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

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 $\xi =5\mathrm{/}4$, in excellent agreement with direct Navier-Stokes simulations [J.,C. McWilliams \emph{et al.}, J. Fluid Mech. {\bf 401}, 1 (1999)].