The influence of bedrock river morphology and alluvial cover on gravel entrainment. Part 2: Modelling critical shear stress

The critical shear stress (τc) at which grains are entrained on a bedrock surface is important for determining how bedrock rivers evolve through changes in sediment cover and bedrock erosion. The difference in τc for grains on bedrock and alluvial surfaces also determines whether a channel may be susceptible to runaway alluviation. Bedrock channel beds can have a wide variety of morphologies, but we do not fully understand how this variation affects τc. Here we address how bedrock morphology alters the grain entrainment parameters of pivoting angle, grain exposure and roughness height z0, and thus τc. In our companion article we used scaled, 3D printed replicas of seven bedrock surfaces to measure grain pivoting angles for four grain sizes. For three surfaces, pivoting angles were also measured with 25–100% sediment cover. In this second article, we combine these pivot angle data with measurements of grain exposure and surface roughness (standard deviation of elevations, σz) to predict τc using a force–balance model. The bedrock topography produces substantial variation in τc; for a given grain diameter (D), a 3.6× range of σz across the surfaces without sediment cover produces up to a 5.1× variation in τc. For comparison, for any single surface, τc varies by up to 2.5× for a fourfold range in grain size. Comparison to previous models with less representation of grain‐scale geometry shows that in our results grains move at lower values of dimensionless critical shear stress (τ*c), and that τ*c decreases more quickly with increasing D/σz. However, direct comparison is difficult because previous relationships are based on a hydraulic roughness length that cannot be easily predicted without hydraulic data. Our results propose a new relationship between D/σz and τ*c, but further development and testing require datasets that combine measurements of flow, τc and grain‐scale geometry.

understand how this variation affects τ c . Here we address how bedrock morphology alters the grain entrainment parameters of pivoting angle, grain exposure and roughness height z 0 , and thus τ c . In our companion article we used scaled, 3D printed replicas of seven bedrock surfaces to measure grain pivoting angles for four grain sizes.
For three surfaces, pivoting angles were also measured with 25-100% sediment cover. In this second article, we combine these pivot angle data with measurements of grain exposure and surface roughness (standard deviation of elevations, σ z ) to predict τ c using a force-balance model. The bedrock topography produces substantial variation in τ c ; for a given grain diameter (D), a 3.6Â range of σ z across the surfaces without sediment cover produces up to a 5.1Â variation in τ c . For comparison, for any single surface, τ c varies by up to 2.5Â for a fourfold range in grain size. Comparison to previous models with less representation of grain-scale geometry shows that in our results grains move at lower values of dimensionless critical shear stress (τ* c ), and that τ* c decreases more quickly with increasing D/σ z . However, direct comparison is difficult because previous relationships are based on a hydraulic roughness length that cannot be easily predicted without hydraulic data. Our results propose a new relationship between D/σ z and τ* c , but further development and testing require datasets that combine measurements of flow, τ c and grain-scale geometry.

K E Y W O R D S
bedrock river, critical shear stress, grain entrainment, pivot angle, surface roughness

| INTRODUCTION
Much literature has focused on the factors determining sediment entrainment and transportation in alluvial rivers, but there is far less understanding of their bedrock counterparts (Wohl, 2015). Understanding grain entrainment in bedrock rivers is critical for predicting both sediment transport rates and the development of sediment cover.
Studies in alluvial rivers have demonstrated how the way in which sediment grains are arranged affects critical shear stress (τ c ); for example, the impact of grain protrusion (Fenton & Abbott, 1977), grain geometry (Carling et al., 1992) and alluvial cover structure (Kirchner et al., 1990).
The presence of exposed bedrock and thin alluvial cover in bedrock channels has been shown to cause sediment transport processes to be different from those in alluvial channels (Chatanantavet & Parker, 2008;Goode & Wohl, 2010;Hodge et al., 2011), but a complete understanding of these interactions is lacking.
Field and flume evidence from bedload tracers suggests that sediment entrainment from bedrock surfaces requires a lower τ c than for the same-size grains on alluvial surfaces (Ferguson et al., 2017;Hodge et al., 2011;Inoue et al., 2014). However, it has been suggested that for grains on rough bedrock surfaces τ c could be higher than for alluvial surfaces (Johnson, 2014). There is also evidence that variations in τ c with grain size may be smaller over bedrock surfaces than in alluvial channels, causing sediment transport to be less size selective than in adjacent alluvial reaches (Ferguson et al., 2017;Hodge et al., 2011).
As well as affecting sediment transport rates, differences in τ c on bedrock and alluvial surfaces also affects the development of sediment cover. If τ c is higher for alluvial patches compared to the surrounding bedrock, then the initiation of sediment cover can cause runaway alluviation, whereby sediment grains encountering the sediment patch become less mobile and the area of sediment cover spreads rapidly (Chatanantavet & Parker, 2008;Demeter et al., 2005;Johnson, 2014).
We expect that at least some of the observed differences in τ c between bedrock and alluvial reaches are because of the impact of the underlying bedrock surface on the grain-scale geometry (e.g., grain pivot angle and exposure), but these effects have not yet been quantified. More generally, the impact of bedrock topography on τ c is often not considered when predicting bedload transport in bedrock channels. Previous attempts have often used a constant dimensionless critical shear stress (τ* c ) and only incorporated the impact of varying bedrock and alluvial roughness on the flow (Bartels et al., 2021;Nelson & Seminara, 2012). Even when the impact of bedrock surfaces on τ* c is considered (e.g., Inoue et al., 2014;Johnson, 2014;Mishra & Inoue, 2020), the focus has still primarily been on the effect of bedrock surface roughness on flow via a hydraulic roughness length.
The aim of this pair of articles was to determine how the properties of bedrock surfaces with and without sediment cover affect the grain-scale geometry of sediment grains (i.e., pivot angle and exposure) and consequently τ c . We address our aim using a novel set of 3D-printed replica bedrock surfaces. In the first article , we reported how surface properties and grain pivot angles vary between surfaces, and with different percentages (0-100%) of alluvial cover; and we assessed relationships between these pivot angles and different methods of quantifying surface roughness.
In this article, we evaluate how variation in bedrock topography affects the entrainment parameters of pivot angle, roughness length and grain exposure. We then use Kirchner et al.'s (1990) grain entrainment model to assess how variation in the parameter values propagates through to variation in τ c . From this, we assess how bedrock topography affects τ c , and identify which parameters are most important to constrain for improved τ c predictions. We then compare our results to entrainment models for bedrock rivers developed by Inoue et al. (2014) and Johnson (2014). By focusing on grain-scale geometry, we isolate the influence of the riverbed morphology on grain entrainment and remove the influence of other factors such as turbulent sweeps and instantaneous pressure gradients in the water column (Schmeeckle et al., 2007;Vollmer & Kleinhans, 2007) and channel slope (Lamb et al., 2008).

| Modelling grain entrainment from alluvial surfaces
To predict τ c at the point of grain entrainment, we use the simple force-based model of Kirchner et al. (1990). At the point when the grain is about to mobilise, the following force balance occurs: where F L is the lift force, ϕ is the grain pivot angle, F W is the grain weight, ρ s is the density of sediment (taken as 2650 kg m À3 ), ρ is the density of water, g is acceleration due to gravity and D is the grain diameter. The model of Kirchner et al. (1990) calculates the shear stress that solves Equation 1.
F D and F L are calculated assuming a logarithmic relationship between flow velocity and height above the bed: where u(z) is the flow velocity at height above the bed z, τ is the boundary shear stress, κ is von Kárman's constant (taken as 0.407) and z 0 is the roughness height. In Kirchner et al.'s (1990) original alluvial application, z 0 is 0.1D 84 (Whiting & Dietrich, 1990). The reference height z = 0 is assumed to be the local mean bed elevation. Equation 2 is only applicable when z > 0; otherwise u(z) = 0. F D is calculated as where w(z) is the width of the grain cross-section at height z; C D is an empirical drag coefficient assumed to be 0.4 (Wiberg & Smith, 1985); p and e are, respectively, grain protrusion and exposure, where protrusion is the height of the grain above the local mean bed elevation and exposure is the height of the grain above the local maximum upstream bed elevation; and F L is calculated as where A is the plan view cross-sectional area of the grain and C L is an empirical lift coefficient assumed to be 0.2 (Wiberg & Smith, 1985).
The boundary shear stress at the threshold of motion, τ c , is calculated by rearranging the preceding equations and assuming that grains have a circular cross-section (for the full derivation, see Kirchner et al., 1990): The original equation of Kirchner et al. (1990) is multiplied by 0.1 so that inputs in SI units produce a value of τ c in Pa. We convert these values to τ* c using τ* c = τ c /(ρ s À ρ)gD. Although more recent entrainment models such as those of Vollmer and Kleinhans (2007) and Lamb et al. (2008) have a more complete treatment of the hydraulics around the grain, our focus is on the impact of grain geometry on entrainment, and so, following the approach of Yager et al. (2018), the simpler Kirchner model is sufficient for this study.

| Modelling grain entrainment from bedrock surfaces
We compare our findings with Inoue et al.'s (2014) and Johnson's (2014) models for entrainment on bedrock surfaces. Both models incorporate differences between alluvial and bedrock surfaces through the effect of bedrock surface roughness. The main effect that is included in both is the effect of surface roughness on the near-bed hydraulics. Inoue et al. (2014) assumed that the grain would entrain through sliding on a smooth surface and did not incorporate the effects of bedrock roughness on pivot angle or grain exposure. At the point of motion: where θ is the bed slope angle and μ f is the static friction coefficient.
This is developed and rearranged to give the following equation for τ* c (see Inoue et al., 2014, for full derivation and justification of parameter values. Note also that the square is missing in eq. 16a in the original paper): where in which a Ã is a dimensionless coefficient relating the local flow velocity to the height above the bed, taken to be 0.65; k sb is the hydraulic roughness height (note that this is different from z 0 ); A 3 is π/6 and A 2 is π/4; k L is the ratio of lift forces to drag forces, taken to be 0.85; and μ f is taken to be 0.75. Inoue et al. (2014) found that their model was able to reproduce values of τ* c measured in a flume experiment, using values of k sb back-calculated from the hydraulic flume data. Johnson (2014) indirectly incorporated grain geometry effects by developing the hiding function of Wilcock and Crowe (2003) for mixedsize alluvial beds. This function was originally designed to represent the increase or decrease in τ c that is experienced by grains that are respectively smaller or larger than the median grain size (D 50 ). Johnson (2014) applied this model to bedrock rivers by scaling τ c as a function of the grain size relative to a measure of the bedrock roughness: where τ Ã cref is a reference dimensionless critical shear stress for grains on alluvium, assumed to be 0.055; and r br is a fitting parameter relating the standard deviation of surface elevations (σ z ) to a length scale equivalent to D 50 in the alluvial scenario. We also use r br to calculate z 0 when applying the Kirchner et al. (1990) model. Johnson (2014) used a range of flume data to determine possible parameter values for r br but did not directly compare the predicted shear stresses to measured values.

| Summary of methods from companion paper
To produce the 3D printed surfaces used in these experiments, we used high-resolution topographic data of exposed bedrock channel beds, collected using terrestrial laser scanning of the River Garry (Scotland) and structure from motion (SfM) photogrammetry from North Wash (Utah, USA). Selected areas of these data were downscaled and processed to produce a 3D printed tile, with maximum horizontal dimensions of 0.27 Â 0.27 m. We started with four surfaces (S1, M1, M2 and R1, where S/M/R refer to smooth/medium/rough).
R1 has a strong directionality to the topography, and so was analysed in two perpendicular orientations (R1 rot being the rotated version). To extend the range of roughness across the surfaces, we also increased the dimensions of R1 and M2 by 100% and printed a section of those to produce R1 x2 and M2 x2 , giving seven surfaces in total. To assess the impact of sediment cover on surface topography and grain entrainment, we added 25%, 50%, 75% and 100% sediment cover of two different grain sizes (11 and 32 mm) to surfaces S1, R1 and R1 x2 , producing a further 24 surfaces with partial or full alluvial cover. The topography of the covered surfaces was measured using SfM photogrammetry ( Figure 1; see Buechel et al., 2022 for details).
To measure pivot angles, each tile was attached to a tilt table. A grid of 81 30 Â 30 mm cells was overlain on the surface, and the pivot angle of each cell was measured three times by dropping a grain into the cell and tilting the table until the grain moved by at least one grain diameter. The pivot angle is the angle of the table at which the grain moved. For the surfaces without sediment cover, the measurements were repeated using four grain sizes (8, 11, 16 and 32 mm). For surfaces with sediment cover, the pivoting grain was the same size as the sediment cover. Further details of these methods are provided in the companion article .
We quantify the roughness of the different surfaces using the standard deviation of elevations, σ z , which is used to calculated z 0 . In the companion article we assessed how pivot angles correlated with surface roughness at different scales. We found that mean pivot angles correlated most with roughness calculated at spatial scales equivalent to the grain size or smaller, and when the roughness was also measured in the pivot direction. However, we do not apply those alternative roughness metrics here for two reasons. First, we need to quantify roughness to predict the impact of channel topography on the flow, which is likely to be different to the impact of the topography on pivot angles. Second, using σ z enables comparison with the work of Johnson (2014).

| Application of Kirchner, Inoue and Johnson models
We use the model of Kirchner et al. (1990) to calculate τ c for grains within each 30 mm measurement cell across all surfaces. To do this we require four parameters from our experiments: grain pivot angles, protrusion and exposure, and z 0 . We use the pivot angles from the experiments reported in Buechel et al. (2022). Grain protrusion is the maximum height of the grain above the mean bed elevation, indicating the vertical position of the grain within the velocity profile. We assume that the base of each grain is at the mean bed elevation within each measurement cell, and so grain protrusion is therefore equal to the grain diameter. It is possible that grains sit within pockets that are lower than the mean bed elevation, and so our vales of τ c may be an underestimate. Setting the base of the grain at an elevation of one σ z 25% 50% 75% 100% S1, 11 mm S1, 32 mm R1, 11 mm R1, 32 mm R1x2, 11 mm R1x2, 32 mm S1 We use the 95th percentile of upstream elevations to calculate exposure rather than the maximum because a grain will be sheltered by a section of the topography rather than a single point, but our results are not sensitive to the exact percentile that is used. For example, using the 90th percentile decreases median τ c for each surface by an average of 2% and a maximum of 7%, and using the 99th percentile increases median τ c by an average of 4% and a maximum of 11%. In both cases, the pattern of median τ c between different surfaces is not much altered. We calculate exposure when the bed is horizontal, and so our values are likely a minimum estimate as up-tilt sheltering could be higher when the bed is tilted.
To model the logarithmic flow profile, we need roughness length z 0 but there is no established method for calculating z 0 in bedrock channels. In alluvial channels, σ z has been shown to be a better predictor of total flow resistance than D 84 , but there is not necessarily a consistent scaling between the two (Aberle & Smart, 2003;Chen et al., 2020;Mishra & Inoue, 2020). Johnson (2014) estimated a bedrock hydraulic roughness length comparable to D 84 as the product of σ z and two scaling factors: r br , which scales bedrock roughness to the D 50 that would produce an equally rough alluvial surface; and r d , which scales D 50 to D 84 . By comparing the measured hydraulic roughness and σ z from flume experiments in bedrock channels and using r br as a fitting parameter, Johnson (2014) determined that r br varies from 1 to 5. Johnson set r d = 2, representing D 84 typically being about twice the size of D 50 in an alluvial bed. We combine this approach with Kirchner et al.'s (1990) use of z 0 = 0.1 D 84 , and so For most model runs, we use r br = 1 and r d = 2. We find that the resulting values of critical shear stress are sensitive to the value of r br .
Although we are primarily interested in the overall patterns, which are not dependent on this value, we also demonstrate this sensitivity by presenting a run using r br = 5.
We ran the Kirchner model for each of the seven printed beds using three different parameterisations, as outlined in Table 1 There is no established way to equate topographic properties to hydraulic roughness lengths, and the limited datasets suggest that there is not a one-to-one correlation between the two (e.g., Chatanantavet & Parker, 2008). Johnson (2014)  3 | RESULTS

| Entrainment parameter values
Before applying the model of Kirchner et al. (1990), we first consider the model parameters: pivot angle, z 0 , and grain exposure (Figure 2).
For the beds without sediment cover, pivot angles (Figure 2b)  Adding sediment cover to surfaces S1 and R1 x2 increases the magnitude and range of pivot angles. However, for R1 sediment cover does not alter pivot angles. On all surfaces with sediment cover, 11 mm cover produces higher pivot angles than 32 mm cover, despite the cover and pivoting grains being the same size.
For the surfaces without sediment cover, z 0 is highest for surface R1 x2 , and lowest for M2 (Figure 2a). For the surfaces with cover, z 0 is similar to z 0 with 0% cover, with the exception of S1 and R1 x2 with 32 mm cover, where the alluvial cover increases the value of z 0 . To compare grain exposure between different grain sizes, exposure is shown relative to grain size, so e/D = 1 is a fully exposed grain.) e/D increases with increasing grain size, and is roughly inverse to z 0 ( Figure 2c). One exception to these patterns is that e/D is similarly high on S1, M1, M2 and M2 x2 , despite M1 and M2 x2 having higher z 0 values. Another exception is that, for most surfaces with alluvial cover, e/D decreases with increasing cover, whereas z 0 remains approximately constant.
3.2 | Application of the Kirchner et al. (1990) entrainment model The first model parameterisation incorporated only our measured variation in pivot angles (Figure 3a). z 0 was held constant between surfaces (at the overall mean value of 1.3 mm) and we assumed zero upstream sheltering (i.e., protrusion and exposure are both equal to grain diameter). Despite a 2.3 times variation in median pivot angles across surfaces, the median value of τ c across all surfaces only varies by 1.8 times, from 1.3 to 2.4 Pa. There is little systematic variation in median τ c by grain size. In alluvial channels, τ* c is typically expected to be around 0.045 (Buffington & Montgomery, 1997;Miller et al., 1977). In comparison, for our data all 95th percentiles of τ c are less than τ c for an 8 mm grain when τ* c equals 0.045. Low percentiles of τ c represent when grains of that size start to be entrained, and so these percentiles could show that bedrock topography has differing impacts on the start and bulk of sediment transport. However, the 5th percentiles and medians of τ c show a similar pattern.
The second and third model parameterisations also include the influence of the surface topography on the flow, and thus on τ c (Figure 3b,c). The second parameterisation incorporates variation in z 0 between surfaces, and the third adds variation in upstream sheltering through grain exposure. In the second parameterisation, including the 4.5 times variation in z 0 between surfaces into the model increases the variation in τ c , with median τ c varying by 5.9 times (Figure 3b).
Median τ c and the range of τ c decrease for surfaces where z 0 is smaller than the average value used in the first parameterisation (M1, M2 x2 , S1 with and without sediment cover). Surfaces with higher than average z 0 (R1 x2 with and without sediment cover) see an increase in the median and range of τ c .
There is a twofold variation in median grain exposure across the different surfaces (Figure 2c). Incorporating this in the third parameterisation further increases the variability in τ c between the surfaces to seven times, because median τ c increases for surfaces with smaller exposure values (primarily R1 and R1 x2 with and without sediment cover; Figure 3c). Under this model parameterisation and across the surfaces without sediment cover, grains of the same size have a variation of up to five times in median τ c (Figure 3c)  Variations in surface topography have a larger impact on τ c than variations in grain size. For uncovered surfaces, median τ c is approximately constant across the different grain sizes, except for R1 and R1 x2 , where it decreases (Figure 3c). On these two surfaces, the difference in τ c between grain sizes increases between the first and third model parameterisations. For surfaces with sediment cover, increasing cover increases τ c for both grain sizes on S1 and R1 x2 , though not for R1. Even at 100% sediment cover, the underlying topography still affects τ c ; different surfaces covered with the same grain size produce different τ c , and R1 x2 continues to have the highest τ c . 3.3 | Comparison to the Inoue et al. (2014) and Johnson (2014)

entrainment models
Plotting τ* c from the third model against D/σ z does collapse the data from all surfaces (with and without sediment cover) into a single trend ( Figure 5), more so than plotting against any single parameter ( Figure 4). This suggests that some parameters compensate for each other. For example, grains on M1 and R1 rot have similar τ* c values. On surface R1 rot , grains are less exposed and z 0 is higher, reducing the flow velocity at a given height, but this is counteracted by lower pivot angles compared to M1. We then explore the impact of the scaling factor, r br , used to convert σ z to z 0 . z 0 affects τ* c because it controls the rate at which flow velocity increases with elevation above the mean bed elevation. Following Johnson (2014), we used a value of 1 to calculate z 0 for our Kirchner et al. (1990) model calculations. Figure 5b shows To produce a predictive relationship, we fit two curves to our data. The first follows the form of Inoue's model (Equation 8), with the relationship where a and b are fitting parameters. For our data when r br = 1 The underlying bedrock topography can have a substantial impact on τ c , even under 100% alluvial cover (Figure 3). Our results show that, across a range of D/σ z from 1.4 to 8.2, the variation in τ c for the same grain size between surfaces is greater than the difference in τ c for different grain sizes on the same surface ( Figure 3c). Of the three parameters, z 0 and grain exposure produce the most difference in τ c between the different surfaces. The importance of z 0 provides support for previous approaches that only considered the impact of roughness on flow rather than grain geometry (e.g., Inoue et al., 2014). The combined impact of the surface topography on the entrainment parameters creates a power relationship between τ* c and D/σ z with an exponent of À1.24 (when r br = 1), such that increasing the size of the sediment grain relative to the surface roughness produces a disproportionate decrease in τ* c . For comparison, in the context of alluvial channels, a power relationship between τ* c and D/D 50 with an exponent of À1 would indicate equal mobility. That is to say, any increase in grain size is balanced by an increase in grain exposure and decrease in pivot angle, such that all grain sizes move at the same τ c (Andrews, 1983;Parker et al., 1982).
Decreases in τ* c with increasing D/σ z may be disproportionately larger in the bedrock setting because of how the bedrock surface affects the entrainment parameters. Grains on a bedrock surface have a greater protrusion (height above mean bed elevation) compared to grains in an alluvial bed which are often at least 50% buried (Hodge et al., 2020;Yager et al., 2018). Consequently, bedrock grains will be more affected by changes in the flow profile and upstream sheltering. As D/σ z increases, there is a strong increase in z 0 and a strong decrease in median e/D, with a less clear trend for pivot angles ( Figure 6). Decreasing z 0 and increasing relative exposure both increase grain mobility. The former changes the velocity profile, bringing higher velocity flows closer to the bed. The latter means that more of the grain's surface area is affected by the flow and so the grain is more sensitive to changes in σ z and hence z 0 . The combined effect of these terms outweighs increases in grain resistance caused by increased grain weight. Such changes may be more rapid compared to the alluvial relationship between τ* c and D/D 50 because in an alluvial bed there is no systematic trend between exposed area and D/D 50 , and protrusion potentially decreases with increasing grain size (Hodge et al., 2020). Consequently, changes in exposure with D/D 50 do not necessarily compensate for changes in grain weight.
Our modelled τ* c values are lower than those predicted by the models of Inoue et al. (2014) and Johnson (2014). However, comparison between our results and Inoue's model is complicated by the model's use of a hydraulic roughness length (k sb ) that we cannot measure directly. Inoue's (2014) model assumes that grains are entrained by sliding rather than pivoting, but our data do agree with the shape of the relationship produced by that model. However, fitting that relationship to our data requires some unlikely parameter values.
Johnson's (2014) model plots above our data, partly because of his assumption that τ* c is 0.055 when D/σ z is one. However, on setting the reference τ* c to 0.02 in line with our modelled data, most of the curve still does not correspond to our data, as the model's rate of decrease in τ* c with increasing D/σ z is not as rapid as in our data ( Figure 5). This slower decrease may be because Johnson's (2014) model is based on Wilcock and Crowe's (2003) equations for hiding behaviour in alluvial beds, in which τ* c may not decrease as rapidly for the reasons outlined above.

| Comparison to bedload tracer measurements
For each exposed surface, our predictions of τ c (Figure 3c) show equal mobility of the four grain sizes, with all sizes moving at low τ c . The only exception is R1 and R1 x2 , where τ c is weakly inverse to grain size.
This equal mobility and low τ c are consistent with bedload tracer data from channels with high bedrock exposure (Ferguson et al., 2017;Hodge et al., 2011), surfaces. A better comparison to R1 is provided by Goode and Wohl's (2010) bedload tracer data from the Ocoee River, which has bedrock ribs. They found that tracer travel distance was further and grain size dependent when the ribs were parallel to flow, and shorter and size independent when ribs were oblique to flow. The difference in travel distances is consistent with our finding that τ c is lower when ribs are parallel (R1 rot ), rather than perpendicular (R1), to the flow direction.
But our data show the opposite pattern in grain size dependence, with less variation in τ c between grain sizes with flow parallel (R1 rot ), compared to flow perpendicular (R1), ribs. The difference could be because in the Ocoee River there was a greater spacing between the ribs relative to the grain size, which was filled with substantial sediment cover. Consequently, when the ribs were flow parallel, sediment entrainment may have been more similar to size-selective entrainment in alluvial channels (Church & Hassan, 1992;Ferguson et al., 1996;Haschenburger, 2013). Goode and Wohl (2010) also found that transport distances were best explained by local-scale bedrock topography and sediment architecture, consistent with our findings in the companion paper that pivot angles are best explained by small-scale roughness.

| Implications for sediment cover development
How τ c changes with increasing cover has implications for predicting the relationship between sediment flux and sediment cover, which is important for modelling channel incision and landscape evolution (Lague, 2010;Sklar & Dietrich, 2004;Turowski, 2021). If sediment cover increases τ c , in turn making sediment grains less mobile, then this positive feedback can produce rapid deposition of sediment cover, known as runaway alluviation (Chatanantavet & Parker, 2008;Demeter et al., 2005;Finnegan et al., 2007). Surfaces S1 and R1 x2 show potential for runaway alluviation, as τ c increases with increasing cover (Figure 3c). In contrast, the surface with intermediate roughness are all clast-smooth (see companion article), but R1 does not show the potential for runaway alluviation. Furthermore, the two surfaces that do, S1 and R1 Â2 , are respectively smoother and rougher than R1, and so the difference in behaviour cannot be explained only through overall roughness σ z . The clast-smooth/clast-rough distinction may therefore be a useful starting point, but does not fully represent the sediment dynamics.
One caveat to consider is that we have focused on the entrainment of individual grains from bedrock and alluvial surfaces (following the approach of Kirchner et al., 1990). When there is partial or full alluvial cover, we have focused on grains that are placed on top of, or adjacent to, the sediment patches. Such locations are where new grains will be deposited, and where erosion of the sediment cover tends to initiate (Hodge & Hoey, 2016). However, as sediment patches expand, newly deposited grains can become surrounded by other grains, decreasing exposure, and increasing pivot angles and thus τ c .
Entrainment of such grains will also be inhibited by the weight of overlying grains (Sanguinito & Johnson, 2012;Yager et al., 2018) and the development of any grain-scale structures Lamarre & Roy, 2008), both of which are not considered in our analysis. Consequently, our reported τ c is not representative of grains in the middle of the patches. It is therefore possible that, for R1, runaway alluviation may still occur because of grains becoming incorporated into patches, even if newly deposited grains are equally likely to be entrained from bedrock or alluvial areas.
The idea of τ c changing as sediment is deposited or eroded on a bedrock surface is similar to that developed by Johnson (2016) for alluvial channels, in which τ c changes as a state function of sediment supply and thus erosion or deposition. In the alluvial example, erosion increases τ c as grains in more mobile positions are preferentially entrained (as also observed by Mao, 2012;Masteller & Finnegan, 2017;Ockelford & Haynes, 2012), and τ c decreases with deposition as grains are deposited in increasingly less stable locations, providing a negative feedback on bed stability. This approach could be extended for bedrock rivers, predicting variation in τ c both with erosion/deposition and the extent of bedrock exposure. However, our results show that bedrock topography still influences τ c even at 100% sediment cover, and so such an approach may also need to consider sediment depth or some other metric that determines the point at which underlying bedrock topography stops influencing sediment transport dynamics. Inoue et al.'s (2014) and Johnson's (2014) models of τ c are a useful starting point for developing a state function, although the model results are sensitive to parameters such as r br that need to be better constrained ( Figure 5). Encouragingly, Mishra and Inoue's (2020) attempt to fit r br using a new flume dataset produced similar values to those identified by Johnson (2014). New methods such as the transform-roughness correlation to predict roughness lengths directly from channel topography (Adams & Zampiron, 2020) would also facilitate the application of these models. Predicting τ c as a function of erosion/deposition and bedrock exposure would be useful for applications including landscape evolution modelling. However, the finding that τ c is most dependent on z 0 and grain exposure, and consequently bed topography at the grain scale is potentially problematic as landscape evolution models cannot reproduce channel properties at this level of detail. To implement such a model, further understanding is needed of whether small-scale channel topography can be predicted from larger-scale factors such as lithology, sediment supply, discharge, channel slope and width.

| CONCLUSIONS
Bedrock surfaces affect τ c of overlying grains by altering pivot angles, grain exposure and local flow conditions in ways that are still difficult to predict. Here we have reported how these parameters vary for grains on a range of bedrock topographies with and without different extents of sediment cover, and we have used the Kirchner et al. (1990) model to predict τ c for these grains. Across a range of D/σ z from 1.4 to 8.2, we find that variations in topography between the surfaces have a larger impact on τ c than variations in grain size, and that values of τ c are primarily controlled by the impact of topography on z 0 and grain exposure. Our values of τ* c are similar to the model of Inoue et al. (2014), though direct comparison is difficult because their model is based on a hydraulic roughness length that cannot robustly be calculated from topographic data. Comparison to Johnson's (2014) model suggests that, as D/σ z increases, τ* c decreases faster than in the comparable scenario of hiding effects in a mixed alluvial bed. Our results provide a relationship to predict τ* c as a function of D/σ z , though this relationship still must be comprehensively tested. Overall, our results show grain-scale geometry can produce substantial variations in τ* c . However, new datasets that combine measurements of flow, τ* c and grain geometry would enable our approaches to be more robustly compared to alternative methods, and hence develop better predictive equations.