Centrifugal instability in non-axisymmetric vortices

We study the centrifugal instability of non-axisymmetric vortices in the presence of an axial flow (w) and a background rotation (Ωz) using the local stability approach. Analytically solving the local stability equations for an axisymmetric vortex with w and Ωz, growth rates for wave vectors that are periodic upon evolution around a closed streamline are calculated. The resulting sufficient criterion for centrifugal instability in an axisymmetric vortex is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities and their derivatives with respect to the streamfunction on a streamline. The new criterion for non-axisymmetric vortices, which converges to the exact criterion of Bayly (Phys. Fluids, vol. 31, 1988, pp. 56-64) in the absence of background rotation and axial flow, is validated by comparisons with numerically calculated growth rates for two different anticyclonic vortices: the Stuart vortex (specified by the concentration parameter ρ, 0 < ρ ≤ 1) and the Taylor-Green vortex (specified by the aspect ratio E, 0 < E ≤ 1). With no axial velocity and finite background rotation, the criterion predicts a lower and an upper threshold of |Ωz| between which centrifugal instability is present. We further demonstrate that the criterion represents an improvement over the criterion of Sipp & Jacquin (Phys. Fluids, vol. 12, 2000, pp. 1740-1748). Finally, in the presence of both axial velocity and background rotation, the criterion is shown to be accurate for large enough ρ and E.