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：我们针对旋转球面浅水方程提出了一种新型高效的向量不变形式间断伽辽金谱元方法（DG-SEM）。理论证明该方法在一般曲线网格上具有能量稳定性，并能离散守恒质量、涡度及线性地转平衡。这些理论成果得益于我们提出的适用于向量不变形式浅水方程的新型熵稳定数值通量。通过在立方球网格上的数值实验验证了上述结论。此外，我们证明了该方法具有鲁棒性——即无需添加耗散项即可稳定运行。熵稳定通量足以控制地转湍流产生的网格尺度噪声，而无需引入人工稳定化处理。