Clustering instability of the spatial distribution of inertial particles in turbulent flows

A theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution is suggested. The reason for the clustering instability is a combined effect of the particles inertia and a finite correlation time of the velocity field. The crucial parameter for the clustering instability is the size of the particles. The critical size is estimated for a strong clustering (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a weak clustering associated with the growth of the second and higher moments. A new concept of compressibility of the turbulent diffusion tensor caused by a finite correlation time of an incompressible velocity field is introduced. In this model of the velocity field, the field of Lagrangian trajectories is not divergence free. A mechanism of saturation of the clustering instability associated with the particles collisions in the clusters is suggested. Applications of the analyzed effects to the dynamics of droplets in the turbulent atmosphere are discussed. An estimated nonlinear level of the saturation of the droplets number density in clouds exceeds by the orders of magnitude their mean number density. The critical size of cloud droplets required for cluster formation is more than $20\hspace{0.5em}\mu \mathrm{m}.$