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

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)].

Final version considerably augmented (7 pages; 3 figures)

References in corpus (1)

• Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy

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• Functional integral derivation of the kinetic equation of two-dimensional point vortices
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