Casimir preserving numerical method for global multi-layer quasi-geostrophic turbulence

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 paper extends this approach to more complex stratified quasi-geostrophic (QG) systems on the sphere. These multi-layered models provide a more comprehensive representation of atmospheric and oceanic dynamics by accounting for vertical variations in density, pressure, and velocity fields. 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.