Quasi-geostrophic shallow-water doubly-connected vortex equilibria and their stability

Hanna Plotka*, David Gerard Dritschel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


We examine the form, properties, stability and evolution of doubly-connected (two-vortex) relative equilibria in the single-layer ƒ-plane quasi-geostrophic shallow-water model of geophysical fluid dynamics. Three parameters completely describe families of equilibria in this system: the ratio γ =L/LD between the horizontal size of the vortices and the Rossby deformation length; the area ratio α of the smaller to the larger vortex; and the minimum distance δ between the two vortices. We vary 0 < γ ≤ 10 and 0.1 ≤ α ≤ 1.0, determining the boundary of stability δ = δC(γ,α). We also examine the nonlinear development of the instabilities and the transitions to other near-equilibrium configurations. Two modes of instability occur when δ < δC: a small -γ asymmetric (wave 3) mode, which is absent for α ≳ 0.6; and a large -γ mode. In general, major structural changes take place during the nonlinear evolution of the vortices, which near δC may be classified as follows: (i) vacillations about equilibrium for γ ≳ 2.5; (ii) partial straining out, associated with the small -γ mode, where either one or both of the vortices get smaller for γ ≲ 2.5 and α ≲ 0.6; (iii) partial merger, occurring at the transition region between the two modes of instability, where one of the vortices gets bigger, and (iv) complete merger, associated with the large-γ mode. We also find that although conservative inviscid transitions to equilibria with the same energy, angular momentum and circulation are possible, they are not the preferred evolutionary path.

Original languageEnglish
Pages (from-to)40-68
Number of pages29
JournalJournal of Fluid Mechanics
Early online date16 Apr 2013
Publication statusPublished - May 2013


  • Contour dynamics
  • Rotating flows
  • Vortex dynamics
  • V-states
  • 2-dimensional vortex
  • Uniform vortices
  • 2 dimensions
  • Numerical algorithms
  • Coherent structures
  • Euler equations
  • Merger
  • Flows


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