Monge–Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem

We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge–Ampére method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis forces. This method can be used to resolve evolving small scale features in the flow. In this paper the method is then applied to the computation of the solution to the Eady problem which is observed to develop large gradients in a finite time. The moving mesh method is shown to work and be stable, and to give significantly better resolution of the evolving singularity than a fixed, uniform mesh.