  We give a new and constructive proof of the existence of global-in-time weak
solutions of the 3-dimensional incompressible semi-geostrophic equations (SG)
in geostrophic coordinates, for arbitrary initial measures with compact
support. This new proof, based on semi-discrete optimal transport techniques,
works by characterising discrete solutions of SG in geostrophic coordinates in
terms of trajectories satisfying an ordinary differential equation. It is
advantageous in its simplicity and its explicit relation to Eulerian
coordinates through the use of Laguerre tessellations. Using our method, we
obtain improved time-regularity for a large class of discrete initial measures,
and we compute explicitly two discrete solutions. The method naturally gives
rise to an efficient numerical method, which we illustrate by presenting
simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates
generated using a numerical solver for the semi-discrete optimal transport
problem coupled with an ordinary differential equation solver.
