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Semi-discrete optimal transport and weather modelling
(Université Herriott-Watt, Edimbourg)
I will present an application of semi-discrete optimal transport theory to simplified models of large-scale rotational flows (weather). In particular, I will discuss the 3D semi-geostrophic equation and the 2D Eady model.
The 3D semi-geostrophic equation is used by researchers at the Met Office in the UK to diagnose problems in simulations of more complicated weather models. It has also attracted a lot of attention in the applied analysis community, e.g., Alessio Figalli's work on the semi-geostrophic equation is listed in his Fields Medal citation. In this talk I will discuss the semi-geostrophic equation in geostrophic coordinates (SG), which is a nonlocal transport equation, where the transport velocity is defined via an optimal transport problem, or equivalently a Monge-Ampère equation. Using recent results from semi-discrete optimal transport theory, we give a new proof of the existence of weak solutions of SG. The proof is constructive and leads to an efficient numerical method.
We illustrate this numerical method by using it to solve the closely related 2D Eady model. By quantizing the initial data, the nonlocal transport equation is approximated by a nonlocal interacting particle system, which can be solved using a standard ODE solver coupled with a semi-discrete optimal transport solver to evaluate the velocity field in the ODE at every time step. Our simulations agree well with simulations of the full incompressible Euler-Boussinesq system, and they demonstrate that the Eady model is capable of predicting weather fronts. The problem of approximating the initial data for the Eady model by a discrete density with the same energy leads to an interesting constrained quantization problem.
This is joint work with Charlie Egan and Beatrice Pelloni (Heriot-Watt University and the Maxwell Institute for Mathematical Sciences), Mark Wilkinson (Nottingham Trent University), Steven Roper (University of Glasgow), Colin Cotter (Imperial College London) and Mike Cullen (Met Office).