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

TL;DR

This paper explores how heavy-tailed jump processes influence large-scale transport in turbulent flows, revealing a transition from anomalous super-diffusion to classical diffusion depending on the tail behavior of the noise.

Contribution

It demonstrates the impact of heavy-tailed versus truncated noise on transport properties, highlighting a transition from anomalous to normal diffusion in turbulent systems.

Findings

Heavy-tailed stable processes cause super-diffusive transport.

Truncated or tempered noise leads to classical diffusion.

Large jumps dominate transport in heavy-tailed scenarios.

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.