Instability of Planetary Flows using Riemann Curvature: a Numerical Study

TL;DR

This paper numerically investigates the instability of ideal planetary zonal flows on a sphere using Riemann curvature, revealing stability conditions, the influence of planetary rotation, and the scaling behavior of perturbations.

Contribution

It applies Riemann curvature-based instability criteria to planetary flows, providing new insights into their stability properties and the effects of planetary rotation.

Findings

Zonal flows are unstable for most perturbations except near resonance.

Sectional curvature scales with the ratio of perturbation wave numbers.

Planetary rotation generally stabilizes the flows.

Abstract

The instability of ideal non-divergent zonal flows on the sphere is determined numerically by the instability criterion of Arnol'd (1966) for the sectional curvature. Zonal flows are unstable for all perturbations besides for a small set which are in approximate resonance. The sectional curvature scales with $m/\ell$ for large total and zonal wave numbers $\ell$ and $m$ of the perturbations. The planetary rotation is stable and the presence of rotation reduces the instability of perturbations.