A Terrain-Following Boussinesq System

Abstract

A long wave model is derived asymptotically from the nonlinear potential theory equations. The flow regime of interest is incompressible, irrotational, and inviscid. Asymptotic analysis leads to a weakly nonlinear, weakly dispersive (variable coefficient) Boussinesq system valid for a wide class of topographies. The mild slope hypothesis is not required and rapidly varying topographies are also considered. In analogy with atmospheric models we use a terrain-following coordinate system. The novelty is that this coordinate system naturally suggests the weighted averaging of terrain-following velocity components, as opposed to the depth-average of horizontal velocity components found in standard shallow water formulations. Furthermore, a Schwarz--Christoffel toolbox is used to provide additional insight on these new results. Regarding applications, the proposed model can be used for studying solitary waves interacting with fine scale inhomogeneities, a theme of great interest. The terrain-following model also presents potential numerical advantages for Boussinesq solvers.