Fractional Brownian motions and enhanced diffusion in a unidirectional wave-like turbulence

We study transport in random undirectional wave-like velocity fields a with non-linear dispersion relations. For this simple model. we have several in teresting findings: (1) In the absence of molecular diffusion the entire fa mily of fractional Brownian motions (FBMs), persistent or anti-persistent c an arise in the scaling limit. (2) Thc infrared cutoff may alter the scalin g limit depending on whether the cutoff exceeds certain critical value or n ot. (3) Small, but nonzero, molecular diffusion can drastically change the scaling limit. As a result, some regimes stay intact; some (persistent) FBM regimes become non-Gaussian and some other FBR regimes become Brownian mot ions with enhanced diffusion coefficients. Moreover, in the particular regi me where the scaling limit is a Brownian motion in the absence of molecular diffusion, the vanishing molecular diffusion limit of the enhanced diffusi on coefficient is strictly larger than the diffusion coefficient with zero molecular diffusion. This is the first such example that we are aware of to demonstrate rigorously a nonperturbative effect of vanishing molecular dif fusion on turbulent diffusion coefficient.