Use of the Entrainment Form of the Exner Equation to Describe Effects of Patchy, Intermittent, Rarefied, Long-Distance Sediment Motions on Steepland Hillslopes

Abstract:

Sediment particle motions on steepland hillslopes often are patchy, intermittent and rarefied. Such motions include particle skittering from rockfall, particle ravel, downslope dispersal of soil material due to tree throw and the activity of fossorial animals, soil slips, and transport by surface flows, particularly following fire. We may envision these as motions that start from the soil surface and then return to it after some finite time. Travel distances can be much larger than the soil thickness. Unless explicitly treated at the transport 'event' scale, the discontinuous qualities of these sediment motions in steeplands are at odds with continuum formulations of the sediment flux and the Exner equation of conservation as normally envisioned. The necessary averaging volumes and timescales are too large. Alternatively, the entrainment form of the Exner equation, when cast in probabilistic terms, provides a framework for describing the averaged effects of patchy, intermittent motions on the land surface. The formulation is mass conserving, "nonlocal" and scale independent, and it therefore does not require an averaging volume. However, there is a tradeoff. One must specify, based on theory or measurements, an ensemble distribution of particle travel distances that is to be considered a signature of the operative transport processes and extant hillslope conditions. Transport events (specifically, sediment travel distances) occurring over a specified interval of time represent "samples" drawn from this ensemble distribution. Predictions of changes in land-surface elevation are then statistically expected values, with associated uncertainty that depends on the frequency of transport. A growing set of field-based and high-resolution DEM measurements suggests that the nonlocal formulation is consistent with observed (one-dimensional) hillslope profiles in steeplands. We are extending the formulation to two-dimensional topography; initial results suggest a rich behavior that is unlike that predicted by local formulations of transport. Because of its general, scale-independent form, the basic formulation is virtually identical to the entrainment form of the Exner equation used in describing bed load sediment transport.