  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
of the order of $r_0$ and thickness $\varepsilon$. We prove that when $r_0=
|\log \varepsilon|^\alpha$, $\alpha>1$, the vorticity field of the fluid
converges for $\varepsilon \to 0$ to the point vortex model, in an interval of
time which diverges as $\log|\log\varepsilon|$. This generalizes previous
result by Cavallaro and Marchioro in [J. Math. Phys. 62, 053102, (2021)], that
assumed $\alpha>2$ and in which the convergence was proved for short times
only.
