Journal Article

Zeitlin Truncation of a Shallow Water Quasi‐Geostrophic Model for Planetary Flow

Cifani, P., Franken, A. D., Caliaro, M., Geurts, B. J.

2024

Overview

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, https://doi.org/10.1175/2008jas2837.1) and Schubert et al. (2009, https://doi.org/10.3894/james.2009.1.2), 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: first, we construct an N‐dimensional Lie‐Poisson system that converges to the continuous one in the limit N → ∞; second, we integrate in time the finite‐dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020, https://doi.org/10.1017/jfm.2019.944). 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. Plain Language Summary We conducted a study on a model that represents the movements of planetary flows. This model has important physical and mathematical properties that are related to its long‐term behavior, which is essential for understanding geophysical turbulence. In this work, we developed a numerical method for simulation that preserves the key mathematical structure of the model through a two‐step process. We applied our method to simulate global atmospheric flow and investigate the impact of varying strengths of planetary rotation. Our findings demonstrate the expected formation of wind patterns known as zonal jets, where stronger winds occur near the equator and weaker winds near the poles. We also present energy spectra that illustrate the influence of planetary rotation on the transfer of turbulent energy, which aligns with existing theoretical predictions found in literature. These results highlight the potential of our numerical method for studying fundamental problems in geophysical fluid dynamics. Key Points We develop a numerical method preserving Casimirs to simulate balanced shallow water flow on the sphere We perform global high‐resolution simulations while accurately accounting for latitude‐dependent effects Our simulations show the formation of robust zonal jets and provide key insights into quasi‐geostrophic turbulence

• Publisher: John Wiley & Sons, Inc,American Geophysical Union (AGU)
• Subject: Atmospheric dynamics / Dynamical systems / Energy / Fluid dynamics / geophysical fluid dynamics / Kinetic energy / Numerical analysis / Shallow water / Simulation / Topography / turbulence / Velocity / Zonal winds