a finite correlation time of the turbulent field was recognized. In [@balkovsky01], the case of particle with Stokes time in the turbulent viscous range was analyzed exploiting the fact that, in this case, the fluid velocity is spatially smooth on the scale of interest for the solid particle. In both [@elperin00] and [@balkovsky01], however, the inertial range structure of the turbulent flow was disregarded altogether. The approach carried on here, allows instead to analyze the production of concentration fluctuations in any regimes of Stokes times, in particular in the inertial range, where qualitatively different behaviors for the concentration fluctuation build-up are observed.

Purpose of this paper is to extend the Kraichnan model to short but finite correlation times, preserving, in a controlled perturbation theory, the Lagrangian structure of correlations, and providing several applications to the transport of particles with and without inertia. The analysis will be confined to a situation of two-dimensional, stationary, homogeneous and isotropic turbulence.

This paper is organized as follows. In section II, the equations determining the extension of the Kraichnan model will be illustrated and their main properties discussed. Section III will be devoted to the dynamics of passive tracers; the self-diffusion and relative diffusion of 