The time rate of bedrock incision (erosion) by mechanical wear Es has been quantified with Eq. (1) by different authors (e.g., Sklar and Dietrich, 2004, 2006; Turowski et al., 2007; Chatanantavet and Parker, 2009) as follows: 1Es=ViIrpo. In Eq. (1), Vi is the volume of bedrock lost per particle impact, Ir is the particle impact rate per unit area per unit time, and po is the fraction of exposed bedrock. The areal fraction of alluvial cover, i.e., the cover factor pc, is thus defined as pc=1-po. A closely related equation to compute erosion is presented in Eq. (2) (e.g., Turowski et al., 2008; Chatanantavet and Parker, 2009). Let β equal a parameter that relates to the fraction of bedrock volume that is lost per particle impact at the end of each saltation; qbt is the capacity bedload transport rate per unit width for a bed fully covered with alluvium, and qbs=pcqbt – the actual bedload transport rate per unit width assuming that particles can only be mobilized from portions of the bed that have an alluvial cover. Then, 2Es=βqbspo=βqbtpc(1-pc). But it is readily seen that the above relation breaks down for throughput load, bedload transport that enters and leaves a reach without depositing on the bed and does not contribute to cover in any meaningful sense. In Sklar and Dietrich (2004) and other works based on their cover model (e.g., Turowski et al., 2007; Lamb et al., 2008), po represents the areal fraction of exposed bedrock and is related to the sediment supply ratio (Eq. 3). In the Zhang et al. (2015) cover model, po represents the fraction of bed elevation at a given cross section that is not covered by alluvium and is instead related to the ratio ηa/Lmr, where ηa is a measure of the thickness of alluvium, and Lmr is a measure of the intrinsic macro-roughness height of the bedrock surface itself (Eq. 4).

Both definitions are presented schematically in Fig. 1. In general, pc is a function of qbs/qbt or ηa/Lmr, and commonly used forms are given by Eqs. (3) and (4). 3pc=1-po=qbsqbtif 0≤qbsqbt<11if 1≤qbsqbt4pc=1-po=ηaLmrif 0≤ηaLmr<11if 1≤ηaLmr Equation (3) or (4) in combination with Eq. (1) must be employed in terms of an appropriate averaging window over which to determine the cover fraction pc and open fraction po=1-pc. For example, Sklar and Dietrich (2006), Gasparini et al. (2007), and Chatanantavet and Parker (2008) assume, explicitly or implicitly, that (a) the averaging window is at least as large as channel width and (b) that pc fluctuates temporally between 0 and 1 within the window. If this were not the case, zones along the reach where pc persistently takes the values 0 and 1 would never be subject to incision, and channel geometry would not change over time in those reaches. However, if these assumptions are met over an appropriate timescale, all the reach would, in the long-term average, erode at the same rate (Sklar and Dietrich, 2004).

In the case of mixed bedrock–alluvial meandering rivers, wherein persistent alluvium deposits may form in, e.g., point bars, the assumptions just described break down. In such rivers, erosion occurs only in areas with transient cover and is not expected to occur in areas that are persistently covered or exposed. Under certain conditions, specific areas of the channel might have little to no probability of being struck by sediment particles, thus limiting the areas that could undergo erosion.

Experiments were conducted in the Kinoshita flume at the Ven Te Chow Hydrosystems Laboratory, University of Illinois at Urbana–Champaign. The flume, shown in Fig. 2b, is 0.60 m wide, 0.40 m deep, 33 m long (along the centerline, not including upstream and downstream tanks), and has a sinuosity of 3.7. All three meander bends are identical and have a down-channel wavelength of 10 m. All results presented herein correspond to experiments conducted with water flowing from right to left as indicated in Fig. 2c, i.e., with the bends skewed in the upstream direction. The flume is a closed system in which water and sediment are recirculated. Readers interested in more specific details about the Kinoshita flume are referred to Abad and Garcia (2009a, b).

The alluvium used in the experiments was crushed walnut shells, which have a specific gravity in the range 1.3–1.4. A bedrock basement was built in the flume using the bathymetry measured by Czapiga (2013), who conducted experiments under fully alluvial conditions using the crushed walnut shells. The Supplement (S3) shows the bathymetry from those experiments. Transverse slopes were measured from the bathymetry, and a relation between the transverse slope and streamwise location was fit to the data. Using it, transverse slopes every 0.5 m were calculated. Based on the computed transverse slopes, cross-sectional bathymetric slices were cut out of foam and placed inside the flume every 0.5 m. Pea gravel was used to fill the flume following the profile established by the foam slices. The region between streamwise stations CS07 and CS23 (Fig. 2c) was filled with gravel to an elevation slightly below the maximum given by the foam slices. This section was then covered with a ∼1 cm layer of concrete and used to create the bedrock surface. We filled the rest of the flume (CS00–CS07 and CS23–CS30) with pea gravel to maintain an average centerline elevation throughout the flume. The size of the gravel was chosen so as to prevent it from being transported by the flow in the experiments. Figure 3c shows the bedrock bed built inside the Kinoshita flume, and Fig. 2a shows its bathymetry. The concrete was painted white to enhance the contrast between the bedrock and the alluvium. The Supplement (S1, S3) has a set of images and diagrams that provide additional information regarding the construction of the bedrock bed inside the flume and the premixed concrete used.

The grain size distributions of the crushed walnut shells, the pea gravel, and the dry concrete mix (including gravel, sand, and cement) are shown in Fig. 3a. The inset figure includes the results of laser scans conducted to measure the as-built bedrock macro-roughness, i.e., the difference between the maximum and minimum elevations according to Zhang et al. (2015).

A Keyence LB-1201 laser (Keyence Corporation, 1992) with sub-millimeter precision (250 µm) was used to scan the bed at five different locations, namely CS10, CS12, CS15, CS17, and CS20 (Fig. 2c). These locations were chosen because they are representative of the bedrock topography at the apices (CS10, CS15, and CS20) and the crossings (CS12 and CS17). A polynomial was fit to the scans, and residual elevations were calculated by subtracting the actual reading from the polynomial. This removed the local topography from the signal. The average residual elevation along the cross sections was calculated and used to estimate the macro-roughness of the bedrock bed. The resulting value (10 mm) is also indicated in Fig. 3a. More details about the scans and the polynomial fit are included in the Supplement (S1).

The percentage of areal alluvial cover was calculated by analyzing time-lapse images of the flume bed. Images were taken, on average, every 10 s (0.1 Hz) during the duration of every run and processed in MATLAB. A region of interest (ROI) was selected for each image series. In this study, the ROI corresponds to the middle bend of the Kinoshita flume, i.e., between streamwise locations 10 and 20 m (Fig. 2).

Images were first converted to grayscale, and then the method of Otsu (1979), as implemented in MATLAB (“graythresh” function), was used to make the images binary. The resulting black (alluvial cover) and white (bedrock) images were used to calculate the percent of areal cover. The fraction of alluvial cover was determined as shown in Eq. (5). 5pcROI=N-∑j=1NpxjN In Eq. (5), pcROI is the percent of areal alluvial cover inside the region of interest, N is the total number of pixels inside the region of interest (i.e., total area), and pxj is the value of the jth pixel in the binary image (white pixels are equal to one and black pixels are equal to zero). The pixel size in the images was approximately 2.8 mm × 2.8 mm. This resolution is not enough to capture individual sediment grains (D50∼1.6 mm). Nevertheless, if a single grain lies in a pixel, ∼25 % of its area would be alluvium, leading to a gray pixel (as seen by MATLAB). Depending on the threshold determined by Otsu's method for the region of interest, a single grain could be enough to be interpreted as alluvial cover. More details regarding the image acquisition and processing are included in the Supplement (S2).

A rectangular bedrock slab was built with the same materials used to build the bedrock basement in the flume. The bedrock slab was built over a piece of foam laid on a floor so as to have no longitudinal or transverse slope. Pea gravel was placed over the foam and a thin layer of concrete was poured over it. It was then painted white to increase the contrast between the bedrock and the alluvium. The purpose of this bedrock slab, which was 0.6 m long by 0.4 m wide, was to measure (i) the relation between areal alluvial cover and sediment mass fraction and (ii) the relation between areal alluvial cover and the ratio of alluvial cover thickness to bedrock macro-roughness. Images of this simple experiment are included in the Supplement (S1).

To quantify the cumulative sediment mass fraction, known weights of sediment were incrementally added to the slab and spread evenly until the bed was fully covered with alluvium. A total of 11 iterations were necessary to cover the bed completely. The total amount of mass used was 646 g. Mass increments used in every iteration are shown in Supplement S1. The cumulative sediment mass fraction was calculated as the cumulative weight of sediment in every iteration divided by the total weight of sediment used to fully cover the bed. Areal alluvial cover was quantified using images and following the approach described in Sect. 2.4. The ratio of alluvial cover thickness to bedrock macro-roughness was quantified by scanning nine cross sections of the bedrock slab with a sub-millimeter-precision Keyence laser (Keyence Corporation, 1992). The cross sections were 4 cm away from each other. The first set of scans were conducted over the bare bedrock slab, and then they were repeated each time that alluvium was added over the slab. The entire process was conducted two times to verify that the results would not change due to any human-induced errors in the measurements or the way in which the alluvium was distributed over the bed after each iteration. After the first set of measurements, the alluvium was initially removed with a brush and then with an air-pressure hose to make sure no grains were left on the slab. Images related to this experiment are included in the Supplement (S1).

Table 1 shows the general experimental conditions used in this study. The flow discharge rate used in all runs was 12.3 liters per second (L s-1), which corresponds to the flow rate used by Czapiga (2013). This flow rate created the alluvial bathymetry used to build the bedrock bed in these experiments. The flow discharge was measured with electromagnetic flowmeters. Given that the sediment recirculating pump only works at a constant discharge of 3.1 L s-1, the main pump was set to have a discharge of 9.2 L s-1.

The volume of sediment inside the Kinoshita flume was modified between runs so as to obtain different reach-averaged areal ratios of alluvial cover. Runs in this study are identified based on this value (Table 2). For example, run pc79 had 79 % of the total bed area covered with alluvium after averaging in space (one wavelength) and time (1 h). The first run conducted was pc79. After this run, we removed sediment from the flume, leading to lower percent of areal alluvial cover (pc) conditions. The values obtained were not planned. The two following runs were pc72 and pc54. Afterwards, all the sediment was removed from the system to run the bare bedrock condition, pc00. The following runs were pc19, pc27, pc38, and pc46, conditions that were achieved after progressively adding sediment to the flume.

We allowed the bed to adjust for at least 8 h between runs. The 60 min we report in the study are after the bed had adapted to the new condition. We computed the alluvial cover statistics throughout the transition from one state to another and once it had reached equilibrium we continued measuring. We report only the values once the system had reached equilibrium for each condition. Water surface slopes were initially calculated by using the water level elevation changes in the upstream and downstream tanks of the Kinoshita flume. Both tanks have a measuring tape glued to the upstream- and downstream-most walls (Fig. 2b). These measuring tapes were used to guarantee that runs always started at the desired water elevation. Before turning on the pumps, desired water elevations were verified, and after the run had started, readings were taken every 20–30 min.

Water surface elevations were also measured with eTapes in runs pc00, pc19, and pc79 (Fernández, 2018). An eTape is a sensor with a resistive output that varies with the level of fluid in which it is immersed. The resistive output of the sensor is inversely proportional to the height of the water. Low water depths correspond to high output resistance. Conversely, high water depths correspond to low output resistance. Details about the eTape installation, calibration, and operation are given in S4, and further information may be found in Fernández (2018). After runs pc79, pc72, and pc54 were finished, we noticed that the water surface slopes in the Kinoshita flume were different depending on if they were calculated for the total length of the flume, i.e., between tanks, or for the middle bend of the flume only. Figure 4 shows an example of the water surface elevations measured with the eTapes (middle zone of the flume) and the measuring tapes (entire flume) for run pc79. To accurately measure the middle-bend water surface slopes in runs pc00–pc46, point gages were placed on the flume at streamwise locations 9 and 21 m (Fig. 2). The slopes calculated with the point gage readings are shown in Table 1. The average ratio of the slopes calculated with the point gages to those calculated with tank elevations in runs pc00–pc46 was used to estimate the slopes in the middle bend of the flume for runs pc54, pc72, and pc79.

The sediment transport rates were measured by collecting material in a box fitted to the diffuser at the upstream end of the flume (Abad and Garcia, 2009b). This box is not shown in Fig. 2. Table 2 shows the average sediment transport rates measured.

Based on Eq. (2), a dimensionless erosion potential Esp may be expressed as a function of the areal fraction of alluvial cover as shown in Eq. (6) below. We use this (Sect. 3.5) to assess the spatiotemporal average erosion potential for the seven experimental conditions with alluvium. 6Esp=pc1-pc At the microscopic level, the value of pc can only take values of zero (exposed bedrock) or one (covered with alluvium). In the context of the areal images obtained during the experiments, this means that pixels may change between white and black throughout the run. This information may be used to quantify erosion potential based on alluvial cover fluctuations.

Bedrock incision can only occur when a particle strikes the bed. If a pixel changes from white to black between consecutive images, it means that sediment particles traveled into the area and struck the bed. If the pixel remains black or white in consecutive images, no strikes occurred; if the pixel changes from black to white, sediment particles have left and thus did not strike the bed. With these definitions, the erosion potential may be quantified by counting the number of times that a pixel changes from white to black, i.e., by quantifying the fluctuations in alluvial cover.

The frequency of strikes (fs) at the jth pixel corresponds to the number of times that the jth pixel has changed from white (pc=0) to black (pc=1) between consecutive images (im) divided by the total number of images (N) in the series (Eq. 7). We use this approach (Sect. 3.6) to assess the erosion potential based on alluvial cover fluctuations for the seven experimental runs containing alluvium. 7fsj=∑i=1Ndpcdimi=-1jN

Figure 5 shows the relation between alluvial cover and sediment supply measured on the bedrock slab and in the Kinoshita flume. Specifically, Fig. 5a shows the relation between areal alluvial cover and cumulative sediment mass fraction measured on the bedrock slab. Figure 5b shows the relation between areal alluvial cover and the ratio of alluvial cover thickness to bedrock macro-roughness measured on the bedrock slab. The thin dashed lines with circle and square markers show the average results of the measurements; the thick dashed lines correspond to a best-fit line, and the dotted lines show the linear relation between variables that has been used by previous authors (e.g., Zhang et al., 2018, 2015; Inoue et al., 2016, 2014; Chatanantavet and Parker, 2009, 2008; Sklar and Dietrich, 2006, 2004). Figure 5c shows the relation between reach-averaged alluvial cover (spatial average measured over one wavelength) and the sediment supply ratio measured in the Kinoshita flume.

The relations between alluvial cover and sediment mass fraction in the bedrock slab (Fig. 5a) and the Kinoshita flume (Fig. 5c) are logarithmic (Eq. 8). The value of the constant a in Eq. (8) below is different between the bedrock slab (a=0.23) and the Kinoshita flume (a=0.14), but the shape of the relation is the same. Previous research has shown that different relations between the percent of cover and sediment supply ratio are valid under certain circumstances (e.g., Turowski and Hodge, 2017; Inoue et al., 2014; Chatanantavet and Parker, 2008), and our results suggest that the relation below is also possible: 8pc=a⋅ln⁡qbsqbt+b. This relation diverges as the sediment supply ratio (term in parentheses) tends towards zero and, as such, does not describe the small sediment flux limit. The relation is similar to those derived by Turowski and Hodge (2017; Eqs. 27 and 31) when the exponential term in their relation is small. Aubert et al. (2016) predict a similar relation to describe the alluvial cover based on direct numerical simulations. In the case of the bedrock slab, the logarithmic relation suggests that, initially, the areal cover increases rapidly with the sediment supply ratio. Once the smaller voids in the bed are filled, more and more alluvium is needed to fully cover the largest roughness elements and further increase pc.

In the case of the Kinoshita flume, the logarithmic relation between alluvial cover and the sediment supply ratio is believed to be due in large part to the formation of point bars and transient alluvial deposits. Initially, a small amount of alluvium covers a proportionately larger area of the bed, but as sediment supply increases, alluvial thickness growth is favored over the areal extent of alluvial cover. As more alluvium accumulates over regions previously covered, additional sediment supplied to the reach tends to deposit at the edge of the existing deposits, thus increasing alluvial cover but at an ever smaller rate.

Figure 5b shows the relation between areal alluvial cover and the alluvial thickness to bedrock macro-roughness ratio. Zhang et al. (2018, 2015) and Inoue et al. (2014) used the assumption that the relation is linear but the results obtained for the bedrock slab suggest that an “S-shaped” (sigmoid curve) relation is more appropriate. A logistic curve, which is a type of sigmoid curve, was fit to the measurements in this study. Equation (9) shows the general logistic function and Eq. (10) shows the one used here. Comparing the two, it may be seen that x=ηa/Lmr and fx=pc(ηa/Lmr); L is the maximum value of the curve corresponding to pcmax⁡=1.0, k is the steepness of the curve, and xo is the x value of the sigmoid curve's midpoint. As shown in Fig. 5b and Eq. (10), the steepness used to fit the sigmoid curve to the measured values was 8 and the midpoint was defined at ηa/Lmr=0.4. 9f(x)=L1+e-k(x-xo)10pcηa/Lmr=1.01+e-8ηa/Lmr-0.4 The function in Eq. (10) is valid between a characteristically low (e.g., 0.05) and a characteristically high (e.g., 0.95) value of ηa/Lmr to avoid unrealistic cover values (Zhang et al., 2018). It is likely that the steepness and midpoint value are associated with some measure of the grain size distribution of the alluvium and the macro-roughness height of the bedrock. In the case of the bedrock and alluvium (Fig. 3a) used in this study, the steepness value corresponds to k∼Lmr/D16 and the midpoint value corresponds to xo∼2.1D84/Lmr. This issue merits further investigation so as to define appropriate relations to calculate the steepness and midpoint value of the sigmoid curve for implementation in numerical models. We discuss the issue of alluvial thickness and alluvial cover further in Sect. 4.4.

Figure 6 shows a snapshot of the middle bend of the Kinoshita flume corresponding to each one of the eight reach-averaged alluvial cover conditions. Similar images for the bedrock slab experiment are included in the Supplement (S1). Links to the videos showing the bed evolution for the different experimental conditions are included in the “Video supplement” section at the end of the paper.

Figure 7 shows the temporal evolution of reach-averaged alluvial cover for all experimental runs. Larger fractions of alluvial cover are associated with fluctuations about the mean value due to the appearance of freely migrating bars as sediment supply increases. Figure 8 shows the maps of alluvial cover for all experimental runs. Darker shades of blue correspond to areas that were covered with alluvium for more than 70 % of the time, and shades of yellow correspond to areas that were covered with alluvium less than 30 % of the time. In regards to the tools and cover effects, the white and black regions in those alluvial cover maps would not experience erosion. No tools (alluvium) are available to erode the bed in the white regions, whereas alluvium completely covered the bed in the black regions, thus protecting it from erosion.

The areal alluvial cover definition in Fig. 1a is based on the assumption that alluvial deposits are transient; i.e., no portions of the bed in the reach remain persistently covered with alluvium or fully exposed. This assumption is not met in meandering channels where persistent alluvial cover deposits form and grow as sediment supply increases, and erosion may only occur in regions where alluvial cover is changing in time, i.e., regions with transient cover.

The alluvial cover maps in Fig. 8 show different percentages of persistently covered or exposed bedrock, as well as regions with transient alluvial cover. The regions with transient alluvial deposits are those over which alluvial cover is changing in time (colored regions in Fig. 8). To delineate and quantify these areas, the following criteria were used: regions with persistent alluvial cover are those in which pc>0.975; regions with persistent exposed bedrock are those in which pc<0.025; and regions with transient alluvial cover are those in which 0.025≤pc≤0.975. Using these criteria, maps of transient alluvial cover were prepared. Figure 9 shows the regions of transient cover (gray), persistent cover (black), and persistently exposed bedrock (white) for each of the eight experimental conditions. The area of the former two regions increases with sediment supply, whereas the area of the latter decreases as sediment supply increases.

Figure 10 shows the reach-averaged percentages of these three regions for all eight experimental conditions. Therein, the yellow dashed line corresponds to the reach-averaged fraction of persistently exposed bedrock, the blue line corresponds to the reach-averaged fraction of persistently covered bedrock, the light blue line corresponds to the reach-averaged fraction of the bed with transient cover, the thick black line corresponds to the sum of the transient and persistent cover fractions, and the black dotted line corresponds to the 1 : 1 line.

The regions of persistent and transient cover increase as a function of reach-averaged alluvial cover. The regions of persistently exposed bedrock decrease concomitantly. In general, both the fraction of the total area with persistent and transient cover grow at a similar rate with increasing reach-averaged pc. The reach-averaged conditions for which transient and persistent cover have similar area ratios are pc=0.27, 0.46, 0.54, and 0.72. The largest differences between persistent and transient cover are observed at pc=0.19, 0.38, and 0.79.

The case pc=0.19 is likely due to the typical sedimentation patterns observed in meandering bedrock channels when alluvial point bars first form. Immediately downstream of the bend apices, i.e., the points of highest curvature, sediment is deposited. In the Kinoshita flume, the apices of bends are located at streamwise locations 9.5, 14.5, and 19.5 m (Fig. 13). Initially, these locations become the upstream-most points of the point bars. Once these deposits have been established and as long as sediment continues to be supplied from upstream, the incoming particles travel above the existing deposit due to decreased resistance from the bed. Under such conditions, persistent alluvial cover is favored over transient alluvial cover.

The case pc=0.38 has a larger portion of the total area with transient cover than with persistent cover. As more sediment was supplied to the system while keeping the initial (no flow) water depth constant (Table 1), the alluvial thickness could not continue to grow indefinitely; rather, the areal extent of alluvial cover grew instead and the water surface slope also increased (Table 2). Sediment particles could no longer be preferentially transported over the alluvial deposits and began to be transported closer to the edge of the existing deposits.

The case pc=0.79 shows a dip in the ratio of transient cover, while the area with persistent cover continues to increase. Although there are no runs with a larger reach-averaged fraction of alluvial cover, it is likely that this trend would be maintained until the bed is completely covered with alluvium. As pc grows, the area ratio of persistently covered regions should increase at a faster rate, and the area ratio of regions with transient cover should decrease rapidly towards zero. Eventually, the channel will not have any area left for the areal extent of alluvial cover to grow so that further deposition promotes increased alluvial thickness instead. Erosion by abrasion would promote the lateral migration of the bedrock river (e.g., Inoue et al., 2017; Shepherd, 1972).

Figure 11 shows the cross-sectional alluvial cover averages for the seven experimental conditions with alluvium. Values were extracted every meter between streamwise locations 10 and 20 m. Therefore, 11 local alluvial cover values were obtained for each experiment. As in the case of the reach-averaged values, these results include persistently covered and exposed portions of the cross section as well as a fraction with transient alluvial cover.

In general, all conditions exhibit similar trends, with local lows in pc at streamwise locations 15 and 19 m and local highs at streamwise locations 11 and 16 m. The regions showing higher local percentages of alluvial cover are located 1.5 m downstream of the bend apices. Point bar deposits are responsible for the higher local value of pc at these locations. On the other hand, the local lows in pc are associated with the points of highest curvature in the reach. Both local lows are within 0.5 m of the bend apices.

Figure 12 shows the ratios of the cross sections that had persistently exposed bedrock (dashed yellow line), persistently covered bedrock (blue line), transient alluvial cover (light blue line), and the ratio corresponding to the sum of persistent plus transient cover (black line) for all experimental conditions but pc00. The ratio of exposed bedrock peaks in the vicinity of the bend apices. Even in the case of reach-averaged pc=0.79, portions of the bed in these areas remain exposed due to high curvature. Except for the cases with reach-averaged pc=0.38 and 0.54, no cross sections have fractions with transient alluvial cover greater than 60 %.

The average fractions of transient alluvial cover at the cross-sectional level have values of 0.10 for pc=0.19, 0.21 for pc=0.27, and between 0.31 and 0.34 for the other experimental conditions. In spite of the local variations in transient alluvial cover, potential erosion is, on average, limited to a rather small portion of the cross section. This is likely due to the combined effects of the sediment supply ratio and local curvature.

Figure 13 shows box plots of cross-sectionally averaged pc normalized with the reach-averaged value. The figure also shows the dimensionless curvature of the Kinoshita flume (black dashed line), the negative value of the curvature (gray dotted line), and the median normalized values of pc (red line). The true (κ) and negative (-κ) centerline curvature signals are shown to better highlight the trend of normalized pc with curvature.

The boxes include information from the seven experiments at each cross section. The median value is indicated by the red line inside the box; the bottom line on each box corresponds to the first quartile (q1); the top line on each box corresponds to the third quartile (q3); whiskers extend to q1-1.5(q3-q1) at the bottom and q3+1.5(q3-q1) at the top; and values lying outside this range are considered outliers and are indicated with a red cross. The cross sections located close to the bend apices, i.e., regions with local high curvature, show normalized pc values below unity, whereas the regions with smaller curvature values show normalized pc values above unity. Normalized, local pc values follow the overall trend of local curvature.

Figure 14 shows the erosion potential (Eq. 6) for all experimental conditions. Regions with higher erosion potential are those for which alluvial cover averages were close to 0.5, in accordance with the parabolic form of Eq. (6). These regions are shown in dark blue in Fig. 13. White regions have no erosion potential due to a lack of tools or the presence of alluvial cover protecting the bed from abrasion. The regions of potential erosion are limited to the areas with transient alluvial cover. In general, their width is a function of sediment supply ratio, with narrower regions associated with smaller sediment supply ratios. Locally, the width of these regions is affected by curvature as well, with narrower regions in areas of high curvature and wider regions in areas of lower curvature.

The region of potential erosion is located closer to the inner bank for lower sediment supply ratios and moves outward as sediment supply increases. Focusing on the region of potential erosion located at the bend apex at streamwise location 14.5 m (see Fig. 2c for location on plots), it is seen that for pc=0.27, the region is located right next to the inside bank, whereas for pc=0.79, the region is much closer to the outer bank.

The results of alluvial cover shown and discussed up to this point correspond to spatial or temporal averages. Nonetheless, Fig. 15 shows the frequency of strikes (fs) for all experimental conditions (Eq. 7). In general, the areas in color in the figure are similar to the areas with transient alluvial cover shown in Fig. 9 and the areas with erosion potential in Fig. 14. Picking out differences in these particular figures is not straightforward, but the videos included in the Supplement illustrate the migrating erosion fronts and suggest that erosion is likely to be driven predominantly by the movement of freely migrating bars. The use of the frequency of strikes associated with fluctuations in alluvial cover provides an improved approach for computing bedrock erosion by abrasion, as discussed below.

Alluvial deposits on the bed of a bedrock river cover it and protect it from abrasion (Gilbert, 1877; Sklar and Dietrich, 1998, 2004). At the microscopic level, a portion of the riverbed can only be covered (pc=1) or exposed (pc=0) in any given instant. Therefore, a notion of transient alluvial deposits becomes necessary to guarantee that in time, pc fluctuates between those end-members and erosion caused by saltating bedload particles is possible (Eq. 2). In the original saltation–abrasion framework of Sklar and Dietrich (2004), transient alluvial deposits were an underlying assumption (Fig. 1a). No temporal averaging window was needed since all portions of the bed had the same probability of being eroded in time.

Turowski et al. (2007), working under the assumption that sediment transport capacity is uniform across a control area of unspecified dimensions, described the cover effect as static or dynamic. Static cover occurs when the amount of sediment supplied to the reach is larger than the transport capacity, and therefore some particles remain immobile on the bed. Dynamic cover occurs when the amount of sediment supplied is smaller than the transport capacity. Sediment particles cover portions of the bed but are mobile. As sediment supply increases, erosion is limited due to more grain–grain collisions than grain–bed interactions. In this framework, the dynamic cover effect reduces erosion by decreasing the impact energy experienced by the bed due to the interactions between grains. Nonetheless, the notion of transient alluvial cover over an unspecified time window is required, and in the long term, all areas of the bed have the same likelihood of being eroded as long as sediment supply is below the transport capacity of the reach.

Chatanantavet and Parker (2008), Inoue et al. (2014), Hodge and Hoey (2016b), and Ferguson et al. (2017) present cases in which sediment particles are being transported over the bed as throughput bedload. These cases challenge the notion of a cover effect because alluvial deposits do not exist at all. They could still be treated as having transient alluvial cover for modeling purposes, but what would be the relevant timescale to characterize it? Chatanantavet and Parker (2008) and Hodge and Hoey (2016b) also observed that throughput load may be unstable; as soon as hydraulic conditions change, runaway alluviation occurred and the same portion of the bed changed from a state of being continuously struck by sediment particles (undergoing erosion) to being protected from further erosion.

The examples above suggest that areas with persistent or transient alluvial cover in a mixed bedrock–alluvial river can only be categorized as such given a specified timescale. The reach-averaged results shown in Figs. 8, 9, and 10 suggest that the areas subject to erosion in mixed bedrock–alluvial meandering rivers are a fraction of the total reach area. In this study, we defined transient alluvial cover as portions of the bed on which the temporal averages of local alluvial cover had values 0.025<pc<0.975 during the time of the experiment (Fig. 8). Based on this definition, areas with persistent alluvial cover or exposed bedrock were also delimited (Fig. 9). In the case of the Kinoshita flume experiments, the areas with transient alluvial deposits occupied less than 50 % of the total reach area, and hence erosion could only occur within a restricted portion of total bed area.

The problem remains in regards to generalizing appropriate timescales for modeling purposes. Our results are based on a constant discharge, but in real rivers, a flood could mobilize all alluvium on the bed of the channel, and within the timescale of the flood, alluvial cover would also be transient (e.g., Turowski and Rickenmann, 2009). The use of temporal averages of alluvial cover has limitations, and our results suggest that characterizing the fluctuations of alluvial cover may be a better approach.

Figure 16 shows a hypothetical example of two cases in which the long-term average of alluvial cover is equal, but the fluctuations in alluvial cover between them are different. Given that erosion by abrasion is driven by the number of times the bed is struck by particles, erosion would only occur in the first case. Erosion would only occur each time the area changes from white to black, i.e., every time a particle moves into the area and strikes the bed upon arrival. This simple example suggests that the use of temporal averages of alluvial cover to calculate erosion may lead to inaccurate results.

The use of a relation such as Eq. 2 with spatiotemporal averages of alluvial cover also has limitations. According to it, the experiment pairs (i) pc=0.19 and pc=0.79, (ii) pc=0.27 and pc=0.72, and (iii) pc=0.46 and pc=0.54 should have very similar, or equal, erosion potentials (Eq. 6) as shown below. i

Esp=0.191-0.19=0.190.81=0.154 and Esp=0.791-0.79=0.790.21=0.166

In spite of characterizing erosion potential with spatiotemporal averages (Fig. 14) or fluctuations (Fig. 15) of alluvial cover, the regions of preferential erosion in our experiments show some characteristics that are worth discussing. In all experiments, the regions of preferential erosion are located at the edges of persistent alluvial cover deposits. Their precise location and width are a function of sediment supply and local curvature.

In general, as sediment supply increases, the areas of preferential erosion moved outwards. Our results suggest that in all cases, inset channels would have been formed at the edge of alluvial deposits, and bank erosion would have only occurred beginning at CS11 (Figs. 2c, 15) for pc54, pc72, and pc79. Downstream of CS15, outer bank erosion would only occur for the cases with pc72 and pc79. Therefore, higher sediment supply is needed for bank erosion to occur, and under low sediment supply, inset channels and outer bedrock benches are likely to form. Figure 17 shows an image of the mixed bedrock–alluvial Shimanto River in Shikoku, Japan, and a sketch of what the cross section might look like with the areas of erosion and no erosion indicated. The reach shown in the image has an alluvial point bar on the inside of the bend, a narrow inset channel at the edge of the point bar, and an exposed bedrock bench on the outside of the bend. The same morphologies have been observed in a smaller-scale stream called Pescadero Creek in California, USA (Fig. 6B in Johnson and Finnegan, 2015). The experiments of Mishra et al. (2018) also show that when sediment supply is low, the alluvial point bar is narrow and an inset channel is eroded at the toe of the point bar, leaving an exposed bedrock bench on the outer part of the bend.

The typical geometry of an alluvial meandering channel cross section is shallow on the inside and deep on the outside. The reach of the Shimanto River shown in Fig. 17 has a different geometry. The deepest portion of the channel is not located on the outer bank. Instead, it is located at the toe of the point bar, which happens to be approximately at the middle of the cross section. It is likely that the narrow inset channel was formed during a long period of decreased sediment supply. During this period, the region of transient alluvial cover was confined to the current width of the channel shown in the image. The outer bedrock bench could potentially be eroded if sediment supplied to the reach from upstream were to be increased and maintained at this increased value for an extended period of time. If this occurred, the point bar would likely extend toward the outer part of the bend, thus moving the area of transient alluvial cover farther into this region. Finnegan et al. (2007) observed similar trends in experiments conducted in a straight flume over an erodible bed, wherein erosion began at the edges of sediment patches and formed longitudinal grooves in the channel. Shepherd and Schumm (1974) also observed that outward bank erosion was possible when bed material was transported at capacity, but when the amount of material in transport was less than the transport capacity, inset channels formed. Similar observations have been made experimentally by Mishra et al. (2018) and numerically by Inoue et al. (2017) and Nelson and Seminara (2011). Even though we did not measure velocities, our observations suggest that the areas with very narrow regions of erosion potential, e.g., between CS14 and CS15, are located at regions of topographically induced high flow velocities in accordance with observations made by Hodge and Hoey (2016b).

The specific links between sediment supply and local curvature, even though suggested by our results, need further investigation to properly parameterize them. It is likely that antecedent curvature and curvature sign also play a role (Fig. 13). Moreover, the use of denser material, e.g., sand, would likely affect the specific locations of the alluvial deposits. However, the main trends observed herein are likely to be general.

The results corresponding to the Kinoshita flume are based on the areal cover of alluvial sediment captured with a camera located above the flume. These observations are related to the framework of Sklar and Dietrich (2004) described in Fig. 1a. The framework proposed by Zhang et al. (2018, 2015) and shown in Fig. 2b relates to the experiment conducted in the small bedrock slab (Supplement S1, Fig. 5) for which the cover is quantified as in Eq. (4). That experiment allowed us to relate the areal cover fraction to the ratio of alluvial thickness to bedrock macro-roughness (Fig. 5b). We obtained an S-shaped relation between these two variables.

This result provides a useful link between the two models (Fig. 1a and b) but is only constrained by geometric variables, specifically the alluvium grain size and the bedrock macro-roughness. Other factors that affect the distribution and size of alluvial deposits are local topography and hydraulic conditions (e.g., Hodge and Hoey, 2016b; Chatanantavet and Parker, 2008; Finnegan et al., 2007; Johnson and Whipple, 2007), grain size and the ratio between grain and bedrock roughness (e.g., Ferguson et al., 2017; Nelson et al., 2014; Johnson 2014; Inoue et al., 2014; Chatanantavet and Parker, 2008), feedbacks between bedrock erosion, sediment deposition, and its effects on hydraulic resistance (e.g., Ferguson et al., 2017; Nelson et al., 2014; Johnson, 2014; Inoue et al., 2014), and channel sinuosity (Shepherd and Schumm 1974; Shepherd, 1972).

The relations between the amount of sediment in the system and alluvial cover in Fig. 5a and c are similar but are not equivalent. Figure 5a is based on the cumulative mass of sediment added to the bedrock slab, whereas Fig. 5c is based on the sediment supply ratio (Eq. 3). Turowski and Hodge (2017) developed an equation to relate the sediment mass on the bed and sediment supply. Their model differentiates between the mass of mobile and stationary bed material and relates them to sediment flux via an entrainment–deposition equation. Their framework could be tested with our dataset. Two particular issues of interest are the following. i.

The area of exposed bedrock is a function of the sediment mass in the system and the probability of incoming particles striking open bed areas. Turowski and Bloem (2015) showed that particle impact energy can be transferred to the bed if the thickness of the alluvial layer, even if static, is small. However, they concluded that the amount of energy transferred to the bed is negligible in comparison to areas where a sediment particle impacts the bed directly; in the long term direct impacts are likely to dominate bed erosion. Therefore, parameterizing these open areas is very important to better model bedrock erosion. Our results show that different areas of the bed have different likelihoods of being eroded. Our dataset could be used to develop a probability function that takes into account the effects of local curvature.

Based on the results of this study, the following two research directions are proposed.

Conduct experiments with the objective of determining appropriate relations to define the steepness and intersection of the sigmoid function for use in numerical models of bedrock erosion based on a framework using the ratio of alluvial thickness to bedrock macro-roughness.