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A Semi-Discrete Optimal Transport Scheme for the Semi-Geostrophic Slice Compressible Model

Théo Lavier, Beatrice Pelloni

Abstract

We develop a semi-discrete optimal transport scheme for the compressible semi-geostrophic equations, a system that plays an important role in modelling large-scale atmospheric dynamics and frontogenesis. Unlike the incompressible case, the compressible equations involve variable density and internal energy, but can be recast into a variational framework that naturally couples the dynamics with an optimal transport formulation. This is done by a change to the so-called geostrophic coordinates, via a transformation inspired by the incompressible case. The discrete version of this variational formulation provides the basis for a numerical particle scheme. The implementation of this scheme presents considerable challenges, due to a non-quadratic cost function and parabolic $c$-Laguerre cells. To address these challenges, we use $c$-exponential charts to construct $c$-Laguerre tessellations efficiently, ensuring conservation of mass and energy while preserving key geometric structures. We analyse the scheme and validate its convergence through numerical experiments, including a single-seed benchmark and error analysis. This work provides a significant new generalisation of existing semi-discrete optimal transport techniques, offering a robust and structure-preserving tool for simulating realistic atmospheric flows.

A Semi-Discrete Optimal Transport Scheme for the Semi-Geostrophic Slice Compressible Model

Abstract

We develop a semi-discrete optimal transport scheme for the compressible semi-geostrophic equations, a system that plays an important role in modelling large-scale atmospheric dynamics and frontogenesis. Unlike the incompressible case, the compressible equations involve variable density and internal energy, but can be recast into a variational framework that naturally couples the dynamics with an optimal transport formulation. This is done by a change to the so-called geostrophic coordinates, via a transformation inspired by the incompressible case. The discrete version of this variational formulation provides the basis for a numerical particle scheme. The implementation of this scheme presents considerable challenges, due to a non-quadratic cost function and parabolic -Laguerre cells. To address these challenges, we use -exponential charts to construct -Laguerre tessellations efficiently, ensuring conservation of mass and energy while preserving key geometric structures. We analyse the scheme and validate its convergence through numerical experiments, including a single-seed benchmark and error analysis. This work provides a significant new generalisation of existing semi-discrete optimal transport techniques, offering a robust and structure-preserving tool for simulating realistic atmospheric flows.
Paper Structure (33 sections, 2 theorems, 112 equations, 6 figures, 1 table)

This paper contains 33 sections, 2 theorems, 112 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Outline of the Paper
  4. Slice Compressible Model
  5. Governing Equations and Physical Assumptions
  6. Vertical Slice and Semi-Geostrophic Approximation
  7. Formulation in geostrophic variables.
  8. The Source Measure $\sigma_t$.
  9. The Target Measure $\alpha_t$.
  10. The Energy Functional and the Optimal Transport Problem.
  11. Semi-discrete optimal transport.
  12. Numerical Method
  13. Optimal Transport Framework
  14. Geometry of the Cost Function
  15. Non-Dimensionalisation
  16. ...and 18 more sections

Key Result

Lemma 2.1

Assume at time $t=0$ that $\sigma_0$ is a probability measure. Then $\sigma_t$ is a probability measure for all $t$, i.e.

Figures (6)

  • Figure 1: The optimal weight $w_*$ derived from the integral, plotted as a function of the parameter $z_2$.
  • Figure 2: The optimal internal energy $E_I$ as a function of the parameter $z_2$.
  • Figure 3: Evolution of particle positions (column 1), velocity (column 2), temperature (column 3), and mass (column 4) over days 2, 4, 7, and 11. Unlike the reference solution Cotter:2025, the front exhibits horizontal drift and elongation due to the definition of $\Pi_0$.
  • Figure 4: Evolution of the relative error in total geostrophic energy (Eq. \ref{['eq:errordefn']}). The error remains bounded, demonstrating the stability of the symplectic-like optimal transport integration.
  • Figure 5: Timestep Error Convergence. Simulations investigating the impact of $h$ were done with $N=2592$. The scheme exhibits first-order convergence due to the non-smoothness of the internal energy field.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Lemma 2.1: $\sigma_t$ is a Probability Measure
  • proof
  • Definition 2.1: $c$-Laguerre tessellation
  • Definition 2.2: Centroid map
  • Lemma 3.1: Loeper's condition
  • proof