Cullen's Stability Principle and Weak Solutions of the Free-surface Semi-geostrophic Equations

TL;DR

This paper proves the global existence of weak solutions for the semi-geostrophic equations with a free boundary, using Cullen's Stability Principle, optimal transport, and Hamiltonian ODE analysis.

Contribution

It provides a rigorous mathematical formulation of Cullen's Stability Principle and establishes the existence of weak solutions in three-dimensional free-boundary semi-geostrophic flows.

Findings

Proved global existence of weak solutions.

Formulated Cullen's Stability Principle rigorously.

Applied optimal transport and Hamiltonian ODE methods.

Abstract

The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimisation argument originally inspired by Cullen's Stability Principle, uses optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of Cullen's Stability Principle in a rigorous mathematical framework.