Singular Vortex Pairs Follow Magnetic Geodesics

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 an magnetic field $B$ that 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 [11] that singular dipoles follow geodesics.