Linear instability criteria for ideal fluid flows subject to two subclasses of perturbations

Abstract

In this paper we examine the linear stability of equilibrium solutions to incompressible Euler's equation in 2- and 3-dimensions. The space of perturbations is split into two classes - those that preserve the topology of vortex lines and those in the corresponding factor space. This classification of perturbations arises naturally from the geometric structure of hydrodynamics; our first class of perturbations is the tangent space to the co-adjoint orbit. Instability criteria for equilibrium solutions are established in the form of lower bounds for the essential spectral radius of the linear evolution operator restricted to each class of perturbation.