Fluid flows with vortices are common observations in daily life. From wash basin vortices to oceanic eddies and cyclones, vortical structures can reside in a wide range of scales. In engineering flows vortices usually form behind solid objects moving in a fluid medium
(e.g. wingtip vortices in aircraft wakes). Instabilities in vortices have hence been studied extensively, owing to their fundamental and applied consequences. Small-scale linear instabilities have characteristic wavelengths smaller than the dominant vortex length scale. These short-wavelength instabilities are a mechanism by which three-dimensional features arise in large-scale two-dimensional flows, thus having implications in turbulent flows. Density stratification is a factor that influences vortex instability characteristics, in geophysical and aircraft wake applications for example.
We present a local stability analysis to investigate the effects of differential diffusion between momentum and density (quantified by the Schmidt number Sc) on the three-dimensional, short-wavelength instabilities in planar vortices with a uniform stable stratification along the vorticity axis. Assuming small diffusion in both momentum and density, but arbitrary values for Sc, we present a detailed analytical/numerical analysis for three different classes of base flows: (a) an axisymmetric vortex, (b) an elliptical vortex, and (c) the flow in the neighbourhood of a hyperbolic stagnation point. While a centrifugally stable axisymmetric vortex remains stable for any Sc, it is shown that Sc can have significant effects in a centrifugally unstable axisymmetric vortex: the range of unstable perturbations increases when Sc is taken away from unity, with the extent of increase being larger for Sc > 1. Additionally, for Sc > 1, we report the possibility of oscillatory instability. In an elliptical vortex with a stable stratification, Sc not equal to one is shown to non-trivially influence the three different inviscid instabilities (subharmonic, fundamental and superharmonic) that have been previously reported, and also introduce a new branch of oscillatory instability that is not present at Sc = 1. We conclude by discussing the physical length-scales associated with the Sc-not-equal-to-one instabilities, and their potential relevance in various realistic settings.