Telegraph-type versus diffusion-type models of turbulent relative dispersion

Properties of two equations describing the evolution of the probability density function (PDF) of the relative dispersion in turbulent flow are compared by investigating their solutions: the Richardson diffusion equation with the drift term and the self-similar telegraph equation derived by Ogasawara and Toh [J. Phys. Soc. Jpn. 75, 083401 (2006)]. The solution of the self-similar telegraph equation vanishes at a finite point, which represents persistent separation of a particle pair, while that of the Richardson equation extends infinitely just after the initial time. Each equation has a similarity solution, which is found to be an asymptotic solution of the initial value problem. The time lag has a dominant effect on the relaxation process into the similarity solution. The approaching time to the similarity solution can be reduced by advancing the time of the similarity solution appropriately. Batchelor scaling, a scaling law relevant to initial separation, is observed only for the telegraph case. For both models, we estimate the Richardson constant, based on their similarity solutions.