Models of bed-load transport across scales: turbulence signature from grain motion to sediment flux

Sediment transport controls the evolution of river channels, playing a fundamental role in physical, ecological, and biogeochemical processes across a wide range of spatial and temporal scales on the Earth surface. However, developing predictive transport models from first principles and understanding scale interactions on sediment fluxes remain as formidable research challenges in fluvial systems. Here we simulate the smallest scales of transport using direct numerical simulations (DNS) to explore the dynamics of bed-load and discover how turbulence and grain-scale processes influence transport rates, showing that their interplay gives rise to a critical regime dominated by fluctuations that propagate across scales. These connections are represented using a stochastic differential equation, and a statistical description through a path integral formulation and Feynman diagrams, thus providing a framework that incorporates nonlinear and turbulence effects to model the dynamics of bed-load across scales.