Evolution of vortex statistics in two-dimensional turbulence

Freely evolving two-dimensional turbulence is dominated by coherent vortices. The density of these vortices decays in time as \ensuremath{\rho}\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\xi}}$ with \ensuremath{\xi}\ensuremath{\approxeq}0.75. A new scaling theory is proposed which expresses all statistical properties in terms of \ensuremath{\xi}. Thus the average circulation of the vortices increases as ${\mathit{t}}^{\ensuremath{\xi}/2}$ and their average radius as ${\mathit{t}}^{\ensuremath{\xi}/4}$. The total energy is constant, the enstrophy decreases as ${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\xi}/2}$, and the vorticity kurtosis increases as ${\mathit{t}}^{\ensuremath{\xi}/2}$. These results are supported both by numerical simulations of the fluid equations and by solutions of a modified point-vortex model.