Structure-preserving numerical methods for global geostrophic turbulence
Arnout Franken is a PhD student in the Department of Multiscale Modeling and Simulation. (Co)Promotors are prof.dr.ir. B.J. Geurts and dr.ir. E. Luesink from the Faculty of Electrical Engineering, Mathematics and Computer Science.
This thesis describes the development of structure-preserving numerical methods for geostrophic flows on the sphere. A geometric derivation of globally defined geostrophic equations reveals their Lie-Poisson structure. A subsequent application of Lie-Poisson discretization results in energy and enstrophy-conserving methods that additionally preserve many higher-order Casimir invariants. The developed methods are used to study geostrophic turbulence in various realistic settings, both in single-layer and multi-layer models.
Chapter 2 introduces the global barotropic quasi-geostrophic (QG) equation as an approximation to the shallow water equations on the sphere. This equation serves as a model for large-scale atmospheric dynamics, generalizing the ß-plane barotropic models from a tangent plane approximation to application on the full sphere. The equation reveals the conservation of any integrated function of potential vorticity (PV), called Casimir invariants. Using the Zeitlin discretization method, a finite-dimensional matrix evolution equation is derived that conserves the numerical representation of the first N monomial Casimirs. An 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 the kinetic energy spectra.
In Chapter 3, a detailed mathematical analysis of the barotropic global QG equation is presented. Starting from a Lagrangian description of the shallow water equations on the sphere, the relevant scaling parameters for geophysical applications are identified. An asymptotic expansion in terms of the Rossby number leads to the identification of a Lagrangian invariant, which is recognized as the potential vorticity of the QG equation. This allowed for the construction of the Lagrangian and Hamiltonian formulations of QG, 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 the preservation of Casimirs and near-conservation of the Hamiltonian.
Chapter 4 extends the numerical investigation of geostrophic turbulence, focusing on the formation of large zonal jets. Simulations were performed to determine the critical latitude for jet formation, which depends on the Rossby number and Lamb parameter. The critical latitude forms a lateral boundary beyond which no jets can form. The results show a confirmation of theoretical estimates for the critical latitude in the most common parameter regimes for geostrophic flows. 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 zonal jets gradually decrease towards the poles.
Chapter 5 introduces the multi-layer quasi-geostrophic equations for global simulations. Starting from the global Boussinesq Primitive Equations, the velocity field is decomposed into non-divergent and divergent parts to derive the stratified QG equation on the sphere. By decomposing the flow domain into horizontal surfaces, the global multi-layer QG equation is derived, extending earlier ß-plane models. A numerical method was then constructed based on the structure-preserving discretization introduced in Chapter 2, leading to a Casimir-preserving isospectral integrator. The capabilities of this method were shown through two simulations, paving the way for further quantitative research on baroclinic effects in global geostrophic flows.
This thesis advances the numerical modelling of geostrophic flows on the sphere through the development of structure-preserving methods. The global barotropic and multi-layer quasi-geostrophic models provide new insights into the dynamics of geostrophic turbulence and zonal jet formation. The developed numerical methods are effective tools for simulating large-scale atmospheric phenomena, contributing to a deeper understanding of the interplay between geostrophic balance, turbulence, and stratification on a global scale. Future research can expand on these methods to investigate more complex and realistic geophysical scenarios, improving predictive capabilities in atmospheric and oceanic sciences.
