Dispersion in Shallow Moment Equations

Abstract

Shallow moment models are extensions of the hyperbolic shallow water equations. They admit variations in the vertical profile of the horizontal velocity. This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables. The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction. Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface. The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given. A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.