*James N. McNair, Annis Water Resources Institute, Grand Valley State University*

Turbulence is often important in the transport of suspended particles in aquatic systems. This fact has long been recognized in the engineering literature on sediment transport dynamics. This literature focuses primarily on particle types that are important determinants of stream channel morphology, such as silt and sand. In ecological applications, the particle types of greatest importance include bacteria, algae, benthic invertebrate larvae (e.g., black fly larvae, photo above), and fine organic debris. Unlike silt and sand, these particles usually are of near-neutral buoyancy, often are motile, usually attach to the bottom of a waterbody by behavioral means or by sticking to biofilms rather than by gravitational settling, and often are capable of moving from the bottom into the water column by behavioral means rather than by passive scouring.

As noted by McNair et al. (1997), a satisfactory theory of turbulent particle transport must address four problems:

*Entrainment problem*how does a particle on the bottom (or bed) become entrained into the water column?*Travel-time problem*how long does it take for an entrained particle at elevation*z*to hit bottom for the first time?*Travel-distance problem*how far does an entrained particle at elevation*z*travel longitudinally before hitting bottom for the first time?*Settlement problem*what determines whether a particle settles when it hits the bottom (i.e., remains there for a positive length of time), rather than immediately bouncing off and returning to the water column?

The particles of interest in ecological applications typically differ from silt and sand in both their vertical transport properties and the factors that determine entrainment and settlement. Moreover, it is often desirable to study the behavior of individual particles rather than simply gross transport phenomena such as sediment loads or net deposition or erosion. For these reasons, the classical engineering theory of turbulent particle transport must be extended in various ways in order to address ecological questions in a satisfactory way.

My work in this area was stimulated by discussions with two colleaguesDr. Denis Newbold and Dr. David Hartcentering on a paper by Denny & Shibata (1989). Denny and Shibata in effect proposed a stochastic diffusion model to approximate the turbulent transport of biological particles such as larvae in the marine surf zone. They specifically addressed the mean hitting time; that is, the expected value of the time required for a particle to hit bottom for the first time from a given initial elevation. In my first set of results (reported in McNair et al. 1997), I began by examining the theoretical foundations of the Denny-Shibata model. I then revised and extended the model in various ways, obtaining a new stochastic diffusion model which I called the Local Exchange Model. I deduced equations governing the mean hitting-time of the Local Exchange Model and analyzed them numerically. I also compared certain quantitative predictions of the Local Exchange Model with the results of published empirical studies that permit estimation of parameters. The vertical distributions of current speed and suspended sediment concentration predicted by the Local Exchange Model were found to agree well with empirical results in cases where the stream bed was reasonably smooth. Hitting-time predictions could not be tested, however, since no data appear to be available on hitting times of appropriate particles.

In 1998–1999, I deduced and numerically analyzed equations governing the probability distribution of the hitting time in the Local Exchange Model. These results (reported in McNair, 2000) show that, for ecologically relevant parameter values, the hitting-time distribution typically is strongly positively skewed,with mode < median < mean. Two important consequences are that the mean is not a satisfactory measure of central tendency, and there typically is a sizable probability that a particles hitting-time will greatly exceed the mean.

Both of these sets of results apply to the hitting time. The corresponding theory for the moments and probability distribution of the hitting distance have also been worked out (McNair & Newbold, 2001). Numerical results indicate that, like the hitting-time distribution, the hitting-distance distribution typically is strongly positively skewed, with mode < median < mean and a sizable probability that a particle's hitting distance will greatly exceed the mean. Thus, if a large number of particles were introduced into the water column at a similar location (e.g., benthic insect larvae hatching from an egg mass), we would expect that a nonnegligible proportion would have hitting distances very much greater than either the most common or the average hitting distance. This longitudinal smearing property is probably important in the dispersal of benthic aquatic larvae.

In McNair (2006), I generalized the Local Exchange Model by deducing a single set of equations addressing the hitting time and distance, as well as the settling time and distance (i.e., the travel time and distance until a particle settles on the bottom for the first time). A key feature of the generalized LEM is that it allows probabilistic settling, where a particle hitting the bed can either settle or immediately reflect back into the water column, each with positive probability. Results for the settling time and distance are broadly similar to those for the hitting time and distance, except that they depend critically on a new parameter that controls the likelihood that a particle encountering the bottom will settle rather than immediately reflect back into the water column. Application of the model to empirical settling-distance data for 14C-labeled FPOM in a stream shows that, in contrast to previous versions of the model, the LEM with probabilistic settling is able to predict the observed distribution with reasonable accuracy. A more-technical summary of the generalized LEM can be found here (PDF).

Most recently, I have been working on applying the LEM to results from a variety of published empirical studies of particle transport and settling in streams and flumes, with particle types including benthic invertebrates, fine particulate organic matter and surrogates, and fine sediment. Most of these studies use the Exponential Settling Model (ESM) to characterize the longitudinal pattern of particle settling. The ESM predicts that if particles are released into a stream or flume, the proportion that have not yet settled will be an exponential function of time or distance downstream and will be independent of the release elevation above the bed. To date, no credible basis in fluid mechanics has been established for this simple model, nor has it been rigorously tested against mechanistic alternative models. The LEM is an alternative to the ESM that is based on classical fluid mechanics and turbulence theory. In collaboration with Denis Newbold of Stroud Water Research Center, I have reviewed properties of the ESM and LEM and compared these with high-quality empirical evidence available in the literature. Most of the evidence contradicts predictions of the ESM and supports the LEM, which predicts that settling immediately downstream from the particle release location typically will be nonexponential and will depend strongly on the release elevation. An interesting implication is that the usual method of estimating nutrient spiraling lengths in streams based on the ESM (with nutrient molecules playing the role of particles) is likely to overestimate spiraling lengths. Results of this ongoing research also show that the LEM predicts settling will eventually become and remain exponential if one looks sufficiently far downstream from the release location, and that this prediction is supported by available empirical evidence. This finding provides a new theoretical justification for far-field exponential settling that is based on plausible fluid mechanics.

**Acknowledgement.** Funding for my research on particle transport and settling during 20062011 was provided by NSF grants DEB-0543134 and DEB-0946811.

McNair, J.N. 2006. Probabilistic settling in the Local Exchange Model of turbulent particle transport. *Journal of Theoretical Biology* **241**: 420–437.

Fingerut, J.T., Hart, D.D. & McNair, J.N. 2006. Silk use enhances benthic invertebrate settlement. *Oecologia* **150**: 202–212.

McNair, J.N. & Newbold, J.D. 2001. Turbulent transport of suspended particles and dispersing benthic organisms: the hitting-distance problem for the Local Exchange Model. *Journal of Theoretical Biology* **209**:351–369.

McNair, J.N. 2000. Turbulent transport of suspended particles and dispersing benthic organisms: the hitting-time distribution for the Local Exchange Model. *Journal of Theoretical Biology* **202**:231–246.

McNair, J.N., Newbold, J.D. & Hart, D.D. 1997. Turbulent transport of suspended particles and dispersing benthic organisms: how long to hit bottom? *Journal of Theoretical Biology* **188**: 29–52.

Denny, M. W. & Shibata, M. F. 1989. Consequences of surf-zone turbulence for settlement and external fertilization. *American Naturalist* **134**: 859–889.

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