Large-scale probability density function for scalar field advected by high Reynolds number turbulent flow

A closed equation for the one-point probability density function (PDF) fora scalar field advected by the three-dimensional random velocity field with arbitrary many spatial/ temporal scales and infrared divergence in the limit of high Reynolds number has been derived, using the functional derivative technique and renormalization theory, together-with the assumption that the turbulent velocity is a homogeneous, isotropic, Gaussian random field. It has been shown that when the spectral parameters of a random velocity field slightly deviate from their Kolmogorov-Obukhov values, the equation for the PDF in the long-time, large-distance limit can be derived exactly and well described by a conventional diffusion theory, while the Lagrangian scaling function describing the large-scale particle displacements in turbulent flow is essentially superdiffusive. The scaling procedure in the limit of high Reynolds number allows us to completely overcome the well known closure problem associated with diffusion term.