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 $N\sim t^{-\xi}$, with $\xi\approx 1$ instead of the value $\xi=4/3$ predicted by a na\"{\i}ve kinetic theory. For short time, we find an effective exponent $\xi\approx 0.7$ consistent with previous simulations and experiments. We show that the mean square displacement of surviving vortices grows as $<x^2>\sim t^{1+\xi/2}$. 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 $\xi=1$. We find that the theoretical relations between kinetic parameters are all in good agreement with experiments.