Clustering and diffusion of particles and passive tracer density in random hydrodynamic flows

The diffusion of particles and conservative, passive tracer density fields in random hydrodynamic flows is considered. The crucial feature of this diffusion in a divergent hydrodynamic flow is the clustering of the conservative, passive tracer density field (in the Euler description) and occasionally of the particles themselves (in the Lagrange description) — a coherent phenomenon which occurs with probability unity and should arise in almost all dynamic scenarios of the process. In the present paper, statistical clustering parameters are described in statistical topography terms. Because of their inertial properties, particles and their concentration field can also cluster in random divergence-free velocity fields, the divergence of the particle velocity field itself being a crucial aspect of such a diffusion. The delta-correlated in time velocity field for fluctuating flow (as, e.g., in the Fokker–Planck diffusion equation for low-inertia particles) is in principle an invalid approximation for the statistical description of particle dynamics, and the diffusion approximation accounting for the finite time correlation radius should instead be used for this purpose.