We describe a mesh-free three-dimensional (3D) numerical scheme for solving the incompressible semi-geostrophic equations, based on semi-discrete optimal transport techniques. These results generalise previous two-dimensional (2D) 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 benchmark problem. To our knowledge, this is the first fully 3D simulation of this particular cyclone, evidencing the model's applicability to atmospheric and oceanic phenomena and offering a novel, robust tool for meteorological and oceanographic modelling.

翻译：本文提出了一种基于半离散最优输运技术的无网格三维数值格式，用于求解不可压半地转方程。该结果推广了先前的二维实现方案。所采用的最优输运方法以其结构保持特性和能量守恒性质著称，在计算效率与数值能量守恒方面表现出色。我们应用该格式对重要基准问题进行了数值模拟。据我们所知，这是对该特定气旋现象的首次完整三维模拟，证明了该模型在大气和海洋现象中的适用性，并为气象学和海洋学建模提供了一种新颖且稳健的工具。