  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.
