Channel Motion as a Random Walk: Erosion Probabilities and Implications for Sediment Residence Time
Abstract
Stochastic models of fluvial systems require an estimate of the erosion probability of sedimentary deposits in order to predict particle trajectories, the distribution of sediment residence time, the time a grain takes to transit the system, and the time to overturn all the sediment in a valley. We can use this information to test the assumptions made in detrital geochronology, to predict the transport and dispersion of solidphase contaminants, and to predict the fate of sudden inputs of sediment. Unfortunately, estimating the erosion probability is difficult. It is possible to determine the average erosion probability if the mean residence time of the sediment in a deposit is known or if the mass of the deposit and the sediment flux out of it can be determined. However, even if these things are known, the average erosion probability does not capture the potential variability in the system. It is more desirable to know the distribution of erosion probability. We examine a simplified fluvial system with a onedimensional, singlethreaded, meandering channel that migrates by point bar deposition and cut bank erosion. There is no net aggradation or incision and overbank deposition is neglected. In this simple model, the probability that a sediment grain in the valley is eroded is equal to the probability that the channel occupies the grain's location. The probability density function (PDF) of erosion is identical to the PDF of channel position. We treat the channel motion as a random walk, allowing us to make predictions about the PDF of channel position for two endmember cases of meandering behavior. When the channel takes small steps relative to the valley width and the motion is symmetrical about the valley axis, the PDF of channel position is a Gaussian centered on the valley axis. Alternately, if the motion of the channel is dominated by meander cutoffs, then the channel location can change by increments that that are similar to the valley size. In this case, the channel position approaches a uniform distribution where all positions in the valley are equally probable. We define the residence time of a sediment grain at a location in the valley as the interval between successive occupations of that location by the channel. This allows us to treat the system as a first passage process and predict the PDF of residence time from the PDF of channel position. For the two end members described above, the results are very different. When the PDF of channel position is a Gaussian, the residence time PDF is a heavy tailed power law. When the PDF of channel position is uniform, the PDF of residence time is exponential.
 Publication:

AGU Fall Meeting Abstracts
 Pub Date:
 December 2007
 Bibcode:
 2007AGUFM.H51E0812B
 Keywords:

 1820 Floodplain dynamics;
 1825 Geomorphology: fluvial (1625);
 1862 Sediment transport (4558);
 3265 Stochastic processes (3235;
 4468;
 4475;
 7857);
 4468 Probability distributions;
 heavy and fattailed (3265)