Fluid
ows 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
ows vortices usually form behind solid objects moving in a
uid 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 in-
stabilities 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
ows, thus having implications in turbulent
ows. Den-
sity stratication is a factor that in
uences vortex instability characteristics, in geophysical
and aircraft wake applications for example.
We present a local stability analysis to investigate the eects of dierential diusion
between momentum and density (quantied by the Schmidt number Sc) on the three-
dimensional, short-wavelength instabilities in planar vortices with a uniform stable strat-
ication along the vorticity axis. Assuming small diusion in both momentum and density,
but arbitrary values for Sc, we present a detailed analytical/numerical analysis for three
dierent classes of base
ows: (a) an axisymmetric vortex, (b) an elliptical vortex, and (c)
the
ow in the neighbourhood of a hyperbolic stagnation point. While a centrifugally sta-
ble axisymmetric vortex remains stable for any Sc, it is shown that Sc can have signicant
eects 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 stratication, Sc not equal to one is shown
to non-trivially in
uence the three dierent 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 phys-
ical length-scales associated with the Sc-not-equal-to-one instabilities, and their potential
relevance in various realistic settings.