Anomalous diffusion due to long-range velocity fluctuations in the absence of a mean flow

Abstract

The dispersion of a tracer in a heterogeneous medium in which the tracer’s velocity has zero mean and a covariance that decays as x^−γ with distance x is studied using nonlocal advection–diffusion theory. If the velocity covariance decays slowly, γ≤2, the tracer’s dispersive motion is non‐Fickian even at long times after its release. Under these circumstances, it is not possible to predict the dispersion by assuming that the tracer samples the velocity fluctuations primarily by molecular diffusion, even if the fluctuations are weak. Instead, we develop a self‐consistent theory in which the tracer samples each velocity fluctuation by the motion resulting from the other fluctuations. It is shown that in cases of anomalous diffusion, the tracer’s mean‐square displacement grows faster than linearly with time—as t^4/(2+γ) for 0<γt(ln t) ^1/2 for γ≡2 as t→∞.