Width evolution of channel belts as a random walk

Channel belts, floodplains and fluvial valley floors form by the mobilization and deposition of sediments during the lateral migration of rivers. Channel-belt width and its temporal evolution is important for the hydraulics, hydrology, and ecology of floodplains, and 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 builds on the assumption that the switch 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 non-standard one-dimensional random walk. 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 first passage time and return to 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. Finally, we derive the distribution of sediment residence times, and show that its right-hand tail follows a power-law scaling with an exponent of -1.5. As such, the mean and variance of ages of sediment deposits in the channel belt do not converge to stable values, 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, and the interpretation of measured sediment ages. Our model predictions compare well to data of sediment-age distributions from various field sites and the temporal evolution channel belts observed in flume experiments. The theoretical description of the temporal evolution of channel-belt width developed herein provides a framework in which observational data can be interpreted, and may serve to connect models designed for long and short timescales.