Stability of a vortex with a heavy core

Abstract

This paper examines the stability of swirling flows in a non-homogeneous fluid. Density gradients are shown to produce two distinct kinds of instability. The first is the centrifugal instability (CTI) which mainly affects axisymmetric, short-axial-wavelength eigenmodes. The second is the Rayleigh–Taylor instability (RTI) which mainly affects non-axisymmetric, two-dimensional eigenmodes. These instabilities are described for a family of model flows for which the velocity law $V(r)$ corresponds to a Gaussian vortex with radius 1, and the density law $R(r)$ corresponds to a Gaussian distribution characterized by a density contrast $C$ and a characteristic radius $b$. A full map in the ($C, b$)-plane is given for the amplification rate and the structure of the most amplified eigenmode. For small density contrasts ($C\,{<}\,0.5$), the CTI occurs only for $b \,{>}\, 1$ and the RTI for $b \,{\lesssim}\, 0.8$. On the other hand, for high density contrasts ($C \,{>}\, 0.5$), a competition between the two kinds of instabilities is observed. From a fundamental point of view, the nature of the instability depends on the local values of $ G^2\,{=}\,{-}r^{-1}V^2R^{-1}{\rm d}R/{\rm d}r$ and the Rayleigh discriminant $ \Phi\,{=}\,r^{-3}{\rm d}(r^2V^2)/{\rm d}r $. CTI occurs whenever $ G^2\,{>}\,\Phi $ somewhere in the flow. For RTI, a necessary condition is that $G^2\,{>}\,0$ somewhere in the flow. By an asymptotic analysis, we show that this condition is also sufficient in the limit $b\,{\rightarrow}\, 0$, $C\,{\rightarrow}\, 0$. This asymptotic analysis also confirms that shear has a stabilizing effect on RTI and that this instability is strictly analogous to the standard RTI obtained in the case where light fluid is situated below heavier fluid in the presence of gravity.