We give an interpretation of the global shallow water quasi-geostrophic equations on the sphere \(\mathbb {S}^2\) as a geodesic equation on the central extension of the quantomorphism group on \(\mathbb {S}^3\) . 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.
