Perturbed Rankine vortices in surface quasi-geostrophic dynamics

Abstract

An analytical dispersion relation is derived for linear perturbations to a Rankine vortex governed by surface quasi-geostrophic dynamics. Such a Rankine vortex is a circular region of uniform anomalous surface temperature evolving under quasi-geostrophic dynamics with uniform interior potential vorticity. The dispersion relation is analysed in detail and compared to the more familiar dispersion relation for a perturbed Rankine vortex governed by the Euler equations. The results are successfully verified against numerical simulations of the full equations. The dispersion relation is relevant to problems including wave propagation on surface temperature fronts and the stability of vortices in quasi-geostrophic turbulence.

Keywords:

• Surface quasi-geostrophic dynamics
• Vortices
• Temperature fronts
• Linear waves

Notes

†A pure contour dynamics method (see e.g. Zabusky et al. 1979, Pullen 1992) appears to be the most natural choice, but this suffers from the logarithmic singularity in the along-front velocity field.