1. Toroidal Geometry Stabilizing a Latitudinal Ring of Point Vortices on a Torus.
- Author
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Sakajo, Takashi and Shimizu, Yuuki
- Subjects
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CURVATURE , *FLUID dynamics , *VORTEX motion , *DYNAMICAL systems , *BIFURCATION theory - Abstract
We carry out the linear stability analysis of a polygonal ring configuration of
N point vortices, called anN -ring, along the line of latitude θ0on a torus with the aspect ratio α . Deriving a criterion for the stability depending on the parameters N , θ0and α , we reveal how the aspect ratio α contributes to the stability of the N -ring. While theN -ring necessarily becomes unstable whenN is sufficiently large for fixed α, the stability is closely associated with the geometric property of the torus for variable α ; for low aspect ratio α∼1 , N=7 is a critical number determining the stability of the N -ring when it is located along a certain range of latitudes, which is an analogous result to those in a plane and on a sphere. On the other hand, the stability is determined by the sign of curvature for high aspect ratio α≫1. That is to say, the N -ring is neutrally stable if it is located on the inner side of the toroidal surface with a negative curvature, while theN -ring on its outer side with a positive curvature is unstable. Furthermore, based on the linear stability analysis, we describe nonlinear evolution of theN -ring when it becomes unstable. It is difficult to deal with this problem, since the evolution equation of theN point vortices is formulated as a Hamiltonian system withN degrees of freedom, which is in general non-integrable. Thus, we reduce the Hamiltonian system to a simple integrable system by introducing a cyclic symmetry. Owing to this reduction, we successfully find some periodic orbits in the reduced system, whose local bifurcations and global transitions for variable αare characterized in terms of the fundamental group of the torus. [ABSTRACT FROM AUTHOR] - Published
- 2018
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