Scaling of Lagrangian structure functions

We use stochastic closure theory and generalized Green-Kubo relations to show that the velocity structure functions have two distinct scaling regimes connected by a passover. Initially, after a brief ballistic (Batchelor) scaling, the structure functions exhibit a Lagrangian scaling regime with no intermittency, and then pass over to a regime with Eulerian scaling, with intermittency. This transition time for the passover region is controlled by the second structure function, through a generalization of Green-Kubo-Obukhov relations. The ultimate time interval of decay seems to be controlled by the scaling of free eddies, analogous to the scaling in the buffer layer of boundary layer turbulence [B. Birnir, L. Angheluta, J. Kaminsky, and X. Chen, Spectral link of the Generalized Townsend-Perry constants in turbulent boundary layers, Phys. Rev. Res. 3, 043054 (2021)2643-156410.1103/PhysRevResearch.3.043054]. The dip observed in the log-derivatives of the structure function [L. Biferale, E. Bodenschatz, M. Cencini, A. S. Lanotte, N. T. Ouellette, F. Toschi, and H. Xu, Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation, Phys. Fluids 20, 065103 (2008)1070-663110.1063/1.2930672], with respect to the second structure function S2, is caused only by the time scales probed by S2. It seems better to take the log-derivative with respect to t, instead of S2, to fully understand the different scaling regimes of Lagrangian turbulence.