Dynamics of Bedload Anomalous Transport: A Nonstationary Lévy Stochastic Process and Upscaling Model

Anomalous bedload transport has been widely observed in both laboratory settings and natural rivers. Previous studies have utilized the fractional advection‐diffusion equation (FADE) model to characterize the observed anomalous bedload transport behaviors, with the model's parameters being determined by the tail distributions of the probability density functions (PDFs) for particle resting times and step lengths. FADE model captures sub‐diffusion and super‐diffusion behavior resulting from the heavy‐tailed PDFs of step lengths and resting times; however, it overlooks the temporal variability of advection and diffusion coefficients throughout the transport process. This study developes a novel microscopic stochastic process that incorporates nonstationary Lévy motion. The scaling limit of the stochastic process leads to the development of a hybrid Hausdorff‐fractional model. In the upscaling model, sub‐diffusion is represented by a time‐metric transformation, while super‐diffusion is characterized by nonlocal particle movements. Applications show that the novel model serves a generalized tool to characterize anomalous bedload transport processes across various scales. The upscaling model bridges the microscopic stochastic particle dynamics and macroscopic bedload transport behavior, and the microscopic stochastic framework provides a comprehensive physical interpretation for each parameter in the macroscopic equations, thereby enhancing the understanding of collective particle dynamics.