Decaying Turbulence and the Dynamics of Diffusing Vortices with Conservation Laws

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.