Pools, riffles and surface roughness in a particle interactions model of steep gravel-to-cobble streams

Abstract

Particle interactions in poorly-sorted sediments arise from the constraints on particle mobility imposed on each sediment particle by the particles surrounding it. Such interactions can be evaluated in terms of the size and position of the top of a given particle and the tops of the neighboring particles. Particle interactions are important in sediment transport in mountain channels, and previous studies in physics suggest that they are important in the development of spatial differentiation of particulate surfaces. Model simulation results from a new particle interactions cellular automata, presented here, suggest that particle interactions lead to nonrandom structuring of the quasi-steady-state sediment surface, and influence the size and spacing of pools in pool-riffle channels. Pool spacing in the model results ranges from one to fourteen bed widths, similar to the range in theoretical predictions and the values measured in pool-riffle channels. The model pools are shallow, generally have an irregular shape in plan view, and are small, typically 1 to 2 meters in nominal length. Trends in pool characteristics were evaluated in a series of model runs by varying one initial condition in the model, while all other variables were held constant. In these variation series, pool length is somewhat greater where the channel bed is wide, and where the sediment is less mobile. In the bed width variation runs, pool spacing increases linearly from one to seven bed widths as the bed width doubles (widths nominally 1.75 to 3 m). Pool spacing shows a non-linear decrease as the particle size distribution varies from a uniformly coarse sediment to encompass a greater fraction of finer sediment, with spacing decreasing from 14 to 1.6 channel widths as the particle size distribution width increases. The spacing of the pools was essentially unvarying at about 3 channel widths as slope and the mobility threshold were varied. Nearest-neighbor analyses of the spatial distribution of cell-scale highs and lows in the detrended channel bed surface were conducted on model surfaces. The analyses indicate that the spacing of high points on the bed is non-random with a tendency toward regularity of the high points, with the preferred spacing in the range of 4 to 12 cells. Although the model can be evaluated as non-dimensional, in this application I assume that each cell is 0.256 by 0.256 meters, so the pool spacing is nominally 1.25 to 3 m. In addition, a tendency to develop clusters that spanned up to 4 cells (nominally 1 m), indicating a tendency toward clustering on a small scale, developed in two runs with very low particle mobility. Formation of such non-random spatial patterns from an initially random surface indicates a process of self-organization, resulting from particle interactions alone, that appears to be an important factor in the formation of the rough, irregular pools and proto-pools that are found in steep, gravel-to-cobble mountain streams. The model bed surface coarsens in all runs, and the degree of coarsening varies between the local deeps and the higher surfaces on the channel bed. The greatest variation in surface particle size occurs with variation in the threshold for motion used in the model. Coarse areas on the model bed overlap deep areas in part, but the degree of overlap is generally small, so that bed tends to differentiate into deep and coarse areas. Such differentiation of deep and coarse areas may be analogous to pools and the steeper, coarser channel units such as are found in pool-riffle and other steep mountain streams. Surface roughness of the model channel bed was characterized using the root-mean-square of the height deviations of the steady-state surface. The roughness of the model bed increases as a power of time, then reaches a maximum value at a value that is stable, on average, during the remainder of the model run. The maximum roughness of the bed surface varies as a power of the model reach length and width. The maximum roughness also varies with other boundary conditions, including the slope, particle size distribution. The roughness also varies with the degree of vertical particle exposure, which is used as the driving variable in the model.