Numerical renormalization group of vortex aggregation in 2D decaying turbulence: the role of three-body interactions

Abstract

In this paper, we introduce a numerical renormalization group procedure which permits long-time simulations of vortex dynamics and coalescence in a 2D turbulent decaying fluid. The number of vortices decreases as $N\sim t^{-ξ}$, with $ξ\approx 1$ instead of the value $ξ=4/3$ predicted by a na\"ıve kinetic theory. For short time, we find an effective exponent $ξ\approx 0.7$ consistent with previous simulations and experiments. We show that the mean square displacement of surviving vortices grows as $ \sim t^{1+ξ/2}$. Introducing effective dynamics for two-body and three-body collisions, we justify that only the latter become relevant at small vortex area coverage. A kinetic theory consistent with this mechanism leads to $ξ=1$. We find that the theoretical relations between kinetic parameters are all in good agreement with experiments.