Autogenic variability and dynamic steady-state in sand-bedded rivers

In sand-bedded rivers, the local physics of sediment transport produces spatially varying topography that evolves unpredictably in time, even when the structure of the stream-bed varies little in a statistical sense. Understanding autogenic adjustments within trains of bedforms under conditions of steady and uniform flow is necessary before we can predict the response of channel morphology to changes in flow conditions, e.g. the stage-discharge relationship. Also, dunes may coalesce to form bars, which are capable of laterally deflecting flow and ultimately modifying the path and shape of a channel. Bedforms are the link between sediment transport and channel morphology in sandy rivers, and their collective interactions maintain a dynamic steady-state on the river bottom. We document the evolution of fields of dunes under steady flow in the N. Loup River, NE, using topographic maps generated from low-altitude aerial photography. The distributions of bedform height, length and migration rate are broad (coefficient of variation 0.5 for each), but remain stationary in time. Individual bedforms, however, undergo substantial deformation during migration, through interactions with neighboring bedforms and the associated spatially varying sediment flux. Cross-correlation techniques show that the spatial/temporal correlation coefficient of the sediment-fluid interface decays exponentially with migration distance and time. Hence, the dunes themselves are inherently unstable objects and become unrecognizable from their original form after migrating a few wavelengths, corresponding here to a distance of 2 m and a time of 1 hour. If bedload is the dominant style of sediment transport, then sediment flux may be treated as responding instantaneously to the flow field. We build a simple mathematical model in which instantaneous sediment flux is computed locally from a combination of bed elevation and slope, and we deduce the general form of a surface evolution equation for bedforms. The goal is to capture the styles and rates of bed adjustments under steady flow, such as splitting and merging of individual bedforms, and to reproduce the spatial variability of bedform shape and size. Many qualitative aspects of bedform geometry and kinematics are reproduced, however, after long model times, a uniform field of periodic bedforms emerges. Nonlocal and stochastic (turbulent) effects of fluid flow are neglected in this treatment, and their inclusion might produce a field of continuously varying bed topography similar to what is observed in the field.