Turbulent super-diffusion as a ballistic cascade

etc15 Tracking Number 180

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. The present work proposes a simple physical phenomenology for the Richardson super-diffusivity in turbulence based on a scale dependent \emph{ballistic} scenario rather than a scale dependent \emph{diffusive} scenario. It is shown that 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 scale-dependent ballistic separations, (ii) it simply relates the Richardson constant to the Kolmogorov constant (and eventually to a ballistic persistence parameter), (iii) it gives a simple physical interpretation of the non-Fickian scale-dependent diffusivity coefficient as originally proposed by Richardson and (iv) a further extension of the phenomenology, taking into account higher order corrections to the local ballisitic motion, gives a robust interpretation of the assymetry between forward and backward dispersion, with an explicit connection to the energy flux accross scales.