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arXiv · 2401.08512

Singular vortex pairs follow magnetic geodesics

Theodore D. Drivas·Daniil Glukhovskiy·Boris Khesin
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Abstract

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.

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Theodore D. Drivas, Daniil Glukhovskiy, Boris Khesin. 2024-01-16. Singular vortex pairs follow magnetic geodesics. https://doi.org/10.1093/imrn%2Frnae106

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