Instability of two-dimensional Taylor-Green Vortices

TL;DR

This paper develops a general mathematical criterion to analyze spectral stability of fluid flows and applies it to fully characterize the instability spectrum of the two-dimensional Taylor-Green vortex.

Contribution

It introduces a new criterion based on holomorphic functions to detect eigenvalue instabilities in Hamiltonian operators, applied here to fluid vortex stability analysis.

Findings

Proves spectral instability of the 2D Taylor-Green vortex.

Identifies conditions for stability of odd perturbations.

Locates and characterizes unstable eigenvalues using computer-assisted methods.

Abstract

For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.