A power-law approximation for fluvial incision by tools and bed coverage processes

The stream-power model is widely used to represent fluvial incision in bedrock channels. The model does not account for the amount of sediment in the channel, which can abrade the channel at low concentrations or armor the channel at high concentrations. Here we use a natural example (Clearwater River, Washington State, USA) and numerical experiments to explore how sediment flux influences bedrock incision at a drainage-wide scale. We have generated numerical landscapes with different uplift patterns using the CHILD numerical model and incision rules that include a tools-and-coverage formulation. We then use regression analysis to fit a power-law function I=K*Am*Sn*, where I is incision rate, S slope, and A drainage area, and K*, m*, and n* are fit parameters. We find that this formulation works very well for the Clearwater and all of our numerical experiments. The function has the same form as the stream-power model, but the parameters are empirically defined (as indicated by the asterisks) and can take on values quite different than those inferred from process-based arguments. The best-fit parameters appear to be constant at the scale of a single drainage, but they vary between drainages depending on the pattern of uplift, and whether or not the landscape has reached steady-state. In all cases, slope-area steepness analysis works well for estimating relative incision rates. Our analysis indicates that, in some cases, m* can be quite low, apparently due to the fact that bed coverage increases with increasing area. We conclude that the power-law formulation provides a good functional representation of fluvial incision, but that there are no universal values for m* and n*. These conclusions have important implications for the size of mountain belts and feedbacks between tectonic uplift and surface processes.