We develop a novel discontinuous Galerkin method for solving the rotating thermal shallow water equations (TRSW) on a curvilinear mesh. Our method is provably entropy stable, conserves mass, buoyancy and vorticity, while also semi-discretely conserving energy. This is achieved by using novel numerical fluxes and splitting the pressure and convection operators. We implement our method on a cubed sphere mesh and numerically verify our theoretical results. Our 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.

翻译：我们提出了一种新颖的间断伽辽金方法，用于在曲线网格上求解旋转热浅水方程（TRSW）。该方法被证明具有熵稳定性，能够守恒质量、浮力及涡量，同时半离散地守恒能量。这一特性通过采用新型数值通量并分离压力项与对流算子实现。我们在立方球网格上实现了该方法，并通过数值实验验证了理论结果。实验表明，该方法在充分发展的湍流状态下具有鲁棒性，可在无需任何耗散的情况下稳定运行。熵稳定通量足以控制地转湍流产生的网格尺度噪声，从而无需引入人工稳定化处理。