Numerical instabilities of spherical shallow water models considering small equivalent depths

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this record.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.Peixoto acknowledges the Sao Paulo Research Foundation (FAPESP) under the grant number 2016/18445-7 and the National Science and Technology Development Council (CNPq) under grant number 441328/2014-8. Thuburn was funded by the Natural Environment Research Council under the “Gung Ho” project (grant NE/1021136/1). Bell was supported by the Joint UK DECC/Defra Met Office Hadley Centre Climate Programme (GA01101)

A Quasi-Hamiltonian Discretization of the Thermal Shallow Water Equations

International audienceThe rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model