We consider the growth rate of a collection of passive tracers moving in the plane under the influence of a random, fluctuating, velocity field. The velocity field we consider is a finite mode approximation to Kolmogorov velocity fields, which are commonly used as models for turbulent diffusion. We show that the diameter of the body of passive tracers grows linearly in time under the influence of these velocity fields.
"Dispersion rates under finite mode Kolmogorov flows analysis." Ann. Appl. Probab. 12 (2) 511 - 532, May 2002. https://doi.org/10.1214/aoap/1026915613