Clustering and coalescence from multiplicative noise: the Kraichnan ensemble

We study the dynamics of the two-point statistics of the Kraichnan ensemble which describes the transport of a passive pollutant by a stochastic turbulent flow characterized by scale invariant structure functions. The fundamental equation of this problem consists of the Fokker-Planck equation for the two-point correlation function of the density of particles performing spatially correlated Brownian motions with scale invariant correlations. This problem is equivalent to the stochastic motion of an effective particle driven by a generic multiplicative noise. In this paper, we propose an alternative and more intuitive approach to the problem than the original one (Gawedzki and Vergassola 2000 Physica D 138 63) leading to the same conclusions. The general features of this new approach make possible to fit it to other more complex contexts.