Stability of Stretched Vortices in a Strain Field

Abstract

The linear stability of an arbitrarily stretched Gaussian vortex is addressed when the vortex of circulation Г and radius δ is subjected to an additional strain field of rate s perpendicular to the vorticity axis. The resulting nonaxisymmetric vortex is analysed in the limit of large Reynolds number R _Г = Г/υ and small strain s ≪ Г/δ ². The helical Kelvin wave resonance mechanism described by Moore & Saffman (1975) is shown to be active. Coupled equations for the Kelvin waves amplitudes are obtained, describing the resonance in presence of diffusion and stretching. The main effect of diffusion and stretching is shown to be stabilizing by limiting in time the resonance. The maximum gain of amplitude of the resonant Kelvin waves is computed in terms of the rescaled strain and stretching rates $s* = s\sqrt {{R_\Gamma }} {\delta ^2}/\Gamma$ and γ* = γR _Г δ ²/Г. The result leads to explicit sufficient instability conditions which could explain various dynamical behaviors of vortex filaments in turbulence.