A new implementation of the geometric method for solving the Eady slice equations

TL;DR

This paper introduces a novel, energy-conserving implementation of the geometric method for solving the semi-geostrophic Eady slice equations, enhancing computational efficiency and enabling better comparison with the full Boussinesq model.

Contribution

It develops a fast, adaptive, energy-conserving particle method based on optimal transport theory for the Eady slice equations, improving numerical solutions and convergence analysis.

Findings

The new implementation is faster and more accurate.

It demonstrates convergence of Boussinesq solutions to semi-geostrophic solutions.

Provides a controlled comparison between models.

Abstract

We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero.