  We consider the movement of a particle advected by a random flow of the form
$\vv+\delta \bF(\vx)$, 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 $|\vx|^{2\alpha-2}$, 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
$O(\delta^{-2})$.
