## 1. Introduction

[2] Bed sediment transport in rivers is scale-dependent, with anomalous scaling observed at early-to-intermediate time scales [*Jerolmack*, 2011; *Nikora et al.*, 2002]. In regular dispersion (Fickian), plumes are symmetric and plume variance grows proportionally to elapsed time. *Einstein* [1937]developed a random walk model for fluvial sediment transport assuming that particle transport events are independent, and both particle jump length and resting time distributions have finite moments. These conditions lead to regular dispersion. However, particle inertia contributes to initial correlated ballistic motion, and hydrodynamic forces can accelerate mobile particles, resulting in super-dispersion where the observed variance grows faster than linearly [*Martin et al.*, 2012]. *Bradley et al.* [2010] extended *Einstein*'s model to include super-dispersion, and showed that mobile particles remain super-diffusive over long timescales in sand-bed rivers. Particles can also be trapped in the bed, leading to sub-dispersion [*Nikora et al.*, 2002]. Trapping timescales range from minutes for turnover of small bedforms to thousands of years for meander migration [*Aalto et al.*, 2008]. On sufficiently long time scales, averaging over the range of variability in particle motion causes a relaxation to regular dispersive behavior. Based on these considerations, we propose three distinct sediment transport regimes: the early time scale, where particles exhibit super-diffusion; the intermediate scale, during which particles exhibit sub-diffusion; and the late time scale with Fickian dispersion.

[3] Here we provide a unified theory for bed sediment transport dynamics in the form of a tempered space-time random walk model. Tempered power law jumps and waiting times lead to a tempered fractional mobile-immobile model that captures particle dynamics in all three transport regimes as well as transitions between regimes (section 2). The microscopic random walk model is used to assign a specific physical meaning to each parameter in the macroscopic equations describing ensemble particle motion. We use this model to unify prior observations of laboratory- and field-scale bed sediment transport dynamics, and show that previously observed scale dependency in particle dispersion behavior is a natural consequence of averaging of the underlying particle motion (section 3). We then discuss the main factors responsible for anomalous transport, the implications for interpreting observations of sediment transport at different scales, and the general utility of the approach for related geophysical and ecological problems (section 4).