SWAN model

The SWAN model [12,18] is based on the spectral action balance equation rather than the spectral energy balance equation. The evolution of the action density N is governed by Komen et al. [13]:

∂N/∂t+∇_x ⃗ ∙[((c_g ) ⃗+U ⃗ )N]+(∂c_θ N)/∂θ+(∂c_σ N)/∂σ=S_tot/σ where, N(σ.θ) is the action density spectrum, x is space, t is time,θ is wave direction, σ is relative frequency, S_tot is the source term total, and c_g represents the propagation velocity.

The first term in the left hand side of the equation represents the local rate of change of action density in time. The second and third terms represent propagation of N in geographic space, accounting for shoaling as the group velocity c_g decreases in shallow water. The

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The first three source terms are significant in deep water and the last three terms are significant in shallow water. The SWAN default values for tuneable parameters are Br=1.75 ×〖10〗^(-3) and C_ds=5.0×〖10〗^(-5). Transfer of wind energy to waves, denoted S_in in SWAN, is described by different authors [7,8,12]. Bottom friction (S_br) also has several different formulations: the empirical JONSWAP model of Hasselmann et al. [19], the drag law model of Collins [20] and the eddy viscosity model of Madsen et al. [21]. Energy dissipation due to depth-induced breaking (S_br) follows Battjes and Jansen [22] whereby the maximum wave height H_max in shallow water depth d is limited by the relation H_max=γd. where γ is the breaking parameter, assumed to be 0.73 in SWAN. In deep water, quadruplet wave–wave interactions dominate the evolution of the wind wave spectra and are represented by the discrete-interaction approximation (DIA) of Hasselmann et al. [23]. In very shallow water, triad wave–wave interactions are important for depths which are small relative to wave height and wave length; they are parameterized by the lumped triad approximation (LTA) of Eldeberky [24] and Eldeberky and Battjes [25]. The

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Full three dimensional Reynold’s averaged Navier-Stokes equations are solved, applying the finite volume method on a standard staggered grid system. Orthogonal grids are used in the horizontal plane and boundary fitted grids with equal number of layers is used in the vertical direction. The projection method with a pressure correction technique is used to solve the governing equations in two distinct steps. In the first step, intermediate velocities are calculated, using the time splitting method for the solution of advection-diffusion, surface level gradient, bed roughness, sponge layer and dynamic pressure gradient terms. For each term, a proper solution method is applied. Leapfrog scheme is used to obtain a second order accuracy in time domain. To ensure local momentum conservativity and monotonicity of the solution, the momentum and mass conservation equations are solved simultaneously. Due to the significance of horizontal advection of horizontal velocity in the simulation of wave propagation in shallow water close to incipient breaking, a Godunov type shock capturing technique is used to solve these terms. In the second step, calculated velocities together with pressure correction terms from the