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César Castilho

Recife, PE, 50740-540 Brazil
Departamento de Matemática, Universidade Federal de Pernambuco


Regis Rodrigues A., Castilho C., Koiller J.
On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero
2022, vol. 27, no. 5, pp.  493-524
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, seen on the representing sphere, 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
Citation: Regis Rodrigues A., Castilho C., Koiller J.,  On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero, Regular and Chaotic Dynamics, 2022, vol. 27, no. 5, pp. 493-524
Koiller J., Castilho C., Regis Rodrigues A.
Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability
2019, vol. 24, no. 1, pp.  61-79
We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid $\mathbb{E}(a,b,c):$ $x^2/a+y^2/b+z^2/c=1, \, a < b < c$. The equations of motion are transported to $S^2 \times S^2$ via a conformal map that combines confocal quadric coordinates for the ellipsoid and sphero-conical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.
Keywords: point vortices, Riemann surfaces
Citation: Koiller J., Castilho C., Regis Rodrigues A.,  Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 61-79

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