Vortices in rotating systems: Centrifugal, elliptic and hyperbolic type instabilities

This paper is devoted to the effects of rotation on the linear dynamics of two-dimensional vortices. The asymmetric behavior of cyclones and anticyclo nes, a basic problem with respect to the dynamics of rotating flows, is par ticularly addressed. This problem is investigated by means of linear stabil ity analyses of flattened Taylor-Green vortices in a rotating system. This flow constitutes an infinite array of contra-rotating one-signed nonaxisymm etric vorticity structures. We address the stability of this flow with resp ect to three-dimensional short-wave perturbations via both the geometrical optics method and via a classical normal mode analysis, based on a matrix e igenvalue method. From a physical point of view, we show that vortices are affected by elliptic, hyperbolic and centrifugal instabilities. A complete picture of the short-wave stability properties of the flow is given for var ious levels of the background rotation. For Taylor-Green cells with aspect ratio E = 2, we show that anticyclones undergo centrifugal instability if t he Rossby number verifies Ro > 1, elliptic instability for all values of Ro except 0.75 < Ro < 1.25 and hyperbolic instability. The Rossby number is h ere defined as the ratio of the maximum amplitude of vorticity to twice the background rotation. On the other hand, cyclones bear elliptic and hyperbo lic instabilities whatever the Rossby number. Besides, depending on the Ros sby number, rotation can either strengthen (anticyclonic vortices) or weake n elliptic instability. From a technical point of view, in this article we bring an assessment of the links between the short-wave asymptotics and the normal mode analysis. Normal modes are exhibited which are in complete agr eement with the short-wave asymptotics both with respect to the amplificati on rate and with respect to the structure of the eigenmode. For example, we show centrifugal eigenmodes which are localized in the vicinity of closed streamlines in the anticyclones; elliptical eigenmodes which are concentrat ed in the center of the cyclones or anticyclones; hyperbolic eigenmodes whi ch are localized in the neighborhood of closed streamlines in cyclones. (C) 1999 American Institute of Physics. [S1070-6631(99)00912-5].