Lagrangian–Eulerian relation and the independence approximation

The relation between Lagrangian and Eulerian functions is basic to the understanding of turbulent diffusion. To explore this relation, the independence approximation was suggested by Corrsin and used by Saffman. Here, a theoretical investigation is made of the Lagrangian–Eulerian relation and the validity of the independence approximation. A renormalized expansion is introduced for this purpose. It is shown that the independence approximation is the first term in a systematic expansion of the Lagrangian function, and is valid when an explicit condition on the turbulence spectrum is satisfied. Correction terms to the independence approximation are derived. The Lagrangian function is then related to the Eulerian function by an integral equation. This equation justifies a result previously derived by Saffman, and is a little more general. In addition, a calculation is made of the probability distribution of a particle displacement, including non-Gaussian corrections. These corrections are consistent with the non-Gaussian form found in numerical simulations by Peskin, and may explain the observation of a curious behavior concerning the independence approximation. Within the context of the independence approximation, it is seen that the Lagrangian correlation time is less than or equal to the Eulerian correlation time.