Anomalous diffusion properties of stochastic transport by heavy-tailed jump processes

Abstract

In this work, we investigate the large-scale transport properties of a passive scalar advected by a turbulent fluid, modelled as a superposition of divergence-free vector fields, each weighted by an independent symmetric $\alpha$-stable-like process. Motivated by recent works showing that complex small-scale spatial structures often lead to Brownian dispersion, we study if this principle persists when the driving noise exhibits heavy-tailed jump statistics. Our numerical results show a clear dichotomy linked with the tail behaviour of the noise. When considering standard $\alpha$-stable processes, very large jumps survive the interaction with the spatial complexity and yield anomalous, super-diffusive transport. In contrast, when the $\alpha$-stable noise is either truncated or exponentially tempered, suppressing extremely long jumps, the transport undergoes a transition to a classical diffusive regime.