Coastal Engineering 56 (2009) 297–312

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Coastal Engineering

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Evolution of waves and currents over a submerged laboratory shoal

Junwoo Choi a, Chae Ho Lim b, Jong In Lee c, Sung Bum Yoon d,⁎a Department of Civil Eng., Hanyang University, 133-791, South Koreab National Oceanographic Research Institute, 400-800, South Koreac Water Resources Research Dept., Korea Institute of Construction Tech., 411-712, South Koread Department of Civil & Environ. Eng., Hanyang University, Ansan, 426-791, South Korea

⁎ Corresponding author. Tel.: +82 31 400 5144; fax: +E-mail address: sbyoon@hanyang.ac.kr (S.B. Yoon).

0378-3839/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.coastaleng.2008.09.002

a b s t r a c t

a r t i c l e i n f o

Article history:

The vertically-integrated Received 5 October 2007Received in revised form 28 August 2008Accepted 10 September 2008Available online 11 October 2008

Keywords:Wave transformationNumerical wave modelWave-induced currentWave-current interactionSubmerged elliptic shoal

effect of interaction between waves and wave-induced currents on wavetransformation over a submerged elliptic shoal was investigated based on numerical simulations of theVincent and Briggs experiment [Vincent, C.L., Briggs, M.J., 1989. Refraction- diffraction of irregularwaves over amound. Journal of Waterway, Port, Coastal and Ocean Engineering, 115(2), pp. 269–284.]. The numericalsimulations were performed using two numerical wave-current model systems: one, a combination of thewavemodel SWANand the currentmodel SHORECIRC, and the other, a combination of thewavemodel REF/DIFand the same currentmodel. A time-dependent, phase-resolvingwave and currentmodel, FUNWAVE, was alsoutilized to simulate the experiment. In the simulations, the developedwave-induced currents defocusedwavesbehind the shoal and brought on awave shadowzone that showed relatively lowwave height distributions. Forthe breaking case of monochromatic waves, the wave heights computed using FUNWAVE showed goodagreement with the measurements and the resulting wave-induced currents showed a jet-like velocitydistribution in transverse direction. And the computed results of the two model combinations agreed betterwith themeasurements than the computed results obtained by neglectingwave-current interaction. However,it was found that for the case in which transverse interference pattern caused by refracted waves was strong,REF/DIF-SHORECIRC did not correctly evaluate radiation stresses, the gradients of which generate wave-induced currents. SWAN-SHORECIRC, which cannot deal with the interference patterns, predicted a jet-likewave-induced current. For breaking randomwave cases, the computed results of the twomodel combinationsand FUNWAVE agreed well with the measurements. The agreements indicate that it is necessary to take intoaccount the effect of wave-induced current on wave refraction when wave breaking occurs over a submergedshoal.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Waves propagating over a submerged shoal are transformed byrefraction, diffraction, shoaling, and wave breaking. In addition, wavesbreaking over the shoal induce a jet-like strong shear current behindthe shoal. Thus, characteristics of wave transformation are altered dueto the breaking wave-induced current. Kirby et al. (1998) and Chenet al. (2000) simulated currents induced by breaking waves behind asubmerged circular shoal by utilizing a fully nonlinear Boussinesqwave model called FUNWAVE. However, it does not appear that thewave-induced current in their simulation appreciably influences thewave field because the wave breaking region is not wide enough, andtherefore the wave-induced current is relatively narrow and weak.While Kirby and Őzkan (1994) developed REF/DIF S, an extendedversion of REF/DIF 1(Kirby and Dalrymple, 1994) using a spectral ap-proach for random waves, and simulated the Vincent and Briggs

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(1989) experiment, they neglected the effect of the wave-inducedcurrent on wave transformation. The REF/DIF (REF/DIF 1 for mono-chromatic waves and REF/DIF S for random waves) model, which is aparabolic wave model including weak nonlinear wave effects, cannottake into account the current field without an extra current module.Yoon et al. (2004) simulated wave transformation due to wave-in-duced currents by using a model system combining REF/DIF S and acurrent model based on the shallow water equations. Indeed, theyshowed that waves breaking over a submerged elliptic shoal induce ajet-like current behind the shoal, and the breaking wave-inducedcurrent defocuses waves. This is called current-induced refraction, andis a counterpart of depth-induced refraction. If the effect of this wave-induced current is neglected, wave height distributions behind theshoal are overestimated (Yoon et al., 2004).

To investigate the effect of wave-induced current on randomwavetransformation over a submerged shoal, Choi et al. (2007) constructedtwo numerical model systems. The first is a combination of REF/DIF Sand SHORECIRC, the latter of which is a nearshore current modeldeveloped by the University of Delaware. It has been verified thatSHORECIRC can simulate nearshore circulations induced by breaking

298 J. Choi et al. / Coastal Engineering 56 (2009) 297–312

waves, although this current model needs a wave-driver model tocalculate the radiation stresses, the gradients of which generate wave-induced currents (Svendsen et al., 2002). SHORECIRC is a more so-phisticated current model than the current model developed by Yoonet al. (2004). The second model system is a combination of SWAN andSHORECIRC. SWAN, developed by Delft University, is a practical wavemodel in many respects. However, it does not include a module tocompute a current field as REF/DIF does not (Booij et al, 2004). In ad-dition, FUNWAVE, a fully nonlinear Boussinesq wavemodel developed(Wei et al, 1995) and modified (Chen et al, 2003) by the University ofDelaware, can be utilized to simulate instantaneous wave surfaces andwave-induced currents without a separate run of wave and currentmodels.

The monochromatic and random wave cases of the Vincent andBriggs (1989) experiment were simulated by using the two combina-tions REF/DIF-SHORECIRC and SWAN-SHORECIRC, as well as theFUNWAVE model. These computed results were compared with eachother and the measurements. Prior to considering the effect of thewave-induced current, simulations of non-breaking cases neglectingthe current effect were conducted. These simulations helped tounderstand the distinction between the different wave models due tothe effect of wave-induced currents.

The numerical models used in this study were developedindependently and have been reliably modified and verified. There-fore, the governing equations and the model closures for eachnumerical model are described very briefly in Section 2. For numericalsimulations, the two wave and current model combinations and theFUNWAVE model were set up corresponding to the experimentalconditions for bathymetry, frequency and directional spectra forrandom wave cases. Descriptions for the model setup are shown inSection 3. The simulations of monochromatic and randomwaves wereconducted using REF/DIF, SWAN and FUNWAVE and the computedresults are compared with the measurements in Section 4 and 5 fornon-breaking and breaking cases, respectively. The results underbreaking wave conditions showed the effects of breaking wave-induced currents on wave transformation. A summary and conclu-sions are presented in Section 6.

2. Numerical models

The REF/DIF (Kirby and Dalrymple, 1994; Kirby and Özkan, 1994)model is utilized to simulate depth-induced and current-inducedrefraction, diffraction, shoaling, and energy dissipation of monochro-matic or random waves. Since this model was developed in para-bolic form under the assumption that waves are mainly traveling in thex-direction, it does not predict the waves reflected directly back to thedirection theycame.However, thismodel candealwith they-directionalinterference patterns (transverse standing waves) due to reflectedwaves from a lateral wall boundary or refracted waves crossing eachother. In the governing equation of the REF/DIF model, the complexwave amplitude An for the nth wave component in an input frequency-directional spectrum is the unknown variable. Note that each of thecomputational components An is characterized by a unidirectionalmonochromatic wave at an initial stage, but can be transformed intomulti-directionalwave components due towave reflection or refraction.And in order to take into account the effect of mean currents, meancurrent velocity U and V should be computed with an extra currentmodel. The properties of each computational component, such as wavenumber kn, the reference wave number given along an incidenceboundary k̄̄n, phase velocity (C)n, and group velocity (Cg)n, are de-termined by the spectrum and the following dispersion relationship:

σ2n ¼ ωn−knUð Þ2¼ gkntanhknh ð1Þ

where σn is the intrinsic frequency that takes into account the Dopplereffect due to currents U, ωn is absolute frequency, g is gravitational

acceleration, and h is local water depth. The directional correctioncoefficients were chosen as a0=1,a1=−0.75, and b1=−0.25, the valuesof whichwere followed by the Booij (1981) approximation. The energydissipation coefficient aBr for wave breaking is given by Thornton andGuza (1983) as

αBr ¼ 3ffiffiffiπ

p4

f B3

γ4h5H5

rms ð2Þ

where f_

is the representative frequency and Hrms the root meansquared wave height. B and γ are the empirical coefficients. For amonochromatic wave condition, REF/DIF 1 (Kirby and Dalrymple,1994) is employed. The energy dissipation due to wave breaking inREF/DIF 1 is given as

αBr ¼ 0 forHhbκBr

αBr ¼ K2h

1−γshH

� �2" #

forHhzκBr

ð3Þ

where K and γs are empirical constants. Measurements by Horikawaand Kuo (1966) showed that a stable value of the wave height H=γshwas approached for waves over a uniform water depth after breakingon a plane slope. They suggested that K=0.15 and γs=0.4. Thebreaking index relation is κBr=0.78 for plane slopes of 0.015–0.15.

SWAN (Booij et al, 1999, 2004) is used for simulating randomwavetransformation due to refraction by topography and current, shoaling,and energy dissipation due to wave breaking and bottom friction(among other reasons). The unknown variable in the governingequation of SWAN is the action density N (i.e., energy density Edivided by its frequency σ in directional (θ) and frequency (σ)spectrum), N(σ,θ)=E(σ,θ)/σ. This wave action density balance equa-tion includes the terms which represent propagation of action ingeographical space, shifting of the relative frequency due to variationsof depths and current in frequency space, and depth-induced andcurrent-induced refraction in directional space. And the source termin the equation represents the effects of generation, dissipation andnonlinear wave-wave interactions. The SWANmodel does not includewave phases so that interference patterns cannot be accounted for,even though this model is capable of simulating wave reflection byobstacles. For the diffraction effect in this model, a phase-decoupledrefraction-diffraction approximation, which is based on themild slopeequation omitting phase information, was suggested by Holthuijsenet al. (2003). In order to consider the effect of mean currents on thewave transformation, current information should be provided as inputto SWAN using an extra current model. The mean rate of energydissipation per unit horizontal area due to wave breaking, Dtot (Battjesand Janssen, 1978), is represented as

Dtot ¼ ρg4

αBJQbσ2π

� �γBJh� �

ð4Þ

where ρ is sea water density, αBJ is the proportionality coefficient ofthe rate of dissipation, Q b is the fraction of breaking waves, σP is themean frequency, and γBJ is the breaker parameter. The fraction ofdepth-induced breakers Q b is determined using

1−Qb

lnQb¼ −8

Etot

γBJh� �2 ð5Þ

where Etot= ∫02π∫0∞E(σ,θ)dσdθ.A current model, SHORECIRC (Svendsen et al., 2002) is utilized to

solve wave-averaged motion such as wave-induced nearshore cir-culations and infragravity waves. The wave-induced motions in thesimulation using SHORECIRC are generated due to excess momentumfluxes which can be evaluated using the wave models. The theoretical

299J. Choi et al. / Coastal Engineering 56 (2009) 297–312

background iswell described in Putrevu and Svendsen (1999) andmanyreferences show the applicability of this model towards the simulationof nearshore currents (Svendsen,1984, Svendsen et al., 2002). Althoughthis model was recently extended into a quasi 3-dimensional versionby using an analytical vertical velocity profile, the depth-integrated2-dimensional version was used in this study. The basic idea of thismodel starts from a decomposition of the total fluid velocity intothree kinds of velocity components:

uα ¼ u0α þ uwα þ Vα ð6Þ

where uVα denotes the turbulent component velocity, uwα the wavecomponent velocity defined so that uwP =0 below trough level, and Vα

is the current velocity. The over-bar ( )P denotes a short wave timeaverage, and the subscripts α and β denote the directions in ahorizontal Cartesian coordinate system (see below). From theReynolds averaged Navier-Stokes equation, the governing equationfor SHORECIRC is derived by using depth-integration and averagingthe equation over a short wind wave period. For the equations, theviscous term is modeled based on the turbulent eddy viscosity con-cept and written as

ταβ ¼ ρmtAVβ

Axαþ AVα

Axβ

� �ð7Þ

where

mt ¼ C1κ

ffiffiffiffiffiffiffifcw2

ru0hþMh

Dρ

� �1=3

þm0 þ ms: ð8Þ

In this equation, C1 and M are empirical coefficients, κ is theKarman constant, fcw is the bottom friction coefficient related to bothwave and current, ν0 is the empirical ambient eddy viscosity, νs is theSmagorinsky eddy viscosity with an empirical coefficient Cs (Smagor-insky, 1963), D is the energy dissipation rate due to wave breaking(equivalent to Dtot in SWAN), and u0 is the magnitude of wave velocityevaluated at the bottom. To evaluate the bottom friction due to bothwave and current, an energy-weightedmeanwave direction is chosenas a representative direction.

The excess momentum fluxes Sαβ due to wave fluctuations (Mei,1989) are expressed as

Sαβ ¼P∫ζ−h0pδαβ þ ρ uwαuwβdz−δαβ

12ρgh2 ð9Þ

where the total water depth h is defined as the sum of mean freesurface ζ

Pand water depth h0 under still water level. The surface

elevation ζ=ζP+η, η is the wave component surface displacement, p is

the pressure, and δαβ is the Kronecker delta. For small ratios of waveamplitude to wave length, δαβ can be approximated as

Sαβ ¼ ∫ζ�−h0

ρuwα u�wβ dz

þ δαβρgη2

2þ ∫ζ−h0

@

@xα∫ζz ρ uwα w�

w dz0dz−∫ζ−h0ρw2wdz

( )ð10Þ

which is called the radiation stress tensor by Longuet-Higgins andStewart (1962, 1964). In REF/DIF and SWAN, the radiation stresses at aspatial grid location (x,y)can be obtained by

Sαβ ¼ ∫ ∫ eαβ θð ÞSm σ ; θð Þ þ δαβSp σ ; θð Þ� �dθdσ ð11Þ

where

eαβ θð Þ ¼ cos2θw sinθwcosθwsinθwcosθw sin2θw

� �ð12Þ

in which θw(x,y) is the wave direction of each wave component in aspatial domain, θ is the wave direction in a directional spectrum, and

using the amplitude distributions An of the wave component n com-puted by using REF/DIF, Sm and Sp are written as

Sm σ ; θð Þ ¼ ρg2

jAnj2Cg

n

Cð Þn

!ð13Þ

Sp σ ; θð Þ ¼ ρg4

jAnj22 Cg

n

Cð Þn−1

!ð14Þ

while using the action density spectrum at each grid point in SWAN

Sm σ ; θð Þ ¼ σN σ ; θð ÞCg σ ; θð ÞC σ ; θð Þ ð15Þ

Sp σ ; θð Þ ¼ σ2N σ ; θð Þ 2Cg σ ; θð Þ

C σ ; θð Þ −1� �

ð16Þ

where a wave direction is defined with respect to the direction ofthe main propagating wave, the x-direction, and θ and σ denotethe direction and frequency of a wave component in the spectraldomain.

FUNWAVE, a time-dependent phase-resolving wave and currentmodel, employs the fully-nonlinear Boussinesq equations derivedby Wei et al. (1995). This model can simulate wave reflections aswell as interference patterns in any direction differently from theabove wave models. And the FUNWAVE model is not only used forsimulating monochromatic and random waves but also capable ofsimulating nearshore hydrodynamics including interaction betweenwaves and currents. Surface elevation ζ and horizontal velocity u⁎

at the water depth z= z⁎=−0.531h0 are the unknown variables in thegoverning equations, where h0 is the still water depth. Anadditional term is added in the momentum equation for partial-rotational motions for a new version (Chen et al., 2003), and iswritten as

V ¼ −vTω1;uTω1ð Þ ð17Þ

where

ω1 ¼ AzTAx

A

Ayj � h0uTð Þ½ � þ zT

A

Ayj � uTð Þ

� �−AzTAy

� A

Axj � h0uTð Þ½ � þ zT

A

Axj � uTð Þ

� � ð18Þ

which allows for second-order effects of vertical vorticity, where j isthe 2-dimensional gradient tensor. For practical application, thegoverning equations require that themodel schemes take into accountphysical effects such as bottom friction, turbulent viscous damping,and wave breaking as well as some numerical needs such as wavegeneration and boundary absorption. For wave generation, FUNWAVEemploys the internal source approach (Wei et al., 1999), which isbased on an ad-hoc source mechanism by Larsen and Dancy (1983). Togenerate random waves corresponding to a given frequency-direc-tional spectrum, the internal source term added to the continuityequation can be written as

f x; y; tð Þ ¼ q xð Þ ∑M

m¼1∑J

j¼1Ds Amj; fm; θj

sin kmysinθj−2πfmt þ /mj ð19Þ

where q(x)=exp[−βs(x−xs)2] is a Gaussian shape function in whichβs=5/(λ/4)2 is a shape coefficient for the source function, which isin turn determined from the peak wave component. The peakwave length is λ and xs is the central location of the source in the

Fig. 1. Location of the submerged elliptic shoal, the centerline at y=12.5 m and thetransect 4 at x=12.2 m (Vincent and Briggs, 1989).

300 J. Choi et al. / Coastal Engineering 56 (2009) 297–312

x-direction. Ds(Amj,km,θj) is the magnitude of the source functionwritten as

Ds Amj; km; θj ¼ 2Amjcosθj 2πfmð Þ2− αs þ 1=3ð Þgk4mh30

h i2πfmkm 1−αs kmh0ð Þ2

h i ffiffiffiffiπβs

qexp − kmcosθjð Þ2

4βs

� ð20Þ

where Amj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E f ; θð Þdfdθ

pis the deterministic amplitude of a

wave traveling in the θj direction with a frequency fm and a wavenumber km corresponding to fm. The random phase independentof frequency and direction is ϕmj, which is uniformly distributedover the interval (0,2π), and αs ¼ −0:390. For energy dissipationdue to wave breaking, FUNWAVE employs the momentum mixingscheme introduced by Kennedy et al. (2000) and Chen et al. (2000). Inaddition, an extra diffusion term is added for the diffusivity that isstrongly localized on the front face of a broken wave. In this schemethe empirical coefficient ηt(I) that controls the onset of wave breakingvaries between 0:35

ffiffiffiffiffiffiffiffigh0

pand 0:65

ffiffiffiffiffiffiffiffigh0

p. FUNWAVE also includes

energy dissipation due to turbulent mixing, based on the Smagorinsky(1963) subgrid approach. This subgrid mixing term defines the eddyviscosity as

ms ¼ cmΔxΔyAUAx

� �2

þ AVAy

� �2

þ12

AUAx

þ AVAy

� �2" #1=2

ð21Þ

in which U and V are the velocity components of the time-averagedcurrent field, cm is the mixing coefficient, and Δx and Δy are the gridspacings in the x and y directions, respectively. In addition, the bottomshear stress is modeled by

Rf ¼fc

h0þζuTjuTj ð22Þ

where fc is the bed shear stress coefficient.

3. Model setup

Vincent and Briggs (1989) performed experiments on monochro-matic waves and frequency-direction spreading randomwaves whichpass over a submerged elliptic shoal by using a directional spectralwave generator (DSWG) in Coastal and Hydraulics Laboratory's wavebasin of the U.S. Army Engineer Research and Development Center.The spectra for random wave generation were formed using TMA(Bouws et al., 1985) for the frequency spectrum and awrapped normaldistribution function for the directional spreading function (Borgman,1984). The experiments included many cases with combinations ofunidirectional, narrow and broad spectra in direction, and monochro-matic, narrow, and broad spectra in frequency. The wave conditionsused for the simulations in this study are shown in Table 1.

For numerical simulations of the Vincent and Briggs (1989) ex-periment, the bathymetry with a submerged elliptic shoal was con-

Table 1Input wave conditions (Vincent and Briggs, 1989)

Input Case ID

Non-breaking case Monochromatic M2Narrow direction Broad frequency N3

Narrow frequency N4Broad direction Broad frequency B3

Narrow frequency B4Breaking case Monochromatic M3

Narrow direction-frequency N5Broad direction-frequency B5

structed as shown in Fig. 1. The computational grids were uniformwith dimensions of 0.1×0.1 m in the spatial domain. The center of theshoal used in their experiment was located at x=6.1 m and y=12.5 min the computational domain with 19 m in the x-direction, which isthe main wave propagating direction, and 25 m in the y-direction. Inorder to avoid erroneous lateral boundary situations due to noincoming wave along the lateral wall boundaries, we appended anextra 10 m in the y-direction of the actual computations using thethree wave models. In utilizing SHORECIRC, a no-flux wall boundarycondition was specified along all lateral boundaries. In addition, forthe simulation performed using FUNWAVE, an additional domain forinternal wave generation and sponge absorption layer were required.Thus, we appended an extra 15 m in the x-direction.

The randomwave spectra for computations were formed using theTMA and the wrapped normal distribution function. For the boundaryinput wave spectra of REF/DIF, 61 directional components and 41frequency components were used. For the energy spectral domain inthe simulations using SWAN, we used 61 directions over the 180°(from −90 to 90°), which indicates that the directional gridΔθ=3°, and41 frequencies in the range from 0.5 to 1.5 Hz. The frequency cutoffsfollowed those of the Vincent and Briggs experiment. The spectralspace consists of 2501 bins. The 3-degree resolution is within therange recommended by Booij et al. (2004) for narrow spectrumcondition (with the 10-degree directional spreading). For FUNWAVE,which is the wave-phase-resolving model that requires higherresolution than the other wave models, 121 directional componentsand 81 frequency components were used to generate incident randomwaves. The parameters for narrow and broad-banded frequency anddirectional spectra in the experiment of Vincent and Briggs are shownin Table 1.

For SWAN, the diffraction option was switched on with the filterwidth around 0.7 m. The 0.7 m filter width was chosen for numericalstability. This is larger than the filterwidth (0.4m) used by Holthuijsenet al. (2003) to achieve the diffraction effect in the simulations of theVincent and Briggs experiment using SWAN. Their spatial grid was0.2×0.2m. Holthuijsen et al. (2003) recommended that the directional

Peak period (s) Significant wave height (cm) γTMA σTMA (degree)

1.3 2.54 – –

1.3 2.54 2 101.3 2.54 20 101.3 2.54 2 301.3 2.54 20 301.3 13.5 – –

1.3 19.0 20 101.3 19.0 2 30

301J. Choi et al. / Coastal Engineering 56 (2009) 297–312

resolutions Δθ of wave spectra should be small compared to theirdirectional spreading to predict reasonable diffraction effects usingSWAN. It should be addressed that the 3-degree directional resolutionin this study is low for quasi-monochromatic waves with an extremelynarrow directional spectrum. However, the resolution higher than thepresent was beyond the limitation of our computer storage system.Moreover, the effect of wind, wave-wave interactions, and the white-capping implemented in the SWAN model were neglected in thesesimulations. The chosen resolution in frequency domain was not highenough to consider the effect of wave-wave interactions in thesesimulations. However, from some trial and error tests, the effect ofwave-wave interactionswas found to be less significant than the effectof wave-current interaction in the simulations for wave-inducedcurrents.

The breaking wave cases of monochromatic and random waves inthe experiment of Vincent and Briggs were simulated mainly to studythe influence of wave-induced currents onwave transformation. Priorto conducting simulations of the breaking waves passing over thesubmerged shoal, simulations of the non-breaking cases, in which thewave-induced current effect can be neglected, were performed also todetermine differences between the REF/DIF, SWAN, and FUNWAVEmodels.

Vincent and Briggs (1989) stated the followings: 36.4 s waveelevation data for the monochromatic waves and 260 s data for thespectral waves were sampled at 50 Hz. A sufficient waiting time wasallowed to elapse in order to permit the slower-traveling highfrequency component waves to travel to the remote transect atx=15.25 m prior to the collection of data after the start of the DSWG.Wave height information was obtained using zero-crossing analysis.For the spectral waves a comparison of the significant wave height H1/3

(obtained from zero-crossing analyses) and the Hmo (obtained fromspectral analyses) value indicated very good agreement, with minordifferences occurring only at the top of the shoal. In addition, it is worthmentioning that the wave height data presented in their study werenormalized by the wave height at “gage 10,” which was located atx=3.05 m and y=20.12 m as shown in Fig. 1. In this paper, theexperimental data were normalized by target wave height at thewavemaker, since the wave height at “gage 10” varies according tothe chosen phase lags between directional components of incidentrandom waves along the wavemaker. This will be discussed in moredetail later.

Accordingly, numerical simulations in this study were performedfor longer than 260 s with a time step Δt=0.02 s for both SHORECIRCand FUNWAVE. In this study the stationary mode was used for theSWANmodel. For thewave and current model combination systems, a

Fig. 2. Wave height computed with (a) REF/DIF, (b) SWAN and (c)

short numerical ramping interval equivalent to approximately 2 peakwave periods was added. During the ramping interval at the initialstage the gradients of radiation stress in the current model increasesgradually from zero to the actual value computed by the wave modelsin order to prevent an abrupt transition error. In the simulation usingFUNWAVE, we allowed the waiting time to include a ramping intervaland the elapsed time needed for the high frequency waves to arrive atx=15.25 m. In the three wave models the computed results of waveheight information were the values of Hmo for the randomwaves andthe values of Hrms for the monochromatic wave. The wave height datashown in this study are the values non-dimensionalized by the waveheight generated at the wavemaker (Yoon et al., 2004).

The radiation stresses generating the wave-induced current inSHORECIRC were obtained from the directional frequency actiondensity spectra (SWAN) or the amplitude distributions of each com-ponent (REF/DIF) as written in Eqs. (10)–(16). In addition, the depth-integrated mean current velocities computed from SHORECIRC wereapplied as input data to simulate wave fields using REF/DIF and SWAN.In other words, a main program controlled the feedback of wave- andcurrent-subprograms, REF/DIF and SHORECIRC or SWAN and SHOR-ECIRC, for wave and current interactions. Under the assumption thatthe rate of wave field change would be slower than the rate of currentfield change, thewave field was computed every 10 s duringwhich thecurrent field was computed with a 0.02 s time-step size forSHORECIRC. This recalculation was repeated for the entire simulationtime, which was around 260 s. Unlike the aforementioned two wavemodels, FUNWAVE is able to solve time-dependent wave surfacemotion and horizontal flow motion at the phase-resolving time level.In other words, an additional current model is not required to solvethe current field in FUNWAVE.

For detailed descriptions of numerical schemes on the models, seeBooij et al. (2004) for SWAN, Kirby and Özkan (1994) for REF/DIF,Svendsen et al. (2002) for SHORECIRC, and Kirby et al. (1998) forFUNWAVE. Most physical or empirical coefficients required for themodels followed default values of each model in these simulations.For the empirical coefficients of the energy dissipation due to wavebreaking, we chose the values as B=1 and γ=0.6 in REF/DIF S, andK=0.14 and γs=0.4 in REF/DIF 1. A breaking index relation κBr=0.68,which is smaller than 0.78 (the suggested value for a plane slope) wasused in REF/DIF 1 for the bathymetry with a submerged shoal. Thedirectional correction coefficients a0=1, a1=−0.75, and b1=−0.25,following the Booij (1981) approximation, were used in REF/DIF. Kirby(1986) showed that the errors in this approximation are small whenθ≤40°. Although the largest angle of the broad band amplitudespectra used for REF/DIF S was around 60°, the wave energy for

FUNWAVE for non-breaking monochromatic wave case M2.

Fig. 3.Wave height distribution computed with REF/DIF, SWAN and FUNWAVE for non-breaking monochromatic wave case M2 along (a) centerline and (b) transect 4.

302 J. Choi et al. / Coastal Engineering 56 (2009) 297–312

40°≤θ≤60° is less than 10% of the total wave energy in the spectra. ForSWAN, the proportionality coefficient of the dissipation rate wasαBJ=1 and the fraction of breaking waves was γBJ=0.8. This value 0.8 isnot the default. However, it is within the range proposed by Battjesand Stive (1985) and close to the values 0.78 of Wood et al. (2001). Infact, this value 0.8 was chosen for closer results to the measurementsafter some trial and error tests. In utilizing SHORECIRC, we set theparameters related to turbulent eddy viscosity as C1=0.2, M=0.2,κ=0.4, fcw=0.02, ν0=0.002, and Cs=0.2, which are within a rangerecommended in the SHORECIRCmanual. In the simulation conductedusing FUNWAVE, the value η Ið Þ

t ¼ 0:35ffiffiffiffiffiffigh

pwas chosen for the

breaking energy dissipation coefficient, the mixing coefficient foreddy viscosity cm=0.15, and the bed shear stress coefficient for energydissipation due to bottom friction fc=0.0015. In order to obtain theeddy viscosity for energy dissipation due to wave breaking andturbulent mixing, the underlying flow field was calculated by aver-

Fig. 4. Wave height computed with (a) REF/DIF, (b) SWAN an

aging the instantaneous fluid velocity over approximately 3 peakwaveperiods.

4. Non-breaking waves

Prior to considering the effect of wave-induced current, to checkthe performance of the different wave models, numerical simulationsneglecting wave-current interaction were conducted for the non-breaking wave conditions.

4.1. Monochromatic waves

Figs. 2 and 3 show the results for the non-breakingmonochromaticwave case M2 of the Vincent and Briggs experiments simulated usingREF/DIF, SWAN, and FUNWAVE. Since the SWAN model deals withonly random waves, it is basically inadequate to simulate monochro-matic waves. To achieve a similar effect from quasi-monochromaticwave simulation using SWAN, the incident waves were approximatedwith a very narrow frequency and directional spectrum (Holthuijsenet al., 2003). In other words, along wave input boundary the energydensity corresponding to the peak wave was set in 3 bins with onefrequency and a very narrow directional distribution among 2501spectral bins. As already addressed, for this quasi-monochromaticwave simulation, the 3-degree directional resolution used in thisstudy is too low to compute wave diffraction effect using SWAN. Theresults of FUNWAVE were the Hrms values obtained by spectralanalyses of the instantaneous free surface time history for 36.4 s afterthe waiting time of 16.9 s.

The horizontal distributions of wave height in Fig. 2 show thefocusing of waves due to depth-induced refraction behind the shoal. Inaddition, Fig. 2 shows some of the characteristics of the wave models.REF/DIF does not deal with wave reflection due to underwatergeometrical changes (Kirby and Özkan, 1994). However, it does deal withtransverse standing waves due to wave refraction in the y-direction. Inother words, the interference patterns due to superposition of wavesmoving in opposite y-directions can be taken into account in REF/DIF.SWAN cannot deal with the interference patterns because the wavephases are neglected. Moreover, SWAN has a limited diffraction effect,which ismodeled dependingon the grid size in the spatial and directionalspace. FUNWAVE manages the interference patterns due to eitherwave refraction or wave reflection. Fig. 3 shows a comparison betweenthe computed results and the measurements along the centerline andthe transect 4. It indicates that the wave heights along the centerlineat y=12.5 m (top) and the transect at x=12.2 m (bottom) computed byRFF/DIF and FUNWAVE agree with the measurements, and confirms thatSWAN is inadequate to simulate monochromatic waves with theinterference patterns.

d (c) FUNWAVE for non-breaking random wave case N4.

Fig. 5. Wave height distribution computed with REF/DIF, SWAN and FUNWAVE for non-breaking random wave case N3 and N4 along (a) centerline and (b) transect 4.

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4.2. Random waves

For numerical simulation of random waves using the REF/DIFmodel the input waves can be prepared by two options. The firstoption is to discretize the whole spectrum of the input waves into a

Fig. 6. Wave height distribution computed with REF/DIF, SWAN and FUNWAVE for no

number of separate wave components (An), each of which has a singlefrequency and a single direction. Each of those computational com-ponents (An) is characterized by a unidirectional monochromatic waveat an initial stage, but can be transformed into multi-directional wavecomponents due to refraction over a variable water depth region.

n-breaking random wave case B3 and B4 along (a) centerline and (b) transect 4.

Fig. 8.Waveheight distributionwith andwithout the current effect andmean current profile(bottom panel) for monochromatic wave case M3 along (a) centerline and (b) transect 4.RD=REF/DIF, RD+SC=REF/DIFandSHORECIRC, SW=SWAN, SW+SC=SWANandSHORECIRC.

Fig. 7. Schematic representation of wave breaking and refraction pattern over a shoal: (a) focusing zone without wave-induced current and (b) shadow zone with wave-inducedcurrent (Yoon et al., 2004).

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Thus, phase interference between the multi-directional wave compo-nents in an individual computational component (An) is still allowedas in the case of monochromatic version of REF/DIF. However, duringthe computation each of the computational components (An) is in-dependently calculated (except for the energy dissipation due tobottom friction and wave breaking) and the phase interference withthe other computational components is neglected. Thus, the waveheight (Hmo) remains constant along the generation line and in thecomputational domain if the water depth is uniform. The secondoption, which needs a slight modification of the original code, is todiscretize only the frequency spectrum (Kirby and Dalrymple, 1994).Each of the discretized components includes implicitly the informa-tion of directional distribution for a given frequency. In this case thephase interference between the waves propagating in differentdirections (with random phases) is possible within each frequencycomponent. Thus, the wave height (Hmo) for the multi-directionalwaves would be non-uniform even along the generation line withconstant water depth (Suh and Dalrymple, 1992). In practice the firstoption is more frequently employed. In the present study the firstoption was selected. In other words, the phase interference betweendifferent computational components (An) was neglected as in the caseof SWAN. On the other hand, the FUNWAVE is a phase-resolvingprimitive-variable model which can deal with randomwaves withoutdividing the variables into separate computational components, thus,full-interactions between wave components are assured. However, inorder to prescribe the incident random waves along the internalwavemaker of FUNWAVE, discretizing the wave spectrum into anumber of components is necessary as in the case of REF/DIF or SWAN.The instantaneous free surfaces for each spectral wave component arecalculated and superposed with phase lags randomly distributed overthe interval of 0 to 2π. As a result, the phase interferences betweendirectional components appear in the wave height (Hmo) along thegeneration line. Thus, it is hard to achieve a uniform distribution ofwave height along the generation line and, in turn, in the computa-tional domain. For the input waves with directional spreading the useof measured wave height at a particular location, e.g. “gage 10”, as areference wave height for the normalization of measured data is notappropriate for this reason.

The numerical results for non-breaking random wave cases areshown in Figs. 4–6, which show wave height distributions computedby REF/DIF, SWAN, and FUNWAVE. The wave height distributions ofFUNWAVE were the Hmo values obtained by spectral analyses of theinstantaneous free surface time history for 260 s after thewaiting timeof 27.3 s.

Fig. 4 presents the results of the N4 case with a narrow frequency-directional spectrum. The horizontal distributions of wave heightshowwave focusing due to depth-induced refraction behind the shoal.

However, the interference patterns in the transverse direction for therandom wave cases are not apparent because a number of spectralwave components with different directions were superposed. Thewave heights calculated using REF/DIF and SWAN are symmetric withrespect to the centerline of y=12.5 m, while this symmetry is not clearfor the case of FUNWAVE. As discussed earlier the phase interferencebetween wave components with different directions along theinternal generation line is responsible for this asymmetry of waveheight distribution in the result of FUNWAVE.

Figs. 5 and 6 show comparisons of the computed and measuredwave heights for the cases of narrow (N3 and N4) and broad (B3 andB4) directional spectra, respectively. The wave heights along thecenter line (y=12.5 m) and the transect 4 (x=12.2 m) calculated using

Fig. 9. Wave height, mean current velocity and mean surface elevation of the 36.4-second simulation computed with REF/DIF and SHORECIRC (top panels) and SWAN and SHORECIRC(bottom panels) for monochromatic wave case M3.

Fig. 10. Wave height distribution and mean current profile computed with FUNWAVE and the combined models for monochromatic wave case M3 along (a) centerline and (btransect 4. RD=REF/DIF, RD+SC=REF/DIF and SHORECIRC, SW=SWAN, SW+SC=SWAN and SHORECIRC.

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)

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various wavemodels show reasonable agreements with the measuredones. Some irregularity in wave heights in the transverse direction isobserved in the calculated results using FUNWAVE, while this ir-regularity disappears in the cases of simulated results using REF/DIFand SWAN as discussed earlier. The measurements also show theirregularity. Even though the details of the irregularity are slightlydifferent from each other, the magnitude of irregularity is in the samelevel. This irregularity in the wave height calculated using FUNWAVEappears from the generation line due to phase interference betweenwave components with different directions. The irregularity ispersistent even though the free surface data of a longer time spanare analyzed to achieve a stationary wave height.

From the comparisons between numerical results and theexperimental measurements it can be concluded that all the wavemodels gave a reasonable prediction for non-breaking random waveswith frequency and directional spectra. In addition it can be inferredthat the non-breaking waves of the present experimental cases werelittle influenced by wave-induced current because the results usingREF/DIF and SWAN without coupling the current module agree withthe experimental measurements and with those using FUNWAVE.

5. Breaking waves

Numerical simulations including wave-current interaction wereconducted for the breaking wave conditions. In the computations thebreaking of waves over a submerged elliptic shoal rapidly increases thegradients of radiation stresses (or excess momentum fluxes) and the

Fig. 11.Wave height, mean current velocity and mean surface elevation of the first 36.4-seco16.9-second waiting time for monochromatic wave case M3.

strong gradients induce a jet-like current behind the shoal. The jet-likewave-induced current defocuses waves behind the shoal. Yoon et al.(2004) explained this mechanism using the refraction diagramdepictedin Fig. 7with neglectingwave diffraction as : The breaking ofwaves overthe shoal induces strong currents in the direction of wave propagation,and the breaking-induced currents, in turn, defocus the convergingwave rays outwards from the central part behind the shoal. If therewerenobreaking-induced current, the convergingwave rayswould cause thewave height to grow again in the focusing zone. However, instead of thefocusing zone, a shadowzone appears there due to defocusedwave rays,and two caustics are formed next to the shadow zone.

5.1. Monochromatic waves

In this section, monochromatic waves transformed due to breakingwave-induced current were investigated using simulations computedby REF/DIF-SHORECIRC and SWAN-SHORECIRC, as well as FUNWAVE.The wave and current fields are transformed by their interactions. Forthe breaking monochromatic wave conditions, REF/DIF 1 was em-ployed instead of REF/DIF S, and the energy density corresponding tothe peak wave was set in one frequency and three directional binsamong 2501 spectral bins for an input boundary spectrum of SWAN.

The computational results of a breaking monochromatic wave case(M3) are presented in Figs. 8–11. These are the results at t=36.4 s inthe SHORECIRC time level, and correspond to the data obtained byanalyzing the instantaneous free surface for 36.4 s after the waitingtime of 16.9 s as mentioned above. Figs. 8 and 9 show results

nd (top panels) and the second 36.4-second (bottom panels) FUNWAVE simulation after

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simulated using the combinations REF/DIF-SHORECIRC and SWAN-SHORECIRC. In Fig. 8 the wave heights computed using REF/DIF andSWAN without coupling with SHORECIRC are also presented. The toppanel of Fig. 8 shows values along the centerline and the two bottompanels along the transect 4. When the effect of wave-induced currentsis excluded, waves are focusing behind the shoal, and thus one localpeak in wave height distribution appears near x=10 m of thecenterline y=12.5 m as conceptually predicted in Fig. 7(a). However,when the current effect is included, the local peak disappears alongthe centerline. Instead two local peaks appear next to the centerline asshown in the middle panel of Fig. 8. The computed results with theeffect of wave-induced currents agree better with experimentalmeasurements for both wave models of REF/DIF and SWAN. However,the accuracy of the models is not satisfactory. Moreover, the currentvelocities calculated using different models do not agree with eachother. The velocity distribution computed with SWAN-SHORECIRC isanalogous to the Gaussian distribution of a jet-like current, while thatcomputed using REF/DIF-SHORECIRC is not.

The horizontal views of these simulations obtained using the twomodel combinations are shown in Fig. 9 presenting wave heighttransformed by wave-current interaction, wave-induced current andmean surface elevation. The wave-induced current simulated usingREF/DIF-SHORECIRC does not showany organized pattern and appearsto be unrealistic, while that calculated using SWAN-SHORECIRC isanalogous to a jet-like current. The discrepancy between two currentpatterns is caused by the difference between the methods to evaluatethe radiation stresses in each wave model. In the REF/DIF 1 model the

Fig.12.Wave height, mean current velocity andmean surface elevation of the 260-second sim(bottom panels) for random wave case N5.

directional information of monochromatic waves is included in thephase of the complex amplitude. The phase interference betweenwaves propagating in different directions is assured in the computa-tion, and the wave amplitude can be easily obtained by taking anabsolute value of the complex amplitude. However, when the angle ofpropagation direction is calculated for the radiation stresses at a gridpoint, a single representative angle is determined without decompos-ing the waves into directional components. Thus, for the transformedwaves with multi-directional wave components, the contributionsfrom each directional component cannot be appropriately reflected onthe calculation of radiation stresses. On the other hand, in the SWANmodel each spectral bin has its ownwave height and angle of directionat any grid location. Since the phase interference between multi-directional wave components in the SWAN model is neglected, thesimulated wave heights are less accurate than those of REF/DIF forthe monochromatic case. However, the SWAN model is free from theuncertainties involved in the determination of wave directions forthe radiation stresses. Judging from the overall pattern of wave-induced currents shown in Fig. 9, the result produced by SWANappears to be more realistic. This topic is further discussed later in thissection by comparing the present results with those simulated usingFUNWAVE.

The computational results of a breaking monochromatic wave case(M3) simulated using FUNWAVE are presented in Figs. 10 and 11. Asalready mentioned, FUNWAVE can solve the instantaneous wave fieldand current field at one phase-resolving time level. Thus, it handlesinterference patterns of standing waves in either the x- or y-direction.

ulation computedwith REF/DIF and SHORECIRC (top panels) and SWAN and SHORECIRC

Fig.13.Wave height, mean current velocity andmean surface elevation of the 260-second simulation computedwith REF/DIF and SHORECIRC (top panels) and SWAN and SHORECIRC(bottom panels) for random wave case B5.

Fig. 14.Wave height distributionwith and without the current effect and mean current profile (bottom panel) for randomwave case N5 and B5 along (a) centerline and (b) transect 4RD=REF/DIF, RD+SC=REF/DIF and SHORECIRC, SW=SWAN, SW+SC=SWAN and SHORECIRC (Choi et al., 2007).

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.

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Note that results simulated using this model always include the wave-induced current effect. Fig. 10 shows the horizontal profile of waveheight andmean current computed by FUNWAVE along the centerlineand the transect 4. The wave height Hrms was obtained by spectralanalyses in the same way that the measurements were obtained andthe mean motions were obtained by averaging their instantaneousquantities over 36.4 s after the waiting time. The results are comparedwith those of the two model combinations, with and without in-cluding the effect of the wave-induced current. Along the transect 4behind the shoal, the wave height distribution computed usingFUNWAVE shows a perfect agreement with the measurements. Sincethe calculated wave heights using FUNWAVE agrees with themeasurements, it can be inferred that the current field calculatedusing FUNWAVE is reliable. The wave-induced current simulatedusing FUNWAVE has a Gaussian profile of a jet-like current which issimilar to that computed with SWAN-SHORECIRC. However, thecurrent profile calculated using REF/DIF-SHORECIRC is totally differentfrom that of FUNWAVE. This confirms that the SWAN model has anadvantage over the REF/DIF model in the evaluation of radiationstresses when the interference between the waves with differentdirections occurs.

Fig. 11 shows the horizontal views of wave height, mean currents,and mean surface calculated using FUNWAVE for breaking mono-chromatic wave case M3. The wave height and mean motions areobtained from numerical results of the first 36.4 s (top panels) and thesecond 36.4 s (bottom panels) after a waiting time of 16.9 s. The waveheight compared to the measurement data was the results from the

Fig.15.Wave height, mean current velocity andmean surface elevation of the 260-second FUN(bottom panels).

first 36.4-second simulation (top panels). The wave height distribu-tions in Fig. 11 show the interference patterns of transverse standingwaves behind the shoal. Additionally, the figure shows that a jet-likewave-induced current is developing, and the waves become defo-cused due to the wave-induced current behind the shoal.

5.2. Random waves

Numerical results from the model systems, REF/DIF-SHORECIRC andSWAN-SHORECIRC, with breaking random wave N5 and B5 conditionsare shown in Figs. 12–14. Figs. 12 and 13 show the horizontaldistributions of wave height, developed wave-induced currents, andmean surface elevation for the cases of N5 and B5, respectively. Thefigures show the results after a 260-second simulation that includes theeffect of wave-induced currents. For the 260-second simulation, eachfield of the wave and current is transformed by their interactions. Asdescribed in Choi et al. (2007) the results show that, when wavebreaking occurs, the waves refracted due to the submerged shoal sufferadditional refraction due to breaking wave-induced currents. Thecurrent-induced refraction defocuses the waves behind the shoal. Ashadow zone occurs and two weak focuses appear due to theconverging waves refracted by the current. For these random wavecases, the influence of the interference patterns of transverse standingwaves is weak.

In Fig. 14 the numerical results using the two model systems arecompared with the measurements for N5 and B5 cases. The figureshows wave heights and current velocity computed by including and

WAVE simulation after 27.3 s waiting time for randomwave case N5 (top panels) and B5

310 J. Choi et al. / Coastal Engineering 56 (2009) 297–312

excluding the effect of the wave-induced currents. In the computationwithout coupling with SHORECIRC, the waves refracted over theshoulder of the shoal are focusing behind the shoal. Thus, as shown inthe left middle panels of Fig. 14, the wave height distribution for N5case along the transect 4, x=12.2 m, has a pattern with a local peak atthe center, y=12.5 m. The same result can be found in Özkan and Kirby(1993). However, when the wave-induced current is included, thepattern of wave height distribution is reversed, and the computationalresults agree with both the measured data of Vincent and Briggs(1989) and the calculated results of Yoon et al. (2004). Goda (2004)conducted the simulations for the same cases of N5 and B5 using theparabolic equation wave model developed by Hirakuchi and Mar-uyama (1986) with some improvements on breaking criteria with thegradational breaker index. From simulations neglecting the effect ofthe wave-induced current, Goda (2004) concluded that his resultswere more accurate than those of Özkan and Kirby (1993) or Yoonet al. (2001) because of the improved breaking criteria in his com-putation. However, the discrepancy between his result and the mea-sured data at the center of transect 4 was not fully explained. In thepresent study it is clearly shown that the wave-induced current is thekey mechanism to resolve this controversial issue.

The results of REF/DIF-SHORECIRC are closer to the measurementsthan those of SWAN-SHORECIRC as mentioned by Choi et al. (2007)but the differences are too small to judge the superiority. Both of theresults show that waves are not focused behind the shoal but ratherrefracted outside of the jet-like wave-induced current.

Numerical simulations of the breaking randomwave cases N5 andB5 were also conducted using FUNWAVE. Fig. 15 shows the horizontaldistributions of wave height, mean current, and mean free surfaceobtained by spectral analyzing or averaging the computed instanta-neous quantities for 260 s after a waiting time of 27.3 s for the narrow-spectral wave condition N5 (top panels) and the broad-spectral wave

Fig. 16. Wave height distribution and mean current profile computed with FUNWAVE and tRD=REF/DIF, RD+SC=REF/DIF and SHORECIRC, SW=SWAN, SW+SC=SWAN and SHORECIRC

condition B5 (bottom panels). Note that thewave shadow zone in eachcase has relatively low wave heights behind the shoal due to wavedefocusing affected by wave-induced currents. The overall distribu-tions of wave height, mean current and mean free surface are close tothe results of the two model systems shown in Figs. 12 and 13.However, the results simulated by FUNWAVE are not symmetric withrespect to the centerline, as discussed for the cases of non-breakingrandom waves.

The profiles of wave height and mean current computed byFUNWAVE along the centerline and the transect 4 are presented inFigs. 16 and 17 for N5 and B5 cases, respectively. These results arecompared with the measurements as well as the results of REF/DIF-SHORECIRC (left panels) and the results of SWAN-SHORECIRC (rightpanels). The figures also show the results excluding the effect of wave-induced current computed by the combined models. The results ofFUNWAVE agree with those of the model systems including the effectof wave-induced current as well as the measurements, and the resultsof SWAN-SHORECIRC are nearly identical to those of FUNWAVE.The results reconfirm that the wave-induced currents preventthe wave focusing behind the shoal. The distribution of the jet-likewave-induced current had been slightly tilted to the left side for the260-second simulation time due to the asymmetry of input waveheight along the generation line as discussed earlier.

6. Conclusions and summary

A jet-like current induced by waves breaking over a submergedshoal changes refraction of waves and the resultant wave transforma-tion influences the wave-induced current. This transformation due towave-current interactions was investigated using numerical simula-tions for breaking monochromatic and random wave cases. Thenumerical simulations were conducted using two combinations of

he combined models for random wave case N5 along (a) centerline and (b) transect 4..

Fig. 17. Wave height distribution and mean current profile computed with FUNWAVE and the combined models for random wave case B5 along (a) centerline and (b) transect 4.RD=REF/DIF, RD+SC=REF/DIF and SHORECIRC, SW=SWAN, SW+SC=SWAN and SHORECIRC.

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wave and current models, REF/DIF-SHORECIRC and SWAN-SHORE-CIRC, as well as the FUNWAVE model.

To check the performance of various wave models, numericalsimulations neglecting the wave-induced currents were conductedfirst for non-breaking wave conditions, and the calculated waveheights were compared with the experimental measurementsreported earlier. The results showed that two wave models, REF/DIFand FUNWAVE, gave good agreements with the experimentalmeasurements for both monochromatic waves and random waves.However, the SWAN model is less accurate for the monochromaticcase, while it gave reasonable agreements for the random cases. This isnatural because this model is originally developed to simulate randomwaves.

For the breaking monochromatic wave case the resulting perfor-mances of these three wave-current interaction models are summar-ized in the followings: When waves were breaking on the top of asubmerged shoal, a strong jet-like current was developed behind theshoal. As a result, the waves refracted towards the center behind theshoal were defocused and two local peaks appeared apart laterallyfrom the center. All three wave-current interaction models gave thesame pattern of wave transformation, and the results agreed qual-itatively well with the measurements. However, when the wave-induced current was intentionally neglected, the wave heightdistribution patterns along the transect 4 were completely reversedfrom the measured ones. Thus, the effect of wave-induced current onthe transformation of waves was confirmed again. The details on theperformance of the three wave-current interaction models, however,were different. The FUNWAVE model gave a perfect agreement withthe measured wave height behind the shoal, and a jet-like current wasgenerated. The SWAN-SHORECIRC model is less accurate in the simu-lation of wave height, but a jet-like current was generated behind theshoal as in the case of FUNWAVE. On the other hand, the REF/DIF-SHORECIRC model does not give an organized jet-like current. Thus,

the wave-current interaction was not accurately reflected in thesimulation of wave field. The poor performance of the REF/DIF-SHORECIRC model is attributed to the uncertainties involved in theevaluation of radiation stresses for the interference pattern of multi-directional waves.

For the breaking randomwave cases the resulting performances ofthe wave-current interaction models are summarized in the follow-ings: Since the incident randomwaves have directional spectrum, theeffect of the phase interference between directional components onthe wave climate behind the shoal is negligible. Thus, the simulatedresults using REF/DIF-SHORECIRC were similar to those using SWAN-SHORECIRC and FUNWAVE. The wave heights simulated using threemodels gave good agreements with the measured data, and the jet-like currents induced by the gradient of radiation stresses of breakingwaves were generated behind the shoal. The performance of the twocombined wave and current models is comparable to that ofFUNWAVE except for the irregularity and asymmetry of wave heightsand currents in the transverse direction. Thus, it can be concluded thatthe wave-current interaction models based on the conventionalradiation stress concept can be successfully applied to simulate therandom waves with directional spectrum.

Based on the numerical results simulated using various wave-current interaction models, it is confirmed again that wave-inducedcurrents play a significant roll in the transformation of waves breakinglocally in the area of interest. For the particular situation whentransverse interference patterns are present due to refraction ofmonochromatic waves, the interference patterns cannot be simulatedin the SWAN-SHORECIRC model and their radiation stresses are notcorrectly evaluated in the REF/DIF-SHORECIRC model. For thissituation models based on primitive variables approach such asFUNWAVE should be a better choice at the expense of computationaleffort. However, for random waves of directional spectra, the twocombined wave and current models worked reasonably well.

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Acknowledgements

We sincerely thank M. J. Briggs of Coastal and Hydraulics Labora-tory of the U.S. Army Engineer Research and Development Center forproviding us with the experimental measurements that were used inour study. We greatly appreciate the constructive comments from thetwo anonymous reviewers on an early version of the manuscript.

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