The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates

M J P Cullen; T Kuna; B Pelloni; M Wilkinson

doi:10.1098/rspa.2018.0787

The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates

Proc Math Phys Eng Sci. 2019 Sep;475(2229):20180787. doi: 10.1098/rspa.2018.0787. Epub 2019 Sep 4.

Authors

M J P Cullen¹, T Kuna², B Pelloni³, M Wilkinson³

Affiliations

¹ Met Office, Exeter, UK.
² Department of Mathematics and Statistics, University of Reading, Reading, UK.
³ Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University, Edinburgh, UK.

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 minimization argument originally inspired by the Stability Principle as studied by Cullen, Purser and others, uses optimal transport techniques 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 the Stability Principle in a rigorous mathematical framework.

Keywords: Wasserstein spaces; optimal transport; semi-geostrophic equations; stabilitycriteria.