A modern approach to the Kelvin-Helmholtz instability on circular vortex sheets

TL;DR

This paper investigates the Kelvin-Helmholtz instability on circular vortex sheets, providing a modern proof of linear instability, analyzing singularity formation, and linking linear instability to wave-breaking in turbulence.

Contribution

It offers a new proof of linear instability for circular vortex sheets and connects linear dynamics to nonlinear wave-breaking phenomena.

Findings

Circular vortex sheets are linearly unstable to Kelvin-Helmholtz instability.

Singularities can develop from analytic initial data in finite time proportional to the square of the vortex radius.

Linear instability may explain wave-breaking observed in nonlinear vortex simulations.

Abstract

We represent the outermost shear interface of an eddy by a circular vortex sheet in two dimensions, and provide a new proof of linear instability via the Birkhoff-Rott equation. Like planar vortex sheets, circular sheets are found to be susceptible to a violent short-wave instability known as the Kelvin-Helmholtz instability, with some modifications due to vortex sheet geometry. This result is in agreement with the classical derivation of (Moore 1974, Saffman 1992), but our modern approach provides greater clarity. We go on to show that the linear evolution problem can develop a singularity from analytic initial data in a time proportional to the square of the vortex sheet radius. Numerical evidence is presented that suggests this linear instability captures the wave-breaking mechanism observed in nonlinear point vortex simulations. Based on these results, we hypothesize that the Kelvin-Helmholtz instability can contribute to the development of secondary instability for eddies in two-dimensional turbulent flow.