An idealised model of turbulent dispersion: two rectangular pulse initial condition

We examine an idealised model of turbulent dispersion which was introduced by Zimmerman and Chatwin. It involves deterministic diffusion in a periodic one-dimensional domain, for a specified initial concentration field. A stochastic element is introduced by sampling at random across the domain, equivalent to random advection by a velocity which does not vary spatially. We consider initial conditions consisting of a single rectangular pulse, displaced a distance X0 from the centre of the domain. We present numerical results for the dependence of the variance, skewness, kurtosis and probability density function (pdf) of concentration on X0 and on time, and we derive the large time asymptotic forms for these quantities. At large time, the pdf is inevitably bimodal, with the peaks at the smallest and largest concentrations. To allow for a range of possible distances between strands of high concentration, as would be expected in a real turbulent flow, we suggest a new model pdf for concentration, obtained by averaging over different values of X0. For a uniform distribution of X0 values, we show that this gives a unimodal concentration pdf at large time, with the peak at the mean concentration.