Analysis of large-scale atmospheric flows

The semi-geostrophic equations are an approximate model used to study the large-scale behaviour of the atmosphere, in particular the formation of atmospheric fronts . We extend the existing analysis of the semi-geostrophic equations to more physically appropriate cases than those previously considered. We prove rigorously the existence of weak Lagrangian solutions of the compressible semigeostrophic system with rigid boundaries, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of stable weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. We also consider the semi-geostrophic equations 'with a free surface boundary condition, which is more accurate than the rigid boundary condition for describing large-scale atmospheric flows. We prove that stable weak solutions of the simpler incompressible system exist in geostrophic coordinates. Finally, we extend this result to the more physically valid compressible form of the equations by rewriting the compressible semi-geostrophic system using pressure as a vertical coordinate. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measuresEThOS - Electronic Theses Online ServiceGBUnited Kingdo