The effect of Gaussian particle‐pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence

The consequences for the theory of concentration fluctuations of different proposals by L. F. Richardson and G. K. Batchelor about the rate of separation of pairs of marked particles (giving rise to non-Gaussian and Gaussian particle separation probability density functions respectively) are explored via two Lagrangian Monte Carlo models. The models are applied to an instantaneous one-dimensional cloud source (which is approximately equivalent to a continuous line source) and in many respects give similar results for concentration fluctuations. The crucial difference is that whereas the non-Gaussian model predicts, in agreement with observation, that at large time the fluctuations remain the same order of magnitude as the mean field (the ratio depending only on the source size), the Gaussian model incorrectly predicts that fluctuations ultimately vanish compared with the mean field. The reason for the failure of the Gaussian model is explored by partitioning the total fluctuations into contributions due to the variation of the distribution of material within the cloud (i.e. in coordinates relative to the centre-of-mass) and due to motions of the cloud as a whole (meandering). It is shown that at large time the Gaussian model accounts only for the meandering contribution to fluctuations, and by smoothing out all internal structure of the doud eliminates relative fluctuations.