A random walk perspective on channel belts and fluvial valleys: model and evidence

The width of channel belts and fluvial valleys and its temporal evolution is important for the hydraulics, hydrology, and ecology of landscapes, and for human activities such as farming, protecting infrastructure, and natural hazard mitigation. Channel belts form by the mobilization and deposition of sediments during the lateral migration of rivers. Similarly, the width of a fluvial valley is set by the river undercutting valley walls and evacuating the resulting sediment. Both channel belt and valley width thus depend on the process of lateral channel migration. We have recently developed a model that predicts a steady-state valley or channel-belt width and its temporal evolution. The model builds on the assumption that the switching of direction of a laterally migrating channel can be described by a Poisson process, with a constant rate parameter related to channel hydraulics. As such, the channel’s lateral migration can be viewed as a non-standard one-dimensional random walk. The model connects channel belt and valley evolution to reach-scale hydraulic parameters. In addition to steady state scaling and the average temporal evolution of valley width, it predicts a range of results on the landscape evolution scale, for example, the age distribution of sediment (equivalent to the distribution of return times to the origin), and bounds on the area that the river is unlikely to migrate beyond (the law of the iterated logarithm). Here, we summarize some key model results and compare model predictions to observations of natural and experimental rivers. First, we demonstrate that a random walk process is a reasonable description of the evolution of channel belt width, because the channel belt width increases with the square root of time and the distribution of return times to the origin has a -3/2 scaling . Second, we argue that the observed downstream scaling of valley or channel-belt width with drainage area is consistent with our model predictions. Third, we show that steady state width of fluvial valleys, as observed in field and experimental data, scales with uplift rate and channel mobility as predicted by the model. Finally, we point out further avenues to test the model and constrain parameters using additional field data.