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- IPS Conference 2012

In this work we explore the short-time behaviour of two Lagrangian tracers, initially separated by the vector R_{0}, in a fully developed turbulent flow in d dimensions. We assume the turbulence to be incompressible, homogeneous, isotropic and stationary.We study the evolution of the correlation functions of the velocity difference, u(t), and the separation, R(t), between the particles. We begin with the spatially smooth velocity, which is relevant beneath the dissipation scale, where u=σ(t)R(t), with σ(t) a random matrix. We show that if < u^{b-m}R^{a-m}(u ·R)^{m}> is an analytic function of t, it is a statistically conserved quantity for the choice a+b=-d and arbitrary m. We expect the result to be valid inside the correlation time of the velocity. This result is a counterpart to the classical result of Furstenberg and Zeldovich that R^{-d} is a martingale at asymptotically long times. For turbulence within the inertial range, where < u^2> ~ R_{0}^{ξ2}, we suggest the generalization of the Furstenberg and Zeldovich moment, < R(t)^{2-ξ2-d}>, and show that while it is not a martingale, it is the only moment whose second time derivative is zero at t=0. Using the Kolmogorov scaling ξ_{2}=2/3, we find that < R(t)^{-2/3} >=R_{0}^{-2/3} -2/27ε t^{3} R_{0}^{-2-2/3}+O(t^{4}) in 2d and < R(t)^{-5/3} > = R_{0}^{-5/3}+14/81 ε t^{3} R_{0}^{-3-2/3}+O(t^{4}) in 3d. This is a direct manifestation of time reversal symmetry breaking using only the separation between the particles, it is thus a counterpart to the Richardson law < R(t)^{2} > ~ t^{3} for short times.

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