Zeitlin truncation of a Shallow Water Quasi-Geostrophic model for planetary flow

TL;DR

This paper introduces a Casimir-preserving numerical method for simulating the Shallow Water Quasi-Geostrophic equation on the sphere, capturing large-scale atmospheric dynamics and jet formation with long-term stability.

Contribution

It develops a Lie-Poisson structure-preserving numerical scheme and demonstrates its effectiveness in simulating planetary atmospheric flows.

Findings

Formation of robust latitudinal jets observed

Decrease in zonal wind amplitude with latitude

Spectral analysis of kinetic energy provided

Abstract

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) and Schubert et al. (2009), 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: firstly, we construct an N-dimensional Lie-Poisson system that converges to the continuous one in the limit $N \to \infty$; secondly, we integrate in time the finite-dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020). 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.