Three-dimensional instability of a multipolar vortex in a rotating flow

Abstract

In this paper, the elliptic instability is generalized to account for Coriolis effects and higher order symmetries. We consider, in a frame rotating at the angular frequency Ω, a stationary vortex which is described near its center r=0 by the stream function written in polar coordinates Ψ=−(r2/2)+p(rn/n)cos(nθ), where the integer n is the order of the azimuthal symmetry, and p is a small positive parameter which measures the strength of the nonaxisymmetric field. Based on the Lifschitz and Hameiri [Phys. Fluids A 3, 2644–2651 (1991)] theory, the local stability analysis of the streamline Ψ=−1/2 is performed in the limit of small p. As for the elliptic instability [Bayly, Phys. Rev. Lett. 57, 2160–2163 (1986)], the instability is shown to be due to a parametric resonance of inertial waves when the inclination angle ξ of their wave vector with respect to the rotation axis takes a particular value given by cos ξ=±4/(n(1+Ω)). An explicit formula for the maximum growth rate of the inertial wave is obtained for arbitrary ξ, Ω, and n. As an immediate consequence, it is shown that a vortex core of relative vorticity Wr (assumed positive) is locally unstable if Ω<−(1+n/4)Wr/2 or Ω>(−1+n/4−p(n−1)/2)Wr/2. The predictive power of the local theory is demonstrated on several vortex examples by comparing the local stability predictions with global stability results. For both the Kirchhoff vortex and Moore and Saffman vortex, it is shown how global stability results can be derived from the local stability analysis using the dispersion relation of normal (Kelvin) modes. These results are compared to those obtained by global methods and a surprisingly good agreement is demonstrated. The local results are also applied to rotating Stuart vortices and compared to available numerical data.