Nonlinear evolution of water waves and their dispersion relation in coastal waters

Preliminary results of the derivation of a new phase-resolving (deterministic) spatio-temporal nonlinear model of water wave evolution in non-deep waters with constant bathymetry are presented in this paper. The model is the first of its kind to include a nonlinear dispersion relation and cubic (four-wave) interactions. Simulations show the importance of nonlinear dispersion for bound wave components which, if not properly accounted for, result in inaccurate transfer of wave energy. We also investigate the relative importance of cubic nonlinearity, as compared to a quadratic (three-wave interactions) one, and show that it is non-negligible. The model provides the first step before the incorporation of these extensions into a phase-averaged (stochastic) formulation, which can then be used as a more accurate nonlinear source term for wave forecasting models.