Quasi-geostrophic shallow-water vortex–patch equilibria and their stability

Abstract

We examine the equilibrium form, properties, stability and nonlinear evolution of steadily-rotating simply-connected vortex patches in the single-layer quasi-geostrophic model of geophysical fluid dynamics. This model, valid for rotating shallow-water flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L D called the “Rossby deformation length” relating the strength of the stratification to that of the background rotation. Here, we generate steadily-rotating vortex equilibria for a wide range of γ = L/L D , where L is the typical horizontal length scale of the vortex. We vary both γ (over the range 0.02 ≤ γ ≤ 10) and the vortex aspect ratio λ (over the range 0 < λ < 1). We find two modes of instability arising at sufficiently small aspect ratio λ < λ c (γ): an asymmetric (dominantly wave 3) mode at small γ (or large L D ) and a symmetric (dominantly wave 4) mode at large γ (or small L D ). At marginal stability, the asymmetric mode dominates for γ ≲ 3, while the symmetric mode dominates for γ ≳ 3. The nonlinear evolution of weakly-perturbed unstable equilibria results in major structural changes, in most cases producing two dominant vortex patches and thin, quasi-passive filaments. Overall, the nonlinear evolution can be classified into three principal types: (1) vacillations for a limited range of aspect ratios λ when 5 ≤ γ ≤ 6, (2) filamentation and a single-dominant vortex for γ ≲ 1, and (3) vortex splitting – asymmetric for 1 ≲ γ ≲ 4 and symmetric for γ ≳ 4.