  We study the time evolution of an incompressible fluid with axial symmetry
without swirl when the vorticity is sharply concentrated on $N$ annuli of radii
$\approx$ $r_0$ and thickness $\epsilon$. We prove that when $r_0= |\log
\epsilon|^\alpha, \,\, \alpha>2$, the vorticity field of the fluid converges as
$\epsilon \to 0$ to the point vortex model, at least for a small but positive
time. This result generalizes a previous paper that assumed a power law for the
relation between $r_0$ and $\epsilon$.
