Zeitlin truncation of a Shallow Water Quasi-Geostrophic model for planetary flow
Journal of Advances in Modeling Earth Systems(2023)
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 →∞; 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.
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Key words
geophysical fluid dynamics,turbulence,simulation
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