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

TL;DR

This work delivers the first mesh-free 3D numerical scheme for incompressible semi-geostrophic flows based on a semi-discrete optimal transport framework, enabling fully 3D simulations of cyclone-and-front dynamics while preserving energy. By coupling Laguerre-cell-based OT with a 4th-order ODE solver (RK4), the method achieves a balance between computational efficiency and energy conservation, and demonstrates long-time evolutions (up to 25 days) of a physically motivated cyclone initial condition. The 2D benchmarks validate the code against established references, and the 3D results showcase realistic cyclone development, front formation, and the influential role of horizontal shear, highlighting the approach as a robust tool for meteorological and oceanographic modelling. The work combines rigorous OT theory, a damped Newton solver, and careful boundary-aware initial-condition construction to extend SG studies into fully three-dimensional, energy-conserving simulations with potential for long-term diagnostics and applications.

Abstract

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.

Figures (10)

• Figure 1: The plot shows the evolution of the relative error in the total energy as defined in Eq. \ref{['eq:errordefn']}. The total energy fluctuates about $2.415\text{e}10$.
• Figure 2: In the first row we display the evolution of the perturbation of the meridional velocity field in the physical space ($\mathcal{X}$). In the second row we display the evolution of the positions of the geostrophic particles (in $\mathcal{Y}$). Finally, in the third row we display the evolution of the perturbation of the temperature field in physical space. Over the course of 10 days we observe the formation and evolution of a weather front.
• Figure 3: A comparison between AB2, Heun, and RK4 when coupled to an optimal transport solver. These two plots demonstrate the trade off between run time, time step size, and maximum relative error in the conservation of the energy. These plots support the idea that a balance can be struck between runtime, step size, and maximum relative error if one wants to run simulations in a "reasonable" amount of time.
• Figure 4: Log-log plot of the change in the Wasserstein-2 error at day 2 with respect to the change in the number of particles (in blue). In orange is a plot of the theoretical best decrease in the discretisation error with respect to the number of particles.
• Figure 5: Log-log plot of the change in the Wasserstein-2 error at day 2 with respect to the change in the size of the timestep (in blue). In orange is a plot of the theoretical best decrease in the error with respect to the timestep size for Runga-Kutta 4.

Theorems & Definitions (1)

• Remark 1.1