Singular vortex pairs follow magnetic geodesics

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 $\epsilon$, 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 $1/{\epsilon }^2$ 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.

14 pages, 1 figure