Curvature and stability of quasi-geostrophic motion

This paper outlines the study of the curvature of the quantomorphism group and its central extension, as well as the quasi-geostrophic equation. By utilizing spherical harmonics and structure constants, a formula for computing the curvature of the $L^2$ metric on the central extension $\widehat{\text{cg}}=\text{cg}{⋉}_{\mathrm{\Omega }}\mathbb{R}$ is derived, where $\text{cg}$ represents the Lie algebra of ${\text{cD}}_{\mu }^s({\mathbb{S}}^2)$. The sectional curvatures of the planes containing $Y_{10}$ and the tradewind current are calculated as special cases. The impact of the Rossby and Froude numbers, as well as the Coriolis effect, on the (exponential) stability of these quasi-geostrophic motions is highlighted. Finally, a lower bound for weather prediction error in a simplified model governed by the tradewind current and the Coriolis effect on a rotating sphere is suggested.