Viscous ring modes in vortices with axial jet

Abstract

In this paper, we show the existence of new families of linear eigenmodes in vortices with axial jet. These modes are viscous in nature and concentrated in a ring around the vortex at the critical radial location rc > 0 where \({m\Omega '_c + kW'_c=0}\) where \({\Omega_c'}\) and \({W_c'}\) are the radial derivative at rc of the angular and axial velocity of the vortex. Using a large Reynolds-number asymptotic approach for an arbitrary axisymmetrical vortex with axial flow, both the complex frequency and the spatial structure of the eigenmodes are obtained for any azimuthal and axial wave number. The asymptotic predictions are compared to numerical results for the q-vortex and a good agreement is demonstrated. We show that for sufficiently large Reynolds numbers, a necessary and sufficient condition of instability of viscous ring modes is that there exists a location rc where \({\Omega_c\Omega_c'[r_c\Omega_c'(2\Omega_c+r_c\Omega'_c)+(W_c')^2]<0}\) and \({W_c'\neq0}\) , which also corresponds to the condition of inviscid instability obtained by Leibovich and Stewartson (J Fluid Mech 126:335–356, 1983).