Development and Validation of a Quasi-Three-DimensionalCoastal Area Morphological Model
Yan Ding1; Sam S. Y. Wang2; and Yafei Jia3
Abstract: A quasi-three-dimensional coastal area morphological �Q3DCAM� model has been developed using the process-based ap-proach. It is for simulating complex multiscale coastal processes, primarily morphodynamic changes of the seabed. This software packagehas integrated three key submodels for simulating irregular wave deformations, nearshore currents, and morphological processes. Thequasi-three-dimensional capability of the depth-averaged model has been developed to consider the vertical flow structure inside the surfzone and the cross-shore movement mechanisms of nearshore currents, e.g., undertow and mass flux. To this end, the calculations of theradiation stresses inside the surf zone have been improved by introducing the nonsinusoidal wave model for surface roller effects due tothe breaking wave. To predict accurately the wave field and the morphological processes near coastal structures, the wave diffractioneffects were included in a multidirectional spectral wave transformation model. The morphodynamic change was modeled by consideringthe sediment transport due to the combinations of waves and currents. These three submodels were validated by simulating threelaboratory experimental cases in regard to: �1� irregular wave deformations over a shoal; �2� longshore currents in a wave basin; and �3�moveable bed evolutions around an offshore breakwater under attack of an incident wave. The numerical results of the morphologicalmodeling confirmed that the Q3DCAM model consisting of the diffraction effects and the surface roller effects is capable of predictingwaves, currents, and morphodynamic changes more accurately than before. Therefore, this validated model can be applied to simulatemore realistic morphological processes in coastal zones including structures.
Understanding morphological processes in coastal zones drivenby wave and current is crucial to coastal sediment management,navigation channel maintenance, and designing erosion protectionstructures. Correct prediction of deposition/erosion process andestimation of sand budget in areas from rivers to coasts/estuariesare key tasks in the regional sediment management. Coastalmorphological changes result mainly from transformation and de-formation of surface gravity waves propagating across the conti-nental shelf to the beach, the wave-induced currents in the surfzone, and longshore and cross-shore sediment movements. In thepast decades, significant progress has been made in the studies ofcoastal processes by means of physical experiments and compu-tational simulations. Especially, with the aid of the advanced
1Research Assistant Professor, National Center for ComputationalHydroscience and Engineering, The Univ. of Mississippi, University, MS38677 �corresponding author�. E-mail: [email protected]
2FAP Barnard Distinguished Professor and Director, The Univ. ofMississippi, University, MS 38677.
3Research Associate Professor, The Univ. of Mississippi, University,MS 38677.
462 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINE
numerical techniques, the simulation of the wave-breaking pro-cess has revealed the details of wave propagation through the surfzone �Liu and Losada 2002�. However, due to the extreme com-plexities of natural morphological processes, the mechanisms ofsediment transport have neither been fully understood nor de-scribed adequately by physical principles and mathematicalanalyses. Direct simulation of long-term �daily to yearly� morpho-logical evolutions in a real-scale coast coupled with irregularwaves and wave-induced currents has been a challenging goal.With the process-based approach having been employed to thedevelopment of the coastal area morphological �CAM� model, thesimulation of morphodynamic changes and shoreline evolutionshas become feasible �e.g., Shimizu et al. 1997; Zyserman andJohnson 2002�. In general, this was accomplished by computingsequentially the wave field, the current field, and the seabedchanges. Then a new bathymetry will be fed back to affect thecomputations of the wave and current fields in the next time step�Fig. 1�. By this iterative procedure going through the wave-current-morphological models, it is possible to simulate the long-term morphological process by using an empirical sedimenttransport model for the fine time-scale morphological process�e.g., Reniers et al. 2004; Ding and Wang 2005�. However, beforethe process-based morphological model can be applied to realisticcoastal problems, it has to be verified and validated systematicallyto find out if: �1� the models have included the most importantprocesses for modeling waves, currents, and sediment transport;�2� each module, which serves as a numerical solver of somephysical variables, can predict the results of the corresponding
variables with reasonable accuracy; and �3� it can run robustly to
simulate long-term morphodynamic changes by selecting a rea-sonable long feedback period Tsed �see Fig. 1�.
According to the existence of different spatial scales,De Vriend et al. �1993� have classified the practical numericalmodels for simulating morphological processes into four types:�1� one-dimensional �1D� longshore coastline models; �2� two-dimensional �2D� cross-shore coastal profile models; �3� 2D hori-zontal morphological models; and �4� fully three-dimensional�3D� local morphological models. 1D coastline models can onlydescribe behaviors of the longshore sediment transport andshoreline evolutions by using the sand budget approach; 2Dcross-shore coastal profile models are able to predict the verticalvariations of coastal profiles, but not the variations of the long-shore sediment transport; 2D horizontal morphological modelscan simulate the morphological variations over a coastal area witha rather wide range of spatial scales �e.g., 100 m2–102 km2� withthe vertical variations of waves and currents ignored. Neverthe-less, only fully 3D morphological models are expected to takeinto account both the vertical and horizontal variations of waveand current �e.g., Lesser et al. 2004�. However due to the time-consuming nature for practical problems, 3D models are restrictedgenerally to predict the temporal–spatial morphological changesin a relatively small near field and in a short duration. As men-tioned above, the horizontal 2D models have the potential to as-sess the bed evolutions in large-scale areas, e.g., tidal inlets andestuaries. Therefore, a quasi-3D model with the features of the 2Ddepth-averaged model and vertical effects of waves and currentswould be a feasible tool for the long-term morphodynamic simu-lations in a large-scale coastal engineering problem. Zysermanand Johnson �2002� have presented a quasi-3D morphologicalprocess model in which an empirical 3D shear stress distributionwas used to take into account a quasi-3D effect of sediment trans-port. But, they used a classical hydrodynamic module to computethe depth-averaged velocity without the consideration of the non-uniformities of vertical current in the momentum equations due tothe surface rolling effect in surf zone. However, the quasi-3D
Fig. 1. Flow chart of feedback system in coastal area morphologicalmodel
coastal area morphological �Q3DCAM� model presented in the
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paper has the following basic characteristics to meet the demandof accuracy and robustness in practical applications: �1� a 2Ddepth averaged current model including the effects of verticalvariations of currents due to the surface rolling of wave breaking;�2� the nonsinusoidal radiation stresses used inside the surf zoneand the shortwave volume flux included in the current model fortaking into account the undertow; and �3� an accurate and effi-cient wave driver �wave spectral model� for computing irregularwave deformations including diffraction, refraction, shoaling,and energy dissipation due to the wave breaking. Therefore, thisQ3DCAM model enables to compute accurately cross-shoresediment transport and morphological changes around coastalstructures.
By means of the process-based approach, the presentQ3DCAM model has integrated systematically the three majorsubmodels for simulating irregular wave deformations, wave-induced currents, and coastal morphodynamic changes. As far asirregular wave models are concerned, wave spectral models aremore efficient than phase-resolving wave models, but omitting thediffraction effects will be a concern in a case with coastal struc-tures. A multidirectional spectral wave transformation �MDSWT�model with the diffraction effect terms proposed by Mase �2001�was therefore used. In order to take into account the 3D featuresof the vertical current structures �e.g., the surface rollers or theundertow currents� in the surf zone, the improved radiationstresses formulae derived from the nonsinusoidal wave assump-tion �Svendsen 1984� were employed in the 2D depth-averagedmomentum equations. Svendsen et al. �2003a, b� showed that thenonsinusoidal wave model could give more accurate nearshorecurrents in surf zone than those without the consideration of thevertical current variations. This Q3DCAM model for simulationof coastal processes has been built in a developed software pack-age called the CCHE2D �Jia and Wang 1999; Jia et al. 2002�,which is a systematically verified and validated tool to analyze2D shallow water flows, sediment transport, and water quality,with natural flow boundary conditions. Similar to the CCHE2Dmodel, the three submodels were discretized in a nonorthogonalgrid system so that the models have high accuracy in simulatingphysical variables in complex coastal zones with irregular coast-lines. A time-marching algorithm proposed by Jia et al. �2002�was used for computing the wave-induced currents. A validatedalgorithm in the CCHE2D for the treatment of wetting and dryingin the computational area was directly used for predicting theshoreline movement �Jia and Wang 1999�.
These three submodels were sequentially validated by simulat-ing three laboratory experimental cases in regard to: �1� irregularwave deformations over an elliptical shoal �Vincent and Bridggs1989�; �2� longshore currents in a wave basin called the large-scale sediment transport facility �LSTF� �Hamilton and Ebersole2001�; and �3� moveable bed evolutions around an offshorebreakwater under attack of an incident wave �Mimura et al.1983�. Modeling the diffraction effect was investigated rigorouslyby computing the irregular wave deformations in the first case. Incomparisons with the measured currents in the second case, it wasconfirmed that the improved radiation stresses could give muchmore accurate currents than those by the classical radiationstresses, in which the sinusoidal wave model applied to all over acoast. The validation results about the morphodynamic changes inthe third case showed that the nearshore currents predicted by theclassical radiation stresses could not convey sediment toward off-shore; however, the currents resulted from the improved radiationstresses could produce reasonable morphodynamic changes in-
cluding sand depositions behind the structure, offshore sand bars,
scours at the structure tips, and shoreline erosions. In summary,this Q3DCAM model consisting of the diffraction effects and thesurface roller effects enables to give better understanding oncoastal processes and accurate predictions of waves, currents, andsediment transport. This model can be further applied to simulatemorphological processes in real-scale coastal zones includingstructures.
Model Descriptions
In the Q3DCAM model, the coastal-process submodels were for-mulated by several partial differential equations. A phase-averaged model with a term representing the diffraction effectswas developed as a fast wave driver to predict statistical variablesof irregular waves such as significant heights, periods, and meandirections in a large-scale coastal zone. Adding the diffractioneffects was to extend the spectral model capability to simulate thewave fields in the lee of coastal structures where the effects maybe dominant. An improved radiation stress model �Svendsen1984� was used to take into account the effects of the variations ofvertical currents due to the surface rolling in the surf zone, inwhich the sinusoidal and nonsinusoidal wave models were ap-plied to represent the radiation stresses inside and outside the surfzone, respectively. Because of the discontinuity in the radiationstress model, a transition model was proposed to calculate thestresses in a transition zone between the deep water zone and thesurf zone. In the wave-induced depth-averaged current model, abed friction stress due to the combined wave and current �Tanakaand Thu 1994� was used. The cross-shore and longshore sedimenttransport rates were calculated by means of a total sediment fluxmodel �Watanabe et al. 1986�. The morphodynamic changes ofthe seabed were then computed by a sediment balance model withthe downslope gravitational effect. Simulations of the morphody-namic changes in one cycle of the coastal processes were imple-mented for a period Tsed �Fig. 1�. Because the fine-scale sedimenttransport process represented by temporal/spatial variations ofsediment flux in the model is much slower than the wave andcurrent processes, this feedback period Tsed could be a relativelong timescale. The temporal/spatial variations of waves and cur-rents were represented by a series of quasi-steady wave and cur-rent fields over the whole period for morphological computation.Therefore this feedback system including the wave–current–morphology interactions becomes efficient to perform a long-termmorphological simulation.
Multidirectional Spectral Wave Transformation Model
The MDSWT model, which produces statistical variables of ir-regular waves such as significant heights, periods, and mean di-rections due to refraction, diffraction, and wave breaking, is basedon a spectral energy balance equation. The variations of waveenergy density S�x ,y ,� , f� in a temporal–spatial–frequencydomain under the attack of multidirectional incident waves isdescribed as
�S
�t+
�vxS
�x+
�vyS
�y+
�v�S
��= Q �1�
where t=time; �=wave direction; x, y=horizontal coordinates;Q=source term which represents generation, wave–wave interac-tion, and energy dissipation due to wave breaking and bottomfriction; and v=energy transport velocity, of which three compo-
nents are
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vx = Cg cos �, vy = Cg sin �, v� =Cg
C�sin �
�C
�x− cos �
�C
�y��2�
where C=wave celerity; and Cg=wave group celerity. The direc-tional spread of the wave energy is frequency dependent, so thedirectional formulations are commonly defined as
S�f ,�� = S�f�D�f ,�� �3�
in which S�f�=1D frequency spectrum; and D�f ,��=directionalspreading function. A number of the 1D wave spectra can befound in the relevant literature. In this study, the TMA spectrum�Texel-Marsden-Arsloe, named after the three data sets used in itsdevelopment� �Bouws et al. 1985� and the Bretschneider–Mitsuyasu �B-M� spectrum �Mitsuyasu 1970; Goda 1998� wereused for simulating the irregular wave deformations in the vali-dation test cases. Similar to the breaking wave dissipation term inThornton and Guza �1983�, Takayama et al. �1991� assumed thatthe probability density function of the irregular breaking waveheight follows a Rayleigh distribution. Therefore, the dissipationterm due to wave breaking can be calculated by using the loss ofenergy flux in a local computational grid. In the computation ofthe wave energy equation, the frequency domain of a wave spec-trum is divided into a set of individual representative wave fre-quencies. An energy loss due to an individual wave breaking iscalculated by taking into account the Goda’s wave breaking cri-terion, i.e.,
Hb
L0= 0.17�1 − exp�− 1.5
�h
L0�1 + 15m4/3��� �4�
where Hb=breaking wave height; L0=deep water wave lengthcorresponding to the significant wave period; h=water depth; andm=beach slope at breaking. The total wave energy at each nodalpoint is obtained through summing up the individual wave energy,which is used to calculate the statistical variables of the irregularwaves.
It has been well known that the energy balance Eq. �1� canpredict correctly the refraction effects of irregular waves, but notthe diffraction effects of the waves generated in the lee of coastalstructures. One of the existing approaches for adding the diffrac-tion effects is to mimic diffraction with spatial or spectral diffu-sion �e.g., Resio 1989; Booji et al. 1997; Mase 2001�. By analogywith a parabolic wave refraction–diffraction equation, Mase�2001� proposed an improved energy balance equation includingthe diffraction effects as follows:
�S
�t+
�vxS
�x+
�vyS
�y+
�v�S
��=
�
2�� �
�y�CCg cos2 �
�S
�y�
−1
2CCg cos2 �
�2S
�y2� + Q �5�
where the new term in the right hand side represents the energydissipation due to the diffraction effects in the alongshore y di-rection, which is implicitly perpendicular to wave direction;�=wave angular frequency; �=empirical coefficient �=2.0–3.0suggested by Mase 2001�. Mase �2001� concluded that Eq. �5�also has the advantages of simplicity and robustness in predictingthe wave conditions in a large-scale coastal area. However, priorto application of the spectral wave to practical wave prediction,
The depth- and shortwave-averaged 2D continuity and momen-tum equations are used for simulating nearshore currents incoastal zones, namely
��
�t+ � · �hu� = 0 �6�
�u
�t+ u · �u = − g � � +
1
�h� · �h�t� −
1
�h� · R +
�S − �b
�h
�7�
where �=water elevation; h=water depth; u=depth- andshortwave-averaged velocity vector in the horizontal co-ordinates; g=gravitational acceleration; �=water density;�t=depth-averaged Reynolds stress; �S=wind stress; �b=seabedfriction stress; and R=radiation stress which represents the net�shortwave-averaged� force the short wave exert on a water col-umn, is defined as �Svendsen et al. 2003a�
R =Zb
�
�pI + �uwuw�dz −1
2�gh2I − �
QwQw
h�8�
where the overbar denotes the time averaging over a shortwaveperiod; z=vertical coordinate; Zb=elevation of the seabed;p=total pressure, I=identity matrix �or Kronecker delta�;uw=horizontal shortwave-induced velocity; and Qw=wave vol-ume flux induced by the short wave motion. As a result, thetime-averaged contribution of the short wave forcing is repre-sented in the mass and momentum equations. The radiationstresses can be calculated using the results of wave heights andwave angles obtained by the above-mentioned wave model Eq.�5�. A generalized form of the radiation stress is
R = Sme + SpI − �QwQw
h�9�
where the tensor e is
e = � cos2 � sin � cos �
sin � cos � sin2 �� �10�
and the scalar Sm and Sp are calculated according to the followingtwo aspects: Outside the surf zone the two terms can be calculatedby the sinusoidal wave assumption, i.e.,
Sm =1
16�gH2�1 +
2kh
sinh 2kh� �11�
Sp =1
16�gH2 2kh
sinh 2kh�12�
where H=wave height; k=wave number for irregular waves; andH=significant wave height. Inside the surf zone, the waves areusually nonsinusoidal long waves �wavelength is greater thanwater depth�. In case of wave breaking, a volume of water inbreakers, the so-called surface roller is carried with the local wavespeed C. Svendsen �1984� proposed improved formulations tocalculate the radiation stresses inside the surf zone
Sm = �gH2C2�B0 +Ah
2 � �13�
gh H L
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Sp =1
2�gH2B0 �14�
where A=surface roller area which may be represented as 0.9H2;B0=wave shape parameter with B0=1/8 being the value for sinu-soidal waves; and L=wavelength. Kaihatu et al. �2002� discussedtwo different approaches to calculate the roller effect. In thestudy, one of them, the “static roller,” is used to describe the rollershape because it is only dependent on local wave properties.Similarly, the wave volume flux Qw outside the surf zone can becalculated by
Qw = B0gH2
Ci� �15�
where i�=unit vector of wave direction= �cos � , sin ��. Inside thesurf zone, the wave volume flux reads
Qw =gH2
C
C2
gh�B0 +
Ah
H2L�i� �16�
Most existing nearshore current models use Eqs. �11� and �12�derived from the sinusoidal wave theory to calculate the radiationstress. However, it has been already known that these classicalradiation stress formulations could not generate accurately near-shore currents inside surf zone when especially wave breaking.The improved radiation stresses consisting of Eqs. �11�–�16� takeinto account the vertical variations of wave breaker structuresinside the surf zone. The 3D features of the cross-shore move-ment mechanisms, e.g., undertow and mass flux, are reflectedaccordingly in the model. Putrevu and Svendsen �1999� furtherderived the comprehensive depth-averaged momentum equationsincluding 3D dispersion terms. Because of the complexities indetermining the 3D dispersion coefficients, this wave-inducedcurrent model only considers the influence of vertical flow struc-tures due to the surface rollers in the surf zone on the radiationstresses.
The bottom friction stress �b can be represented as ashortwave-averaged combination of wave and current, namely
�b = �Cfu + ub�u + ub� �17�
where ub=wave orbital velocity at the bottom; and Cf =frictioncoefficient due to combination of wave and current. The frictionlaw of the combined wave and current proposed by Tanaka andThu �1994� is used to estimate the friction coefficient in the dif-ferent flow regimes, i.e., rough turbulent flow, smooth turbulentflow, and laminar flow. The friction coefficient is generally givenas follows:
Cf = 0.5fc + �fc�fwcos � + 0.5�fw �18�
where fw and fc=friction coefficients due to wave and current,respectively; �=coefficient due to nonlinear interactions of wavesand currents; and �=angle between wave orthogonal and currentvector. One may refer to Tanaka and Thu �1994� for more detailsof calculating the friction coefficients in different flow regimes.
In Eq. �7� the depth-averaged Reynolds stress �t can be repre-sented as a model of turbulence closure. The Boussinesq eddy-viscosity approximation is used for formulating the turbulencestresses
�t = �e��u + ��u�T �19�
where ve=eddy viscosity coefficient; and superscriptT=transpose of a tensor. The present nearshore current model
provides users with two eddy viscosity turbulence models pertain-
ing to the characteristics of waves. One eddy viscosity model isthe Longuet-Higgins model �Longuet-Higgins 1970� which hasbeen proven to be effective in simulations of uniform longshorecurrents
e = Nl�gh �20�
where N=empirical coefficient �=0.001–0.01�; and l=distancefrom shoreline toward offshore. Another is the Larson–Krausmodel �Larson and Kraus 1991�, which is suitable for simulationsof circulations in the coasts with installation of structures
e = UWH �21�
where =empirical coefficient �=0.1–3.0�; and UW=magnitudeof wave orbital velocity at the bottom.
In general, the computational domain in a coastal zone is sur-rounded by the four boundaries: a shoreline, an offshore bound-ary, and two open cross sections in the cross-shore direction.Inside the domain, island shorelines or offshore structure bound-aries may be present. The known values of velocities or dis-charges can be imposed on the corresponding cross sections in thecross-shore boundaries. The impermeable condition of currentscan be used on shorelines. The known water elevation at offshoreboundary is needed to be specified as a reference value or a valuevarying with time. For the initial conditions for velocities andwater elevations, the cold start �starting from a static state� isgenerally utilized to initiate the simulations of the wave-inducedcurrents. If the incident wave is constant, the stable �or conver-gent� currents can be obtained after a period of time marching incomputing velocities and water elevations.
Sediment Transport and Morphodynamic ChangeModels
The variation of the seabed elevation Zb is calculated by consid-ering the local sediment balance and the downslope gravitationaltransport
�Zb
�t= − � · q +
�
�x��qx
�Zb
�x� +
�
�y��qy
�Zb
�y� �22�
where q= �qx ,qy�=local sediment transport rate; and �=empiriccoefficient. The bed evolution is described by a divergence termat the right-hand side and the other two terms for the anisotropicdownstream gravitational effect. De Vriend et al. �1993� pointedout that this slope-related transport mechanism enables a coastalprofile to reach the equilibrium bed topography; otherwise themorphodynamic simulation only based on the law of mass bal-ance will encounter an inherent instability of bed evolution. Thelocal sediment transport rate has two contributions from wave andcurrent �Watanabe et al. 1986�
q = qw + qc �23�
where qw and qc=local sediment transport rates due to wave andcurrent, respectively
qw = AwFD
�m − �c
�gubi� �24�
where �m=maximum bottom shear stress under the action ofwave and current
�m = �Cfub2 �25�
�c=critical shear stress
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�c = ��s − ��gd c �26�
where �s=density of sand; d=grain diameter; c=critical Shieldsparameter �=0.11 for fine sand 0.06 for rough sand�; andAw=empirical coefficient, which has a form proposed by Shimizuet al. �1997�, i.e.,
Aw =W0�0.5fw�0.5Bw
�1 − ���s� − 1��s�gd�0.5 �27�
where s�=�s /�−1; �=porosity of sand; W0=settling velocity; andBw=empirical coefficient of the wave-induced sediment transportrate. Shimizu et al. �1997� suggested that the value of Bw is 7.0for laboratory experimental cases, 3.0–5.0 for field cases. FD rep-resents the direction function �+1 for onshore, −1 for offshore�.The sediment transport rate qc due to mean currents has a similarform to qw
qc = Ac
�m − �c
�gu �28�
where Ac=empirical coefficient. Shimizu et al. �1997� suggestedthat the value of Ac could be approximately ten times as much asthat of Aw. In this study, the mean currents driven by the improvedradiation stresses were used to calculate the local sediment trans-port rate qc. The cross-shore sand movement mechanisms due toundertow and mass flux were included accordingly in the mor-phodynamic change model.
Numerical Approaches in Nonorthogonal MeshSystem
For simulation of the coastal morphological processes, the inte-grated numerical models have been developed to solve the abovementioned four partial differential equations, i.e., the energy bal-ance Eq. �5�, the depth-averaged continuity Eq. �6�, the momen-tum Eq. �7�, and the seabed level evolution Eq. �22�, together withtheir boundary conditions. A special numerical discretizationmethodology called the efficient element method �Jia and Wang1999� was used for discretizing the above four equations in anonorthogonal grid system. This numerical model is therefore ca-pable of simulating the morphological processes in coastal zoneswith complex coastlines.
As shown in Fig. 1, the morphodynamic modeling was imple-mented sequentially. First, the energy balance Eq. �5� describingirregular wave deformations was solved by means of the para-bolic approximation, by which the waves were assumed to have aprincipal propagation direction from offshore toward onshore.The calculations of the waves were therefore carried out line byline from offshore to onshore. In this study, the reflection effect ofthe waves in the negative x direction was neglected �Ding et al.2003�. Second, a velocity correction method �Jia et al. 2002� wasused to decouple and solve the continuity Eq. �6� and the momen-tum Eqs. �7�. And a time-marching algorithm proposed by Jiaet al. �2002� was employed for computing the nearshore currentfield with a time interval �called the current time step�. The statusof current field may be steady, quasi-steady, or unsteady, in ac-cordance with the status of incident wave. Then, the term of thedownslope gravitational transport in the bed level evolutionEq. �22� was calculated implicitly; and the term of the bed changewas calculated by means of the Eulerian forward scheme. How-ever, the time interval for simulating the morphodynamic changecould be different from the current time step. The bed levels were
calculated at each morphological time step by updating the local
sediment transport rate due to the variations of bed levels andbottom frictions. Finally, after a computational duration Tsed forsimulating the morphodynamic changes of the seabed, the wavesand currents were updated according to the computed new bedlevels. This integrated process-based model therefore consideredthe interaction of wave, current, and sediment transport by adjust-ing the period Tsed for controlling the frequency of this feedbackprocesses. This consideration is important to facilitate the long-term morphological processes calculation. In addition, to predictthe shoreline changes due to morphodynamic changes in thecoastal zone, a moving boundary treatment is capable of handlingthese complex and dynamic wetting and drying processes. Thewetting and drying treatment procedure checked locally the situ-ations of flows and the seabed levels at each time step and grid. Atypical criterion of critical water depth was given to activate agrid �wet� or to freeze an element �dry� in a computationaldomain.
Validation of Q3DCAM Model
As above mentioned, this Q3DCAM model consists of the threesubmodels for predicting wave, current, and morphologicalchange. These three submodels were sequentially validated bysimulating three laboratory experimental cases in regard to: �1�irregular wave deformations over an elliptical shoal carried out byVincent and Bridggs �1989�; �2� longshore currents in a wavebasin called the LSTF conducted at the U.S. Army EngineerResearch and Development Center’s Coastal and HydraulicsLaboratory �USA-ERDC’s CHL� by Hamilton and Ebersole�2001�; and �3� moveable bed evolutions around an offshorebreakwater under attack of an incident wave conducted byMimura et al. �1983�. The first case was to test the wave model’scapability for predicting irregular wave heights, periods, and di-rections; the second case was to test both the wave model and thecurrent model, and especially to confirm the advantages of theimproved radiation stresses in taking into account the surfaceroller effects; the third case was to validate the morphodynamicchange models through the wave–current–morphological interac-tion on simulating the seabed changes around an offshore break-
Fig. 2. Measured versus predicted wave spectra and directional sprfunctions D�� , f� at �m=10 and 30
water.
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Validation of Wave Model
In order to validate the MDSWT model, the distributions of ir-regular waves over an elliptical shoal were computed. This ellip-tical shoal in an experimental flume �Vincent and Briggs 1989�had a major radius of 3.96 m, a minor radius of 3.05 m, and amaximum height of 30.48 cm�0.3048 m at the center. The regionoutside the shoal was of constant depth �45.72 cm�0.4572 m�. Inthe experiments, a directional spectral wave generator producedincident irregular wave conditions which were described by theTMA spectrum with the mean wave directions identical to theaxis of the minor radius. The wave heights along several sectionsin several cases with different spectral parameters were measured.This well-known benchmark experimental case was investigatedintensively by a number of modelers by means of the phase-resolving method �e.g., Panchang et al. 1990� and the phase-averaged method �e.g., Mase 2001�. In this study, two selectedcases with different directional spreading spectra were computedby the MDSWT model. One was the narrow directional spreadingspectrum �called Case N1�; another was the broad directionalspreading spectrum �Case B1�. The input parameters for generat-ing the identical TMA frequency spectrum in the two cases werespecified as: the incident significant wave height H0=7.75 cm, thepeak period of wave spectrum Tp=1.3 s, the alpha constant ca-pable of adjusting variance �=0.0144, the peak enhancement fac-tor �=2. Fig. 2�a� compares the predicted frequency spectrum bythe MDSWT model with that measured by Vincent and Briggs�1989�. The TMA frequency spectrum reproduced very well theincident wave conditions.
A Fourier series representation for the wrapped normal spread-ing function D�� , f� in the TMA spectrum was used for producingtwo different wave spreading conditions, i.e., a narrow spreadingin the Case N1 and a broad spreading in the Case B1 �Vincent andBriggs 1989�
D��, f� =1
2�+
1
��j=1
J �exp�−1
2�j�m�2�cos�j�� − �m� � �29�
where �m=mean wave direction; J=total number of terms in theseries; and �m=spreading parameter which determines the width
functions: �a� frequency spectra at �=2; �b� directional spreading
eading
of the directional spreading. For the narrow spreading function,
�m=10°; the broad spreading �m=30°. In both the two cases, 50harmonics, i.e., J=50, were chosen to represent the directionalspreading function. Fig. 2�b� compares the two measured andpredicted directional spreading functions, respectively. It showsthat the two functions almost replicated the directional spreadingin the two different TMA spectra.
Meantime, because the B-M spectrum is also commonly usedto study irregular wave deformations in engineering applications,the performance of the B-M spectrum was also investigated in thetwo cases. To do so, the Mitsuyasu-type directional spreadingfunction �or “cosine squared” function� �Mitsuyasu et al. 1975�was employed in the B-M spectrum, i.e.,
D��, f� =22s−1
�
�2�s + 1���2s + 1�
cos2s�� − �m
2� �30�
where �=gamma function; and s=spreading function assumed tovary with wave frequency f , peak frequency f p, and a peak valueof s denoted as smax
Fig. 3. Normalized wave heights �Hs /H0� for narrow and broad direcon figures represents outline of elliptical shoal�
Fig. 4. Comparisons of normalized wave heights between computati�=0.0�
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s = �smax�f/f p�5 when f � f p
smax�f/f p�−2.5 when f � f p� �31�
In this study, two values of smax were selected, in whichsmax=75 represented the narrow spreading for Case N1, andsmax=10 the broad spreading for Case B1.
In the computations of the wave fields, the lower and upperfrequency bounds were set to 0.5 and 10 Hz, respectively. Thefrequency interval 0.095 �i.e., 101 frequency bins� and the angleinterval 5.0° �i.e., 37 directional bins between −90 and +90°�were adopted. The spatial increment 10 cm was used in the hori-zontal coordinates to create a uniform mesh. Due to physically nowave breaking, the terms of breaking wave effects were omittedin the simulation. At first, the wave deformations in the two caseswere computed without the consideration of the diffraction effect��=0.0�. The distributions of the normalized significant waveheights �Hs /H0� in Case N1 and B1, which were computed bymeans of the TMA spectra without the diffraction, are shown in
spreading generated by TMA spectra without diffraction �dashed line
d measurements �dashed dot lines represent measurement transacts;
Figs. 3�a and b�, respectively. The waves generated by the narrowdirectional spreading spectrum in Case N1 show a stronger con-vergent region behind the shoal than that by the broad one in CaseB1. In Figs. 4�a and b�, the normalized wave heights along threetransects obtained by the TMA and the B-M spectra in Case N1and Case B1 are, respectively, compared with the correspondingmeasured wave heights. Overall, the numerical results in Fig. 4show that both the TMA and the B-M spectrum predicted waveheights with quite good accuracy. However, due to ignoring thediffraction effect, the computed wave heights in the lee of theshoal in the narrow directional spreading spectrum �Case N1� aremuch higher than the measurements. Then this influence of dif-fraction behind the shoal needs to be concerned.
To take the diffraction effect into account correctly, the � valuehas been calibrated under the wave conditions of the narrow andbroad directional spreading spectra. According to the range of thevalue suggested by Mase �2001�, the computations of the wavefields with different � values from 0.5 to 2.5 were implemented.These numerical simulations with diffraction terms were quitestable and robust because of its virtual effect of diffusion in the
Fig. 5. Comparisons of normalized wa
Fig. 6. Comparisons of normalized wave heights
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equation. The unstable phenomenon pointed out by Holthuijsenet al. �2003� never happened in the simulations. Figs. 5�a and b�show, respectively, comparisons of the normalized significantwave heights along three transects between the measurements byVincent and Briggs �1989� and the computations. The inclusion ofthe diffraction term in the wave equation indeed improved thepredictions of wave heights right behind the shoal. This termhowever has a little influence on the wave heights in the broaddirectional spreading spectrum �Case B1�. The transverse profilesof the significant wave heights in the second transect�x=9.14 m� obtained from different � values are plotted inFigs. 6�a and b� for Cases N1 and B1, respectively. Similarly,the diffraction made more improvement on the predictions of thewave heights in the narrow directional spectrum than that in thebroad one. From Fig. 6�a�, a reasonable � value, i.e., �=1.5, wasfinally suggested for studying the wave deformation around theshoal, which is less than that value ��=2.5� obtained by Mase�2001�. From the present results, that value of �=2.5 underesti-mated the wave heights in most places far from the shoal.
The current module in the Q3DCAM model has been validated bysimulating the wave-induced longshore current in a wave basincalled the LSTF, which was built at USA-ERDC’s CHL �Hamil-ton and Ebersole 2001�. The facility consists of a 30-m cross-shore, 50-m longshore, 1.4-m deep basin, and includes wavegenerators, a pumped recirculation system, and an instrumenta-tion bridge. Hamilton and Ebersole �2001� reported two compre-hensive test cases conducted on a concrete beach with straightand parallel contours �1:30 slope� which were to verify the facili-ty’s capability to generate the desired longshore uniform currentsunder regular and irregular incident wave conditions. The currentswere driven by long crested waves, which were generated by fourpiston-type wavemakers. The concrete beach had a longshoredimension of 31 m and a cross-shore dimension of 21 m. Theconstant still water depth at the offshore was 0.677 m. The com-putational domain in the study covered the parallelogram-likewave basin in the facility �30 m cross shore and 31 m longshore�,and was uniformly divided into nonorthogonal quadrilateral gridswith a mesh size of 149�156 �Fig. 7�. The irregular incidentwave case was chosen to validate the quasi-3D current model,which was named Test-8E in Hamilton and Ebersole �2001�. Inthe case, a TMA spectrum was used to define the spectral shape,in which the spectral width parameter � was 3.3. The TMA spec-trum generated an irregular wave condition by which the offshoresignificant wave height H0 was 0.233 m, the peak period Tp 2.5 s,and the incident angle 10.0°.
The irregular wave deformation in the wave basin was com-puted by using the MDSWT model with the parameterized TMAspectrum. Due to physically negligible diffraction effects in thebeach, the empirical parameter � was set to 0.0. The energy dis-sipation term in the wave equation was considered through thewave breaking criterion in Eq. �4�. The computed normalized sig-nificant wave heights �Hs /H0� and mean wave directions �waverays� are shown in Fig. 7. The wave breaker line in the figure wasdetermined by the wave breaking criterion of the saturated break-ers, in which the breaking wave height was assumed to have alinear relation with the water depth, i.e., Hb=�h, where �=0.75.The value of the empirical constant � was the same as that mea-sured by Hamilton and Ebersole �2001�. The comparisons of the
Fig. 7. Computed distribution of normalized significant wave heightand direction. Grid in upper left corner shows closeup view of part ofnonorthogonal mesh; profile at right side shows bathymetry in whichbeach slope is 1:30.
cross-shore wave height profiles between the measurements by
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Hamilton and Ebersole �2001� and the computations with andwithout the wave breaking effects are shown in Fig. 8. The com-puted wave heights with the wave breaking effects are in goodagreement with the measurements.
The computed significant wave heights and mean wave angleswere used to calculate the radiation stresses, and then to finallycompute the wave-induced currents. In this study, according tothe wave features varying in the cross-shore direction, the radia-tion stresses outside the surf zone were computed by using thesinusoidal wave model; the stresses inside the surf zone by em-ploying the nonsinusoidal wave model. However, if one woulduse wave breaking line to define the boundary of the surf zone�e.g., the breaker line in Fig. 7�, and would directly apply the twodifferent wave models to compute the stresses inside and outsidethe surf zone, the discontinuity might happen in the cross-shoreprofiles of the radiation stresses and eventually could result in adiscontinuous longshore current profile. This problem is essen-tially caused by a nonphysical gap in the values of the roller areaA which jump from zero to a finite value at the boundary of thesurf zone. Therefore, it is necessary to introduce a transition zonebetween the surf zone and the deep water zone. Actually, Svend-sen et al. �2003b� already pointed out this discontinuity existing inthe radiation stress and the currents; and they further proposed an
Fig. 8. Comparison of significant wave heights Hs �lower part showsbeach profile�
Fig. 9. Conceptual diagram of transition zone and calculation ofrolling area
approach to smooth two wave parameters, i.e., roller area A andphase speed C over a transition zone. But, the detailed implemen-tation and the definition of the transition region were not clear.
In this study, a transition zone illustrated in Fig. 9 was definedto connect the deep water zone with the surf zone: Assuming thatthe wave breaking happens at the water depth hb, from the loca-
Fig. 10. Computed wave-induced longshore currents and circulationsin LSTF in case of �1=25.0 �cross sections named Y19, Y23, Y27,and Y31 are four of measured sections in experiments�
Fig. 11. Comparisons of longshore currents at cross section
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tion of the wave breaker xb, the transition zone is supposed toextend toward offshore to x0b. The width of the transition zone isassumed to be �1hb, where �1 is an empirical parameter needs tobe calibrated in the application. Inside the transition zone, it canbe assumed that the roller area A varies continuously from zero atthe deep water to the 0.9H2 inside the surf zone
A = 0.9�1.0 +x − xb
�1hb�H2 �32�
The wave-induced currents were computed in the computa-tional domain covered the LSTF wave basin. The irregular waveproperties, i.e., significant wave height, mean wave direction, andsignificant wave period, were used to calculate the radiationstresses and the wave volume fluxes inside and outside the surfzone. The Longuet-Higgins model in Eq. �20� was employed asthe turbulence model in the case, in which the empirical coeffi-cient N was 0.001. The recirculation system was simulated bysetting the inflow in the left opening boundary and the outflow inthe right opening boundary �Fig. 10�. The measured long-shore-averaged velocities by Hamilton and Ebersole �2001� werespecified on the inflow boundary. The so-called open boundarycondition was imposed on the outflow boundary. Fig. 10 alsoshows the circulations illustrated by the streamlines in the upperpart and the longshore current profiles in six transects which wereprojected from the velocity field computed in the nonorthogonalgrid points. The computed flow pattern reproduced very well theuniform longshore currents and the circulations in the facility. Toconfirm the performance of the two models for calculating the
, Y23, Y27, and Y31 where locations are shown in Fig. 10
radiation stresses inside and outside the surf zone, the longshorecurrent profiles computed by the classical sinusoidal wave modelwere compared with that by the wave model with the roller effect.The four pictures from Figs. 11�a–d� show comparisons of thecurrent profiles between the measurements and simulations in thefour transects, i.e., Y19, Y23, Y27, and Y31, respectively, ofwhich the locations are illustrated in Fig. 10. The currents by theclassical sinusoidal wave model without the roller effect �longdash lines� were always underestimated. Although the currentprofiles inside the surf zone �short dash lines� by the improvedradiation stresses �without the transition zone, i.e., �1=0.0� wereproduced better than the sinusoidal wave model, the unexpecteddiscontinuity appeared in all the four profiles. In contrast, theimplementations of the transition zone improved the longshorecurrent profiles quite well. The parameter values of the �1 in Eq.�32� were calibrated, and only the results of the currents at the �1
values 25.0, 40.0, and 50.0 are shown in Fig. 11. A calibratedvalue of �1=40.0 �bold solid lines� was finally found to predictthe most accurate longshore currents both inside and outside thesurf zone. This result indicates that the transition zone in theoffshore was approximately 8 m wide.
Validation of Morphological Change Model
This integrated Q3DCAM model has been validated systemati-cally by simulating the morphodynamic changes due to theinteraction of wave and current in a movable bed laboratory ex-periment conducted by Mimura et al. �1983�. This experiment
Fig. 12. Computed wave heights �unit:cm� and directions after 6 h ofbreakwater installation. Assumed transition zone is surrounded bytwo dashed dot lines.
Fig. 13. Comparison of breaking wave heights between simulationand measurement
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Fig. 14. Computed currents at t=6 h by: �a� classical radiationstresses; �b� improved stresses
Fig. 15. Computed bed changes after 6 h by: �a� classical radiationstresses; �b� improved stresses; and �c� measured bed changes �cm�
was carried out in a wave basin being 14 m long, 7.5 m wide, and0.42 m deep. The beach with 1/20 slope was initially coveredwith 10-cm thick sand, which had a uniform diameter of 0.2 mmand the density of 2.65 g/cm3. The incident irregular wave with5.7 cm height and 0.9 s period attacked normally the beach forapproximately 12 h. Then, an offshore breakwater of an iron platewith 1.5 m long and 0.5 m height was installed approximately atthe wave breaking line �1.8 m offshore from the initial shoreline�.The experiment of the morphodynamic changes lasted more than12 h after the installation of the offshore breakwater. The simula-tion by this numerical model was started just from the installationof the structure and terminated after 6 h. The measured beachtopography at the initial time of the installation was used forcreating a computational mesh with a 10-cm uniform spatial in-crement. This measured initial bathymetry used for the followingsimulations of the morphodynamic changes was not symmetrical.The wave heights and directions have been computed by usingthe B-M spectrum as the incident wave spectrum. To include thewave diffraction in the lee of the breakwater, the calibrated em-pirical parameter � in the wave Eq. �5� was 2.5. Fig. 12 shows thedistributions of the wave heights and the directions computedafter 6 h of the structure installation. Fig. 13 compares the com-puted breaking wave heights with the measurements by Mimuraet al. �1983�, for which the breaking wave index � was 0.65. Thesimulated breaking wave heights are in good agreement withthe measured ones. The transition zone between the surf zone andthe deep water was defined as the area from the breakwater to thesecond wave breaking line near the shoreline. Thus, the parameterof the zone width �1 was set to 20.0. In Fig. 12, a transition zonefor computing the currents at t=6 h is illustrated as the area sur-rounded by two dashed dot lines.
By using the abovementioned two radiation stress models, twocases with and without the surface roller effects were imple-mented to simulate the morphological processes around thebreakwater. The Larson–Kraus eddy viscosity model in Eq. �21�was chosen to calculate the turbulence stress with the valueequal to 0.3. The time step for simulating the currents was 1.0 s.Each quasi-steady current field in the domain under the action ofa steady wave field was obtained after approximately 1,000 stepsof time marching computations. The time step for simulations ofthe seabed evolutions was 0.5 s. The local sediment transport ratecoefficient Bw in Eq. �27� was 7.0 suggested by Shimizu et al.�1997�. The critical water depth for checking the wetting anddrying process was set to 0.2 mm. The empirical coefficient � inEq. �22� was 5.0. Through a number of test runs, the feedbackperiod Tsed was finally set to 5 min, namely, the simulations of
Fig. 16. Computed local sediment transport rates and bed levels
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waves and currents were repeated after every 5 min of the seabedevolution computations. Figs. 14�a and b� show two computedcurrents at t=6 h by the classical radiation stresses and the im-proved stresses, respectively. Mimura et al. �1983� had roughlymeasured currents at some locations cross the surf zone by whichthey proved the existence of the circulations behind the structure.Unfortunately, they did not carry out an overall survey of currentsin the coast. Nevertheless, some important differences betweenthe two currents computed by the two radiation stresses can beremarked: �1� the improved stresses produced clearly four circu-lations in Fig. 14�b� behind the structure and offshore, but theclassical stresses only detected two behind the breakwater inFig. 14�a�; �2� the improved stress generated the longshore cur-rents near the first breaking line �almost parallel to the breakwa-ter� more strongly than the classical stresses did; and �3� theonshore currents washing the two tips of the structure shown inFig. 14�b� were only captured by the improved stresses. The off-shore current patterns illustrated in Fig. 14�b�, which are virtuallyconnected with the appearance of undertow flow in the surf zone,play an important role in simulating the evolutions of scoursaround the structure and bars in the offshore.
Consequently, two types of the seabed changes over the 6 hwave attack were computed by using, respectively, the two differ-ent current fields. Two snapshots of the bed changes at t=6 h areshown in Figs. 15�a and b�, respectively. In addition, the mea-sured bed change distribution shown in Fig. 15�c� was obtainedby comparing the initial bathymetry with the bed form after 6 hmeasured by Mimura et al. �1983�. The bed changes in the ex-periment have exhibited abundant sediment depositions behindthe breakwater and developing sand bars in the offshore; it alsoshowed the severe shoreline erosions and local scours at the tipsof the structure. The results of bed changes shown in Fig. 15�b�by the improved radiation stresses precisely reproduced thesechanges of the coastal topography both behind the structure andoffshore. The local sizes of the depositions and erosions onshoreand offshore are quite similar to the observation, although thestrength of shoreline erosion in the simulation is not as strong asthe experiment. However, the bed changes shown in Fig. 15�a� bythe classical stresses could not match the overall morphodynamicchanges; the scours and the offshore bars were almost missed, butonly the depositions behind the structure.
Furthermore, Figs. 16�a and b� presented the seabed levels andthe total sediment transport rates at t=6 h obtained by the tworadiation stresses. It has been known that both of the sedimentfluxes can generate the tombolo-like topography behind thebreakwater. However, because the classical radiation stress model
t t=6 h by: �a� classical radiation stresses; �b� improved stresses
did not include the mechanisms of undertow current and massflux in the surf zone, the sediment transport in Fig. 16�a� wasconfined inside the lee side of the structure, and thus the sedi-ments were barely transported offshore. In contrast, the local sedi-ment fluxes shown in Fig. 16�b� almost followed the directions ofcurrents obtained by the improved radiation stresses shown inFig. 14�b�. The strong cross-shore sediment transport outside thesurf zone oriented offshore was obtained. This effect resultedfrom the inclusion of the 3D features of the currents by consid-ering the surface rolling effect in the surf zone. This morphologi-cal model therefore was also capable of producing the offshoresand bars and the local scours at the tips of the breakwater asshown in Fig. 15�b�. Figs. 17�a and b� compares the contour linesof bed elevations at −2.0 and −3.0 cm behind the breakwater att=6 h computed by the improved radiation stresses with the mea-sured contour lines, respectively. Although the shoreline erosionsare underpredicted due to lack of knowledge of the sedimenttransport in the swash zone, the overall agreement between thesimulations and measurements is quite reasonable.
Conclusions
In this paper, a newly developed quasi-3D costal area morpho-logical model �Q3DCAM� is presented, which consists of threekey submodels for simulating irregular wave deformations, near-shore currents, and morphodynamic changes. According to ourliterature review at present, this study is the first investigation tovalidate systematically the morphological processes model byconsidering the surface roller effect in the surf zone using thenonsinusoidal radiation stresses. The wave diffraction effects aretaken into account in the wave spectral equation. To remedy thediscontinuity in the radiation stresses between the surf zone anddeep water region, the concept for modeling the transition zone isproposed and tested.
These three submodels were sequentially validated by simulat-ing three laboratory experimental cases. First, the effectiveness ofthe diffraction term was confirmed by simulating irregular wavedeformations over a shoal in an experimental flume �Vincent andBriggs 1989�. Second, improvements on the radiation stresseswere mainly concerned with the hydrodynamic model in order totake into account the 3D features of currents induced by the sur-face roller effect, e.g., undertow current and mass flux in the surfzone. The nonsinusoidal wave formulations derived by Svendsen�1984� were then adopted to improve the accuracy of the radiationstresses. To remedy the discontinuity problem in the computedcurrents, a special approach has been proposed to consider thetransition zone in the calculation of the roller area. The simulation
Fig. 17. Comparisons of cont
results about the longshore currents generated in the LSTF facility
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at the USA-ERDC-CHL showed that the inclusion of the rollereffects and the transition zone could remarkably improve the pre-dictions of currents. Finally, the modeling for the morphodynamicchanges in an experimental setup of a coast with the installationof a breakwater �Mimura et al. 1983� was conducted. Because ofthe inclusions of the 3D flow features in the improved radiationstresses, the computed currents could correctly transport sedi-ments onshore and offshore in a complex bathymetry with thestructure installation. Therefore, this morphological processmodel was finally applied to the reproduction of complicated pat-terns of morphodynamic changes around the structure includingsand depositions behind the structure, offshore sand bars, scoursat the structure tips, and shoreline erosions. Numerical resultshave demonstrated that this Q3DCAM model is capable of simu-lating seabed changes including coastal structures. Thus, it canhelp researchers and engineers improve their understanding ofmorphological processes driven by waves and currents and fur-ther support coastal sediment management and planning of prac-tical coastal structures.
In order to achieve higher accuracy in the predictions of mor-phological processes for engineering applications, in the near fu-ture additional research to include other mechanisms of sedimenttransport, e.g., suspended sediment, transport in the swash zone,nonequilibrium sediment transport of nonuniform sediment sizeclasses, etc. are to be carried out.
Acknowledgments
This work is a result of research sponsored partially by the U.S.State Department Agency for International Development underAgreement No. EE-G-00-02-00015-00 and the University ofMississippi. Special appreciation is expressed to Dr. MustafaAltinakar and three anonymous reviewers for their criticalcomments.
Notation
The following symbols are used in this paper:A � roller area;
Booji, N., Holthuijsen, L. H., Doorn, N., and Kieftenburg, A. T. M. M.�1997�. “Diffraction in a spectral wave model.” Proc., 3rd Int. Symp.Ocean Wave Measurement and Analysis WAVES 97, ASCE, NewYork, 243–255.
Bouws, E., Gunther, H., Rosenthal, W., and Vincent, C. L. �1985�. “Simi-larity of the wind wave spectrum in finite depth water, 1-Spectralform.” J. Geophys. Res., 90�C1�, 975–986.
De Vriend, H. J., Zyserman, J., Nicholson, J., Roelvink, J. A., Pechon, P.,and Southgate, H. N. �1993�. “Medium-term 2DH coastal area mod-eling.” Coastal Eng., 21�1–3�, 193–224.
Ding, Y., and Wang, S. S. Y. �2005�. “Tests of capability and reliability ofa model simulating coastal processes.” EWRI 2005—Impacts of Glo-bal Climate Change, Proc., 2005 World Water and EnvironmentalResources Congress, R. Walton, ed., ASCE, Anchorage, Alaska.
Ding, Y., Wang, S. S. Y., and Jia, Y. �2003�. “Numerical studies on simu-lations of waves and nearshore currents in non-orthogonal mesh sys-tem.” Proc., Int. Conf. on Estuaries and Coasts, Zhejiang Institute ofHydraulics and Estuary, Hanzhou, China, 719–726.
Goda, Y. �1998�. “An overview of coastal engineering with emphasis onrandom wave approach.” Coastal Eng., 40�1�, 1–21.
Hamilton, D. G., and Ebersole, B. A. �2001�. “Establishing uniform long-shore currents in a large-scale sediment transport facility.” CoastalEng., 42�3�, 199–218.
Holthuijsen, L. H., Herman, A., and Booji, N. �2003�. “Phase-decoupledrefraction-diffraction for spectral wave models.” Coastal Eng., 49�4�,291–305.
Jia, Y., and Wang, S. S. Y. �1999�. “Numerical model for channel flowand morphological change studies.” J. Hydraul. Res., 125�9�,924–933.
Jia, Y., Wang, S. S. Y., and Xu, Y. C. �2002�. “Validation and applicationof a 2D model to channel with complex geometry.” Int. J. Comput.Eng. Sci., 3�1�, 57–71.
Kaihatu, J. M., Shi, F., Kirby, J. T., and Svendsen, I. A. �2002�. “Incor-poration of random wave effects into a quasi-3D nearshore circulationmodel.” Coastal Engineering 2002, Proc., 28th Int. Conf. on CoastalEngineering, J. M. Smith, ed., Cardiff, Wales, U.K., 747–759.
Larson, M., and Kraus, N. C. �1991�. “Numerical model of longshorecurrent for bar and trough beaches.” J. Waterway, Port, Coastal,Ocean Eng., ASCE, 117�4�, 326–347.
Lesser, G. R., Roelvinka, J. A., van Kester, J. A. T. M., and Stelling, G.S. �2004�. “Development and validation of a three-dimensional mor-phological model.” Coastal Eng., 51�8–9�, 883–915.
Liu, P. L.-F., and Losada, I. J. �2002�. “Wave propagation modeling incoastal engineering.” J. Hydraul. Res., 40�3�, 229–240.
Longuet-Higgins, M. S. �1970�. “Longshore currents generated by ob-liquely incident wave.” J. Geophys. Res., 75�33�, 6778–6789.
Mase, H. �2001�. “Multidirectional random wave transformation modelbased on energy balance equation.” Coastal Eng., 43�4�, 317–337.
Mimura, N., Shimizu, T., and Horikawa, K. �1983�. “Laboratory study onthe influence of detached breakwater on coastal change.” CoastalStructure ’83, Proc. Specialty Conf. on Design, Construction, Main-tenance and Performance of Coastal Structures, J. R. Weggel, ed.,ASCE, Arlington, Va., 740–752.
Mitsuyasu, H. �1970�. “On the growth of spectrum of wind-generated
waves �2�—spectral shapes of wind waves at finite fetch.” Proc., 17th
Japanese Conf. on Coastal Engineering, 1–8 �http://cive.gifu-u.ac.jp/~kobat/kaiko/kaiko17/k17001.html� �Nov. 9, 2005� �in Japanese�.
Mitsuyasu, H., Suhaya, T., Mizuno, S., Ohkuso, M., Honda, T., and Riki-ishi, K. �1975�. “Observations of the directional spectrum of oceanwaves using a cloverleaf buoy.” J. Phys. Oceanogr., 5, 750–760.
Panchang, V. G., Briggs, M. J., and Pearce, B. R. �1990�. “Numericalsimulation of irregular wave propagation over shoal.” J. Waterway,Port, Coastal, Ocean Eng., ASCE, 116�3�, 324–340.
Putrevu, U., and Svendsen, I. A. �1999�. “Three-dimensional dispersionof momentum in wave-induced nearshore currents.” Eur. J. Mech.B/Fluids, 18�3�, 409–427.
Reniers, A. J. H. M., Roelvink, J. A., and Thornton, E. B. �2004�.“Morphodynamic modeling of an embayed beach under wave groupforcing.” J. Geophys. Res., 109, C01030.
Resio, D. T. �1989�. “A steady-state wave model for coastal applications.”Coastal Engineering 1988, Proc., 21st Int. Conf. Coastal Engineer-ing, B. L. Edge, ed., ASCE, New York, 929–940.
Shimizu, T., Kumagai, T., and Watanabe, A. �1997�, “Improved 3-Dbeach evolution model coupled with the shoreline model �3D-SHORE�.” Coastal Engineering 1996, Proc., 25th Conf. on CoastalEngineering, B. L. Edge, ed., ASCE, Orlando, Fla., 2843–2856.
Svendsen, I. A. �1984�. “Mass flux and undertow in a surf zone.” CoastalEng., 8�4�, 347–365.
Svendsen, I. A., Haas, K., and Zhao, Q. �2003a�. “Quasi-3D nearshore
476 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINE
circulation model SHORECIRC—Version 2.0.” Technical Rep. Pre-pared for Center for Applied Coastal Research, Univ. of Delaware,Newark, Del.
Svendsen, I. A., Qin, W., and Ebersole, B. A. �2003b�. “Modelling wavesand currents at the LSTF and other laboratory facilities.” CoastalEng., 50�1–2�, 19–45.
Takayama, T., Ikeda, N., and Hiraishi, T. �1991�. “Practical computationmethod of directional random wave transformation.” Report of thePort and Harbour Research Institute, Vol. 30, No. 1, Japan Ministryof Transport, Nagase, Yokosuka, Japan, 22–67 �in Japanese�.
Tanaka, H., and Thu, A. �1994�. “Full-range equation of friction coeffi-cient and phrase difference in a wave-current boundary layer.”Coastal Eng., 22�3–4�, 237–254.
Thornton, E. B., and Guza, R. T. �1983�. “Transformation of wave heightdistribution.” J. Geophys. Res., 88�C10�, 5925–5938.
Vincent, C. L., and Briggs, M. J. �1989�. “Refraction-difraction of irregu-lar waves over a mound.” J. Waterway, Port, Coastal, Ocean Eng.,115�2�, 269–284.
Watanabe, A., Maruyama, K., Shimuzi, T., and Sakakiyama, T. �1986�.“Numerical prediction model of three-dimensional beach deformationaround a structure.” Coast. Eng. Japan, 29, 179–194.
Zyserman, J. A., and Johnson, H. K. �2002�. “Modelling morphologicalprocesses in the vicinity of shore-parallel breakwaters.” Coastal Eng.,45�3�, 261–284.
