at a scale $l$ within the inertial range, have characteristic velocity $\sim l^\frac{1}{3}$ and decay time $\sim l^{-\frac{2}{3}}$ along fluid trajectories. Hence, in a time $t$ the velocity of a fluid parcel will change by an amount of the order of that of a fluctuation with that lifetime, i.e. by $t^\frac{1}{2}$. If the fluctuations were not advected by the flow, the fluid parcel would see the fluctuation only for the time $\sim l^{-1}$ it takes to cross it. The variation of the fluid parcel velocity in a time $t$ would be therefore $\sim t^\frac{1}{3}$.

Concerning relative diffusion, this process is determined by vortices with the size of the fluid parcel separation at the given time. If these vortices were fixed in space, their effect on relative diffusion would be proportional to the crossing time by the fluid parcels, which is determined by the large scale properties of the flow. In other words, if time correlations were given in an Eulerian reference frame, the process of relative diffusion would not depend solely on the inter-particle distance and on the velocity difference, but also on the total velocity.

Given the difficulty in defining a velocity field with Lagrangian statistics, a successful strategy 