A physics-based model for fluvial valley width

Abstract

The width of fluvial valley floors is a key parameter to quantifying the morphology of mountain regions. Valley floor width is relevant to diverse fields including sedimentology, fluvial geomorphology, and archaeology. The width of valleys 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 valley walls. Here, we derive a physically based model that can be used to predict valley width and test it against three different datasets. The model applies to valleys that are carved by a river migrating laterally across the valley floor. We conceptualize river migration as a Poisson process, in which the river changes its direction stochastically at a mean rate determined by hydraulic boundary conditions. This approach yields a characteristic timescale for the river to cross the valley floor from one wall to the other. The valley width can then be determined by integrating the speed of migration over this timescale. For a laterally unconfined river that is not uplifting, the model predicts that the channel-belt width scales with river flow depth. Channel-belt width corresponds to the maximum width of a fluvial valley. We expand the model to include the effects of uplift and lateral sediment supply from valley walls. Both of these effects lead to a decrease in valley width in comparison to the maximum width. We identify a dimensionless number, termed the mobility–uplift number, which is the ratio between the lateral mobility of the river channel and uplift rate. The model predicts two limits: at high values of the mobility–uplift number, the valley evolves to the channel-belt width, whereas it corresponds to the channel width at low values. Between these limits, valley width is linked to the mobility–uplift number by a logarithmic function. As a consequence of the model, valley width increases with increasing drainage area, with a scaling exponent that typically has a value between 0.4 and 0.5, but can also be lower or higher. We compare the model to three independent datasets of valleys in experimental and natural uplifting landscapes and show that it closely predicts the first-order relationship between valley width and the mobility–uplift number.