  The statistics of Lagrangian pair dispersion in a homogeneous isotropic flow
is investigated by means of direct numerical simulations. The focus is on
deviations from Richardson eddy-diffusivity model and in particular on the
strong fluctuations experienced by tracers. Evidence is obtained that the
distribution of distances attains an almost self-similar regime characterized
by a very weak intermittency. The timescale of convergence to this behavior is
found to be given by the kinetic energy dissipation time measured at the scale
of the initial separation. Conversely the velocity differences between tracers
are displaying a strongly anomalous behavior whose scaling properties are very
close to that of Lagrangian structure functions. These violent fluctuations are
interpreted geometrically and are shown to be responsible for a long-term
memory of the initial separation. Despite this strong intermittency, it is
found that the mixed moment defined by the ratio between the cube of the
longitudinal velocity difference and the distance attains a statistically
stationary regime on very short timescales. These results are brought together
to address the question of violent events in the distribution of distances. It
is found that distances much larger than the average are reached by pairs that
have always separated faster since the initial time. They contribute a
stretched exponential behavior in the tail of the inter-tracer distance
probability distribution. The tail approaches a pure exponential at large
times, contradicting Richardson diffusive approach. At the same time, the
distance distribution displays a time-dependent power-law behavior at very
small values, which is interpreted in terms of fractal geometry. It is argued
and demonstrated numerically that the exponent converges to one at large time,
again in conflict with Richardson's distribution.
