On Turbulent Particle Pair Diffusion

Richardson's theory of turbulent particle pair diffusion [Richardson, L. F. Proc. Roy. Soc. Lond. A 100, 709--737, 1926], based upon observational data, is equivalent to a locality hypothesis in which the turbulent pair diffusivity $(K)$ scales with the pair separation $(\sigma_l)$ with a 4/3-power law, $K\sim \sigma_l^{4/3}$. Here, a reappraisal of the 1926 dataset reveals that one of the data-points is from a molecular diffusion context; the remaining data from geophysical turbulence display an unequivocal non-local scaling, $K \sim \sigma_l^{1.564}$. Consequently, the foundations of pair diffusion theory have been re-examined, leading to a new theory based upon the principle that both local and non-local diffusional processes govern pair diffusion in homogeneous turbulence. Through a novel mathematical approach the theory is developed in the context of generalised power law energy spectra, $E(k)\sim k^{-p}$ for $1<p\le 3$, over extended inertial subranges. The theory predicts the scaling, $K(p)\sim \sigma_l^{\gamma_p}$, with $\gamma_p$ intermediate between the purely local and the purely non-local scalings, i.e. $(1+p)/2<\gamma_p\le 2$. A Lagrangian diffusion model, Kinematic Simulations [Kraichnan, R. H., Phys. Fluids 13, 22-31, 1970; Fung et al., J. Fluid Mech. 236, 281-318, 1992], is used to examine the predictions of the new theory all of which are confirmed. The simulations produce the scalings, $K\sim \sigma_l^{1.545}$ to $\sim \sigma_l^{1.570}$, in the accepted range of intermittent turbulence spectra, $E(k)\sim k^{-1.72}$ to $\sim k^{-1.74}$, in close agreement with the revised 1926 dataset.