On nonlinear stability of the regular vortex systems on a sphere

Kurakin, L. G.

doi:10.1063/1.1764432

Cited by 25 publications

(27 citation statements)

References 25 publications

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“…But the case of N = 7 (called the Thomson heptagon) was noticed as a peculiar configuration dividing stability and instability, which eventually turns out to be neutrally stable after several decades (see Boatto & Cabral (2003) for details). The ring of point vortices problem is generalized later and is investigated analogously for various underlying surfaces such as a sphere (Polvani & Dritschel 1993;Boatto & Cabral 2003;Boatto & Simo 2004;Kurakin 2004;Boatto & Simo 2008) or a cylinder (Souliere & Tokieda 2002;Montaldi et al 2003). In the same spirit, here we study the stability of a vortex ring on the spheroid as an application and also a concrete example of previous mathematical formulation.…”

Section: Latitudinal Ring Of Point Vortices On Ementioning

confidence: 99%

“…This problem originated from the classical work of Thomson (1883), in which the stability of point vortices on the edge of a regular n-gon is analysed in the plane. This study was subsequently generalized and has been applied to various underlying surfaces such as a cylinder (Souliere & Tokieda 2002;Montaldi et al 2003) and a sphere (Polvani & Dritschel 1993;Boatto & Cabral 2003;Boatto & Simo 2004;Kurakin 2004;Boatto & Simo 2008). The stability of the point vortex ring on a spheroid will be analysed by comparing it with the case of a sphere.…”

Section: Introductionmentioning

confidence: 99%

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Latitudinal point vortex rings on the spheroid

Kim

2010

Proc. R. Soc. A.

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Point vortex motion on the surface of a spheroid is studied. Exact dynamical equations from the corresponding Hamiltonian are constructed by computing the conformal metric which induces a modified stereographic projection. As a concrete example, the motion of point vortices at the same latitude (called the point vortex ring) is investigated as an extension of the sphere case. The role of eccentricity to the stability of the rotating motion is analysed. The influence of a pole vortex is also discussed.

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Section: Latitudinal Ring Of Point Vortices On Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Latitudinal point vortex rings on the spheroid

Kim

2010

Proc. R. Soc. A.

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“…The expression (9) indicates that λ If ξ p η p < 0, then λ p is pure imaginary due to (9). Therefore we have the following stability proposition.…”

Section: Introductionmentioning

confidence: 94%

“…They showed that the range of the nonlinear stability coincided with that of the linear stability. Kurakin [9] recently considered the nonlinear stability in the sense of Routh and complemented the stability analysis of the N -ring on the sphere.…”

Section: Introductionmentioning

confidence: 99%

High-dimensional heteroclinic and homoclinic connections in odd point-vortex ring on a sphere

Sakajo

2005

Nonlinearity

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We consider the motion of the N -vortex points that are equally spaced along a line of latitude on sphere with fixed pole vortices, called "N -ring". We are especially interested in the case when the number of the vortex points is odd. Since the eigenvalues that determine the stability of the odd Nring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional. Hence, it is generally difficult to describe the global structure of the manifolds. In this article, based on the linear stability analysis, we propose a projection method to observe the structure of the iso-surface of the Hamiltonian, in which the orbit of the vortex points evolves. Applying the projection method to the motion of the 3-ring and 5-ring, we characterize the complex evolution of the unstable odd N -ring from the topological structure of the iso-surface of the Hamiltonian.

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“…While the N -ring is one of the simplest models to approximate the evolution of such coherent structures, it is also one of relative equilibria of (1.1) and (1.2) discussed in [26]. Linear and nonlinear stability of the N -ring has been extensively investigated [3,5,20,37].…”

Section: Introductionmentioning

confidence: 99%

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

Sakajo

Yagasaki

2008

J Nonlinear Sci

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We study the motion of N point vortices with N ∈ N on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N = 5n or 6n with n ∈ N, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N -ring. Utilizing a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N -ring is shown to be chaotic.

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On nonlinear stability of the regular vortex systems on a sphere

Cited by 25 publications

References 25 publications

Latitudinal point vortex rings on the spheroid

Latitudinal point vortex rings on the spheroid

High-dimensional heteroclinic and homoclinic connections in odd point-vortex ring on a sphere

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

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