In the idealised model of turbulent dispersion introduced by Zimmerman and Chatwin (1995), the probability density function (pdf) of concentration becomes bimodal at large times, with peaks at the smallest and largest concentrations. That model only has one spatial dimension, but here I extend the model to two and three spatial dimensions. I also extend to two and three dimensions the pdf calculation method that was given by Mole and Yeun (2007). I use this method to derive large-time analytical solutions for the pdf, and to show that in two and three dimensions the weight of the pdf shifts from extreme concentrations towards the mean concentration. In the typical three-dimensional case, the large-time pdf is unimodal with the peak at the mean concentration. Copyright © 2010 John Wiley & Sons, Ltd.