fluid parcels, including the expression for the constants involved, will be determined; the effect of finite diffusivity will be discussed and the Batchelor constant for a passive scalar injected at large scale in the flow will be calculated. In section IV, the transport properties of a heavy particle with Stokes time longer than the Kolmogorov time will be studied, focusing on the relation between the correlation time for the fluid velocity sampled by the particle, and its Lagrangian and Eulerian counterparts. Section V will be devoted to calculation of the concentration fluctuations arising from compressibility of the heavy particle flow. In section VI, the bias introduced by inertia in the sampling of fluid velocity by solid particles (non-ergodic effects) will be analyzed. Section VII will be devoted to conclusions.

**II. Finite correlation time extension of the Kraichnan model**

A two-dimensional random velocity field with Eulerian correlation times scaling like the eddy turn-over time of a real turbulent flow can be obtained very simply, writing appropriate Langevin equations for the Fourier components of the vorticity field: $$\partial_tq_\k(t)+\gamma_kq_\k(t)=h_k\xi_\k(t)
\eqno(2.1)$$ where $q_\k(t)$ is the (space) Fourier transform of the vorticity: $$q(\x,t)=\nabla_\perp\cdot\u(\x,t),
\eqno(2.2)$$ $[$for the generic vector ${\bf v}$, we indicate ${\bf v}_\perp=(-v_2,v_1)]$, and 