A new implementation of the geometric method for solving the Eady slice equations

• Published in: Journal of computational physics Vol. 469; p. 111542
• Main Authors: Egan, C.P., Bourne, D.P., Cotter, C.J., Cullen, M.J.P., Pelloni, B., Roper, S.M., Wilkinson, M.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 15.11.2022; Elsevier Science Ltd
• Subjects: Adaptive time-stepping; Approximation; Boussinesq equations; Computational physics; Discretization; Eady model; Frontogenesis; Geometric method; Semi-discrete optimal transport; Semi-geostrophic; Transport theory; Geometric method; Adaptive time-stepping; Eady model; Semi-discrete optimal transport; Semi-geostrophic; Frontogenesis
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111542

•The geometric method is recast in the language of semi-discrete optimal transport.•We develop a fast adaptive time-stepping algorithm.•Our algorithm uses the latest results from numerical optimal transport.•Numerical results validate the semi-geostrophic approximation. We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations, which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence 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.