  Tracers in a turbulent flow separate according to the celebrated $t^{3/2}$
Richardson--Obukhov law, which is usually explained by a scale-dependent
effective diffusivity. Here, supported by state-of-the-art numerics, we revisit
this argument. The Lagrangian correlation time of velocity differences is found
to increase too quickly for validating this approach, but acceleration
differences decorrelate on dissipative timescales. This results in an
asymptotic diffusion $\propto t^{1/2}$ of velocity differences, so that the
long-time behavior of distances is that of the integral of Brownian motion. The
time of convergence to this regime is shown to be that of deviations from
Batchelor's initial ballistic regime, given by a scale-dependent energy
dissipation time rather than the usual turnover time. It is finally argued that
the fluid flow intermittency should not affect this long-time behavior of
relative
