Three-dimensional stability of a Rankine vortex with radial density stratification

In this work, we present a 3D linear stability analysis of a radially stratified Rankine vortex of radius, r=a. The flow is assumed to be inviscid and the density distribution axisymmetric with a single density jump at radial location r=r_j. In an earlier study [1], it was shown that a light-cored vortex, i.e. ρ₁ < ρ₂, may become unstable while heavy-cored vortex, i.e. ρ₁ > ρ₂, may be stabilized. Instabilities were attributed to a linear wave-interaction mechanism between the density mode at r_j and a Kelvin mode at r=a. In this study, we extend the 2D stability results to include three-dimensional perturbations. For step density jumps at arbitrary radial locations outside the vortex core, we derive the complete dispersion relation analytically. The special asymptotic limit of r_j→a and ρ₁ = ρ₂ recovers the Kelvin's dispersion relation for a homogenous Rankine vortex. The dispersion relation also captures one family of continous spectrum modes of the Rankine vortex recently discovered [2]. Like the 2D case, we again find that a light-cored vortex can be unstable and a heavy-cored vortex stable.

References:

1. Dixit & Govindarajan, J. Fluid Mech., 679, 2011, 582-615.

2. Roy & Subramanian, J. Fluid Mech., 741 , 2014, 404-460