On the Stability of Vortices in an Ideal Gas

Abstract

Two distinct mechanisms of three-dimensional instability in compressible planar vortices in an ideal gas are presented. Both mechanisms have been obtained with the geometrical optics (WKB) stability theory which consists in studying the evolution of short-wavelength disturbances localized along the trajectories of the vortex. The first one corresponds to parametric resonances arising when a vortex is periodically compressed; the resulting instabilities are localized in the core of the vortex. On the contrary, in the second case, which corresponds to the generalization to compressible flows of the Rayleigh stability criterion for centrifugal instability, the growing perturbation surrounds the vortex at a given radius. In the latter case, the structure of the corresponding discrete eigenmodes may be described exactly, thus complementing and improving the WKB theory.