Structure-preserving numerical methods for global geostrophic turbulence

This thesis presents the development of structure-preserving numerical methods for geostrophic flows on the sphere. A geometric derivation of the geostrophic equations on a sphere reveals a Lie-Poisson structure, enabling discretization techniques that conserve energy, enstrophy, and higher-order Casimir invariants. These methods are applied to study geostrophic turbulence in both single-layer and multi-layer models, advancing simulations of large-scale oceanic and atmospheric dynamics.
The global barotropic quasi-geostrophic (QG) equation is introduced as an approximation to shallow water dynamics on the sphere, generalizing ß-plane models. This equation conserves any integrated function of potential vorticity (PV), or Casimir invariants. Using the Zeitlin discretization, a finite-dimensional matrix evolution equation is constructed that maintains the first 𝑁 monomial Casimirs. High-resolution simulations reveal the effects of geostrophy on zonal jet formation, showing attenuation in polar regions and anisotropy in kinetic energy spectra.
The QG equation’s Hamiltonian and Lagrangian formulations reveal the existence of Casimir functionals, validated through simulations of unforced, undissipated geostrophic turbulence that develop stable zonal jets. An extension of these simulations explores jet formation dependence on the Rossby number and Lamb parameter, identifying a critical latitude beyond which jets do not form. Under weak rotation and strong stratification, the amplitude and width of jets decrease towards the poles rather than showing a clear boundary.
Additionally, a multi-layer quasi-geostrophic model is derived from the Boussinesq Primitive Equations, extending previous ß-plane models to the full sphere. This approach introduces a Casimir-preserving isospectral integrator for stratified flows, with simulations demonstrating its utility for studying baroclinic effects in global geostrophic turbulence.
These structure-preserving methods contribute to the numerical modeling of geostrophic flows, contributing insights into the dynamics of zonal jet formation and turbulence on a global scale. They provide robust tools for simulating atmospheric phenomena and open pathways for future research on complex geophysical dynamics in atmospheric and oceanic sciences.