McNair, J.N., and J.D. Newbold. 2001. Journal of Theoretical Biology 209:351–369.
The local exchange model developed by McNair et al. (1997) provides a stochastic diffusion approximation to the random-like motion of fine particles suspended in turbulent water. Based on this model, McNair (2000) derived equations governing the probability distribution and moments of the hitting time, which is the time until a particle hits the bottom for the first time from a given initial elevation. In the present paper, we derive the corresponding equations for the probability distribution and moments of the hitting distance, which is the longitudinal distance a particle has traveled when it hits the bottom for the first time. We study the dependence of the distribution and moments on a particle’s initial elevation and on two dimensionless parameters: an inverse Reynolds numberM̂(a measure of the importance of viscous mixing compared to turbulent mixing of water) and the Rouse number ŝ (a measure of the importance of deterministic gravitational sinking compared to stochastic turbulent mixing in governing the vertical motion of a particle). We also compute predicted hitting-distance distributions for two published data sets. The results show that for fine particles suspended in moderately to highly turbulent water, the hitting-distance distribution is strongly skewed to the right, with mode<median<mean. Because of the distribution’s thick upper tail, there is a significant probability that a particle’s hitting distance will greatly exceed the mean. The results also show that the position of the mode depends strongly on a particle’s initial elevation but, compared to the median or mean, is relatively insensitive to ŝ. These results are broadly similar to those obtained by McNair (2000) for the hitting-time distribution, but the distribution and moments of the hitting distance are noticeably more sensitive toM̂than are the corresponding properties of the hitting time. Comparison of predicted hitting-distance distributions with data of Cushing et al. (1993) on settling distances of fine particulate organic matter in natural streams supports the view that such particles commonly fail to settle the first time they hit the stream bed.