Decaying Turbulence and the Dynamics of Diffusing Vortices with Conservation Laws

TL;DR

This paper models the decay of vortices in 2D turbulence, deriving scaling laws for vortex dynamics and merging processes, with results aligning well with numerical simulations and experimental observations.

Contribution

It introduces a novel model incorporating conservation laws to describe vortex diffusion and merging in decaying 2D turbulence, deriving new scaling laws.

Findings

Scaling law for vortex number: N ~ (t/ln(t))^(-2n/(3n-4)) for n>2

Vortex number scales as N ~ t^{-2} for n=2

Model predictions agree with numerical simulations and experiments

Abstract

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.