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Singular vortex pairs follow magnetic geodesics
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Singular vortex pairs follow magnetic geodesics
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We consider pairs of point vortices having circulations $\Gamma_1$ and $\Gamma_2$ and confined to a two-dimensional surface $S$. In the limit of zero initial separation $\varepsilon$, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the ``singular vortex pair" moves as a single charged particle on the surface with a charge of order $1/\varepsilon^2$ in a magnetic field $B$ which is everywhere normal to the surface and of strength $|B|=\Gamma_1 +\Gamma_2$. In the case $\Gamma_1=-\Gamma_2$, this gives another proof of Kimura's conjecture (Kimura 1999) that singular dipoles follow geodesics.
Forward citations
Cited by 4 Pith papers
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Collective Dynamics of Vortex Clusters in Compact Fluid Domains: From Pair Interactions to a Quadrupole Description
Vortex clusters on doubly periodic domains show collective rotation and breathing captured at leading order by a complex quadrupole moment derived from pair interactions via the Schottky-Klein prime function.
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Dissipative Vortex Binaries in Compact Fluid Domains with Geometric Corrections
Dissipative vortex binaries on a torus show outward spiraling for same-sign pairs, geometry-induced angular drift for dipoles, and a finite-time nonlinear chirp with frequency scaling as the square of itself for unequ...
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Dissipative Vortex Binaries in Compact Fluid Domains with Geometric Corrections
Dissipative vortex binaries on a torus admit closed-form local solutions, geometry-induced dipole drift, and a finite-time chirp with ω̇ ∝ ω² unlike EM or gravitational inspirals.
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Collective Dynamics of Vortex Clusters in Compact Fluid Domains: From Pair Interactions to a Quadrupole Description
On the flat torus, small co-rotating vortex clusters reduce at leading order to a complex quadrupole whose real and imaginary parts set rotation-rate corrections and slow size breathing.
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