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Modelling shoaling and breaking waves on a mild sloping beach

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Maria Kazakova (Rouen)

The mathematical modelling of coastal wave is a quite challenging issue since it is difficult to describe in the
same model the dispersive effects in the shoaling zone and the energy dissipation in the surf zone. In this
study we propose a new model for waves propagation over a mild slopping topography. With Teshukov's hypothesis of
weakly sheared flows an asymptotic model is derived from the full Navier-Stokes system. Shearing and turbulence
effects in breaking waves are taken into account by a third variable called enstrophy.
In the absence of enstrophy, the system reduces to the classical equations of Green-Naghdi. Since the
model is dispersive and not hyperbolic, the enstrophy equation can replace conveniently the energy equation for
the numerical resolution. The equations were numerically solved with the strategy of Le Métayer et al. (2010). The
scheme is rewritten for the new variables and allows us to use a hybrid method which consist in the resolution of
a hyperbolic system by a Godunov-type method and an elliptic equation. The numerical simulations were successfully
compared to the experimental data of Hsiao et al. (2008). This is a joint work with Gaël Richard (IRSTEA, Grenoble).