Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

The well-known Rayleigh criterion is a necessary and sufficient condition for inviscid centrifugal instability of axisymmetric perturbations. We have generalized this criterion to disturbances of any azimuthal wavenumber m by means of large-axial-wavenumber WKB asymptotics. A sufficient condition for a free axisymmetric vortex with angular velocity Ω(r) to be unstable to a three-dimensional perturbation of azimuthal wavenumber m is that the real part of the growth rate \[σ (r) =-{ i}mΩ(r)+√-φ(r)\] is positive at the complex radius r=r₀ where ∂ σ (r)/∂ r=0 , i.e. \[φ'(r_0) =-2{ i}mΩ'(r_0)√-φ(r_0),\] where φ=(1/r³)∂r⁴Ω²/∂ r is the Rayleigh discriminant, provided that some a posteriori checks are satisfied. The application of this new criterion to various classes of vortex profiles shows that the growth rate of non-axisymmetric disturbances decreases as m increases until a cutoff is reached. The criterion is in excellent agreement with numerical stability analyses of the Carton & McWilliams (1989) vortices and allows one to analyse the competition between the centrifugal instability and the shear instability. The generalized criterion is also valid for a vertical vortex in a stably stratified and rotating fluid, except that ϕ becomes φ=(1/r³)∂r⁴(Ω+Ωb)²/∂ r , where Ωb is the background rotation about the vertical axis. The stratification is found to have no effect. For the Taylor–Couette flow between two coaxial cylinders, the same criterion applies except that r₀ is real and equal to the inner cylinder radius. In sharp contrast, the maximum growth rate of non-axisymmetric disturbances is then independent of m.