Passive tracer in a slowly decorrelating random flow with a large mean

Abstract

We consider the movement of a particle advected by a random flow of the form $\vv+δ\bF(\vx)$, with $\vv\in\R^d$ a constant drift, $\bF(\vx)$ -- the fluctuation -- given by a zero mean, stationary random field and $\delta \ll 1$ so that the drift dominates over the fluctuation. The two-point correlation matrix $\bR(\vx)$ of the random field decays as $|\vx|^{2α-2}$, as $|\vx|\to+\infty$ with $\alpha <1$. The Kubo formula for the effective diffusion coefficient obtained in \cite{kp79} for rapidly decorrelating fields diverges when $1/2\le \alpha <1$. We show formally that on the time scale ${\delta }^{-1/\alpha }$ the deviation of the trajectory from its mean $\by(t)=\vx(t)-\vv t$ converges to a fractional Brownian motion $B_{\alpha }(t)$ in this range of the exponent $\alpha$. We also prove rigorously upper and lower bounds which show that $\E[|\by(t)|^2]$ converges to zero for times $t\ll {\delta }^{-1/\alpha }$ and to infinity on time scales $t\gg {\delta }^{-1/\alpha }$ as $\delta \to 0$ when $\alpha \in (1/2,1)$. On the other hand, when $\alpha <1/2$ non-trivial behavior is observed on the time-scale $O({\delta }^{-2})$.

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• Averaging versus Chaos in Turbulent Transport?
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• A Simple Passive Scalar Advection-Diffusion Model