Numerical instabilities of spherical shallow water models considering small equivalent depths

Abstract

Shallow water models are often adopted as an intermediate step in the development of atmosphere and ocean models, though they are usually tested only with fluid depths relevant to barotropic fluids. Here we investigate numerical instabilities emerging in shallow water models considering small fluid depths, which are relevant for baroclinic fluids. Different numerical instabilities of similar nature are investigated. The first one is due to the adoption of the vector invariant form of the momentum equations, related to what is known as the Hollingsworth instability. We provide examples of this instability with finite volume and finite element schemes used in modern quasi-uniform spherical grid based models. The second is related to an energy conserving form of discretization of the Coriolis term in finite difference schemes on latitude-longitude global models. Simple test cases with shallow fluid depths are proposed as a means of capturing and predicting stability issues that can appear in three-dimensional models using only twodimensional shallow-water codes.