Stability of a vortex in radial density stratification: role of wave interactions

Abstract

We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the 'stable stratification' of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh-Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood number At show growth rates scaling as At 1/3 close to the critical radius, and At 1/2 further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-cored vortex, consistent with recent work which found Gaussian profiles to be stable. For sharp gradients, it is argued that wave interaction can be supported due to the presence of quasi-modes. Probably for the first time, a quasi-mode which decays exponentially is shown to interact with a neutral wave to give exponential growth in the combined case. We finally study the nonlinear stages using viscous direct numerical simulations. The initial exponential instability of light-cored vortices is arrested due to a restoring centrifugal buoyancy force, leading to stable non-axisymmetric structures, such as a tripolar state for an azimuthal wavenumber of 2. The study is restricted to two dimensions, and neglects gravity.

Interaction of a vortex ring with a single bubble: bubble and vorticity dynamics

“…The sharp change in viscosity and density at the interface could also induce instabilities. This is discussed for example in Dixit & Govindarajan (2011) and Govindarajan & Sahu (2014), the latter showing that a vortex with a light core having a 'stable density' stratification can still be unstable. Apart from these single core instabilities, the influence of the neighbouring oppositely signed core can also be important, as it overlays a strain field on the main vortex, as discussed for example by Leweke & Williamson (1998) in the case of the 'cooperative' instability of vortex pairs.…”

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

“…Conventional wisdom indicates that stable stratification and linear shear, both modally stable individually, would not give rise to any instabilities when coupled. The exception to this rule occurs when interfacial waves, riding on density discontinuities in a stably stratified scenario, resonantly interact via shear [29][30][31]. For a uniformly stratified shear flow, the scenario in this analysis, no such exponential instabilities exist.…”

Azimuthal shear instability of a liquid jet injected into a gaseous cross-flow

“…Therefore, for the purposes of the current work, which correspond to a dense jet injected into a lighter medium, the jet periphery will always be potentially unstable to the centrifugal Rayleigh-Taylor (CRT) type of azimuthal instability. Studies on the stability of vortices in which density decreases monotonically in the radial direction have demonstrated the possibility of the CRT instability (Fung 1983;Sipp et al 2005;Dixit & Govindarajan 2011). This type of instability is analogues to the typical Rayleigh-Taylor instability; however, while the latter instability can exist without flow inertia, the CRT instability owes its existence to the rotational motion of the mean flow.…”