A process-based model for fluvial valley width

The width of fluvial valley-floors is a key parameter to quantifying morphology in mountain regions. It is important in diverse fields including sedimentology, fluvial geomorphology, and archaeology. Valley-floor width has been argued to depend on climatic and tectonic conditions, on the hydraulics and hydrology of the river channel that forms the valley, and on sediment supply from the valley walls. Yet, so far, a physically-based model that can be used to predict valley width is lacking. Here, we derive such a model. As has been done before, we assume that valleys are formed by the erosion of the valley walls by a river migrating across the valley floor. We conceptualize river migration as a Poisson process, in which the river changes its direction stochastically, at a rate determined by hydraulic boundary conditions. This approach yields a characteristic timescale of migration into a particular direction. The valley width can then be determined by integrating the speed of migration over this timescale. We develop equations for the timescale and the migration speed and predict that an unconfined valley or channel belt width scales with the square of the flow depth of the river. We expand the model to arrive a single equation that also includes the effects of uplift and lateral hillslope sediment supply. Both of these effects lead to a decrease in valley width in comparison to the unconfined width. The model predicts that at low uplift rates, the valley width tends to the unconfined width, and at high uplift rates to the channel width, and connects these two limits by a logarithmic equation. As a consequence, valley width increases with increasing drainage area, with a scaling exponent that typically lies in the range between 0.4 and 0.8, but can also be lower or higher. Finally, we compare the model to three data sets of valleys in uplifted regions and show that it closely predicts the first order relationship between valley width and uplift rate.