We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.

翻译：我们提出了一种新颖且高效的间断伽辽金谱元方法（DG-SEM），用于求解向量不变形式的球面旋转浅水方程。我们证明，在一般曲线网格上，该DG-SEM方法具有能量稳定性，并能离散地守恒质量、涡度及线性地转平衡。这些理论结果的实现得益于我们提出的针对向量不变形式浅水方程的新型熵稳定数值DG通量。我们通过立方球网格上的实验验证了这些结果。此外，我们还证明了该方法的鲁棒性，即无需任何耗散即可稳定运行。熵稳定通量足以控制地转湍流产生的网格尺度噪声，无需额外的人工稳定化处理。