About the second order moment of the Lagrangian velocity increments L. Biferale 1 and A.S. Lanotte 2 1 Dept. Physics, Univ. Rome Tor Vergata (Italy), Tech. University of Eindhoven (The Netherlands) & Kavli Institute for Theoretical Physics, UCSB, Santa Barbara (CA) 2 CNR-ISAC, Str. Prov. Lecce-Monteroni, 73100 Lecce (Italy) & Kavli Institute for Theoretical Physics, UCSB, Santa Barbara (CA) We discuss a parameterization for the scaling behavior of the second moment of Lagrangian velocity increment in statistically isotropic and homogeneous 3D turbulence, and we compare it with data from Direct Numerical Simulation at Reynolds number up to Re λ ≃ 300. I. BATCHELOR PARAMETERIZATION FOR THE LAGRANGIAN SECOND ORDER STRUCTURE FUNCTION This is the follow up of the discussion [1], we had about the scaling behavior of the second order moment of velocity incements measured along tracers trajectories in statistically isotropic and homogeneous (HIT) three dimensional turbulence. These are: S (i) 2 (τ ) ≡〈[v i (t + τ ) − v i (t)] 2 〉 , (1) where v i (t) is the i−th component of the turbulent Lagrangian velocity field. A Kolmogorov-like scaling (K41) for the Eulerian velocity increments of statistically homogeneous and isotropic turbulence, translated into time domain would give - for any velocity component-, the linear prediction S 2 (τ )= C 0 〈ǫ〉τ , where ǫ is the kinetic energy dissipation rate, and C 0 is an order one constant. Observations suggest that in 3D HIT, C 0 ∈ [6.0; 7.0] [2]. The point is the following: verify or not if the poor scaling behavior currently observed in the data, at largest Reynolds number, is indeed consistent with a second order Lagrangian structure function scaling linearly in the Lagrangian inertial range. In particular, we refer to the absence of a clear plateau in the compensated second order structure function, S 2 (τ )/(ǫτ ) [2, 3]. Any possible parameterization for the time behavior of S 2 (τ ) has to reproduce the three following regimes:      S 2 (τ ) ∼ τ 2 τ ≪ τ η , S 2 (τ ) ∼ τ z2 τ η ≪ τ ≪ T L , S 2 (τ ) ∼ const. τ ≫ T L , (2) where τ η is the Kolmogorov time scale, and T L is the large scale Lagrangian eddy turn-over-time. If we assume, a K41 scaling in the inertial range of time scale, then z 2 = 1. We also recall that by dimensional argument we have T L /τ η ∝ Re λ . We propose a Batchelor-like parameterization [4, 5], adding a saturation function for large τ in order to reproduce the decorrelation for times larger than the eddy turn over: S 2 (τ )= C 0 τ 2 (c 1 τ 2 η + τ 2 ) (2-z 2 ) 2 (1 + c 3 τ/T L ) -z2 , (3) where c 1 and c 3 are order one dimensionless constant. In figure 1, we show the results when we take T L /τ η =0.1Re λ [2]. It turns out that the effect of finite Reynolds number induced by the large scale saturation are big, and that a plateau develops only for very large Reynolds numbers, currently unreacheable. One can of course play with the parameterization in order to modify the the transitions from viscous to inertial, and from inertial to integral ranges: in particular, by changing the functional form of the denominator in eq.(3) and of the saturation factor, these transitions can be made sharper or smoother. Hence, it is probable that what we see in data (absence of a clear and well developed platueau), is just a finite Reynolds effect that in the Lagrangian Inertial range is more important than in the Eulerian case (remind that L/η ∝ Re 3/4 and T L /τ η ∝ Re 1/2 ). Also, in order to be consistent with an exponential decay for velocity correlation function one needs to sligtly refine the functional form of the saturation factor for large times. We remark that in 2D turbulence in the inverse energy cascade regime, both the small scale and the large scale saturation functions have to be changed to take into account of forcing effects (at small scales) and of Ekman friction