Anomalous Diffusion in fluid flows

A random walk where the distribution of the individual step sizes has a power law tail and an infinite variance is called a Lévy flight. In contrast to ordinary diffusive processes, the growth of the variance of an ensemble of random walkers that undergo Lévy flights can be anomalous, so that the asymptotic growth of the variance need not be linear as in the case of an ordinary diffusive process. A question of considerable interest is the role of Lévy flights in the description of physical phenomena.

Experiments by Weeks, Solomon, Urbach and Swinney at the Center for Nonlinear dynamics in the Univ. of Texas have found Lévy flights and anomalous transport of a passive tracer in a 2D planetary-type flow. They observe this anomalous behavior in situations where there are no KAM surfaces. We propose a model in which the Lévy flights are due to the sticking of the tracers near the walls where the fluid has a no-slip boundary condition. From our analysis, we derive an effective PDE for the tracer density, and analyzing the PDE we obtain anomalous diffusion for tracers in the slow.

Figure 4: Incompressible 2-D flows

Fig. 4 is a schematic representation of tracer trajectories in 2-D time independent/periodic incompressible flows. In a ``random'' flow, the vortices jiggle around, and there are no coherent smooth streamlines in contrast to these two cases.

To numerically study this problem, we also construct a ``random'' incompressible flow. The tracers are advected by this flow, and we get the tracer dynamics by a map

(x_n+1, y_n+1) = M(x_n, y_n,$$\theta_{n}^{}$$)

where M is an area-preserving map depending on a parameter $\theta_{n}^{}$. $\theta_{n}^{}$ is taken to be random. This map can be thought of as a symplectic integrator for the random flow.

The tracers in this model spread superdiffusively with an exponent 3/2, in good agreement with the measured experimental value. We also obtain exponents for the lifetime distribution of the tracers near and away from the walls that are in good agreement with their experimentally measured values.

The figure shows the variance of the tracer distribution that we obtain from numerical simulations of our model, and a power-law with exponent gives a very good fit for the large time behavior.