Upscaling Sediment‐Flux‐Dependent Fluvial Bedrock Incision to Long Timescales

Fluvial bedrock incision is driven by the impact of moving bedload particles. Mechanistic, sediment‐flux‐dependent incision models have been proposed, but the stream power incision model (SPIM) is frequently used to model landscape evolution over large spatial and temporal scales. This disconnect between the mechanistic understanding of fluvial bedrock incision on the process scale, and the way it is modeled on long time scales presents one of the current challenges in quantitative geomorphology. Here, a mechanistic model of fluvial bedrock incision that is rooted in current process understanding is explicitly upscaled to long time scales by integrating over the distribution of discharge. The model predicts a channel long profile form equivalent to the one yielded by the SPIM, but explicitly resolves the effects of channel width, cross‐sectional shape, bedrock erodibility, and discharge variability. The channel long profile chiefly depends on the mechanics of bedload transport, rather than bedrock incision. In addition to the imposed boundary conditions specifying the upstream supply of water and sediment, and the incision rate, the model includes four free parameters, describing the at‐a‐station hydraulic geometry of channel width, the dependence of bedload transport capacity on channel width, the threshold discharge of bedload motion, and reach‐scale cover dynamics. For certain parameter combinations, no solutions exist. However, by adjusting the free parameters, one or several solutions can usually be found. The controls on and the feedbacks between the free parameters have so far been little studied, but may exert important controls on bedrock channel morphology and dynamics.Plain Language Summary: Bedrock erosion by rivers is driven by the impact of moving sediment particles, chipping away tiny pieces of rocks in their passage. Sediment transport occurs infrequently, during floods. Over thousands of years, this slow process shapes the river, sometimes leading to the creation of spectacular landforms such as gorges. Mechanistic models of fluvial bedrock erosion explicitly take into account the effects of moving sediment particles, while models used for long time scales do not. Here, the connection between mechanistic and long‐term models is made explicit by integrating a mechanistic model over the entire distribution of floods, yielding solutions for the long‐term erosion rate and the channel bed slope. Some of these solutions are similar to those used previously, but other solutions are also possible, showing the rich dynamic behavior that rivers can exhibit. The solutions make explicit the role of lithology, channel width, and discharge variability, which were previously hidden in a single lumped calibration parameter.Key Points: Analytical solution from explicit upscaling of a sediment‐flux‐dependent fluvial bedrock incision model to long time scales. The model includes solutions similar to those obtained in the stream power paradigm, in addition to other possible solutions. The model explicitly resolves forcing behavior and highlights potential dynamic feedbacks that have so far not been considered.GF