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Vortex pairs on a triaxial ellipsoid and Kimura's conjecture

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  • 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.

    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

    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

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

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

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

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

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

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