Perturbations of vortex ring pairs

TL;DR

This paper investigates the dynamics and stability of co-axial vortex ring pairs, analyzing how their circulations influence classical orbit phase diagrams and perturbation-induced instabilities.

Contribution

It extends previous work by explicitly analyzing vortex pairs with equal and opposite circulations, and examines their perturbation stability in detail.

Findings

Same-sign circulations lead to small-wavelength instabilities.

Opposite-sign circulations can cause large-wavelength instabilities.

Instability behavior depends on vortex circulation sign and wavelength.

Abstract

We study pairs of co-axial vortex rings starting from the action for a classical bosonic string in a three-form background. We complete earlier work on the phase diagram of classical orbits by explicitly considering the case where the circulations of the two vortex rings are equal and opposite. We then go on to study perturbations, focusing on cases where the relevant four-dimensional transfer matrix splits into two-dimensional blocks. When the circulations of the rings have the same sign, instabilities are mostly limited to wavelengths smaller than a dynamically generated length scale at which single-ring instabilities occur. When the circulations have the opposite sign, larger wavelength instabilities can occur.