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JOURNALS // Regular and Chaotic Dynamics // Archive

Regul. Chaotic Dyn., 2022 Volume 27, Issue 5, Pages 493–524 (Mi rcd1177)

This article is cited in 1 paper

Alexey Borisov Memorial Volume

On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero

Adriano Regis Rodriguesa, César Castilhob, Jair Koillerc

a Departamento de Matemática, Universidade Federal Rural de Pernambuco, 52171-900 Recife PE, Brazil
b Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540 Recife PE, Brazil
c Departamento de Matemática, Universidade Federal de Juiz de Fora, 36036-900 Juiz de Fora MG, Brazil

Abstract: We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium on a genus zero surface with an arbitrary metric. It consists of transferring the calculations to the round sphere in $\mathbb{R}^3$, with a conformal factor, and exploring the Möbius invariance of the conformal structure, so that the equilibria, <i> seen on the representing sphere</i>, appear in the north/south poles. Three example problems are analyzed: $i)$ For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability. We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). $ii)$ A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. $iii)$ The stability, to a restricted set of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.

Keywords: point vortices, symplectic geometry, Hamiltonian systems.

MSC: 76B47, 76M60, 34C23, 37E35

Received: 31.10.2021
Accepted: 26.07.2022

Language: English

DOI: 10.1134/S156035472205001X



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