Superballistic motion in a ‘‘random-walk’’ shear flow

We investigate the motion of a random walker that is driven by a ‘‘random-walk’’ shear flow. This unidirectional two-dimensional flow is defined by a velocity field in the x direction, which depends only on the transverse position y, and whose magnitude v=${\mathit{v}}_{\mathit{x}}$(y) is given by the displacement of a random walk of y steps. For this model, the root-mean-square longitudinal displacement of a diffusing particle that is passively carried by the flow increases as ${\mathit{t}}^{5/4}$. In a single configuration of the random shear, the probability distribution of displacements is bimodal, while the distribution function averaged over many configurations has a single cusped peak at the origin. As a consequence, the configuration-averaged probability that a walk is at x=0 decays more slowly than the ${\mathit{t}}^{\mathrm{-}5/4}$ dependence that would be expected on the basis of single-parameter scaling. The large-distance decay of the average probability distribution is also found to be anomalously slow. These unusual features can be explained on the basis of a scaling argument together with an effective-medium-type approximation. Our results are confirmed by numerical simulations.