Alluvial cover on bedrock channels: applicability of existing models

Several studies have demonstrated the importance of alluvial cover; furthermore, several mathematical models have also been introduced to predict the alluvial cover on bedrock channels. Here, we provide an extensive review of research exploring the relationship between alluvial cover, sediment supply and bed topography of bedrock channels, describing various mathematical models used to analyse the deposition of alluvium. To test one-dimensional theoretical models, we performed a series of laboratory-scale experiments with varying bed roughness under simple conditions without bar formation. Our experiments show that alluvial cover is not merely governed by increasing sediment supply and that bed roughness is an important controlling factor of alluvial cover. A comparison between the experimental results and the five theoretical models shows that (1) two simple models that calculate alluvial cover as a linear or exponential function of the ratio of the sediment supplied to the capacity of the channel produce good results for rough bedrock beds but not for smoother bedrock beds; (2) two roughness models which include changes in roughness with alluviation and a model including the probability of sediment accumulation can accurately predict alluvial cover in both rough and smooth beds; and (3), however, except for a model using the observed hydraulic roughness, it is necessary to adjust model parameters even in a straight channel without bars.


Introduction
Economic growth worldwide has fuelled the demand for the construction of straightened river channels, sabo dams, the 70 collection of gravel samples for various research, etc., leading to a decline in sediment availability and alluvial bed cover. Sumner et al. (2019) reported that the straightening of the Yubari River, which was carried out to improve the drainage of farmland, caused the bedrock to be exposed and the knickpoint to migrate upstream. Also, construction of a dam in the upstream section of Toyohira river in Hokkaido -Japan, decreased the sediment availability to the downstream section contributing to the formation of a knickpoint (Yamaguchi et al. 2018). Sediment availability plays a very important role in 75 controlling landscape evolution and determining the morphology of rivers over geologic time (Moore 1926;Shepherd 1972), and has two contradicting effects on bedrock-bed, known as the Tools and Cover effects (Gilbert, 1877;Sklar and Dietrich, 1998). Sediment acts as a tool for erosion by increasing the number of impacting-particles that erode the bedrock bed, known as tools effect. As sediment availability increases, the sediment starts settling down on the river bed providing a cover for the bed underneath from further erosion, known as the cover effect. In the last 20 years, various field-scale (Turowski et al., 2008b;80 Turowski and Rickenmann, 2009;Johnson et al., 2010;Jansen et al., 2011;Hobley et al., 2011;Cook et al., 2013;Inoue et al., 4 channel's alluvial cover. Yanites et al. (2011) studied the changes in the Peikang River in central Taiwan triggered by the thick sediment cover introduced by landslides and typhoons during the 1999 Chi-Chi earthquake. Their results show slowed or no incision in high transport capacity and low transport capacity channels. Cook et al. (2013) suggested that rapid knickpoint propagation are dominantly controlled by the availability of bedload. Their field surveys of the bedrock gorge cut by the Daán River in Taiwan showed that the channel bed was not eroded in the absence of coarse bedload, despite floods and available 95 suspended sediment. Izumi et al. (2017) showed that sediment transport and bedrock abrasion lead to the formation of cyclic steps, and Scheingross et al. (2019) suggested that undulating bedforms like cyclic steps grow to become waterfalls and knickpoints.
Sediment availability also controls the width of bedrock channel. Finnegan et al. (2007) conducted laboratory-scale experiments and studied the interdependence between incision, bed roughness and alluvial cover. Their results indicated that 100 alluvial deposition on the bed shifted bedrock erosion to higher regions of the channel or bank of the channel, and suggested that the sediment supply rate controls the thalweg width of bedrock channel. Similar findings were noted in flume studies conducted by Johnson and Whipple (2010). They have shown the importance of alluvial cover in regulating the roughness of the bedrock bed by providing a cover for the local lows and thereby inhibiting the erosion and focusing erosion on local highs.
Field observations also show that channels with higher sediment supply to capacity ratio are expected to be wider as alluvial 105 cover shifts erosion from bed to banks of the channel (Beer et al. 2016;Turowski et al., 2008a andWhitbread et al., 2015). Inoue et al. (2016) and Inoue and Nelson (2020) showed the formation of several longitudinal grooves at low sediment supply to capacity ratio. As the sediment supply increases, one of the grooves attracts more sediment supply and progresses into a comparatively straight, wide and shallow inner channel which further progresses into a more sinuous, deeper inner channel (Wohl and Ikeda, 1997;Shepherd and Schumm,1974). 110 Some studies have credited the seasonally and climatically driven higher sediment supplies during floods to be the driving force for bedrock meander and strath terrace formation (De Vecchio et al., 2012;Hancock & Anderson, 2002). Periods of higher sediment supply promote lateral erosion and strath terrace formation, whereas periods of lower sediment supply lead to vertical erosion and steep slip-off slopes (e.g., Fuller et al. 2009;Inoue et al., 2017a). Mishra et al. (2018) showed that in the bend, lateral abrasion followed a monotonically increasing linear relationship with sediment feed rate. Fuller et al. (2016) 115 performed laboratory-scale experiments and established the importance of bed-roughness in determining lateral erosion rates because high roughness scatters the direction of bedload transport, increasing the frequency with which it collides with the wall.
There have been advances in theoretical and numerical methods mimicking, reproducing and predicting the morphodynamics of laboratory scale and field-scale observations. A majority of traditional bed-erosion models are classified as the stream power 120 and shear stress family of models (cf. Shobe et al., 2017;Turowski, 2018) (e.g., Howard, 1994Whipple and Tucker, 1999), in which bed erosion is a function of discharge and bed-slope. These models, however, cannot describe the role of sediment in controlling the bed dynamics. Several models remedy this shortcoming by considering the tools and cover effect of sediment supply Dietrich, 1998, 2004;Turowski et al., 2007;Chatanantavet and Parker, 2009;Hobley et al., 2011;Inoue et al., 2017aInoue et al., , 2017bShobe et al., 2017;). 125 5 was introduced by Sklar and Dietrich (1998;. According to the saltation-abrasion model proposed by Sklar and Dietrich (1998;, the alluvial cover increases linearly with the ratio of sediment supply to sediment transport capacity ⁄ . In contrast, Turowski et al. (2007) proposed a model that considered the cover effect as an exponential function of the ratio of 135 sediment flux to sediment transport capacity. The model uses a probabilistic argument i.e. when sediment supply is less than the capacity of the channel, grains have an equal probability of settling down over any part of the bed. Also, the deposited grains can be static or mobile.
The erosion formula including the above model was able to reproduce the relationship between the sediment mass and the erosion rate observed in the rotary-abrasion mill experiment performed by Sklar and Dietrich (2001). However, subsequent 140 experiments using a straight channel pointed out a phenomenon that cannot be reproduced by the above models. Chatanantavet and Parker (2008) conducted laboratory-scale experiments in straight concrete bedrock channels with varying bedrock roughness and evaluated bedrock exposure with respect to sediment availability. In their experiments, alluvial cover increased linearly with increasing sediment supply in case of higher bed roughness, whereas in case of lower bed roughness and higher slopes, the bed shifted abruptly from being completely exposed to being completely covered. This process of the bedrock bed 145 suddenly becoming completely alluvial from being completely exposed is known as rapid alluviation a.k.a run-away alluviation (Hodge and Hoey, 2016a). Rapid alluviation was also observed in the laboratory scale experiment conducted by Hodge and Hoey (2016a;2016b) in a 3D printed flume of natural stream Trout Beck, North Pennies-U.K. Their first set of experiments focused on quantifying hydraulic change with varying discharge, suggesting that hydraulic properties fluctuate more during higher discharge. Their second set of experiments (Hodge and Hoey, 2016b) concentrated on quantifying the 150 sediment dynamics for varying discharge and sediment supply. They supplied 4 kg and 8 kg sediment pulse to the channel and observed a similar alluvial pattern in both cases suggesting that the deposition of sediment on the bed may not only depend on the amount of sediment supplied but may be strongly influenced by the bed topography and roughness. Inoue et al. (2014) conducted experiments by excavating a channel into natural bedrocks in Ishikari River, Asahikawa, Hokkaido -Japan. They conducted experiments with different combinations of flow discharge, sediment supply rate, grain size and roughness. Their 155 experiments advocated that the dimensionless critical shear stress for sediment movement on bedrock is related to the roughness of the channel. Their experiments also showed that in the case when the alluvial cover is smoother than the bedrock, with an increase in alluvial cover, the hydraulic roughness in a mixed bedrockalluvial bed decreases.
Besides, the simple models described above cannot capture the sediment mass in a channel that changes due to sediment supply and runoff because they do not conserve sediment mass. Lague (2010) employed the Exner equation to calculate alluvial 160 thickness with respect to average grain size d. Their model, however, lacks the tools effect for bed erosion. Recently, Johnson (2014) and Inoue et al. (2014) proposed reach-scale physically-based models that encompass the effects of bed roughness in addition to mass conservation. Inoue et al. (2014) also conceptualised 'Clast Rough' and 'Clast Smooth' bedrock surfaces. A bedrock surface is clast-rough when bedrock hydraulic roughness is greater than the alluvial bed hydraulic roughness (supplied sediment), otherwise, a surface is clast-smooth i.e. when the bedrock roughness is lower than the alluvial roughness. Inoue et 165 al. (2014) and Johnson (2014) clarified that the areal fraction of alluvial cover exhibits a hysteresis with respect to the sediment supply and transport ratio in a clast smooth bedrock channel. They described that along with rapid alluviation, perturbations in sediment supply can also lead to rapid entrainment. Whether the bed undergoes rapid alluviation or rapid entrainment is determined by the bed condition when perturbations in sediment supply occur. If the perturbations occur on an exposed bed, it undergoes rapid alluviation, conversely, when perturbations happen on an alluviated bed, it undergoes rapid entrainment. 170 Zhang et al. (2015) proposed a macro-roughness saltation-abrasion model (MRSA) in which cover is a function of alluvial thickness and macro-roughness height. Nelson and Seminara (2012) proposed a linear stability analysis model for the formation of alternate bars on bedrock bed. Inoue et al. (2016) expanded Inoue et al. (2014) to allow variations in the depth and width of alluvial thickness in the channel cross-section. They further modified the numerical model (Inoue et al., 2017a) and implemented the model to observe changes in a meander bend. Turowski and Hodge (2017) generalized the arguments presented by Turowski et al. (2007) and Turowski (2009) and proposed a reach-scale probability-based model that can deal with the evolution of cover residing on the bed and the exposed bedrock. Turowski (2018) proposed a model and linked the availability of cover in regulating the sinuosity of the channel. Shobe et al. (2017) proposed the SPACE 1.0 model for the simultaneous evolution of an alluvium layer and a bedrock bed. These models utilise the entrainment/deposition flux for sediment mass conservation. 180 Hodge and Hoey (2012) introduced a reach-scale Cellular Automaton Model that assigned an entrainment probability to each grain. The assigned probability of each grain was decided by the number of neighbouring cells containing a grain. If five or more of total eight neighbouring cells contained grain, the grain was considered to be a part of the cover, otherwise, it was considered an isolated grain. They suggested that rapid alluviation occurred only in cases when isolated grains were more than the cover on the bed. Also, they advised a sigmoidal relationship between ⁄ and 1 − . Aubert et al. (2016) proposed 185 a Discrete-Element Model where they determined from the velocity distribution of the grains. If the velocity of a grain is 1/10 th or lower than the maximum velocity, the grain settles as cover on the bedrock surface. The model, however, cannot deal with non-uniform velocity fields and hence cannot predict results for varying alluvial cover.
Except for the Lagrangian description models that track individual particles (i.e., Hodge and Hoey, 2012;Aubert et al., 2016), the Eulerian description models are roughly classified into four categories; the linear model proposed by Dietrich 190 (1998, 2004), the exponential model proposed by Turowski et al. (2007), the roughness models proposed by Nelson and Seminara (2012), Inoue et al. (2014), Johnson (2014), Zhang et al. (2015) and the probabilistic model proposed by Turowski and Hodge (2017). In this study, we focus on a detailed study of the similarities and differences among the Eulerian description models proposed by Sklar and Dietrich (2004), Turowski et al. (2007), Inoue et al. (2014), Johnson (2014) and Turowski and Hodge (2017). These one-dimensional models have already been compared to experiment with bars (Chatanantavet and Parker, 195 2008) and experiments with irregular roughness arrangement Hoey, 2016a, 2016b;Inoue et al., 2014), but a test in one-dimensional flow fields has not been performed. In this study, we compare the efficacy of these models from comparisons with our experimental results without bars with relatively regular roughness distribution. In addition, we apply the roughness models (Inoue et al., 2014;Johnson, 2014) to the experiments conducted by Chatanantavet and Parker (2008) in order to discuss the effect of bar formation on alluvial cover in a mixed bedrock -alluvial river. 200

Linear Model
When the sediment supply is larger than the transport capacity, the bedrock eventually becomes completely covered by alluvial material and the alluvial cover ratio is equal to 1. If there is no sediment supply, the sediment deposit disappears and eventually, the bedrock bed becomes completely exposed and is equal to 0. Sklar and Dietrich (2004) linearly connected 205 these two situations, and proposed a linear model to include the cover effect in their saltationabrasion model; where, is the mean areal fraction of alluvial cover, and are the volume sediment supply rate per unit width and transport capacity, respectively. 210

Exponential Model
When the dimensionless mass of sediment on the bed * is increased by a small amount * , a fraction of this amount will fall on exposed bedrock and cover it. Hence, (1 − ) = − * , where, is a dimensionless cover factor parameter and determines whether sediment deposition is more likely on covered areas for < 1 and deposition on uncovered areas for > 7 1. Integration gives = 1 − exp(− * ). Turowski (2007) assumed that the * is equal to the ratio of sediment supply to 215 capacity, and derived the following exponential model using a probabilistic argument;

Macro Roughness Model
The experimental results of Inoue et al. (2014) motivated their mathematical model formulating the interaction between alluvial 220 cover, dimensionless critical shear stress, transport capacity and the ratio of bedrock hydraulic roughness to alluvial hydraulic roughness. They calculated the total hydraulic roughness height ( ) as a function of alluvial cover: where is the total hydraulic roughness height of bedrock channel, is the cover fraction calculated as proposed by Parker et al. (2013) that depends on the ratio ⁄ where ɳa is the alluvial cover thickness and is the bedrock macro-roughness 225 height (i.e. topographic unevenness of the bed). and (= 1 ~ 4 d, here set to 2) represent the hydraulic roughness height of bedrock and alluvial bed respectively. The total transport capacity per unit width in Inoue et al.'s model is calculated as follows: where is a bedload transport coefficient taken as 2.66 in this study, * and * are the dimensionless shear stress and dimensionless critical shear stress, is the specific gravity of the sediment in water (1.65), is the gravitational acceleration and is the particle size. In this model, is back-calculated from Equations (3), (4) and (5) under the assumption that the sediment supply rate and the sediment transport capacity are balanced in a dynamic equilibrium state (i.e., = 0 ⁄ in Exner's mass conservation equation). 235 The sensitivity analysis of bedrock roughness and sediment supply rate conducted by Inoue et al. (2014) showed that for a given sediment supply, the deposition (Pc) is higher when bedrock roughness is larger. They also showed that if sediment supply rate is larger than the transport capacity of bedrock bed, the clast-smooth surface shows a sudden transition from completely exposed bedrock to completely alluvial, i.e., clast-smooth surfaces show rapid alluviation. 240

Surface Roughness Model
Johnson (2014) proposed a roughness model using the median diameter grain size. They also calculated the hydraulic roughness using the aerial alluvial cover fraction.
where = 2 is a coefficient and # is called a non-dimensional alluvial roughness representing variations in topography. 245 For a fully alluviated bed, ksa=2d.
where is the transport capacity per unit width for sediment moving on purely alluvial bed and is the transport capacity per unit width for sediment moving on purely bedrock bed. * is the dimensionless critical shear stress for grains on bedrock portions of the bed. 260 The models proposed by Inoue et al. (2014) and Johnson (2014) may seem rather similar in that they estimate the transport capacity of a mixed alluvialbedrock surface. However, both models opt for different approaches when it comes to estimating hydraulic roughness. The model by Inoue et al. (2014) uses the observed hydraulic roughness, but the model by Johnson (2014) calculates the hydraulic roughness from the roughness (topographic unevenness) of the bed surface. The model by Inoue et al. (2014) needs measurements of observed bedrock hydraulic roughness, and the model by Johnson (2014) needs topographic 265 bedrock roughness. In the model by Inoue et al. (2014), the macro roughness of the bed acts only when converting the alluvial layer thickness to the alluvial cover ratio. The macro roughness affects the temporal change of the alluvial cover ratio but does not affect the alluvial cover ratio in the dynamic equilibrium state. In addition, in the model by Johnson (2014), first, the sediment transport capacities for the bedrock and alluvial bed are separately calculated, then total transport capacity is estimated using . Whereas, in the model by Inoue et al. (2014), first, the total hydraulic roughness height is calculated using 270 , then total transport capacity is estimated using the total hydraulic roughness.

Probabilistic Model
Turowski and Hodge (2017) proposed a probability-based model for prediction of cover on bedrock channels and investigated the distribution of sediment on the bedrock. Because they mainly focused on the transformation between a point of view considering sediment masses and one considering sediment fluxes, they did not treat the interaction between the alluvial cover 275 and the bed roughness. However, there is a possibility to capture the effects of bedrock roughness on the alluvial cover by adjusting the probability of grain entrainment and deposition included in the model. They defined as the probability that a grain will settle on the exposed bed and used a power-law dependence of on the exposed area (1 − ), taking the form = (1 − ) , here is a model parameter. Similar to the exponential model (Turowski, 2007), integrating (1 − ) = − * , They further introduced the mass conservation equation and derived the following equation.
where 0 * is the dimensionless characteristic sediment mass obtained as follows: Their model also provides two other analytical solutions and potentially other variables (Equations 30, 31 in Turowski and 285 Hodge, 2017), however, we employed Equation 13 in this study as the equation has the highest flexibility of and is likely to be able to include roughness feedbacks.
We hereafter refer to Sklar and Dietrich (2004)

Experimental Flume
We conducted experiments to measure how sediment cover developed over surfaces of different roughnesses and different sediment fluxes. The experiments were conducted in a straight channel at the Civil Engineering Research Institute for Cold Region, Sapporo, Hokkaido, Japan. The experimental channel was 22 m long, 0.5 m wide and had a slope of 0.01. The width-295 depth ratio was chosen to achieve no-sandbar condition (i.e., small width-depth ratio, 6.1 to 8.3 in our experiments). Chatanantavet and Parker (2008) conducted several flume experiments with sandbar conditions (i.e., large width-depth ratio, 11 to 30 in their experiments) and suggested that the alluvial cover increases linearly to the ratio of sediment supply and transport capacity of the channel when the slope is less than 0.015. The formation of bars strongly depends on the widthdepth ratio (e.g., Kuroki and Kishi, 1984;Colombini et al., 1987). Generally, neither alternate bars nor double-row bars are 300 formed under conditions with width-depth ratios < 15.
In this study, we investigated the influence of bedrock roughness on the alluvial cover under conditions where the slope and width -depth ratios were small compared to the experiments of Chatanantavet and Parker (2008).

Bed characteristics and conditions
The channel bed consisted of hard mortar that was not eroded by the bed load supplied in this experiment. In order to achieve 305 different roughness conditions, the bed in Gravel30 was embedded with gravel of particle size 30 mm, Gravel50 was embedded with 50 mm gravel, and Gravel5 was embedded with 5 mm gravel.
We performed an additional 2 cases with net-installation on the riverbed. The net was made of plastic. An installed net on the riverbed can trap sediment during high flow, eventually protecting the bed from further erosion from abrading sediment (Mutsuura et al.,2015). A net of mesh size 30 mm X 30 mm was installed on the bed in Net4 and Net2. The height of the net 310 was 4 mm and 2 mm respectively. Figure 1 shows the experimental channel bed of all 5 runs.

Measurement of observed bedrock roughness 320
In order to measure the initial bed roughness (before supplying sand), a water discharge of 0.03 m 3 /s was supplied, and the water level was measured longitudinally at every 1 m at the centre of the channel. The hydraulic roughness height for bedrock (ksb) was calculated using Manning -Strickler relation and Manning's velocity formula.
where is the Manning's roughness coefficient and is the average velocity ( = ⁄ where is the water discharge, is the channel width, is the water depth), is the energy gradient. Several previous studies have suggested that in bedrock rivers the Manning's nm value can depend on the discharge (Heritage et al., 2004;Hodge and Hoey, 2016a), but in our experiments, the discharge is held constant between the different runs.
In order to compare the hydraulic roughness height and the riverbed-surface unevenness height, the riverbed height before 330 water flow was measured along a 1-metre length (12 m to 13 m) with a laser sand gauge. The measurements were taken longitudinally at every 5 mm. The measurements were taken at three points: 0.15 m away from the right wall, the centre of the channel, and 0.15 m away from the left wall. The standard deviation representing the topographic roughness was obtained by subtracting the mean slope from the riverbed elevation and then calculating the standard deviation of the remaining elevations 2010). 335

Measurement of dimensionless critical shear stress on bedrock
To measure the dimensionless critical shear stress of grains on completely bedrock portion, i.e. * , 30 gravels of 5mm diameter each, were placed on the flume floor at intervals of 10 cm or more to make sure that there was no shielding effect between the gravels (there was shielding effect due to unevenness of the bedrock). Next, water flow was supplied at a flow discharge that no gravel moved, and was slowly increased to a flow discharge at which all the gravels moved. The water level 340 and the number of gravels displaced were measured and recorded for each flow discharge. These measurements were performed for all the 5 bedrock surfaces.
We calculated the dimensionless shear stress * (= ⁄ ), here is the specific gravity of the submerged sediment (1.65).
We defined the critical shear stress * is the weighted average of * using the number of displaced gravels.

Measurement of Alluvial cover 345
In order to perform the main set of experiment, different amounts of gravel (5mm, hereafter called as sediment) was supplied manually at a constant rate while the flow rate was kept constant at 0.03 m 3 /s. The alluvial cover ratio was measured once equilibrium state was achieved. Once the areal fraction became stable in qualitative observations and the variation of hydraulic roughness of mixed alluvialbedrock bed calculated from the observed water depth decreased despite sediment being supplied, we considered that the experiment has reached its equilibrium state. Equilibrium conditions were achieved after 2~4 350 hours of sediment supply. The sediment supply amounts and other experimental conditions for various cases are provided in Table 1. Each run has multiple cases, each with different sediment supply and time duration. Each case was performed until the became constant. The gravels were supplied from Run-0 of no sediment to Run-4~5 of completely alluvial cover. The Run-0 with no sediment supply in each run represents the bedrock-roughness measurement experiment explained in section 2.3. 355 For each roughness condition, initially, we supplied sediment at the rate of 3.73x10 -5 m 2 /s and observed the evolution of . A sediment supply rate of 3.73x10 -5 m 2 /s is used as it was measured in the flume with complete alluvial bed and it is in good agreement with the calculated value obtained from Equation 4. If ≈ 1, the sediment supply was approximately reduced by 1.5 times in the subsequent run, and then the sediment supply was further reduced to 2 times and 4 times in subsequent runs (example: Gravel30, Gravel50 and Net4). In roughness conditions where sediment supply of 3.73x10 -5 m 2 /s resulted in ≈ 0, 360 the sediment supply was increased by 1.25 or 1.5 times and 2 times in the subsequent runs (example: Gravel5 and Net2).
However, for ease of understanding, we will present each experimental run in ascending order of sediment supply rate.
The alluvial cover was calculated at the end of the experiment, using black and white photographs of the flume by taking the ratio of the number of pixels. The dark/black colour represented sediment cover while white represented exposed bedrock. The water level was measured and recorded every hour at the centre of the channel, to calculate the hydraulic roughness during 365 and at the end of the experiment. The cross-sectional profile of the channel bed was measured with a laser sand gauge at 11 longitudinal intervals of 1 m from 10 m to 15 m from the downstream end before and after each run. We calculated the alluvial thickness from the difference between the two data.

3.1
Here, ksb represents the hydraulic roughness height of purely bedrock bed, ksb/d is the relative roughness of the bedrock bed, qbs represents sediment supply rate, Pc is the alluvial cover, D is the water depth, U is the depth-averaged velocity, Fr is the Froude number (= u/(gD) 0.5 ), ks/d is the ratio of hydraulic roughness height to grain size. Figure 2 shows the relationship between the hydraulic roughness height of bedrock bed and the topographic roughness height of bedrock bed . This figure suggests that Gravel30 with 30 mm sized embedded gravel, has the largest hydraulic roughness and Gravel5 with 5 mm sized embedded gravel has the lowest hydraulic roughness. Gravel50 embedded with 50 380 mm gravel has large topographical roughness error bars for the reason that, the large gravels were embedded randomly in the bed, resulting in unintended longitudinal spatial variation in the unevenness of the channel bed. The error bars here represent the minima, average and maxima of the calculated standard deviation of measurements taken along the left wall, centre and right wall of the channel, as mentioned in section 2.3. Although the hydraulic roughness tends to increase with an increase in topographical roughness, it has a large variation. This variation is due to the fact that the hydrological roughness height does 385 not only depend on the topographical roughness but also on the arrangement of the unevenness.  Figure 3 shows the channel bed after the experiments of the Gravel30 series (Gravel30-1, Gravel30-2, Gravel30-3 and 395

Relative roughness of the bedrock bed, sediment supply and alluvial cover
Gravel30-4) with the highest relative roughness of the bedrock bed (ksb/d). Figure 4 shows the channel bed after the experiments of the Gravel5 series (Gravel5-1, Gravel5-2, Gravel5-3) which has the lowest relative roughness of the bedrock bed. In these two figures, we can compare Gravel30-4 and Gravel5-1 with equal sediment supply rates. The bed in Gravel30-4 is completely covered with sediment whereas the bed in Gravel5-1 has almost no accumulated sediment on the bed. Figure 5 shows the relationship between alluvial-cover fraction Pc and sediment supply per unit width . is obtained by 400 dividing the sediment-covered area by the total area of the channel from photographs. The value of is 1 for a completely covered channel and 0 for a completely exposed bedrock bed. In Figure 5, if we compare Gravel30-4, Gravel50-4, Gravel5-1, Net4-4 and Net2-1, the cases with equal sediment supply rate of 3.73 × 10 -5 m 2 /s, it can be observed that alluvial-cover fraction is increasing with an increase in the bedrock roughness. Moreover, in Gravel30 series, Gravel50 series and Net4 series with high relative roughness of the bedrock bed ⁄ (ratio of the hydraulic roughness height of bedrock bed to the grain size 405 ), is roughly proportional to the sediment supply rate qbs. However, in Gravel5 series and Net2 series, which have lower ⁄ (relative roughness of the bedrock bed), shows hardly any increase when qbs is low (Gravel5-0, Gravel5-1, Gravel5-2, Net2-0, Net2-1) and when sediment supply ( ) increases (Gravel5-3, Net2-2), the bedrock suddenly transitions to completely alluvial bed. In clast-smooth bedrock (i.e., Gravel5 and Net2), it is possible to supply more sediment flux than qbca because qbcb (transport capacity on completely bedrock bed) is larger than qbca (transport capacity on a completely alluvial bed). 410   Figure 6 shows the change in relative roughness in a mixed alluvialbedrock channel i.e. ⁄ with time in Gravel30 and Gravel5 series. In Gravel30 series with a higher initial relative roughness, relative roughness decreased due to the increase in 425 alluvial deposition and cover. In Gravel5 series which has a lower initial relative roughness, relative roughness increased due to the increase in alluvial deposition and cover. The relative roughness nears ~2 for both Gravel30-4 and Gravel5-3 in which the alluvial cover fraction approaches 1.  Figure 7 shows the variation in with respect to relative roughness. In cases with lower initial relative roughness, for example: Gravel50 and Net2, the relative roughness is increasing with an increase in . Whereas, in cases with higher initial 435 relative roughness, Gravel30, Gravel5 and Net4, an increase in reduces the relative roughness. Besides, irrespective of the initial relative roughness, the bed becomes completely alluvial as ≈ 1 and its relative roughness becomes a similar value (i.e., 1 to 4). Several studies in the past have suggested that when the bed consists of uniform grain size, the hydraulic roughness height for such a gravel bed is 1 to 4 times the grain diameter (Inoue et al., 2014;Kamphuis, 1974;Parker, 1991) which is also the case in our experiments as shown in Figure 7. 440

Relationship between gravel layer thickness and alluvial cover fraction
The ratio of the alluvial thickness to macro-roughness L is not used in the model comparison in this study. However, we 445 experimentally investigate ⁄ because various numerical and theoretical models have predicted alluvial cover as a function of relative alluvial thickness (Zhang et al., 2015;Inoue et al., 2014;Parker et al., 2013;Tanaka and Izumi, 2013;Nelson and Seminara, 2012) here, is the average thickness of the alluvial layer calculated from the total flume area instead of the area of sediment 450 patches, is the macro-roughness height of the bedrock bed. Parker et al. (2013) define as the macroscopic asperity height of rough bedrock rivers (≈2 ). Tanaka and Izumi (2013) and Nelson and Seminara (2012) define as the surface unevenness of alluvial deposits on smooth bedrock river (≈ ). In this study, we define = 2 + so that it can cope with both smooth and rough bedrocks. Figure 8 shows the relationship between relative gravel layer thickness ⁄ and alluvial cover ratio. The figure confirms that the alluvial cover ratio of the experimental result can be efficiently evaluated by 455 Equation (15).   Figure 9 shows the relationship between the ratio of the hydraulic roughness height of bedrock bed to the grain size ( ⁄ : referred to as the relative roughness of the bedrock bed in section 3.2) and the dimensionless critical shear stress over bedrock bed * . In this figure, we compare the results obtained from Inoue et al. (2014) (Eq. 5) and Johnson (2014) (Eq. 10) 465 with the experimental results in this study, experimental results of Inoue et al. (2013) (the same channel and grain size as this study, but with a smoother bedrock bed) and Inoue et al. (2014) (the channel excavated in Ishikari river).

Relative Roughness of the bedrock bed and dimensionless critical shear stress
According to Figure 9, the non-dimensional critical shear stress depends on the relative roughness of the bedrock bed to the power of 0.6. Besides, the results obtained from Eq. (5)

Predicting experimental results of alluvial cover ratio using the models
For the purpose of model comparisons with experimental results, we first calibrate the model parameters included in the exponential model, the surface roughness model and the probabilistic model to minimize RMSD (root mean square deviation) of cover between experimental data and the model. We do not calibrate the linear and macro-roughness models as they do not include free model parameters.

20
The parameter in the exponential model implies that the probability of sediment deposition in uncovered areas (Turowski et al, 2007) can vary with the roughness of the bedrock. The parameter # in the surface roughness model (Johnson, 2014) represents the change in alluvial roughness that varies with the cover. When # = 1, it means the alluvial hydraulic roughness is proportional to the grain diameter size and is independent of the cover fraction. For our calculations, we have used # = 4 as applied in Johnson (2014). The parameter in the surface roughness model is used to calculate the hydraulic bedrock 495 roughness from the topographic roughness . This value can be back-calculated from the experimental results (Fig. 2), but using the back-calculated value (i.e., using the observed instead of the calculated ) did not minimize the RMSD of cover. Hence we adjusted to minimize the RMSD of cover. The parameter is introduced to express the relation between the deposition probability and the cover ratio exponentially and can vary with bedrock roughness. The parameter 0 * represents the dimensionless value of sediment mass at sediment transport capacity. Although this parameter is calculated from Equation 500 14, the experimental results could not be reproduced only by adjusting . Hence we adjusted both and 0 * by trial and error to minimise the RMSD of cover. Table 2 provides the calibration values.  Figure 10 shows the comparison among experimental results presented in this paper, Sklar and Dietrich's linear model (2014)  505 and Turowski et al.'s exponential model (2007). In order to calculate the model results for Figure 10, we altered the ratio of ⁄ by 0.01, 0.02, 0.03 and so on. This Figure suggests that the linear model is generally applicable to rough bed with relative roughness of the bedrock bed (ksb/d) of 2 or more, but not to smooth bed with relative roughness of the bedrock bed (ksb/d) less than 2 (Gravel30, Gravel50 and Net4). As suggested by Inoue et al. (2014), in this study, "clast-smooth bed" refers to the bed with roughness less than the roughness of supplied gravel and "clast-rough bed" stands for the bed with roughness 510 more that the roughness of the supplied gravel. The exponential model is also more suitable for a clast-rough bed. Johnson's surface-roughness model (2014). In roughness models, ⁄ (= ⁄ in dynamic equilibrium state in the roughness models) is calculated with a given at intervals of 0.01. It shows that the macro-roughness model proposed by Inoue et al. (2014) can predict the increasing alluvial cover for cases with high relative roughness of the bedrock bed (ksb/d), 515 as well as the rapid alluviation and hysteresis (green shaded region) for cases with lower relative roughness of the bedrock bed (Gravel5 and Net2), without adjusting the roughness. The surface-roughness model proposed by Johnson (2014) also shows good agreement in predictions of alluvial cover and rapid alluviation and hysteresis if are adjusted.

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As mentioned earlier, the major difference between the macro-roughness model (Inoue et al., 2014) and the surface-roughness model (Johnson, 2014) is the way the transport capacity is calculated. In the case of the surface-roughness model (Johnson, 520 2014), first, the transport capacities for bedrock ( ) and alluvial bed ( ) are separately calculated, then the total transport capacity ( ) is calculated for a range of cover fractions ( ). Hence, in cases when * < * < * , the transport capacity over bedrock portion = 0 and thereby the bedrock roughness hardly affects the alluvial cover fraction which can also be the reason for inconsistency between the surface-roughness model (Johnson, 2014) results and experimental study for Gravel30 and Net4 in Figure 11. Whereas, in the case of the macro-roughness model (Inoue et al., 2014), the critical shear stress takes 525 into account the value of total hydraulic roughness, which depends on cover fraction, alluvial hydraulic roughness and bedrock hydraulic roughness. Hence, even when * is smaller than * , the bedrock roughness tends to affect the cover fraction. The macro-roughness model (Inoue et al., 2014) is more capable of dealing with clast-rough surfaces. whereas it is as low as ~3 for other runs.
In Figure 11, in Gravel5 and Net2 series with relatively smooth beds, rapid alluviation occurred because the transport capacity 535 over bedrock is larger than that over alluvial bed . The reverse-line slopes produced by macro-roughness and surfaceroughness models depict similar hysteresis relationship between alluvial cover and sediment supply. The shaded portion shows that as ⁄ increases, the cover does not increase unless it reaches a threshold ( ⁄ > 1 , i.e. sediment supply rate is higher than transport capacity over fully exposed bed), after which the cover increases abruptly, showing rapid alluviation.
The shaded portion, however, is unstable between = 0 and = 1, i.e. it shows the hysteresis of rapid alluviation and rapid 540 entrainment. If becomes smaller than , will decrease until = 0 (rapid entrainment). For the bed to become alluviated again, must reach a condition where ⁄ > 1 , in which case rapid alluviation will happen again. This phenomenon has also been observed in sufficiently steep channels, for slopes greater than 0.015 by Chatanantavet and Parker (2008). Hodge and Hoey (2016b) also suggested a similar relationship between sediment cover and sediment supply. However, our study shows that rapid alluviation occurs irrespective of the slope steepness, if roughness of the bed is less than the 545 roughness of supplied gravel, i.e. when relative roughness of the bedrock bed is less than 2.
In a channel without bars and with a relatively regular roughness distribution (i.e., a channel close to a one-dimensional flow field), the macro-roughness model (Inoue et al., 2014) is the most suitable because it can predict alluvial cover ratio without adjusting the parameters. When the observation of hydraulic roughness is difficult, it is useful to obtain the hydraulic roughness from the topographical roughness like the surface roughness model (Johnson, 2014). However, accurate prediction of hydraulic 550 roughness should not only take into account the bedrock topographic roughness but also the arrangement of bed unevenness.
For example, in Figure 2, the topographic roughness of Gravel50 is higher than that of Gravel30, but hydraulic roughness of Gravel50 is lower than that of Gravel30. Ferguson et al. (2019) argued that the standard deviation of exposed bed is an effective way of roughness estimation, however, their finding is for a relatively smooth bedrock. Also, in order to deploy models on field-scale, they must take into account bank-roughness and its effects on shear stress and other hydraulic parameters (Ferguson 555 et al., 2019). Prediction of hydraulic roughness from topographic roughness requires further work.
Another solution is to use the probabilistic model (Turowski and Hodge, 2017). The probabilistic model proposed by Turowski and Hodge (2017) could reproduce experimental results but the model needed optimisation of and 0 * to minimize the RMSD. Small means that the deposition probability gradually decreases with increasing alluvial cover, in contrast, large means that the deposition probability rapidly approaches zero with increasing alluvial cover. The model, however, does not 560 emulate the hysteresis for clast-smooth beds. In this case, we may need to use different probability functions for entrainment and deposition. In addition, 0 * calculated physically from Equation (14) is 0.04 (alluvial bed) to 0.06 (smoothest bedrock, i.e., Gravel5) in this experiment, which is significantly different from the adjusted 0 * . Because the model does not include the effects of bed roughness yet, further alterations to take into account the effect of the probability of grain entrainment and deposition can greatly extend the applicability of the model. How to link and 0 * with topographic roughness is a future 565 issue.

The Effects of Bar formation on Alluvial Cover
For investigating the influence of bed roughness and bar formation on the alluvial cover, we also compared the experimental results of Chatanantavet and Parker (2008) with the model results of the physically based models including interaction between 570 roughness and alluvial cover (i.e., Inoue et al., 2014;Johnson, 2014). Chatanantavet and Parker (2008) conducted experiments in a metallic straight channel with three different types of bedrock bed surfaces namely Longitudinal Grooves (LG), Random Abrasion Type 1 (RA1) and Random Abrasion Type 2 (RA2), where RA1 is smoother than RA2. They performed various cases for each type with varying slope range of 0.0115 -0.03. They also varied the sediment supply rate and grain size (2 mm and 7 mm). The major difference between their experiment and our experiments is the width -depth ratio. The width -depth 575 ratios of their experiments were 11 -30 and thus allowed for the formation of alternate bars. In contrast, the width -depth ratios of our experiments were 6.1 -8.3, as a result, alternate bars usually cannot develop. Although we can see alternate alluvial patches in Figure 5, their thickness was less than 1 cm, and the patches did not progress to alternate bars with large wave height. Figure 13 shows the comparison among the two models and Chatanantavet and Parker's experiment (2008). The experimental 580 conditions are taken from Table 1 of Chatanantavet and Parker (2008). Because the two models do not include the 2-D effects caused by bar formation, we adjusted in the macro-roughness model in addition to in the surface model. In the case of the surface-roughness model, # = 4 is used, the bedrock surface roughness required for calculations is taken as mentioned in Table 1 Johnson (2014), rbr is adjusted to minimize RMSD of cover between experiments and the model. In the case of the macro-roughness model by Inoue et al. (2014), is adjusted to minimize RMSD of cover. The two models can accurately 585 predict the cover fraction and rapid alluviation for the experimental study conducted by Chatanantavet and Parker (2008 (Table 3). 590 In Table 3, when we compare the observed with the adjusted in the roughness models proposed by Inoue et al. (2014) and Johnson (2014), the adjusted ksb strongly depends on observed in our experiments without alternate bars ( Figure 14a).
Whereas, the adjusted is not dependent on the observed in case of experiments with alternate bars conducted by Chatanantavet and Parker (2008) (Figure 14b). This suggests that bedrock roughness has a smaller effect on the alluvial cover in case of mixed alluvialbedrock rivers with alternate bars. In such rivers, the bed slope may affect the alluvial cover fraction 595 ( Figure 11c) because bar formation process depends on the slope as well as the width-depth ratio (e.g., Kuroki and Kishi, 1984).
The roughness models are adjusted to produce the experimental results with alternate bars by fine-tuning and values which must be determined by trial and error method. While this method can be applicable to laboratory-scale experiments, the model calibration is unfeasible for a large-scale channel or natural rivers. In general, the formation of alternate bars is barely 600 reproduced with a one-dimensional model as introduced in this study. In the future, research to incorporate the effects of bars into a one-dimensional model, or analysis using a two-dimensional planar model (e.g., Nelson and Seminara, 2012;Inoue et al., 2016Inoue et al., , 2017 is expected.    Table  2). Note that there is no adjustment of in macro-roughness model.   Chatanantavet and Parker (2008); RA1 is Random Abrasion type 1, RA2 is Random Abrasion type 2 and LG is Longitudinal grooves, respectively. The for the surface roughness model and the for macro roughness model are adjusted to minimize RMSD of the alluvial cover (see Table 3). Figure 13a represents runs 2-C1 to 2-C4, Figure 13b represents runs 2-E1 to 2-E3, Figure 13c represents runs 3-A1 to 3-A5, Figure 13d represents runs 3-B1 to 3-B5, Figure 13e represents runs 1-B1 to 1-B4 (Chatanantavet and Parker 2008, Table 1

Summary
Here we provide a review of models and studies focused at discovering the interaction between alluvial cover and bed 645 roughness. For evaluating the previous models, we conducted laboratory-scale experiments with multiple runs of varying bed roughness and sediment supply. The experimental results show that the change in alluvial cover with the sediment supply rate is controlled by bedrock roughness to a great extent. When the bedrock hydraulic roughness is higher than the hydraulic roughness of the alluvial bed (i.e., clast-rough bedrock), the alluvial cover increases proportionally with the increase in sediment supply and then reaches an equilibrium state. However, in cases where bedrock roughness is lower than the roughness 650 of the alluvial bed (i.e., clast-smooth bedrock), the deposition is insignificant unless sediment supply exceeds the transport capacity of the bedrock bed. When sediment supply exceeds transport capacity, the bed is abruptly covered by sediment and quickly reaches a completely alleviated bed.
We have also implemented the previous models for alluvial cover, i.e., the linear model proposed by Sklar and Dietrich (2004), the exponential model by Turowski et al. (2007), the macro-roughness model by Inoue et al. (2014), the surface-roughness 655 model by Johnson (2014) and the probabilistic model by Turowski and Hodge (2017) to predict the experimental results. The linear model and exponential model are inefficient for cases with a clast-smooth bedrock specifically, they cannot predict the rapid-alluviation. The macro-roughness model (Inoue et al. 2014) and surface-roughness model (Johnson, 2014) can predict the rapid-alluviation and hysteresis for clast-smooth bedrock as well as the proportionate increase in alluvial cover for clastrough bedrock. Although the macro-roughness model (Inoue et al. 2014) was able to reproduce the observed alluvial cover 660 ratio without adjusting the parameters, the surface roughness model needs parameter adjustments. The probabilistic model by Turowski and Hodge (2017) also needs parameter adjustments to make it sensitive to rapid alluviation in clast-smooth bed, however, it does not reproduce the hysteresis. Connecting model parameters with roughness parameters is an exciting challenge in the future.
We also tested the macro-roughness model (Inoue et al. 2014) and surface-roughness model (Johnson, 2014) for their capability 665 to predict the experimental results observed by Chatanantavet and Parker (2008), in which the bedrock surface has alluvial alternate bar formations. Both models required significant parameter adjustments to reproduce the alluvial cover fraction. The two models do not include the 2-D effects caused by variable alluvial deposition and formation of bars on bedrock. Although models that extended the roughness model into two-dimensional planes (e.g., Nelson and Seminara, 2012;Inoue et al., 2016) will be able to capture bar formation in a bedrock river, these models require long calculation time. Building a simpler model 670 that can predict alluvial cover fraction with bar formation represents another exciting challenge in the future contributing to a better understanding of long-time evolution of natural bedrock channel.
Author Contribution: Both authors contributed equally to the manuscript.

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Acknowledgements: Data used in this publication is available in this paper itself or available in the papers referred (Chatanantavet andParker, 2008 andJohnson 2014). In proceeding with this research, we received valuable comments from Professor Yasuyuki Shimizu, Professor Norihiro Izumi, and Professor Gary Parker. We would like to express our gratitude here. The authors would also like to thank Jens M Turowski, Rebecca Hodge and an anonymous referee for their constructive feedback that helped improve the earlier version of this paper. 680