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A mathematical model is advanced to simulate dynamically and spatially varied shallow water flow and soil detachment, transport, and deposition in rills. The model mimics the dynamic process of rill evolution, including variable rates of sediment redistribution along the bed and changes in local bed morphology. The sediment source term in the model uses a point scale, probabilistic relationship based on turbulent flow mechanics and a recently developed sediment transport relationship for rills based on stream power. The interdependent feedback loops between channel bed morphology, local flow hydraulics, and local scour and deposition, within the framework of the full hydrodynamic equations with inertial terms, constitute a mathematical model with the capacity to represent spatial variability and temporal evolution of the rill. Finite elements were applied to numerically solve the hydrodynamic and sediment continuity equations. A series of laboratory flume experiments were performed to evaluate the model. Initial bed slopes were 3, 5, and 7% with step increases of water inflow rates of 7.6, 11.4, and 15.2 L min−1. The soil material used in the flume was a kaolinitic, sandy‐clay loam. The rill model equations were solved for increasingly complex cases of spatial and temporal variabilities. The model followed measured patterns of morphological changes as the rill evolved, which suggests that the feedback loops in the model between erosion, bed morphological changes, and hydraulics were adequate to capture the essence of rill evolution.

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Eroding rills evolve morphologically in time and space. Most current studies on rill erosion use spatially averaged soil erosion data, providing little information on soil erosion dynamics. A method is proposed to use rare earth elements (REEs) to trace sediment distribution in eroding rills. Laboratory flume simulation experiments were conducted at three flow rates (2, 4 and 8 litres minute À1 ) and five slope gradients (5, 10, 15, 20 and 25 ) with three replicates of each treatment. The rills, of 8 m length, were subdivided into 10 equal segments of 0.8 m length and 0.1 m width, with a different REE applied to each segment. We derived computational formulae for estimating the distribution of eroded amounts along the rills. The actual erosion distribution along rills was then estimated with the data from the experiments. The precision of the REEs for tracing rill erosion was analysed. The results showed that sediment concentration increased with rill length, but the increased rate (the slope of the curve) flattened gradually. Sediment yields increased with slope gradients and flow rates, but the slope gradients had a greater effect on sediment concentration than flow rates, and greater flow rates caused more rill erosion and soil loss under the same slope gradient. The results also demonstrated the feasibility of using REEs to trace the dynamic processes of rill erosion.

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