  In this letter, I solve a model for the dynamics of vortices in a decaying
two-dimensional turbulent fluid. The model describes their effective diffusion,
and the merging of pairs of vortices of same vorticity sign, when they get too
close. The merging process is characterized by the conservation of energy and
of the quantity $Nr^n$, where $r$ is the mean vortex radius, and $N$ their
number. $n=4$ corresponds to a constant peak vorticity, and $n=2$ to a constant
kurtosis. I found the scaling laws for various physical quantities ($r$,
enstrophy, kurtosis...), and for instance, it is shown that $N\sim
(t/\ln(t))^{-\frac{2n}{3n-4}}$ for $n>2$, and $N\sim t^{-2}$ for $n=2$, in good
agreement with extensive numerical simulations. I also discuss some recent
experiments in view of these results.
