Spectral stability of inviscid columnar vortices

TL;DR

This paper proves that a broad class of inviscid columnar vortices are spectrally stable against three-dimensional perturbations, extending previous results limited to planar or axisymmetric cases, with implications for atmospheric and engineering flows.

Contribution

It establishes spectral stability of columnar vortices for general velocity profiles under three-dimensional perturbations, using a novel homotopy and spectral analysis approach.

Findings

Columnar vortices are spectrally stable to 3D perturbations.

The stability applies to common models in atmospheric and engineering contexts.

The proof uses integral identities and critical layer analysis.

Abstract

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry. Our result applies to a large family of velocity profiles, including the most common models in atmospheric flows and engineering applications. The proof is based on a homotopy argument, which allows us to concentrate in the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis, where unstable eigenvalues can be excluded using integral identities and a careful study of the so-called critical layers.