Probabilistic settling in the Local Exchange Model of turbulent particle transport

The Local Exchange Model (LEM) is a stochastic diffusion model of particle transport in turbulent flowing water. It was developed mainly for application to particles of near-neutral buoyancy that are strongly influenced by turbulent eddies. Turbulence can rapidly transfer such particles to the bed, where settlement can then occur by, for example, sticking to biofilms (e.g., fine particulate organic matter, or FPOM) or attaching to the substrate behaviorally (e.g., benthic invertebrates). Previous papers on the LEM have addressed the problems of how long (time) and far (distance) a suspended particle will be transported before hitting the bed for the first time. These are the hitting-time and hitting-distance problems, respectively. Hitting distances predicted by the LEM for FPOM in natural streams tend to be much shorter than the distances at which most particles actually settle, suggesting that particles usually do not settle the first time they hit the bed. The present paper extends the LEM so it can address probabilistic settling, where a particle encountering the bed can either remain there for a positive length of time (i.e., settle) or immediately reflect back into the water column, each with positive probability. Previous results for the LEM are generalized by deducing a single set of equations governing the probability distribution and moments of a broad class of quantities that accumulate during particle trajectories terminated by hitting or settling on the bed (e.g., transport time, transport distance, cumulative energy expenditure during transport). Key properties of the settling-time and settling-distance distributions are studied numerically and compared with the observed FPOM settling-distance distribution for a natural stream. Some remaining limitations of the LEM and possible means of overcoming them are discussed.