Kumar et al. (2012) Ocean Modeling (doi: /j.ocemod )

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1 Implementation of the vortex fore formalism in the Coupled Oean- Atmosphere-Wave-Sediment Transport (COAWST) modeling system for inner shelf and surf zone appliations Nirnimesh Kumar 1*, George Voulgaris 1 John C. Warner 2, Maitane Olabarrieta 2 1 Department of Earth and Oean Sienes 2 U.S. Geologial Survey Marine Siene Program Coastal and Marine Geology Program University of South Carolina 384 Woods Hole Road Columbia, SC 29208, USA Woods Hole, MA nkumar@geol.s.edu; Ph: ; jwarner@usgs.gov; Ph: ; gvoulgaris@geol.s.edu; Ph: molabarrieta@usgs.gov; Ph: * Corresponding Author Abstrat The oupled oean-atmosphere-wave-sediment transport modeling system (COAWST) enables simulations that integrate oeani, atmospheri, wave and morphologial proesses in the oastal oean. Within the modeling system, the three-dimensional oean irulation module (ROMS) is oupled with the wave generation and propagation model (SWAN) to allow full integration of the effet of waves on irulation and vie versa. The existing wave-urrent oupling omponent utilizes a depth dependent radiation stress approah. In here we present a new approah that uses the vortex fore formalism. The formulation adopted and the various parameterizations used in the model as well as their numerial implementation are presented in detail. The performane of the new system is examined through the presentation of four test ases. These inlude obliquely inident waves on a syntheti planar beah and a natural barred beah (DUCK 94); normal inident waves on a nearshore barred morphology with rip hannels; and wave-indued mean flows outside the surf zone at the Martha s Vineyard Coastal Observatory (MVCO). Model results from the planar beah ase show good agreement with depth-averaged analytial solutions and with theoretial flow strutures. Simulation results for the DUCK 94 experiment agree losely with measured profiles of ross-shore and longshore veloity data from Garez-Faria et al. (1998, 2000). Diagnosti simulations showed that the nonlinear proesses of wave roller generation and wave-indued mixing are important for the aurate simulation of surf zone flows. It is further reommended that a more realisti approah for determining the ontribution of wave rollers and breaking indued turbulent mixing an be formulated using nondimensional parameters whih are funtions of loal wave parameters and the beah slope. Dominant terms in the ross-shore momentum balane are found to be the quasi-stati pressure gradient and breaking aeleration. In the alongshore diretion, bottom stress, breaking aeleration, horizontal advetion and horizontal vortex fores dominate the momentum balane. The simulation results for the bar / rip hannel morphology ase learly show the ability of the modeling system to reprodue horizontal and vertial irulation patterns similar to those found in laboratory studies and to numerial simulations using the radiation stress representation. The vortex fore term is found to be more important at loations where strong flow vortiity interats with the wave-indued Stokes flow field. Outside the surf zone, the three-dimensional model 1

2 simulations of wave-indued flows for non- breaking waves losely agree with flow observations from MVCO, with the vertial struture of the simulated flow varying as a funtion of the vertial visosity as demonstrated by Lentz et al. (2008). Keywords: vortex-fore, wave-urrent interation, COAWST, ROMS, SWAN, radiation stress, three-dimensional, modeling, rip urrent, littoral veloities, nearshore irulation, bottom streaming 2

3 Introdution Coupling of rapidly osillating surfae gravity waves to slowly varying urrents reates unique flow patterns in both inner shelf and surf zone environments. The effet of mean urrents on surfae gravity waves is exhibited as a Doppler shift in wave frequeny, aompanied with a hange in phase speed. Conversely, the effet of rapidly osillating surfae gravity waves on mean flow is manifested through the provision of additional momentum and mass flux to the mean flow. This oupling is usually aommodated by averaging the fast osillations over longer time sales and provides a mehanism for the inlusion of the so alled Wave Effet on Currents (WEC). Wave-urrent interation an be expressed in an Eulerian referene frame by assuming an analyti ontinuation of the wave-indued veloities above the air-sea interfae (e.g., Garrett, 1976; MWilliams et al., 2004; Smith, 2006; Newberger and Allen, 2007a). A Generalized Lagrangian Mean (GLM) Framework (Andrews and MIntyre, 1978ab; Ardhuin et al., 2008) provides a formulation whih is generally equivalent to the Eulerian mean, with a physial interpretation of the veloity above the wave trough level as a quasi-eulerian veloity. Finally, the alternate Generalized Lagrangian Mean framework (Andrews and MIntyre, 1978a; Groeneweg and Klopman, 1998; Ardhuin et al., 2008b) is a distint approah whih works with the Lagrangian mean veloity. This may also be obtained using a vertially moving average (Mellor, 2003; 2005; 2008; 2011). The prognosti variables in Eulerian and Lagrangian mean flow equations are the quasi-eulerian mean and Lagrangian mean veloity, respetively. The quasi-eulerian mean veloity is the differene between Lagrangian mean veloity and the wave pseudo-momentum (e.g., Jenkins, 1989; Bennis et al., 2011). A detailed desription of available Eulerian, GLM and alternative GLM averaging tehniques along with their advantages and disadvantages in identifying the role of fast osillations on mean irulation an be found in Ardhuin et al. (2008) and Bennis et al. (2011). In the present work we onentrate on the threedimensional momentum equations and their representation in an Eulerian referene frame. The terms orresponding to WEC in the mean flow equations an be represented as the gradient of radiation stress tensor or as Vortex Fore (VF, Craik and Leibovih, 1976). The radiation stress tensor is defined as the flux of momentum due to surfae gravity waves (Longuet-Higgins and Stewart, 1964), while the VF representation splits the wave-averaged effets into gradients of a Bernoulli head and a vortex fore. The Bernoulli head is an adjustment of pressure in aommodating inompressibility (Lane et al., 2007), while, after wave averaging, the vortex fore is a funtion of wave-indued Stokes drift and flow vortiity. Reently, a number of three-dimensional, hydrostati 1 oean models have been developed and used to study wave-urrent interation. Newberger and Allen (2007a) using an Eulerian framework, added wave foring in form of surfae and body fores, and depth- averaged VF terms in the Prineton Oean Model (POM), whih has evolved into Nearshore POM. MWilliams et al. (2004, hereinafter MRL04) developed a multi-sale asymptoti theory for the evolution and interation of urrents and surfae gravity waves of finite depth. This method 1 Non-hydrostati oean models have also been developed and used to study wave-urrent interation. The models resolving non-hydrostati pressure have been used to study propagation of shortwaves on a deep basin, internal waves and tides (Kanarska et al., 2007), internal solitary wave shoaling and breaking, lok-exhange problem (Lai et al., 2010), and wave propagation, shoaling and breaking in the surf zone (Zijlema et al., 2011). Nevertheless, most of the model simulations onduted using non-hydrostati models are validated against laboratory test ases, while hydrostati oean models are utilized for majority of realisti nearshore field experiments. 3

4 separates urrents, long waves and surfae gravity waves on the basis of differenes in their spatial and temporal variation, as a funtion of the wave slope. The work of MRL04 was implemented in UCLA ROMS and extended for appliations within the surf zone by Uhiyama et al. (2010, hereinafter U10). The VF formalism based model presented by U10 separates the effets of wave foring into onservative (Bernoulli head and vortex fore) and non-onservative (wave dissipation indued aeleration) ontributions. A separation between onservative and non-onservative fores is pertinent as the former has a known vertial distribution, while the latter an presently only be expressed with an empirial vertial profile. U10 presented the non-onservative wave forings with a depth-limited wave dissipation alulated using the formulations of Thornton and Guza (1983) and Churh and Thornton (1993), bottom streaming and a wave roller model based on Reniers et al. (2004a). The turbulene losure model is K-profile parameterization (KPP) with additional mixing due to wave breaking following Apotsos et al. (2007). The irulation model is oupled to a spetrum-peak WKB (Wentzel Kramer Brillouin) wave refration model. The methodology presented by MRL04 and U10 an be extrapolated for modeling with nesting omponents that requires seamless simulation of proesses simulated at a variety of sales for water depths from the deep oean to the surf zone. Suh appliations inlude the development of rip urrent predition system (e.g., Voulgaris et al., 2011), sediment transport studies (e.g., Kumar et al., 2011b), and nearshore water quality predition systems (Grant et al., 2005) amongst others. Thus it is imperative that these types of models are made available to the sientifi ommunity through the upgrade of existing publi domain modeling systems. As a primary step in this diretion we implement the VF approah to the three-dimensional oean model ROMS, oupled with the wave model Simulating Waves Nearshore (SWAN), within the framework of the COAWST modeling system (Warner et al., 2010). This modeling system allows for simulating wave driven flows and sediment transport in the intertidal region (see Kumar et al., 2011b) through wetting and drying algorithms, a apability not available in the U10 model implementation. Furthermore, the U10 model uses a KPP parameterized mixing sheme that fails to aurately represent the mixing in bottom boundary layer and in nearshore regions where the surfae and bottom boundary layer interat (Durski et al., 2004). The implementation of VF formalism into the COAWST modeling system is onduted with signifiant modifiations to the method of U10 whih inludes: (a) Enhaned mixing implementation using the Generi Length Sale (GLS) sheme with the addition of waveindued mixing in the form of surfae boundary ondition (Feddersen and Trowbridge, 2005). (b) Improved vertial struture of depth-limited wave dissipation indued aeleration that sales with the wave height. This osh-based distribution is limited to the surfae layer in deeper water, while in shallow waters (where the water depth is similar to wave height) the dissipation effet is delivered to the depth of water olumn influened by wave propagation () Improved implementation of wave dissipation input whih is provided by the wave driver model diretly rather than being estimated loally using empirial formulations (Thornton and Guza, 1983; Churh and Thornton, 1993); (d) Inorporation of bottom streaming using multiple formulations and wave-indued tangential flux at the surfae as an option. The objetives of this manusript are: (a) to desribe the implementation of the VF formalism; (b) validate the model using analytial, laboratory and field observations appliable to both surf zone and inner shelf environments; and () provide a set of standard test ases for model omparisons. The versatility and general appliability of the model presented here are demonstrated over a number of ases that not only inlude ommonly used ase (see U10, Newberger and 4

5 Allen, 2007b) of obliquely inident waves on a planar and barred beah, but extend beyond to inlude appliations of the model for the study of omplex flow regimes developed in a nearshore barred beah with rip hannels as well as for the study of wave-indued flow fields outside the surf zone in the inner shelf. We ompare depth-averaged flow fields from VF representation to those obtained using a two-dimensional numerial model based on radiation stress representation, and identify the role of different foring terms. A omparison to threedimensional flows from a radiation stress representation has been avoided in absene of any selfonsistent theory (see Ardhuin et al., 2008b; Bennis et al., 2011; Kumar et al., 2011a) for the same. The outline of the paper is as follows. The model formulation is presented in setion 2, while its numerial implementation is desribed in setion 3. The new model apabilities are demonstrated in setion 4 through presentation of four test ases that over both surf zone and inner shelf proesses: (a) obliquely inident waves on a syntheti planar beah; (b) obliquely inident waves on a natural, sandy, barred beah (DUCK 94 experiment); () nearshore barred morphology with rip hannels; and (d) struture of undertow observed on the inner shelf. Disussion on differenes and similarities of flow struture derived by expressing WEC using VF and radiation stress representations are shown in setion 5 followed by a summary and onlusion setion. 2 Model Formulation The oean omponent of COAWST is the Regional Oean Modeling System (ROMS), a three-dimensional, free surfae, topography following numerial model, whih solves finite differene approximation of Reynolds Averaged Navier Stokes (RANS) equations using hydrostati and Boussinesq approximation with a split expliit time stepping algorithm (Shhepetkin and MWilliams, 2005; Haidvogel et al., 2008; Shhepetkin and MWilliams, 2009). ROMS inludes several options for several model apabilities, suh as various advetion shemes (seond, third and fourth order) and turbulene losure models (e.g., Generi Length Sale mixing, Mellor-Yamada, Brunt-Väisälä frequeny mixing, user provided analytial expression and K-profile parameterization). As Shhepetkin and MWilliams (2009) state, urrently there are four variations of ROMS-family odes. In this ontribution we use a version based on the Rutgers University ROMS whih was first introdued by Haidvogel et al. (2000) and subsequently any referene to ROMS denotes this partiular version. 2.1 VF Equations The VF approah was implemented following the onventions desribed in MRL04 and based on the formulation as presented in U10, with several key modifiations that are appliable for this partiular modeling system. Terms orresponding to wave effet on urrent are assembled and shown on the right hand side of the equations presented below. Boldfae typesets are used for horizontal vetors only, while the vertial omponent is represented by a normal typeset so that a three-dimensional urrent vetor is designated as (u, w). Following the above onventions the model equations an be written as: u ' w u fˆ w v ' u w u u zu F D u J F [1] t z z z 5

6 g K z 0 z w u 0 z ' ' St 1 w C soure w v w Є u u St t z z z z 2 z z where (u, w) and (u St, w St ) are Eulerian mean and Stokes veloities, respetively. At this stage it is pertinent to point out that the veloities presented in this paper are the quasi-eulerian mean veloities. This veloity is defined as the Lagrangian mean veloity minus the Stokes drift (Jenkins, 1989). Below the wave troughs it is very nearly equal to the one that is measurable by a fixed urrent meter. Above the wave troughs, it provides a smooth extension of the veloity profile all the way to the mean sea level, as assumed in the MRL04 theory. For onsisteny purposes with the notation of MRL04 and U10, in the remaining of the paper these quasi- Eulerian mean veloities are referred to as Eulerian mean veloities. f is the Coriolis parameter, φ is the dynami pressure (normalized by the density ρ 0 ); F is the non-wave, non-onservative fore; D represents the diffusive terms (visosity and diffusion); (J, K) is the VF and is the lower order Bernoulli head as desribed in MRL04 (see Se. 9.6 in MRL04); F w is the sum of momentum flux due to all non-onservative wave fores; is any material traer onentration; C soure are traer soure/sink terms and Є is the wave-indued traer diffusivity. An overbar indiates time average, and a prime indiates turbulent flutuating quantity. ρ and ρ 0 are total and referene densities of sea water; g is the aeleration due to gravity; and v and v θ are the moleular visosity and diffusivity, respetively. The vertial oordinate (z) range varies from h( x) z ˆ, where and ˆ, are the mean and quasi-stati sea (wave-averaged) level omponents, respetively. All wave quantities are referened to a loal wave-averaged sea level, z ˆ. The three-dimensional Stokes veloity (u St, w St ) is defined for a spetral wave-field as: u St where h(x) is the resting depth; E is the wave energy; is the phase speed of the waves; k is the wave number vetor and k is its magnitude, and and are the normalized vertial lengths defined as: 2E osh 2 () z sinh 2 St St w z u dz ' z h k k( h ) kd; and k z h [3] [2] 6

7 where D ( h ) is the wave-averaged thikness of the water olumn. Finally, the wave energy (E), phase speed (), and intrinsi frequeny ( ) are given by: 1 2 E gh 8 [4] k g k tanh where H is the root mean square wave height. The VF (J, K) and the Bernoulli head ( ) terms shown in Eqn. 1 are expressed as: st st u J zˆ u zˆ u f w z K u st u z [5] 2 z 2 ' 2 ( ) 2 2 z h H 16 ksinh ' ' sinh k z z dz where k. u, and ẑ is the unit vetor in vertial diretion. Physially, the Bernoulli head term represents an adjustment in the mean pressure to aommodate for the presene of waves (Lane et al., 2007) and the VF terms represent an interation between Stokes drift and vortiity of the mean flow. The wave-indued traer diffusivity in Eqn. 1 is defined as: 1 Hsinh Є 8 t sinh [6] while the quasi-stati sea level omponent is given by: 2 ˆ Patm Hk [7] g 8 sinh 2 0 The term ˆ ontains inverse barometri response due to hanges in atmospheri pressure (P atm ) and a wave-averaged set-up/set-down (with respet to the still water). It is important to note that in the above formulations the total ontribution of Bernoulli head has been separated in two parts of different order eah. The higher order ontribution is a quasi-stati balane between mean pressure, mean surfae elevation and the wave stresses, whih is absorbed here as a part of the quasi-stati sea level omponent ( ˆ ) in Eqn. 7. The lower order quasi-stati balane has been expressed as in Eqn. 5. When using a spetral wave model suh as SWAN, wave height ( H H sig / 2 ) and bottom orbital veloity (u orb ) values are provided after integrating the energy over all frequenies and diretions, while wavenumber (k), wave diretion and, frequeny (f) are those orresponding to the peak frequeny. Wave dissipation due to depth-limited breaking, whiteapping and bottom frition is also provided by SWAN. 7

8 Boundary Conditions The kinemati and pressure boundary onditions are given as: w h h 0 u h w ˆ t t St u U u ˆ ˆ ˆ ˆ [8] g g ˆ St where U is the depth-averaged Stokes veloity and is the wave-averaged foring surfae boundary ondition (see setion 9.3 in MRL04) defined as: ˆ 2 tanh 2 H osh2 osh2 kz ' dz ' sinh 2 2 z ˆ z h ' h z 8 [9] 2ktanh ˆ The equations orresponding to barotropi mode have not been presented here for brevity, although details an be found in U10. 3 Numerial implementation For implementation into the modeling system, the equations presented in setion 2 are expressed in a mass flux form. The wave-indued terms are no longer retained to the right hand side and the lower order Bernoulli head beomes part of the dynami pressure. First we define the following quantities: u l, l u, u St, St ˆ where is the omposite sea level, is the sum of Bernoulli head and the dynami pressure, while to lowest order the Lagrangian mean veloity (u l, ω l ) is represented as the sum of Stokes (u St, ω St ) and Eulerian mean veloities (u, ω). s is the vertial Eulerian mean veloity in a sigma oordinate system. The ontinuity and momentum balane equations in horizontal (, ) orthogonal urvilinear and vertial ( s ) terrain following oordinate system are: l l l H z H zu H zv s 0 [10] t mn n m s mn 8

9 St St z z z z z H H u H v H u H v u u u u u t mn n m n m ACC HA st fv fv u u H H z s mn s mn mn mn St s s z VA COR StCOR H z st 1 v 1 u St u Hv z s n n m s mn z [11] PG HVF w z z z ' ' uw and H H H D v u mn mn mn s H z s BF BA+RA+BtSt+SuSt HM VM FCurv St St z z z z z H H u H v H u H v v v v v v t mn n m n m ˆ u ACC HA st fu fu v v H H z s mn s mn mn mn St s s z VA COR StCOR H z st 1 v 1 u St v Hu z s m n m s mn z [12] PG HVF H H H D v v mn mn mn s H z s w z z z ' ' vw BF BA+RA+BtSt+ SuSt HM VM FCurv ˆ v 9

10 for the 'x' and 'y' ( and ) diretions, respetively; where m -1 and n -1 are the Lamé metri oeffiients, and H z is the grid-ell thikness. The vertial sigma oordinates s varies from -1 at the bottom to 0 at the free surfae. F, is the non-wave body fore; D, D w w represents the parameterized momentum mixing terms; and F w, is the momentum flux from non-onservative wave terms desribed in 3.1 below. In a Cartesian oordinate system m and n are unity and the urvilinear terms ( ˆ u, ˆ v ) beome zero. In the momentum balane equations (Eqn. 11 and 12), the first term on the left hand side is the loal aeleration (ACC), seond to fifth terms onstitute the horizontal advetion (HA), sixth and seventh terms are vertial advetion (VA), eighth and the ninth term represent Coriolis (COR) and Stokes-Coriolis (StCOR) fores, respetively. On the right hand side of the momentum balane equations, the first term is the pressure gradient (PG), and the ombination of the seond and the third term is the horizontal vortex fore (HVF). The non-wave body fore (BF) is the fourth term, while the ontribution of breaking and roller aeleration, and bottom and surfae streaming is represented olletively by the fifth term (BA+RA+BtSt+SuSt). Horizontal (HM) and vertial mixing (VM) are sixth and the seventh terms, respetively. The last term on the right hand side is the urvilinear metri term given by Eqn. 16 and 17. The geopotential funtion derived after depth integrating the vertial momentum balane equation (see Eqn. 1) is: 0 ˆ g g g K H zds s 0 [13] The urvilinear terms ˆ u and ˆ v in Eqns. (11) and (12), respetively are: ˆ u H st z u v v H z u v v m n m n ˆ v St 1 St 1 H z u v v m n H st z u v u H z u v u m n m n St 1 St 1 H z u v u m n The vertial motion past sigma surfaes is given by: The vertial mass flux through the sigma surfaes is alulated as: s l l ' 1 l U V z h W ds mn h t 1 [17] U l H u l / n, V l H v l / m, andw l l / mn are grid-ell volume fluxes. where l s z t l l w z z s s [14] [15] u z [16] 10

11 274 The three-dimensional traer equation is: s St H z H zu H zv H zu Csoure t mn n m s mn n St St Hv z s ' ' v v Є w m s mn s H z s s H z s [18] 275 where Є is the wave-indued traer diffusivity as defined as in Eqn Eqns. (11), (12) and (18) are losed by parameterization of the Reynolds stresses and 277 turbulent traer fluxes as: KM u KM v KH u ' w' ; v' w' ; ' w' H z s H z s H z s [19] 278 where K M is the momentum eddy visosity and K H is the eddy diffusivity. Along with the 279 kinemati and pressure boundary onditions (Eqn. 8), the surfae wind and bottom stresses are 280 presribed as vertial boundary ondition for the Reynolds stresses given as: K H where, τ s s, s and b b, b M z K H K H M K H z M z M z u s v s u s v s s0 s0 s 1 s 1 s s,, t b b,, t τ are surfae wind stress and bottom stress, respetively. Although many different methods are available to inorporate bottom stress in ROMS (see Warner et al., 2008a), in the present appliation we use the simple quadrati drag method or the wave-urrent interation method of Madsen (1994). The horizontal momentum, ontinuity and traer equations as well as the geopotential funtion along with the boundary onditions (Eqns ) are solved to obtain the Eulerian mean veloity (u, ω) and omposite sea level ( ) as the prognosti variables. The wave parameters required for alulating the Stokes veloities, WEC terms, and momentum flux from non-onservative wave foring terms, F w (see next setion for details) are provided through oupling to the wave model (SWAN). SWAN reeives information about sea surfae elevation, bathymetri hange, and a irulation field from ROMS to determine the effet of urrents and total water depth on wave propagation. In turn, ROMS reeives information on surfae and bottom wave parameters (height, orbital veloity, period, wavelength and diretion), wave dissipation due to bottom frition, wave breaking, and whiteapping for non-onservative WEC proesses. This exhange of information between the irulation and wave models ours at user defined intervals in a two-way oupling senario. One-way oupling of data feeding from wave to irulation model an be used if the impat of urrents on wave field is negligible, or simply from the wave model to the oean for proesses suh as enhaned bottom stress,, t,, t [20] 11

12 omputations. A detailed disussion about model oupling an be found in Warner et al. (2008ab). 3.1 Parameterization of non-onservative wave foring, F w Waves propagating towards the shoreline lose energy through three different dissipation bf wap mehanisms: (a) bottom frition ( ); (b) whiteapping ( ); and () depth-indued wave breaking ( b ). The energy lost by these proesses is inluded in the momentum balane (Eqns. 11 and 12) through the non-onservative wave foring/aeleration term F w : w w F w, B bf B sf B wb B bf B sf B wap B b B r [21] where B bf and B sf are aelerations due to bottom and surfae streaming, respetively, while B wb denotes aelerations due to wave breaking. The latter is further deomposed to aelerations due to whiteapping (B wap ), depth-limited wave breaking (B b ) and wave roller (B r ). It is important to point out that the ontribution of bottom frition ( bf ) is manifested in the form of bottom streaming, while the wave breaking indued aeleration ( b ) is further divided into depth-limited breaking and roller ontribution (see next few paragraphs). The model options used to ativate these formulations within the COAWST modeling system are listed in Table Bottom streaming (B bf ) term Interation of waves with the sea bed leads to wave dissipation due to frition within the wave boundary layer. Three different bottom frition formulations are available in SWAN that are based on: (a) empirial formulations (JONSWAP) by Hasselmann et al. (1973); (b) the drag law model of Collins (1972); and () eddy visosity model of Madsen et al. (1988). These bf formulations an be used to alulate for a spetral wave field. In addition, the option for dissipation due to bottom drag using the parameterization presented by Reniers et al. (2004b) as bf in U10 has also been implemented (see Table 1). This option estimates using: 321 bf 1 w 3 w H * z 0 0 fw uorb ; uorb ; fw 1.39 w 2 2sinhkD u orb [22] w where, u orb is the bottom orbital veloity and f w is the wave frition fator (Soulsby, 1995). 322 Dissipation of wave energy in the wave boundary layer auses the instantaneous, 323 osillatory wave bottom orbital veloities (u and w ) to be slightly in phase from quadrature 324 ausing a wave stress (bottom streaming) in the wave bottom boundary layer, along the diretion 325 of wave propagation (Longuet-Higgins, 1953; Phillips, 1977; Xu and Bowen, 1994; Lentz et al., ). This stress an be provided as a bottom stress or a body fore. We have implemented two 327 approahes to allow the effets of bottom streaming on the mean flows. First, following U10, the 328 effet of bottom streaming in momentum balane is aounted for by using the wave dissipation 329 due to bottom frition with an upward deaying vertial distribution. 330 bf where f (z) is a vertial distribution funtion given by: B bf bf kf bf ( z) [23] 12

13 with k wd being a deay length whih is a funtion of wave bottom boundary layer thikness ( δ w) and given by: where a wd is an empirial onstant (=1 in here) and δ w is a funtion of semi-orbital exursion ( A ), Nikuradse roughness ( k n ) and bottom roughness length ( z 0 ). w orb z As a seond approah of bottom streaming, the method of Xu and Bowen (1994) is implemented (see Table 1) where: ' ' H k z 2z B bf u w 2 z sin z z os z os ze e 8sinh kh [27] 0 with β /2K m, where K m is the eddy visosity. The first method is more suitable when the vertial resolution of the model is not high enough to resolve the wave bottom boundary layer while the seond method is preferred for simulations that use high vertial resolution Surfae streaming (B sf ) term Similar to the onept of bottom streaming, at the surfae of the water olumn the waveindued stress develops a thin visous boundary layer known as surfae streaming (Longuet- Higgins, 1953; Xu and Bowen, 1994 and Lentz et al., 2008). This ontribution to nononservative wave foring is parameterized as (Xu and Bowen, 1994): 2 KmH B sf k k oth( kh) [28] 2 and it is implemented as a surfae boundary ondition (see setion 4). The effet of surfae streaming an be interpreted in a similar manner as that of wind stress ating on the oean surfae (Weber et al., 2006). This effet may not be signifiant in a dynami environment like the surf zone, but ould be signifiant outside the surf zone as shown by Lentz et al. (2008) Wave Breaking (B wb ) terms f bf osh osh Non-onservative wave foring due to wave breaking is traditionally defined only in a depth-averaged form (Longuet-Higgins, 1964; Smith, 2006). Newberger and Allen (2007a) implement the fore due to depth indued breaking (B b ) as a surfae stress, while Walstra et al. (2000), U10 and Kumar et al. (2011a) implement it as a surfae intensified body fore through the development of ad-ho vertial distribution funtions. In the present work, we use a surfae intensified distribution of B wap, B b and B r as in Kumar et al. (2011a). Whiteapping indued aeleration (B wap ) h k k wd z kwd z dz [24] a δ [25] wd wd w w w A u / ; k 30z [26] orb orb n 0 13

14 Whiteapping an our in any water depth (van der Westhuysen et al., 2007; Jones and Monismith, 2008) as a response to wave steepening. Presently SWAN provides many different expressions for alulation of wave dissipation due to whiteapping (e.g., Rogers et al., 2003; van der Westhuysen et al., 2007). The assoiated aeleration is given as: B wap wap kf 0 where wap b is the dissipation from SWAN, and f () z is the vertial distribution funtion suh that: b FB 2 f z ; FB osh z h FBdz H [30] h Bathymetry indued breaking and aeleration (B b ) Depth limited wave breaking dissipation ( b ) is omputed in SWAN using a spetral version of the bore model based on Battjes and Janssen (1978), whih depends on the ratio of wave height to water depth (see, Eldeberky and Battjes, 1996). Alternative empirial relationships for depth-indued breaking have been provided by Thornton and Guza (1983) and Churh and Thornton (1993) and have been added as options (see Table 1). These formulations are: 3 b 3 Bb fp 7 0g H [31] D b Bb f b p 3 H H 0g H 1 tanh D D [32] bd where, H is the root mean square wave height; f p is the wave frequeny; g is the aeleration due to gravity; B b and γ b (the ratio of wave height to water depth) are empirial parameters. The aeleration due to the depth-limited breaking dissipation is: r b b (1 α ) b B kf ( z) [33] 0 where, α r is the perentage of wave dissipation involved in reation of wave rollers (desribed b in details below), and f () z is a vertial distribution funtion, where we have deided to use the same funtion as defined in Eqn. 30. Wave Rollers and Roller Aeleration (B r ) Within the surf zone the spatial distribution of wave dissipation is dominated by wave breaking that depends on bathymetry, but it is further modified due to the ation of wave rollers. Wave rollers at as storage of dissipated wave energy, whih is gradually transferred to the mean flow ausing a lag in the transfer of momentum (Svendsen, 1984; Nairn et al., 1990). Warner et al. (2008a) and Haas and Warner (2009, hereinafter HW09) demonstrated the implementation and appliation into the ROMS model of a roller formulation based on Svendsen (1984). However, U10 presented a wave roller model whih is similar to that of Reniers et al. (2004a) b ( z) [29] 14

15 and Stive and DeVriend (1994). To provide additional apabilities, we also implemented this time dependent advetive roller model into the COAWST system. The equations for evolution of wave rollers are similar to spetral wave evolution equation and an be represented as: r b r r r [34] t r where, is the roller energy density; is the phase speed of the primary wave, b is the wave r dissipation; is the roller dissipation rate; is the wave frequeny and r is an empirial parameter denoting the ontribution of wave dissipation in reation of wave roller (see below). The roller energy density is related to roller energy by: r r E [35] The phase speed of the primary wave is given by: 2 u k k [36] where, u is the mean veloity, and k is the wave number. The roller dissipation rate is: r r g sin E [37] where, is the phase veloity (Eqn. 4) and sin ( 0.1) is an empirial onstant (Reniers et al., 2004a). As suggested by Tajima and Madsen (2006) and U10, the quantity r in Eqn. 34 an vary between 0 and 1, providing a ontrol on the amount of wave energy expended for the reation of wave rollers. This hoie of r would be ontingent upon wave breaking type (i.e., spilling, plunging, surging) whih in turn depends on beah slope and type (Short, 1985). The ontribution of wave rollers in form of aeleration is given by: r r b B kf ( z) [38] 0 Combining Eqns. 29, 33 and 38 and after some re-organization the total aeleration ontribution of the wave breaking term is written as: r b r wap 1 α wb b B kf ( z) [39] Mass flux due to wave rollers The ontinuity equation (Eqn. 10) aommodates the mass flux due to Eulerian and Stokes transport. Wave rollers also ontribute to assoiated mass flux inreasing the total Stokes transport (Svendsen, 1984; Reniers et al., 2004a). The roller Stokes transport is given by: r r E U r k k [40] 0 0 and the total Stokes and roller transport beomes: r r EE U St k k [41]

16 The vertial profile of U r has a distribution similar to that of the Stokes veloity. Sine the effet of wave rollers is usually limited to the surfae, a surfae intensified distribution (e.g., HW09) may be more suitable. Simulations onduted using Stokes vs. surfae intensified distribution provide similar results, so in the present implementation we use a Stokes veloity type distribution. Table 1. COAWST options available for the omputation of non-onservative wave fores (for details see Se. 3.2). VF options in the COAWST Modeling system. Proess Swith Name Desription Referenes WDISS_WAVEMOD Wave dissipation from wave model (SWAN) SWAN manual Thornton and WDISS_THORGUZA Wave-dissipation using Eq. 31 Wave Guza (1986) Dissipation Churh and WDISS_CHURTHOR Wave-dissipation using Eq. 32 Thornton (1993) Roller Model Momentum transfer due to nononservative fores Streaming Wave indued mixing ROLLER_RENIERS ROLLER_SVENDSEN WEC_BREAKING WEC_ROLLER WEC_WCAP BOTTOM_STREAMING_YU BOTTOM_STREAMING_XU _BOWEN SURFACE_STREAMING TKE_WAVEDISS 3.3 Enhaned mixing due to wave breaking Solve the roller evolution equation (Eqn ) to alulate roller dissipation. α r value is provided by as an input parameter Calulate roller area and roller energy using Svendsen (1984) formulations Momentum ontribution due to wave breaking (Eqn. 33) Momentum ontribution due to rollers (Eqn. 38) Momentum ontribution due to whiteapping (Eqn. 29) Calulate wave dissipation due to bottom frition (Eqn. 22) and the ontribution to momentum balane (Eqn ) Calulate ontribution of bottom streaming to momentum balane using Eqn. 27 Calulate surfae streaming ontribution to momentum balane using Eqn. 28 Compute TKE ontribution due to wave breaking using Eqn. 47, whih is implemented as a surfae boundary ondition (Eqn. 44, 45) to solve GLS model Reniers et al. (2004); Uhiyama et al. (2010) Svendsen (1984) Warner et al. (2008) This paper This paper This paper Uhiyama et al. (2010) Xu and Bowen (1994) Xu and Bowen (1994) Feddersen and Trowbridge (2005) 16

17 Wave breaking indued dissipation leads to mixing of momentum in the water olumn (Agrawal et al., 1992). In surf zone this enhaned mixing an also be responsible for sediment resuspension in the water olumn (Voulgaris and Collins, 2000). The vertial sale of this mixing an be empirially related to the wave height (Rapp and Melville, 1990), whih for shallow waters is usually of the same order as the water depth. This leads to a region in the water olumn of overlapped mixing due to wave breaking and turbulene from the bottom layer (Feddersen and Trowbridge, 2005). Following Umlauf and Burhard (2003) as implemented in Warner et al. (2005), a generalized expression for transport of turbulent kineti energy (k) and generi length sale (ψ) an be written as: k KM k u k P B t z k z [42] KM k u 1 P 3 B 2 Fwall t z z k where P and B are the shear and buoyany prodution, respetively, k and are turbulene Shmidt numbers for k and, respetively, and F wall is a wall funtion. 1, 2 and 3 are oeffiients defined in detail in Warner et al. (2005). The generi length sale (ψ) is defined as: where, 0 is a numerial onstant; m, n and p are speified to relate to a turbulent quantity. The turbulene due to injetion of surfae flux of TKE is given as surfae boundary onditions (Craig and Banner, 1994; Feddersen and Trowbridge, 2005): 435 where w is the downward TKE flux due to breaking waves. The surfae boundary ondition for 436 due to wave breaking is (Carniel et al., 2009): У u * In deep waters, 3 w s, where * u s is the frition veloity; and w is a parameter that depends on the sea state, with a typial value of w 100 (Carniel et al., 2009). In the surf zone, У w. z 0 is the surfae roughness or the surfae mixing length. For breaking wave onditions, the surfae roughness is provided using the losure model of Staey (1999): z H [46] where w 0.5. In the surf zone, part of the wave dissipation ontributes to the flux of momentum (i.e., 1α r b wap ), while the remaining amount (i.e., α r b ) is expended for the reation of wave KM k z p 0 m n k l [43] KM k k z w p 0 m1 m k z0 z n k У t nk z z p 0 m n n w 17 [44] [45]

18 rollers. Furthermore, part of the wave and roller dissipation ( r ) also ontributes to turbulene mixing within the surf zone. Feddersen and Trowbridge (2005) assume that 25% of wave energy dissipation goes into the water olumn as TKE, while Jones and Monismith (2008) use a value of 6%. In the present work the ontribution of wave dissipation as surfae flux of TKE is expressed through an empirial oeffiient w whih an be manually adjusted (see setion 4.2.3) based on data availability. The surfae flux of TKE is therefore: 1 α r b r wap w w [47] In order to onserve the total ontribution to momentum balane due to wave dissipation, the amount of wave dissipation introdued as surfae flux of TKE is subtrated from Eqn Model Simulations The modeling system with the VF formalism desribed above is applied to idealized and realisti surf zone and inner shelf environments to study the spatial variation and vertial struture of ross-shore and longshore flows. Four simulations are presented in detail, provided as standard test ases. The first two ases onsist of reation of alongshore urrents and undertow due to oblique inidene of spetral waves on a planar and a natural, barred beah assuming alongshore uniformity. The third ase introdues three-dimensionality in the domain and flow development as it simulates a nearshore barred morphology interrupted by rip hannels. The fourth ase is designed to demonstrate the appliability of the model for inner shelf appliations and simulates wave-indued ross-shore flows in the inner shelf. For all ases, an orthogonal oordinate system is defined so that x and y represent the ross-shore and longshore diretions, respetively with positive x towards the open oean. Correspondingly, positive ross-shore veloity values indiate offshore direted flow. 4.1 Test Case 1: Obliquely inident waves on a planar beah The effet of VF formalism is examined through simulations for obliquely inident waves on a planar beah. This ase has been previously disussed by HW09 and Kumar et al. (2011a) using depth dependent radiation stress formulations based on Mellor (2003) and Mellor (2008), respetively, and by U10 using a VF based model. In the simulations presented here, we use our implementation of the VF formulations whih utilizes a different vertial distribution of wave dissipation and turbulene losure sheme than U10 (Eqns. 30 and 42). The model domain has a ross-shore (x) width of 1,180 m and an alongshore (y) length of 140 m, with a 20 m grid resolution. The resting water depth varies from 12 m at the offshore boundary to 0 m at the shoreline. The vertial domain onsists of 30 equally distributed vertial layers. The boundary onditions are periodi in the alongshore (i.e., north and south boundaries) and losed at the shoreline. At the offshore side we use Flather radiation ondition (Flather, 1976) for free surfae and Neumann boundary onditions for barotropi and barolini veloities (inluding boundary ondition for Stokes veloities). The effet of earth rotation is not inluded. The bottom stress has been formulated using a quadrati bottom drag with a d value of The turbulene losure sheme is Generi Length Sale (GLS, k-ε) as desribed in Warner et al. (2005). Wave foring is provided by SWAN, whih propagates an offshore JONSWAP wave spetrum with a root mean square wave height (H) of 1.4 m, a peak period of 10 s and a 10 18

19 angle of inidene. The barotropi and barolini time steps used are 0.16 and 5 s, respetively. Effets of wave rollers, wave breaking indued mixing and bottom streaming are not inluded for this simulation, whih is onsistent with Kumar et al. (2011a), HW09 and U10. Uhiyama et al. (2009) showed that in the presene of wave and urrent flutuations, the mean ontinuity balane at steady state an be integrated in the ross-shore diretion to yield a St balane between barotropi Eulerian mean and Stokes veloities (i.e., u u ). This information along with the wave parameters and dissipation due to wave breaking ( b ) an be used to solve for sea-surfae elevation and barotropi longshore veloity using the following approximate equations: Sxx gh d u x x V [48] Sxy bky d V v x where V is magnitude of the barotropi veloity vetor, while S xx and S xy are onshore and longshore omponents of onshore radiation stress Figure 1. Obliquely inident waves on a planar beah simulated using the VF, the RS 2D model and analytial solution (see Eqn. 48). Cross-shore distribution of (a) root-mean-square wave height (H) and water depth (h); (b) sea surfae elevation, ζ and depth-indued wave dissipation (ε b ). The results from the VF model simulation are ompared to the analytial solutions of and v obtained using Eqn. 48, with a d value idential to that used in the numerial simulation, as well as to results from the depth-averaged, Lagrangian, radiation stress based model presented in Warner et al. (2008a, hereafter referred to as RS 2D ). In the latter model the wave foring was provided as depth-averaged radiation stress (i.e., similar to Longuet-Higgins, 1970a and b) Wave parameters and sea-surfae elevation 19

20 Figures 1a and b show the wave height, depth-indued dissipation and sea surfae elevation. Wave shoaling ours in the region 500 to 1,000 m; inshore of this region the waves start breaking in the depth-limited environment (Fig. 1a). Depth-indued dissipation ( b, Fig. 1b) remains zero during wave shoaling. Inshore of x= 500 m, b inreases monotonially to a maximum value of 0.07 m 3 s -3 at x= 300 m, and then dereases gradually to zero at the shoreline. This depth-indued wave dissipation is the wave foring whih ontributes to the momentum flux (Eqns. 33 and 12), leading to reation of longshore urrents. Estimates of from VF, RS 2D and the analytial solution (Eqn. 48) are in lose agreement as shown in Figure 1b, with a slight differene at the oastline most likely due to lateral mixing or frition. Outside the surf zone, in the wave shoaling region, the mean sea level dereases (wave set-down), while within the surf zone, the mean sea level inreases (wave set-up) as shown in Figure 1b Nearshore flows Vertial variability of Eulerian mean and Stokes veloities from the VF simulation are shown in Figure 2. Inside the surf zone (x<500; Fig. 2a) the Eulerian mean ross-shore flow is inshore near the surfae and offshore direted lose to the sea bed. This vertial segregation of the ross-shore flow reates a irulation ell within the surf zone with downward and upward direted vertial veloities (see Fig. 2), onsistent with field observations of ross-shore veloity profiles for barred (Garez-Faria et al., 2000), planar (Ting and Kirby, 1994) and laboratory (Roelvink and Reniers, 1994) beahes. Outside the surf zone the veloity is weakly offshore throughout the entire water olumn. These results are also onsistent with U10, regardless of the differenes in turbulene losure shemes and vertial distribution of wave dissipation. Depth-averaging the ross-shore Eulerian mean veloities shown in Figure 2a, we obtain veloities (Fig. 2g) that are equal in magnitude and opposite in sign to the depth-averaged Stokes veloity. This balane is indiative of a steady state solution ahieved by the model and mass flux onservation. The longshore veloity (Fig. 2b) attains its maximum value of approximately -1 ms -1 at x= 250 m and dereases to zero at the oastline and towards offshore. Vertially, the veloity shows maximum value at the surfae and slightly lower values near the sea bed. Depth averaging these veloities, we find that the maximum alongshore veloity from the VF simulation is further inshore in omparison to the analytial solution, whih shows a maximum value at x= 300 m, at the same loation as the maximum b (Fig. 1b). This differene is mainly due to the inlusion of vertial visous mixing, horizontal advetion and VF leading to spreading and distribution of the momentum flux in the surf zone, something not inluded in the simplified analytial solution of Eqn. 48. Comparison to results obtained by RS 2D simulations are disussed separately in Setion 5. The ross-shore Stokes veloity (Fig. 2d) is one and two orders higher than the longshore (Fig. 2e) and vertial Stokes veloity (Fig. 2f), respetively. Close to the sea surfae, ross-shore veloity varies from zero at the offshore boundary to a maximum value of ~ ms -1 at the loation of maximum wave breaking (i.e., x= 300 m), dereasing with inreasing water depth. Further inshore of this position, the ross-shore veloity redues to zero. Longshore veloity is weaker in strength, but shows a distribution similar to that of the ross-shore Stokes veloity. Sine the vertial Stokes veloity is alulated as divergene of horizontal mass flux (Eqn. 2), at the loation of maximum breaking, the vertial Stokes veloity is zero. Inshore of this point, the 20

21 veloity is positive with a maximum value at the surfae, dereasing with inreasing water depth. Offshore of the break point, the veloity is negative and downwards direted, with a vertial struture similar to other Stokes veloity omponents. The vertial Stokes veloity has similar 1 magnitude ( ms ) but opposite sign to the vertial Eulerian mean flows (Fig. 2) Figure 2. Cross-shore setions of Eulerian (a, b and ) and Stokes (d, e and f) veloities from the VF model. (a) ross-shore (u); (b) longshore (v); and () vertial (w) Eulerian veloities. (d) ross-shore (u st ); (e) alongshore (v st ); (f) vertial (w st ) Stokes veloities; Cross-shore distribution of (g) depth-averaged, ross-shore Eulerian veloity ( u ) and Stokes veloity ( St u ); and (h) depth-averaged, alongshore ( v ) 21

22 Eulerian veloity for obliquely inident waves on a planar beah simulated using the VF, the RS 2D model and analytial solution (see Eqn. 48) Three-dimensional momentum balane The relative ontribution of the ross-shore (x) and longshore (y) momentum balane terms are desribed here, using the nomenlature as shown in Eqn. 11 and 12 orresponding to aeleration (ACC), horizontal and vertial advetion (HA and VA), Coriolis fore (COR), Stokes-Coriolis fore (StCOR), pressure gradient (PG), horizontal VF (HVF), horizontal and vertial mixing (HM and VM), and breaking and roller aeleration (BA and RA). Though the ontribution of vertial vortex fore (K, Eqn. 1) an be analyzed separately as a part of the geopotential funtion (Eqn. 13), in this work we have added it to the HVF term, as its importane is negligible in all the ases disussed here. In the ross-shore diretion (Fig. 3), sine earth rotation and roller efet were not onsidered, the RA, COR and StCOR terms are zero. The horizontal advetion (HA, Fig. 3b), horizontal vortex fore (HVF, Fig. 3) and vertial advetion (VA, Fig. 3f) terms are negligible. The balane is mainly between three terms: BA, VM and PG. Within the surf zone (x<350m), the wave breaking aeleration (BA, Fig. 3a) term is the largest with a high value at the sea surfae, sharply dereasing to a negligible value below 1 m under the surfae. A signifiant portion of the BA ontribution is balaned by a relatively strong vertial mixing (VM, Fig. 3e) whih is enhaned lose to surfae layer. At water depths where BA beomes negligible, the VM hanges sign and beomes negative. At the loation where waves start breaking (i.e., 350 m< x < 500 m), the ontribution of pressure gradient (PG, Fig. 3d) is negligible, but inreases toward the shoreline, with a vertially uniform distribution. Close to the sea surfae, both PG and VM terms add to balane the BA ontribution while further below the balane is mainly between PG and VM, with the latter term also beoming vertially uniform. It is important to note that omparing the present momentum balane to that obtained from simulations using models based on depth varying radiation stress (e.g., Kumar et al., 2011a) and quasi-3d models suh as SHORECIRC (e.g., HW09) we find that in the VF formulation, the VM term is responsible for vertially redistributing the BA and balaning PG. In the former two models the primary balane ours between vertially uniform PG and almost vertially uniform radiation stress ontribution. The major terms in alongshore momentum balane are BA, HA, HVF, VM and VA, while PG is negligible. BA (Fig. 3g) is dominant only in the surfae layer within the wave breaking zone where signifiant part of it is balaned by the VM term (Fig. 3k). Further below the sea surfae (> 1 m), VM hanges sign from positive to negative, and when added to VA and HVF the sum balanes HA (Figs. 3h, i, k and l). HA and VA terms (Figs. 3h and l) show opposite signs over the entire water olumn, whih an be attributed to vertial segregation of ross-shore veloity (Fig. 2a) and hange in the gradient, inshore and offshore of the loation of maximum undertow. The HVF term (Fig. 3i) is zero at the loation of maximum longshore veloity (as v/ is zero, see Eqn. 12), and has opposite signs on either side of this point. Overall, at loations inshore of the longshore flow maximum (x < 260 m) the sum of BA, HA and HVF is balaned by the sum of VA and VM near the sea surfae; lose to the bed, the sum of HVF, VM and VA is balaned by HA. Similar balanes also our at loations further offshore (x > 260 m). 22

23 Balane of vertially-integrated three-dimensional momentum balane The vertially averaged ross-shore momentum balane terms (Fig. 4a) show a balane between PG and BA similar to that presented analytially by Bowen et al. (1968) for a planar beah. The ontribution of the remaining terms (inluding HVF) is negligible. Vertial integration of the longshore momentum terms (Fig. 4b) shows a primary balane between the BStr (vertial integral of VM term, Eqn. 20) and the BA terms. A seondary balane ours between the HVF and the HA terms. The horizontal vortex fore (HVF) term is positive seaward of the loation of maximum longshore urrent and beomes negative inshore that loation, and the horizontal advetion (HA) is of similar magnitude as the HVF but of opposite sign. This seondary balane suggests a balane between Stokes and anti-stokes (Eulerian mean) flows; however, these terms do not anel out ompletely due to differenes in vertial struture of Stokes and Eulerian mean flows (see Se 5) Figure 3. Vertial and horizontal ross-shore distribution of the various ross-shore (x) and longshore (y) momentum balane terms. Cross-shore terms: (a) x- breaking aeleration (x-ba) ; (b) x-horizontal advetion (x-ha); () x-horizontal vortex fore (x-hvf); (d) x- pressure gradient (x-pg); (e) x- vertial mixing (x-vm); and (f) x-vertial advetion (x-va); Longshore terms: (g) y- breaking aeleration (y-ba) ; (h) y-horizontal advetion (y-ha); (i) y-horizontal vortex fore (y-hvf); (j) y- pressure gradient (y-pg); (k) y- vertial mixing (y-vm); and (l) y-vertial advetion (y-va)term. 23

24 In the present modeling framework, the terms ontributing to the total pressure gradient fore ( from Eqn. 13 = P tot, i.e., gradient of dynamially relevant kinemati pressure), exluding the vertial vortex fore (K), an be deomposed into two terms that desribe individual ontributions from the Eulerian non-wec (P ) and WEC (P we ) ontributions. The latter an be further divided into a quasi-stati response (P qs ), a Bernoulli head (P bh, see Eqn. 5) and a surfae pressure boundary orretion (P p ) (see Table 2): tot we qs bh p P P P ( P ) P P P g ˆ g dz g g h 0 Table 2. List of the omponents whih onstitute the total pressure gradient fore (Eqn. 49) Individual Terms Desription tot P Total pressure gradient fore. Contribution of both WEC and non-wec terms P Non-WEC urrent ontribution we P qs P bh P p P WEC ontribution qs P Quasi-stati response (Eqn. 7) bh P Bernoulli head (Eqn. 5) p P Surfae pressure boundary orretion (Eqn. 9) Analysis of the individual omponents of pressure gradient fore (PG, Fig. 4) show that major ontribution to P totx is from the non-wec response of the system to wave breaking, (i.e., P x ). Outside the surf zone, quasi-stati response (P qs ) and P x balane eah other whih ause the wave set-down at this loation. The terms orresponding to Bernoulli head and dynami surfae boundary orretion are negligible for this planar beah ase. 4.2 Test Case 2: Obliquely inident waves on a natural, barred beah (DUCK 94 Experiment) In this test ase we simulated wave-indued urrents for a natural, barred beah orresponding to the DUCK 94 experiment (Gallagher et al., 1998; Elgar et al., 1997; Feddersen et al., 1998). Simulations are ompared to data olleted on Ot 12 th, 1994, when strong veloities were observed in the surf zone due to waves generated by winds assoiated with the passage of a low-pressure storm system (Garez-Faria et al., 2000). During this period, both waves and winds were direted towards the southwest generating a longshore flow down-oast (i.e., towards southeast, see Garez-Faria et al., 1998). The measured bathymetry, shown in Figure 5a, originates with the shoreline at x=0 and the nearshore bar loated near x=130 m. The model domain is assumed alongshore uniform with a ross-shore (x) width of 780 m and a horizontal resolution of 2 m. The water depth varies from 0 m at the shoreline to 7.26 m at the offshore boundary. A tidal elevation of 0.70 m was added to the water level and assumed onstant over the simulation period (simulations with tidal variability did not show substantial hanges in the model results). The vertial dimension is disretized with 32 equally distributed layers. The boundary onditions are periodi in the alongshore (i.e., north and south boundaries) and losed at the shoreline. At the offshore end we 24 [49]

25 use Flather radiation ondition (Flather, 1976) for free surfae and Neumann boundary onditions for barotropi and barolini veloities (inluding boundary ondition for Stokes veloities). Effet of earth rotation is not inluded. Bottom stress due to the ombined ation of waves and urrents is estimated using a benthi boundary layer formulation (Madsen, 1994) as desribed in Warner et al. (2008). Weak horizontal momentum diffusion of the order 0.05 m 2.s -1 is also applied to obtain smooth solutions. The turbulene losure sheme used is Generi Length Sale (GLS, k-ε). Wind stress foring of 0.25 and 0.16 Nm -2 is imposed in the ross-shore and longshore diretions, respetively. Wave foring is provided by SWAN, whih propagates an offshore JONSWAP wave spetrum with a signifiant wave height of 2.3 m, a peak period of 6 s and a 13 angle of inidene. The model simulation is arried out for a period of 3 hours with a barolini and barotropi time stepping of 3.0 and 0.1 s, respetively Figure 4. Cross-shore variation of depth-averaged (a) ross-shore; (b) longshore momentum balane terms; and () deomposed PGF terms in ross-shore as desribed in Eqn. 49. Ten different simulations were arried out in order to identify the behavior of wave rollers and wave-indued mixing. The simulations are designated as Run # (where # is the simulation number) and the differenes between individual Runs are listed in Table 3. Run 1 is onduted using the two-dimensional (x-y), depth-averaged, Lagrangian, radiation stress based model (RS 2D, i.e., no vertial distribution of wave foring or flows, and the wave foring is depth-averaged radiation stress ontribution as in Longuet-Higgins, 1970a), while for Runs 2 to 25

26 we use the vortex fore formulation as desribed in this paper (VF). Run 2 does not inlude the effet of wave rollers and wave-indued mixing. Runs 3 to 6 do inlude the effet of wave rollers but eah run assumes a different fration (Eqn. 34) of depth-indued dissipation ( b ) being used for roller generation. Finally, Runs 7 to10 are used to distinguish the ontribution of wave-indued mixing. Table 3. Model onfiguration for different DUCK 94 simulations. RS 2D refers to simulations onduted using depth-averaged, radiation stress based model, while simulations done using the VF formalism are referred to as VF. α r is the oeffiient whih determines the perentage of wave breaking indued dissipation ontributing to reation of wave rollers (Eqn. 34), while εw is the perentage of total dissipation going as turbulent kineti energy (Eqn. 43 and 47). Desription of Model Runs for DUCK 94 Experiment Run # Model Effet of Surfae α Formulation Wave Rollers r TKE εw 1 RS 2D OFF VF OFF 0 OFF 0 3 VF ON 0.25 OFF 0 4 VF ON 0.50 OFF 0 5 VF ON 0.75 OFF 0 6 VF ON 1.00 OFF 0 7 VF ON 0.5 ON VF ON 0.5 ON VF ON 1.0 ON VF ON 1.0 ON Wave parameters and sea-surfae elevation In this setion we first examine two runs: (a) Run 1 (radiation stress based, depthaveraged model with no rollers) and (b) Run 2 (baseline experiment using VF model without wave rollers and mixing), to ompare the flow pattern simulated by a two and three-dimensional model. Measured (Elgar et al., 1997) and simulated H are in a lose agreement throughout the profile (Fig. 5a), despite the fat that the wave solution does not aount for the effet of urrents (one-way oupling). Depth-limited wave breaking, as exhibited through the wave dissipation (ε b ) parameter, takes plae predominantly over the bar-rest and then a seond time lose to the shoreline (Fig. 5b). Over the bar-trough (60 m < x < 100 m), the wave dissipation is negligible, as shown by the relatively stable wave height along this region (Fig. 5a).The overall trend of sea surfae elevation ( ) for both Runs 1 and 2 is a wave set-down outside and wave setup inside the surf zone (Fig. 5b). At the bar-rest and further shoreward, from Run 1 shows a ontinuous inrease, unlike Run 2, whih suggests slight derease at these loations due to dominant ontribution of Bernoulli head (see setion 5) Nearshore flows 26

27 The ross-shore profiles of depth-averaged, ross-shore and longshore veloities from Runs 1 (RS 2D, no vertial flow distribution) and 2 (VF, no rollers/mixing, vertially averaged veloities) are shown in Figures 5 and d. Although the depth-averaged ross-shore veloities from Runs 1 and 2 are idential (Fig. 5), the longshore veloities show signifiant differene both in terms of ross-shore variability and magnitude (Fig. 5d). Strongest longshore veloity from Run 1 ours at the bar-rest and at loations lose to the shoreline, whih does not agree with the observations. On the other hand, maximum longshore urrent from Run 2 is at a loation inshore of the bar-rest, and is in better agreement with measured veloities. This inshore shift of the maximum longshore urrent is due to vortex fore and mixing due to shear, details of whih are provided in setion and Figure 5. Obliquely inident waves on a barred beah simulated using the VF model (no roller model, i.e., Run 2) and the RS 2D (Run 1) model (see Table 3). Cross-shore distribution of: (a) root mean square wave height (H rms ) from SWAN (solid blak line), observed wave height (from Elgar et al., 1997; grey irles) and water depth (h). (b) Sea surfae elevation (ζ ) and depth-indued wave dissipation (ε b ). () Depthaveraged, ross-shore Eulerian veloity, ( u ). (d) Depth-averaged, longshore Eulerian veloity ( v ) along with observed veloity (from Feddersen et al., 1998; grey irles). 27

28 Effet of wave roller Wave roller generation is ontrolled through the parameter α r, whih defines the fration of wave dissipation allowed to at as the soure term in the roller evolution equation (Eqns. 33, 34 and 37). When α r =0, no wave rollers are inluded, while when α r =1 the total of the depthindued dissipation ( b ) is used as a soure for the reation of wave rollers. The roller dissipation is alulated empirially (Eqn. 37) whih ontributes to roller aeleration in the momentum balane along with breaking aeleration (Eqn. 39). Five simulations with no wave-indued mixing and α r values of 0.25, 0.5, 0.75 and 1 (Runs 2 to 6, respetively, see Table 3) were arried out and the total dissipation (= (1 r ) b r ) for eah run is shown in Figure 6a. When α r =0, maximum depth-indued dissipation is observed at the bar-rest and lose to the shoreline. As the value of α r inreases, the ontribution of breaking dissipation dereases and the ontribution of roller dissipation inreases. The advetion of wave rollers with a speed equal to the phase speed of the surfae gravity waves leads to an onshore movement of the total dissipation peak (Fig. 6a). For α r =1, the total dissipation dereases at the bar-rest and lose to the shoreline, and inreases in the bar-trough region, providing a wider distribution of the energy lost by breaking waves. Physially this mehanism modifies the setup in the transition zone (Nairn et al., 1990), reates a delay in the transfer of energy from wave breaking to the mean flow (Reniers and Battjes, 1997; Ruessink et al., 2001) and aounts for the assoiated mass flux in the diretion of wave propagation (Svendsen, 1984). In the next three sub-setions we desribe the physial impat of wave rollers in modifying the ross-shore profile of barotropi flows, ross-shore profile and vertial struture of ross-shore and longshore urrent, vertial profile of eddy visosity and turbulent kineti energy. The simulated flows are also ompared to field measurements of ross-shore and longshore veloities. In absene of any other foring mehanism and under steady state onditions, the vertially averaged Stokes flow is balaned by an opposing Eulerian mean flow (Uhiyama et al., 2009). In absene of wave rollers (Fig. 6b) this flow is strongest at the loation of wave breaking (i.e., bar-rest and at the shoreline). As the ontribution of wave rollers inrease, the rollers ontribute an onshore direted mass flux, leading to a stronger return flow in the offshore diretion (Fig. 6b). Changes in wave roller ontribution also affet the ross-shore variation of the depth averaged longshore urrents (see Fig 6). When α r =0 (Run 2), the maximum longshore veloity is predited just inshore of the bar-rest. Inreasing the wave roller delays the transfer of energy from waves to mean flow, leading to a more uniform distribution of flow within the areas inshore and offshore of the bar-trough ( m). When α r =1, relatively stronger longshore veloity is modeled inshore of the bar-trough. Simulated profiles of ross-shore and longshore veloity from Runs 2, 4 and 6 (i.e., VF based model with α r =0, 0.5 and 1, respetively, see Table 3) are ompared to observations (Garez-Faria et al., 1998, 2000) at seven different ross-shore loations spanning the region between the bar-trough and rest (Figs. 7a and b). The normalized root mean square (rms) errors (defined same way as in Newberger and Allen, 2007b) for eah simulation and ross-shore loation are listed in Table 4. The observed ross-shore veloities (Fig. 7a) show a strong vertial shear at the bar-trough and bar-rest regions, reating a irulation pattern with inshore direted flows at the surfae and offshore direted undertow lose to the bed. Simulated veloity profiles from Runs 2, 4 and 6 (VF based model with α r =0, 0.5 and 1, respetively) show similar general pattern. When α r =0 (Run 2), the veloity shear is strongest over the bar-rest, while when α r =1 (Run 6) veloity 28

29 shear inreases at the bar-trough region (Fig. 7a). It is also shown that the undertow strength inreases with an inreased roller ontribution due to additional return flows generated to ompensate for the inreased mass flux due to rollers. Overall, Run 6, a ase where the entire wave dissipation is onverted to wave rollers (i.e, α r =1), shows the best agreement with the measured ross-shore veloities as revealed by their least rms error values Figure 6. Cross-shore variability of (a) total dissipation (breaking + roller dissipation) and depth-averages of three-dimensional (b) ross-shore, u and () longshore veloity, v estimates, for different values of α r (Runs 2-6, Table 3). The measured longshore veloity is highest in the bar-trough region and gradually dereases on either side (Fig. 7b). When the roller effet is not onsidered (i.e., α r =0, Run 2), the longshore veloity maximum ours in the region between the bar-trough and rest (x ~ 110 m). As the roller ontribution inreases to 50% (i.e., α r =0.5, Run 4), this loal maximum is shifted further inshore at x= 100 m (Fig. 7b). When the total dissipation is used to generate wave rollers (α r =1, Run 6), the longshore veloity peak moves inshore to x~ 80 m, with a smoother distribution of veloity in the bar-trough region. Veloity strength over the bar-rest dereases from 0.7 ms -1 for α r =0, to 0.5 ms -1 for α r =1. The offshore veloity (x > 200 m) values do not hange signifiantly by hanging the roller ontribution as roller/breaking dissipation offshore of the bar-rest is negligible (Fig. 6a). 29

30 Interestingly, using the mean normalized rms error from all seven stations, the results from Run 2 (VF model with no effet of rollers/wave-indued mixing) show the best overall agreement with the observations. Considering the variability observed in model performane at different ross-shore loations, it is lear that inlusion of wave rollers provides better agreement of longshore and ross-shore flows at the bar-rest and bar-trough region, but at loations further offshore, simulations with no rollers/mixing effets show a better agreement to observed profiles. These findings suggest that inlusion of proesses like wave rollers requires areful definition of the amount of wave-dissipation responsible for driving the wave roller model (i.e., value of α r ). Furthermore, it appears more sensible for this value to be a funtion of the ross-shore position within the surf zone (see Cambazoglu and Haas, 2011) Figure 7. Comparison of model results (Runs 2, 4 and 6; i.e., VF model with α r= 0, 0.5 and 1, respetively) with observed vertial profiles (grey squares) of ross-shore (a) and longshore (b) veloities. Vertial grey lines indiate profile measurement loations and zero value for eah profile (Data from Garez-Faria et al. 1998; 2000). Vertial struture of eddy visosity (), K v and turbulent kineti energy (d), TKE from model simulations at the same ross-shore loations as the veloities. 30

31 The effet of wave rollers on the vertial distribution of vertial mixing (K v ) and turbulent kineti energy (TKE) is also examined using the same runs as those desribed in the previous paragraphs. At α r =0 (Run 2), the strongest veloity shear is observed offshore of the bar-rest (Fig. 7a), whih orresponds to an inreased region of TKE prodution and inreased K v levels (Figs. 7 and d). As the roller ontribution inreases, the veloity shear at the bar-rest redues, while an inrease in veloity shear further inshore is observed. This is refleted by a derease in TKE and K v (Figs. 7 and d) at the bar-rest and spreading of the TKE in the region between bar-rest and trough. Subsequently the vertial mixing within the bar-trough region also starts inreasing. Overall, the roller ontribution modifies the shear prodution and assoiated TKE and K v, by moving the entire pattern further inshore and dispersing the breaking indued energy transformation more uniformly within the surf zone. It is interesting to point out that the K v values obtained in the present ase are almost twie the magnitude of those used by U10 in their simulations. This ours beause the GLS mixing utilizes the ambient flow field to reate the shear prodution and assoiated eddy visosity profiles, while in U10 the K v values were derived using a K-profile parameterization. These differenes in K v values also explain the small differenes between results obtained by U10 and the present work. Effet of wave-indued mixing Wave-indued mixing is provided as a surfae flux of TKE in the GLS turbulene losure sheme (see Eqns , also Feddersen and Trowbridge, 2005), ontrolled by the empirial parameter εw that modifies the ontribution from the breaking and roller dissipation. Feddersen and Trowbridge (2005) suggested a value of 0.25, while Jones and Monismith (2008) used a value of 0.06 for their simulations. We arried out a limited in sope sensitivity analysis by using εw = 0, 0.01 and 0.05 for Runs 6, 9, and 10, respetively, all of whih orrespond to VF model based simulation with roller ontribution of α r =1.0 (see Table 3). Simulated profiles of rossshore and longshore veloity from these runs are ompared to field measurements (Figs. 8a and b), and the normalized rms errors are shown in Table 4. Simulations onduted using a εw = 0.25 (not shown here) signifiantly inrease the K v and vertially mix the entire water olumn, destroying the vertial struture in ross-shore and longshore veloities. The surfae TKE flux inreases the total TKE within the water olumn (Fig. 8d), developing a maximum value at the bar-rest and the shoreward boundary where the total dissipation is greatest. The vertial mixing (K v ) shows a orresponding inrease (Fig. 8). When εw = 0 (i.e., Run 6), K v values of approximately 0.03 m 2.s -1 are found over the bar-rest; and for εw values of 0.01 and 0.05, these values subsequently inrease to 0.05 and 0.06 m 2.s -1 respetively (Fig. 8). Similar inreases in K v are also seen for loations further offshore of the bar-rest and over the bar-trough. In general, inreasing the surfae TKE flux begins to destroy the vertial shear and the assoiated irulation pattern observed in the ross-shore and longshore veloities (Figs. 8a and b). In omparison to Run 6 (VF model with α r =1.0 and no wave mixing), the simulated rossshore veloity profiles from Run 9 and 10 (VF model with α r =1.0, εw = 0.01 and 0.05, respetively) show higher rms errors at loations within the region between bar-trough and barrest, and smaller errors at the station further offshore (see Table 4). The omparison of simulated and measured longshore veloity profiles (Fig. 8b) suggests that enhaned wave mixing (Runs 9 and 10) redues the flow magnitude. This redution deteriorates the agreement of model results to field observations at most of the measurement positions (see Table 4). Simulations onduted using α r =0.5 (not shown here) have shown similar response to that 31

32 disussed here. We feel this is a typial response in ross-shore and longshore veloity field to inreased mixing. In Figures 7a, 7b, 8a and 8b the simulated and measured ross-shore and longshore veloity profiles are ompared. Overall, the normalized rms errors obtained in these simulations vary between 0.54 and 0.66 for the ross-shore veloities and 0.20 to 0.3 for the longshore veloities. These values are similar to those of Newberger and Allen (2007b) and at times slightly higher than those shown by U10 ( and for ross-shore and longshore veloities, respetively). Nevertheless, our simulations show that the model is apable of reating realisti veloity profiles in a surf zone environment. In the remainder of the presentation, we fous on results from Run 6 (VF model with α r =1 and no wave mixing) as these simulated profiles show the best agreement to the observed ross-shore and longshore veloity measurements at the majority of the loations Figure 8. Comparison of model results (Runs 6, 9 and 10; VF model with rollers, α r =1 and wave-indued mixing with εw =0, 0.01 and 0.05, respetively) with observed vertial profiles (grey squares) of ross-shore (a) and longshore (b) veloities. Vertial grey lines indiate profile measurement loations and zero value for eah profile (Data from Garez-Faria et al. 1998; 2000). Vertial struture of eddy visosity (), K v and turbulent kineti energy (d), TKE model simulations at the same ross-shore loations as the veloities. 32

33 Longshore Cross-shore Kumar et al. (2012) Oean Modeling (doi: /j.oemod ) n i1 2 dij m ijk Table 4. Normalized root mean square error nrms ( jk, ) n 2 for the ross-shore d i1 ij and longshore veloity estimates for DUCK 94 for various loations aross the profile and the different model simulations (Runs 2-10, see Table 3). d ij and m ijk represent measured (from Garez-Faria et al., 1998, 2000) and model estimated veloity values at the 7 ross-shore loations (j) and various elevations (i) above the sea bed (for measurement loations see Fig. 10). Station 1 is losest to the shoreline. Numbers in bold typefae indiate minimum values. RUN # STN # Normalized Root Mean Square Error Analysis Mean Mean Cross-shore and Vertial struture of Eulerian mean and Stokes Veloity The horizontal and vertial distribution of the ross-shore veloity for Run 6 (VF model with α r =1 and no wave mixing) is shown in Figure 9a. As disussed previously, at the loation of wave breaking, vertial segregation of flow ours with an inshore direted flow at the surfae and offshore direted flow at the bottom. Maximum strength of this undertow ours at the barrest and lose to the shoreline, while relatively weaker values are found in the bar-trough. Outside the surf zone, flow through signifiant part of the water olumn is direted offshore with a maximum flow at the bottom layer, dereasing monotonially to a small onshore direted value at the sea surfae (Fig. 9a). Maximum longshore veloity (Fig. 9b) ours over the bar-trough with a smooth variation in the trough-rest region due to the effet of wave rollers. The strongest flow ours at the surfae, dereasing with an inrease in the water depth. Further offshore of the bar-rest, longshore veloity dereases signifiantly, and most of the modeled longshore flow is 33

34 wind driven. The vertial veloity (Fig. 9) is direted downwards inshore of the bar-rest and upwards offshore of the bar-rest (x=130m). This pattern along with inshore flows at the surfae and offshore direted flow in the enter of the water olumn reates an antilokwise irulation ell pattern whih is similar to that found in the planar beah ase presented in setion 4.1 (see Fig. 2) Figure 9. Cross-shore setions showing horizontal and vertial variability of Eulerian and Stokes veloity omponents for Run 6 (VF model with wave rollers, α r =1 and no wave mixing). (a) ross-shore (u); (b) longshore (v); and () vertial (w) Eulerian veloities; (d) Cross-shore (u st ); (e) longshore (v st ); and (f) vertial (w st ) Stokes veloity. 34

35 The vertial distribution of wave-indued Stokes drift follows a osh (2kz) distribution, with strongest flow near the surfae and weakest flow near the sea bed. Maximum ross-shore and longshore veloities our over the bar-rest and at very shallow waters further inshore (Figs. 9d and e). The ross-shore veloity is stronger than the longshore veloity, while the vertial Stokes veloity (Fig. 9f) is of similar strength as its Eulerian mean ounterpart. As the flux divergene of longshore and ross-shore Stokes veloities is zero over the bar-rest, the vertial Stokes flow hanges sign at this point. The upward and downward flow struture in the present ase is opposite in sign to Eulerian mean flows (Fig. 9). Presene of a vertial struture in water depth < 1m, also onfirms presene of a vertially varying VF Three-dimensional momentum balane The ross-shore and vertial variation of momentum balanes for the VF simulation with wave roller ation enabled ( r 1) and no wave mixing (Run 6) is shown in Figure 10. In the ross-shore diretion the horizontal momentum balane (see Eqns. 11 and 12) is dominated by the roller aeleration (BA, Fig. 10a), pressure gradient (PG, Fig. 10d) and vertial mixing (VM, Fig. 10e). The horizontal advetion (HA, Fig. 10b), horizontal vortex fore (HVF, Fig. 10d) and vertial advetion (VA, Fig. 10f) terms are insignifiant. The BA is surfae intensified (Fig. 10a) with strongest values ourring at loations where total wave dissipation is maximum. At the surfae layer, the BA is balaned by the sum of VM and PG (Figs. 10a, and e), while further below (D > 1 m), BA beomes negligible and PG is balaned by VM (Fig. 10e). Similar balane is also observed at the shoreward boundary. This ross-shore momentum balane is similar to that observed for the planar beah example in setion Analysis of the longshore momentum balane shows that with the exeption of PG all remaining terms (i.e., BA, VM, HA, VA and HVF) are signifiant. The sum of BA and HA terms (Figs. 10g and h) is balaned by the sum of VM, VA and HVF (Figs. 10k, h and l, respetively). BA (Fig. 10g) is strongest in the surfae layer over the bar-rest/trough region and near the shoreline and balaned primarily by the HVF term (Fig. 10i). It is notieable that at these loations of strong BA ontribution, VM takes its smallest values. However near the surfae and in the region between the bar-rest and shoreline, the VM term beomes more signifiant. In addition, near the bed the VM term is largest over the bar-rest and together with HVF (Fig. 10i) balane HA (Fig. 10h). It is notieable that over the bar-rest BA is balaned mainly by HVF, in the absene of a bar (see planar beah ase) BA is balaned by VM. At this stage it is important to point out that a traditional alongshore momentum balane in a radiation stress approah suggests that gradient of radiation stress ( S xy / x) is balaned by VM (see HW09). In the present ase, a summation of HA (Fig. 11h), HVF (Fig. 11i) and VA (Fig. 11i) is small and dominant balane is between BA and VM at most of the ross-shore loations, i.e., similar to radiation stress approah. However, HA and HVF do not ompletely anel eah other and have a net-ontribution in modifying the flow pattern (see Se. 5) Balane of vertially-integrated three-dimensional momentum balane The two-dimensional momentum balane in the ross-shore diretion (Fig. 11a) demonstrates a balane between pressure gradient (PG) and the breaking /roller aeleration (BA) terms. In the longshore diretion the major ontributors are vortex fores (VF), horizontal advetion (HA), breaking aelerations (BA) and bottom stress (BStr), as was the ase for a 35

36 planar beah (Fig. 4b). It is notieable that due to non-planar variation in bathymetry in this ase, the relative ontribution of eah term is different than that found for the planar beah ase, and the HA and HVF (Fig. 11b) are not symmetrial anymore Figure 10. Cross-shore and vertial distribution of the terms ontributing to the ross-shore (x) and longshore (y) momentum balane for Run 6 (VF model with wave rollers, α r =1 and no wave mixing). Cross-shore terms: (a) x-breaking aeleration (x-ba) ; (b) Eulerian, x-horizontal advetion (x-ha); () x- horizontal vortex fore (x-hvf); (d) x- pressure gradient (x-pg); (e) x-vertial mixing (VM); (f) x-vertial advetion (VA); and alongshore terms: (g) y-breaking aeleration (y-ba); (h) Eulerian, y-horizontal advetion (y-ha); (i) y-horizontal vortex fore (y-hvf); (j) y-pressure gradient (y-pg); (k) y-vertial mixing (y-vm); and (l) y-vertial advetion (y-va). 36

37 Deomposing the pressure gradient fore into individual omponents (Eqn. 49) shows that the Eulerian response, P x is the major ontributor (Fig. 11). Unlike the planar beah, the Bernoulli head (P bhx ) plays an important role over the bar-rest and further inshore. This ours beause Bernoulli head is dependent upon the veloity shear, and in this example high veloity shear is present in the region between the trough and rest of the bar. The quasi-stati response (P qsx ) also beomes dominant at the bar-rest and adds to P bh, while the surfae pressure boundary orretion (P px ) is negligible Figure 11. Cross-shore variation of depth-averaged (a) ross-shore and (b) longshore momentum balane terms. () Deomposed PGF terms in ross-shore as desribed in Eqn. 49 for Run 6 (VF model with wave rollers, α r =1 and no wave mixing). 37

38 Test Case 3: Nearshore barred morphology with rip hannels This ase investigates the dynamis of a barred beah bathymetry that develops rip urrents for normally inident waves. The appliation is based on a laboratory sale experiment and is similar to a ase demonstrated in HW09, with a few major differenes: (a) in HW09 the wave driver was a monohromati wave model (REF/DIF), while here we use a spetral wave model (SWAN); (b) the HW09 domain was idential to the laboratory experiments while our domain has been saled by a fator of 20 (kinemati similarity, Hughes, 1993) to reate more realisti field onditions (similar sale as Aagaard et al., 1997; Mamahan et al., 2005; ); and () bottom frition due to ombined ation of waves and urrents (Madsen, 1994, also see setion 4.2.1) is used instead of a logarithmi bottom drag. The bathymetry domain (Fig. 12a) is an idealized version of that used by Haller et al. (2002) and Haas and Svendsen (2002). The saling of the domain by a length sale, NL= 20, leads to a maximum depth of 10 m, a nearshore bar of 1.20 m loated 80 m off the oastline, ross-shore domain width of 292 m and alongshore length of 524 m. To avoid interation of rip hannel flow with the lateral boundaries, the domain was extended laterally by 80 m in either diretion. Rip hannels are spaed 184 m apart and the hannel width is 36.4 m whih makes the ratio of hannel width to rip urrent spaing 0.2, a value onsistent with those found in the field (e.g., Huntley and Short, 1992). The model grid has a horizontal resolution of 2 m in both diretions and onsists of 20 equally spaed sigma layers. The boundary onditions at shoreline, offshore boundary and lateral ends are no flow onditions (i.e., losed boundary onditions at the oast, lateral boundaries and offshore) and are the same as the laboratory experiments of Haller et al., (2002). Sine the effet of wave rollers is important in a surf zone environment (see setion 4.2.2), we use a α r =0.5 to allow for 50% ontribution of roller aeleration to momentum balane. In order to maintain realisti onditions, enhaned mixing due to wave breaking is also onsidered with a εw =0.02. At the offshore boundary, SWAN is fored with 1.0 m waves (H sig ) with peak period of 6.3 s, and diretional spreading of 8 propagating perpendiular to the shoreline. From these values, SWAN omputes a wave spetrum based on a JONSWAP distribution. The spetral resolution is 20 frequeny bands in the frequeny range between 0.04 Hz and 1 Hz, and 36 diretional bins of 10 eah from 0 to 360. A depth indued breaking onstant of γ b =0.6 is hosen to aount for depth limited wave breaking (Battjes and Janssen, 1978; Eldeberky and Battjes, 1996), while the eddy visosity model of Madsen (1988) for bottom frition indued wave attenuation is used with a bottom frition roughness length sale of 0.05 m. Beause of the high spatial resolution of the domain a time step of 0.5 s is used for both ROMS and SWAN 2 while the oupling between the two models takes plae every 5 s. Comparisons are shown after 1 hour of simulation time. In order to make our results omparable to those presented in HW09 and Kumar et al. (2011a) a relatively higher horizontal mixing oeffiient (0.20 m 2 s -1 ) has been used that leads to relatively stable flows. The ability of the model to simulate the unstable harater of rip urrents (e.g., Haas and Svendsen, 2002) is demonstrated through the 2 A time step of 0.5 s leads to a CFL number of ~1. Trial runs with larger time steps in SWAN (1, 2, 3 and 5 s) when ompared with the 0.5 s time step run revealed overall RMS differenes in wave height of 0.34, 0.83, 0.81 and 0.66% respetively, while the RMS differene in vortiity was 0.009, 0.013, and 0.044% respetively. These differenes beome larger for smaller water depths (1.12, 2.30, 2.23 and 1.96% for wave height and 0.031, 0.045, 0.072, and 0.16 for vortiity for water depths less than 0.5m) This suggests that although the internal limiter in SWAN (Ris, 1999) is effetive in making the wave model stable, it does not ompletely eliminate inauraies in the wave results due to large time steps, but the overall differenes are found to be relatively small. 38

39 presentation of a ase where a lower, more realisti horizontal mixing oeffiient is used (0.05 m 2 s -1 ) Wave parameters and sea surfae elevation At the rip hannel loations, wave - rip urrent interation (Fig. 12b) auses a loal inrease of wave steepness. Greater water depths at the hannel loations allow for further inshore propagation of these inoming waves, whih finally start breaking at x~ 50m. On the other hand, waves propagating over the bar start breaking at x~70 m, beome stable (25 m < x < 65 m) and then break again near the shoreward boundary (x < 25 m). The differene in wave breaking pattern over the hannel and the bar reates a lateral differene in breaking indued wave set-up at these two loations Figure 12. Rip hannel ase. (a) Bathymetri domain; (b) signifiant wave height (ontours) and diretion (arrows); and () vortiity vetor after 1 hour of model simulation. Blak arrows in () show the depth averaged, Eulerian veloity vetor. The white line in () shows veloity strength of 0.5 ms -1. The solid white lines in (a) show the transets along whih ross-shore and longshore momentum balanes are desribed in Figures 15 to Nearshore Flows Differenes in sea surfae elevation due to wave set-up drive mean flow patterns. Higher wave-setup at the bar than the hannel reates feeder urrents direted towards the latter whih results in a onfluene of flow from both sides leading to the development of the outgoing rip urrent (Fig. 12). Close to the shoreline, the wave set-up pattern reverses, as the larger waves within the rip hannel break further inshore; this reates a higher wave set-up inshore of the hannel in omparison to loations inshore of the bar. The waves in the latter loation have 39

40 already dissipated due to wave breaking over the bar. This wave-setup gradient auses alongshore flows inshore the bar, direted away from the hannel (see Fig. 12). Overall a primary irulation pattern develops with outgoing feeder urrents from the rip hannel and return flow over the bar, and a seondary irulation pattern lose to the shoreline, with inshore flows direted towards the shoreline and longshore veloity direted away from the rip hannel (Fig. 12). Further offshore, the strength of the rip urrent gradually dereases until it beomes negligible. These simulated results are onsistent with the laboratory studies onduted by Haller et al. (2002), Haas and Svendsen (2002) and the modeling work of Haas et al. (2003), Yu and Slinn (2003) and Kumar et al. (2011a). Flow vortiity vetor ontours (Fig. 12) show two vortex patterns inshore and offshore of the rip hannel, orresponding to the seondary and primary irulation patterns, respetively. Eah vortex pattern onsists of a pair of vorties of opposite signs, suggesting opposite irulation tendenies Figure 13. Vertial struture of ross-shore Eulerian veloity at (a) the enter of rip hannel and (b) over the bar. Results derived from VF 3D based model simulations. () Comparison of normalized model derived ross-shore veloity with normalized data from Haas and Svendsen, 2002 (key: symbols and and grey and blak lines denote data and model results at the enter and 8 m off the hannel, respetively). Vertial profiles of Eulerian mean ross-shore veloities in the rip hannel and over the bar are shown in Figures 13a and b, respetively. At loations inshore of the rip hannel (x < 40 m, Fig. 13a) the flow is direted inshore from surfae layer to the middle of the water olumn, while weak offshore direted flow is seen at the bottom layer. Inshore flow is strongest at the surfae (~0.3 ms -1 ) and dereases with depth. Within the rip hannel and further offshore (40 m < x < 100 m) the flow is direted seaward. Strongest offshore direted flow (of the order of 0.7 ms -1 ) ours over the rip hannel at x ~ 70 m and lose to the middle of the water olumn with a 40

41 monotoni derease in magnitude with inreasing or dereasing water depth. Inlusion of horizontal visous mixing and wave-indued enhanement in mixing redues the horizontal and vertial shear in veloity by dispersal of momentum, providing smoother solutions. In omparison to flows observed within the rip hannel, the flow field is relatively weaker over the bar (Fig. 13b). Wave breaking ours over the bar-rest and at the shoreward boundary. Undertow in the bottom layer with a magnitude of ~ 0.3 ms -1 is observed at both breaking loations, while in the surfae layer flow is direted towards the shore (Fig. 13b). Overall, the veloity profile observed over the bar is similar to that disussed earlier for the DUCK 94 simulations. Our saled numerial experiment onditions orrespond to Test B of Haller et al. (2002) and Test R of Haas and Svendsen (2002). We use the results of those lab experiments to provide a semi-quantitative omparison between the measured and modeled vertial struture of the ross-shore veloity field. For this omparison we use all of the bin averaged veloities from Test R (see Fig. 11 in Haas and Svendsen, 2002) and for all reported loations (Fig. 13). The measured and simulated veloities are normalized by the orresponding maximum ross-shore veloity at the enter of the rip hannel, respetively. The simulated normalized ross-shore urrent vertial struture from the model simulation agrees well with the experimental data (Fig. 13 and Table 5). Inside the hannel, rip urrent speed is greatest just below the level of the barrest and dereases toward the surfae and bed. However no experimental data are available near the surfae. Just offshore the bar, the normalized data from the model simulation show a paraboli profile with stronger veloities at the enter of the water olumn, while the experimental data suggest vertially dereasing magnitude of veloity in the water olumn. The rms error (normalized by maximum observed value, as in Sheng and Liu, 2011) is small within the rip hannel (5.8%, Table 5) but inreases for loations further offshore (Table 5). The overall rms error is 12.7%. Table 5. The RMS error (normalized by the maximum observed value) for the simulated ross-shore veloities for nearshore barred beah with rip hannels (se 4.3) Unstable rip urrent flow STN # RMS Error (%) Cross-shore Vel Overall 12.7 Rip urrents are unstable in nature (Haas and Svendsen, 2002), and proesses like vortex propagation and vortex shedding have been observed both in numerial simulations and field experiments (see Yu and Slinn, 2003; Haller and Dalrymple, 2001; MaMahan et al., 2005; Reniers et al., 2009). The importane of these vorties lays in the fat that they interat with the 41

42 inoming wave-indued Stokes drift and reate a strong VF in the longshore diretion (negligible in ross-shore diretion as v St is almost zero), whih may play a relevant part in the maintenane and advetion of these vorties Figure 14. Example of unstable rip urrent onditions simulated using with a linear bottom frition (μ=0.002 m) and a horizontal mixing of 0.05 m 2 s -1. Snapshots of vortiity and depth-averaged, Eulerian veloity vetor for six different time steps with a time interval of 20 min. Only the omputational domain in the viinity of the rip hannel is shown here. The dynamis of a barred beah with rip hannels for normally inident waves are investigated for the same model domain as in Fig. 14, and same offshore wave onditions. Unlike the previous simulation (Se 4.3), in this ase we use a linear bottom drag formulation with a drag oeffiient of ms -1 (Yu and Slinn, 2003) and a horizontal mixing oeffiient of 0.05 m 2 s -1. Snapshots of vortiity vetor and mean flow (Fig. 14) show the evolution of flow vortiity over the omputational domain. The diretion of rip urrent is at an angle to the rip hannel, and its strength hanges over time. It is also interesting to see that the vortiity pattern has a periodiity of approximately 60 minutes, whih agrees with previous model simulations of rip urrents (Yu and Slinn, 2003) Three-dimensional momentum balane The three-dimensional momentum balane is presented along a ross-shore and a longshore transet. The ross-shore transet is defined by a line that passes through the enter of a vortex (i.e., y = 180 m, Fig. 12), as this is the region where the VF ontribution is most signifiant. This transet is midway on the slope between the bar and the rip hannel. The alongshore transet is at loation x=70m and it passes through the enter of the rip hannel (see Fig. 12). The horizontal ross-shore momentum balane has a pattern whih is similar to that presented for the planar and barred beah ases. PG, BA, HA and VM are the dominant terms, 42

43 while HM and HVF are negligible (Figs. 15a to f). The BA term beomes important at x< 90 m, a loation where wave breaking has just initiated within the rip hannel while the majority of the waves break further inshore (Fig. 15a). As in the other ases, the influene of BA is limited to the sea surfae and is balaned by VM. Sine the domain is not alongshore uniform, advetion beomes important. This is shown in Fig. 15b, where the HA ontribution is signifiant on the shoreward side of the bar and when is added to VM, the sum balanes the PG term Figure 15. Cross-shore distribution of vertial profiles of ontributing terms in ross-shore (x)- and longshore (y) momentum balane at y=180m (see Fig. 12 for transet loation). Cross-shore terms: (a) x-breaking aeleration (x- BA); (b) Eulerian, x-horizontal advetion (x-ha); () x-horizontal vortex fore (x-hvf); (d) x-pressure gradient (x- PG); (e) x-vertial mixing (x-vm); (f) x-horizontal mixing (x-hm); Longshore terms: (g) y-breaking aeleration (y-ba); (h) Eulerian, y-horizontal advetion (HA); (i) y-horizontal vortex fore (y-hvf); (j) y-pressure gradient (y- PG); (k) y-vertial mixing (y-vm); and (l) y-horizontal mixing (y-hm). 43

44 The longshore momentum balane analysis shows that PG, HA, HM, HVF and VM are the important terms while BA (Fig. 15g) is negligible. The feeder urrent developed near the rip hannel (Fig. 15) is driven by pressure gradients (PG, Fig. 15j) due to differenes in wave setup levels over the bar and the hannel loation, respetively. This PG term is stronger in the viinity of the bar and it is balaned predominantly by the HVF term (Fig. 15i) whih is stronger near the sea surfae and dereases toward the sea bed. It is near the bed where the positive values of VM (Figs. 15k) add to HVF to balane PG. Near the surfae the negative values of VM add to PG to balane HVF. Finally, HA (Fig. 15h) is half the strength of PG and has similar magnitude and opposite sign as of HM (Fig. 15l) Figure 16. Longshore distribution of vertial profile of ontributing terms in ross-shore (x) and longshore (y) momentum balane at x=70m (see Fig. 12 for transet loation). Cross-shore terms: (a) x-breaking aeleration (x- BA); (b) Eulerian, x-horizontal advetion (x-ha); () x-horizontal vortex fore (x-hvf); (d) x-pressure gradient (x- PG); (e) x-vertial mixing (x-vm); (f) x-vertial mixing (x-vm); Longshore terms: (g) y-breaking aeleration (y- BA); (h) Eulerian, y-horizontal advetion (y-ha); (i) y-horizontal vortex fore (y-hvf); (j) y-pressure gradient (y- PG); (k) y-vertial mixing (y-vm); and (l) y-horizontal mixing (y-hm); 44