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Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity
- P. G. POTYLITSIN, W. R. PELTIER
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- Journal:
- Journal of Fluid Mechanics / Volume 387 / 25 May 1999
- Published online by Cambridge University Press:
- 25 May 1999, pp. 205-226
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We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.
Stratification effects on the stability of columnar vortices on the f-plane
- P. G. POTYLITSIN, W. R. PELTIER
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- Journal:
- Journal of Fluid Mechanics / Volume 355 / 25 January 1998
- Published online by Cambridge University Press:
- 25 January 1998, pp. 45-79
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We consider the stability with respect to three-dimensional perturbations of columnar vortices on the f-plane and as a function of the strength of a stabilizing density stratification parallel to the axis of the vortex. We seek to understand the dynamics of the processes through which such a vertically oriented barotropic vortex may be destabilized. As models of the basic vorticity distribution we consider both Kelvin–Helmholtz vortices in shear and ‘Kida-like’ vortices in strain. In the case of rotating unstratified flow, an isolated anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations by small values of the background rotation, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as expected on the basis of the Taylor–Proudman theorem. The dominant instability mechanism which drives the destruction of anticyclonic vortices in the presence of slow background rotation may be understood to constitute a three-dimensional inertial (centrifugal) instability. Through explicit analysis we show that sufficiently strong density stratification stabilizes the two-dimensional columnar structures to disruption by this and additional modes of instability that exist even in the absence of rotation. We furthermore demonstrate that there exists a second fundamental mode of instability in the presense of background rotation which affects only anticyclonic vortex columns whose cross-sections are elliptical. Only when the ellipticity of the vortex is sufficiently high does this mode dominate the centrifugal mode. The process whereby anticyclonic vortices may be selectively destroyed appears to provide a possible explanation of an asymmetry that is sometimes observed to be characteristic of the atmospheric von Kármán vortex streets that are observed in the lee of oceanic islands. The anticyclonic branch of the street often seems to be absent. More generally, the centrifugal mechanism for the selective destruction of anticyclones discussed herein very clearly explains a number of recent results obtained from both laboratory experiments and numerical simulations.