Zeitlin Truncation of a Shallow Water Quasi-Geostrophic Model for Planetary Flow

In this work, we consider a Shallow-Water Quasi Geostrophic equation on the sphere, as a model for global large-scale atmospheric dynamics. This equation, previously studied by Verkley (2009, https://doi.org/10.1175/2008jas2837.1) and Schubert et al. (2009, https://doi.org/10.3894/james.2009.1.2), possesses a rich geometric structure, called Lie-Poisson, and admits an infinite number of conserved quantities, called Casimirs. In this paper, we develop a Casimir preserving numerical method for long-time simulations of this equation. The method develops in two steps: first, we construct an N-dimensional Lie-Poisson system that converges to the continuous one in the limit N → ∞; second, we integrate in time the finite-dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020, https://doi.org/10.1017/jfm.2019.944). We demonstrate the efficacy of this computational method by simulating a flow on the entire sphere for different values of the Lamb parameter. We particularly focus on rotation-induced effects, such as the formation of jets. In agreement with shallow water models of the atmosphere, we observe the formation of robust latitudinal jets and a decrease in the zonal wind amplitude with latitude. Furthermore, spectra of the kinetic energy are computed as a point of reference for future studies.