Mean-field theory for a passive scalar advected by a turbulent velocity field with a random renewal time

Mean-field theory for turbulent transport of a passive scalar (e.g., particles and gases) is discussed. Equations for the mean number density of particles advected by a random velocity field, with a finite correlation time, are derived. Mean-field equations for a passive scalar comprise spatial derivatives of high orders due to the nonlocal nature of passive scalar transport in a random velocity field with a finite correlation time. A turbulent velocity field with a random renewal time is considered. This model is more realistic than that with a constant renewal time used by Elperin et al. [Phys. Rev. E 61, 2617 (2000)], and employs two characteristic times: the correlation time of a random velocity field (Formula presented) and a mean renewal time (Formula presented). It is demonstrated that the turbulent diffusion coefficient is determined by the minimum of the times (Formula presented) and (Formula presented). The mean-field equation for a passive scalar was derived for different ratios of (Formula presented) The important role of the statistics of the field of Lagrangian trajectories in turbulent transport of a passive scalar, in a random velocity field with a finite correlation time, is demonstrated. It is shown that in the case (Formula presented) the form of the mean-field equation for a passive scalar is independent of the statistics of the velocity field, where (Formula presented) is the characteristic time of variations of a mean passive scalar field.