The copyright line for this article was changed on 18 JUN 2015 after original online publication.
Sediment transport in cobble-boulder rivers consists mostly of fine sediment moving over a coarse, immobile bed. Transport rate depends on several interrelated factors: boundary shear stress, the grain size and volume of fine sediment, and the configuration of fine sediment into interstitial deposits and bed forms. Existing models do not incorporate all of these factors. Approaches that partition stress face a daunting challenge because most of the boundary shear is exerted on immobile grains. We present an alternative approach that divides the bed into sand patches and interstitial deposits and is well constrained by two clear end-member cases: full sand cover and absence of sand. Entrainment from sand patches is a function of their aerial coverage. Entrainment from interstices among immobile grains is a function of sand elevation relative to the size of the immobile grains. The bed-sand coverage function is used to predict the ratio of the rate of entrainment from a partially covered bed to the rate of entrainment from a completely sand-covered bed, which is determined using a standard sand transport model. We implement the bed-sand coverage function in a morphodynamic routing model and test it against observations of sand bed elevation and suspended sand concentration for conditions of nonuniform fine sediment transport in a large flume with steady uniform flow over immobile hemispheres. The results suggest that this approach may provide a simple and robust method for predicting the transport and migration of fine sediment through rivers with coarse, immobile beds.
The beds of many mountain and canyon rivers consist of coarse grains that are transported only in rare events. Most of the sediment passing through such rivers is composed of much smaller grains—generally sand and fine gravel. Transport of the moveable sediment depends strongly on the size of both immobile and mobile grains. Observation and theory for streambeds that are composed entirely of transportable grains are of limited use in guiding estimates of transport over coarse, immobile beds. This has led to interest in developing methods for understanding flow and transport in these cases [Wiberg and Smith, 1991; Nelson et al., 1991; Topping, 1997; Grams and Wilcock, 2007; Yager et al., 2007, 2012; Tuijnder, 2010; Wren et al., 2011; Tuijnder and Ribberink, 2012; Papanicolaou et al., 2011; Kuhnle et al., 2013]. As with any river transporting sediment, an ability to predict rates of transport is a key component for understanding geomorphic development, sediment yield, and local habitat.
Efforts to address the transport of fine sediment over a coarse, immobile bed must address two general phenomena. The first is the effect of exposed immobile grains on the near-bed flow and transport field. The second is the tendency of the fine sediment to gather into a variety of low-relief bed forms, including stripes, patches, and dunes [Mclean, 1981; Nezu and Nakagawa, 1989; Kleinhans et al., 2002; Tuijnder et al., 2009]. The dunes may be of barchan shape when the amount of fine sediment is relatively small and coalesce into increasingly connected dune fields as the amount of fine sediment on the bed increases. The mode of transport for sediment moving over immobile beds can include bed load, suspended load, or a mixture of the two, depending on the sediment size relative to the strength of the flow.
Many efforts to estimate the transport of fine sediment over coarse, immobile grains have focused on the role of coarse grains in exerting form drag and thereby reducing the stress available to move the fine sediment [Topping, 1997; Yager et al., 2007, 2012; Kuhnle et al., 2013]. Grams and Wilcock  presented an alternative model for sand entrainment from these conditions that are based entirely on the relative elevation of fine sediment among the immobile grains. We extend that approach here because we believe it offers a simple and robust method for predicting fine sediment transport over coarse, immobile beds. It supplements (rather than replaces) existing transport relations using a function whose limiting behavior is well constrained. The work of Grams and Wilcock  focused on the effect of sediment bed elevation on grain entrainment in conditions of equilibrium transport. Here, we expand the approach to include the tendency of fine sediment to coalesce into bed forms causing entrainment to be derived from both sand patches and regions of interstitial sand.
The modeling approach presented in this paper is based on relations among the volume of fine sediment available on the bed, the proportional distribution of fine sediment into bed forms (patches) and grain interstices, and the thickness of the sand in the interstices. These relations were developed from observations of fine, suspendable sand transport in conditions of steady, uniform flow and nonuniform transport (aggrading and degrading bed) over a bed of large immobile hemispheres attached to a flume floor. Different volumes of sand storage and different flow velocities produced a range of bed configurations, including sand stripes, isolated barchans, and large sand patches. The experimental results were used to develop new relations for the fraction of bed area covered by sand patches, the fraction of the bed with interstitial sand, and the average depth of sand in each region. We employ these relations in a new bed-sand coverage function to model suspended sediment transport in conditions of partial fine sediment cover. This function treats entrainment from the parts of the bed covered by sand bed forms with a traditional entrainment function and entrainment from interstitial spaces in coarse patches as a function of local sand bed elevation using the relation of Grams and Wilcock . We evaluate the function by implementing it in a routing model that includes stress calculations, an advection-diffusion model of the suspended sediment distribution, and sediment mass balance. The model predictions are compared with the experimental observations. Although the bed-sand coverage function is developed and tested here for the case of sand that is fully suspendable, we suggest that the broader conceptual model may be applied to cases of bed load as well.
2 Controls on Sand Entrainment in a Coarse-Bedded River
For a homogeneous bed without coarse, immobile grains, the primary controls on fine sediment transport are the bed stress and sediment grain size. Transported sediment can collect into bed forms, and various approaches have been developed to account for their effect on the flow and transport field [e.g., Engelund and Hansen, 1967; Smith and McLean, 1977; Brownlie, 1983; McLean et al., 1999]. Entrainment from fine-sediment patches and interstices among large immobile grains introduces additional complications to the entrainment problem. Exposed large grains will exert drag on the flow, reducing the mean stress acting on the fine, moveable sediment. Wakes shed by the coarse grains also introduce large local velocity and pressure excursions that can dominate fine sediment entrainment [Schmeeckle et al., 2007]. When fine sediment coalesces into bed forms, sediment entrainment can occur from either bed forms or from the interstices of the coarse grains. The relative importance of these two locations will depend on the overall volume of fine sediment in the system, with greater prevalence of bed forms as fine sediment content increases. Finally, because immobile grains occupy part of the bed, there is less mobile sediment per unit bed area.
Exposed coarse grains extract momentum from the flow and, depending on their size and packing arrangement, may carry most or all of the bed stress locally [Wiberg and Smith, 1991; Nelson et al., 1991; Mignot et al., 2009; Wren et al., 2011]. Flow in the interstices over the mobile sediment will be strongly dominated by wakes shed by the coarse grains. In some configurations, the fraction of the total stress acting on the fine sediment based on a drag partition may be very small, making such partitioning difficult, if not impossible to apply. The structure and intensity of the turbulent wakes vary depending on the elevation of the sand relative to the coarse grains. Wren et al.  studied the turbulence characteristics of flow over immobile gravel substrate partially filled with mobile sand and observed a decrease in the near-bed Reynolds stress as the mean elevation of sand-filling interstices increased. Thus, in a bed that includes a few large patches of mobile sediment as bed forms and many small patches of mobile sediment in interstitial pockets, entrainment is derived from regions of very different local stress characteristics. A bed stress calculation targeted to a specific entrainment situation (i.e., sand at a particular elevation among immobile grains) will not apply to the entire bed. For these reasons and because these methods do not account for the presence of bed forms, we focus our attention on the effects that the distribution and exposure of mobile sediment have on transport.
When the sand bed elevation in large grain interstices is at or close to the tops of the coarse grains, the effect of the immobile grains should be small and sand entrainment rates should approach those that would be predicted for a sand bed. When the sand elevation is low relative to the coarse grains, the amount of sand available for transport and its exposure to the flow will decrease, approaching zero entrainment at sufficiently low sand elevations. The rate of entrainment of sand from pockets between coarse grains should, therefore, depend directly on sand bed elevation. This suggests that, in the absence of bed forms, the effect of sand bed elevation on entrainment may be captured by a functional relation that increases from zero entrainment at low sand elevations to the entrainment rate of a fully sand-covered bed as sand bed elevation rises to the tops of the coarse grains.
Grams and Wilcock  made observations of near-bed suspended sediment concentrations in equilibrium transport conditions (equal sediment influx and efflux) over a bed of immobile grains partially covered by mobile sand. The conditions of partial sand cover included areas of the bed with sand partially filling interstitial spaces and areas of the bed with sand completely filling interstitial spaces. They compared observed sand entrainment rates to entrainment rate predicted for a fully sand-covered bed and developed a sand elevation correction function. The sand elevation correction ε is the ratio of the observed rate of sand entrainment Es from among coarse, immobile grains to the rate of entrainment that would be predicted over a completely sand-covered bed :
Based on the experimental data, Grams and Wilcock  defined the sand elevation correction as a function of the normalized sand elevation among the coarse, immobile grains ;
where zsi is the average sand bed elevation among the immobile grains and rb is the average height of the immobile grains (Figure 1). Sand elevation and immobile grain height are measured from the base of the interstitial spaces between the immobile grains. The logistic function shown in Figure 1 is given by
with parameters ξ = 18, Z* = 0.2, and η = 1.1. The function approaches 0 as the bed becomes void of mobile sediment at and approaches 1 as the bed becomes covered with fine sediment at .
The shape of the sand elevation correction indicates that entrainment rates are small relative to sand bed conditions when the bed elevation is low. This implies that low sand bed elevations are maintained by small entrainment rates of fine sediment from among coarse grains at flows that would produce much higher entrainment rates over a full sand bed. At slightly higher sand bed elevations, the rate of entrainment increases rapidly. This suggests that as sand begins to accumulate on the bed, entrainment rate increases and effectively limits or prevents the further accumulation of sand in this intermediate range of bed elevations. Two of the observations indicated that entrainment rates may even exceed the rate expected for a fully sand covered bed (Figure 1). This behavior leads to the prediction that bed elevations in this range () are unstable, which was observed in the experiments reported by Grams and Wilcock . However, because the observations in this range of elevations were limited, Grams and Wilcock  constructed the sand elevation correction such that entrainment rate equals that of a completely sand-covered bed for .
Sand entrainment from a coarse bed is further complicated by the organization of the sand bed into bed forms. Possible bed configurations include longitudinal sand stripes, isolated barchan dunes, migrating dunes, and larger patches of sand-covered bed [Mclean, 1981; Nezu and Nakagawa, 1989; Kleinhans et al., 2002; Tuijnder et al., 2009]. When bed forms are present, sand entrainment is derived from a mix of sand-covered patches and areas of sand contained at different elevations within the interstitial pockets of exposed coarse grains. The observations used to produce the sand elevation correction shown in Figure 1 were made in a narrow flume that suppressed the development of sand patches [Grams and Wilcock, 2007]. A model of sand entrainment from a bed that includes complex topography and bed forms requires a prediction of the proportion of the bed occupied by those bed forms as well as submodels for entrainment rates from the sandy patches and from the interstitial pockets among coarse exposed grains. In these conditions, sand entrainment occurs from a mix of local bed conditions, which may be characterized as sand patches for which and interstitial regions, or nonpatches, where . According to Figure 1, we might expect that nonpatch areas of the bed with relatively small will also have small ε, such that bed forms, if present, will produce most of the sand entrainment.
3 Experiments of Nonuniform Sand Transport Over a Coarse Bed
3.1 Experimental Design and Procedures
The flume experiments were designed to encompass a broader variety of transport conditions and bed configurations than were observed in the equilibrium transport experiments used to develop the sand elevation correction (equation (3)) of Grams and Wilcock . The nonuniform transport experiments were designed to simulate the migration of a sand pulse through a coarse-bedded channel, thereby producing conditions of partial sand cover over immobile grains. The most important scaling properties are those that determine the near-bed flow and sediment entrainment conditions and those that contribute to the development of a spatially variable bed configuration representative of natural conditions. To achieve the latter objective, the experiments were conducted in a much larger channel than used in the Grams and Wilcock  experiments. The 2.74 m wide and 84 m long experimental channel at the St. Anthony Falls Laboratory of the University of Minnesota provided near field-scale conditions (Figure 2).
The bed surface was covered with 15 cm diameter concrete hemispheres arranged in closest packing across the full width of the channel over a 40 m test section. This bed arrangement was chosen to simulate a well-sorted natural cobble bed while providing the advantage of known geometry that allows for conversion among depth, volume, and areal coverage of fine sediment for a given fine sediment depth. The flow depth over the tops of the hemispheres was 60 cm for all runs, in order to allow a spatially integrated region in the flow above the near-bed region and to provide sufficient depth for the development of a range of bed forms [Southard, 1991]. Two flow rates were used, both of which produced Rouse numbers less than 1 such that the dominant mode of transport was suspension. The Rouse number is the ratio of particle settling velocity ws to shear velocity u∗,
where k is the von Karmen constant. Froude numbers were less than 1 to ensure lower regime flow conditions and bed forms (Table 1). The bed slope was 0.0005 for all experimental runs.
Table 1. Streamflow and Stress Characteristics of the Experimental Runs
The average water surface slope for all runs at the same mean velocity.
The Rouse number was calculated using the feed sediment grain size of 0.13 mm, common to all runs.
The Froude number was calculated using the flow depth of 60 cm, common to all runs.
Each experimental run included a 90 min segment with sediment feed followed by one to four additional segments with no sediment feed (Table 2). Fine sediment with a median diameter of approximately 0.13 mm (Figure 3) was fed at the upstream end of the test section. The same feed sediment was used for each run. The initial bed condition for Runs 1, 2, and 5 consisted of bare hemispheres. Prior to Runs 3 and 4, a layer of mobile sediment with a median diameter of 0.34 mm (Figure 3) was spread evenly to a thickness of 4 cm (). The initial sediment bed was added to introduce a coarser fraction of mobile sediment and to simulate the effect of a fine sediment pulse (the 0.13 mm D50 sand feed) introduced to a winnowed bed (the 0.34 mm D50 seeded bed). The flow rates and sediment feed rates resulted in nonuniform transport conditions such that sediment accumulation occurred in some run segments and sediment evacuation occurred in others.
Table 2. Summary Characteristics of Experimental Runs
Initial bed state: N indicates no sediment bed at the beginning of run. Y indicates that 4.4 cm of sand with D50 = 0.34 mm was distributed uniformly among coarse grains before the first segment of the run.
no bed forms
3.1.2 Flow Measurements
Flow was controlled by a motor-driven headgate that diverted water from the Mississippi River. This mechanism did not allow precise control of inflow volume, resulting in slightly variable flow rates between run segments (Table 2). The flow rate of each segment was measured by collecting outflow for a measured interval in large volumetric tanks. Flow structure and the near-bed shear stress were characterized by velocity profiles measured with an acoustic Doppler velocimeter (ADV). Each velocity profile consisted of eight measurement positions located 1, 2, 3.5, 6, 10, 15, 25, and 40 cm above the top of a roughness element (hemisphere). These profiles were measured at five cross-channel positions 45.5, 91.5, 137, 183, and 228.5 cm from the left sidewall, respectively. For the low flow conditions, measurements at all five cross-channel positions were made at six locations along the flume: 1.2, 3.0, 5.1, 7.0, 10.9, and 15.1 m upstream from the downstream end of the test section (hereafter, all streamwise positions are referenced by distance upstream from the downstream end of the test section). Four additional velocity profiles were collected at the center-channel position at distances of 19.1, 23.0, 31.1, and 37.2 m. For the high flow condition, velocity profiles were collected at all five cross-channel positions at streamwise positions of 1.2, 3.0, 5.1, and 7.0 m. Center-position profiles were also collected at 10.9, 15.1, 19.1, 23.0, and 31.1 m. All measurements were made at a sample rate of 50 Hz for 1 min or longer with a SonTek 16 MHz MicroADV. This instrument has a sampling rate of up to 50 Hz and a velocity resolution of 0.01 cm/s up to 250 cm/s. Reported accuracy is 1% of measured velocity. The sampling volume is centered 4.67 cm below the probe tip, and the sampling volume is 0.09 cm3.
The data collected by the ADV were processed to remove spikes then analyzed to determine velocity profiles and turbulence characteristics [Grams, 2006]. The turbulent Reynolds stresses were calculated in units of shear velocity,
and u is the instantaneous downstream velocity, w is the instantaneous vertical velocity, and n is the number of instantaneous measurements. Primes denote instantaneous deviations from the mean velocity and overbars represent time averages. We used the velocity measurements sampled at 50 Hz to approximate the instantaneous velocities in equation (6).
3.1.3 Suspended Sediment Sampling
Direct measurements of sediment concentration were made during each run segment by collecting samples via fifteen 3.18 mm internal diameter stainless steel Pitot tubes. The tube inlets were positioned 5.17 m upstream from the downstream end of the test section at three cross-channel positions and at five elevations above the hemisphere tops. The sample tubes were connected to plastic tubing with outlets positioned outside the channel at an elevation such that the samples were collected isokinetically (water velocity in the tube inlet matched the ambient stream velocity at that position in the channel). Three rakes of five tubes each were positioned 44, 136, and 227 cm from the left sidewall. Within each rake, tube inlets were positioned 1, 2, 6, 15, and 40 cm above the hemisphere tops. Samples were collected in 2 to 4 sample events in each run segment (Table 2). Sample durations typically ranged from 60 to 90 s, depending on the flow rate. Each sample was analyzed for suspended sediment concentration, and selected samples were analyzed for grain size.
3.1.4 Measurement of Bed Topography and Bed Material Sampling
Each run segment was concluded by shutting off the water supply and dropping a tailgate that stopped the flow in less than 30 s. This procedure was used to stop transport as quickly as possible thereby preserving the bed configuration. The flume was then drained slowly to minimize disturbance of the bed. This method was generally successful, and minor reworking of the bed was observed only in the downstream few meters of the test section. Once drained, the entire bed was photographed from above. The area of the bed covered by patches of sand at the conclusion of each run segment was measured from the overhead photographs. The areas were measured by tracing sand patch boundaries on digital images and computing the total area within the digitized polygons. Sand patches were defined as areas of the bed where the immobile grains were either completely covered with sand or nearly covered such that only isolated tops (< 0.5 cm) of the hemispheres were exposed. Between sand patches, sand is present in the interstitial spaces among the immobile grains. We refer to these as interstitial or “nonpatch” areas of the bed. The sand depth above the elevation of the base of the hemispheres was measured using a ruler at locations on a grid defined by 34 streamwise positions at 1.2 m intervals and 18 or 19 transverse positions (Figure 4). The transverse positions were located in the spaces between adjacent hemispheres. In each space, sand depth was measured by probing through the sand to the bed in the interstitial space immediately downstream from the upstream hemisphere and in the interstitial space immediately upstream from the downstream hemisphere. There are 18 interstitial spaces in the even-numbered rows and 19 spaces in the odd-numbered rows. The measurements were made in alternating odd and even-numbered rows to avoid bias in cross-stream position. A measurement set consisted of 630 measurement pairs, or 1260 individual measurements. The measured sand bed elevations were grouped into measurements of sand elevation in the sand patches and nonpatch areas.
3.2 Observations of Bed Configuration
The condition of the bed following each run segment was characterized by the mean elevation of sand on the bed and by the distribution of sand bed elevations between bed forms (sand patches) and interstitial areas. During the feed segment of each run, the channel aggraded throughout the length of the 40 m test section. The average elevation of sand among the hemispheres was typically from 2 to 4 cm (Table 2). Deposition was greatest in an accumulation zone in the upstream 10 m where the hemispheres were largely buried (Figure 5). Downstream from this region of nearly complete sand cover, the hemispheres were partially covered by the mobile sediment. Bed-sand coverage was not uniform but organized in flow parallel sand stripes, isolated barchan dunes, or a combination of these two bed form types (Figure 5). In Runs 1, 2, and 5, sand stripes were the dominant bed form. Runs 4 and 5 included both barchan dunes and stripes (Table 2). Sand stripes and isolated barchan bed forms have been identified in laboratory and field studies as characteristic of low sediment supply conditions [McCulloch and Janda, 1964; Allen, 1968; Carling et al., 2000a; Carling et al., 2000b; Kleinhans et al., 2002; Tuijnder et al., 2009]. For example, Figure 6 shows sand stripes observed on the bed of the Colorado River, indicating that the bed forms observed following the experimental runs are consistent with bed configurations observed in the field for conditions of partial fine sediment cover.
During the nonfeed run segments, the deposits in the accumulation zone slowly eroded. Some downstream migration of the deposit occurred, but the primary mechanism of erosion was entrainment into suspension. The bed configuration also evolved during the evacuation run segments. In runs dominated by sand stripes, the number of stripes decreased and the individual stripes became narrower as the sediment evacuated (Figure 5a). In runs that included both sand stripes and dunes, the stripes decreased in number and coalesced resulting in transitional sand stripe and barchan bed forms (Figure 5b). The transitional bed forms further coalesced into isolated barchans (Figure 5c), which then decreased in size and abundance as evacuation of fine sediment progressed. The only run segment that lacked bed forms (Run 5B) was for conditions of near-complete sand evacuation.
3.3 Incorporating Bed Configuration in a Sand Entrainment Model
The persistence of bed forms and a bimodal distribution of bed elevations are consistent with the observation of Grams and Wilcock  that a condition of sand uniformly filling interstitial spaces to a height above roughly half the height of the roughness elements, while not completely burying them, is unstable. Instead, sand in the interstices is typically at an elevation less than or equal to half the roughness height, while the balance of the sand coalesces into fine-sediment bed forms. This motivates the effort to define a fine sediment transport model that distinguishes between interstitial and bed form sediment. We, therefore, modify equation (1) such that the total entrainment from the bed Etotal is equal to the sum of the entrainment from patch Ep and nonpatch Enp areas of the bed,
where each term is a relative entrainment rate (i.e., the predicted entrainment rate normalized by the entrainment rate expected for a sand bed ).
We assume that sand is entrained from sand patches at the same rate per unit area as would occur over a completely sand-covered bed. Therefore, the relative entrainment rate is equal to the fraction of the bed covered by sand patches. To be easily implemented in a sediment-routing model, a function for the fraction of bed covered by sand patches would ideally be related to the total amount of sand stored in the bed, because a routing model must track sediment volume anyway. For the flume experiments, the fraction of the bed surface covered by sand bed forms Fs varies consistently with the mass of sand stored in the bed, as represented by the mean thickness of sand on the bed (Figure 7). Interestingly, the same trend holds for all bed configurations, including sand stripes, barchan dunes, and transitional bed forms. We compare these results to the experiments of Tuijnder , which provide the necessary information on the area of channel covered by sand bed forms, the volume of sediment on the bed, and the immobile bed size. Even though there are several important differences in the Tuijnder  experiments (they used natural gravel for the coarse grains, the fine sediment moved as bed load, and the difference in size between the mobile and immobile sediment was much smaller), the values of Fs and fall within the range of our observations.
We use a logistic relation,
to represent the variation of Fs and . Parameters for this fit were obtained by least squares (R2 = 0.91), and the function provides a plausible trend for Fs > 0.7, for which observations are not available. In particular, the trend is defined such that the entire bed is not covered by sandy bed forms (Fs < 1) when = 1 and that Fs approaches one asymptotically. Although the form of the function is plausible and fits the available observations well, further observations are needed to test and expand this relation for a wider range of conditions.
The nonpatch area of the bed is given by Figure 7 as (1 − Fs). Entrainment of sand from nonpatch areas requires an estimate of the sand bed elevation among the immobile grains. Again, such a relation is most useful if defined relative to the overall storage of sediment in the bed. Figure 8 shows the mean bed elevation of the nonpatch fraction of the bed zsi plotted as a function of the mean elevation of the entire bed for each run segment. The variation of zsi with is consistent across all bed configurations in the experiments (Figure 8). The trend is also consistent with the observations of Tuijnder .
A simple relation between zsi and
fits the observations well (least squares with R2 = 0.85) and provides the constraints that zsi = 0 at = 0 and that zsi approaches 0.4 at large . Again, we have no direct observation of the form of the function beyond = 0.7. However, the flume observations indicate that most individual values of in nonpatch areas are smaller than 0.4, making it unlikely that the mean value could exceed 0.4. This is consistent with previous observations of the apparent instability of values in the range 0.4 < < 1.0 [Grams and Wilcock, 2007]. In any event, the relative volume of sand in nonpatch areas becomes small for exceeding 0.7 (Figure 9a), so uncertainty in the value zsi/rb will have a small effect on sand entrainment and routing for these values.
The relative entrainment rates from sand patches and nonpatch areas are combined in the final bed-sand coverage function (Figure 9b). The relative entrainment from sand patch areas is given identically by the relation for Fs in Figure 7,
because entrainment rate in patch areas is taken to be the same per area as from a sandy bed.
The relative entrainment from the nonpatch areas is given as the product of (1 − Fs) from Figure 7 and the logistic Grams and Wilcock  sand elevation correction function.
where ε is evaluated using equation (3) with zsi/rb determined from equation (9).
The sum of patch (equation (10)) and nonpatch (equation (11)) functions give the bed-sand coverage function (Figure 9b), which is fit with a logistic function of the same form as equation (3) with a new set of parameters,
Note that is the mean sand elevation for the entire bed, including both sand patches and nonpatch areas, while equation (3) is evaluated based on the mean sand elevation for interstitial sand only. Comparison of the bed-sand coverage function in Figure 9b with Figure 1 shows that the bed-sand coverage function incorporating both patch and nonpatch relative entrainment is shifted toward larger values of than the sand elevation correction function of Grams and Wilcock , reflecting the influence of sand storage in patch areas.
4 Nonuniform Sand-Routing Model
The bed-sand coverage function was implemented in a morphodynamic numerical model designed to predict evolving sand bed elevations, grain size, and suspended sand concentrations. A 2-D advection-diffusion model was used to handle conditions of unsteady suspended transport and nonuniform bed coverage. The main components of the numerical model include characterization of the vertical flow structure and bed stress, calculation of the sand entrainment rate for a full sand bed, application of the bed-sand coverage function to account for partial sand cover, calculation of the vertical profile of suspended sediment concentrations, and conservation of sand mass to track the evolving sand-bed coverage.
4.1 Parameterization of Streamflow and Bed Stress
The nonuniform transport experiments were conducted under conditions of steady uniform flow with constant fixed bed roughness. We calculated the stress near the bed as shear velocity u* from bulk flow properties as,
where g is the gravitational acceleration, R is the hydraulic radius, and S is the water surface slope (Table 1). To examine the stress regime near the bed, we compared this calculation of stress with profiles of Reynolds stress, both of which were nondimensionalized as the ratio of shear velocity to depth-averaged velocity. The stress decreased linearly from the upper region down to approximately 0.10h above the hemispheres, where h is flow depth as measured from the hemisphere tops (Figure 10). From this height down to about 0.05h, the mean shear velocity is constant and equivalent to about 0.085U, where U is depth-averaged velocity. Closer to the bed, the measured Reynolds stress decreases. Voulgaris and Trowbridge  reported that Reynolds stresses computed from ADV measurements within 3 cm of the bed often deviate from expected flow characteristics. Errors in ADV measurements near the bed resulting from shear within the sampling volume have also been reported by Kim et al. . Therefore, the Reynolds stresses measured within the segment of constant stress between 0.10h and 0.05h are likely the most reliable estimate of average bed stress. Based on these observations, the bed stress for both flow conditions is approximated as a function of the depth-averaged velocity
This approximation is in good agreement with the stress calculated from the bulk flow measurements, which was equivalent to u∗ = 0.086U and u∗ = 0.080U for the low and high flow conditions, respectively.
Specification of the velocity profile is required to compute the profile of suspended sediment concentration. The velocity profile was observed to be consistent at all transverse and alongstream positions, and a mean profile was used in the model. The profile is composed of separate log linear segments for the lower 10% of the flow and the upper 90% of the flow. The two logarithmic profiles are joined smoothly with a linear fit:
where u is velocity at elevation z above the top of the bed roughness elements.
4.2 Entrainment From a Sand Bed
Sand entrainment rate from a sand bed is needed for locations covered by sandy bed forms and as the reference transport rate for the sand elevation correction function. We used the Garcia and Parker  entrainment relation
where A = 1.3 × 10−7 is an empirically fit constant, λ is a straining parameter related to the standard deviation of the grain size distribution, and Zm,j is a similarity variable given by
The Garcia and Parker  model is based on grain stress calculated using the Einstein stress decomposition [Einstein and Barbarossa, 1952] as presented by Engelund and Hansen , detailed below. The particle settling velocity ws,j by grain size j is calculated with the relation of Dietrich  for typical quartz grains. The exponent m represents the degree to which the mobility of grains of a given size Dj is affected by its relationship to the median of the size distribution D50. Garcia and Parker  use m = 0.2, which suppresses the effect on Zm,j of the ratio Dj/D50. The grain Reynolds number Rp,j is given by
where s is the specific gravity of the sediment and g is the gravitational acceleration.
The Einstein stress decomposition is based on an analogy between the flow of interest and a hypothetical flow that has the same slope and depth-averaged velocity but lacks the drag exerted by bed forms. The analogy is extended here to include the drag exerted by the coarse, immobile grains in addition to that exerted by bed forms. The depth-averaged velocity is related to the bed stress through a standard resistance relation
where ks is the Nikuradse equivalent grain roughness associated with the total bed stress. By the analogy of the Einstein decomposition, a similar relation holds for the hypothetical flow without immobile grains:
where h′ is the flow depth due to grain stress and is the grain stress roughness height. Here is used. The slope S of the hypothetical flow is also the same as that of the flow in question, thus,
Given the total stress and depth-averaged velocity, equations (20) and (23) can be solved iteratively for the grain stress. The grain stress is not assumed to apply to any particular point on the bed but is taken to provide a consistent scaling between the flow velocity, depth, and stress-producing entrainment from a full sand bed. The entrainment predicted by equation (16) is then adjusted for the appropriate bed sand thickness and configuration using the bed-sand coverage function (equation (12)).
4.3 Sand Transport and Bed Sediment Continuity
The representation of the vertical distribution of suspended sediment concentration is a critical aspect of accurately predicting the total suspended sediment flux. The governing equation for suspended sediment mass balance is the 2-D form of the advection-diffusion equation,
where c is suspended sediment concentration, x is the streamwise coordinate axis, z is the vertical coordinate axis, is time average streamwise velocity, and Kz is a coefficient of diffusion for suspended sediment. The vertical distribution of suspended sediment concentration may be obtained from equation (24) by representing the sediment diffusivity as eddy diffusivity and integrating over the flow depth with appropriate boundary conditions. The common analytical solution to equation (24) neglects downstream advection, which forces the requirement that the concentration profile is in equilibrium with the local boundary concentration. To evaluate the validity of this assumption, we applied the analytical solution for equation (24) of Stansby and Awang , which predicts the adjustment for developing concentration profiles when clear flow abruptly encounters a sand-covered bed. Applied to our experiments, the adjustment time is about 24 s, corresponding to a distance of about 12 m. Because our experiments included a zone of about 5 m of nearly complete sand cover with decreasing cover downstream, this is probably an underestimate of the actual adjustment time. To permit prediction of changes in bed elevation for channel segments on the order of 1 to 2 m in length, we chose to implement a numerical solution to equation (24), thereby, retaining the term for downstream advection. We used the boundary conditions of vanishing concentration at the free surface
and a flux boundary condition at the bed [e.g., Parker, 1978].
Exchange of sediment between the flow and the bed is given by the equation for bed sediment mass continuity
where λp is the porosity of sediment in the bed and qT is the total streamwise sediment flux. In the laboratory experiments, flow strength (shear velocity) and grain size were scaled such that the dominant mode of transport was by suspension. We also noted that sediment was redistributed from the accumulation zone at the upstream end of the test section by deflation, not bed form migration, further indicating suspension as the dominant mode of transport. For these reasons, bed load transport was neglected and suspended load transport qs = qT.
4.4 Computational Procedure
Equation (24) was solved numerically using an upwind finite difference scheme based on the method presented by Patankar . Matrix coefficients were solved for implicitly using an efficient tridiagonal algorithm. Sediment diffusivity Kz was approximated as eddy diffusivity KE, which is related to the stress and the velocity gradient by
The bed stress is related to the total stress for a linear stress distribution according to
Equations (28) and (29) were combined to solve for KE using the empirical bed stress (equation (14)) and velocity profile (equation (15)).
The flux at the bed is provided by the predicted entrainment rate (equation (16)), with the bed-sand coverage function (equation (12)) applied when the average sand bed elevation is less than the bed roughness height. Changes in the volume of sand on the bed are calculated from the divergence in the transport according to equation (27). Conversion between bed sand volume and sand bed elevation is made assuming that the sand uniformly fills the interstitial spaces of the hemispheres. The upstream boundary condition is the feed concentration during the feed segment of the run and zero concentration during subsequent segments. The initial conditions of the bed were no bed sediment for the simulations of Runs 1, 2, and 5. The initial conditions for simulations of Runs 3 and 4 included a 4.4 cm layer of evenly distributed sediment with a size distribution corresponding to that of the seeded bed.
Accurate prediction of fine sediment entrainment and transport in suspension requires a model that allows the bed grain size to evolve in concert with the transport [Rubin and Topping, 2001, 2008]. Grain size evolution was included in the model by the use of multiple size fractions and conserving mass for each fraction. The grain size distribution was divided into seven classes based on the sedimentological φ scale, where φ = − log 2D. The midpoint (on the logarithmic φ scale) of each size fraction is used as the representative size for that fraction. Each fraction is 0.5φ in size with midpoint values between 0.5φ (0.71 mm) and 3.5φ (0.088 mm). Each size fraction is treated independently in the model, and grain size interactions are incorporated using the straining parameter λ and sorting parameter m of the original Garcia and Parker  entrainment formulation (equation (16)).
Implementation of bed sediment continuity for mixed sizes requires consideration of the bed sediment mixing depth. Grains from within the mixing depth, or active layer, are eligible for entrainment in proportion to their presence in the grain size distribution. Particles in the substrate, beneath the active layer, are eligible for entrainment only after erosion has occurred and the bed elevation has changed. In these experiments, the bed sediment was well mixed to the depth of the bed roughness elements, which are of approximately the same scale as the bed forms [Grams, 2006]. Because average bed elevations in these experiments rarely exceeded 4 cm and never exceeded 6 cm, the entire sediment bed was always within the active layer.
4.5 Comparison of Numerical Predictions and Experimental Observations
Two sets of numerical simulations were conducted for each of the five experimental runs. In the first set, the bed-sand coverage function was used as described above. In the second set, bed-sand coverage function was not used and, therefore, entrainment rate did not depend on in any way (ε = 1 for all , and entrainment occurred from the full bed area).
Comparison of predicted and observed sand bed elevations is given in Figure 11. Immediately apparent is a zone at the upstream end of the test section in which the thickness of the predicted deposit exceeds that of the observed. The cause of this difference is that the flow field at the entrance to the sediment bed was disturbed by concrete blocks (visible in Figure 2), which were emplaced to prevent excess deposition at the location of the sediment feed. The increased turbulence behind these blocks enhanced entrainment in the first few meters of the flow. We chose not to calibrate the 1-D flow model to accommodate this local effect. Our goal was to use the simplest possible, uncalibrated flow and transport model so we could directly explore the effect of using the bed-sand coverage function in predicting sand routing. We focus our attention primarily on the downstream 30 m of the 40 m domain.
For Run 1, the model predicted accumulation of about 10 cm of sand at the upstream end of the domain and decreasing sand elevation downstream. Predicted sand elevations in the central and downstream parts of the domain approximately matched observed bed elevations when the bed-sand coverage function was used and were lower than observed when the coverage function was not used. Following the feed segment of the simulation, a downstream thinning profile of predicted bed elevations was maintained as sand elevations decreased throughout the model domain. Using no bed-sand coverage function, sand evacuated rapidly and was nearly exhausted from the downstream part of the domain after about 150 min of run time. Using the bed-sand coverage function, the model predicted that sand persisted throughout the channel for the entire run time (660 min). The model predicted a similar pattern for Run 2, which had a larger amount of feed sediment. The final segment of Run 2 used an increased water discharge which produced greater sand removal than Run 1.
Runs 3 and 4 included an initial 4.4 cm thickness of sand coarser than the supply. In Run 3, the predicted and observed sand elevations generally agree, although the model predicts a more uniform distribution of sand than was observed. The predicted sand bed elevations are the same with or without the bed-sand coverage function. Use of the function has a weak effect because the mean sand bed elevation remained large throughout the run such that entrainment rates were rarely modified by the function (Figure 9). Run 4 used a similar initial bed and a higher discharge than Run 3. The influence of the bed-sand coverage function is evident in Run 4E, where sand bed elevations are lower than the other runs (Figure 11). The sand elevations predicted using the bed-sand coverage function match the observed values well, and the uncorrected entrainment values are too large, producing excessive sand evacuation.
Run 5 used an empty initial bed and the higher flow rate. The simulation using the bed-sand coverage function predicts the observed sand elevation well, and the simulation that does not use the function again produces excessive sand evacuation.
Observed and modeled mean bed elevations at the conclusion of each run segment are plotted in Figure 12. As described above, model predictions using the bed-sand coverage function slightly over predict bed elevations and model predictions without the coverage function greatly under predict bed elevations at the low range of the observations (values of less than about 0.4).
The coupled entrainment-transport model is also compared with observations of suspended sediment concentration. Suspended sediment samples were collected at one streamwise position and three cross-channel positions during each run segment. The samples are compared with the modeled profile for the corresponding simulation time, rounded to the nearest 10 min. (Figure 13). In most cases, modeled concentrations are within the range of variability of the measured concentrations. When the observed and modeled concentrations match near the bed, the rest of the profile tends to be in agreement, even though the shape of the profiles is variable (Figure 13a). This suggests that the advection-diffusion component of the transport model is adequately predicting the concentration profiles. Most of the larger discrepancies in Figure 13b are for measurements made during the first segment of the run, either during or immediately following the sediment feed. In these cases, the predicted concentrations tend to lag behind the observations. In these cases, the match between observed and predicted profiles improves in subsequent run segments.
The bed-sand coverage function is an approach to predict net sand entrainment from sand patches and interstices among large grains that are based on the amount of sand on the bed and the proportion of sand that is organized into bed forms. The function is coupled with a standard sand entrainment relation to provide a smooth transition from modeling entrainment from a sand bed to a bed that is partially covered with sand. When implemented in a morphodynamic sediment entrainment and routing model, the approach provides good predictions of bed evolution and suspended sediment concentration.
The relative contribution to net entrainment from different parts of the bed varies as a function of the average sand elevation (Figure 9). Entrainment from sand patches occurs at the same rate per unit sand area as sand entrainment from a fully sand covered bed. When the immobile grains are barely exposed, most entrainment is derived from the large sand patches. As average sand elevation decreases, sand patches occupy a progressively smaller fraction of the bed area and entrainment from patches declines. Entrainment from areas of the bed not covered by sand patches is estimated by the sand elevation correction of Grams and Wilcock . As sand elevations drop, the Grams and Wilcock  sand elevation correction incorporates the effects of reduced net entrainment from interstitial spaces owing to decreasing sand supply and enhanced entrainment per unit area of exposed sand. The enhanced entrainment likely results from increased turbulence in the wakes among the coarse roughness elements [Wren et al., 2011]. As the quantity of sand continues to decrease and the sand patches become smaller, an increasing proportion of the entrainment is derived from the areas between sand patches. Relatively high entrainment rates are maintained owing to the enhanced entrainment per unit area of exposed sand from among the interstitial areas.
While net entrainment rates predicted by the bed-sand coverage function are always less than the rate over a sand bed (Figure 9), they are not as low as would be predicted if entrainment was directly proportional to the area of exposed sand. Entrainment rates per unit area of exposed sand are up to 3 times the rate for a sand bed (Figure 14). Enhanced entrainment from interstitial areas is consistent with the distribution of sand bed elevations observed in these experiments and also consistent with other experimental studies [McLean, 1981; Nickling and McKenna Neuman, 1995; Grams and Wilcock, 2007; Wren et al., 2011]. Grams and Wilcock  reported difficulty creating a bed with average sand elevations in the range of 0.5rb–1.0rb that did not have large bed forms. We did observe average bed elevations in this range in the main channel experiments, but only because the width of the channel allowed the segregation of the bed into bed forms and areas with sand cover less than about 0.5rb. Thus, interstitial pockets tend to be nearly full or less than about half full. Nickling and McKenna Neuman  observed a similar phenomenon when they measured accelerated sand entrainment as immobile roughness elements became exposed in a deflating sand bed during aeolian transport.
Particle entrainment rates that are a function of the elevation of the sand surface relative to the height of immobile grains also provide an explanation for the organization of sand patches, consistent with theories that have been proposed to explain the formation of sand stripes. McLean  estimated bed stress from velocity profiles measured over sand stripes and the areas of rough bare bed between sand stripes. He reported lower stresses over the stripes than the adjacent rough bed, composed of 2 to 4 mm immobile grains. Lower stress, and relatively lower rates of entrainment, over the sand patches helps to maintain the patches, whereas relatively higher rates of entrainment from elevations less than the height of the immobile grains result in rapid deflation to a low sand bed elevation as those grains become exposed. Relatively large immobile grains, such as those used in these experiments, allow some interstitial storage of fine sediment below the region of accelerated entrainment.
The organization of the bed into patches of mobile sediment and patches of immobile sediment or bedrock has also been observed in bedrock channels. In a study of bed load-dominated bedrock channels in southern Utah, Johnson et al.  reported that the channels tended to be organized into reaches of nearly complete sediment cover and reaches nearly devoid of sediment cover. This suggests that the relations that we have developed between mobile sediment cover, mobile sediment thickness, and mobile sediment transport may apply across a wide range of sediment-supply-limited systems.
Despite enhanced entrainment from interstitial areas, it is the change in mobile sediment supply that exerts the strongest control on sand entrainment rates. Most of the decrease in entrainment that occurs as average sand elevation drops is caused by the decline in the fraction of the bed covered by sand patches (Figure 9). This is consistent with studies that have shown that transport scales strongly with sediment supply across a range of conditions for both sand- and gravel-bedded streams. Vericat et al.  showed that patches of mobile gravel control gravel transport rates. Kuhnle et al.  scaled bed load transport of mobile sand using the cumulative probability distribution of gravel elevations, which when evaluated at the mean elevation of sand among coarse grains represents the fraction of bed covered by sand. In an investigation of the transport of mobile fine gravel among large roughness elements for conditions analogous to steep mountain streams, Yager et al.  also found that supply, expressed by the average height of the mobile sediment, was an important control on transport rates.
Thus, the observations that connect the quantity of mobile sediment on the bed, the fraction of the bed covered by bed forms, and the relative thickness of sediment in the sand patches and interstitial areas are important and useful outcomes of this work. The simple relation between average sand elevation and the fraction of the bed surface covered by bed forms (Figure 7) is a key component in the bed-sand coverage function presented here but could also be applied in a range of models as an index of mobile sediment supply. An important aspect of the relation between the fraction of the bed covered by bed forms (sand patches) and average sand elevation is the lack of dependence on bed form configuration. This lack of dependence on bed form type indicates that it is the quantity of the mobile sediment as expressed by the area of sand exposed on the bed, not the specific configuration of the sediment, that is the dominant control on entrainment.
6 Broader Application of the Bed-Sand Coverage Function
Ultimately, the rate of entrainment of sediment into motion depends on the details of the grain-to-fluid and grain-to-grain interactions. Understanding of the physics of these interactions at the grain scale [e.g., McEwan and Heald, 2001; Schmeeckle et al., 2007] is important for describing the processes of particle entrainment and may lead to the development of more fundamental models for particle entrainment and transport. Currently, however, the force required to mobilize a particular particle from at rest among other particles of various size is never precisely known, and it is difficult to estimate the grain stress acting on fine interstitial sediment when most of the bed stress acts on immobile grains. Thus, modeling at the scale of natural streams and rivers requires generalization and approximation.
Because the bed-sand coverage function and its component functions are largely constrained by geometric relations among fine sediment volume, elevation, and areal extent, we argue that they are general in concept. Further testing is required, however, to verify the particular relations developed from our experiments conducted under a specific and limited range of conditions. Thus, application of the approach to natural river channels or flume experiments with a bed composed of natural grains is a logical next step in evaluating the approach. Joint measurements of the fraction of bed covered by mobile sediment, the thickness of the mobile sediment layer, and the size of the mobile and immobile grains could be used to test individual model components. Although the relation between mean fine sediment elevation and fraction of the bed covered by bed forms (equation (8)) spans a range of mean bed elevations, it is based only on our experimental conditions and compared only with the limited observations of Tuijnder . Thus, additional observations of fine sediment thickness coupled with observations of the extent of sand patches for beds with different immobile grain sizes would either support or result in modifications to equation (8). The relation between fine sediment thickness among the interstitial areas and mean fine sediment thickness for the entire bed (equation (9)) also requires further evaluation for a broader range of immobile grain sizes.
Although we have tested the bed-sand coverage function for the condition of suspendable sand, the underlying reasoning behind the function applies equally to sediment moving as bed load. That is, the rate of fine sediment entrainment should vary from 0 to that of a fine-sediment bed over the range 0 < < 1. In the case of fine sediment that moves predominantly as bed load, the transport function used for a full fine sediment bed would be different from the suspended sediment relation we use here, but the reduction in fine sediment transport should follow a similar trend as the sediment elevation decreases relative to the roughness. The specific empirical elements of the formulation, represented by Figures 7 and 8, require further confirmation for a broader range of conditions, as discussed above. But we note that the experiments of Tuijnder  are for conditions of bed load transport, and these data follow the trends in those figures well.
The model developed here specifically applies to the case in which the riverbed is composed entirely of coarse, immobile material. The transport of fine sediment acts to produce a variable fine sediment elevation, depending on the volume of fine sediment stored in the reach, including the case in which most or all of the coarse elements are covered. The approach may not be well suited to cases in which the size of coarse, immobile grains is large relative to the flow depth and the density of those grains is smaller, leaving broad permanent patches of fine sediment in between. This would include many step -pool channels and streams in which the supply of coarse sediment is never large enough to fill more than a small fraction of extended void spaces between large grains.
This study describes a novel set of experiments designed to investigate relations among mobile sediment supply, bed configuration, and suspended sediment transport rate. The experimental results show that when a limited supply of mobile sediment is transported over a bed of coarser, immobile grains, the available fine sediment does not gradually and uniformly fill interstitial spaces. These results are consistent with the findings of Grams and Wilcock , who reported that fine sediment uniformly filling interstitial spaces is an unstable condition. Instead, the available sand coalesces into bed forms which typically take the form of barchan dunes, sand stripes, or a combination of these forms. Similar bed form configurations have been observed in previous studies of fine sediment transport in supply-limited conditions.
Predicting sediment entrainment from these bed conditions is, therefore, a problem of predicting the fraction of bed covered by bed forms, the fraction of the bed where sand occurs only in interstitial spaces, and then a problem of predicting entrainment rates separately for each of these regions. While it is necessary to consider the dynamics of fine sediment entrainment from interstitial regions of the bed, it is of equal or greater important to consider the entrainment from the fine sediment patches. The results from this study demonstrate that an approach that divides the bed into fine sediment patches and nonpatches is possible and practical, because there are consistent relations among sand content (mean sand elevation) and other bed characteristics. Most important is the relation between sand content and the fraction of the bed surface covered by bed forms. This relation allows for implementation of separate entrainment formulations for each region of the bed. For the conditions we studied, this relation did not depend on bed form configuration, suggesting that the quantity (exposed area) of mobile sediment on the bed is more important than the specific configuration of the sediment in determining entrainment rates.
We also found a consistent relation between sand content (mean sand elevation) for the entire bed and sand content for the fraction of the bed not covered by bed forms. This relation allows prediction of the sand elevation in interstitial areas, which is required to predict entrainment from nonpatches. We incorporated these relations in a bed-sand coverage function. The function is simple and robust with well-constrained limiting behavior, making it an attractive alternative to approaches in which the boundary stress is partitioned between the coarse, immobile sediment and the fine transportable grains. The bed-sand coverage function is used in conjunction with a standard transport model to provide the entrainment from the fine sediment bed forms and the entrainment rate in the limit of a bed completely covered with fine sediment. In areas of the bed not covered by sandy bed forms, the model uses the local entrainment function developed by Grams and Wilcock .
The form of the bed-sand coverage function indicates that as sand bed elevation decreases from a level corresponding to the tops of the coarse grains to about half the coarse grain elevation, the effect on sand entrainment due to the reduction in bed area covered by sand is more than balanced by enhanced entrainment from interstitial areas. Local rates of sand entrainment exceed the rate that would occur over a sand bed by up to a factor of 3. Accelerated particle entrainment from sand bed elevations below the height of immobile roughness elements is consistent with observations of increased shear stress as sand elevations drop [Wren et al., 2011], observations of aeolian transport among immobile roughness elements [Nickling and McKenna Neuman, 1995], and observations of near-bed stress distributions for beds with sand stripes [McLean, 1981].
constant = 1.3 × 10−7 in Garcia and Parker  entrainment relation
time average concentration of suspended sediment
median grain diameter
dimensionless rate of entrainment of sand from nonpatch areas of the bed
dimensionless rate of entrainment of sand from sand patches on the bed
dimensionless rate of entrainment of sand from the bed
predicted rate of entrainment for a sand-covered bed
dimensionless rate of entrainment from patch and nonpatch regions of the bed
fraction of bed area covered by sand bed forms
flow depth due to grain stress, in stress decomposition
von Karmen's constant = 0.4
coefficient of diffusion for suspended sediment
turbulent eddy viscosity
Nikuradse equivalent grain roughness associated with the total bed stress
skin friction roughness height
constant = 0.2 in Garcia and Parker  entrainment relation
number of instantaneous velocity measurements
total streamwise sediment flux
streamwise suspended sediment flux
particle Reynolds number for mobile sediment
height of immobile coarse grains
energy gradient (water surface slope)
specific gravity of sediment
mean depth-averaged velocity
instantaneous streamwise velocity
instantaneous deviation from time average streamwise fluid velocity
time average streamwise fluid velocity
shear velocity near the bed
skin friction shear velocity near the bed
instantaneous vertical fluid velocity
instantaneous deviation from time average vertical fluid velocity
similarity variable in Garcia and Parker  entrainment relation
parameter in sand elevation correction function
elevation above bed
mean sand bed elevation
mean elevation of sand in interstices not covered by sand bed forms
average sand bed elevation normalized by the height of the immobile grains
sand elevation correction
parameter in sand elevation correction function
straining parameter in Garcia and Parker  entrainment relation
porosity of sediment in the bed
density of fluid
= − log2D
average boundary shear stress
boundary shear stress
parameter in sand elevation correction function
This work was funded by the Glen Canyon Dam Adaptive Management Program administered by the U.S. Bureau of Reclamation. Additional support for the laboratory experiments was provided by the St. Anthony Falls Laboratory of the University of Minnesota and the STC program of the National Science Foundation via the National Center for Earth-surface Dynamics under agreement EAR-0120914. David J. Topping, Douglas J. Jerolmack, and three anonymous reviewers provided constructive and insightful comments that are gratefully acknowledged. Stephen M. Wiele provided helpful contributions to the experimental design and model development. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. government.