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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

Morphodynamics of river bed variation with variable bedload step length