Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2010-06-16
Phys. Rev. E 84, 056317 (2011)
Physics
Condensed Matter
Statistical Mechanics
Final version considerably augmented (7 pages; 3 figures)
Scientific paper
10.1103/PhysRevE.84.056317
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)].
Chavanis Pierre-Henri
Sire Clément
Sopik Julien
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