Turbulent "dynamo" in the passive scalar problem

A turbulent magnetic dynamo can be considered as the evolution of a vector field in a turbulent fluid flow. The problem of evolution of scalar fields (e.g., number density of small particles) in a turbulent fluid flow is similar to the turbulent magnetic dynamo. The. dynamo instability results in generation of magnetic field. The most important effect which can cause a generation of mean magnetic field in a turbulent fluid flow is the α-effect: α = -(1/3)〈τu · (Δ × u)〉, where u is the turbulent velocity field with the correlation time τ. A similar instability in the passive scalar problem results in formation of large-scale inhomogeneous structures in a spatial distribution of particles due to the β-effect: β = 〈τu_p(Δ · u_p)〉, where u_p is the random velocity field of the particles which they acquire in a turbulent fluid velocity field. The effect is caused by inertia of particles which results in divergent velocity field of the particles. This results in additional turbulent nondiffusive flux of particles. The mean-field dynamics of inertial particles are studied by considering the stability of the equilibrium solution of the derived evolution equation for the mean number density of the particles in the limit of large Péclet numbers. The resulting equation is reduced to an eigenvalue problem for a Schrödinger equation with a variable mass, and a modified Rayleigh-Ritz variational method is used to estimate the lowest eigenvalue (corresponding to the growth rate of the instability). This estimate is in good agreement with obtained numerical solution of the Schrödinger equation. Similar effects arise during turbulent transport of gaseous admixtures (or light noninertial particles) in a low-Mach-number compressible fluid flow. The discussed effects are important in planetary and atmospheric physics (cloud formation, pollutant dynamics, preferential concentration of particles in protoplanetary disks and also planetesimals in them).