Geodesic interpretation of the global quasi-geostrophic equations
Klas Modin, Ali Suri
TL;DR
This work links the global SWQG equations on the sphere to geodesic flow on the central extension of the quantomorphism group on $\mathbb{S}^3$ by developing a weak Riemannian metric and a geometric Euler-Arnold framework. It proves local and global well-posedness for the associated geodesic equation and introduces a conserved quantity that drives a priori bounds, enabling global existence for Sobolev indices $s>2$. A central extension is employed to incorporate the Lamb parameter $\gamma$ and topographic effects, revealing that large $\gamma$ can induce positive sectional curvature along certain flows (notably the trade-wind current), hence conjugate points and stability features within the dynamics. The results illuminate how curvature and symmetry (via Hopf fibration, quantomorphisms, and central extensions) shape the well-posedness and stability of global QG dynamics, forging a rigorous bridge between geophysical fluid models and infinite-dimensional geometric analysis.
Abstract
We give an interpretation of the global shallow water quasi-geostrophic equations on the sphere $\Sph^2$ as a geodesic equation on the central extension of the quantomorphism group on $\Sph^3$. The study includes deriving the model as a geodesic equation for a weak Riemannian metric, demonstrating smooth dependence on the initial data, and establishing global-in-time existence and uniqueness of solutions. We also prove that the Lamb parameter in the model has a stabilizing effect on the dynamics: if it is large enough, the sectional curvature along the trade-wind current is positive, implying conjugate points.