[9] The Q2D-morfo model is a nonlinear morphodynamic model for large scale shoreline dynamics. A cartesian frame with horizontal coordinates*x*, *y* and upward vertical coordinate *z* is used, where *y* runs along the initial mean shoreline orientation. The nearshore region is represented by a rectangular domain, 0 < *x* < *L*_{x}, 0 < *y* < *L*_{y}. The unknowns are the moving shoreline, *x* = *x*_{s}(*y*, *t*), and the changing bed level, *z* = *z*_{b}(*x*, *y*, *t*). The dynamic equation for the bed level is the sediment mass conservation:

where is the depth integrated sediment flux and the bed porosity factor is included for convenience in . The shoreline position is determined by interpolating between the cells with *z*_{b} > 0 and the cells with *z*_{b} < 0.

#### 2.1. Sediment Transport

[10] Nearshore 2D depth average models compute the sediment flux from the wavefield and the mean hydrodynamics (currents). In contrast, the present model computes the sediment transport directly from the wavefield via parameterizations without determining the mean hydrodynamics. It is this important simplification that makes this model capable of performing large scale simulations with a reasonably low computational cost. The dynamics of small scale surfzone features like rhythmic bars and rip currents can however not be reproduced but this simplification seems reasonable in the context of large scale shoreline modeling.

[11] The sediment flux in the model is decomposed as

The first term represents the littoral drift, which is due to the alongshore current driven by the breaking waves in case of off-normal wave incidence. It is evaluated by first computing the total sediment transport rate, i.e., the cross-shore integrated flux, with an extended version of the CERC formula [*Komar*, 1998]. The formula has been adapted to include a second term introduced by *Ozasa and Brampton*, 1980, which represents the contribution of alongshore gradients in wave height to the alongshore transport,

where *H*_{b}(*y*) is the root mean square wave height at breaking, *α*_{b} = *θ*_{b}(*y*) − *ϕ*(*y*) is the angle between wavefronts at breaking and the coastline and *β* is the mean surfzone slope. The constant *μ* is proportional to the empirical parameter *K*_{1} of the original CERC formula. This parameter controls the magnitude of the transport and the default value *μ* = 0.2 m^{1/2} s^{−1} roughly corresponds to *K*_{1} = 0.7. The constant *r* = *K*_{2}/*K*_{1}, where *K*_{2} is the empirical parameter of the second term. The default value of *r* = 1 is used, which is equivalent to *K*_{2} = *K*_{1}. Then, the sediment flux is computed by multiplying the total transport rate by a normalized shape function *f*(*x*), qualitatively based on the cross-shore profile of the alongshore current [*Komar*, 1998]:

with

where *L* = 0.7*X*_{b}(*y*) and *X*_{b}(*y*) = *x*_{b}(*y*) − *x*_{s}(*y*) is the width of the surfzone. The point of breaking, *x*_{b}(*y*), is the most offshore point where *H*(*x*, *y*) ≥ *γ*_{b}*D*(*x*, *y*). *D* is the water depth and *γ*_{b} is the breaking index (the ratio wave height to water depth at breaking).

[12] The orientation of the coast, *ϕ*, is represented by the mean orientation of the bathymetric contours in the surfzone with respect to the *y*-axis rather than the orientation of the coastline itself. This seems more appropriate because it is this orientation that actually affects the waves at breaking. It is computed as

where the average is computed within a rectangular box with a cross-shore length*L*_{box}, an alongshore length 2 ∗ *L*_{box}, where *L*_{box} = *B* ∗ *X*_{b} and the default value of the constant *B* is 2.

[13] *Ashton and Murray* [2006a]explored other formulas for breaking-wave-driven transport and found that all formulas show the potential for shoreline instability but that they may predict somewhat different shoreline responses under the same conditions. An interesting study by*List and Ashton* [2007]demonstrated that the cross-shore integrated alongshore transport computed with a process-based wave, circulation, and sediment transport model showed patterns along an undulating shoreline similar to the transport computed directly from the wavefield with the CERC formula. Even though they did not compute morphological evolution, the process based model predicted the potential for high angle wave instability, confirming that the present simplified approach using the CERC formula can be used for the exploration of HAWI. However, care must be taken when mathematical models like the CERC formula are used for quantitative predictions of alongshore transport and the resulting shoreline change [*Cooper and Pilkey*, 2004]. In this study we only look at the qualitative behavior and the use of CERC formula therefore seems valid.

[14] The second term in equation (2)is a parametrization of cross-shore sediment transport processes. We assume that, on a relatively long timescale, these processes drive the cross-shore profile to an equilibrium profile*z*_{be}, so that

where *z*_{be}(*x*, *y*) = *Z*(*x* − *x*_{s}(*y*)) is the assumed equilibrium profile and *γ*_{x}is a cross-shore diffusivity coefficient. The third term inequation (2) is an alongshore diffusive transport that suppresses the growth of small scale noise,

The physical basis for the coefficients *γ*_{x} and *γ*_{y} is the diffusivity caused by wave breaking. Thereby, they depend on the wave energy dissipation and their order of magnitude has been estimated by using the expression for momentum mixing due to wave breaking [*Battjes*, 1975],

where *M*is a non-dimensional constant (*O*(1)), is the wave energy dissipation per time and area unit, *ρ* is the water density and *H* is the root mean square wave height. We assume that *γ*_{x} and *γ*_{y} scale with *v*_{t}, with *H* = *H*_{b} in equation (9). The order of magnitude of can be estimated as the total energy flux entering the surfzone divided by the cross-shore length,

where *g* is the gravity acceleration and *c*_{gb} is the group celerity at breaking, computed with the shallow water assumption ( ). An estimation for the morphodynamic diffusivity is therefore,

where *ϵ*_{x}is a non-dimensional constant and a similar expression is used for*γ*_{y} with the constant *ϵ*_{y}. The shape function,

has a cross-shore distribution with a maximum in the surfzone and it decays to almost zero at the depth of closure,*D*_{c}. *X*_{1} controls the position of *D*_{c} and is defined as *X*_{1} = *C* ∗ *X*_{b}, where the default value of the constant *C* is 2 and *X*_{b} varies with the wave height. *L*_{d} controls the length scale of the decay until *X*_{1} and offshore of this point the shape function tends to a residual value controlled by *b.* The choice of the default values for *ϵ*_{x} and *ϵ*_{y} is based on the characteristic diffusion time, *T*_{d} ∼ *L*_{d}^{2}/*γ*_{x} and *L*_{d}^{2}/*γ*_{y}, of a bathymetric feature with a characteristic length *L*_{d} and the dimensional values of *γ*_{x} and *γ*_{y} are in the same order of magnitude as those corresponding to the bedslope transport in surfzone morphodynamic models [*Garnier et al.*, 2008].

[15] Notice that the cross-shore equilibrium profile is assumed to be perpendicular to the initial shoreline rather than to the evolving local shoreline orientation. Consistently, the flux given byequation (7) is assumed to be in the direction of the *x*-axis. The inaccuracy introduced by this approximation is not significant in the present application since changes in shoreline orientation do not exceed about 13°.

#### 2.2. Waves

[16] For the computation of the sediment transport the wave height and direction at breaking are needed. The wave module computes the wavefield in the domain using the wave height, period and angle given at the offshore boundary, the dispersion relation,

the equation for wave number irrotationality,

and the wave energy conservation,

Here *ω* = 2*π*/*T*_{p} is the radian frequency, *T*_{p} is the peak period, is the wave number vector, *c*_{g} is the group celerity and *θ*is the angle of the wave crest with respect to the y-axis. This approach takes into account refraction and shoaling, but it neglects diffraction and dissipation by bottom shear stresses. Dissipation by breaking is not included because the wavefield is only needed up to breaking. The wavefield is computed every time step Δ*t*_{w} = 1 day. There is also an option to compute the wavefield with a more detailed external wave model but this increases the computational cost and studies that included more hydrodynamic processes showed that this did not change the qualitative behavior of HAWI [*Uguccioni et al.*, 2006; *List and Ashton*, 2007].

#### 2.3. Realistic Range of Wave Angles

[17] In theory, any wave angle is possible in infinitely deep water. However, the angle between wavefronts and coastline decreases as water depth decreases because of wave refraction. This poses an upper bound on the wave angles that are realistic at the offshore boundary of the model domain. Wave refraction depends on the wave period and shorter wave periods allow for larger angles at a given water depth.

[18] For any wave period this can be determined by assuming *θ* = *θ*_{∞} at an offshore water depth, *D*_{∞}, and refracting the waves up to the water depth of the offshore boundary, *D*_{0} = *D*(*L*_{x}). The angle *θ*_{0} is found by solving equations (13) and (14). The latter reduces to the Snell law, *k*_{o} sin *θ*_{0} = *k*_{∞} sin *θ*_{∞}, by assuming rectilinear and parallel depth contours. Taking, for example, *D*_{∞} = 250 m, and *θ*_{∞} 90*°*, the angle *θ*_{0} at *D*_{0} gives the maximum incidence angle allowed at such depth. The results for the maximum angle as a function of wave period and *D*_{0} are shown in Figure 1. Larger water depths in deep water, *D*_{∞}, give the same results for a wave period not larger than about 20 s.

#### 2.4. Boundary Conditions

[19] The boundary condition,

is assumed at the shoreline, where *ϕ*_{s} is the angle between the shoreline and the *y*-axis. This means that the swash zone slope relaxes to an equilibrium slope*β*_{s}. If the swash slope is smaller than the equilibrium slope, sediment is transported from the wet cells to the dry cells and the shoreline advances seaward. If the swash slope is steeper, the dry beach is eroded and the shoreline retreats. The coefficient *γ*_{s} is related to the relaxation time *T*_{s} by *γ*_{s} ∼ (Δ*x*)^{2}/*T*_{s}, where Δ*x* is the grid size.

[20] At the offshore boundary, *x* = *L*_{x}, it is assumed that the bathymetry relaxes to the equilibrium bathymetry within a certain decay distance *λ*_{x} from the boundary. At the lateral boundaries (*y* = 0, *L*_{y}) the diffusive transport is assumed to be zero and the sediment flux is controlled by the wave driven alongshore transport (equation (3)). In this sense, an open boundary condition is used, so that sediment is not necessarily conserved within the domain and the bathymetry can evolve freely. The wave driven alongshore transport depends on the local values of *H*_{b}, *θ*_{b} and *ϕ.* The value of *ϕ* at the lateral boundaries is however not obvious because it is the average surfzone orientation within a rectangle and the bathymetry outside the domain is unknown. If *ϕ* is determined by only using interior cells a positive feedback between the surfzone orientation and gradients in *Q* can arise leading to a numerical instability that causes strong accretion or erosion at the boundary. Therefore, the following boundary condition is used:

This is consistent with an exponential decay to zero of *ϕ*far from the domain. The e-folding length of the decay is set to*λ*_{y} = 500 m and this has proven to lead to realistic behavior at the boundaries. In order to check the sensitivity to the lateral boundary condition, we varied the alongshore length of the domain and the area where the initial random perturbations were imposed. The results were qualitatively similar and showed that the sand waves traversed the downdrift boundary freely. The present lateral boundary condition was preferred over a periodic boundary condition because the latter would lead to artificial behavior where sand waves, leaving the domain at the downdrift boundary, would enter at the updrift boundary and interact with the beginning of the sand wavefield. Even though the general properties of the sand waves would be similar, this would lead to different dynamics and the wavelength of the sand waves would not be allowed to evolve freely (only dividers of the length of the domain).

#### 2.5. Numerical Implementation

[21] The set of equations is discretized in space by standard finite differences on a staggered grid. Equation (1)is discretized in time by a second order Adam-Bashforth explicit method. The use of an explicit method gives a Courant-Friedrichs-Lewy stability condition of the type

based on the morphological diffusivity which is roughly proportional to *H*^{3/2}. Numerical experiments show that *c* ∼ 0.13 m^{−1/2} s.

[22] When the shoreline deviates from a line parallel to the *y*-axis, jumps occur in the shoreline position. If these jumps in the shoreline position become larger than two grid cells, the sediment transport at the shoreline is not correctly evaluated. This leads to an unrealistic evolution of the shoreline and therefore there is a limitation on the maximum shoreline angle*ϕ*_{s}: |tan *ϕ*_{s}| ≤ 2Δ*x*/Δ*y.* On the other hand, since the wave propagation equations are hyperbolic, it is required that a wave ray entering a cell from its offshore boundary does not exit trough a lateral boundary. This results in a constraint on Δ*x*/Δ*y* opposite to that based on the shoreline angle: Δ*x*/Δ*y* < (tan *θ*)^{−1}. It is numerically found that Δ*x*/Δ*y* ≤ 1 can be used for *θ*_{0} < 55° but for waves up to *θ*_{0} ≃ 89°, Δ*x*/Δ*y* ≤ 0.25 is required.