An Entropy Stable Discontinuous Galerkin Method for the Spherical Thermal Shallow Water Equations

We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive the buoyancy variance which is convex and defines an entropy functional, and which must be preserved in order to preserve model stability at the discrete level. Our spatial discretization is provably entropy and energy stable, and the fully discrete method conserves mass, buoyancy, and vorticity. This is achieved by using novel entropy stable numerical fluxes, the summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilization.