Stability of Long's vortex at large flow force

Abstract

Long's self-similar vortex is known to have two solutions for each supercritical value of the flow force. Each of those solutions is shown here to have a double structure if the flow force is large. We then investigate the inertial instabilities of one of those large-flow-force limit solutions, and find them to be related to the instabilities of the Bickley jet in one régime. However, the swirl in the vortex becomes important for long waves, very strongly modifying the sinuous and varicose Bickley modes. We find in particular that the asymptotic results obtained agree well with our numerical solutions for the sinuous mode, but not for the varicose mode, the difficulty in the latter case being apparently due to mode jumping. The asymptotics show a varicose long-wave neutral mode for positive azimuthal wavenumber, and two such modes for negative wavenumbers. The upper neutral sinuous mode occurs at much, larger wavenumber than in the Bickley case, and its structure is also presented. The study overall is aimed at providing a basis for the investigation of strongly nonlinear effects.