Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere

TL;DR

This paper derives a geometric, structure-preserving model of quasi-geostrophic flow on the sphere, enabling stable long-term simulations that capture key features like jet formation.

Contribution

It introduces a novel geometric derivation of the quasi-geostrophic equations on the sphere and develops structure-preserving numerical methods for simulation.

Findings

Preservation of Casimir invariants during simulations

Stable long-term simulation of quasi-geostrophic flow

Emergence of longitudinal jets on the sphere

Abstract

We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation.