Clustering and coalescence from multiplicative noise: the Kraichnan ensemble

TL;DR

This paper investigates the behavior of particle clustering and coalescence in turbulent flows using the Kraichnan ensemble, introducing an intuitive approach to analyze two-point statistics under multiplicative noise.

Contribution

It presents a new, more intuitive method for analyzing the Kraichnan ensemble, applicable to complex stochastic systems involving multiplicative noise.

Findings

The approach reproduces known results of particle clustering.

It offers a flexible framework for complex turbulent flow analysis.

The method simplifies understanding of two-point correlation dynamics.

Abstract

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 in 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 leading to the same conclusions. The general features of this new approach make possible to fit it to other more complex contexts.