Instabilities of two-dimensional inviscid compressible vortices

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Abstract
The linear stability and subsequent nonlinear evolution and acoustic radiation of a planar inviscid compressible vortex is examined. Linear-stability analysis shows that vortices with smoother vorticity profiles than the Rankine vortex considered by Broadbent & Moore (1979) are also unstable. However, only neutrally stable waves are found for a Gaussian vorticity profile. The effects of entropy gradient are investigated and for the particular entropy profile chosen, positive average entropy gradient in the vortex core is destabilizing while the opposite is true for negative average entropy gradient.The linear initial-value problem is studied by finite-difference methods. It is found that these methods are capable of accurately computing the frequencies and weak growth rates of the normal modes. When the initial condition consists of random perturbations, the long-time behaviour is found to correspond to the most unstable normal mode in all cases. In particular, the Gaussian vortex has no algebraically growing modes. This procedure also reveals the existence of weakly decaying and neutrally stable waves rotating in the direction opposite to the vortex core, which were not observed previously.The nonlinear development of an elliptic-mode perturbation is studied by numerical solution of the Euler equations. The vortex elongates and forms shocklets; eventually, the core splits into two corotating vortices. The individual vortices then gradually move away from each other while their rate of rotation about their mid-point slowly decreases. The acoustic flux reaches a maximum at the time of fission and decreases as the vortices move apart.

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