Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space arXiv.org / hal Abstract. In this paper we review the history and current state-of-the-art in modelling of long nonlinear dispersive waves. For the sake of conciseness of this review we omit the unidirectional models and focus especially on some classical and improved Boussinesqtype and Serre-Green-Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

International audienceA new derivation of completely nonlinear weakly-dispersive shallow water equations is given without assumption of flow potentiality. Boussinesq type equations are derived for weakly nonlinear waves above a moving bottom. It is established that the total energy balance condition holds for all nonlinear dispersion models obtained here

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Nonlinear-dispersive shallow water equations on a sphere are obtained. These equations can be used in simulation of large-scale propagation of long waves in the problems of atmosphere and ocean dynamics taking into account the Earth rotation and wave dispersion.

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Nonlinear dispersive shallow water equations on a sphere are obtained without using the potential flow assumption. Boussinesq-type equations for weakly nonlinear waves over a moving bottom are derived. It is found that the total energy balance holds for all obtained nonlinear dispersive equations on a sphere.