Vortex pairs on a triaxial ellipsoid and Kimura's conjecture

Adriano Regis Rodrigues; César Castilho; Jair Koiller

doi:10.3934/jgm.2018007

Article Contents

2018, Volume 10, Issue 2: 189-208. Doi: 10.3934/jgm.2018007

This issue Previous Article Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems Next Article A note on the normalization of generating functions

Vortex pairs on a triaxial ellipsoid and Kimura's conjecture

1.
Departamento de Matemática, Universidade Federal Rural de Pernambuco, Recife, PE CEP 52171-900 Brazil
2.
Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE CEP 50740-540 Brazil
3.
Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG CEP 36036-900 Brazil

* Corresponding author
* Corresponding author

Received: October 2016

Revised: December 2017

Published: May 2018

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Abstract

We consider the problem of point vortices moving on the surface of a triaxial ellipsoid. Following Hally's approach, we obtain the equations of motion by constructing a conformal map from the ellipsoid into the sphere and composing with stereographic projection. We focus on the case of a pair of opposite vortices. Our approach is validated by testing a prediction by Kimura that a (infinitesimally close) vortex dipole follows the geodesic flow. Poincaré sections suggest that the global flow is non-integrable.

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Mathematics Subject Classification: Primary: 76B47, 34C28; Secondary: 53Z05.

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Figure 2. Scheme for the double branched covering of the torus over the ellipsoid. See Proposition 3

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Figure 1. Lines of curvature of the triaxial ellipsoid. Cuts along the top and bottom segments joining the umbilical points results (topologically) on an open cylinder. One could as well make the cuts sidewise. Coordinates $(\lambda_1, \lambda_2)$ cannot be made global

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Figure 5. Poincaré map. Prolate, nearly symmetrical $a = 1, \ b = 1.1, \ c = 9, \ H = -40$

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Figure 6. Poincaré map. Prolate $a = 1, \ b = 2, \ c = 9, \ H = -36$

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Figure 7. Poincaré map. Prolate $a = 1, \ b = 4, \ c = 9, \ H = -60$

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Figure 3. Nearly spherical example $a = 1, \ b = 1.01, \ c = 1.02$

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Figure 4. Ellipsoid $a = 1, \ b = 6, \ c = 9.$

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