Variable depth and the validity of Boussinesq-type models

Abstract

The main issue of this report is comparison and verification of long wave models with emphasis on variable depth effects. A selection of dispersive long wave theories are reviewed. Included are the so-called standard Boussinesq equations and the much celebrated counterpart where the velocities are prescribed at an optimal vertical position, namely at 0.531 times the depth down from the equilibrium surface. The latter option, referred to as improved Boussinesq equations, displays substantially improved dispersion properties on constant depth. We also address formulations with the velocity potential as primary unknown and where approximations linked to a mild bottom slope have been invoked. One of the potential formulations are modified in a very simple way to yield dispersion properties equal to the "improved Boussinesq equations". First the different Boussinesq models are assessed for constant depth propagation. Then testing is extended to wave motion in an idealized bathymetry with two horizontal planes joined by a smooth slope. The length of the waves incident on the slope as well as the slope steepness are systematically varied. It turns out that the new potential formulation, with improved dispersion properties, performs well even in presence of steep bottom gradients.