Curvature and stability of quasi-geostrophic motion

TL;DR

This paper investigates the curvature of the quantomorphism group and its central extension, deriving formulas using spherical harmonics, and examines how physical parameters influence the stability of quasi-geostrophic flows on a sphere.

Contribution

It provides a new formula for the curvature of the $L^2$ metric on the central extension of the quantomorphism group and analyzes stability effects of physical parameters on quasi-geostrophic motions.

Findings

Derived a curvature formula for the central extension of the quantomorphism group.

Calculated sectional curvatures for specific flow planes including the tradewind current.

Highlighted the influence of Rossby and Froude numbers on flow stability and weather prediction errors.

Abstract

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.