Transition to instability of the leapfrogging vortex quartet

TL;DR

This paper analytically determines the precise critical point at which a leapfrogging vortex quartet transitions from stable to unstable motion, confirming a previously hypothesized value involving the golden ratio.

Contribution

The authors provide the first exact analytical proof of the critical stability transition point in the leapfrogging vortex quartet system.

Findings

Transition occurs at lpha = \u03c6^{-2}

Critical value matches previous numerical hypotheses

Analytical proof confirms stability-instability boundary

Abstract

The point vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter $\alpha$ defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value $\alpha = \phi^{-2}$, where $\phi$ is the golden ratio. This value had been hypothesized based on careful numerics by Toph{\o}j and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.