A Semi-Discrete Optimal Transport Scheme for the 3D Incompressible Semi-Geostrophic Equations

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.

💡 Research Summary

The paper introduces a novel mesh‑free numerical scheme for solving the three‑dimensional incompressible semi‑geostrophic (SG) equations by exploiting semi‑discrete optimal transport (OT) techniques. The SG system, a second‑order reduction of the full Euler equations, is known for its ability to capture frontogenesis and cyclogenesis in large‑scale, rotation‑dominated flows. Traditional SG implementations have been limited to two dimensions because the change of variables between physical and geostrophic coordinates breaks down after a finite time in three dimensions. Brenier and Benamou’s reformulation of SG as a coupled optimal‑transport problem overcomes this limitation by representing the fluid configuration as a transport map from a continuous source measure (the normalized Lebesgue measure on the physical domain) to a discrete target measure consisting of N particles in geostrophic space.

In the semi‑discrete setting the target measure is (\alpha_N(t)=\frac{1}{N}\sum_{i=1}^N\delta_{z_i(t)}). The optimal transport map (T) is characterized by a Laguerre (or power) diagram ({L_i}{i=1}^N) whose cells satisfy (\int{L_i}dx=1/N). The ODE governing particle motion follows from the SG velocity law (v