  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.
