A stochastic model for particle sorting and related phenomena

Abstract

A number of factors contribute to the variability in particle size distribution with position, or sorting, that is often observed in natural channels. Several of these factors are readily examined by using stochastic models of particle movement along a streambed. If particle position and particle size are treated as jointly distributed random variables, then sorting is reflected in the fact that the conditional distribution of size given position depends on the value of position. This conditional distribution is most easily derived by first postulating the form of the conditional distribution of position given size and of the marginal size distribution. Further, the effect of mixing particles from two or more distinct sources can be considered by examining the joint distribution of three random variables: size, position, and source. This is useful in predicting size distribution downstream from the point where a tributary enters a channel. Longitudinally varying transport conditions, which also contribute significantly to sorting, are conveniently modeled with a nonhomogeneous Poisson process. Here it is assumed that the particle motion consists of a sequence of steps of random length, each step followed by a rest period of random duration. It is the number of particle rest positions as a function of distance from the initial position that obeys a Poisson process, so the proposed model allows for systematic change of the step length and rest duration distributions with position downstream.