Width evolution of channel belts as a random walk

Abstract

Channel belts form through the mobilization and deposition of sediments during the lateral migration of rivers. Channel-belt width and its temporal evolution are important for the hydraulics, hydrology, and ecology of landscapes, as well as for human activities such as farming, protecting infrastructure, and natural hazard mitigation. Yet, we currently lack a comprehensive theoretical description of the width evolution of channel belts. Here, we explore the predictions of a physics-based model of channel-belt width for the transient evolution of channel belts. The model applies to laterally unconfined channel belts in foreland areas as well as to laterally confined channel belts in mountain settings (here, channel-belt width equals valley floor width). 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 lateral migration of the channel can be viewed as a nonstandard one-dimensional random walk. In other words, at each river cross section the river randomly moves either to the left or right at a given time. The model predicts three phases in the growth of channel belts. First, before the channel switches direction for the first time, the channel belt grows linearly. Second, as long as the current width is smaller than the steady-state width, growth follows an exponential curve on average. Finally, there is a drift phase, in which the channel-belt width grows with the square root of time. We exploit the properties of random walks to obtain equations for the distance from a channel that is unlikely to be inundated in a given time interval (law of the iterated logarithm), distributions of times the channel requires to return to its origin and to first arrive at a given position away from the origin, and the mean lateral drift speed of steady-state channel belts. All of the equations can be directly framed in terms of the channel's hydraulic properties, in particular its lateral transport capacity that quantifies the amount of material that the river can move in lateral migration per unit time and channel length. The distribution of sediment age within the channel belt is equivalent to the distribution of times to return to the origin, which has a right-hand tail that follows a power-law scaling with an exponent of -3/2. As such, the mean and variance of ages of sediment deposits in the channel belt do not converge to stable values over time but depend on the time since the formation of the channel belt. This result has implications for storage times and chemical alteration of floodplain sediments, as well as the interpretation of measured sediment ages. Model predictions compare well to data of sediment age distributions measured at field sites and the temporal evolution of channel belts observed in flume experiments. Both comparisons indicate that a random walk approach adequately describes the lateral migration of channels and the formation of channel belts. The theoretical description of the temporal evolution of channel-belt width developed herein can be used for predictions, for example, in hazard mitigation and stream restoration, and to invert fluvial strata for ambient hydraulic conditions. Further, it may serve to connect models designed for either geological or process timescales.