We introduce a structure preserving discretization of stochastic rotating shallow water equations, stabilized with an energy conserving Casimir (i.e. potential enstrophy) dissipation. A stabilization of a stochastic scheme is usually required as, by modeling subgrid effects via stochastic processes, small scale features are injected which often lead to noise on the grid scale and numerical instability. Such noise is usually dissipated with a standard diffusion via a Laplacian which necessarily also dissipates energy. In this contribution we study the effects of using an energy preserving selective Casimir dissipation method compared to diffusion via a Laplacian. For both, we analyze stability and accuracy of the stochastic scheme. The results for a test case of a barotropically unstable jet show that Casimir dissipation allows for stable simulations that preserve energy and exhibit more dynamics than comparable runs that use a Laplacian.

翻译：我们提出了一种保结构的随机旋转浅水方程离散化方法，并通过能量守恒的Casimir（即势涡度）耗散进行稳定。由于通过随机过程模拟亚网格效应会注入小尺度特征，通常导致网格尺度噪声和数值不稳定性，因此随机方案的稳定化通常是必要的。这类噪声通常通过拉普拉斯算子的标准扩散来耗散，但必然也会消耗能量。本研究对比分析了采用能量守恒的选择性Casimir耗散方法与基于拉普拉斯算子的扩散方案的效果。针对这两种方法，我们分析了随机方案的稳定性和精度。在正压不稳定急流测试案例中，结果表明Casimir耗散既能实现稳定模拟，又能保持能量，并且比使用拉普拉斯算子的对比实验展现出更丰富的动力学特性。