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A first-order autoregressive (AR(1)) model is analysed to investigate the relationship between a time dependent random variable Γ t and its derivative . This is motivated by the problem of the turbulent dispersion of a gas cloud or plume in the atmosphere, where the square of the derivative, , is also of interest. It is shown that in general Γ t and D t are correlated, with their correlation proportional to the skewness of Γ t. The probability distributions of the innovations and of Y t and D t, and the joint distributions of Γ t and Y t, and of Γ t and D t, are derived for a number of assumed distributions for Γ t. The latter include the normal, exponential, gamma (of integer order), uniform, and sum of uniform distributions. The lack of time reversibility of the AR(1) model is apparent in the asymmetric distributions of the derivative when Γ t has the exponential or gamma distribution. Comparisons are made with some experimental observations. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the idealised model of turbulent dispersion introduced by Zimmerman and Chatwin (), 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 (). 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. [ABSTRACT FROM AUTHOR]