Stability theory of an elliptical vortex

Abstract

The process of transition from the laminar to the turbulent flow regime is intimately related to the emergence of both three-dimensional structures and small-scale excitations. It has been proposed that the instabilities of two-dimensional vortices having elliptical streamlines constitute a relevant mechanism that is capable of giving rise to excitations on all scales. The present work is devoted to a detailed linear stability analysis of such vortices (for both the viscous and the inviscid cases). This is conveniently done by transforming the pertinent equations to a set of natural variables in which some symmetries of the problem are evident. A variable related to the pressure is shown to satisfy a Hill equation. The mechanism of instability for a small eccentricity of the vortex is shown to be related to a spectral degeneracy of the circular vortex. The case of large eccentricity is solved by asymptotic methods. One obtains a logarithmic dependence of the rate of growth of the leading instability as a function of the eccentricity. The set of eigenstates is shown to be complete, despite the fact that the pertinent equations are not self-adjoint, and the rates of growth of unstable modes are given by universal functions (in the inviscid limit) of an effective radial wave number that is identified. The degree of localization of an initially localized perturbation is studied and some implications of the results are discussed.