  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.
