Two-particle dispersion in 2D inverse cascade turbulence and its telegraph equation model

Abstract

How the distance between two fluid particles advected by a turbulent flow evolves in time is one of the fundamental questions in turbulence research. The final goal of this two-particle dispersion problem is to describe the probability density function (PDF) of the dispersion P(r, t), which gives probability to have a pair of particles whose relative distance is r at time t. L.F. Richardson made the first attempt to phenomenologically derive an equation of P(r, t)

$$\partial_{t}P = \partial_{r}[r^{d-1} K (r)\partial_{r} (P/r^{d - 1})]$$

where d is the spatial dimension and the diffusion coefficient is given by the inertial range scaling as $K(r) \propto \epsilon^{1/3}r^{4/3}$ with the energy dissipation rate $\epsilon$.