Inserted: 9 sep 2020
Last Updated: 9 sep 2020
Using semi-discrete optimal transport, we construct 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 of an existing result is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations, and it naturally gives rise to an effective numerical method. We show that discrete solutions of SG in geostrophic coordinates may be characterised by trajectories satisfying an ordinary differential equation, and we derive explicit formulas for two such solutions. To illustrate the utility of our method, we present 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.