Effects of a large-scale mean circulating flow on passive scalar statistics in a model of random advection

We study the effects of a large-scale mean circulating flow on passive scalar statistics in turbulent advection using a two-dimensional lattice model. The incompressible advecting velocity field consists of a large-scale circulation plus fluctuations. The latter are modeled by a random Gaussian field that has a finite correlation time but is statistically independent at different lattice points. In the presence of the large-scale flow, we find that the profiles of both the mean and the rms fluctuation of the passive scalar are modified significantly, and have close resemblance to those observed experimentally in turbulent convection. The one-point probability density functions (PDFs) of the passive scalar at the two sides are affected by the large-scale flow and become more skewed. There is, however, not much change for the PDFs at other locations. Furthermore, the shape of the PDFs of the scalar increments remains the same with or without the presence of the large-scale flow, supporting the idea that large-scale effects can be filtered out by studying increments.