Determination of stable branches of relative equilibria of the $N$-vortex problem on the sphere

TL;DR

This paper develops a computer-assisted framework to identify and prove the stability of relative equilibria in the $N$-vortex problem on the sphere, focusing on configurations with latitudinal rings and vortices at poles, for $5 \,\leq\, N \leq 12$.

Contribution

It introduces a novel computational approach to rigorously determine and analyze the stability of specific vortex configurations on the sphere, overcoming analytical challenges for multiple rings.

Findings

Proved stability of several vortex equilibria configurations.

Determined enclosures for relative equilibria for $5\leq N\leq 12$.

Developed a framework applicable to complex vortex arrangements.

Abstract

We consider the $N$-vortex problem on the sphere assuming that all vorticities have equal strength. We investigate relative equilibria (RE) consisting of $n$ latitudinal rings which are uniformly rotating about the vertical axis with angular velocity $\omega$. Each such ring contains $m$ vortices placed at the vertices of a concentric regular polygon and we allow the presence of additional vortices at the poles. We develop a framework to prove existence and orbital stability of branches of RE of this type parametrised by $\omega$. Such framework is implemented to rigorously determine and prove stability of segments of branches using computer-assisted proofs. This approach circumvents the analytical complexities that arise when the number of rings $n\geq 2$ and allows us to give several new rigorous results. We exemplify our method providing new contributions consisting in the determination of enclosures and proofs of stability of several equilibria and RE for $5\leq N\leq 12$.