ALBERT

Beschreibung: Here we consider the 1-D morphodynamics of an erodible bed subject to bedload transport. Fluvial bed elevation variation is typically modeled by the Exner equation which, in its classical form, expresses mass conservation in terms of the divergence of the bedload sediment flux. An entrainment form of the Exner equation can be written as an alternative description of the same bedload processes, by introducing the notions of an entrainment rate into bedload and of a particle step length, and assuming a certain probability distribution for the step length. This entrainment form implies some degree of non-locality which is absent from the standard flux form, so that these two expressions, which are different ways to look at same conservation principle (i.e. sediment continuity), may no longer become equivalent in cases when channel complexity and flow conditions allow for long particle saltation steps (including, but not limited to the case where particle step length has a heavy tailed distribution) or when the domain of interest is not long compared to the step length (e.g. laboratory scales, or saltation over relatively smooth surfaces). We perform a systematic analysis of the effects of the non-locality in the entrainment form of Exner equation on transient aggradational/degradational bed profiles by using the flux form as a benchmark. As expected, the two forms converge to the same results as the step length converges to zero, in which case non-locality is negligible. As step length increases relative to domain length, the mode of aggradation changes from an upward-concave form to a rotational, and then eventually a downward-concave form. Corresponding behavior is found for the case of degradation. These results may explain anomalously flat aggradational long profiles that have been observed in some short laboratory flume experiments.

Beschreibung: The 1-D saltation-abrasion model of channel bedrock incision of Sklar and Dietrich, in which the erosion rate is buffered by the surface area fraction of bedrock covered by alluvium, was a major advance over models that treat river erosion as a function of bed slope and drainage area. Their model is, however, limited because it calculates bed cover in terms of bedload sediment supply rather than local bedload transport. It implicitly assumes that as sediment supply from upstream changes, the transport rate adjusts instantaneously everywhere downstream to match. This assumption is not valid in general, and thus can give rise unphysical consequences. Here we present a unified morphodynamic formulation of both channel incision and alluviation which specifically tracks the spatiotemporal variation of both bedload transport and alluvial thickness. It does so by relating the cover fraction not to a ratio of bedload supply rate to capacity bedload transport, but rather to the ratio of alluvium thickness to a macro-roughness characterizing the bedrock surface. The new formulation predicts waves of alluviation and rarification, in addition to bedrock erosion. Embedded in it are three physical processes: alluvial diffusion, fast downstream advection of alluvial disturbances and slow upstream migration of incisional disturbances. Solutions of this formulation over a fixed bed are used to demonstrate the stripping of an initial alluvial cover, the emplacement of alluvial cover over an initially bare bed and the advection–diffusion of a sediment pulse over an alluvial bed. A solution for alluvial-incisional interaction in a channel with a basement undergoing net rock uplift shows how an impulsive increase in sediment supply can quickly and completely bury the bedrock under thick alluvium, so blocking bedrock erosion. As the river responds to rock uplift or base level fall, the transition point separating an alluvial reach upstream from an alluvial-bedrock reach downstream migrates upstream in the form of a "hidden knickpoint". A solution for the case of a zone of rock subsidence (graben) bounded upstream and downstream by zones of rock uplift (horsts) yields a steady-state solution that is unattainable with the original saltation-abrasion model. A solution for the case of bedrock-alluvial coevolution upstream of an alluviated river mouth illustrates how the bedrock surface can be progressive buried not far below the alluvium. Because the model tracks the spatiotemporal variation of both bedload transport and alluvial thickness, it is applicable to the study of the incisional response of a river subject to temporally varying sediment supply. It thus has the potential to capture the response of an alluvial-bedrock river to massive impulsive sediment inputs associated with landslides or debris flows.

Beschreibung: Here we consider the 1-D morphodynamics of an erodible bed subject to bedload transport. Fluvial bed elevation variation is typically modeled by the Exner equation, which, in its classical form, expresses mass conservation in terms of the divergence of the bedload sediment flux. An entrainment form of the Exner equation can be written as an alternative description of the same bedload processes, by introducing the notions of an entrainment rate into bedload and of a particle step length, and assuming a certain probability distribution for the step length. This entrainment form implies some degree of nonlocality, which is absent from the standard flux form, so that these two expressions, which are different ways to look at same conservation principle (i.e., sediment continuity), may no longer become equivalent in cases when channel complexity and flow conditions allow for long particle saltation steps (including, but not limited to the case where particle step length has a heavy tailed distribution) or when the domain of interest is not long compared to the step length (e.g., laboratory scales, or saltation over relatively smooth surfaces). We perform a systematic analysis of the effects of the nonlocality in the entrainment form of the Exner equation on transient aggradational/degradational bed profiles by using the flux form as a benchmark. As expected, the two forms converge to the same results as the step length converges to zero, in which case nonlocality is negligible. As step length increases relative to domain length, the mode of aggradation changes from an upward-concave form to a rotational, and then eventually a downward-concave form. Corresponding behavior is found for the case of degradation. These results may explain anomalously flat, aggradational, long profiles that have been observed in some short laboratory flume experiments.