Browsing by Author "Egan, Charlie"

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    Semi-discrete optimal transport methods for the semi-geostrophic equations
    (Heriot-Watt University, 2022-11) Egan, Charlie; Bourne, Associate Professor David; Pelloni, Professor Beatrice; Wilkinson, Mark
    We develop a framework for using semi-discrete optimal transport theory to rigorously study the semi-geostrophic equations, which model large scale atmospheric flows and frontogenesis. Our framework is based on the geometric method of Cullen and Purser (1984) – an energy-conserving Lagrangian discretisation of the semi-geostrophic equations. Within this framework, we give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations in geostrophic coordinates, obtaining improved time-regularity for a large class of discrete initial measures. Our proof is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. We give explicit examples of solutions to the discrete system and we demonstrate how they can be used to approximate Eulerian solutions of the semi-geostrophic equations. Our proof naturally gives rise to a new and efficient implementation of the geometric method, which we use to solve the semi-geostrophic Eady slice equations – a formal low Rossby number approximation of the Eady-Boussinesq vertical slice equations. Our implementation combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Since the geometric method is an energy-conserving discretisation, it is desirable to initialise this scheme with discrete approximations of a given initial condition that have a specified energy. We prove that this is possible for a wide class of initial conditions. Our numerical results support the conjecture that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero.