nonlinear equations for one-dimensional wave transformation on porous beds. Since these models include the leading order of nonlinearity, they are able to generate higher harmonics on the shallow water region. However, the inherent dispersivity is weak and consequently the frequency-dependent wave decomposition beyond submerged breakwaters cannot be reproduced. Following the approach by Madsen et al. (1991) for impermeable beds, Cruz et al. (1997) derived a set of Boussinesq equations over a porous bed of arbitrary thickness with an underlying solid bottom of arbitrary depth. After determining the governing equations and boundary conditions for the three-dimensional wave motion, assuming incompressible and irrotational flow in both the fluid and permeable layer, a set of time-dependent, vertically-integrated equations is

182 /. J. Losada

derived containing the leading order of nonlinearity. The weak dispersivity of Boussinesq-type equations is corrected by adding dispersion terms to the basic momentum equations and matching the resulting dispersion relation with that of appropriate theory.

The equations of motion inside the porous layer include resistance terms following the approach by Sollitt and Cross (1972). The permeable material parameters were extrapolated from values tabulated in Sollitt and Cross (1972).

Numerical results are obtained for wave propagation on a horizontal bottom of uniform thickness with good agreement.

Liu and Wen (1997) derived a two-dimensional fully nonlinear, diffusive and weakly dispersive set of equations for long wave propagation in a shallow porous medium. Using analytical perturbation solutions as well as numerical solutions, the one-dimensional equations are used to study the tide-induced free surface fluctuations in a porous region and the transmission and reflection of solitary waves by a rectangular porous breakwater. The last is calculated considering the Boussinesq approximation carrying out a linearization process inside the porous breakwater to convert the nonlinear resistance formula to a Darcy-type resistance. Numerical results are compared with experimental data (Vidal et al, 1988) observing an excellent agreement.

5.3. Diffraction by porous structures

Diffraction of waves by a solid breakwater has received a considerable amount of attention. However, the role of breakwater porosity on the wave diffraction process has not been addressed until recently. Based on the linear potential wave theory, Yu (1995) developed a porous breakwater diffraction model. This model was extended to waves of arbitrary incidence (Yu and Togashi, 1996; Mclver, 1999), but requires that the breakwater be thin compared to the incident wave length. Lynett et al. (2000) presents a model based on depth-integrated equations suitable for weakly nonlinear, weakly dispersive transient waves propagating in both variable-depth open water and porous region. In this first work, the model is applied to analyze solitary wave interaction with vertically-walled porous structures in a horizontal bottom.

In the open water region, the model employs the generalized Boussinesq equations presented originally by Wu (1981). The equations are expressed in terms of the free surface displacement ( and the depth-averaged velocity potential <j>. In dimensional form, the equations are given as:


^ | + V.[(C + W ] = 0 , (42)

^ + 2 ( W + K - ^ V . ( W ) + - - V ^ = 0, (43)

where /i is the local water depth, g is gravity and V = (d/dx,d/dy) the horizontal gradient. Using 6, the depth-averaged velocity u can be calculated by:

u = V<j>. (44)

These equations are valid only for weakly nonlinear and dispersive waves, limiting the application to waves satisfying 0{A/h) = 0(kh) <C I, where A/h is a nonlinearity parameter and (kh)2 a dispersion parameter.

This model is coupled with a specific model for the porous region. In the porous region, a truncated version of Liu and Wen's (1997) Boussinesq-type equations is used. Consistency with the water region model requires the truncation of the equations retaining only weakly nonlinear effects and depth averaging. Denoting -ip as the depth-averaged piezometric head, K as the hydraulic conductivity, ne as the effective porosity of the porous material, the governing equation can be expressed as:

A__V [ (C + / l ) - v c ] - y ^ v 2 C = o,

h?

Velocity in the porous region is given as:

u = -KViP . (47)

The resolution of the problem requires reflective and radiation conditions at the exterior boundaries and matching conditions along the interface between the fluid and porous region.

The matching conditions are given as:

C | + = C I - , u\+=u\-, (48)

n - V < | + = u - V C | - , V - ( n - u ) | + = V - ( n - i l ) | _ , (49)

where the sign denotes opposite sides of the interface and n is the unit normal vector. Velocities are evaluated using Eqs. (46) and (47).

(45)

(46)

184 /. J. Losada

Using a high-order finite difference scheme, the numerical model is used for both ID and 2D problems. Considering solitary wave interaction with a permeable vertical structure considerably simplifies our understanding of the wave diffraction-transmission process.

To validate the model and compare the different mechanisms, a comprehensive set of experiments was performed in a wave tank with a porous and impermeable breakwater perpendicular to solitary wave incidence. Water depth, wave height and gravel diameter were varied and free surface was recorded in a dense grid covering the region in front of the structure and in the shadow zone.

Results show that the model predicts wave height, wave form and arrival time excellently. Furthermore, the model and experimental results are useful for evaluating the differences between the wave field in the shadow zone comparing the porous breakwater case with the solid breakwater case. Figure 4 presents the results of the numerical simulation of the interaction of a solitary wave passing a solid (left) and a porous (right) detached breakwater. The snapshots correspond to spatial profiles of a normally incident solitary wave with A/h = 0.1 interacting with a breakwater, with a length of 5 water depths and a width of 80 water depths. For the porous breakwater, the scaled rock size has the following characteristics Dso/h = 0.2, ne = 0.5 m ap = 1100, and (3p = 0.81.

The first two snapshots (a) and (b) show the solitary wave approaching the detached breakwater. In Figs. 4(c) and 4(d), the wave height at the front face is at a maximum, while Figs. 4(e) and 4(f) show the reflected waves at the beginning of diffraction behind the breakwater. The analysis of the shadow zone in Figs. 4(e), 4(f), 4(g), and 4(h) shows that less energy is diffracting to form a wave with a circular crest line in the porous breakwater case than in the solid breakwater case. The main difference between both cases is that most of the energy that diffracts in the solid breakwater case to form this circular wave, diffracts into the transmitted wave front in the porous breakwater case. This is due to the fact that in the shadow zone of the porous breakwater, wave diffraction occurs in two ways. A wave with circular crest lines is created since part of the wave energy is diffracting into the calm water behind the transmitted wave in the same form as diffraction occurs behind a solid breakwater. The second part of diffraction takes place because wave energy diffracts into the transmitted wave front from the incident wave front due to the discontinuity of wave amplitude. The relative importance of each of the mechanisms will depend on the incident wave characteristics, breakwater geometry and permeable material characteristics.


JJ

)

(a)

m

(c)

in

m

n

(b)

(d) • 1

(f)

Ijj

Fig. 4. Numerical simulation of solitary wave interaction with detached impermeable and permeable breakwaters.

186 /. J. Losada

6. Short Wave-Averaged Flow

6.1. Introduction

6.2. Time and depth-averaged equations

The continuity and Navier-Stokes equations describing the fluid flow may be written in terras of the instantaneous vertical velocity component w* and the viscous stress tensor r,*.- as:

du* dw* _

dxi dz

du* d(u*u*) d(u*w*)

dt dxi dz

di-P^ij+rti} ^ dr* jz

dxi dz J = 1,2 (51)

dw* , d(w*u*) , d(w*2)] d[-p* + pgz] dr* dr*zz

dt dxi dz dz dxi dz

where the viscous shear stress tensor can be expressed in terms of the instantaneous velocity field,

Uj

« i

A

= Ui+Ui,

= Ui + ui, = Ut + u\,

Inside the porous medium, the velocity and pressure fields can be decomposed according to Eq. (14). Each of these components can be resolved in a mean component (depth and time-averaged value) and a deviation from the mean,

(54)

where Ui is the seepage current, \n is the seepage oscillatory flow, U? is the spatial fluctuation of the current, U\ is the temporal fluctuation of the current, u\ is the spatial fluctuation of the oscillatory flow and u\ is the temporal fluctuation of the oscillatory flow.

In order to derive the time and depth-averaged equations, Losada (1996) carried out the following operations,

• The continuity equation, Eq. (50), is averaged over a finite volume of porous medium and in a time scale smaller than the characteristic wave period resulting in a continuity equation in terms of the seepage components.


• This equation is integrated from the bottom z = — h to the phreatic surface z = j). Applying Leibniz's rule using the kinematic boundary conditions and averaging over a wave period yields,

on dim±m = dt dxi ' K '

where

"'-TC {ah £«*)"• <56)

Following a similar procedure with the momentum equations (see Losada, 1996 for details) and assuming that no correlation exists between the seepage flow and the fluctuations and that the correlation between the mean and oscillatory flow is also very weak, the following equations can be obtained,

d[pUj{fj + h)} dfj d t

— m — = ~"{T]+h)dVi-d^i {Sij+5«+5«>

d p(fj + h)(U!U° + U*U*) +TJ+Rj, (57) dx,

where

P 1 _ S^ = / (pSij + puiUj) dz- - pg(rj + h)2Sij , (58)

J-h 2'

(59) S^ = / pufuj dz , J — h

%,= T pS$dz, (60) J —h

Sij is the (i,j) component of the stress tensor representing the excess of momentum flux due to seepage magnitudes. S*j are S\j the stress tensor components due to the spatial and temporal fluctuations. These two terms may be considered lateral mixing mechanisms induced by the temporal and spatial fluctuations due to pore irregularities.

The horizontal force term due to the oscillatory motion T; is given as:

Ti = (pU-pg(n + h))^-Pg(h + n)§^. (61)

188 /. J. Losada

The frictional terms on the free surface F(x,y,z) = 0 and bottom B(x,y, z) = 0 have the following expression,

Ro = -^-.fnjdz + Vf\VF\+^\WB\. (62)

Equations (55) and (57) may be simplified under certain circumstances to obtain other equations that are important for practical applications. Assuming horizontal bottom, neglecting viscous forces (Rj = 0), and using Eq. (55), the momentum equations can be expressed as:

d f dfj\ d2fj 1 d2

dxi \ dxi) dt2 p dxidxj

S^Sij + S^+Slj + lUfUj + UUJl}. (64)

Equation (64) governs the forced long wave motion in a porous medium.

6.3. Applications

The application of the time and depth-averaged equations requires expressions for the wave-averaged quantities, mass flux, radiation stress, etc. in terms of wave height, wave period and water depths. Assuming irrotational flow and based on the potential <j) derived from Sollitt and Cross (1972) theory, expressions correct to the second order have been derived for porous media flow in terms of the seepage velocity. Mendez et al. (2000) provides general expressions of mass transport, radiation stress and energy flux for wave propagation inside or above a porous layer in terms of some shape functions defined accordingly.

Using an approximation, Mendez et al. (2000) applied a simplified ID version of Eq. (57) considering the forcing of the radiation stress of the seepage component in order to analyze the mean water level variations induced by the presence of a submerged permeable breakwater.

Figure 5 presents numerical and experimental results of wave height and mean water level variation for a submerged permeable breakwater. A description of the experimental conditions can be found in Rivero et al. (1998). Results show a modulation of the mean water level in front of the structure imposed by reflection. A set-down can be clearly observed at the front slope or above the breakwater crest just before the maximum wave height is reached before breaking. After breaking, a slightly modulated set-up is clearly observed leewards the structure in both the experimental and numerical results. Results


0.80

0.60

0.20 -

0.00

4.00

2.00

0.00

^ * ^ = * ^ * •J»«L»

Hr= 0.435 m T = 4.0 s r\ = -1.5 cm

L <V 1 0

V

----

2.5E-05

- • |R|

- - • mi

m2

0.0,

0.07

0.07

Cf 0.3

f=1.30

V

V

S=l

0°, f=1.24

-60° . f=1.34

9.0 x(m)

Fig. 5. Wave height and mean water level variation in a submerged breakwater under breaking conditions.

point out that the set-up induced by a permeable submerged breakwater is due to radiation stress gradients induced by dissipation associated with both breaking and friction. In general, the set-up due to breaking is much more important than the one produced by the porous material. However, the relative magnitude of the different contributions requires further exploration due to the lack of roller effects, flow separation or turbulent effects in this first approach to the problem.

Baquerizo and Losada (1998) used Eqs. (55) and (57) to analyze how dissipation inside an infinitely long vertical breakwater in a constant water depth generates an along-breakwater current inside the structure. For an obliquely incident wave train, dissipation of energy inside the pores produces a radiation stress gradient which is balanced by the frictional and the diffusive terms in

190 /. J. Losada

the equation. The current driven inside the porous medium is transferred by turbulent diffusion seawards and leewards the structure.

The solution of the equations requires an expression for the radiation stresses associated with the fluctuations. Baquerizo and Losada (1998) based on a Boussinesq approach give approximated expressions introducing eddy viscosity coefficients which take into account the effect, over the mean velocity, of the turbulence induced by the spatial and temporal fluctuations respectively. Approximate expressions for these terms need to be explored further.

Losada et al. (1998) analyzed the mean flows in vertical rubble-mound structures. Using an analytical approach based on Sollit and Cross (1972) theory, it is shown that waves impinging on rubble mound breakwaters and seawalls induce a mean flow within the breakwater analogous to the so-called undertow within the surf zone. The mean flow is expressed in terms of a mean stream function which is analytically derived and is based on the mass flux balance between the incident, reflected and transmitted waves. It can be shown that the rapid decrease in the Eulerian mass transport results in a mean vertical velocity that capable of inducing a mean flow that exits mainly at the toe of the structure. The higher the reduction in mass flux, the stronger the mean return current.

z/h

*F in a vertical rubble-mound breakwater

|m|=0.20 _

Mean horizontal velocity profile

-m M3

V :*#*

Return flow

n If • * • • • • i

-|m|=0.50

-]m|=p.20

-•|iii=0.U,-

Fig. 6. Streamlines and mean horizontal velocity profiles in a finite breakwater, b/h = 1, kh = 0.34, and ne = 0.4.


The upper panel of Fig. 6 shows the calculated streamline patterns for a finite porous breakwater. The admittance m = neT/(s — if) where V is the complex wave number in the porous medium takes into account the hydraulic characteristics of the structure. Increasing friction results in diminishing admittance.

The lower panel presents a qualitative plotting of the horizontal mean velocity profiles for different admittances. The results indicate that for \m\ = 0.5, the smallest friction considered, the influence of the presence of the structure on the mean velocities is almost negligible. However, decreasing admittance turns out in higher return flow due to increasing mass flux decay in z =0. For 177i| = 0.14, the velocity profile presents a large curvature and showing a net flux in the direction of wave propagation near the top of the structure.

7. Modeling Based on the Navier- Stokes Equations

In order to overcome most of the limitations associated with previous models, important efforts have been made in the last few years to develop a tool capable of successfully simulating the free surface of breaking waves on permeable structures. Three succesful examples are the models SKYLLA (van Gent, 1995), VOFbreak (Troch and de Rouck, 1998) and COBRAS (Liu et al, 1999, 2000), all based on 2D Navier-Stokes equations for the fluid and porous regions and making use of the VOF method to track the free surface.

COBRAS (COrnell BReAking wave and Structure) has been initially developed to track the free surface movement and to describe the turbulence generated by the wave breaking process on slopes (Lin and Liu, 1998a, 1998b).

The breaking waves numerical model is based on the Reynolds Averaged Navier-Stokes (RANS) equations. Although the model has only been applied to two-dimensional problems, the complete three-dimensional formulation will be given herein. For a turbulent flow in the fluid region, the velocity field and pressure field can be divided into two parts: The mean (ensemble average) velocity and pressure and (u$) and (p), the turbulent velocity and pressure u\ and p'. Thus,

Ui = (ui) +u'i, p= (p)+p' (65)

in which i = 1,2,3 for a three-dimensional flow. If the fluid is assumed incompressible, the mean flow field is governed by the Reynolds Averaged

192 /. J. Losada

Navier-Stokes equations,

9(uj)

dxi 0,

dt (uj) %)=_19(p)+ { ld{rtj) djuty)

dxj p dxi l p dxj dxj

(66)

(67)

in which p is the density of the fluid, gi the zth component of the gravitational

acceleration, and mean molecular stress tensor (r^) = 2p{<7ij) with p being

the molecular viscosity and (cr^) = \{ ^" ' ' + QJ ) the rate of strain tensor of

the mean flow. In the momentum equation as in Eq. (67), the influence of the

turbulent fluctuations on the mean flow field is represented by the Reynolds

stress tensor /o(w^u'). Many second-order turbulence closure models have been

developed for different applications which have been summarized in a recent re

view article (Jaw and Chen 1998a, 1998b). In the present model, the Reynolds

stress p(u^u') is expressed by a nonlinear algebraic stress model (Shih et al.,

1996; Lin and Liu (1998a, 1998b):

p(uiuj) = ^pkSij -Cap-It2 (d(m) 9{UJ)

dxi dxi

~P~2

+ C2

Ci d(ui) d(ui) d{uj) d(ui)

dxi dxj dxi dxi

d(uj) d{uj) _ 1 d{ut) d(ui)

dxk dxk 3 dxf. dxk

2 d(ui) d(uk)

3 dxk dxi

+ C, fd(uk) d{uk)

V dxi dxj

1 d(ui) djm) 5

3 dxk dxf. lJ (68)

in which Cd, C\, C2, and C3 are empirical coefficients, Sij the Kronecker delta, v((d^L)2) ^ n e dissipation rate k = ^{u'iii'i) the turbulent kinetic energy, ande

of turbulent kinetic energy, where v = p/ p is the molecular kinematic viscosity.

It is noted that for the conventional eddy viscosity model C\ = C2 = C3 = 0 in

Eq. (68) and the eddy viscosity is then expressed as«( = Cd — - Compared with

the conventional eddy viscosity model, the nonlinear Reynolds stress model in

Eq. (68) can be applied to general anisotropic turbulent flows.


The governing equations for k and s are modeled as (Rodi, 1980; Lin and Liu, 1998a, 1998b):

dk . . dk d

dt dxj dxj i\crk vt + v

de de

m + ^dx-_d_ dxj

djuj) dxj

vt_

-a

dk dxj

de

dx-j

« « ; • >

d(uj)

dxj (69)

•Cu\rH d(ui) 8(UJ)

dxj dxi

2e k (70)

in which o^, a£, C\e, and Cie are empirical coefficients. In the transport equation for the turbulent kinetic energy, Eq. (69), the left-hand side term denotes the convection while the first term on the right-hand side represents the diffusion. The second and the third term on the right-hand side of Eq. (69) are the production and the dissipation of turbulent kinetic energy respectively.

The coefficients in Eqs. (68) to (70) have been determined by performing many simple experiments and enforcing the physical realizability; the recommended values for these coefficients are (Rodi, 1980; Lin and Liu, 1998a, 1998b):

Cd =

C2 = -

1 3 V 7.4 + 5 m a x

1

C i

C,

1 185.2+ D2

(71) 58.5 + D ^ ' 370.4 + D ^ x '

C l £ = 1.44, C2e = 1.92, <7fc = 1.0, <7e = 1.3,

where 5, max — 7 max | 1^1 (repeated indices not summed) and Dn

- maxf! Appropriate boundary conditions need to be specified. For the mean flow

field, the no-slip boundary condition is imposed on the solid boundary and the zero-stress condition is required on the mean free surface by neglecting the effect of airflow. For the turbulent field near the solid boundary, the log-law distribution of mean tangential velocity in the turbulent boundary layer is applied so that the values of k and e can be expressed as functions of distance from the boundary and the mean tangential velocity outside the viscous sublayer. On the free surface, the zero-gradient boundary conditions are imposed for both k and e , i.e., §^ = | p = 0. A low level of k for the

194 /. J. Losada

initial and inflow boundary conditions is assumed. The justification for this approximation can be found in Lin and Liu (1998a, 1998b).

In the numerical model, the RANS equations are solved by the finite difference two-step projection method (Chorin, 1968). The forward time difference method is used to discretize the time derivative. The convection terms are dis-cretized by the combination of central difference method and upwind method. The central difference method is employed to discretize the pressure gradient terms and stress gradient terms. The VOF method is used to track the free surface (Hirt and Nichols, 1981). The transport equations for k and e are solved with the similar method used in solving the momentum equations. Detailed information can be found in Kothe et al. (1991), Liu and Lin (1997), and Lin and Liu (1998a, 1998b).

The mathematical model described above has been verified by comparing numerical results with either experimental data or analytical solutions. The detailed descriptions of the numerical results and their comparison with experimental data can be found in Liu and Lin (1997) and Lin and Liu (1998a, 1998b). The overall agreement between numerical solutions and experimental data was very good.

The flow in the porous domain is described using Eqs. (21), (22), and (11). In order to couple the flow inside and outside the permeable structure, Liu

et al. (1999) applies continuity of the mean and averaged velocity and pressure across the interface of porous media and outside flow. Strictly speaking, the outside mean (ensemble-averaged) flow is not equivalent to the spatially-averaged flow in porous media since the latter may still contain turbulent fluctuations. However, as previously explained, these turbulent fluctuations are in general negligible.

Please note that the turbulence model is not solved in porous media. Therefore, the evaluation of the turbulence kinetic energy needs a special treatment. For details, see Liu et al. (1999).

The model is suitable to analyze wave interaction with emerged or submerged permeable or solid structures, both for breaking or nonbreaking conditions.

Figure 7(b) shows the comparison of calculated and experimental time histories of free surface displacement before and after the wave passes a trapezoidal porous structure, Fig. 7(a). Reasonably good agreements are obtained. In Fig. 7(c), the vertical and horizontal components of the velocity at two locations are shown. Both experimental and numerical results show that because


0.6

0.5

0.4

It 0.3

0.2

0.1

measurement sections

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 x(m)

(a)

free surface displacement at x=0.24 m (e), 0.33 m (f), 0.42 m (g), and 1.42 m (h)

E u

E o

E o

E

10 0

-10 20 10 0

-10 20 10 0

-10 20 10

(f)

(g)

/ \

A

i\

(b) Fig. 7. (a) Wave transformation above a trapezoidal submerged breakwater. Location of the free surface gauges, (b) Free surface displacement at different locations. Comparison between experimental and numerical results, (c) Horizontal and vertical velocity time histories at x = 0.24 and at different depths. Experimental and numerical results.

196 /. J. Losada

u & v at x=0.24 m and y=0.30 m (a), 0.16 m (b), 0.12 m (c), 0.10 m (d) and 0.08 m (e)

- 0 . 5 ' ' ' = — ' 3 ' -=> 4 6 8 10 12 14

t ( m )

(c)

Fig. 7. (Continued)

of the presence of the structure, the horizontal velocity near the structure is enhanced while the vertical velocity is reduced. The agreement is again quite good.

8. Conclusions

Very important progress has been achieved in the field of modeling wave and permeable structure interaction in the last years. The modeling is based on the coupling of two models describing the flow acting on the structure and through the porous structures.

Several new equations including the resistance forces in the porous medium have been derived covering an ample range of applications. The accuracy of the modeling of wave and permeable structure interaction relies greatly on some constants depending on the flow. Even if some progress has been carried out to determine predictive formulas for these constants under oscillatory flow conditions, uncertainty is still present especially if these models are considered to be an alternative to physical modeling in the design of coastal structures.

Equations for wave and structure interaction have evolved in parallel with equations for wave-propagation in fluids. Modified Boussinesq equations or other kind of shallow water equations are currently available. However, the


application by the engineering community is limited. From the engineering point of view, there is still additional work to do in order to include the effect of permeable structures in the wave propagation modeling.

For wave and permeable structure interaction where the numerical domains are relatively small, models based on the RANS equations seem to be the most suitable. However, this kind of model requires further work and validation before being useful for engineering applications.

9. Future Work

Further research on the determination of predictive expressions for the porous flow parameters under oscillatory flow conditions is needed. It is clear that in the near future, models solving the Navier-Stokes equations have an enormous potential to analyze wave and permeable structure interaction. Therefore, COBRAS should be improved in several aspects.

The enforcement of the continuity of the mean and averaged velocity and pressure across the interface results in unrealistic velocity information just outside the porous media. Just at the porous media surface, the flow forms turbulent jets and wakes that in most practical applications will be quickly mixed within a short distance. If detailed velocity information at locations very close to the porous surface is needed, the present approximation results should be improved. This could be the case for the development of structure stability models where the forces on individual units have to be determined. Therefore, the modeling of turbulence inside the structure should be explored. Furthermore, different turbulence closure models need to be considered.

The extension of COBRAS to three-dimensions would imply a considerable improvement to the current model.

Acknowledgment

The financial support from the Spanish Comision Interministerial de Ciencia y Tecnologia (CICYT) through grant MAR99-0653 is gratefully acknowledged.

References

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List of Symbols

/ = hydraulic gradient p = fluid density g — gravitational acceleration po = the effective pressure ho = the vertical distance from the selected datum ud = discharge velocity K = l/ap (m/s) permeability coefficient ap = empirical coefficient Kp (m2) = intrinsic permeability Re = ucDc/v = Critical Reynolds number uc = characteristic discharge velocity Dc = characteristic length scale of the porous media v = molecular viscosity bp = empirical coefficient with dimension (s2/m2) ne = the porosity ap = nondimensional coefficient (3P = nondimensional coefficient cp — empirical coefficient (s2/m) 7 = nondimensional coefficient accounting for the added mass fl = resistance force due to laminar flow fff = resistance force due to turbulent flow / / = resistance force due to inertia KC = Keulegan-Carpenter number u* = ith component of the instantaneous velocity in the pores p* = instantaneous effective pressure Ui = ith component of the seepage velocity u\ = ith component of the spatial perturbation u\ = ith component of time perturbation Cj = dimensionless turbulent coefficient s = co-factor accounting for added mass / = linearized friction coefficient u> = wave frequency h = water depth b = porous structure width $ = velocity potential fo = energy dissipation function associated with wave breaking

202 I. J. Losada

To = T.R + iTi = complex wavenumber in the porous medium C = free surface displacement Hrms = mean-root-square wave height Tp = peak period hef = effective water depth 4> = depth-averaged potential u = depth-averaged potential tp = depth-averaged piezometric head 5ij = Kronecker delta k = turbulent kinetic energy £ = dissipation rate of turbulent kinetic energy Ui = seepage current Ui = seepage oscillatory flow U* = spatial fluctuation of the current U\ = temporal fluctuation of the current u\ = spatial fluctuation of the oscillatory flow u\ = temporal fluctuation of the oscillatory flow Sij = components of the radiation stress due seepage magnitudes Sfj = components of the radiation stress due to the spatial fluc

tuations Sjj = components of the radiation stress due to the temporal fluc

tuations

DESCRIPTIVE HYDRODYNAMICS OF LOCK-EXCHANGE FLOWS

HARRY YEH and KIYOSHI WADA

Salt-water gravity currents and internal bores are created in a horizontal flume with a lock-exchange device, i.e., by lifting a partition gate that initially separates fresh water from salt water. Using laser-induced fluorescein-dye illumination in the laboratory, qualitative characteristics and behaviors of the flows are examined. In the second set of the experiments, gravity currents and internal bores are first generated in a channel of finite breadth to establish its quasi-steady condition. Then, the established current is let off to spread from the end of the channel to open environment. The spreading flow pattern and mixing around the end of the channel is examined and interpreted based on vortex dynamics.

1. Introduction

Gravity currents are gravity-driven flows by the fluid-density difference from that of its surroundings. When a similar flow advances into a quiescent two-layer fluid of which the thinner layer has the same fluid density as the advancing current and distinct wave breaking is formed at the leading wave, the flow is called an internal bore. Gravity currents and internal bores are investigated in a horizontal laboratory tank with a lock-exchange device, i.e., by lifting a partition gate that initially separates fresh water from salt water. Many researchers have investigated the fundamental flow characteristics of gravity currents and internal bores. What we wish to present here is the clarification of features and mixing mechanisms associated with gravity-current and internal-bore phenomena, and the differences between the two flows. Furthermore, observing the sudden expansion of the current to open environment, the three-dimensional flow patterns and resulting mixing mechanisms are described. Features and hydrodynamics of such flow phenomena are discussed particularly in terms of vortex dynamics.

Based on the assumptions of an inviscid fluid and a two-dimensional flow field between a pair of infinite parallel horizontal plates (no free surface

203

204 H. Yeh & K. Wada

involved), Benjamin (1968) established the lowest-order theory for gravity currents flowing on a horizontal bed. Combining the depth-integrated linear-momentum equation and continuity, Benjamin derived that:

^ = ( 2 ^ i ) ( l - § ) g'h (1 + 1)

(1)

where, as shown in Fig. 1, h is the current depth, d is the total flow depth, which is the spacing between the two parallel horizontal plates, U is the speed of the front, F is the densimetric Froude number, g' = (p2 — Pi)d/P\ is the buoyant acceleration, where p\ and p2 are the fluid densities of the ambient and the current respectively, and g is the acceleration of gravity. Benjamin (1968) ironically found that the inviscid theory breaks down except for in a very special case; the only possible solution to satisfy Eq. (1) and the Bernoulli theorem is h/d — 1/2 and F = 2 - 1 / 2 . The steady-state condition of h/d > 1/2 is not possible, while the condition of h/d < 1/2 requires energy dissipation by wave breaking. Benjamin's solution of this very special case implies that energy dissipation associated with turbulence must play an essential role in determining the dynamics of gravity currents. Not only for that, turbulence that intrinsically generated must cause mixing of two fluids that make up the gravity currents. Figure 1 summarizes the features of an inviscid gravity current.

Several extensions of Benjamin's work by incorporating energy dissipation are proposed by Wilkinson and Wood (1972), Chu and Baddour (1977), Wood and Simpson (1984), Denton (1990), and Klemp et al. (1994). These models

' ' * •> ' ' £ <f r S * £ £. -» -* ' ' '

1

h /7I/3 i

• / / /—7—7—7—7—7—7—7-7-7—7—?—r

Fig. 1. A schematic of inviscid gravity current that is equivalent to the model made by Benjamin (1968). For the inviscid model, the height of gravity current is one-half of the channel height and the leading edge contacts at 7I"/3 with the bed, which is basically the Stokes 120° corner flow.

Descriptive Hydrodynamics of Lock-Exchange Flows 205

are based on the mixing-layer formation. The model by Wood and Simpson assumes that energy dissipation occurs only in the lower layer, whereas Denton's model assumes energy dissipation in the upper layer. Using a perturbation analysis, Jirka and Arita (1987) showed that the head formation of a gravity current becomes a density wedge (without a head formation) in the case of momentum deficit relative to the inviscid case. A steady gravity current can be maintained only if there is a momentum surplus.

In his book, Simpson (1987) summarized that, as a gravity current advances, a characteristic head is formed at the leading edge of the gravity current, which is approximately twice as deep as the following flow, H « 2/i, as depicted in Fig. 2. The characteristics of the head are considered to control the entire flow behavior, e.g., mixing, velocity of the advancement, and its profile. The head profile is sensitive to the Reynolds number (R = Uh/is), the densimetric Froude number (F = U/y/g'h), and the ambient flow conditions (Note that in the Reynolds number, v is the kinematic viscosity of the fluid). There are according to Simpson (1987) two types of instabilities that are responsible for mixing associated with gravity currents: (1) billows which roll up in the region of velocity shear above the advancing front and (2) a complex shifting pattern of "lobes and clefts" located on the face of the head. Simpson suggested that the billow formation on the upper surface of the head is similar to the Kelvin-Helmholtz instability. He also indicated that the formation of "lobes and clefts" develops from the lighter fluid that is overrun by the gravity current's foremost leading edge. The foremost leading edge of the current is located slightly above the bed; approximately 1/8 of the total height of the head. Hence, some ambient fresh water is trapped under the head and

Fig. 2. A sketch of a typical gravity current in a laboratory channel. The forefront nose is approximately 1/8 of the total height of the head H. Because of this feature, the lighter fluid can be entrapped under the nose, which causes the formation of flow pattern "lobes and clefts" on the front face of the head. The broken line indicates the location of laser sheet used for Fig. 6.


entrained into the current from the bottom. This trapped lighter fluid is ad-vected upward through the denser fluid due to buoyancy to form a complicated shifting pattern of "lobes and clefts" on the front face of the head.

The majority of previous work was to investigate the phenomena in a two-dimensional channel, i.e., the mean variations in the horizontal direction perpendicular to the current direction were neglected. Investigation remains insufficient for the mixing processes of two-layer flows disturbed by "vertically intruding" objects which evidently cause three-dimensional disturbances. Such a mixing process occurs, for example, at irregular estuarine side boundaries, jetties, piles and the like, and appears to be effective and efficient in mixing. It is noted that locally created vertical mixing often causes horizontal density gradients which may drive a transverse circulation in the entire water body (Fischer, et al, 1979; Maxworthy and Monismith, 1988); this process could potentially be dominant for mixing in some estuaries.

In this paper, we first discuss fundamental features of gravity currents and internal bores. Then, three-dimensional mixing characteristics associated with sudden expansion of gravity current and internal bore in a two-dimensional channel into open environment are examined. Strong three-dimensional disturbances are observed with flow visualization and the dominant flow characteristics are identified. Flow behaviors associated with the expansion should be similar to the disturbance caused by "vertically intruding" objects.

2. Experimental Facilities

Two different experimental apparatuses were used to examine a variety of flow behaviors associated with gravity currents and internal bores. First, traditional experiments for lock exchange flows were performed in a 16.2 in long, 0.61 m wide, and 0.45 m deep tank. The tank was initially divided into two separate chambers with the aluminum gate: The front chamber is 8.9 m long and the back chamber is 7.3 m long. For the gravity current experiments, uniformly mixed salt water filled the back chamber, while fresh water filled the front chamber. For the internal bore experiments, a thin layer of salt water was placed beneath the fresh water in the front chamber, and the fluid density of the thin layer was identical to that of the salt water in the back chamber. A sharp interface was established by introducing salt water slowly through the diffuser along the bottom against the side wall.

Throughout a series of experiments, the total water depths of both front and back chambers are set at 36 cm. A gravity current or an internal bore


was created as the lock-exchange flow by lifting the aforementioned aluminum gate vertically to 20 cm from the bed. This partial gate opening is a similar generation scheme as that used by Wood and Simpson (1984), which minimizes free-surface disturbance created at the gate and controls the current depth to approximately 9 cm.

Horizontal and vertical laser sheets were employed to resolve the flow behavior of gravity currents and internal bores. A 4-watt Argon-ion laser beam is shone onto the oscillating mirror, which produces a laser sheet. The laser sheet illuminates the fluid dyed with disodium fluorescein (C20H10O5N2) in the flow. The dye is previously mixed with the saltwater. The Argon-ion laser operates in the 457.9-528.7 nm wavelength range, i.e., the green spectrum. Fluorescein dye is excited by the green spectrum and becomes a brilliant yellow-green. This flow visualization technique is often referred to as the laser-induced fluorescence (LIF) technique.

Observations by the LIF technique were made in the front-chamber area 4.5 m downstream of the gate, approximately 50 times the average saltwater flow depth. All the transient disturbances caused by the gate motion should have sufficiently subsided in the area of the observation and measurements. It is also noted that the length of the back chamber is long enough so that flows in the observation area are not disturbed by the reflection from the back-chamber end wall.

The conductivity-temperature instrument was used which measures the electrical conductivity and temperature of a solution. An estimate of the spatial resolution is a 1 mm diameter sphere. There is 1 mm separation between the conductivity sensor and the thermistor bead, which implies that the two sensors are reading nearly the same parcel of fluid. The traversing system was used at its maximum traveling velocity in order to obtain an "instantaneous" profile of the flow. At the maximum velocity, the traversing system takes about 2 s to travel down and up through the water column. With the data-scanning rate of 400 scans/s, the spatial resolution of the probe is approximately 1 mm. More detailed discussions of this laboratory apparatus are given in Grandinetti (1992).

The second set of experiments was designed to examine the flow behaviors and characteristics associated with flow expansion of established gravity currents to a much wider reservoir. This series of experiments was performed in a 3.0 m long, 1.2 m wide, and 0.9 m high glass-panel water tank. Within the tank, a false partition wall was placed parallel to the tank sidewall to form a


(a) Elevation View (A-A Section)

Scanner . . . . . ; • • ' • - ' - ' .

vyZtCvyA Q I » A I > fa, Laser\ W. (Unit:cm)

V

Iho

t

I Mirror |

f^Laser \ - /~ jsheet^ \ .

Gate

z ' Fresh Water J Salt Water [

X 300

J8 II

•o ,

t

(b) Plane View

i 180

Partition Wall 9° »

CM II n

Not to scale

Fig. 3. Schematic views of the experimental set up: (a) elevation view and (b) plan view. Note that the origin of a Cartesian coordinate system with the right-hand rule is taken at the end of the intersection between the partition wall and the channel bed.

0.22 m wide and 1.80 m long confinement as shown in Fig. 3. Gravity currents and internal bores were also generated by the lock-exchange mechanism, i.e., by lifting the gate to form a flow pattern guided by the confinement. The generated currents were then expanded into open environment at the end of the confinement. Note that the expansion took place on only one side of the channel.

In order to describe the locations and directions, we define the coordinate system as shown in Fig. 3. The origin is taken at the bottom of the channel at the tip of the partition wall. Taking Cartesian coordinates from the origin with the right-hand rule, the x-axis points horizontally in the direction extending the partially confined partition wall, the y-axis points horizontally in the direction toward the opposite wall and the z-axis points upward. We conveniently term the partition wall (at y = 0) the "right" wall and the fully extended wall (at y = 0.22 m) the "left" wall. The tip of the partition is identified by the location (x = 0,y = 0). The directions of fluid rotation are described based


on the right-hand rule of the coordinate system. We also set the time origin (t = 0 s) when the leading edge of the current passes at the position x = 0. The resulting flow patterns were visualized using the laser-induced fluorescent (LIF) technique.

3. Basis for Interpretations of Flow Images

The primary interest of hydrodynamics associated with gravity currents and internal bores is their ability to generate turbulence at the interface and subsequently to induce mixing. Hence, it is necessary to review briefly the mechanisms of turbulence generation. Turbulence is rotational flows although the inverse may not be always true: There are many examples for laminar rotational flows. On the other hand, there is no turbulent flow that is irrotational. On this basis, there are two possibilities to generate turbulence associated with gravity currents. First, the fluid is initially rotational (vortical) by some reason, and turbulence is generated by stretching and bending of pre-existing vortex tubes. Under the condition of lock exchange flows, both fluids (saline and fresh water) separated by the lock are considered to be quiescent and initially irrotational. The other possibility is that fluid rotation is created within the fluid domain during its propagation. The fluid rotation then leads to turbulence by stretching and bending the created vortex tubes.

For Newtonian mechanics, there must be a force to accelerate a fluid parcel, which is the statement of the conservation of linear momentum. Likewise, the conservation of angular momentum requires torque to create rotation of a fluid parcel when it is initially irrotational. In other words, if there is no torque, then fluid remains irrotational. From this fundamental point of view, we pay careful attention to torque that arises from lock-exchange flows.

Fluid rotation can be conveniently measured by flow circulation T, which is defined by:

T= fudx= fuj-dA, (2) J c J s

where u is the fluid velocity, u> is the vorticity (curl u), x is the position vector of a closed integration contour c, and A is the area vector of the surface s whose boundary is a single closed curve c. The rate of change in circulation T following a fluid parcel that make up the curve c can be found to be (e.g., see Lighthill, 1987):

ft -Liix)-*1-!.&•*')•*• (3)


— (Vp x Vp) + - V2w - — (Vp x pV2u) dA, (4)

where r^- is the stress tensor, p is the fluid density, and D/Dt is the material derivative. In Eq. (3), we used the Cauchy equation with the conservative body force (e.g., see Serrin, 1959).

For incompressible Newtonian fluids with uniform viscosity, using the Stokes theorem, Eq. (3) can be written as:

DT

Dt j s |_p' ' ' • ' p (f

where p is the pressure and fi is the dynamic viscosity of the fluid. The second term on the right-hand side of Eq. (4) represents the diffusion of vorticity from fluid parcels adjacent to the boundary of the integration surface s. Hence, this does not represent vorticity creation within the fluid domain, but represents vorticity transfer from the surroundings (or the boundaries) due to viscous diffusion.

In the case of inviscid fluids, flow circulation can be produced within the fluid domain whenever the fluid is displaced from a state in which the pressure gradient Vp and the density gradient Vp are parallel, i.e., Eq. (4) can be modified to be:

f = / s > P x V p W A . (5)

This is often called Bjerknes' theorem (see, for example, Lamb, 1932, Article 166-a). Equation (5) represents the time rate of change in circulation by torque acting on a fluid parcel: This torque is often termed "baroclinic torque": Physical interpretation of baroclinic torque was discussed in, for example, Yeh (1995).

The last term in Eq. (4) represents net viscous force acting upon a fluid particle. It is emphasized that, under the influence of a conservative body-force field, rotational motion can be produced from an initially irrotational state only by two mechanisms: Baroclinic torque and the interaction of density gradient and viscous shear force, and both mechanisms require the existence of density gradients. The latter rotation creation mechanism is termed as "viscous-shear torque" (Yeh, 1991, 1995).

If we further assume the fluid to be inviscid and homogeneous, then Eq. (5) is reduced to the Kelvin theorem:

Equation (6) states that flow circulation T around a closed curve c moving with the fluid remains constant.


Now at this point, it is evident that fluid rotation can be created within the fluid domain only by the action of baroclinic torque or/and viscous shear torque. Consider a typical flow pattern resulted from a lock-exchange flow moving with the velocity U. To compare the two mechanisms of fluid-rotation creation associated with lock exchange flows, first, we note that the gradient of fluid density Vp always points in the direction normal to the interface. Hence, the order of magnitude of the ratio of viscous shear torque l/p2Vp x /xV2u to baroclinic torque l/p2Vp x Vp along the frontal interface is:

/uV2Ma_b

Va_6p ' l >

where the subscript a-b indicates the component in the direction along the interface a-b as shown in Fig. 4.

* A / — 7 — 7 7 7—7—7—7 7—7 7 T—f T

L

Fig. 4. A definition sketch for the front of an internal bore. Fluid rotation is induced at the front due to the presence of sharp density gradient at the interface by baroclinic torque.

Since the divergence of viscous-shear stresses, /xV2u represents a net viscous force per unit volume acting on a fluid parcel and the force must be continuous across the salt-and-fresh water interface, the value of /iV2u at the interface can be evaluated either in the fresh or salt water domain. We estimate this value based on the assumption of thin-boundary-layer analogy in the domain of fresh water; the velocity outside of the boundary layer can be considered to be small at the front in the stationary reference frame (the same is not true in the salt-water domain and the thin-boundary layer analogy cannot be justified. Consequently, it is difficult to estimate the values of /LtV2u by using the saltwater velocity field). The Laplacian of the water velocity along the interface


72 . _. ,.d2Ua-b _,^f pU

can be estimated by:

/ i V ^ a _ 6 « P~-^ » O ( - ^ — ) , (8) a ; \ i c o s a /

where ua_b « U/ cos a is the water velocity parallel to the interface in the coordinates moving with the current, a is the angle of the front face from the horizontal, £ points in the direction normal to the interface as shown in Fig. 4, U is the propagation speed, and we used the boundary-layer thickness to be 0{y/vi) where v is the kinematic viscosity of the water {y = p/p). Along the frontal interface (a-b in Fig. 4), the time scale t can be estimated by t « L/U in which L is the length of the front as shown in Fig. 4, hence Eq. (8) can be written as:

pU2

/ J W - 6 » O -f . (9) \ L cos a J

The gradient of pressure field along the frontal interface may be estimated based on the assumption of hydrostatic pressure field, which is:

dri Va-bpK, pg—tt pg sin a, (10)

where r] is the elevation of the interface and £ points in the direction along the interface a-b as shown in Fig. 4. The ratio of baroclinic torque to viscous shear torque is then written as:

MW-6 W U2 W U2

(11) Va-bp gL sin a cos a gh* cos2 a '

where h* is the height of the front, i.e., h* m L tan a (see Fig. 4). It is emphasized that g is the acceleration of gravity but not the reduced gravity g ^ g' = gAp/p, that often arises from the analyses of stratified flows. Our estimate in Eq. (11) demonstrates that with the exception of forced flows such as a jet by maintaining the interface slope very small, the ratio of viscous shear torque to baroclinic torque is extremely small. For example, the front face of typical lock-exchange flow with Ap/p = 0.002, p = 1,000 kg/m3 and fj, = 0.001 kg/m-s, the densimetric Froude number U/yfgH is approximately unity, hence U2/gh w U2/gh* is 0.002. If the interface slope is less than 25 degrees, the baroclinic torque is more than 400 times greater than the viscous shear torque.

Even without the aforementioned quantitative estimate, it can be argued that the magnitude of viscous shear torque is usually small at the interface. It


is because the inflection point of the velocity profile of a two-layer flow is often located at or very close to the density interface. At the inflection point where the curvature of the velocity profile vanishes, i.e., V 2 u = 0, hence, there is no viscous shear torque. It is now fair to say that the dominant mechanism for the creation of fluid rotation for free lock-exchange flow is baroclinic torque at the salt-fresh water interface. The flow characteristics and behaviors that we will present in this article will be primarily interpreted based on this concept.

4. Features of Gravi ty Cur ren t s

Figures 5 and 6 show our laser-induced-fluorescence flow images in a longitudinal vertical plane at the center of the tank and in a horizontal plane 10.5 cm above the bed respectively. For the images shown in Fig. 6, the location of the laser illumination plane is depicted in Fig. 2. Only the salt water (p2 — 1002 kg/m3) was dyed with fluorescein and appears as bright regions having been illuminated by the Argon-ion laser sheet. The Reynolds number and the densimetric Froude number of this flow are R = 1450 and F = 0.995 respectively.

The flow images in Fig. 5 show the formation of billows on the front face; these small-scale billows are found to be three-dimensional. This can be verified in Fig. 6 where the front's pattern in the horizontal plane is irregular. This is different from the features of a typical Kelvin-Helmholtz instability in a plain shear flow that is, for example, created by a splitter plate; the billows

(a) (b)

Fig. 5. Gravity-current flow structure in the longitudinal vertical plane, the Reynolds number R = 1450 and the internal Froude number F = 0.995. (a) Laser-induced fluorescence shows the irregular front face and the ascension of entrapped fresh water and (b) the inverted roll-up under the nose of the leading front. Time interval between (a) and (b) is 5.6 s. Note that the faint horizontal band inside the gravity current shown in the photographs is the reflection from the diffuser.


associated with the Kelvin-Helmholtz instability are at least initially two-dimensional, i.e., the roll-up features of billows are long-crested. Instead, the three-dimensional front formation observed here is consistent with the "lobes and clefts" pattern indicated by Simpson (1987).

There are several intriguing features in Figs. 5 and 6 that need to be addressed. The underside entrainment of the lighter fluid at the front of the gravity current is clearly seen in Fig. 5. In Fig. 5(a), the dark streaks being flared near the very front along the bed show the ascension of entrapped fresh water due to its buoyancy. Figure 5(b) shows an underside entrainment; the inverted roll-up and entrainment of the ambient fluid under the "nose" can be identified.

The leading edge is irregular as seen in Fig. 6, including the formation of counter-rotating, mushroom-shaped eddies (Fig. 6(a)), which are presumably associated with the formation of U-shaped vortex loops on the front face. In the flow following the front (see Fig. 6(b)), the pattern is irregular, although it appears to be an alternating pattern of salt and fresh water rows in the propagation direction indicating periodic formation of large-size eddies.

5. Features of Internal Bores

Our experimental results in Fig. 7 show that the flow characteristics of the gravity current and internal bore are significantly different from each other, even when the pre-existing front layer of the denser fluid for the internal bore is thin (0.5 cm for Fig. 7(b), i.e., approximately 1.4 percent of the total depth while Figs. 7(c) and 7(d) are 1 cm and 2 cm respectively). Note that the Reynolds number and the densimetric Froude number of the flows shown in Fig. 7 are (a) R = 1450 and F = 0.995 and (b) R = 1440 and F = 1.00, (c) R = 1730 and F = 1.29 and (d) R = 1230 and F = 0.861 respectively. Unlike a gravity current, the front face of an internal bore is smooth, and turbulence is generated at the rear side of the head. Figure 7 shows that the thinner the initial front layer, the steeper the front face and the earlier the formation of billow roll-ups. Contrary to the case of gravity currents, the billow formation of an internal bore resembles that of the Kelvin-Helmholtz instability, which appears to be two-dimensional at least initially, i.e., the flow pattern is uniform in the direction transverse to the flow.

The flow pattern of the internal bore (the case shown in Fig. 7(d)) is further examined by introducing a small red dye (Rhodamine) patch in the initial salt-water layer as shown in Fig. 8. It is evident from the time sequence of


photographs that the initially undisturbed fluid of the thin layer moves along the front face (saline and fresh water interface) and the bottom boundary; i.e., it looks like the dyed fluid is coating the advancing bore head.

The distinct differences in the features between gravity currents and internal bores might be explained by Simpson's (1987) "lobes and clefts" formation at a gravity current's foremost leading edge. Such formations are not possible in an internal bore because ambient fresh water cannot be trapped along the bottom where the denser fluid already occupies. Further explanation can be given based on the vorticity creation mechanisms given by Eq. (4). Recall that there are only two mechanisms to create fluid rotation within the fluid domain (excluding at solid boundary surfaces). These are baroclinic torque (the first integrand term in Eq. (4)) and viscous-shear torque (the last integrand term) both of which require the presence of density gradient. It is emphasized that the second integrand in Eq. (4) represents the transfer of fluid rotation by vorticity diffusion from fluid parcels adjacent to the boundary of the integration surface, and does not represent creation of fluid rotation within the fluid domain. Also note that it was shown in Eq. (11) by an order-of-magnitude analysis that viscous-shear torque plays an insignificant role compared with the role of baroclinic torque.

By following the red-dye fluid patch shown in Fig. 8, the leading bore front consists of fluid parcels previously located in the quiescent thin layer in front of the bore. Hence, in the case of the internal bore, the fluid parcels along the interface are initially quiescent and irrotational. Fluid rotation must be created at the interface by baroclinic torque: The pressure gradient must be close to that of hydrostatic condition while the density gradient is normal to the interface as shown in Fig. 9. Once fluid rotation was created by baroclinic torque, the rotationality is advected with the fluid motion (Helmholtz's theorem) and diffuses by viscous effects. This property manifests itself in the formation of billows behind the ridge of the bore front.

On the other hand, for a gravity current, the fluid parcels along the interface are advected from inside of the advancing current as shown in Fig. 9(a). The fluid within the current is already vortical due to turbulence induced by (a) wave breaking behind the head, (b) entrapped fresh water under the leading edge or nose (see Fig. 5), and/or (c) the bottom (no-slip) boundary condition. Hence, fluid parcels along the interface are originally vortical. This vortically perturbed flow is affected further by the creation of fluid rotation by baroclinic torque at the front interface, which results in three-dimensional


Vp Vp

- 7 — 7 — v — 7 — 7 — r - -?—s / s s >

(a)

Vp Vp

/ / -r—r (b)

Fig. 9. A sketch of flow patterns at the front of (a) gravity current and (b) internal bore. Fluid-rotation is induced along the front face by baroclinic torque due to misalignment of the density gradient Vp that is perpendicular to the interface and the pressure gradient Vp that is approximately vertical for hydrostatic condition.

and complicated patterns at the interface of the head as shown in Figs. 5 and 6. The nature of inherently vortical flow at the gravity-current head can be an explanation for "lobe-and-cleft" formations as well as the reason for three-dimensional small-scale billow formations.

It appears in Figs. 7 and 8 that the vortex formation behind the head of the internal bore resembles that of a separation eddy associated with sudden flow expansion of the fresh water along a fictitious saline wall (i.e., the saline-fresh water interface). Furthermore, roll-ups form a periodic pattern of turbulent regions at somewhat constant intervals. The periodic features of turbulence patches might be related to the intermittent nature of the flow. It is conjectured that turbulence patches are formed by the "generation-advection" cycle of the roll-up formation. As soon as the roll-up is formed, the large vortex of the roll-up is advected behind the head, then the next roll-up is created, and this process is repeated. Yeh and Mok (1990) demonstrated that turbulence-patch formations behind bores at the air-water interface also result from the generation-advection cycle. In order to test this hypothesis, the periodic turbulence-patch formation is quantified by the frequency of its


generation; the generation frequency can be measured directly from the video images of our experiments. This measured frequency should be related to the frequency associated with the excursion time of a fluid parcel traveling around the vortex. Assuming the vortex size 5 (say the diameter of the orbital motion) is comparable with the difference in height between the head and the current depth behind the head, and using the vortex-velocity scale to be the bore propagation velocity U, then the time scale for roll-up is found to be Tr w irS/U. The vortex size S as indicated in the sketch shown in Fig. 9(b) (the actual length scale of the vortex is somewhat smaller than this estimation as seen in Figs. 7 and 8) and the propagation speed U were measured directly from the time-lapse flow images. Computed frequencies l/Tr shown in Fig. 10 are in a very good agreement with the measured frequency from the video images; note that the error bars in the figure represent the 90% confidence limit for the frequency measurements. The excellent agreement in Fig. 10 supports the conjecture that the periodic behavior of the turbulence patches

1.0

•o 0.8 o> t3 'S a> a. 0.6 f?

& 0.4 c ID 3 CT 0)

* 0.2

0.0 1 I T I I I

0.0 0.2 0.4 0.6 0.8 1.0 frequency (Hz) measured

Fig. 10. Predicted and measured generation frequencies of the large-scale eddies: O, gravity current (Fig. 7(a)); x , internal bore with ho = 0.5 cm (Fig. 7(b)); A, internal bore with ho = 1.0 cm (Fig. 7(c)); o, internal bore with ho = 2.0 cm (Fig. 7(d)). The error bars represent 90% confidence limits. The prediction is based on the hypothesis that each eddy is formed and advected at the rate of the excursion time for a fluid parcel travelling around the roll-up.


the three-dimensional variations appears to be plausible for the present case. To see this more clearly, consider the time scales of the vortex motion and the buoyancy effect. The buoyancy time scale TJ, can be estimated as the traversing time for a fluid-parcel to ascend through the vortex length scale S by the buoyancy force: Tb « y/5/g'. Assuming the vortex size 5 is comparable with the difference in height between the head and the current depth behind the head (see Fig. 9(b)) and using the buoyant acceleration g' = (p2 — Pi)g/pi, the time scale for the buoyancy effect is found to be T(, w 1.2 s for the case shown in Fig. 12. The roll-up time scale can be estimated as before: An excursion time for a fluid parcel to travel around the vortex. Using the vortex velocity scale to be the bore propagation velocity U, the time scale for roll-up is found to be Tr « nS/U = 3.8 s. This rough estimate demonstrates that the order of computed magnitudes of the buoyancy-effect time scale and of the vortex-motion time scale is comparable, which supports the explanation for three-dimensional variations discussed the above. Rather than the manifestation of longitudinal (streamwise) vortex formations in the Kelvin-Helmholtz instability, the gravitational instability is more likely the one that causes the transverse perturbation of vortices that appears in Fig. 12.

Typical density-profile data of an internal bore are presented in Fig. 13. The results in Fig. 13(a) show the mixing phase caused by overturning billows; the small-scale density inversions in the figures can be interpreted as the presence of small active eddies. Approximately 10 s after the passage of the head, these small-scale density inversions are diminished (Fig. 13(b)). At this point, the region of overturning billows is considered to be mixed, i.e., appreciable fluid mixing has already taken place in the vicinity of the head. The results appear to be consistent with our discussion made earlier, indicating that fluid mixing at the head is immediate and efficient in comparison to mixing in the current far behind the head. In fact, Figs. 13(c) and 13(d) show no density inversion that means no significant fluid entrainment being taken place.

Note that two separate mixing processes are represented by the appearance of two distinguishable gradients in the density profile plot (Fig. 13(d)). The upper portion of the profile has a uniform density gradient caused by large-scale mixing of overturning billows. As mentioned, this mixing process takes place directly behind the head. In the second gradient, the mixing process is attributed to the continual shear flow created by the fast-moving dense water under the nearly stationary mixed-fluid region. This fast-moving fluid layer is vortical due to the boundary-layer effect, which gradually erodes the layer, as


is

le

s '

n-

v«' V*

997 998 999 1000 1001 1002 1003

p (kg/m3)

(a)

15

3. 10 .c

-a

5-

S*.

^

0 997 998 999 1000 1001 1002 1003

p (kg/m3)

(b) Fig. 13. Time sequence of the density profile for an internal bore shown in Fig. 8, R = 1230, F = 0.861; (a) t = t0 , (b) t = t0 + 11 s., (c) t = t0 + 21 s. and (d) t = to + 32 s. The small-scale density inversions shown in (a) due to active eddies are diminished in 11 s. as shown in (b). This indicates that appreciable fluid mixing took place quickly. Thereafter, the density profile remains relatively unchanged.


15

? io-

a. T3

V \ .

* - . .

• ^

0 997 998 999 1000 1001 1002 1003

p (kg/m3)

(c)

15

-4—' a* o

5-

v.. " N

^

— 1 1 1 1 1 1 1 1 1 I 1 1

997 998 999 1000 1001 1002 1003

p (kg/m3)

(d)

Fig. 13. {Continued)


do the combined effects of shear instability and small-scale eddies within the lower layer fluid. It is also emphasized that the immediate mixing appears to take place for the fluid initially placed in front of the internal bore, while the fluid in the core of the internal bore tends to remain unmixed as shown in Fig. 11.

6. Flow Expansion of Gravity Currents and Internal Bores

Three-dimensional mixing characteristics associated with sudden expansion of a gravity current and an internal bore into open environment are examined using the second experimental apparatus (see Fig. 3). In order to demonstrate the effects of expansion, we first repeat the experiments in the second apparatus for the gravity current and internal bore in the confined narrow channel by blocking the expansion. It is evident in Fig. 14 that the similar flow patterns are obtained as those shown in Fig. 7. Note that the Reynolds number and the densimetric Froude number of the flows shown in Fig. 14 are (a) R = 1700 and F = 0.73 for the gravity current and (b) R = 1800 and F = 0.78 for the internal bore respectively. Effects of spreading of gravity current and internal bore can been seen clearly by comparing the longitudinal flow profiles in Figs. 14 and 15. Figure 15 shows the flow patterns at y/d = 0.17 where d is the total flow depth (0.15 m) and the location of x — 0 is indicated with the mark T in the figure (see Fig. 3 for the coordinates used in the discussion). The leading face of the gravity current has a complex flow pattern, while the internal bore has a smooth front face followed by the formation of roll up. While the difference in the head characteristics between the gravity current and the internal

(a)

•trKv

(b)

Fig. 14. Longitudinal profiles of (a) gravity current (R = 1700, F = 0.73) and (b) internal bore (R = 1800, F = 0.78) in a channel with a uniform width. Note that the features are similar to those shown in Fig. 7.


f

T

(b)

Fig. 15. Longitudinal profiles of (a) gravity current (R = 1700, F = 0.73) and (b) internal bore (R = 1800, F = 0.78) along the vertical plane at y/d = 0.17 when they spread out into open environment. Note that the location of the end of the false partition wall is marked by V in (b).

bore remains the same, the leading head heights are substantially reduced due to spreading. Behind the leading head, the flow depth is grossly depressed and then the mixed fluid is lifted up to the free surface at the location near x = 0 (Fig. 15).

The variations of the depressed flow depth behind the head are plotted in Fig. 16 for the gravity current (the results for internal bores are similar and not shown here). The depth of the two-dimensional gravity current in the confined channel remains constant at hm\n/d « 0.4. On the other hand, once the gravity current spreads into open environment (x > 0), the depth behind the head decreases substantially to hmjn/d w 0.1. The reduction of the depth occurs first near the confined wall (y m 0) and the effect propagates toward the left wall of the channel. Along y/d — 0.17, the depth decreases less than hm\x\/d ~ 0.1 and gradually increases to 0.1, i.e., overshooting the reduction, while along y/d = 1.30, near the left wall, the reduction is gradual without the overshoot.

The flow patterns of the gravity current and internal bore in the horizontal plane at z/d = 0.27 are visualized a.nd presented in Figs. 17 and 18 respectively.

* * * * /JTP?

(a)


0.6

*^

0.3

0

the tip of thei partition wall (x=0)\

(y/d) O 1.30 • 0.73 © 0.17 + -0.3 ffl 2-D (y/d=0.73)

0 £i JC Jbl C&-

Fig. 16. Variations of the depressed water depth behind the leading head of the gravity current. The depth /imi„, as identified in Fig. 15, is normalized with the total flow depth d, which is plotted against the location of the leading head.

f=1.2s f = 11.8 s

f = 7.1s t= 18.9 s

Fig. 17. Time sequences of the flow patterns of gravity current (Fig. 15(a), R = 1700, F = 0.73) in the horizontal plane (the X-AJ plane) at the height z/d = 0.27. The location of the end of the partition wall (x = Q,y = 0) is marked by • . Note that the time origin (t — 0 s) is set at when the leading edge of the current passes at the position x = 0 (the end of the false partition wall).


( = 0.0s t~ 13.7 s

A

r = 8.0s

Fig. 18. Time sequences of the flow patterns of internal bore (Fig. 15(b), R = 1800, F = 0.78) in the horizontal plane (the x-y plane) at the height z/d = 0.27. The location of the end of the partition wall (x — 0,y = 0) is marked by • . Note that the time origin (! = 0 s) is set at when the leading edge of the current passes at the position x — 0 (the end of the false partition wall).

In Fig. 17, the gravity current shows the complex leading front pattern just as in Fig. 6. On the other hand, the leading front of the internal bore is smooth as shown in Fig. 18 (also see Fig. 12). Time sequences of Figs. 17 and 18 clearly show that once the front leaves the confined channel, the fresh water inflow is introduced from the region y < 0 into the channel.

While the current is advancing within the confined channel (x < 0), the fresh water and salt water are simply exchanged within the confined channel just like the flow in a two-dimensional channel as shown in Fig. 14. Once the front escaped from the confined channel (x > 0), the fresh water intrudes from the side (y < 0) around the end of the right wall to supplement the volume loss due to the saline current outflow into open environment. This fresh water intrusion from the side around the end wall causes the sudden flow-depth reduction behind the leading head as well as inducing the strong three-dimensional mixing and upwelling near the end wall as shown in Fig. 15. For the internal bore, the roll-up (wave breaking) behind the head manifests itself as a white streak line in Fig. 18. Observation of the time evolution in Fig. 18 suggests that the roll-up is not aligned to the direction of the front, but the direction is oblique to the spreading current. Recall that the roll-up and the front are usually parallel in a two-dimensional channel as shown in Fig. 12.

t- 16.1s


o

3

90

75

60

45

30

G J - - -0 • d Q----H-Q

•'": . t

, \ l . B . ( * „ =

-o

A 1 G.C.

\ I.BJ0\=lcm) O.Scm)

i , i

30 40 50 60 70 ?(S) 80

Fig. 25. Temporal variations of the side interface slope f3 as indicated in Figs. 23 and 24; • , gravity current; D, internal bore with ho = 0.5 cm; o, internal bore with ho = 1.0 cm. The peak locations of dominant oscillations are marked by: — • — for the gravity current;

D for internal bore with ho = 0.5 cm o for internal bore with ho = 1.0 cm.

The time variations of the side slope (3 at x = 2 cm as shown in Figs. 23 and 24 are plotted in Fig. 25. Note that the side slope (3 is much steeper, (3 > 45°, than the front slope as shown in Fig. 14. Also noted that, unlike the front slope of the gravity current being much steeper than that of the internal bore, the values of /3 for gravity current and internal bore are comparable. It is partly because the location of the measurements is right after the end wall (x = 2 cm) and its outflow momentum disallows the flow to spread at this point. The slope (3 for both cases fluctuates at comparable rates, the frequency 0.1 ~ 0.15 Hz. This pulsating side slope might be related to the eddy formation due to baroclinic torque on the side face. The eddy formation process is not continuous but discrete, i.e., the process repeats its creation and advection. This creation-advection cycle may have caused the distinct fluctuation of the side slope.

Using Eq. (5) and based on the flow-visualization results described herein, the following flow patterns can be inferred for the gravity current and internal bore when they spread out from the confined channel. Once the current exits the channel into open environment, the effects evidently first appear near y « 0. Near the exit point x w 0, the side interface is nearly vertical and the fluid there appears to be vortical due to advection of fluid from the boundary layer along the tank's side wall (Figs. 23 and 24): The initial vorticity at the


Fig. 26. Generation and transport of fluid rotation near the exit to open environment. While the vortical flow along the side-wall boundary is advected to the open environment, the creation of vertical salt and fresh water interface at the exit induces fluid rotation in the negative x-direction due to baroclinic torque. The created fluid rotation is stretched and bent by the fresh-water inflow to form a vortex tube, which influences the roll-up pattern of gravity current and internal bore as shown in Figs. 17 and 18.

sidewall must be in the negative z-direction. At the same time, once the current exits from the channel, the vertical side interface induces strong baroclinic torque l /p 2Vp x Vp, which creates the vorticity in the negative ^-direction and causes slump of the interface in the negative y-direction and induction of fresh-water intrusion in the positive y-direction. Such fluid-rotation patterns near the exit (x = 0 and y = 0) is depicted in Fig. 26. The induced freshwater intrusion from the right provides the dominant supply into the channel to compensate the saline loss. Hence, the flow pattern near the exit becomes three-dimensional. At the same time, the created vortical flow at the side interface due to baroclinic torque is advected with this fresh-water intrusion,


and then vortex tubes are being bent and stretched. This fresh-water advection adjacent to the saline interface is initially in the positive ^-direction due to the initial saline-flow momentum, then is bent inward (in the y-direction) and accelerated due to the flow convergence into the channel. Hence, the initial fluid rotation in the negative ^-direction is bent and stretched toward the negative y-direction as depicted in Fig. 26. This bending causes the roll-up behind the internal bores shown in Figs. 18 and 22, and also the rotation causes the fresh-water down welling, resulting the substantial reduction of the saline depth behind the head (see Figs. 15 and 16). This can explain why the roll-up and the depth reduction propagate toward the y-direction, because the roll-up is caused by the fresh-water intrusion from the right-hand side, instead of the roll-up from the leading head (Note that in the two-dimensional channel, the roll-up is caused by the accumulation of baroclinic torque generated along the front face).

Inside of the channel near the right wall, we observed strong mixing and upwelling (see Figs. 15, 19, and 20). This upwelling must be due to the

Fig. 27. Schematic flow pattern near the exit. The upwelling of the mixed fluid occurs by the secondary current generation due to the fresh-water separation eddy formation (see Figs. 15, 19, and 20).


formation of the secondary current caused by flow separation of fresh-water inflow around the tip of the right wall. The inflow velocity is fast near the surface and it decreases in depth: The inflow velocity must diminish just above the saline outflow. Hence, just as the separation flow pattern in an open-channel flow (Yeh, et al., 1988), the separation eddy is larger near the surface and smaller near the fresh-water-and-saline interface as depicted in Fig. 27. The flow divergence of the upper fresh water causes the upwelling of the lower fluid. Nonetheless, the mixed fluid that appears in the upwelling must be originated at the side interface near the exiting point. Note that active mixing near i/fsO can be seen in Figs. 23 and 24: The mixing is due presumably to baroclinic torque as well as the advection of vortical fluid from the side wall as depicted in Fig. 26. Such mixed fluid may be transported by the fresh-water inflow to the area of the upwelling.

7. Conclusion

It is reconfirmed that the gravity current near the leading head is significantly different from that of the internal bore. When the gravity current or internal bore spreads from the two-dimensional channel into open environment, the flow pattern of each case is similar except at the leading front. The leading front of the gravity current is complex, while that of the internal bore is smooth. The complex flow pattern associated with the spreading appears to be influenced significantly by the upper fresh-water inflow to the channel and by the fluid rotation generated by baroclinic torque at the flow exit. The vigorous mixing and upwelling near the flow exit within the confined channel must be caused by the fresh-water inflow.

Acknowledgment

The photographs presented herein were taken at the University of Washington by C. Grandinetti, S. Lingel and K. Wada. The support from the Washington Sea Grant Program is acknowledged.

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