Casimir preserving numerical method for global multilayer geostrophic turbulence

TL;DR

This paper introduces a structure-preserving numerical method for multi-layer quasi-geostrophic models on the sphere, enabling accurate long-term simulations of geophysical turbulence by conserving physical invariants.

Contribution

It extends Lie-Poisson discretization techniques to multi-layer QG systems on the globe, ensuring structure preservation and robustness for long-term geophysical flow simulations.

Findings

The method accurately conserves energy and enstrophy over long simulations.

Benchmark tests show robustness and structure-preserving properties.

Results improve understanding of high-order conserved quantities in geophysical flows.

Abstract

Accurate long-term predictions of large-scale flow features on planets are crucial for understanding global atmospheric and oceanic systems, necessitating the development of numerical methods that can preserve essential physical structures over extended simulation periods without excessive computational costs. Recent advancements in the study of global single-layer barotropic models have led to novel numerical methods based on Lie-Poisson discretization that preserve energy, enstrophy and higher-order moments of potential vorticity. This work extends this approach to more complex stratified quasi-geostrophic (QG) systems on the sphere. In this work, we present a formulation of the multi-layer QG equations on the full globe. This allows for extending the Lie-Poisson discretization to multi-layer QG models, ensuring consistency with the underlying structure and enabling long-term simulations without additional regularization. The numerical method is benchmarked through simulations of forced geostrophic turbulence and the long-term behaviour of unforced multi-layered systems. These results demonstrate the structure-preserving properties and robustness of the proposed numerical method, paving the way for a better understanding of the role of high-order conserved quantities in large-scale geophysical flow dynamics.