Channel topography in gravel-bedded rivers interacts with the local flow and sediment transport fields to produce “forced patches,” which are temporally stable areas on the bed that display similar grain size and sorting. In an effort to enhance our empirical understanding of the mechanisms responsible for the formation of forced patches, we conducted a near-field scale flume experiment in which a large (55 m long, 2.74 m wide), straight, sediment recirculating flume was provided a constant water discharge and a unimodal sediment mixture ranging in size from 2 to 45 mm. A sequence of alternate bars developed and became essentially fixed in space, producing quasi-steady state bed topography over which we made measurements of local near-bed velocity and sediment transport. The bed developed temporally and spatially persistent forced patches with a general pattern of coarse bar tops and fine pools, which we characterized by visual mapping and with the application of an automated image processing procedure to a high-resolution (1 × 1 mm) digital elevation model of the bed surface. The boundary shear stress field, calculated from velocity measurements and with a quasi-three-dimensional hydraulic model (FaSTMECH), displayed substantial variability across the bar unit. Bed surface grain size did not correlate with local boundary shear stress; instead, topographically forced divergences in the boundary shear stress field were matched by divergences in the sediment transport field. The resultant cross-stream sediment flux was size-selective that in turn forced a bed surface textural response leading to coarse bar tops and fine pools. Our observations suggest that size-selective cross-stream bed load transport is a mechanism responsible for the development of forced bed surface patches in gravel bed channels that have topographically forced heterogeneous flow fields.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 The development of persistent sorting patterns and forced patches should depend on a number of controlling factors, such as stage-dependent flow fields, channel morphology, including bed topography and planform shape, local sediment supply, the grain size distribution of available sediment, and channel obstructions. However, we still lack a quantitative description of mechanistic processes that explicitly connect these variables, particularly for gravel bed channels where excess shear stress is low and the sediment is not fully mobile. Parker and Klingeman  suggested that gravel beds are armored because equal mobility requires intrinsically less mobile coarse grains to be overrepresented on the bed surface. Paola and Seal  proposed that in rivers that typically have spatially variable shear stress fields, bed patchiness arises in response to this spatial variability, creating equal mobility patches of different sizes that differentially pass finer sediment and lead to downstream fining. Lisle and Hilton  attributed the coarse bars and fine pools they observed in their relatively straight study reaches to “downstream winnowing,” where fine particles winnow out of coarse patches and are deposited somewhere downstream in a fine patch; this requires the coarse fraction to be at least temporarily immobile. Their observations of high shear stress in coarse zones and low stress in fine zones (at low flow) are consistent with this interpretation. Much of the riffle-pool literature [e.g., Gilbert, 1914; Leopold and Wolman, 1960; Keller, 1971; Lisle, 1979; Thompson et al., 1996, 1999; Thompson and Hoffman, 2001; Milan et al., 2001; Heritage and Milan, 2004; Harrison and Keller, 2007] suggests that in straight reaches with pools and riffles, a “velocity reversal,” where the near bed velocity (and presumably, the boundary shear stress and sediment transport capacity) in the pool exceeds that of the riffle at high discharge but does not exceed it at low discharge, is responsible for the development of coarse riffles and fine pools. Building on this work, MacWilliams et al.  suggested pool-riffle sequences are maintained through “flow convergence routing,” in which the flow field downstream of a channel constriction converges in riffles at low flow and in pools at high flow, and Sawyer et al.'s  field work indicated that this process occurs in large gravel bed rivers. Lisle and Hilton  suggested that stage-dependent effects were responsible for pools filling with sand, the depth of which can serve as a proxy for sediment supply. While these studies have shed light on mechanisms important for the development of forced bed surface patches, none present simultaneous, detailed measurements of stage-dependent flow fields, bed load transport fields, and surface grain size patterns to fully support an explanatory hypothesis.
 Sediment supply has been shown to be a critical control on the surface structure of gravel bed rivers. Dietrich et al.  suggested that armoring is largely a result of insufficient sediment supply, and Nelson et al.'s  reanalysis of their data and new gravel-only data showed that reductions in sediment supply lead to an expansion of coarse fixed patches, and they found sediment supply to be a primary control on the dynamics and occurrence of free patches. Lisle et al. [1991, 1993] also observed that in a flume with bar topography, reducing the sediment supply led to changes in bed surface sorting and emergence of bars. Kinerson's  study of several field sites in northern California suggested that bed surface patchiness is correlated with sediment supply, and Yarnell et al.'s  analysis of field and flume studies found that streams with intermediate relative sediment supply values exhibited the greatest habitat heterogeneity (corresponding to bed surface patchiness). Buffington and Montgomery's [1999b] analysis of several flume studies found that the median grain size of the bed surface varied inversely with sediment supply. Hydraulic roughness elements like bank irregularities and large woody debris also can cause the bed surface to become finer than it otherwise would be in their absence [Buffington and Montgomery, 1999c].
 Planform geometry and cross-channel bed topography interact with the flow and sediment transport fields to produce bed sorting and forced patches. Early models for sorting in channel bends [Kikkawa et al., 1976; Bridge, 1977; Allen, 1978] assumed that equilibrium channel conditions occur when the local cross-stream effect of particle weight is exactly balanced by drag from secondary circulations. Consequently, these models predict that particles will travel along lines of equal depth such that there is no net cross-stream transport, and the resulting bed surface will exhibit coarse particles in deep water where the inward component of shear stress is greatest and the outward slope is the highest, and the bed will be fine in shallow water where the inward shear and cross-stream slope are lowest. Parker and Andrews  developed a model combining a simple representation of flow processes in a bend with the assumption that sorting is solely a result of the cross-streambed slope, arriving at an analytical solution that predicts coarse sediment follows a path from the inside bank to the outside bank of a bend apex. This model was later incorporated into computer simulations of channel migration [Sun et al., 2001a, 2001b]. In contrast with earlier models, Dietrich and Smith , Dietrich , and Dietrich and Whiting  built upon detailed field measurements of flow and sediment transport to develop a hypothesis that in fully mobile beds, equilibrium occurs when shear stress divergence is balanced by sediment transport divergence. Their observations indicated that shoaling of the flow over the point bar as the shear stress declines forces all sediment toward the pool, but counteracting secondary flows (and effects of currents along oblique dune fronts) preferentially transport finer sediment inward. Net transport of sediment toward the pool matches increasing boundary shear stress, but the finer sediment is shifted toward the inside bank due to differential transport by inward-directed secondary flows [Parker and Andrews, 1985; Ikeda, 1989; Bridge, 1992]. While these mechanisms have explained observations in sand-bedded channels, it is not clear whether the same reasoning could be used to predict sorting in gravel bed rivers.
 It is tempting to hypothesize that channels locally maintain “threshold” conditions, in which the bed surface locally maintains a grain size distribution (or median grain size) that is just able to be mobilized during a characteristic flood. A standard parameter for grain mobilization is the Shields stress, τ*,
which attains a critical value τc* at initial motion. Here τ is the boundary shear stress, ρs is the density of the sediment, ρ is the density of water, g is the gravitational constant, and D is the particle diameter. If threshold conditions are achieved and τc* is a constant for all grain sizes, spatial variations in boundary shear stress would be linearly reflected in the surface grain size distribution [Lisle et al., 2000]. This assumption is used to design channels [Lane, 1955] and can be used on a watershed scale to make predictions about what the reach-averaged median grain size would be in the absence of external roughness or sediment supply effects [e.g., Power et al., 1998; Buffington et al., 2004]. However, threshold conditions are unlikely to be the norm, as suggested by Lisle et al.'s  study where, at several natural channels, they measured bed topography, mapped the surface grain size distribution, and modeled the bed shear stress at bankfull discharge, ultimately finding no correlation between local shear stress and bed grain size. Alternatively, Dietrich  and Dietrich and Whiting  argued that in gravel bedded meanders with low excess boundary shear stress and low sediment supply, shear stress divergences might be balanced by divergences in sediment transport achieved primarily though bed grain size adjustment rather than erosion or deposition. In this case bed surface patchiness develops to moderate local bed mobility such that the divergences of stress and sediment flux are in balance. The predictive capability of this hypothesis has yet to be verified due to a limited number of well-developed quantitative linkages between topographically controlled stress fields, the sediment transport field, and the bed surface grain size distribution.
 Here we attempt to improve our empirical understanding of these linkages using observations collected during a near field scale flume experiment. We use detailed measurements of the bed surface topography, grain size, and the flow and sediment transport fields over a gravel bed of quasi-steady alternate bars that produced persistent topography and forced patches. Our observations suggest that topographically forced boundary shear stress divergences are matched by size-selective sediment transport divergences. The resultant changes in local sediment supply are accommodated by local changes in bed surface grain size, forcing the development of patches that allow the maintenance of quasi-steady morphology.
 The observations presented in this paper were collected during a flume experiment conducted in the main channel at St. Anthony Falls Laboratory (SAFL) at the University of Minnesota in Minneapolis, MN. The experiment was part of the larger “StreamLab06”suite of experiments designed to examine the interactions between geomorphology, nutrient cycling, and biomass accumulation. The large scale of the flume enabled us to make local measurements of the bed surface, flow field, and sediment transport field over a bar sequence.
2.1. Flume Setup and Experimental Procedure
 The main channel at SAFL is a 2.75 m wide, 80 m long (with a 55 m long test section) rectangular cross-section concrete research channel that has recently been refurbished and reinstrumented. Initially, a homogeneous, flat bed was created by filling the flume with a gravel mixture composed of grains 2–45 mm in diameter with a median grain diameter of 11 mm (Figure 1a). Water discharge was held constant throughout the experiment at 400 ± 20 l/s.
 Sediment was recirculated throughout the experiment. Bed load exiting the flume was collected in five side-by-side sediment traps, which in total spanned the flume width. Each sediment trap was connected to a load cell, which sampled sediment discharge continuously in 1 s intervals. An impeller pump in the floor pit where the sediment traps were located would then return the sediment to the head of the flume (see Marr et al.  for more information on the flume facilities and operation).
 Prior theoretical [Blondeaux and Seminara, 1985; Colombini et al., 1987; Lanzoni and Tubino, 1999] and experimental [e.g., Lanzoni, 2000] work on alternate bars suggests that bar development requires a large width-to-depth ratio and full sediment mobility. Wilcock and McArdell  suggested that in a sediment mixture of a grain size range similar to ours, full mobility of all grain sizes occurs when the Shields stress (equation (1)) is approximately twice the critical Shields number (assumed to equal 0.045). Considering these suggestions, we chose a design bed slope (0.013) and discharge (0.4 ± 0.02 m3/s) that resulted in a width-to-depth ratio of 20 and a dimensionless shear stress ratio (τ*/τc*) of about 2 (see Table 1 for a summary of the hydraulic conditions during the experiment). Although most gravel bed rivers have slopes smaller than 0.013 [Parker et al., 2007], the width of our flume necessitated a rather steep slope to simultaneously achieve the desired excess stress and large width-to-depth ratio necessary for bar formation. Details regarding the measurement of shear stress are presented in section 2.5. To further enhance the development of alternate bars, we constricted the flow at the flume entrance to 2/3 of the channel width, creating a flow perturbation at the inlet that allowed bars to form freely downstream.
Table 1. Hydraulic Parameters
0.4 ± 0.02
Bed and water surface slope
Mean depth (cm)
Shear stress (Pa)
 We ran the flume with constant discharge and sediment recirculation until the bed topography and bed load flux exiting the channel reached approximately steady state conditions. The water surface elevation was monitored using an ultrasonic water level sensor that traversed the flume on a mechanized cart that ran along rails above the flume walls. The water surface elevation was repeatedly measured along five longitudinal profiles spaced 500 mm apart in the cross-stream direction with the center transect located along the center of the flume. Measurements in the along-channel direction were obtained at 5 mm intervals. The mechanized cart was also equipped with an acoustic echo sounder that could measure bed surface elevation subaqueously; however, as bed topography developed, the flow became too shallow in some places for this instrument to function, so we relied upon dry bed topographic scans (described in section 2.3) to obtain bed surface elevation data.
2.2. Flow and Sediment Transport Measurements Over a Bar Sequence
 After the flume developed about a quasi-steady state topography consisting of about 1.5 alternate bar sequences, we made a series of local near-bed velocity, bed surface grain size, and sediment transport measurements over a bar sequence. We made measurements at 11 cross sections, spaced every three meters from x = 14.5 m to x = 44.5 m (we use x and y as downstream and cross-stream coordinates, respectively), with five points in each cross section spaced 50–60 cm apart (Figure 2a). At each point, we used a Pygmy current meter to measure the local velocity 3 cm above the bed. A Pygmy meter was chosen because initial attempts to measure the velocity with an acoustic Doppler anemometer (ADV) produced unreliable results. Velocity measurements were taken for an average of 55–60 s [U.S. Bureau of Reclamation, 2001]. Care was taken to ensure the Pygmy meter was oriented orthogonal to the flow, the direction of which was estimated visually with the aid of a string tied to the measurement rod. Although Pygmy meters generally are accurate to better than 3.5 per cent when used between about 6 and 30 cm above the bed, they may suffer from additional errors when placed closer to the bed [Hubbard, 1988]; therefore, in the following analysis we assume a 10 per cent uncertainty in measured near-bed velocity. After each velocity measurement, a photograph of the bed was taken with a digital camera placed in a waterproof housing that was attached to a pole. A scale bar was attached to the pole such that it typically appeared along the top of the underwater photographs, which generally covered an area of 7.1 cm by 5.3 cm. After the experiment, these photographs were analyzed in the lab to characterize the local bed grain size at the time of the velocity measurement. The b axis of each grain (as visible in the photographs, see Bunte and Abt  or Graham et al.  for a discussion of the possible effects of grain shape and imbrication on grain size distributions derived from photographs) was digitized by hand and the resulting area-by-number grain size distributions were converted to volume-by-weight distributions with the Kellerhals and Bray  voidless cube model. Finally, in addition to the local velocity measurements and bed photographs, we measured the local bed load transport at each point on the grid by holding a miniature Helley-Smith-type of sediment sampler (similar to the traps used by Dietrich and Smith ), which had a square opening with 5 cm sides and a nylon mesh bag attached to the downstream end to collect the sediment. The sampler was pointed directly upstream, and samples were taken for 60 s, although in some high-transport locations samples were shorter because the bag became filled with sediment. Each sediment sample was bagged, dried, and sieved. Although the bed surface topography appeared to be unchanging, we worked quickly and completed all of the local velocity and bed load transport measurements in under 5 h.
2.3. Bed Topography Measurements and Bed Surface Patchiness Characterization
 After all of the measurements taken while the flume was running were completed, the flume was rapidly drained, preserving the bed state as it existed at the end of the experiment. Bed surface patches were mapped by hand, wherein the bed was classified (by eye) into patches of similar grain size and sorting [e.g., Nelson et al., 2009]. Four patch types were defined: fine (median grain size D50 of 5.9 mm, geometric standard deviation σ of 1.79), medium-fine (D50 = 8.3 mm, σ = 1.84), medium-coarse (D50 = 12.1 mm, σ = 1.92), and coarse (D50 = 14.7 mm, σ = 2.11). Figure 1a presents the volume-by-weight grain size distributions of these patch types. Eight bed surface samples were collected throughout the StreamLab06 experiments, in which 30 × 30 cm areas of the bed were painted and all painted particles were collected and sieved. Kellerhals and Bray's  method was then used to convert the area-by-weight grain size distributions to the volume-by-weight distributions commonly used in empirical bed load transport formulas. The painted areas were located so that they fell entirely within a particular patch on the hand-drawn map and so that all mapped facies types were represented by physical samples. The samples were averaged to obtain characteristic grain size distributions for each patch type.
 In addition to the patch maps, the quasi-equilibrium dry bed surface was characterized with a high-resolution 1 × 1 mm topographic scan generated by a range-finding laser attached to the motorized cart that traversed the bed. The 1 × 1 mm digital elevation model (DEM) covered a 24 m swath of the flume, from x = 18 m to x = 42 m. A coarser, 10 × 10 mm DEM was also generated, covering essentially the entire flume test section from x = 9 m to x = 55 m. A digital camera was also attached to the cart and a sequence of photographs of this larger region of the bed were taken.
2.4. Automated Bed Surface Grain Size Measurements
 We also characterized the spatial variability of the bed surface grain size distribution through an automated image analysis procedure. Following the methodology of Graham et al. [2005a, 2005b], we used a shaded relief map generated from the 1 × 1 mm digital elevation model created from the high-resolution laser scan as an input image for automated grain extraction (similar to McEwan et al. ). Although the resolution (on a pixels per millimeter basis) of the 1 × 1 mm DEM was not as high as the resolution of the bed photographs taken from the cart, we found the DEM provided more reliable results than the photographs; that is, the resultant grain size distributions better matched those from surface samples, because the DEM image displayed stronger contrast between grains, the variability within grains due to lithology was eliminated, spatial registration was exact, and there were no perspective issues to correct. Grain size statistics for the portion of the flume covered by the 1 × 1 mm DEM were calculated on a 20 × 20 cm grid in the following manner: A 30 × 30 cm window was centered at each grid point and the hill shade image was cropped to that window size. Graham et al.'s [2005a] method was then applied to the 30 × 30 cm neighborhood and the local D50 was calculated from its output. While the reader should refer to Graham et al. [2005a] for a complete description of their method, a summary of the technique and the parameters we used is described below.
Graham et al.'s [2005a] method essentially uses pixel intensity to delineate individual sediment grains. It does this by dividing the image into regions, where each region represents a single grain. Because grain boundaries tend to be darker than the grains themselves, the method uses the intensity to detect grain edges. Initially, a median filter is applied to the image (in our case, the 30 × 30 cm crop of the hill shade DEM) to help blur out image noise and grain imperfections. A morphological bottom-hat filter using a disk structuring element is then applied to the image to enhance the contrast. A double-threshold method is then used to determine which pixels are likely grain edges. Here two threshold intensities are selected; one threshold corresponding to very dark pixels (which almost certainly are grain edges) and a second, lower threshold corresponding to somewhat dark pixels that are potentially grain edges. Features in the lower threshold image that overlap those in the higher threshold image are then classified as edges. The resulting binary image is then divided into regions using the watershed segmentation algorithm [Vincent and Soille, 1991]. Because the watershed segmentation can often produce oversegmented images, Graham et al. [2005a] perform an h-minima transform to suppress small pits prior to the application of the watershed algorithm. Figure 3 shows an example portion of our hill shade DEM at various stages in the grain identification process.
 Once the grains have been identified as regions, an ellipse is fit to each region and the minor axis of the ellipse is measured (in pixels). Graham et al. [2005a] found this technique gives a reasonably accurate estimate of the true b axis of each grain. The resulting list of grain diameters is then converted to mm using the known image scale (in our case, 1 pixel = 1 mm), and grain size distributions are generated. Importantly, the grain size distributions must be concatenated at an appropriate level to avoid spurious measurement of image noise [Graham et al., 2005a, 2005b, 2005c]. We concatenated our grain estimated grain size distributions at 4 mm because our experiment involved only gravel and since visual observation (and comparison with surface samples) suggested that the automated results were reasonable to that level.
 We found that the h-minima transform had positive or negative consequences on the accuracy of the algorithm depending on the type of surface being analyzed. Without the h-minima transform, coarse surfaces (such as the one displayed in Figure 3a) would be oversegmented by the watershed algorithm, and inclusion of a 1 pixel h-minima transform improved the accuracy considerably (e.g., Figure 3f). However, finer bed surfaces were better characterized without the h-minima transform, since the watershed algorithm tended not to oversegment in these cases. While the importance of the h-minima transform is likely related to the 1 mm/pixel scale of our DEM, it does mean that the use of a single h-minima parameter for analysis of the entire DEM will produce either too fine distributions for coarse areas (if h-minima = 0) or too coarse distributions for fine areas (if h-minima = 1). Nevertheless, the overall pattern of sorting is essentially the same either way, so for the remainder of this paper, grain size data generated from the automated procedure are presented with h-minima = 1. Table 2 shows the parameters we used in the image analysis algorithm.
Table 2. Parameters Used in the Automated Grain Size Analysis
Median filter size
Structuring element disk radius
2.5. Shear Stress Calculations
 Local boundary shear stress was estimated in two ways. First we used the single velocity method described by Dietrich and Whiting , in which shear stress is estimated from a single measurement of the near bed velocity. The law of the wall describes the vertical profile of downstream velocity near the boundary:
where u(z) is the velocity at height z above the bed, z0 is the roughness height, κ is von Karman's constant (κ = 0.4), and u* is the shear velocity, which is defined as
Here τ is the boundary shear stress and ρ is the fluid density. Using the law of the wall, the boundary shear stress can be estimated from a single velocity measurement u(z) as
The roughness height, z0, is commonly parameterized to scale with the coarse grains on the bed; for instance:
Here D84 is the grain size for which 84% of the grains on the bed surface are finer. Following Leopold and Wolman , Dietrich and Whiting  suggested A = 0.1 is a reasonable scaling factor, which we use here. Wilcock  found that the single velocity method for estimating u* was accurate to within about 7%, using a roughness height of z0 = 0.095D90, which is essentially equal to z0 = 0.1D84 used here and z0 = 0.09D84, found by Wiberg and Smith . Using this formulation, we estimated the local shear stress on our grid of velocity measurements using the measured velocity at z = 3 cm above the bed, with a D84 estimated from the grain size distributions derived from the underwater photographs.
 Despite its simplicity, the single-velocity method has been shown to produce more reasonable and reliable estimates of boundary shear stress than other more complicated techniques. Whiting and Dietrich  found that the single-velocity method outperformed a method using the slope of the velocity profile because the latter method projected well into the flow and included large-scale drag, causing the boundary shear stress to be overestimated. They also pointed out that the near-bed velocity measurement can avoid upstream influences of features such as dunes or bank irregularities, suggesting that instead this method captures primarily the effects of local grain roughness essential to sediment transport. The usefulness of incorporating the local grain size into stress calculations has more recently been confirmed by Sime et al. , who found that an unconstrained velocity gradient estimate of shear stress leads to greater error than a grain size constrained profile. The single-velocity method relies upon the assumption of a logarithmic near-bed velocity profile, which may not occur in areas of high relative roughness (h/D84 > ∼5) [Wiberg and Smith, 1991; Bathurst, 2002; Katul et al., 2002]. Since our flume developed shallow flow over coarse bar tops, the velocity profile there may differ from logarithmic and thus the single-velocity method may overestimate or underestimate the local boundary shear stress, but in the absence of velocity profiles over the bar top we proceed with the assumption of a logarithmic profile.
 The discrete spacing of velocity measurements, and the lack of data on velocity (and boundary shear stress) direction, suggests that an additional method of modeling the entire boundary shear stress field, constrained by measurements, is in order. We used a quasi-three-dimensional hydrodynamic model, FaSTMECH (Flow and Sediment Transport and Morphological Evolution of Channels, available online at http://wwwbrr.cr.usgs.gov/gstl/project-MDSWMS.html), which was developed by the U.S. Geological Survey and has been described in a number of studies [e.g., Lisle et al., 2000; Kinzel et al., 2009]. For a full description of the model, the reader is directed to the work by Nelson and McDonald  and the USGS online documentation, but, in summary, the model solves the full vertically averaged and Reynolds-averaged momentum equations and it incorporates a vertical structure submodel that computes the vertical velocity profile along the streamlines of the vertically averaged flow and the secondary flow across those streamlines. The vertically averaged equations are cast in a channel-fitted coordinate system and are derived and presented by Nelson and Smith . In its computation of the depth-averaged solution, the model assumes steady, hydrostatic flow and models turbulence with an isotropic eddy viscosity. The vertical submodel takes the solution to the depth-averaged model and an assumed eddy viscosity structure [Rattray and Mitsuda, 1974] to calculate the three-dimensional solution. While a fully three-dimensional model might be capable of calculating more reliable estimates of boundary shear stress than a two-dimensional model like FaSTMECH, three-dimensional models inherently require detailed measurements of the three-dimensional flow field for calibration and are highly sensitive to the method of specifying topographic complexity [Lane et al., 1999]. Given the difficulty we encountered in measuring the 3-D velocity field with an ADV (Section 2.2), the 2-D modeling capabilities of FaSTMECH are appropriate for this exercise.
 The momentum equations are solved over a curvilinear orthogonal finite difference grid, constructed in the graphical user interface MD_SWMS (Multidimensional Surface Water Modeling System) with which FaSTMECH is distributed [McDonald et al., 2005]. The Cartesian grid we used to model the SAFL flume had 8 cm node spacing in both the x and y directions. Elevation data from the 10 × 10 mm DEM were interpolated onto the grid points.
 We provided the model a constant discharge upstream boundary condition of 0.4 m3/s, and we specified a constant downstream water surface elevation and a constant lateral eddy viscosity of 0.0019 m2/s. We used spatially variable roughness in which the drag coefficient Cd was estimated to be a function of the flow depth computed from constant roughness conditions and the local D84 estimated from the automated image analysis [e.g., Nelson et al., 2003]. We calibrated the roughness by matching the model-predicted water surface elevation to the five longitudinal water surface profiles collected by the ultrasonic water level sensor during the experiment, and by comparing the model-predicted flow velocity 3 cm above the bed to the pygmy meter velocity measurements; the Cd parameter ranged from 0.0034 to 0.0142. The calculated shear stress field was similar to one calculated with a constant roughness with Cd = 0.009, but the variable roughness solution did a better job matching our near-bed velocity measurements. After calibration, the root-mean-square error in our calculated water surface elevation was 0.0146 m and the root-mean-square error between the calculated and measured near-bed velocities was 0.11 m/s. The model's convergence (the per cent deviation between the predicted discharge at a grid cross section and the specified discharge) was between ±0.07 per cent.
 The single-velocity method provides a useful check on the FaSTMECH output. However, because the model provides a continuous estimate of the entire stress field, including the magnitude and direction of the local stress vectors, we focus on and use those estimates throughout our subsequent analysis.
3.1. Bed Surface Topography and Sorting
 Alternate bars began to form shortly after the experiment began and they immediately started to migrate downstream; however, their rate of migration slowed as the bar amplitude became more pronounced and after about 13 h the bars appeared visually to have stopped migrating. Repeat scans of the water surface indicated that there was little to no change in the water surface (Figure 4b), which corroborated the visual observations that a quasi-steady state condition had developed.
 The establishment of quasi-steady conditions was further verified through analysis of the sediment flux exiting the downstream end of the flume (Figure 4a). For 15 h leading up to the bed load and velocity measurements and the 5 h during which the measurements were taken, the total bed load flux collected in the downstream weigh pans generally ranged from about 0.5 to 2.5 kg/s with occasional deviations from 0.0 to 3.0 kg/s (Figure 4a). In spite of this variability, which is an inherent feature of sediment transport [e.g., Pitlick, 1988; Nelson et al., 2009; Singh et al., 2009], a linear regression of the time series has a slope of 10−5 kg/s/h, indicating that quasi-steady conditions persisted throughout this period. The four sediment recirculation samples taken during this time (Figure 4a) had very similar grain size distributions, the average of which is presented in Figure 1b. All grain sizes from the bulk sample were present in the recirculation samples. Interestingly, if Parker's  surface-based bed load transport algorithm is provided with the flume-averaged boundary shear stress (τ = 17.2 Pa) and the bulk sediment grain size distribution to characterize the surface, the algorithm calculates a bed load grain size distribution virtually identical to our averaged recirculation samples (Figure 1b), suggesting that Parker's  algorithm is well suited for modeling our experiment.
 The quasi-steady state bed developed about 1.5 complete bar-pool sequences. Bar heads were located at approximately x = 11 m, x = 35 m, and x = 55 m (Figures 2a and 2b), indicating a bar wavelength of about 44 m. The furthest upstream bar's location was forced by the placement of sandbags at the flume entrance, and the furthest downstream bar was almost completely developed before reaching the flume outlet. The bar tops were about 25 cm higher than the adjacent pools (Figure 2a). By the time we began to make velocity and sediment transport measurements, the bed configuration was essentially unchanging.
 In general, the bed surface on bar tops was coarse, and the bed surface in pools was fine (Figures 2c and 2d). Surface samples of the bar top and pool (Figure 1a) indicate that the D50 of the bar top was 14.6 mm and the D50 of the pool was 6.6 mm. The hand-drawn patch map (Figure 2d) shows that while there is some local variability in grain size pattern, this general pattern of coarse bars and fine pools predominated in our system.
 The map of median grain size generated from the automated image analysis (Figure 2c) shows essentially the same pattern as the hand-drawn patch map. This internal consistency is encouraging as it suggests that both methods are effectively capturing the overall sorting structure that developed on the bed. The estimated D50 of the bar tops from the automated procedure is generally around 14–18 mm, which is consistent with the sieved surface sample. The automated procedure overestimates the D50 of the pools to be around 11–13 mm, probably due to the inclusion of the h-minima threshold everywhere (as discussed in section 2.4). Table 3 shows the number of cells from each class of the automated image analysis that are co-located in each class of the hand-drawn patch map, indicating that in general there is good agreement between the two methods. The overall pattern of coarse bars and fine pools prevails regardless of the method used to assess it.
Table 3. Number of Cells From the Automated Grain Size Analysis (Figure 2c) Located in Each Patch Class on the Hand-Drawn Patch Map (Figure 2d)
Automated D50 (mm)
 The sorting maps in Figures 2c and 2d are useful for visualizing the overall sorting patterns on the flume bed, but in reality the transition from one patch to another may not be as sharp as the boundaries on the maps suggest. Figure 5a illustrates in detail how the automated image analysis map and the 1 × 1 mm DEM compare with each other. In general, a visual assessment shows that areas of the bed that are obviously coarse or fine are placed in the appropriate patch category, although the precise location of the boundary depends upon the details of the classification method (in this case, the value of the automated D50 at the transition between classes). So the discrete boundaries in Figures 2c and 2d might more accurately be thought of as fuzzy boundaries rather than sharp transitions from one patch to another. Research into more sophisticated characterization of objective bed surface patch delineation is ongoing.
3.2. Flow Field
 The model-predicted stress field is spatially heterogeneous (Figure 6a). The zone of maximum shear stress shifts from the left side of the channel (looking downstream) at the upstream end of the flume toward the right side of the flume by about x = 25 m. The large bar on the left side of the channel between x = 20 and x = 35 m directs most of the flow to the right, producing reach-maximum calculated stresses of around 30 Pa in the adjacent pool. Downstream of that bar, the developing bar on the right side of the channel directs flow, and the zone of maximum shear stress, toward the left wall. The oscillations visible in the stress map in Figure 6a are not numerical instabilities; they appear instead to be a result of small-scale bed forms present in the scanned topography used to create the computational mesh. Although the water was too turbid to make direct bed surface observations during the experiment, according to the unified bed form phase diagram presented by Best , dunes or bed load sheets can be expected for our hydraulic conditions and sediment mixture.
Figure 7a compares the model-predicted water surface elevation with the observed profiles for the three middle longitudinal profiles. There is excellent agreement, except at about x = 31 m where the echo sounder measured a small (probably transient) wave that was not calculated by the steady state model. Figures 7b–7e show comparisons between the velocity measured 3 cm above the bed with the pygmy meter and the near-bed velocity computed with FaSTMECH's vertical profile submodel, as well as comparisons between the boundary shear stress calculated with the single velocity measurements and the FaSTMECH-computed boundary shear stress. The error bars on the velocity measurements represent 10% uncertainty, and those on the single-velocity stress measurements are estimated through Gaussian error propagation assuming 10% uncertainty in the measured velocity u(z), 30% uncertainty in the D84 estimated from the underwater photographs, and uncertainties of 5 mm for z and 0.02 for A. The agreement in near-bed velocity is generally quite good, and in general there is reasonable agreement between the two methods of computing boundary shear stress despite the uncertainty in many of the parameters used in the single-velocity calculation (equation (4)). Because boundary shear stress is modeled as a function of the square of the depth-averaged velocity [Nelson et al., 2003], one should expect it to be more difficult for the model to match stress estimates than velocities.
3.3. Sediment Transport Field
Table 4 presents the total downstream flux Qs for each section, which were computed from the five measurements taken across each section as Qs = qsdy, where W is the width of the channel and y is the cross-stream coordinate. The mean value of the measurements, 1258 g/s, is close to the average sediment flux exiting the flume of about 1400 g/s (Figure 4a).
Table 4. Total Downstream Sediment Flux Measured at Each Cross Section (qsdy)
x Coordinate (m)
Total Downstream Sediment Flux (g/s)
Coefficient of variation
 There was substantial cross-stream variation in the downstream-oriented sediment transport we measured. Figure 6a presents cross-sectional profiles of the downstream sediment transport qs. To facilitate comparison between cross sections (using a method previously adopted in other studies [e.g., Dietrich and Smith, 1984]), the data have been corrected such that the total downstream transport at each section equals the mean value of 1258 g/s from Table 4. The coefficient of variation (standard deviation/mean) of these measurements was 0.22, which, considering the brief duration of measurements and considerable cross-stream variability, we think makes such a correction reasonable for purposes of comparison (Dietrich and Smith  observed a coefficient of variation of 0.25 in the downstream sediment flux in their field study of Muddy Creek). Downstream sediment transport tended to follow the zone of maximum shear stress (Figure 6a), with the highest measured values in the high-stress pools. Although the bed load measurements were sieved, we present only the total flux data since it is unlikely our samples were large enough to get accurate grain size distributions of the bed load [e.g., Dietrich and Whiting, 1989].
 Sediment was not moving solely in the downstream direction. Because we oriented our bed load samplers directly upstream, thereby sampling only the downstream component of the local sediment flux qs, we can calculate the local cross-stream sediment flux qn if we take advantage of a steady state assumption. The generalized grain size specific two-dimensional conservation of bed sediment equation can be written as [Parker, 2008]
where λp is the bed porosity, η is the bed elevation, t is time, fli is the fraction of the ith grain size transferred between the surface and the subsurface during aggradation or degradation, La is the active layer thickness [e.g., Hirano, 1971], Fi is the fraction of the ith grain size on the bed surface, and qi,s and qi,n are the volume bed load transport rates per unit width of the ith grain size in the downstream (s) and cross-stream (n) directions. Following the right-hand rule, we consider the s coordinate to be positive in the downstream direction and the n coordinate to be positive toward the left bank. If we assume steady state conditions, the time derivatives in equation (6) go to zero and we arrive at a relationship between the downstream and cross stream fractional transport rates:
This equation can be integrated to solve for the cross-stream sediment flux:
We used equation (8) with our measured downstream bed load transport rates qi,s (corrected to integrate to the mean total flux at each cross section) to calculate the cross-stream bed load transport rates qi,n necessary for steady state. Doing so requires the additional boundary condition of zero flux at the flume walls: qi,n = 0 at n = −W/2 and W/2. We evaluated ∂qi,s/∂s with a forward difference calculation, so the first cross section for which we were able to calculate cross-stream flux was cross section 2 at x = 17.5 m. Note that in our experiment, the (s, n) coordinates can be replaced with (x, y) because we are working in a straight flume.
 The calculated cross-stream sediment fluxes (summed over all grain sizes) are presented in Figure 6c. In the upstream portion of the flume, where there is a prominent bar on the left side of the channel (facing downstream), cross stream transport is primarily to the right. Going downstream, as the bar is passed (and another bar begins to form on the right side of the channel), at around x = 28 m, the cross-stream sediment flux changes direction and becomes primarily oriented toward the left bank. This is more readily illustrated in Figure 6d, where we have plotted the ratio of the integrated cross-stream flux Qn (Qn = qndn) to the integrated downstream flux Qs (which, because we are using the corrected data, is the mean value for each section) as a function of distance downstream. In the upstream portion of the channel, there is net transport to the right, and downstream of the bar there is net transport to the left. The magnitude of the Qn/Qs ratio is ±18%, which is in the range of Dietrich and Smith's  observations at Muddy Creek.
 This change in direction of net cross stream transport coincides nicely with the cross-stream component of shear stress predicted by the FaSTMECH model (Figure 6b). The model predicts shear stresses oriented to the right in the upstream portion of the channel, changing direction toward the left downstream of x = 28 m. The internal consistency between our measured and computed sediment transport rates and the computed shear stress field is an encouraging sign that our observations and modeling are capturing the essential dynamics of our system.
 Our flume observations provide an opportunity to examine how topography and the flow and sediment transport fields interact to produce forced patches and persistent sorting. Our observations do not suggest that our flume experiment demonstrated “threshold” conditions; like Lisle et al. , we did not find a simple correlation between the local median bed surface grain size and the local shear stress. Instead, our results show that in places the shear stress and grain size appear to be positively correlated (both stress and grain size increasing or decreasing), negatively correlated (stress increasing while grain size decreases, or vice versa), or insignificantly correlated. We propose that this is a direct consequence of the particular dynamics of the size-selective nature of mixed-grain size gravel transport interacting with topographically induced steering of the flow [e.g., Clayton and Pitlick, 2007].
Figure 8 shows FaSTMECH-calculated shear stress and surface D50 estimates from the automated grain size analysis along a profile at position y = 0.8 m from the left bank (Figure 2a). Most of this profile is in an area of the bed classified in the coarser (darker) facies on the hand-drawn patch map (Figure 2d), and since the h-minima = 1 parameter worked well in the coarser areas of the bed (Figure 3f), the D50 values presented in Figure 8 should be reasonable estimates of the magnitude of the actual surface D50. Since much of the transect falls in the same patch type, the variation in D50 illustrated by the automated analysis, which can have a significant impact on local sediment transport rates, evidently was not detectable by eye. Along the y = 0.8 m profile, between x = 18 m and x = 23 m the boundary shear stress is essentially constant whereas the median grain size increases. Then from x = 23 m to x = 29 m the boundary shear stress drops dramatically and the grain size declines. From x = 29 m to x = 41 m the boundary shear stress increases, but the grain size progressively declines.
 An important piece missing from Figure 8 is the cross-stream component of the sediment transport field, which covaries with the boundary shear stress field and the surface grain size such that quasi-steady topography and sorting patterns are produced. In their description of fixed bars that developed in a high-gradient flume, Lisle et al.  suggested that their stationary bars formed due to deposition of coarse material at the front of the bar preventing bar erosion and causing flow and sediment transport to be directed around the bar due to shoaling and “topographic steering” [e.g., Dietrich and Smith, 1984; Smith and McLean, 1984; Nelson and Smith, 1989]. Figure 6 shows that, in our flume, cross-stream bed load transport away from zones of decreasing boundary shear stress and toward zones of increasing shear stress occurred and this could lead to a dynamically stable bed topography and surface grain size distribution.
Figure 9 places the observations of Figure 8 in the context of stress-grain size-transport rate relationships built into Parker's  surface-based bed load transport model. The procedure used to make the contours of total bed load flux in Figure 8 is similar to that presented by Nelson et al. , in which we compute the total bed load transport rate predicted by the algorithm as a function of boundary shear stress and surface geometric mean grain size for synthetic bed surface distributions that have a wide range of mean grain sizes but constant geometric standard deviation (here, σ = 1.7, close to the bulk standard deviation of 1.8). The use of thousands of synthetic grain size distributions in this analysis allows us to fully explore the effects of changes in grain size statistics on predicted sediment transport rates. Also shown on Figure 8 are the shear stress (from FaSTMECH) and observed bed surface geometric mean grain size (from the automated image analysis) for four locations along the y = 0.8 m transect shown in Figure 8, which when connected show the path the transect followed in stress-grain size-flux space. Between x = 18 m to x = 23 m, lateral sediment transport driven by topographically induced cross-stream shear stress (Figure 6b) reduces the amount of sediment transport necessary to maintain steady conditions. Because the shear stress at x = 23 m is virtually the same as that at x = 18.5 m, this reduced sediment transport is achieved through bed coarsening (this is further elucidated in the following paragraph). From x = 23 m to x = 29.25 m, the shear stress declines dramatically due to thinning of the flow and flattening of the water surface slope associated with the slowing and shoaling of the flow around the bar such that essentially no sediment transport is occurring on the top of the bar, and by this point all of the sediment that was being transported upstream has been transported laterally so that it is accommodated by flux through the pool. From x = 30 m to x = 41 m, the boundary shear stress increases, and here there is a lateral sediment influx driven by cross-stream component of the shear stress oriented toward the left bank (Figure 6b). Here convergence of boundary shear stress is balanced by convergence in sediment transport, and the bed accommodates this increasing sediment flux by becoming finer.
 The development of coarse bars and fine pools in our experiment suggests that, along a path moving up the bar, the material moving as cross-stream sediment transport became finer, preferentially shuttling fine sediment into the pools. Unfortunately, because our bed load samples were small, we are unable to present reliably accurate measurements of the grain size distribution of the bed load and therefore cannot experimentally verify that the cross-stream transport became finer along a profile progressing up the bar. However, Parker  and Wilcock and Crowe's  mixed-grain size bed load transport models can be used to show how, theoretically, the cross-stream flux should be becoming more size selective. Wilcock and McArdell  suggested that the ratio of the fraction of the ith grain size in the load, pi, to the fraction of the ith grain size on the bed surface, Fi, can be used to assess whether the bed is in a state of full or partial mobility. If pi/Fi is constant (i.e., equal to 1.0) for all grain sizes Di, the system is in a state of perfect surface-based equal mobility [Parker, 2008]. However, when the ratio becomes smaller as grain size increases, the relative transport of coarse grains is reduced so that partial or selective transport is occurring, or that fine grain size fractions were transported at rates higher than proportionally represented on the bed surface (perhaps due to high supply of fine material), giving the appearance of a cascading trend. Greater deviations from constant pi/Fi indicate more size-selective conditions.
 To explore this for our experiment, we used the FaSTMECH-computed boundary shear stress and the observed (from automated image analysis) bed grain size distribution to calculate bed load transport from Parker  and Wilcock and Crowe's  formulas. The algorithms use the shear stress and bed surface grain size distribution as inputs to compute the total transport rate qT and the grain size distribution of the load pi, which can then be used with the local bed surface grain size distribution Fi to compute pi/Fi for each grain size Di. This type of analysis normally compares conditions of partial versus equal mobility per grain size fraction at different flow levels [e.g., Wilcock and McArdell, 1993; Clayton and Pitlick, 2007], whereas for this study, transport conditions are compared spatially, over changes in x. Figure 10 shows computed pi/Fi plots for selected distances downstream along the y = 0.8 m transect where the bed surface is coarser than the median size of the bed material (11 mm). Both transport expressions predict that where the boundary shear stress is relatively low compared to the observed grain size of the surface (x = 26 and 30 m in Figure 10) that deviation from equal mobility is greatest and conversely where the boundary shear stress is relatively large compared to the surface grain size (x = 20 and 37 m in Figure 10), equal mobility is more closely approximated (though not reached). This effect is more apparent in Parker's  algorithm than Wilcock and Crowe's  algorithm, particularly for coarse size fractions, perhaps because Wilcock and Crowe's  algorithm incorporates a more complex hiding function whose exponent is a function of Di/Dg, and because it was developed with the intention of capturing mobilization effects of sand on coarse grains while our experimental conditions were sand-free. Nevertheless, where the boundary shear stress rapidly declines (as flow shoals over the top of the bar between x = 20 m and x = 30 m (Figures 2, 6, and 8) and the bed is coarse, both algorithms suggest that the strongest tendency to transport mostly the finer fraction occurs. Along this path the boundary shear stress is oriented toward the opposite bank (Figure 6b) and the mobilized finer sediment is transported across the channel away from the zone of decreasing boundary shear stress (Figure 6c). This suggests that the topographically induced divergence of boundary shear stress leads to differential mobilization of the finer sediment, coarsening of the bed and equilibrium topography, i.e., the formation of a forced bed-surface patch.
 Our interpretation is generally in agreement with Lanzoni and Tubino's  description of the implications of heterogeneous sediment on alternate bar formation, which emerged from their analytical stability analysis. Their theoretical results suggested that mixed-size sediments might appreciably inhibit bar migration, and that selective transport leads to coarse bars. The alternate bars that developed in Lisle et al.'s [1991, 1993] experiments also formed coarse bar tops and became stationary, which they attributed to bar head stabilization by coarse sediment and preferential incision in the pools. Our bars formed sorting patterns similar to those of Lisle et al. [1991, 1993], but it is difficult to determine precisely why our bars stopped growing and migrating since, in addition to strong sorting, the upstream bar's location was forced by a physical constriction of the flow at the flume inlet.
 The contradictory sorting patterns observed in straight and curved channels suggests that the presence of curvature induces a shift in the dominant sorting mechanism. In bends, channel curvature creates a strong secondary circulation directed inward at the bed, toward the point bar [Smith and McLean, 1984]. For example, following Parker and Andrews , Lanzoni and Tubino  suggest that cross-stream sediment flux, qi,n, can be expressed as:
Here qi is the sediment flux vector of the ith grain size computed by, say, Parker's  model, χ is the angle the boundary shear stress vector makes with the s axis, β is the width to depth ratio, τg* is the Shields stress computed for Dg, fh (Di/Dg) is a reduced hiding function, ∂η/∂n is the cross-stream bed slope, and r is a parameter that varies from 0.5 to 1.5. The term on the right-hand side inside the braces represents the effects of friction, lateral bed topography, and hiding effects. The sin χ term represents the effects of the transverse component of velocity and the effects of curvature-induced secondary circulations. Blondeaux and Seminara  suggested that sin χ can be expressed as:
where u and v are the components of the velocity vector in the s and n directions, h is the flow depth, ro is the radius of curvature, and a is a scaling parameter ranging from 7 to 12 [e.g., Engelund, 1974; Allen, 1978; Bridge and Jarvis, 1982; Dietrich and Smith, 1983]. In straight channels, the h/(r0 + n) term is zero, but in bends it can become quite important. This secondary circulation is opposed by the outward gravitational force on particles provided by the cross-stream slope of the point bar. For fine particles, the effect of the secondary circulation exceeds the gravitational force, causing them to be preferentially transported into a reduced shear stress zone over the bar and transported away. Larger particles, however, roll downslope into the higher shear stress zone in the pool [Parker and Andrews, 1985; Clayton and Pitlick, 2007]. Curvature effects may be less important in straight channels, which should have weaker secondary circulations and will develop coarse bars and fine pools because the finer sediment is preferentially transported laterally away from the bar via topographic steering.
 The actual importance of curvature on bed sediment sorting for particular location will of course depend on the details of local channel plan form, bed topography, and flow regime. Local sorting also is tightly coupled to the details of the bar shape, since the shape of the bar will determine where flow accelerations or decelerations, and consequently shear stress increases or decreases, occur. The shape of our bars resulted in flow deceleration over the bar top, but it is possible that had the upstream end of our bar been wider, the flow could have accelerated and the stress could have increased over the bar, resulting in a different bed surface sorting pattern [e.g., Whiting and Dietrich, 1991].
 Unlike gravel bedded channels, sediment transport in sand-bedded channels typically occurs under high excess shear stress conditions in which all grain sizes are fully mobile [Parker, 2008]. In these channels, one would expect the coarse sediment to track the zone of highest shear stress, so even relatively straight, alternate bar reaches should have fine bars and coarse pools [Dietrich and Smith, 1984].
 Observations of the surface texture of gravel bed rivers generally indicate that in curved channels the point bar is finer than the adjacent pool, in straight channels alternate bars are coarser than pools, and bars in both straight and curved channels tend to have a coarse to fine texture gradient along the bar top in the downstream direction. Our experiment shows how this pattern of forced bed surface patchiness is a consequence of interactions between channel topography, the flow field, and the sediment transport field.
 In our experiment, we developed bed topography and a pattern of forced bed surface patchiness similar to the field case of a straight channel with alternate bars. Our channel reached quasi-steady conditions with an overall sorting pattern of coarse bar tops and fine pools. This pattern did not track the calculated boundary shear stress field; that is, the coarse bar top was not an area of high boundary shear stress and the fine pool bed was not an area of low boundary shear stress.
 The development of alternate bars in our channel produced a spatially variable shear stress field. Thinning and shoaling of the flow over the bar caused the stress over the bar to decrease and flow convergence in the pool produced locally high boundary shear stress there. Sediment was transported laterally off of the bar and into the pool due to the cross-stream component of the flow field. This lateral transport became increasingly size-selective toward the downstream end of the bar as the shear stress on the bar declined, and it caused the local sediment supply along the bar to progressively decrease, which was accommodated locally through bed surface coarsening. Downstream of the bar, an influx of lateral sediment supply due to local stress convergence allowed the bed to fine. Thus, topographically forced divergences in the boundary shear stress field are matched by divergences in the sediment transport field, which in turn were accommodated by changes in bed surface texture allowing the channel to maintain quasi-steady state conditions.
scaling parameter for helical flow.
parameter relating grain size to roughness height.
grain size for which 50 and 84% of the grain size distribution is finer.
geometric mean bed surface grain size.
fraction of the ith grain size on the bed surface.
fraction of the ith grain size transferred between the surface and the subsurface during aggradation or degradation.
lateral hiding function.
active layer thickness.
cross-stream coordinate (channel fitted).
cross-stream sediment flux per unit width of the ith grain size fraction.
downstream sediment flux per unit width of the ith grain size fraction.
cross-stream sediment flux per unit width.
downstream sediment flux per unit width.
total cross-stream sediment flux.
total downstream sediment flux.
radius of curvature.
downstream coordinate (channel-fitted).
downstream coordinate (Cartesian).
cross-stream coordinate (Cartesian).
height above the bed.
angle the boundary shear stress vector makes with the s axis.
von Karman's constant.
geometric standard deviation.
boundary shear stress.
nondimensional Shields stress.
Shields number for initial motion.
nondimensional Shields stress computed with Dg.
 This work was supported financially by the National Center for Earth-surface Dynamics and by a National Science Foundation Graduate Research Fellowship (to P.A.N.). Jeff Marr provided critical assistance during the experiments, and the large group of staff and researchers involved with the Streamlab06 project were very helpful. This work also benefited from discussions with Christian Braudrick, Toby Minear, Stefano Lanzoni, and Giovanni Seminara. Carl Legleiter provided very useful code that helped in processing FaSTMECH output. Constructive reviews from two anonymous reviewers, Jonathan Nelson, and Associate Editor John Pitlick improved the clarity of the manuscript.