  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
$\hat{\cg}=\cg\ltimes_\Omega\mathbb{R}$ is derived, where $\cg$ represents the
Lie algebra of $\cD^s_\mu(\mathbb{S}^2)$. 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.

Differential Geometry