Acoustic destablilization of vortices

A line vortex which has uniform vorticity 2Ω0 in its core is subjected to a small two-dimensional disturbance whose dependence on polar angle is eimθ. The stability is examined according to the equations of compressible, inviscid flow in a homentropic medium. The boundary condition at infinity is that of outgoing acoustic waves, and it is found that this capacity to radiate leads to a slow instability by comparison with the corresponding incompressible vortex which is stable. Numerical eigenvalues are computed as functions of the mode number m and the Mach number M based on the circumferential speed of the vortex. These are compared with an asymptotic analysis for the m = 2 mode at low Mach number in which it is found that the growth rate is (π/ 32) M4Ω0 in good agreement with the numerical results.