Structure-preserving numerical methods for global geostrophic turbulence

Arnout Franken is a PhD student in the Department of Multiscale Modeling. (Co) Promoters – Prof. dr.ir. B.J. Geurts et al. E. Lusink from the Faculty of Electrical Engineering, Mathematics and Computer Science.

This thesis describes the development of structure-preserving numerical methods for geostrophic flows on a sphere. The geometric derivation of the globally defined geostrophic equations shows their Lie-Poisson structure. Subsequent application of Lie-Poisson discretization leads to the creation of energy and enstrophy conservation methods that additionally preserve many higher-order Casimir invariants. The developed methods are used to study geostrophic turbulence under various realistic conditions, both in single-layer and multi-layer models.

Chapter 2 introduces the global barotropic quasigeostrophic equation (QG) as an approximation to the shallow water equations on a sphere. This equation serves as a model of large-scale atmospheric dynamics, generalizing barotropic β-plane models from the tangent plane approximation to application to the entire sphere. The equation reveals the conservation of any integrated potential vorticity (PV) function, called Casimir invariants. Using the Tseitlin discretization method, a finite-dimensional matrix evolution equation is obtained that preserves the numerical representation of the first N Casimir monomials. The efficient solver is used for high-resolution simulations, demonstrating the attenuation of zonal jets in polar regions due to geostrophic effects and highlighting the anisotropic nature of jet formation in kinetic energy spectra.

IN Chapter 3, a detailed mathematical analysis of the barotropic global KG equation is presented. Based on the Lagrangian description of the shallow water equations on a sphere, appropriate scaling parameters for geophysical applications are determined. The asymptotic expansion in terms of the Rossby number leads to the identification of the Lagrangian invariant, which is recognized as the potential vorticity of the CG equation. This made it possible to construct Lagrangian and Hamiltonian CG formulations, revealing the existence of Casimir functionals. Numerical simulations of geostrophic turbulence in the absence of any forcing or dissipation show the development of stable zonal jets and confirm Casimir conservation and near-Hamiltonian conservation.

Chapter 4 extends the numerical study of geostrophic turbulence with a particular focus on the formation of large zonal jets. Simulations were carried out to determine the critical latitude of jet formation, which depends on the Rossby number and the Lamb parameter. The critical latitude forms a lateral boundary beyond which jets cannot form. The results show confirmation of theoretical estimates of the critical latitude in the most common regimes of geostrophic current parameters. However, the results also show that a clear critical latitude does not appear in weak rotation and strong stratification regimes; instead, the amplitude and width of the zonal jets gradually decrease toward the poles.

Chapter 5 presents multilayer quasi-geostrophic equations for global modeling. Starting from the global primitive Boussinesq equations, the velocity field is decomposed into nondivergent and divergent parts to obtain the stratified KG equation on the sphere. By decomposing the flow domain into horizontal surfaces, a global multilayer CG equation is obtained, extending earlier ß-plane models. A numerical method was then constructed based on the structure-preserving discretization presented in Chapter 2, resulting in the creation of a Casimir-preserving isospectral integrator. The capabilities of this method were demonstrated through two simulations, paving the way for further quantitative studies of baroclinic effects in global geostrophic flows.

This thesis advances the numerical modeling of geostrophic flows on a sphere through the development of structure-preserving methods. Global barotropic and multilayer quasi-geostrophic models provide new insights into geostrophic turbulence dynamics and zonal jet formation. The developed numerical methods are an effective tool for modeling large-scale atmospheric phenomena, contributing to a deeper understanding of the interaction of geostrophic balance, turbulence and stratification on a global scale. Future research could extend these methods to explore more complex and realistic geophysical scenarios, improving predictive capabilities in atmospheric and ocean sciences.