A wave-breaking model for the depth-semi-averaged equations

We propose an efficient model for the description of the three-dimensional (3-D) evolution of breaking water waves in the nearshore region. A fundamental property of the model is its intrinsic ability to account for the 3-D dynamics of vorticity and the energy dissipation induced by wave breaking. In particular, the vorticity evolution is achieved through the use of mollified operators, an approach similar in spirit to that adopted in smoothed particle hydrodynamics. Further, since the model is based on depth-semi-averaged equations with a core structure similar to that of nonlinear shallow-water equations, it takes advantage of well-known numerical methods for hyperbolic equations, while permitting computation of local flows. Finally, the model relies on a limited number of tunable parameters and a very simple breaking criterion. All the above aspects allow for a simple and reliable representation of the main features of wave breaking at the time and spatial scales typical of the nearshore wave dynamics. A number of benchmarks are used to explore the properties of the model, which is tuned only once for all cases. Wave height decay rates are well described for both sloshing (thin) and shoaling (thick) spillers, and a good description is also provided of the vorticity field. A final run of an impulsive wave over a submerged breakwater is used to illustrate the representation of the 3-D vorticity dynamics.