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We develop a novel and efficient discontinuous Galerkin spectral elementmethod (DG-SEM) for the spherical rotating shallow water equations in vectorinvariant form. We prove that the DG-SEM is energy stable, and discretelyconserves mass, vorticity, and linear geostrophic balance on general curvlinearmeshes. These theoretical results are possible due to our novel entropy stablenumerical DG fluxes for the shallow water equations in vector invariant form.We experimentally verify these results on a cubed sphere mesh. Additionally, weshow that our method is robust, that is can be run stably without anydissipation. The entropy stable fluxes are sufficient to control the grid scalenoise generated by geostrophic turbulence without the need for artificialstabilisation.

Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations