Effective merging dynamics of two and three fluid vortices: Application to two-dimensional decaying turbulence

TL;DR

This paper develops a kinetic theory for vortex merging in 2D decaying turbulence, showing three-body interactions dominate at low densities and deriving decay laws that match simulations.

Contribution

It introduces a kinetic framework for vortex merging, highlighting the importance of three-body interactions over two-body processes in turbulence decay.

Findings

Two-body vortex merging becomes inefficient at low vortex density.

Three-body vortex merging leads to a decay law n ~ t^{-1}.

The decay exponent in 3D geostrophic turbulence is predicted as 5/4, matching simulations.

Abstract

We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density $n\ll 1$. When the more efficient three-body vortex mergings are considered {(involving vortices of different signs)}, we show that $n\sim t^{-\xi}$, with $\xi=1$. We generalize this argument to three-dimensional geostrophic turbulence, finding $\xi=5/4$, in excellent agreement with direct Navier-Stokes simulations [J.\,C. McWilliams \emph{et al.}, J. Fluid Mech. {\bf 401}, 1 (1999)].