We describe a mesh-free three-dimensional numerical scheme for solving the incompressible semi-geostrophic equations based on semi-discrete optimal transport techniques. These results generalise previous two-dimensional implementations. The optimal transport methods we adopt are known for their structural preservation and energy conservation qualities and achieve an excellent level of efficiency and numerical energy-conservation. We use this scheme to generate numerical simulations of an important cyclone benchmark problem. To our knowledge, this is the first fully three-dimensional simulation of the semi-geostrophic equations, evidencing semi-discrete optimal transport as a novel, robust numerical tool for meteorological and oceanographic modelling.

翻译：本文描述了一种基于半离散最优输运技术的无网格三维数值方案，用于求解不可压缩半地转方程。该结果推广了先前的二维实现。我们所采用的最优输运方法以其结构保持和能量守恒特性而闻名，并实现了卓越的效率和数值能量守恒水平。我们利用该方案对一个重要的气旋基准问题进行了数值模拟。据我们所知，这是首次对半地转方程进行完全三维的模拟，证明了半离散最优输运是一种新颖、稳健的用于气象学和海洋学建模的数值工具。