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

Abstract

This paper is devoted to the effects of rotation on the linear dynamics of two-dimensional vortices. The asymmetric behavior of cyclones and anticyclones, a basic problem with respect to the dynamics of rotating flows, is particularly addressed. This problem is investigated by means of linear stability analyses of flattened Taylor–Green vortices in a rotating system. This flow constitutes an infinite array of contra-rotating one-signed nonaxisymmetric vorticity structures. We address the stability of this flow with respect to three-dimensional short-wave perturbations via both the geometrical optics method and via a classical normal mode analysis, based on a matrix eigenvalue 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 various levels of the background rotation. For Taylor–Green cells with aspect ratio E=2, we show that anticyclones undergo centrifugal instability if the 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 here defined as the ratio of the maximum amplitude of vorticity to twice the background rotation. On the other hand, cyclones bear elliptic and hyperbolic instabilities whatever the Rossby number. Besides, depending on the Rossby number, rotation can either strengthen (anticyclonic vortices) or weaken 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 agreement with the short-wave asymptotics both with respect to the amplification 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 concentrated in the center of the cyclones or anticyclones; hyperbolic eigenmodes which are localized in the neighborhood of closed streamlines in cyclones.