On Nonlinear Shoaling Properties of Enhanced Boussinesq Models

In this paper, we investigate the nonlinear properties of Boussinesq models. In particular, we consider the wave shoaling obtained in physical regimes which go from linear to weakly nonlinear, to the wave breaking limit. For a given asymptotic accuracy in terms of dispersion and nonlinearity, we consider two families of models: the first depending on derivatives of the velocity, the second on derivatives of the volume flux. We show that, while linear dispersion and linear shoaling characteristics are strongly dependent on the type of dispersive terms introduced, when approaching the nonlinear regime the only influencing factor is whether the model is in amplitude-velocity of amplitude-flux form. We investigate these two alternative formulations of several known models, and propose a new model with a compact differential form, and the same linear characteristics of the model of Nwogu. The nonlinear shoaling properties of the models are investigated numerically showing that inside one given family, all the models have almost identical behaviour.