Does a particle swept by a turbulent liquid diffuse?

• We study the long time mean square displacement (MSD) of a particle swept by a random flow.
• We obtain the MSD for a family of random flows.
• For ordinary turbulence which is a member of that family we find that the swept particle super-diffuses.
• The exponent characterizing the super-diffusion is 6/5.
• We explain why it is difficult to distinguish experimentally the slight super-diffusion from diffusion.

We consider a particle swept by ordinary   turbulent flow and find that its mean square displacement (MSD) is not linear in the traversed time, TT, which would correspond to diffusion, but rather it is proportional to TT raised to the power 6/5, for large times. This behavior stems from correlations of the velocity field, on length scales much larger   than the largest scale corresponding to the inertial range. We derive our results using two different methods. The first employs an analytic self consistent path integral formulation for obtaining the statistics of the trajectory of a swept particle, given the correlations of the flow. The second involves a numerical solution of an integro-differential equation for the MSD. The conditions which make it possible to observe this behavior are discussed. A by-product of our considerations is the general large TT behavior of the MSD for a large family of flows, which includes ordinary turbulence as a special case.