Lévy dynamics of enhanced diffusion: Application to turbulence

We introduce a stochastic process called a L\'evy walk which is a random walk with a nonlocal memory coupled in space and in time in a scaling fashion. L\'evy walks result in enhanced diffusion, i.e., diffusion that grows as ${\mathrm{t}}^{\mathrm{\ensuremath{\alpha}}}$,\ensuremath{\alpha}>1. When applied to the description of a passive scalar diffusing in a fluctuating fluid flow the model generalizes Taylor's correlated-walk approach. It yields Richardson's ${\mathrm{t}}^{3}$ law for the turbulent diffusion of a passive scalar in a Kolmogorov -(5/3) homogeneous turbulent flow and also gives the deviations from the (5/3) exponent resulting from Mandelbrot's intermittency. The model can be extended to studies of chemical reactions in turbulent flow.