Optimal Transport Techniques for Atmospheric Dynamics

This thesis develops a comprehensive theoretical and numerical framework for the semi-geostrophic equations of atmospheric dynamics, grounded in the theory of semi-discrete optimal transport. First, we present a novel, mesh-free, 3D numerical scheme for the incompressible semi-geostrophic equations. This method is structurally energy-conserving, enabling robust, long-term simulations, as demonstrated by the first fully 3D simulation of a twin cyclone using a semi-discrete optimal transport approach. Second, we establish a major theoretical result: the global-in-time existence of weak solutions for the compressible semi-geostrophic equations with compactly supported, measure-valued initial data. This significantly generalizes previous results by leveraging a particle discretization strategy and a dual formulation of the underlying energy minimization problem. Finally, we provide a rigorous justification for the frameworks applicability to physically relevant settings by verifying the necessary geometric conditions for rectangular domains, which are not covered by standard c-convexity assumptions. As a first step towards simulating the compressible system, the final chapter derives a 2D slice model and its particle-based discretization, establishing the foundation for future 2D and 3D compressible simulations.Together, these contributions provide a unified and powerful approach for the analysis and simulation of large-scale atmospheric flows.

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