Passive scalar transport in a random flow with a finite renewal time: Mean-field equations

A mean-field equation for a passive scalar (e.g., for a mean number density of particles) in a random velocity field (incompressible and compressible) with a finite constant renewal time is derived. The finite renewal time of a random velocity field results in the appearance of high-order spatial derivatives in the mean-field equation for a passive scalar. We considered three models of a random velocity field: (i) a velocity field with a small renewal time; (ii) the Gaussian approximation for Lagrangian trajectories; and (iii) a small inhomogeneity of the velocity and mean passive scalar fields. For a small renewal time we recovered results obtained using the $\delta$-function-correlated in time random velocity field. The finite renewal time and compressibility of the velocity field can cause a depletion of turbulent diffusion and a modification of an effective drift velocity. For a compressible velocity field the form of the mean-field equation for a passive scalar depends on the details of the velocity field, i.e., the universality is lost. For an incompressible velocity field the universality exists in spite of the finite renewal time. Results by Saffman [J. Fluid Mech. 8, 273 (1960)] for the effect of molecular diffusivity in turbulent diffusion are generalized for the case of a compressible and anisotropic random velocity field. The obtained results may be of relevance in some atmospheric phenomena (e.g., atmospheric aerosols and smog formation).