Theoretical analysis of the zigzag instability of a vertical co-rotating vortex pair in a strongly stratified fluid

Abstract

A long-wavelength stability analysis of two co-rotating Gaussian vertical vortices in an inviscid strongly stratified fluid is conducted for vortices separated by a large distance b compared to their radius a (b ≫ a). This analysis predicts and explains the zigzag instability found by a numerical stability analysis in a companion paper (Otheguy, Chomaz & Billant, J. Fluid. Mech. vol. 553, 2006, p. 253). The zigzag instability results from the coupling between the bending perturbations of each vortex and the external strain that one vortex induces on the other S = Γ/2 π b2, where Γ is the circulation of the vortices. The analysis predicts that the maximum growth rate of the instability is twice the strain S and that the most unstable vertical wavelength λ scales as the buoyancy length, defined by LB = Γ/πaN, multiplied by the ratio b/a, i.e. λ ∝ Fhb, where Fh = Γ/πa2N is the horizontal Froude number. The asymptotic results are in very good agreement with the numerical results.