  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.
