Turbulent pair dispersion as a ballistic cascade phenomenology

Abstract

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 article proposes a simple physical phenomenology for the time evolution of the mean square relative separation in turbulent flows, based on a scale dependent ballistic scenario rather than a scale dependent diffusive. This phenomenology accurately retrieves most of the known features of relative dispersion ; among others : (i) it is quantitatively consistent with recent numerical simulations and experiments (both for the short term Batchelor regime and the long term Richardson regime, and for all initial separations at inertial scales), (ii) it gives a simple physical explanation of the origin of the super diffusive t^3 Richardson regime which naturally builts itself as an iterative process of elementary short-term-scale-dependent ballistic steps, (iii) it shows that the Richardson constant is directly related to the Kolmogorov constant (and eventually to a ballistic persistence parameter) and (iv) in a further extension of the phenomenology, taking into account third order corrections, it robustly describes the temporal asymmetry between forward and backward dispersion, with an explicit connection to the cascade of energy flux across scales.