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

TL;DR

This paper develops linear instability criteria for ideal fluid flows by analyzing two subclasses of perturbations based on vortex line topology, providing new spectral bounds for stability analysis.

Contribution

It introduces a geometric classification of perturbations in fluid flows and derives instability criteria specific to each class, advancing understanding of flow stability.

Findings

Instability criteria established via spectral radius bounds.

Perturbation classes linked to vortex line topology.

Results applicable to 2D and 3D Euler flows.

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.