Turbulent pair dispersion as a ballistic cascade process

Since the pioneering work of Richardson in 1926, later refined by Batchelor and Obukhov in 1950, it is predicted that the rate of separation of pairs of fluid elements in turbulent flows with initial separation at inertial scales, grows ballistically first (Batchelor regime), before undergoing a transition towards a super-diffusive regime where the mean-square separation grows as t^3 (Richardson regime). Richardson empirically interpreted this super- diffusive regime in terms of a non-Fickian process with a scale dependent diffusion coefficient (the celebrated Richardson's ``4/3rd'' law). However, the actual physical mechanism at the origin of such a scale dependent diffusion coefficient remains unclear. I will present a  simple physical phenomenology for the Richardson super-diffusivity in turbulence based on a scale dependent ballistic scenario rather than a scale dependent diffusive scenario. This phenomenology elucidates several aspects of turbulent dispersion: (i) it gives a simple physical explanation of the origin of the super diffusive t^3 Richardson regime as an iterative cascade of successive scale-dependent ballistic separations, (ii) it is quantitatively consistent with most recent numerical simulations for pair dispersion, (iii) it shows that the Richardson constant is directly related to the Kolmogorov constant (and eventually to a ballisitic persistence parameter), (iv) it gives a simple physical interpretation of the non-Fickian scale-dependent diffusivity coefficient as originally proposed by Richardson and (v) a simple extension of the model quantitatively predicts the long-term temporal asymmetry of relative dispersion in turbulence with no other additional parameters. Besides, the present phenomenology builds an explicit connection between the Lagrangian problem of pair dispersion and the usual picture of energy cascade of turbulence in the Eulerian framework.