Turbulent particle pair diffusion: A theory based on local and non-local diffusional processes

A re-Appraisal of the Richardson's 1926 dataset [Richardson, L. F. Proc. Roy. Soc. Lond. A 100, 709±737, (1926)] displays an unequivocal non-local scaling for the pair diffusion coefficient, K ∼ σ ₁ ^4/3 , quite different to the previously assumed locality scaling law ∼ σ ^4/3 i , where σi is the pair separation. Consequently, the foundations of turbulent pair diffusion theory are re-examined here and it is shown that pair diffusion is governed by both local and non-local diffusional processess inside the inertial subrange. In the context of generalised energy spectra, E(k) ∼ k-P for 1 < p ≤ 3, the new theory predicts two non-Richardson regimes depending on the size of the inertial subrange: (1) in the limit of asymptotically infinite subrange, non-local scaling laws is obtained, K ∼ σγ i , with γ intermediate between the purely local and the purely non-local scalings, i.e. (1+p)/2 < γ ≤ 2; and (2) for finite (short) inertial subrange, local scaling laws are obtained, K < γ ^{(1+.p)./ 2} . The theory features a novel mathematical approach expressing the pair diffusion coefficient through a Fourier integral decomposition.