RESEARCH ARTICLE
10.1029/2021WR030952
Key Points:• N ew model for the morphodynamics
of a bedrock channel due to saltationabrasion and wet-dry weathering• The model is able to replicate theformation of a deep inner channel atchannel center and flat bedrock benchon the channel margins• Weathering and abrasion indirectlyinteract with each other throughchanges in channel morphology
Correspondence to:T. Inoue,
inouetakuya0317@gmail.com
Citation:

Sumner, T., Mishra, J., Inoue, T., &Shimizu, Y. (2022). Numerical studyof the interaction between wet-dry
weathering and bedload induced
abrasion. Water Resources Research,
58, e2021WR030952. https://doi.
org/10.1029/2021WR030952
Received 30 JUL 2021
Accepted 2 MAR 2022
Author Contributions:Conceptualization: Takuya Inoue
Investigation: Tamaki Sumner, Jagriti
Mishra, Takuya Inoue
Methodology: Tamaki Sumner, Takuya
Inoue
Software: Tamaki Sumner, Takuya
Inoue, Yasuyuki Shimizu
Supervision: Yasuyuki Shimizu
Visualization: Takuya Inoue
Writing – original draft: Tamaki
Sumner, Jagriti Mishra
Writing – review & editing: Jagriti
Mishra, Takuya Inoue© 2022. American Geophysical Union.
All Rights Reserved.

SUMNER ET AL.

Numerical Study of the Interaction Between Wet-Dry
Weathering and Bedload Induced Abrasion
Tamaki Sumner1, Jagriti Mishra2, Takuya Inoue3, and Yasuyuki Shimizu4

1
Suiko Research Co. Ltd., Sapporo, Japan, 2Jawaharlal Nehru University, New Delhi, India, 3Hiroshima University, HigashiHiroshima, Japan, 4Hokkaido University, Sapporo, Japan

Abstract Erosion and weathering have always been considered as key factors regulating landscapemorphodynamics. In this study, we implemented a numerical model that incorporates the effect of abrasion bysaltation as well as the effect of slaking by wet-dry weathering. Our numerical results indicate abrasion andweathering acting together on bedrock beds, that is, abrasion leads to a deep inner channel, and weathering
leads to flat benches on the channel margins. We also inquired how changes in the water discharge, alluvial
cover, bedrock strength and weathering depth define the abrasion and weathering rates. We found that changes
in water discharge can control the weathering rate and depth and thereby the channel shape. Alluvial cover acts
as tools or cover to determine the abrasion rates. Our results show that bedrock strength minimally changes thechannel cross-section while weathering depth controls the valley depth. Since weathering and abrasion occur
at different heights in the channel cross-section, there is little opportunity for weathered bedrock to be abraded.
However, weathering and abrasion interact indirectly through changes in channel morphology. The internal
channel incision lowers the elevation of wet-dry weathering zone, and the formation of the bedrock bench
reduces the bedload transport rate, which drives saltation-abrasion.
1. Introduction
Bedrock erosion determines the shape of the landscape and its response to various tectonic and climatic activities and perturbations (Berlin & Anderson, 2007; Crosby et al., 2007; Wobus et al., 2006; Wohl & David, 2008;
Zaprowski et al., 2001). Bedrock erosion can be driven by various processes such as abrasion caused bysuspended and bedload sediment impacts, plucking, weathering and cavitation (Hancock et al., 1998; Sklar &Dietrich, 2004; Whipple, 2004; Whipple et al., 2000). Hall et al. (2012) pointed out the inconsistency in definitions and distinction between erosion and weathering. Ollier (1984) suggested abrasion as a constituent elementof physical weathering while Easterbook (1999) advised the removal of rocks by falling rocks as mechanicalweathering. However, Montgomery (2004) treated weathering and erosion as different processes and Murphy
et al. (2018) suggested weathering as a process that weakens the surface layer of the channel whereas erosion is
the process by which the surface layer of the channel is removed. In this study, we follow the definitions given
by Murphy et al. (2018).
Besides the inconsistency in definition, researchers have also debated the role of rainfall (or precipitation) in
dictating the erosion rates in bedrock channels. In a study conducted by Burbank et al. (2003) in the Greater
Himalayas in Nepal (∼20 km length with rainfall variation >5 times), they found that erosion rates at geologicaltimescales (>10 5 years) did not vary at detectable rates. On the contrary, Reiners et al. (2003) observed that longterm erosion rates (>10 6–10 7 years) in the Cascades mountains in Washington chases the mean annual precipitation rate. These contrasting studies imply that precipitation and landform morphology are related in numerous
distinct ways producing diverse landform topography.
Several previous studies have advocated weathering as a key factor responsible for shaping the channel morphology (Han et al., 2014; Hancock et al., 2011; Menking et al., 2013; Montgomery, 2004; Mudd & Furbish, 2004;
Turowski et al., 2009; Whipple et al., 2000). Rempe and Dietrich (2018) conducted field-scale studies andsuggested that deep-weathered rocks hold moisture that can be used by plants during the drought season. Fieldscale studies conducted by Menking et al. (2013) point toward weak weathered rocks as a central cause of incisionin the Kohala Peninsula in Big-Island, Hawaii. Schanz and Montgomery (2016) and Johnson and Finnegan (2015)
have also suggested weathering (or slaking) as one of the most important factors contributing to lateral erosion
and width adjustment by the bedrock channel. The numerical modeling performed by Hancock et al. (2011) also

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Figure 1. Kushiro river-Hokkaido—Japan (N43.4691, E144.4735). (a) Cross-sectional profiles observed in 2002 and 2016.
Note that the bottom of the inner channel was not measured in 2016 because the flow velocity was too fast. Hence, the innerchannel may be deeper. (b) Deep inner channel with flat bedrock bench on the edges.

showed that wet-dry weathering increases the erodibility of the channel by flow-induced bedrock erosion andthat the bedrock bed in the channel center erodes slower than the channel margins. In areas prone to flooding, theexposed bedrock bed becomes increasingly susceptible to weathering and therefore prone to erosion during highflows (Stock et al., 2005) or, floods may not “cut” bedrock but remove clasts already weakened by weathering(Retallack & Roering, 2012). Langston and Temme (2019) conducted a field-scale study on the Colorado Front
range—Rocky Mountains—USA, post a large flood. They ascribed slaking of weak weathered shale responsible
for lateral erosion in the channel.
In recent years, several studies on the relationship between wet-dry weathering and bedload impact abrasion
have been conducted. Small et al. (2015) took rock samples from the riverbed and margins and showed that thesaltation-abrasion rate of margin samples is higher than that of bed samples under the same sediment transportconditions. Inoue, Parker, & Stark (2017) studied the effect of wet-dry weathering on samples collected from 8
different rivers in Japan and suggested an exponential decrease in the tensile strength of the bedrock concerning
each wet-dry cycle. They advocated that weathering increases the erodibility of the channel by saltation-abrasion.
However, as sediment mainly travels on the riverbed, the abrasion rate of riverbanks might not increase withweathering and weakening of the riverbanks. To understand the effect of weathering on abrasion rates in bedrock
channels, we need to understand the cross-sectional distribution of weathering degree, bedload transport rate andalluvial cover fraction.
Figure 1 shows cross-sectional profiles and a photograph of the Kushiro River in Hokkaido-Japan. A deep innerchannel can be noted near the center of the channel, and bedrock benches with flat and smooth surfaces canbe noticed near the riverbank (referred to as channel-bed margins in this study). To the best of our knowledge,
hitherto, there are no examples available of saltation-abrasion leading to flat and smooth bedrock bed. In fact,
Aydin et al. (2011) conducted laboratory-scale experiments on granites and found water flow and abrasion as
the leading cause of roughness in samples. The experimental results of Wohl and Ikeda (1997) and Inoue andNelson (2020) also showed longitudinal grooves form on the smooth bedrock surface along sediment transport
pathways. Therefore, it is likely that the bedrock benches have been eroded by the flood removal of well-weathered clasts, rather than been abraded by bedload particles. Although similar assumption has been reported in thefield observations at Pescadero Creek, California, USA (Johnson & Finnegan, 2015), the interaction between
wet-dry weathering and saltation-abrasion and the resulting topography are not known in detail. In this study, we
implement a numerical model and try to answer the following questions: (a) How is bedrock channel topography
caused by saltation-abrasion different from the one caused by wet-dry weathering? (b) How can parameters like
sediment supply rate, bedrock strength, fluctuation of water discharge affect the weathering rate, abrasion rate
and bedrock channel topography?

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2. Numerical Model
The equations governing the flow field are the two-dimensional depth-averaged equations for mass and momentum conservation. Incisional/alluvial morphodynamics is calculated by employing the framework proposed byInoue et al. (2014, 2016). The differences between our model and the model proposed by Inoue et al. (2014, 2016)pertain to the wet-dry weathering model presented in Section 2.2. Here, we describe the equations for bedrockabrasion and weathering that are highly relevant to the discussion of this study. Other basic equations aredescribed in Appendix A.
2.1. Bedrock Abrasion
The bedrock erosion rate is estimated using a 2D version of the model proposed by Chatanantavet andParker (2009);√
𝜕𝜕𝜕𝜕𝑏𝑏22= −𝛽𝛽 𝑞𝑞bx
+ 𝑞𝑞by(1 − 𝑝𝑝𝑐𝑐 )
(1)
𝜕𝜕𝜕𝜕

𝐴𝐴

where 𝐴𝐴𝑏𝑏 represents the bedrock elevation,
𝐴𝐴
𝐴𝐴
𝐴𝐴 is the abrasion coefficient,𝐴𝐴
𝐴𝐴𝑐𝑐 is the areal fraction of alluvial cover,in the𝐴𝐴𝐴𝐴 and y-direction, respectively. Inoue, Parker, and𝐴𝐴𝐴𝐴bx and 𝐴𝐴by are the bedload transport rates per unit width 𝐴𝐴
Stark (2017) showed that the abrasion coefficient for bedrock𝐴𝐴
𝐴𝐴 can be estimated as follows:−2
𝛽𝛽 = 𝐶𝐶(𝑑𝑑∕𝑘𝑘𝑠𝑠 )0.5 𝜎𝜎TN(2)where 𝐴𝐴 =𝐴𝐴10−4, 𝐴𝐴 is the grain𝐴𝐴size, 𝐴𝐴𝑠𝑠 is the hydraulic roughness height
𝐴𝐴 and 𝐴𝐴𝑇𝑇 𝑇𝑇 is the rock tensile strength.
𝐴𝐴
The areal fraction of alluvial 𝐴𝐴
cover 𝐴𝐴𝑐𝑐 is calculated as a function of the alluvial layer thickness blanketing the
bedrock (Inoue et al., 2014; Mishra & Inoue, 2020; Zhang et al., 2015). This model was introduced by Zhang
et al. (2015) and is known as the Macro-Roughness Saltation Abrasion Alluviation model, (MRSAA), where𝐴𝐴
𝐴𝐴𝑐𝑐
is given as:𝜂𝜂𝑎𝑎⎧ 𝜂𝜂𝑎𝑎𝑓𝑓 𝑓𝑓𝑓𝑓 0 ≤≤1⎪ 𝐿𝐿mr𝐿𝐿
mr𝑝𝑝𝑐𝑐 = ⎨
(3)
𝜂𝜂𝑎𝑎
⎪ 1
>1
𝑓𝑓 𝑓𝑓𝑓𝑓⎩
𝐿𝐿mrwhere 𝐴𝐴𝑎𝑎 is the thickness of the alluvial layer𝐴𝐴
𝐴𝐴 and 𝐴𝐴mr is the bedrock macro-roughness (e.g., the standard deviationof bedrock topographic height).
Change in bed surface elevation is obtained as follows:𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑎𝑎 𝜕𝜕𝜕𝜕𝑏𝑏
=
+
(4)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

where 𝐴𝐴 is the elevation of the top of the river bed, and subscripts a and b represent alluvium and bedrock, respec𝐴𝐴
tively. We assume that the debris of bedrock abrasion is small enough to be treated as wash load, and do notconsider an increase in sediment supply caused by bedrock bed and bank abrasion in our model.
The unevenness of the riverbed (Li et al., 2021) and the flow meandering via bar formation (Mishra et al., 2018)
cause sediments to collide with the sidewalls, leading to lateral erosion of bedrock channel (Inoue et al., 2021).
Although the increase in channel width due to lateral erosion may reduce the effect of sidewall resistance, lateralerosion is not considered in the model proposed in this study due to the complexity of modeling, for example, how
to handle overhanging walls and how to move the inner channel sidewall grid in the horizontal plane.

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Figure 2. Conceptual image of elevation change due to wet-dry weathering. The initial bedrock strength𝐴𝐴
𝐴𝐴T0 decreases due
to wet-dry weathering. When the bedrock strength
𝐴𝐴 after 𝐴𝐴 wet-dry𝐴𝐴cycles 𝐴𝐴TN becomes less than the critical bedrock strength
𝐴𝐴TC, the bedrock bed turns into fine clasts. The clasts are removed as wash load, forming a new bed represented by the redlines (the shaded red area represents
𝐴𝐴Δ𝜂𝜂𝑏𝑏𝑏𝑏). The bedrock strength is initialized
𝐴𝐴 to 𝐴𝐴T0
𝐴𝐴 from 𝐴𝐴TN .

2.2. Wet-Dry Weathering
In this study, we implement a model that emulates the physics and approach proposed by Inoue, Parker, andStark (2017) where the rock strength is associated with the number of wet-dry cycles. The rock strength decreases
exponentially with the number of wet-dry cycles.
𝜎𝜎TN= exp (−𝛼𝛼dw 𝑁𝑁)
(5)
𝜎𝜎T0

where
𝐴𝐴
𝐴𝐴TN is bedrock strength𝐴𝐴after 𝐴𝐴 wet-dry cycles,𝐴𝐴
𝐴𝐴T0 is the initial bedrock strength,
𝐴𝐴
𝐴𝐴dw is the wet-dry
weathering coefficient calculated
𝐴𝐴 as 𝐴𝐴dw = 𝐶𝐶dw (𝑊𝑊a0 ∕𝜎𝜎T0 ) where𝐴𝐴
𝐴𝐴dw is a constant (0.21)
𝐴𝐴 and 𝐴𝐴a0 is the initialnormalized rate of water absorption,
𝐴𝐴 and 𝐴𝐴 is the number of wet-dry cycles. By substituting the calculated𝐴𝐴
𝐴𝐴TN
into Equation 2, the increase in erodibility due to weathering is expressed.
Figure 2 shows a sample cross-sectional grid used for the numerical calculations. Grid spacing is 0.5 m in thelateral direction and 1 m in the longitudinal direction. A section of the grid is bedrock bed where the water level
increases during high flow and decreases during low flow season, making this area prone to wet-dry weathering
cycles. The bedrock bed below the low-water level (water level during the low-flow season) always remains
submerged. As the weathering progresses, the bedrock cracks and eventually turns into fine clasts. Then the fineclasts are removed by water flow. In our model, bedrock bed changes to clasts when the bedrock strength
𝐴𝐴
𝐴𝐴TN ,
decreases below the critical bedrock strength
𝐴𝐴
𝐴𝐴TC. When the bedrock strength
𝐴𝐴
𝐴𝐴TN reaches the critical bedrock
strength
𝐴𝐴
𝐴𝐴TC, from Equation 5, the critical wet-dry𝐴𝐴cycle 𝐴𝐴𝑐𝑐 is indicated as follows:
(
)
𝜎𝜎TC
1
log
𝑁𝑁𝑐𝑐 = −
(6)
𝛼𝛼dw
𝜎𝜎T0

Although the size of the clasts decreases with an increasing wet-dry cycle, for simplicity, we assume that the
bedrock changes to very fine clasts at once and the clasts are small enough to be treated as wash load. When waterflows over the fine-clasts area, the riverbed is lowered. The change in the riverbed height is determined by theweathering depth removal
𝐴𝐴Δ𝜂𝜂bw. The bedrock strength of the freshly appearing bedrock is set to the initial bedrock
strength𝐴𝐴
𝐴𝐴𝑇𝑇 0. The weathering depth removal
𝐴𝐴Δ𝜂𝜂bw is only calculated in the vertical direction.

3. Numerical Experiments
3.1. Numerical Conditions
We have used a v-shaped 10 (longitudinal direction)𝐴𝐴×46 (transverse direction) meters grid as shown in Figure 2.
The longitudinal bed slope is 0.003 (average value near the site shown in Figure 1) and the transverse bed slope
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Figure 3. (a) Two-stage hydrograph used in Run 0. Low flow is an average discharge (15 m 3/s), and high flow is the annual
highest discharge in the Kushiro River from 1969 to 2018. (b) Cyclic hydrograph used in Runs 1–19.

is 0.2 (average value of the left and right slopes in the 2002 cross-sectional profile in Figure 1). We have usedperiodic boundary conditions for flow discharge and sediment transport rate, that is, the outflow is circulatedback into the simulations grid as inflow. The calculation channel is the same as an infinitely long straight channel. Since we use a completely straight channel with periodic boundary conditions, the numerical results for the
cross-sectional profile are the same along the longitudinal direction.
The flow discharge at the study site shown in Figure 1 was observed only in 2004. The flow discharge observedin 2004 was 67% of the flow discharge observed at the Shibecha discharge gauging station 25 km downstreamof the Kushiro River. Therefore, we use 67% of the flow discharge observed at the Shibecha station from 1969
to 2018 (50 years) as the flow discharge at the study site. In Run 0, we use a two-stage hydrograph with an average discharge (15 m 3/s) and an annual highest discharge (Figure 3a). The 50-year average of the annual highest
discharge is 136 m 3/s. The duration of flood is set as 15 hr, including 5 hr rising limbs, 5 hr peak and 5 hr recession limb. Since the bedrock is not eroded during low flow, the period of a low flow is shortened to 5 hr. It is
crucial to set low-flow periods to identify repeatedly wet-dry zones.

𝐴𝐴

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The bedrock of the Kushiro River is composed of mudstone. We collected the bedrock samples of the Kushiro
River and investigated the bedrock strength, the water absorption rate and the changes after wet-dry weatheringof fresh bedrock using the same method as Inoue, Parker, and Stark (2017). The tensile strength of fresh bedrock
𝐴𝐴T0 was 0.13 MPa and the water absorption
𝐴𝐴 rate 𝐴𝐴a0 was 0.5. Also, the fresh bedrock sample of the Kushiro

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Table 1
Numerical Conditions for Sensitivity Analysis
InitialWeathering
Initial𝐴𝐴
alluvial
depth
bedrock
thicknessremovalstrength(cm)
𝐴𝐴 (cm)𝐴𝐴𝑏𝑏𝑏𝑏 𝐴𝐴(Mpa)𝐴𝐴𝑇𝑇 0

Case

High flow discharge (m 3/s)
Qmax

Run 0

Annual highest discharge

30

5

0.13

Run 1

100

30

5

0.13

Run 2

150

30

5

0.13

Run 3

200

30

5

0.13

Run 4

250

30

5

0.13

Run 5

300

30

5

0.13

Run 6

150

0

5

0.13

Run 7

150

10

5

0.13

Run 8

150

20

5

0.13

Run 9

150

40

5

0.13

Run 10

150

50

5

0.13

Run 11

150

30

1

0.13

Run 12

150

30

9

0.13

Run 13

150

30

5

0.17

Run 14

150

30

5

0.20

Run 15

150

30

5

0.30

Run 16a

150

30

5

0.40

Run 17

a

150

30

5

1.00

Run 18a

150

30

4

0.40

Run 19

150

30

2.3

1.00

10.1029/2021WR030952

River turned into fine clasts and mud after one cycle of being completely
dry and then re-soaking in water for several hours. Therefore, we set thecritical bedrock strength
𝐴𝐴
𝐴𝐴TC as 0.06 MPa, which is the value obtained when
𝐴𝐴 and 𝐴𝐴 = 1 are substituted into Equation 5. The weath𝐴𝐴T0 =𝐴𝐴0.13, 𝐴𝐴a0 = 0.5
ering depth removal
𝐴𝐴Δ𝜂𝜂bw is set at 5 cm, same as Hancock et al. (2011).
In the experiments conducted by Inoue, Parker, and Stark (2017) using 4
natural bedrock samples and 1 concrete sample (Figure 11 in Inoue, Parker, &Stark, 2017), they showed that the abrasion coefficient𝐴𝐴
𝐴𝐴 is roughly 0.0001.
In this 𝐴𝐴
study, 𝐴𝐴 is set as 0.00004 excluding the results of the concrete sample.
Since the bottom of the inner channel could not be observed, we assume themacro-roughness of the bedrock surface to be 30 cm and the initial alluvial
thickness at the channel center to be 30 cm. The particle𝐴𝐴size (𝐴𝐴 ) is taken as
30 mm, which is the average grain size observed by the Hokkaido Regional
Development Bureau near the study site. In general, the hydraulic roughness
height of an alluvial bed is taken𝐴𝐴to be 1 ∼ 4𝑑𝑑 (Inoue et al., 2014). Here, thealluvial hydraulic roughness
set to 2.5𝑑𝑑 (median𝐴𝐴
𝐴𝐴sa is 𝐴𝐴
𝐴𝐴 of𝐴𝐴1𝑑𝑑 to 4𝑑𝑑 ), that is,
75 mm. To simplify our simulations here, the hydraulic roughness height for
bedrock𝐴𝐴
𝐴𝐴sb is taken to be equal to that of an alluvial
𝐴𝐴 bed 𝐴𝐴sa .
To understand the effect of different input parameters on the channel topography, we performed 19 sets of numerical experiments (see Table 1). In Runs
1–5, we used cyclic hydrographs and changed the high flow discharge asshown in Figure 3b. Other conditions are the same as Run 0. In Run 2 andRuns 6–10, we changed the initial alluvial thickness of the channel center toanalyze the influence of sediment transport rate and alluvial cover. In Run
2 and Runs 11–19, we changed the weathering depth removal
𝐴𝐴Δ𝜂𝜂bw and theinitial bedrock strength𝐴𝐴
𝐴𝐴T0 to investigate the effects of lithology.

In our model, the volume of bedrock erosion decreases with increasing theinitial bedrock strength as the erosion rate depends on the negative squarea
The total time of the two-stage hydrograph is 7,200 hr in Runs 16 and 18,
of bedrock strength (Equation 2). Also, the volume of weathering removal
26,800 hr in Runs 17 and 19. The time of the two-stage hydrograph in otherdecreases with increasing the initial bedrock strength because the criticalruns is 1000 hr.
wet-dry cycle required for bedrock removal𝐴𝐴
𝐴𝐴𝑐𝑐 increases (Equation 6). For
Note. Bold characters indicate changes from Run 2.
example,𝐴𝐴
𝐴𝐴𝑐𝑐 is roughly 1 𝐴𝐴when 𝐴𝐴T0 = 0.13 MPa (Run𝐴𝐴 2), 𝐴𝐴𝑐𝑐 is roughly 7.2
when 𝐴𝐴T0 = 0.4 (Run 16 and 18),
when 𝐴𝐴T0 = 1.0
𝐴𝐴𝐴𝐴 and 𝐴𝐴𝑐𝑐 is roughly 26.8 𝐴𝐴
Mpa (Runs 17 and 19). For this reason, the channel shape in Runs
𝐴𝐴 16 ∼ 19with harder bedrocks barely evolved in 1,000 hr (50 wet-dry cycles). To keep the amount of weathering removal
similar to that of Run 2, we increased the total time of the two-stage hydrograph for Runs 16, 18 and Runs 17, 19
to 7,200 hr and 26,800 hr respectively.
a𝐴𝐴

If all the conditions other than the bedrock strength and weathering depth removal are the same, the abrasion ratedepends
𝐴𝐴 on 𝐴𝐴𝑇𝑇−2 and the weathering rate depends
𝐴𝐴 on Δ𝜂𝜂bw ∕𝑁𝑁𝑐𝑐 in our system and the weathering/abrasion ratio canbe approximately expressed
𝐴𝐴 as Δ𝜂𝜂bw ∕𝑁𝑁𝑐𝑐 𝜎𝜎𝑇𝑇−2. We set up Runs 18 and 19 to compare the channel morphology whenΔ𝜂𝜂bw ∕𝑁𝑁𝑐𝑐 𝜎𝜎𝑇𝑇−2 is the same and the bedrock strength is different.
3.2. Numerical Results
In Run 0, which uses a two-stage hydrograph with the annual highest discharge and average flow discharge, thecenter of the cross-sectional profile suffers vertical erosion due to bedload induced abrasion, forming a deep innerchannel (Figures 4a and 4b). Although 60%–70% of the inner channel bed is covered by alluvium, the channel
bed margins are completely exposed, that is, no bedload transport (Figure 4c). This suggests that the channel bed
margins do not erode due to bedload collisions. In contrast, as the inner channel always remains submerged, it is
not vulnerable to wet-dry weathering, only the channel-bed margins are weathered and removed by water flow
(Figures 4b and 4d). The results of Run 0 clearly show that flat bedrock benches and a deep inner channel are

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Figure 4. Numerical results of two-stage hydrograph with annual highest discharge and average discharge (Run 0). Crosssectional profiles of (a) alluvial bed, bedrock and water level, (b) erosion depth of weathering removal and abrasion, (c)alluvial cover fraction at 1000 hr and (d) number of wet-dry cycles at 1000 hr.

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formed as weathering and abrasion occur in different zones. Since almost no sediment deposits were observed on
the Kushiro River benches, it is unlikely that the benches were formed by abrasion by bedload.
Note that, Run 0 is performed to show that the benches are formed by weathering, not to reproduce the detailed
bed shape of the Kushiro River. However, it is important to show the detailed differences and the reasons forthem, so we describe them here. According to the calculation results, the width of the left and right benches
is 6.0 m (total 12 m). The survey data of the Kushiro River shown in Figure 1 shows that the width of the left
bench is 2.4 m and the width of the right bench is 9.5 m (total 11.9 m). The width of the inner channel is 5 m insimulation and 11.2 m in observation. The depth of the inner channel is, on average, 3 m in simulation and 1.3 m
in observation. That is, the eroded area is almost the same, but the aspect ratio is different. The flow discharge
and the sediment supply rate given in the simulation is different from the actual values (temporal variations of theflow discharge and the sediment supply rate have not been observed at the study site), which may have affected
the simulated channel shape. Also, the Kushiro River is slightly meandering, which may have caused lateralerosion in the inner channel. Lateral erosion is not included in the presented study model.
We show the high and low water level at the early (T = 0 and 10 hr) and the last stage (T = 920 and 930 hr) in
Figure 4a. The high flows at 0 and 920 hr are the same, roughly 150 m 3/s. The low flows at 10 and 930 hr are thesame, 15 m 3/s. Since the water level lowers as the inner channel erodes, weathering zones and bedrock benches
also proceed vertically over time.
In Runs 1–5 with different high flows (Qmax), as the high flow increases, the inner channel becomes deeper(Figure 5a) and the abrasion area (i.e., abrasion volume per unit longitudinal length) increases (Figure 5b). Thisis because a higher discharge rate increases the bedload transport rate, that is, the tools for abrasion (Figure 5c). In
addition, the bedrock benches become wider because the wet-dry zones become wider as the high flow increases(Figure 5a). Thus, the weathering area (i.e., weathering removal volume per unit longitudinal length) increases
with increasing high flow (Figure 5d). Although higher discharge increases both abrasion and weathering volume,the increased rate of abrasion is larger than that of weathering (Figure 5e).
In Runs 2, 6, 7, 8, 9 and 10 with different initial alluvial thickness, the change in cross-section is quite complicated (Figure 6). When the alluvial thickness is zero or 0.5 m, the inner channel does not develop because of notools and no exposure (Figure 7a). As the initial alluvial thickness increases from 0.1 to 0.4 m (Runs 2, 7, 8 and 9),
the depth of the inner channel decreases (Figure 6) because the cover effect increases (Figure 7a). Since the initialtransverse shape is V-shaped, the initial cover effect decreases outward from the channel center and the width of
the inner channel expands in proportion to the increase in the initial alluvial thickness. As a result, the relationshipbetween the abrasion area (i.e., abrasion volume per unit longitudinal length) and the alluvial thickness has a peaksimilar to the experiments performed by Sklar and Dietrich (2001) (Figure 7b).
Interestingly, the relationship between the weathering area (i.e., weathering removal per unit length) and theinitial alluvial thickness shows a similar curve as the abrasion area (Figures 7c and 7a). This mechanism can be
explained by the water surface widths at the high and low flows shown in Figure 7d. When the alluvial thickness
is zero, the elevation of the riverbed bottom will not be lowered because abrasion by sediment does not occur. In
addition, when the alluvial thickness is 0.5 m, the bedrock is fully covered with sediment, so there is no riverbed
degradation due to cover effects. Therefore, in these two cases, V-shape gradually shifts to a flat topography by
wet-dry weathering, that is, the difference between bench elevation and riverbed bottom elevation decreases.
However, the water surface widths at high and low flow eventually become the same, the wet-dry weatheringzone vanishes. When the alluvial thickness is 0.1–0.4 m, the elevation of the wet-dry weathering zone lowers
as the inner channel develops, consequently increasing weathering removal volume. However, when the innerchannel abrasion is too quick as Run 2 (initial alluvial thickness = 0.2 m), the width of high flow decreases.
Thus, the weathering removal area is slightly reduced (Figure 7c). This indicates that the abrasion rate affects theweathering removal rate through changes in channel topography.
We varied the weathering depth removal
𝐴𝐴Δ𝜂𝜂bw in Runs 2, 11 and 12. When the weathering depth removal issmaller in the𝐴𝐴case Δ𝜂𝜂𝑏𝑏𝑏𝑏 = 1 cm, weathering is slow and the vertical incision of the channel is dominant. In
case(Δ𝜂𝜂𝑏𝑏𝑏𝑏 = 9 cm), weathering is faster, however, the incision of the inner channel is slower
𝐴𝐴 the Δ𝜂𝜂𝑏𝑏𝑏𝑏 is higher
𝐴𝐴(Figure 8). Because the lowering of bedrock benches reduced the bed shear stress in the inner channel, the

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Figure 5. Numerical results with different high flows. (a) Cross-sectional profiles of bedrock elevation and wet-dry cycles.
The relationship between high flow and (b) abrasion area, (c) averaged bedload transport rate at 1000 hr, (d) weathering area,(e) abrasion/weathering ratio.

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Figure 6. Cross-sectional profiles of alluvial bed, bedrock and water level with different initial alluvial thickness, (a) 0.0 m,(b) 0.1 m, (c) 0.2 m, (d) 0.3 m, (e) 0.4 m and (f) 0.5 m.

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Figure 7. Numerical results with different initial alluvial thickness. The relationship between initial alluvial thickness and (a)abrasion area, (b) averaged alluvial cover fraction in the inner channel at 1000 hr, (c) weathering area, (d) water surface width
at low flow (990 hr) and high flow (1000 hr).

bedload transport rate (i.e., tools effect) decreases. This indicates that the weathering removal rate affects theabrasion rate through changes in channel topography.
In Runs 2, 13, 14 and 15 (Figure 9), we varied the initial bedrock strength as mentioned in Table 1. When theinitial bedrock strength is high, the inner channel incises slowly because the erodibility calculated from Equation 2 decreases. Also, the bedrock benches lower slowly because the number of wet-dry cycles reaching thecritical bedrock strength calculated from Equation 6 increases.
In Runs 2, 16 and 17 (Figure 10a), we compare the effects of the initial strength on the development of inner channels when the erosion depth due to weathering removal is the same. The stronger the bedrock, the shallower the
inner channel. This is because the reduction rate of saltation abrasion by the increase of bedrock strength calculated from Equation 2 is larger than that of weathering removal calculated from Equation 5. The erosion depth due
to weathering removal are consistent for all three runs around y = ±13 m but differ near the inner channel. If theinner channel develops slowly, the difference in water surface width between high and low flows becomes small,preventing wet-dry weathering near the inner channel. A similar phenomenon is observed in Figures 6e and 6f,
where the inner channels are less developed due to the cover effect.
Generally, as the bedrock strength𝐴𝐴Δ𝜂𝜂𝑏𝑏𝑏𝑏 is likely to decrease. Since𝐴𝐴
𝐴𝐴T0 increases, the weathering depth removal
the relationship between bedrock strength and weathering depth has not been elucidated, we𝐴𝐴 set 𝐴𝐴𝐴𝐴
T0 and Δ𝜂𝜂𝑏𝑏𝑏𝑏 of−2
Runs 2, 18 and 19 sois the same value. Our results show that the weathering/wear ratio for Runs
𝐴𝐴 that Δ𝜂𝜂bw ∕𝑁𝑁𝑐𝑐 𝜎𝜎T02, 18, and 19 are approximately the same. This suggests that the lithology has a limited effect on the cross-sectional shape if the above equation is satisfied (Figures 10b and 10c).

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Figure 8. Numerical results with different weathering depth removal
𝐴𝐴Δ𝜂𝜂bw. (a) Cross-sectional profiles of bedrock elevation.
The relationship between weathering depth removal
𝐴𝐴Δ𝜂𝜂bw and (b) abrasion area, (c) weathering area.

4. Discussion
In our numerical results, the abraded bed incises and exposes bed-margins making it prone to drying in low-flowseason and then wetting again during the high-flow season. As a result, the surficial layer of the channel suffersfrom slaking and weathering and gets eroded easily during the high flow season. As the exposed surficial layer
erodes by weathering removal, the bedrock benches form. The formation of bedrock benches decreases thewater-depth and bedload flux, thereby decreasing the vertical erosion rate (Figure 9). Similar patterns were
found by Schanz and Montgomery (2016) in their field-scale studies conducted on Willapa and Nehalem river
basins, Pacific Northwest, USA. They found that lithologies that are hard to weather form narrower valleys.
They suggested that this wetting-drying cycle followed by slaking events during high flow can be a cause for theformation of fluvial terraces in the study area. Retallack and Roering (2012) also advocated the same mechanism
responsible for the formation of wave-cut benches in Sunset bay and Days creek in the USA. Also, Menking()1∕𝑛𝑛
et al. (2013) proposed a mathematical model
where𝐴𝐴
𝐴𝐴𝑠𝑠 = 𝐼𝐼∕𝑘𝑘𝑚𝑚𝑞𝑞 𝐾𝐾
𝐴𝐴𝐴𝐴𝑠𝑠 is the slope at the unit area (channel
steepness),𝐴𝐴
𝐴𝐴 represents incision𝐴𝐴rate, 𝐴𝐴𝑞𝑞 is precipitation rate coefficient,𝐴𝐴
𝐴𝐴 is rock type constant
𝐴𝐴 and, 𝐴𝐴
𝐴𝐴 and 𝐴𝐴 are
non-universal, positive exponents. According to their model, an increase in precipitation𝐴𝐴rate (𝐴𝐴𝑞𝑞 ) decreases the

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Figure 9. Numerical results with different initial bedrock strength. (a) Cross-sectional profiles of bedrock elevation at
T = 1000 hr. The relationship between initial bedrock strength and (b) abrasion area, (c) weathering area.

channel steepness when incision rates are uniform and all else is held equal. Gilbert (1877) also suggested that
hard to erode lithologies achieve steeper channels to maintain topographic equilibrium. These results suggest thatweathering does not directly influence the abrasion rate, instead, weathering in synergy with other model parameters affect the abrasion rate of bedrock. In addition, our numerical results show that the elevation of bedrockbenches lower with developing the inner channel. Weathering and abrasion interfere with each other throughchanges in channel morphology.
We also performed numerical experiments to see how water discharge can affect the weathering rate or abrasion
rate. As shown in Figure 5, in the case of a high discharge rate the inner channel suffers from higher abrasion
and incises vertically as the bedload transport rate is higher. Whereas in the case of low water discharge, verticalerosion is minimal and channel margins also show a decrease in elevation. Collins et al. (2016) conducted a fieldscale study in the West Fork Teanaway River in the USA. Their study indicated higher incision rates in weathering prone channel section during peak flows, which, according to them can be the mechanism responsible forthe planation of straths. We interpret that this could be the reason for flat bench formations on the Kushiro river
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Figure 10. Cross-sectional profiles of calculated bedrock elevation (a) for different initial bedrock strength and constantweathering depth removal
𝐴𝐴Δ𝜂𝜂bw, (b) for decreasing weathering depth removal
𝐴𝐴Δ𝜂𝜂bw with increasing initial bedrock strength.
(c) Weathering-abrasion ratio.

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bed margins. During low flow, the repeated wet-dry weathering leaves the bed margins prone to erosion. The
exposed bed margins suffer from higher erosion rates during stronger stream flows resulting in bedrock benches.
In addition, we confirmed that Run 0, which uses a hydrograph based on the flow discharge of the Kushiro River,has a rounded riverbed margin shape (Figure 4a) compared to other cases (e.g., Figure 5a). It indicates that thechannel bed topography is affected by the change in the weathering position due to the irregularity of the flow
discharge fluctuation. This suggests that rivers with irregular flow fluctuations form less clear bench channels.
In our numerical experiments, we explore how initial alluvial thickness or sediment deposition on the bedrock bed
can affect the erosion rate when weathering removal and abrasion are acting together. As the initial alluvial cover
increases, the inner channel tries to increase its width and the vertical incision decreases (Figures 6b–6f), except
for the case with no sediment. A higher alluvial cover acts as a cover protecting the bed against abrasion. As aresult, the abrasion volume has a peak against the sediment volume in the channel. Several field-scale and laboratory-scale studies have established this relationship between alluvial cover and incision rates (Cowie et al., 2008;
Finnegan et al., 2007; Gilbert, 1877; Inoue et al., 2014; Johnson & Whipple, 2010; Johnson et al., 2010; Lambet al., 2015; Mishra et al., 2018; Mishra & Inoue, 2020; Sklar & Dietrich, 1998).

𝐴𝐴
𝐴𝐴

In natural rivers, the weathering depth removal
𝐴𝐴Δ𝜂𝜂bw may change depending on the bedrock strength or the
bedrock materials (e.g., mud, silt, sand, gravel and so on). Thus, we tested how the weathering depth removal
When Δ𝜂𝜂bw decreases, weathering/abrasion ratio decreases and theΔ𝜂𝜂bw influences the channel shape (Figure 8). 𝐴𝐴
narrow channel without flat benches forms (Figure 8). We also tested how changes in initial bedrock strength(𝐴𝐴T0) can affect the weathering and erosion rates. When the weathering depth removal
𝐴𝐴Δ𝜂𝜂bw is constant, theratio of weathering/abrasion increases with increasing bedrock strength (Figure 10). In general, however, theweathering depth removal
𝐴𝐴Δ𝜂𝜂bw is expected to decrease as the bedrock strength increases. That is, the two effects
described above are likely to cancel each other out. This suggests that, if all conditions other than bedrock
strength are the same, bedrock strength affects the channel development rate, but may not significantly affect thechannel cross-sectional shape. These results support the findings based on field data (Wohl & David, 2008), thatis, erosional resistance is not the primary control on scaling relations for channel geometry.
For a more accurate analysis of the relationship between channel cross-section and lithology, it is necessary
to investigate the relationship between strength and weathering depth removal
𝐴𝐴Δ𝜂𝜂bw. Moreover, in the case ofsandstone and conglomerate, weathering clasts can increase bed load and alluvial cover, which is not consideredby our model for mudstone. The transportation of weathered clasts and its impact on channel morphology is an
exciting challenge for the future.
The simulations in this study are conducted at a low width-depth ratio where no alluvial bars form in an innerchannel. As the scale of the river increases and the width-depth ratio exceeds 15, alluvial bars will form on the
bedrock and the erosional geometry will become more complex (e.g., Inoue et al., 2016). Also, the model doesn'ttake into account the effect of sidewalls on flow resistance in the incised narrow channel. In a wider river bed
channel, the sidewalls will not have a determining effect on the near-bed velocities of the channel. However, innarrow channels resembling the ones in our calculation, the resistance from sidewalls will lead to lower near-bed
velocities, and thence, lower shear stresses. The decreased near-bed velocities will stimulate sediment deposition,
thereby inhibiting the incision rates Note that the influence of the sidewalls is strong only in very narrow innerchannels such as run 7 shown in Figure 6b. In the inner channel formed in run 2 (Figure 6d), which is the basis of
our simulations, the effect is not so strong (see Appendix B).

5. Conclusion
Here we show the results of a numerical model that accounts for both abrasion due to bedload collisions andwet-dry weathering in a physically realistic way, as follows.
1. Abrasion leads to a deep inner channel, and weathering leads to flat benches on the channel margins
2. There is little chance that the weathered bedrock will be abraded because abrasion occurs at the bottom of theinner channel and weathering occurs on the channel margins

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3. However, weathering and abrasion indirectly interact with each other through changes in channel morphology,
that is, the development of the inner channel lowers the elevation of the bedrock bench, and the formation of
the bedrock bench reduces the abrasion of the inner channel in synergy with streamflow and water level

Appendix A
A1. Flow Field
We deployed 2-Dimensional depth-averaged continuity of mass and momentum equations to calculate the water
discharge:𝜕ℎ 𝜕(ℎ𝑈𝑥 ) 𝜕(ℎ𝑈𝑦 )
=0
+
+
(A1)
𝜕𝑡
𝜕𝑥
𝜕𝑦
( 2)
𝜕(ℎ𝑈𝑥 𝑈𝑦 )𝜕(ℎ𝑈𝑥 ) 𝜕 ℎ𝑈𝑥
𝜕𝐻 𝜏𝑥
(A2)
+
+
= −gh
−
+ 𝐷𝑥
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜌
(
)
𝜕(ℎ𝑈𝑦 ) 𝜕(ℎ𝑈𝑥 𝑈𝑦 ) 𝜕 ℎ𝑈𝑦2
𝜕𝐻 𝜏𝑦
(A3)
+
+
= −gh
−
+ 𝐷𝑦
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜌

where 𝐴 is the water depth,𝐴𝐴
𝐴𝐴
𝐴𝐴 is𝐴𝐴time,𝐴𝐴𝐴𝐴 and 𝐴𝐴 are orthogonal coordinates,
𝐴𝐴
𝐴𝐴𝐴𝐴𝑥𝑥 and 𝐴𝐴𝑦𝑦 depth-averaged velocities in𝐴𝐴 the𝐴𝐴𝐴𝐴 and 𝐴𝐴 directions,𝐴𝐴
𝐴𝐴 is the gravitational acceleration,𝐴𝐴
𝐴𝐴 is the water surface𝐴𝐴level, 𝐴𝐴
𝐴𝐴𝑥𝑥 and 𝐴𝐴𝑦𝑦 are the bed shear
stresses 𝐴𝐴
in the𝐴𝐴𝐴𝐴 and 𝐴𝐴 direction respectively,𝐴𝐴
𝐴𝐴 is the water density,
𝐴𝐴 and 𝐴𝐴𝐴𝐴𝑥𝑥 and 𝐴𝐴𝑦𝑦 are the diffusion terms in the
momentum balance equations.
The shear stresses
𝐴𝐴
𝐴𝐴
𝐴𝐴𝑥𝑥 and 𝐴𝐴𝑦𝑦 are calculated as
√
√
(A4)
𝜏𝜏𝑥𝑥 = 𝐶𝐶𝑓𝑓 𝑈𝑈𝑥𝑥 𝑈𝑈𝑥𝑥2 + 𝑈𝑈𝑦𝑦2 , 𝜏𝜏𝑦𝑦 = 𝐶𝐶𝑓𝑓 𝑈𝑈𝑦𝑦 𝑈𝑈𝑥𝑥2 + 𝑈𝑈𝑦𝑦2

where 𝐴𝐴𝑓𝑓 is the river-bed resistance coefficient, determined using the Manning–Strickler formula,
𝐴𝐴
(
)
1∕6
𝑘𝑘𝑠𝑠
𝑔𝑔
𝐶𝐶𝑓𝑓 = 1∕3
(A5)
√
ℎ
7.66 𝑔𝑔

Here, 𝐴𝐴𝑠𝑠 is the hydraulic roughness height𝐴𝐴
𝐴𝐴 and 𝐴 is the flow depth. In this study, we assume (e.g., Nelson &Seminara, 2012) a linear relationship between the hydraulic roughness height and areal alluvial-cover fraction pc(Parker et al., 2013):
𝑘𝑘𝑠𝑠 = 𝑝𝑝𝑐𝑐 𝑘𝑘sa + (1 − 𝑝𝑝𝑐𝑐 ) 𝑘𝑘sb
(A6)where 𝐴𝐴sa is the hydraulic roughness height of a completely alluvial bed
𝐴𝐴
𝐴𝐴 and 𝐴𝐴sb is the hydraulic roughness heightof a complete bedrock bed.
The diffusion parameters
𝐴𝐴
𝐴𝐴𝐴𝐴𝑥𝑥 and 𝐴𝐴𝑦𝑦 can be expressed as:()()
𝜕𝜕𝜕𝜕𝑥𝑥𝜕𝜕𝜕𝜕𝑥𝑥𝜕𝜕𝜕𝜕𝜈𝜈𝑡𝑡 ℎ𝜈𝜈𝑡𝑡 ℎ+𝐷𝐷𝑥𝑥 =
(A7)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
(
)
(
)
𝜕𝜕𝜕𝜕𝑦𝑦𝜕𝜕𝜕𝜕𝑦𝑦𝜕𝜕𝜕𝜕𝜈𝜈𝑡𝑡 ℎ+𝜈𝜈𝑡𝑡 ℎ𝐷𝐷𝑦𝑦 =
(A8)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝐴𝐴Here, 𝐴𝐴𝑡𝑡 is the coefficient of eddy viscosity given by𝜅𝜅𝜈𝜈𝑡𝑡 = 𝑢𝑢∗ ℎ(A9)6where 𝐴𝐴 is the von Kármán constant𝐴𝐴
𝐴𝐴 and 𝐴𝐴∗ represents the shear velocity.

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A2. Sediment Transport
We calculated the alluvial cover thickness above bedrock using a form of sediment conservation that takes into
account the sediment stored in the bedload layer as well as the sediment deposited on the bed (Inoue et al., 2014;
Inoue, Yamaguchi, & Nelson 2017):)(
𝜕𝜕𝜕𝜕𝑎𝑎𝜕𝜕𝜕𝜕bx 𝜕𝜕𝜕𝜕by𝜕𝜕𝜕𝜕𝑏𝑏=0+ (1 − 𝜆𝜆)𝑝𝑝𝑐𝑐++
(A10)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

where 𝐴𝐴𝑏𝑏 represents the volume of bedload per unit𝐴𝐴area, 𝐴𝐴 is the porosity of the alluvium,𝐴𝐴
𝐴𝐴
𝐴𝐴𝑎𝑎 is the thickness of the
alluvial layer
in the𝐴𝐴𝐴𝐴 and y-direction, respectively.
𝐴𝐴 and 𝐴𝐴𝐴𝐴
bx and 𝐴𝐴by are the bedload transport rates per unit width 𝐴𝐴
In alluvial rivers,
𝐴𝐴
𝐴𝐴𝑏𝑏 is equal to the saturation volume of bedload per unit𝐴𝐴area, 𝐴𝐴bs (i.e., the volume at bedloadtransport capacity). However, in a mixed alluvial—Bedrock channel,
𝐴𝐴 to 𝐴𝐴bs. Hence,𝐴𝐴
𝐴𝐴𝑏𝑏 is not necessarily equal
𝐴𝐴
𝐴𝐴bx
𝐴𝐴 and 𝐴𝐴by are represented as follows:)
𝑉𝑉𝑏𝑏⎧ 𝑉𝑉𝑏𝑏 (() ⎪ 𝑉𝑉bs 𝑞𝑞bcx , 𝑞𝑞bcy 𝑓𝑓 𝑓𝑓𝑓𝑓 0 ≤ 𝑉𝑉bs ≤ 1
𝑞𝑞bx , 𝑞𝑞by = ⎨(A11)()
𝑉𝑉⎪
𝑞𝑞bcx , 𝑞𝑞bcy 𝑓𝑓 𝑓𝑓𝑓𝑓 𝑏𝑏 = 1⎩
𝑉𝑉bs

where 𝐴𝐴bcx
𝐴𝐴
𝐴𝐴 and 𝐴𝐴bcy represent the bedload transport capacity rates𝐴𝐴in the𝐴𝐴𝐴𝐴 and y directions, respectively.
These capacity components are determined as follows:

𝑈𝑈𝑦𝑦
𝑈𝑈
𝑞𝑞bcx = √ 𝑥𝑥
𝑞𝑞bcs − √𝑞𝑞bcn
(A12)
2
2
𝑈𝑈𝑥𝑥 + 𝑈𝑈𝑦𝑦
𝑈𝑈𝑥𝑥2 + 𝑈𝑈𝑦𝑦2
𝑈𝑈𝑦𝑦
𝑈𝑈
𝑞𝑞bcy = √𝑞𝑞bcs + √ 𝑥𝑥
𝑞𝑞bcn
(A13)
𝑈𝑈𝑥𝑥2 + 𝑈𝑈𝑦𝑦2
𝑈𝑈𝑥𝑥2 + 𝑈𝑈𝑦𝑦2

where 𝐴𝐴bcs
𝐴𝐴
𝐴𝐴 and 𝐴𝐴bcn represent the bedload transport capacity per unit width in the streamwise and transverse directions respectively,
𝐴𝐴 and 𝐴𝐴𝐴𝐴𝑥𝑥 and 𝐴𝐴𝑦𝑦 are depth-averaged velocities𝐴𝐴in the𝐴𝐴𝐴𝐴 and 𝐴𝐴 directions.
The bedload flux in the streamwise direction𝐴𝐴𝐴𝐴bcs can be predicted using the empirical relation of Meyer-Peterand Müller (1948):√
(A14)
𝑞𝑞bcs = 8(𝜏𝜏∗ − 𝜏𝜏∗𝑐𝑐 )1.5 𝑠𝑠𝑔𝑔 𝑔𝑔𝑔𝑔 3

where,
𝐴𝐴
𝐴𝐴∗ is the Shields number,
𝐴𝐴
𝐴𝐴∗𝑐𝑐 is the critical Shields number, sg is the specific gravity of submerged sediment,
𝐴𝐴 and 𝐴𝐴 is the particle size.𝐴𝐴Here 𝐴𝐴∗ is calculated as follows:
𝐶𝐶𝑓𝑓 𝑈𝑈 2𝜏𝜏∗ =(A15)
𝑠𝑠𝑔𝑔 𝑔𝑔𝑔𝑔√
𝐴𝐴
where 𝐴𝐴 = 𝑈𝑈𝑥𝑥2 + 𝑈𝑈𝑦𝑦2 . The critical Shields number
𝐴𝐴𝐴𝐴∗𝑐𝑐 is calculated as follows:(A16)
𝜏𝜏∗𝑐𝑐 = 0.03(𝑘𝑘𝑠𝑠 ∕𝑑𝑑)0.6
Although this equation is obtained from experiments on bedrock channels (Mishra & Inoue, 2020), when substituting the roughness height𝐴𝐴of 2.5 𝐴𝐴 for alluvial𝐴𝐴beds, 𝐴𝐴∗𝑐𝑐 roughly becomes the general value for alluvial beds, thatis, 0.052. Therefore, we apply this equation to both bedrock and alluvial beds.
The transverse bedload
𝐴𝐴 flux 𝐴𝐴bcn can be estimated from the relation proposed by Hasegawa (1989), or equivalently
that of Mosselman and Crosato (1991):()√
𝜏𝜏∗𝑐𝑐 𝜕𝜕𝜕𝜕𝑢𝑢bn
𝑞𝑞bcn = 𝑞𝑞bcs−
(A17)
𝑢𝑢bs
𝜇𝜇𝑠𝑠 𝜇𝜇𝑘𝑘 𝜏𝜏∗ 𝜕𝜕𝜕𝜕SUMNER ET AL.

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Water Resources Research𝐴𝐴10.1029/2021WR030952

where (𝑢𝑢bs , 𝑢𝑢bn ) represent the along-streamline and cross-streamline vector components of near-bed velocity,
𝐴𝐴
𝐴𝐴𝐴𝐴∕𝜕𝜕𝜕𝜕 is the local riverbed slope in the transverse direction,
𝐴𝐴
𝐴𝐴𝑠𝑠 is a coefficient of static friction of sediment (here
set to 0.7)
𝐴𝐴 and 𝐴𝐴𝑘𝑘 is a coefficient of kinematic friction of sediment (0.7). Since the flow model calculates thedepth-averaged velocities
𝐴𝐴
𝐴𝐴 of𝐴𝐴𝐴𝐴bs to 𝐴𝐴bn is calculated using the(𝑈𝑈𝑥𝑥 , 𝑈𝑈𝑦𝑦 ) but not the near-bed velocities, the ratio
following equation, which includes the influence of secondary flow:
( )
𝑢𝑢bn
ℎ
= 𝑁𝑁∗
(A18)
𝑢𝑢bs
𝑟𝑟𝑠𝑠

where
𝐴𝐴
𝐴𝐴∗ is a parameter dependent on the strength of the secondary flow
𝐴𝐴 and 𝐴𝐴𝑠𝑠 is the radius of streamline
curvature.
The parameter𝐴𝐴
𝐴𝐴bs that is, the volume at bedload transport capacity takes the following form:
√
2
2
𝑞𝑞bcs
+ 𝑞𝑞bcn(A19)
𝑉𝑉bs =
𝑢𝑢𝑠𝑠

where 𝐴𝐴𝑠𝑠 is the bedload saltation velocity calculated as following (Sklar & Dietrich, 2004):𝐴𝐴√(A20)
𝑢𝑢𝑠𝑠 = 1.56(𝜏𝜏∗ ∕𝜏𝜏∗𝑐𝑐 − 1)0.56 𝑠𝑠𝑔𝑔 gd

Appendix B
Here, we show the simulation results considering the effect of sidewalls on flow resistance in the incised narrow
channel. The effect of the side walls is added to the second term on the right-hand side of Equation A7. We
explain how to consider the effect using discretized equations near both sidewalls. It should be noted that inreal phenomena, three-dimensional flow occurs due to the compound open-channel flow structure (e.g., Xieet al., 2013). However, since our model is a depth-averaged model, the effect of three-dimensional flow is nottreated.
In the left bank of the inner channel (see Figure B1),
𝐴𝐴 if 𝐴𝐴𝑗𝑗+1 − 𝜂𝜂𝑗𝑗 > 𝑧𝑧𝑤𝑤 ,
)
[
]
(
ℎ𝑗𝑗
𝑈𝑈𝑥𝑥𝑥𝑥𝑥+1 − 𝑈𝑈𝑥𝑥𝑥𝑥
𝑈𝑈𝑥𝑥𝑥𝑥𝑥 − 𝑈𝑈𝑥𝑥𝑥𝑥𝑥−1𝜕𝜕𝜕𝜕𝜕𝜕− 𝜈𝜈𝑡𝑡𝑡𝑡𝑡−1
𝜈𝜈𝑡𝑡 ℎ 𝑥𝑥 =𝑎𝑎𝑤𝑤 𝑐𝑐𝑤𝑤 𝑈𝑈𝑥𝑥𝑥𝑥𝑥 |𝑈𝑈𝑥𝑥𝑥𝑥𝑥 | + (1 − 𝑎𝑎𝑤𝑤 ) 𝜈𝜈𝑡𝑡𝑡𝑡𝑡
(B1)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
Δ𝑦𝑦
Δ𝑦𝑦
Δ𝑦𝑦

where 𝐴𝐴𝑤𝑤 is the friction coefficient of the sidewall𝐴𝐴
𝐴𝐴 and 𝐴𝐴𝑤𝑤 is the critical bank height of the inner channel at which
the resistance of the sidewall is taken into account, set to 1 m in this study. The subscript J denotes the grid
number from the right𝐴𝐴end. 𝐴𝐴𝑤𝑤 is the correction term of bank height to water depth, which is expressed by thefollowing equations,⎧ 𝜂𝜂𝑗𝑗+1 − 𝜂𝜂𝑗𝑗 when 𝜂𝜂𝑗𝑗+1 − 𝜂𝜂𝑗𝑗 < 1
⎪
ℎ𝑗𝑗
ℎ𝑗𝑗
𝑎𝑎𝑤𝑤 = ⎨(B2)𝜂𝜂𝑗𝑗+1 − 𝜂𝜂𝑗𝑗
⎪1
when
≥1
⎩
ℎ𝑗𝑗

In the right bank of the inner channel (see Figure B1),
𝐴𝐴 if 𝐴𝐴𝑗𝑗−1 − 𝜂𝜂𝑗𝑗 > 𝑧𝑧𝑤𝑤,
)
[
(
]
ℎ𝑗𝑗
𝑈𝑈𝑥𝑥𝑥𝑥𝑥+1 − 𝑈𝑈𝑥𝑥𝑥𝑥
𝑈𝑈𝑥𝑥𝑥𝑥𝑥 − 𝑈𝑈𝑥𝑥𝑥𝑥𝑥−1
𝜕𝜕𝜕𝜕𝑥𝑥
𝜕𝜕
𝜈𝜈
−
−
𝑎𝑎
−
𝑎𝑎
𝜈𝜈
ℎ
=
𝜈𝜈
𝑐𝑐
𝑈𝑈
|𝑈𝑈
|
)
(1
(B3)
𝑡𝑡
𝑡𝑡𝑡𝑡𝑡
𝑤𝑤
𝑡𝑡𝑡𝑡𝑡−1
𝑤𝑤 𝑤𝑤 𝑥𝑥𝑥𝑥𝑥
𝑥𝑥𝑥𝑥𝑥
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
Δ𝑦𝑦
Δ𝑦𝑦
Δ𝑦𝑦
⎧ 𝜂𝜂𝑗𝑗−1 − 𝜂𝜂𝑗𝑗 when 𝜂𝜂𝑗𝑗−1 − 𝜂𝜂𝑗𝑗 < 1
⎪
ℎ𝑗𝑗
ℎ𝑗𝑗
𝑎𝑎𝑤𝑤 = ⎨(B4)𝜂𝜂𝑗𝑗−1 − 𝜂𝜂𝑗𝑗
⎪1
when
≥1
⎩
ℎ𝑗𝑗

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Figure B1. Description of the parameters near the inner channel sidewalls.
𝐴𝐴
𝐴𝐴𝑥𝑥 is the depth averaged flow velocity in thestreamwise direction,
𝐴𝐴
𝐴 is the flow depth
𝐴𝐴 and 𝐴𝐴 represents the bed elevation.

Assuming that the horizontal distribution of streamwise velocity follows a logarithmic velocity distribution near
the sidewall,𝐴𝐴
𝐴𝐴𝑤𝑤 is given by the following equation,)2(
1 30Δ𝑦𝑦(B5)𝑙𝑙𝑙𝑙
𝑐𝑐𝑤𝑤 = 1∕𝜅𝜅
𝑘𝑘sw

where,
𝐴𝐴Δ𝑦𝑦 is the grid size in the transverse direction,𝐴𝐴
𝐴𝐴sw is the roughness height of the sidewall.
We test the effect of sidewall resistance on the channel evolution under the conditions of Runs 2 and 7 (seeTable 1). In Run 7, where the narrowest channel is formed, the channel depth decreases by about 3 m due to theeffect of the sidewall resistance. In Run 2, which is the base of our simulations, the channel width is wider thanthat in Run 7, so the effect of the sidewall resistance is limited and the decrease in channel depth is only 1 m(Figure B2).

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Figure B2. Effects of sidewall resistance on the channel evolution in Runs 2 and 7.

Data Availability Statement
All the data are made available via the following repository: https://zenodo.org/record/5144814#.YQJZ0477RPY.

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Acknowledgments
We would like to acknowledge theconstructive suggestions received from
Dr. Ellen Wohl and three anonymous
reviewers. Jagriti Mishra wants tothank Ramanujan fellowship no.
RJF/2021/000168 for financial assistance.

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10.1029/2021WR030952

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