 incompressible component of the velocity
field and $\phi$ the potential of the irrotational component. 
The averages in the asymptotic expressions
(\ref{marco1}) or (\ref{marco2}) are easy to 
measure and 
can be used to estimate whether the ballistic
drift is present or not.

Finally, we investigate time-independent potential 
velocity fields, i.e. satisfying
\be
\label{rotzero}
{\bm v}({\bm x}) = -{\bm \partial}\phi({\bm x}),
\end{equation}
where $\phi$ is the potential.
The stationary solution of (\ref{boh}) for this class of flows 
is the Boltzmann distribution (see, e.g., Ref.~\cite{Risken})\,:
\be
\label{Gibbs}
m={\exp\left(-{\phi\over \kappa}\right)\over 
\langle \exp\left(-{\phi\over \kappa}\right)\rangle }.
\end{equation}
It follows from periodicity (or homogeneity) that
\be
\label{merde}
{\bm V}_b \propto \,\langle{\bm v}\exp\left(-{\phi\over \kappa}\right)\rangle
\,=\,\kappa\langle {\bm \partial}\left(\exp\left(-{\phi\over \kappa}\right)\right)
\rangle \,=\,0.
\end{equation}
Ballistic effect is therefore impossible  
for time-independent potential flow. 
Note that the condition of time-independence is essential 
for this result. Indeed, if $\phi$ depends on time the 
solution of (\ref{boh}) is not
the Boltzmann distribution.
Furthermore, let us consider the one-dimensional flow
\be
\label{esempio}
v(x,t)=\cos (x-\kappa\,\omega\, t) + a\cos (x+\kappa\, \omega\, t).
\end{equation}
Using (\ref{gener}) in Appendix~A, it is easy to check that 
the first coefficient of the perturbative expansion in $1/\kappa$
of the ballistic velocity is
\be
\label{timedep}
V_b=\left({\omega\over 1+\omega^2} {1-a^2\over 2}\right){1\over \kappa}
+O\left({1\over \kappa^2}\right). 
\end{equation}
The fact that (\ref{timedep}) does not vanish illustrates the fact that
ballistic transport is possible for time-dependent potential flow.
One can also easily check that the expression (\ref{Vbdelta}), valid for
compressible $\delta$-correlated flow,
does not generally vanishes for potential flow.

\section{Effective diffusivities for compressible flow}
\label{s:diff}

In the previous Section we have considered advective effects
in large-scale transport by compressible flow. It is physically 
clear that diffusive effects will also be present. In order to 
systematically treat them, multi-scale techniques can again be used. 
To capture both advective and diffusive effects, two 
independent slow times 
$T_1=\epsilon t$ and $T_2=\epsilon^2 t$ are 
needed. The solvability condition
at order $\epsilon$ gives the advective effects, as shown in 
Section~\ref{s:ball},
while at order $\epsilon^2$ we also include diffusive effects (see 
Ref.~\cite{Complex}). For the flows where the ballistic velocity ${\bm V}_b$
vanishes, no advective time is needed. 
The formalism is then the same as in Section~\ref{s:ball}, 
with the only
difference that the slow time is now defined as $T=\epsilon^2 t$.
Since there are just technical differences between the two
cases ${\bm V}_b\neq 0$ and ${\bm V}_b=0$, we shall treat in detail the latter 
and give only the final results for
the former.

As in Section~\ref{s:ball} we denote the fast variables by
${\bm x}$, $t$ and the slow space variable is 
defined as ${\bm X}=\epsilon{\bm x}$.
The analogous of equation (\ref{slow}) for the derivatives
is now\,:
\be
\label{slow2}
\partial_i \mapsto \partial_i + \epsilon \nabla_i \,\,\,,
\qquad \partial_t \mapsto \partial_t + \epsilon^2 \partial_T \,\,\,,
\end{equation}
The solution $\Theta$ of the Fokker-Planck equation (\ref{FP})
is again sought in the form of a series in $\epsilon$.
When (\ref{slow2}) and (\ref{expansion}) are inserted
into (\ref{FP}), a hierarchy of equations of the form (\ref{proto}) is
recovered. For the analysis of turbulent diffusion the following
three equations are needed\,:
\begin{eqnarray}
\label{ger1}
{\cal A} \Theta^{(0)} &=& 0\,, \\
\label{ger2}
{\cal A} \Theta^{(1)} &=& - {\bm v}\cdot\nabla \Theta^{(0)}
+2\kappa{\bm \partial}\cdot\nabla\Theta^{(0)}\,, \\
\label{ger3}
{\cal A} \Theta^{(2)} &=& -\partial_T \Theta^{(0)} - {\bm v}\cdot\nabla \Theta^{(1)}
+2\kappa{\bm \partial}\cdot\nabla\Theta^{(1)} + \kappa\nabla^2\Theta^{(0)}.
\end{eqnarray}
The operator ${\cal A}$ is defined by (\ref{proto}). Using  
the linearity 
of (\ref{ger1}), we can express the solution as in (\ref{measure}), 
where the field $m$
satisfies (\ref{boh}). We can then insert (\ref{measure}) into (\ref{ger2}).
Using again the linearity of the operator ${\cal A}$, we obtain:
\be
\label{evai}
\Theta^{(1)} = {\bm \chi}\cdot\nabla\langle \Theta^{(0)}\rangle + \, m\langle\Theta^{(1)}\rangle.
\end{equation}
where the vector field ${\bm \chi}$ satisfies
\be
\label{beep}
\partial_t {\bm \chi} + \underline{{\bm \partial}}\cdot
\left(\underline{{\bm v}} {\bm \chi} \right) - \kappa\,
\partial^2 {\bm \chi} = -{\bm v}\,m + 2 \kappa\, {\bm \partial}\,m.
\end{equation}
The underline in the l.h.s. of (\ref{beep}) 
is meant to make it clear that in the scalar product the components of 
${\bm \partial}$ have to be contracted only with those of ${{\bm v}}$, and not
of ${\bm \chi}$. 
As expected, the solvability condition 
for (\ref{beep}) requires ${\bm V}_b=0$ (which 
we have supposed to be true). In the cases when ${\bm V}_b$ does not vanish, 
the calculations with the two-times formalism give the following modified
equation 
\be
\label{beep2}
\partial_t {\bm \chi} +\underline{{\bm \partial}}
\cdot\left(\underline{{\bm v}} {\bm \chi} \right) - \kappa\,
\partial^2 {\bm \chi} = -m\,\left({\bm v}-{\bm V}_b\right) + 2 \kappa\, {\bm \partial}\,m,
\end{equation}
where ${\bm V}_b$ is given by (\ref{Vb}). Eq.~(\ref{beep}) is clearly 
a particular case of (\ref{beep2}). 

The equation for the large scales dynamics
emerges as the solvability condition for (\ref{ger3})
\be
\label{diffusione}
\partial_T\langle\Theta^{(0)}\rangle = 
\kappa^E_{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}
\langle\Theta^{(0)}\rangle,
\end{equation}
where the effective diffusivity tensor is
\be
\label{eddykappa}
\kappa^E_{\alpha\beta} = \kappa \delta_{\alpha\beta}
-{1\over 2}\left[\langle v_{\alpha}\chi_{\beta}\rangle + 
\langle v_{\beta}\chi_{\alpha}\rangle
\right].
\end{equation}

When ${\bm V}_b\neq 0$, the effective equation for the mean value of the 
field becomes
\be
\label{diffusione2}
\partial_T\langle \Theta^{(0)}\rangle + {\bm V}_b\cdot\nabla\langle\Theta^{(0)}\rangle = 
\kappa^E_{\alpha\beta}\nabla_{\alpha}\nabla_{\