  The diffusive properties in velocity fields whose small scales are
parameterized by non $\delta$-correlated noise is investigated using multiscale
technique. The analytical expression of the eddy diffusivity tensor is found
for a 2D steady shear flow and it is an increasing function of the
characteristic noise decorrelation time $\tau$. In order to study a generic
flow ${\bm v}$, a small-$\tau$ expansion is performed and the first correction
${\cal O}(\tau)$ to the effective diffusion coefficients is evaluated. This is
done using two different approaches and it results that at the order $\tau$ the
problem with a colored noise is equivalent to the white-in-time case provided
by a renormalization of the velocity field ${\bm v} \mapsto \tilde{\bm v}$
depending on $\tau$. Two examples of 2D closed-streamlines velocity field are
considered and in both the cases an enhancement of the diffusion is found.
