Mean-field theory for a passive scalar advected by a turbulent velocity field with a random renewal time - art. no. 026304

Mean-field theory for turbulent transport of a passive scalar (e.g., partic les and gases) is discussed. Equations for the mean number density of parti cles advected by a random velocity field, with a finite correlation time, a re derived. Mean-field equations for a passive scalar comprise spatial deri vatives of high orders due to the nonlocal nature of passive scalar transpo rt in a random velocity field with a finite correlation time. A turbulent v elocity 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. [Ph ys. Rev. E 61, 2617 (2000)], and employs two characteristic times: the corr elation time of a random velocity field tau (c), and a mean renewal time ta u. It is demonstrated that the turbulent diffusion coefficient is determine d by the minimum of the times tau (c) and tau. The mean-field equation for a passive scalar was derived for different ratios of tau/tau (c). The impor tant role of the statistics of the field of Lagrangian trajectories in turb ulent transport of a passive scalar, in a random velocity field with a fini te correlation time, is demonstrated. It is shown that in the case the form of the mean-field equation for a passive scalar is independent of the stat istics of the velocity field, where tau (N) is the characteristic time of v ariations of a mean passive scalar field.