A nonlocal theory of sediment buffering and bedrock channel evolution

Abstract

Bedrock erosion in mountain river channels ultimately sets the erosion rate of the surrounding hillslopes and the rate of sediment supply to the channels. The supply of coarse bed sediment acts as a dampening effect on further erosion by depositing an alluvial cover that temporarily obscures the bedrock. For landscapes where the residence time of the alluvial bed cover is comparable to the timescale of bedrock incision, coarse sediment supply and transport generate a strong negative feedback on fluvial downcutting and the coupled process of hillslope‐channel erosion is inherently self‐buffering. Here we study a simple model of self‐buffered bedrock channel erosion that incorporates the spreading of bed sediment cover downstream in a way that allows for a broad‐tailed, power law probability distribution of transport velocities of bed sediment over the long‐term. This leads us to consider a nonlocal transport law (fractional advection) parameterized by a scaling exponent 0 ≤ α < 1 which collapses to local advection for α → 1. For strong sediment buffering, we find that nonlocality 1 − α has a direct control on the power law scaling of channel slope S with upstream area A, giving S ∼ A−(1−α)/2 at steady state. Empirical observations of slope‐area scaling are consistent with α < 1 and nonlocal transport. In general, the model predicts linear, logarithmic, or power law stream profiles depending on the extent of buffering, the degree of nonlocality, and the scaling of the bedrock erosion law. It also predicts, somewhat counterintuitively, that bed cover should thicken with distance x downstream slower than linearly as xα, i.e., the more nonlocal the bed sediment spreading process (α → 0), the slower the bed cover increases downstream. We deduce that long‐range, heterogeneous transport of coarse sediment in mixed bedrock‐alluvial rivers may be a key element of landscape scaling and an important factor in landscape dynamics.