Latitudinal point vortex rings on the spheroid

Kim, Sun-Chul

doi:10.1098/rspa.2009.0597

Cited by 7 publications

(14 citation statements)

References 24 publications

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“…Boatto [ 21 ] extended Kimura's analysis to prove that positive and negative curvatures have, respectively, a destabilizing and a stabilizing effect on the linear and nonlinear stability of a ring of identical vortices. More recently, Kim [ 22 ] appeared to have further extended the analysis to surfaces of variable curvature, specifically the ellipsoid of revolution, adapting the analysis of the sphere carried out by Boatto and Simó in a preprint [ 18 ]. A general theory for compact surfaces of variable curvature, however, appears to be lacking.…”

Section: Introductionmentioning

confidence: 99%

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The motion of point vortices on closed surfaces

Dritschel

Boatto

2015

Proc. R. Soc. A.

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We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech. 157, 95–134 (doi:10.1017/S002211208800308810.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255, 35–64 (doi:10.1017/S002211209300238110.1017/S0022112093002381)).

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Section: Introductionmentioning

confidence: 99%

“…We illustrate aspects of the vortex dynamics for several surfaces, and revisit the stability of a ring of vortices. We correct the previous analysis of Kim [ 22 ], who omitted the contribution due to variable surface curvature and used the incorrect area form (i.e. the incorrect Hamiltonian coordinates).…”

Section: Introductionmentioning

confidence: 99%

The motion of point vortices on closed surfaces

Dritschel

Boatto

2015

Proc. R. Soc. A.

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“…The ellipsoid's symmetry was used to reduce the dimension of the problem. In 2010 Kim [20] obtained the full equations for any ellipsoid of revolution. Several other surfaces of revolution were considered in [10].…”

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confidence: 99%

Vortex pairs on a triaxial ellipsoid and Kimura's conjecture

Rodrigues¹,

Castilho²,

Koiller³

2018

Journal of Geometric Mechanics

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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.

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“…Keeping with the application to vortex dynamics, it would be interesting to emulate the various systems on a sphere considered by various authors [2,3,7,8,[15][16][17] but on a toroidal surface. It would be of interest to analyse the possible similarities and differences of the exhibiting behaviours of the vortical systems between compact surfaces of differing genus.…”

Section: Discussionmentioning

confidence: 99%

Green's function for the Laplace–Beltrami operator on a toroidal surface

Green

Marshall

2013

Proc. R. Soc. A.

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Green's function for the Laplace-Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky-Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.

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Latitudinal point vortex rings on the spheroid

Cited by 7 publications

References 24 publications

The motion of point vortices on closed surfaces

The motion of point vortices on closed surfaces

Vortex pairs on a triaxial ellipsoid and Kimura's conjecture

Green's function for the Laplace–Beltrami operator on a toroidal surface

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