Lateral Erosion of Bedrock Channel Banks by Bedload and Suspended Load

Bedrock rivers carry large amounts of fine sediment in suspension. We developed a mechanistic model for erosion of bedrock channel banks by impacting bedload and suspended load particles that are advected laterally by turbulent eddies (advection‐abrasion model). The model predicts high lateral erosion rates near the bed, with rates decreasing up to the water surface. The model also predicts greater erosion within the suspended load layer than the bedload layer for many typical sediment supply and transport conditions explored. We compared the advection‐abrasion model with a previously derived model for lateral erosion of bedrock banks by bedload particles deflected by stationary bed alluvium (deflection‐abrasion model). Erosion rates predicted by the deflection‐abrasion model are lower, except within limited conditions where sediment is transported near the threshold of motion and the bed is near fully covered in sediment. Both processes occur in bedrock rivers at the same time, so we combined the advection‐abrasion and deflection‐abrasion models and found that the lateral erosion rate generally increases with increasing transport stage and relative sediment supply for a given grain size. Application of our combined‐abrasion model to a natural bedrock river with a wide distribution of discharge and supply events, and mixed grain sizes, indicates that finer sediment dominates the lateral erosion on channel banks in low sediment supply environments and can be as important as coarser sediment in high sediment supply environments.

because they are no longer dominated by the effect of gravity, instead they are strongly influenced by turbulence (Bagnold, 1973;Naqshband et al., 2017). Within the suspension regime, sediment is transported in a near-bed bedload layer with high sediment concentrations and a more dilute suspended-load layer above, with active interchange of these two layers (S. R. McLean, 1991;Rouse, 1937). To incorporate the change in sediment transport from bedload to suspension, the saltation-abrasion model has been reformulated in terms of near-bed sediment concentration instead of particle hop length (referred to as the total-load model hereafter; Lamb, Dietrich, & Sklar, 2008). The total-load model predicts higher impact velocity of suspended particles at higher transport stages and hence nonzero erosion rates within the suspension regime, consistent with observations in laboratory experiments (Scheingross et al., 2014).
Previous investigations of the mechanics of lateral erosion of bedrock channel banks have focused on abrasion by bedload particles. Fuller et al. (2016) conducted a set of flume experiments with a non-erodible bed covered with protruding roughness elements and erodible banks composed of weak concrete, and documented lateral erosion by saltating bedload impacts deflected by bed roughness. Using this mechanism, Turowski (2018Turowski ( , 2020) developed a reach-scale lateral erosion model that treated gravel bars as a source of roughness to deflect bedload particles but did not explicitly model the deflection process. Another lateral erosion model directly captured the energy transfer of bedload particles deflected by bed roughness to impact the banks (Li et al., 2020(Li et al., , 2021; referred to as the deflection-abrasion model hereafter). These lateral erosion models have been used to explore bedrock channel width and bed slope dynamics in response to changes in grain size and sediment supply rate (Li et al., 2021) as well as adjustment timescales to achieve steady-state bedrock channel morphology (Turowski, 2020). One weakness of deflection-abrasion models is that they predict lateral erosion concentrated only in the lower part of bedrock riverbanks, creating undercut banks without eroding the overhanging upper portion of the channel bank (Li et al., 2020). While overhanging banks have been observed in laboratory experiments that included only bedload (Cao et al., 2022;Fuller et al., 2016;Mishra et al., 2018), overhangs are not commonly observed in natural bedrock rivers. Overhanging banks might be removed in natural rivers due to collapse, particularly in highly fractured, weak bedrock (Curran, 2020). In addition, overhanging banks may also be rare because the suspended load can be advected by turbulent eddies to erode channel banks above the bedload layer height.
Using field observations, it has been suggested that erosion by suspended sediment is likely responsible for the formation of sculpted banks of slot canyons (Carter & Anderson, 2006;Richardson & Carling, 2005;Wohl, 1993Wohl, , 1998, creation of flutes and scallops on boulders protruding high into the flow (Whipple et al., 2000), and wear of bedrock banks above the bedload layer height and near the water surface (Beer et al., 2017;Hartshorn et al., 2002). To explore the efficacy of suspended load on eroding bedrock channel banks in comparison to bedload, we developed a mechanistic model for lateral riverbank erosion due to impacts of bedload and suspended load particles advected by turbulent eddies (referred to as the advection-abrasion model hereafter). We start by deriving the advection-abrasion model, building on the total-load model for vertical incision by bedload and suspended load (Lamb, Dietrich, & Sklar, 2008). Next, we combine the advection-abrasion and deflection-abrasion models for a complete model that captures both processes of lateral erosion. We investigate the sensitivity of predicted lateral erosion rates within the bedload layer and the suspended load layer to variation in grain size, water discharge, sediment supply and change slope. Finally, we discuss the implications of the combined-abrasion model for the relative importance of finer and coarser sediment in eroding bedrock channel banks.
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3 of 25 . The total load model uses conventional suspended transport theory to describe the vertical distribution of suspended sediment, based on the Rouse-Vanoni eddy viscosity concept (e.g., Rouse, 1937;Vanoni, 1946). For a suspended sediment particle to impact the bank, it must first be accelerated toward the bank by turbulent eddies, but then the particle trajectory, due to momentum, must deviate from the fluid trajectory that necessarily turns at the bank. The degree to which the turbulent diffusion of suspended sediment deviates from the diffusion of fluid momentum is quantified by the ratio, β, of sediment diffusivity (ε s ) to fluid diffusivity (eddy viscosity, ε f ) in the Rouse number (Coleman, 1970;de Leeuw et al., 2020;van Rijn, 1993). The ratio β can differ from unity for a host of reasons, such as when fine particles dampen turbulence at high concentrations and particles cluster outside vortices (Muste et al., 2005). Larger particles can enhance turbulence (β < 1) and have their own momentum, imparted from the fluid, resulting in trajectories that may deviate from local flow directions.
Another condition for a suspended sediment particle to impact the channel bank is that it must have sufficient momentum to push away the fluid in between the particle and the bank, which becomes viscously dominated as the gap distance becomes small (Joseph & Hunt, 2004). This condition is readily satisfied for sand-sized particles, as demonstrated in laboratory investigations of vertical bedrock erosion by suspended sediment (Scheingross et al., 2014). Those experiments show that bed-impact velocities far exceed the gravitational settling velocity because the particles detach from eddies as the fluid is forced to turn at the boundary, and the particles have sufficient momentum to overcome the viscous drag (Scheingross et al., 2014).
We assume that flow is hydraulically rough, as is the case for flow in bedrock rivers carrying suspending sediment. We assume that turbulence intensity follows the Nezu and Nakagawa (1993) scaling relation where turbulence intensity decreases with height above the bed. This relation does not consider turbulence caused by bank roughness and is most applicable when bedrock banks are relatively smooth and where channel width-to-depth ratios (e.g., >10) are large (e.g., Vanoni & Brooks, 1957). These assumptions likely apply in most bedrock channels, but may be violated in narrow bedrock canyons where bedrock banks can dominate the friction along the boundary (e.g., Li, Venditti, Rennie, & Nelson, 2022). There is no established relation to predict turbulent fluctuations due to bank roughness at this time and a different turbulence model could be used in the future when it is developed for narrow bedrock canyons. Such a model would likely have a more uniform vertical distribution of turbulence, and therefore might predict more enhanced wall erosion far above the bed than in the model developed here.

General Expression
Following the formulation of the saltation-abrasion model (Sklar & Dietrich, 2004) and total-load model (Lamb, Dietrich, & Sklar, 2008) for vertical erosion, the lateral erosion rate predicted by the advection-abrasion model, E a , can be written as a product of two terms: the eroded volume per particle impact, V, and the impact rate per unit area and time, I, = (1) The eroded volume per particle impact, V, is proportional to the kinetic energy of the impacting particles (Sklar & Dietrich, 2004) where Y is Young's modulus of elasticity of the bedrock, k v is the dimensionless bedrock strength coefficient, σ T is the tensile yield strength, ρ s is the sediment density, D is the median grain size and v p is the impact velocity normal to the banks. Equation 2 has been verified over a wide range of conditions (e.g., Beer & Lamb, 2021). The time-averaged impact rate, I, is proportional to the product of v p and the volumetric sediment concentration, c, (Lamb, Dietrich, & Sklar, 2008) where Ψ is a dimensionless coefficient describing the portion of particles near the banks that are advected toward the banks. There are no field or laboratory observations to constrain Ψ, so we choose a value of 0.5, which LI ET AL.

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4 of 25 means that the amount of particles moves toward and away from the wall are equal as would be expected for near isotropic turbulence in the cross-stream direction (Nezu and Nakagawa, 1993). The volume of nominally spherical sediment grains V p (=πD 3 /6) is used in Equation 3 to convert the mass flux into the volumetric flux. Substituting Equations 2 and 3 into Equation 1 yields To estimate the lateral erosion rate in Equation 4 directly from control variables in bedrock rivers, we determine local hydraulic conditions from the law of the wall, and then develop expressions for lateral impact velocity and sediment concentration.

Local Hydraulic Conditions
Local hydraulic conditions are calculated from five input variables: volumetric water discharge: Q w , channel width: W, channel slope: S, particle diameter: D, and boundary roughness height: z 0 . For turbulent flow in open channels, we approximate the semi-logarithmic vertical distribution of velocity above the bed from the law of the wall, and the downstream flow velocity, u(z), is calculated as where κ is von Karman's constant (κ = 0.41), z is height above the river bed, and z 0 is a function of the boundary roughness (Henderson, 1966). We assume that the law of the wall is applicable near the wall and that the velocity profile is semi-logarithmic throughout the full depth, which is commonly observed in rivers. This is a simplification that allows the model to be one dimensional. Local downstream velocity near the wall may differ from the semi-logarithmic velocity profile due to wall drag or form drag (Li, Venditti, Rennie, & Nelson, 2022) and there might be reduced velocity near the water surface due to secondary circulation (e.g., Coles, 1956;Gelfenbaum & Smith, 1986). We ignore these minor deviations from the semilogarithmic velocity profile because they have negligible effects on the bulk hydraulic calculations for which we use Equation 5.
Following Lamb, Dietrich, and Sklar (2008), we set z 0 = nD 84 /30 with empirical coefficient n = 3 (D 84 is the 84th percentile of the surface bed material). This parameterization of the hydraulic roughness can be modified in natural bedrock rivers with partially covered beds by sediment, and with protruded bedrock roughness on the bed and banks (Beer et al., 2017;Finnegan et al., 2007;Inoue et al., 2014;Johnson, 2014;Johnson & Whipple, 2010;Li, Venditti, Rennie, & Nelson, 2022). While flow within complex roughness (gravel or bedforms) deviates from the law of the wall (e.g., Nikora et al., 2001Nikora et al., , 2004, flow above the roughness layer is almost universally logarithmic S. R. McLean & Nikora, 2006) and the velocity distribution in hydraulically rough flows typically exends all the way to the boundary (Garcia, 2008;Nezu & Nakagawa, 1993;van Rijn, 1990). The shear velocity is calculated from the hydraulic radius, R, and channel bed slope, S, as * = √ assuming steady and uniform flow, where R = WH/(W + 2H), and H is water depth. The depth-averaged flow velocity is estimated by integrating Equation 5 Depth-averaged flow velocity, U, and flow depth, H, are solved by combining Equations 6 and 7 with continuity, Q w = WHU. The non-dimensional shear stress, τ*, is obtained from shear velocity, * = ( * ) 2 .
where δ b = ρ s /ρ w − 1 is non-dimensional buoyant density, ρ w is water density, and g is the gravitational acceleration, neglecting form drag.
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Impact Velocity
The particle impact velocity, v p , on the banks is induced by lateral turbulent flow velocity fluctuations, v′, near the banks that advect particles into the banks. Turbulent fluctuations embedded in open channel flows are a complex phenomenon. Although numerical and experimental work has been done to explore the complex three-dimensional turbulence structure in open channel flows (e.g., Nezu, 2005;Venditti et al., 2013), this level of complexity is arguably not warranted for developing a model for bedrock channel change over geomorphic timescales. For simplicity, we follow Lamb, Dietrich, and Sklar (2008) and assume that the probability density function, P, of lateral flow velocity fluctuations v′ is a Gaussian distribution (Nezu and Nakagawa, 1993) where the mean lateral flow velocity is zero, and σ v is the standard deviation of lateral flow velocity fluctuations and is approximated by the turbulent intensity. Nezu & Nakagawa (1993) proposed an exponential expression to describe the vertical variation of lateral flow turbulence intensity σ v in open channel flows, * = − ∕ where D v is a dimensionless coefficient that has a typical value of 1.63 (Nezu & Nakagawa, 1993). Variation in D v depends on channel boundary roughness (Carling et al., 2002;Knight & Shiono, 1990;Soulsby, 1981;Sukhodolov et al., 1998Sukhodolov et al., , 2006Tominaga et al., 1989) and flow depth relative to boundary roughness Lamb, Dietrich, & Venditti, 2008). However, the range of variations in D v is narrow (1.39-1.89) and no improvement in scaling σ v with depth-averaged velocity is found as compared with u* . For simplicity, we elected to use D v = 1.63 reported by Nezu & Nakagawa (1993).
The particle impact velocity fluctuations normal to the banks, ′ , are calculated from the acceleration of a particle moving toward the banks laterally due to the fluid drag force, which is given by conservation of linear momentum as where C 1 (0.45) is the drag parameter and A p is the cross-sectional area of the particle perpendicular to v′.
To solve Equation 11 analytically, we assume v′ is constant over an eddy turnover timescale, t l . Under this assumption, d ′ ∕d is approximated by −d where the boundary condition ′ ( = 0) = 0 is applied and particles are assumed to be spheres (i.e., A p /V p = 1.5/D). Equation 12 incorporates particles that are accelerated by turbulent eddies to obtain enough lateral momentum to impact on the banks, but does not consider small particles that follow the flow and hence cause no erosion. The degree to which a particle follows the flow is controlled by the ratio of particle response time t p and eddy turnover time t l (e.g., Crowe et al., 1985;Fessler et al., 1994). Particles with very small t p /t l (<0.001-0.01;  act as passive tracers following the flow. There is currently no established relation to predict the degree to which the particle follows the flow from the ratio t p /t l . In our simulations, particles with a very small ratio of particle response time and eddy turnover time are already viscously damped. Therefore, we use the viscous damping effect to eliminate small particles that follow the flow and cannot impact on the banks. The mean lateral particle velocity is then found by combining Equations 9 and 12 and integrating ′ over all possible velocity fluctuations. However, the lateral erosion rate scales with the cube of individual particle velocity ′ (Equation 4), not the mean lateral particle velocity. Therefore, following Lamb, Dietrich, and Sklar (2008), we define an effective impact velocity, v p , by combining Equations 9 and 12 and nonlinear averaging, as where the upper limit of integration of 6σ v is chosen to include near 100% of the possible fluctuations and the lower limit of integration, ′ min , is chosen to distinguish between impacts that cause erosion and impacts that are viscously damped. Viscous damping is a function of the particle Stokes number S t , such that the minimum particle velocity ′ min can be given as where S t is the particle Stokes number (∼100; Joseph & Hunt, 2004;Schmeeckle et al., 2001), and η is the kinematic viscosity of the fluid (10 −6 m 2 s −1 ). The lower limit of integration ′ min can be obtained from ′ min using the relation between them in Equation 12.
To solve Equation 13, the eddy turnover time t l needs to be specified. Turbulent energy is extracted from the mean flow in the production subrange, and is transferred from macroscale (energetic) eddies to microscale (dissipative) eddies in the energy-cascade process. The rate at which energy is transferred from macroscale structures to microstructures ϵ (i.e., the energy dissipation rate) can be characterized by the fluctuating velocity σ v over an integral length scale l (e.g., Batchelor, 1953;Mouri et al., 2012;Nezu & Nakagawa, 1993;Vassilicos, 2015), where C ϵ is a constant (∼1) originating from the Richardson-Kolmogorov cascade under the assumption that turbulence is at equilibrium (Kolmogorov, 1941a(Kolmogorov, , 1941b, and the time scale associated with l is t l = l/σ v . Assuming the turbulent energy is in local equilibrium (turbulent kinetic energy generation, G, equals turbulence dissipation, ϵ), ϵ is obtained as a function of the distance from the bed, z, using the law of the wall (Grinvald, 1974;Nikora & Smart, 1997) Combining Equations 15 and 16 with t l = l/σ v , t l is solved as

Sediment Concentration
To calculate the vertical distribution of sediment concentration, we partition the supplied sediment flux into a bedload layer and a suspended load layer. In doing so, we explicitly acknowledge that within the suspension regime, when the threshold to suspend particles is exceeded, both bedload and suspended load occur with the division defined by the saltation layer height. We assume that the sediment within the bedload layer is well mixed (De Leeuw et al., 2020;Lamb, Dietrich, & Sklar, 2008;S. R. McLean, 1991) and use the Rouse-Vanoni equation as an approximation of the vertical distribution of sediment concentration above the bedload layer where c b is the volumetric concentration within the bedload layer, h s is the bedload layer height, P = w f /βκu* is the Rouse number (Rouse, 1937), β is a dimensionless coefficient and w f is the particle settling velocity. The bedload layer height, h s , is predicted from the empirical relation developed by Sklar and Dietrich (2004), where * is the value of τ* at the threshold of sediment motion.
The coefficient β is a dimensionless factor that accounts for differences between the diffusivities of momentum and sediment, typically assumed to be a constant of order unity (e.g., Coleman, 1970;van Rijn, 1990). Previous studies have found that β varies with the ratio of settling velocity w f to shear velocity u* and flow resistance coefficient (e.g., De Leeuw et al., 2020;van Rijn, 1993), concentration levels, stratification, river slope (Wright & Parker, 2004), and particle flow resistance coefficients (e.g., De Leeuw et al., 2020). We elect to use the best-fit one-parameter model for β that is calibrated against field data proposed by De Leeuw et al. (2020), which is = 2.44( * ∕ ) −0.55 . We follow Ferguson and Church (2004) to calculate the particle settling velocity w f , where C 2 = 18 and C 3 = 1 are constants set for natural sediment.
Following Lamb, Dietrich, and Sklar (2008), the volumetric concentration within the bedload layer, c b , is calculated from continuity as where q is the total volumetric flux of sediment per unit channel width traveling as both bedload and suspended load, u s is the longitudinal saltation velocity and χ is the integral relating suspended sediment flux to the parameters of the flow and sediment concentration within the bedload layer, which we calculate according to Lamb, Dietrich, and Sklar (2008).
The longitudinal saltation velocity u s is estimated from Sklar and Dietrich (2004).

Composite Expression for the Advection-Abrasion Model
Substituting Equations 13 and 18 into Equation 4 yields the composite expression for the advection-abrasion model, where the vertical profile of the lateral erosion rate E a (z) is divided into two layers: the bedload layer (z ≤ h s ) and the suspended load layer (z > h s ).

Combined-Abrasion Model
We propose that lateral erosion should be modeled as a combination of the deflection-abrasion and advection-abrasion mechanisms. To do that, we briefly review the deflection-abrasion model and combine it with the advection-abrasion model (referred to as the combined-abrasion model hereafter).
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10.1029/2022JF006806 8 of 25 Li et al. (2021) developed an expression for the lateral erosion rate assuming uniformly distributed alluvium, E d ,

Expression for the Deflection-Abrasion Model
where = 0.75 1 ∕( ) is the drag deceleration coefficient, q s is the bedload supply rate, q t is the bedload transport rate, y max is the maximum distance between the point of deflection and the channel bank above which deflectors will not cause lateral erosion, h max is the maximum impact height on the bank, l s is the saltation length, and v o is lateral particle velocity after being deflected by the alluvium surface.
Bedload transport rate, q t , is estimated from Fernandez Luque and Van Beek (1976), The distance, y max , is calculated from the minimum velocity that causes erosion (Li et al., 2021) The saltation length, l s , is predicted from the empirical relation developed by Sklar and Dietrich (2004) = 8.0 The maximum height of erosion, h max , is given as where = (1 − ∕ ) is the gravitational acceleration coefficient and w o is the vertical particle velocity after being deflected by the alluvium surface (Li et al., 2021). The velocities, w o and v o , can be obtained from the momentum transfer at the point of deflection, where C r is the restitution coefficient (set to 0.9) that describes the loss of particle momentum during the collision between bedload particle and alluvium surface, n x , n y and n z are downstream, lateral and vertical component of the unit vector that is normal to the alluvium surface at the point of deflection, respectively (n x = 0.30q s /q t − 0.54; n y = 0.25q s /q t − 0.58; n z = 0.40q s /q t + 0.50; Li et al., 2021), and w s is vertical saltation velocity that is estimated from empirical relations by Lamb, Dietrich, and Sklar (2008).
where h d is the height of the bedload particle during collision with the deflector. The ratio, h d /h s , is a function of transport stage * ∕ * and the ratio of bedload sediment supply to bedload transport capacity q s /q t (Li et al., 2021) ℎ ∕ℎ = 0.83 −0.68( * ∕ * −1) + 0.11 ∕ + 0.06 The lateral erosion rate predicted by the deflection-abrasion model can be obtained by substituting Equations 26-33 into Equation 25. The deflection-abrasion model does not account for the possibility of lateral erosion by suspended load abrasion, assuming that the lateral erosion rate is zero if u*/w f (Equation 25).

Expression for the Combined-Abrasion Model
The deflection-abrasion models and the advection-abrasion models predict erosion rates on different parts of the channel banks. The deflection-abrasion model focuses on erosion by saltating bedload particle impacts and 9 of 25 hence predicts erosion near the bottom of the banks. The advection-abrasion model considers the sediment concentration for the whole water column, and thus predicts erosion over the whole bank. To combine these two models, we calculate the cross-sectional area eroded per unit time to obtain an areal erosion rate from each model, which we add to get the total eroded cross-sectional area per unit time.
The cross-sectional area eroded per unit time for the deflection-abrasion model, E cd , is equal to the product of the vertically averaged erosion rate, E d , and the maximum height above the bed over which erosion occurs, h max = ℎmax The cross-sectional area eroded per unit time for the advection-abrasion model, E ca , is the sum of the eroded cross-section area per unit time within the bedload layer, E cb , and the suspended load layer, E cs , where E cb and E cs are obtained by integrating E a (z) over the thickness of the bedload layer, h s , and suspended layer,H−h s , respectively. The total cross-sectional area eroded per unit time on the banks is the sum of E cd and E ca , To explore the behavior of the advection-abrasion and deflection-abrasion models over a wide range of parameter space, we nondimensionalize areal erosion rates by multiplying by 2 ∕ 1.5 2.5 (Lamb, Dietrich, & Sklar, 2008;Li et al., 2020Li et al., , 2021Sklar & Dietrich, 2004). In this parameter space, the non-dimensional erosion rates (deflection-abrasion model: * ; advection-abrasion model: * ; and combined-abrasion model: * ) are a function of transport stage, * ∕ * , and relative sediment supply, q/q t , for a given grain size. For given transport stage, * also depends on flow depth because the vertical distributions of sediment concentration and impact velocity are a function of flow depth (Equation 24). In contrast, the deflection-abrasion model only considers the abrasion by saltating bedload particles and does not depend on flow depth (Equation 26).

Model Results
Both the advection-abrasion model and the deflection-abrasion model predict that the lateral erosion rate depends on four principal variables: water discharge Q w , channel slope S, sediment supply q, and grain size D for a given rock strength. To explore the distribution of lateral erosion on channel banks and the relative importance of lateral erosion within the bedload and the suspended load layers, we vary each of these four variables independently. We also explore the variation of the nondimensionalized erosion rates predicted by the advection-abrasion model, the deflection-abrasion model, and the combined-abrasion model over parameter space defined by transport stage and relative sediment supply.
We selected Black Canyon in the Fraser River, British Columbia as the reference field site to help specify model input variables (Table 1; Li, Venditti, Sklar, & Lamb, 2022). The Fraser River annual hydrograph has peaks within the range between ∼6,070 and ∼12,900 m 3 /s over the past 20-years at Hope, British Columbia, the nearest gauging station to Black Canyon. Coarse sediment supply to the gravel bed reach that starts at Hope, British Columbia, at the downstream end of bedrock canyons of the Fraser River is 0.3-0.4 Mt/yr (megatons per year) (A. D. Nelson and Church (2012) based on D. G. , Church et al. (2001), Church (2009), andFerguson et al. (2015)), and the annual sediment supply to the river, including gravel, sand, silt and clay is ∼17.5 Mt/a (D. G. . Here, we choose a value of 0.35 Mt/a as the coarse sediment supply in Black Canyon because that is how much is transmitted downstream to the gravel bed reach. Ferguson and Church (2009) found that a discharge of 7,000 m 3 /s operating 15% of the year transported the same amount and size distribution of sediment as the 20-year hydrograph, so we selected the combination of these values as our reference discharge and intermittency. An active bar upstream of Black

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Canyon has grain diameters that range between 6 and 471 mm with a median size of 195 mm and the 84th percentile of the grain size of 267 mm. The representative discharge can transport grains up to 285 mm in diameter as bedload and size up to 11 mm diameter in suspension. To compare the relative importance of bedload and suspended load, we consider two grain sizes: 10 mm diameter gravel and 195 mm diameter cobbles, which are carried as suspended load and bedload, respectively.

Influence of Discharge
We explored the effect of varying water discharge while holding sediment supply, grain size and channel slope set to constant values for the reference field site. Like the deflection-abrasion model, the advection-abrasion model predicts an undercut erosional shape on channel banks, where lateral erosion is concentrated near the bottom of channel banks and decreases progressively up to the water surface (Figures 1a and 1b). The maximum erosional height on channel banks increases with increasing discharge due to the increase in water depth (Figures 1a  and 1b). The maximum E a within the undercut zone increases with increasing discharge due to the increase in transport stage and hence impact velocity, but starts to decline at high discharge when the increase in impact velocity is outpaced by the decrease in near-bed sediment concentration, as more sediment is held in the upper water column for 10-mm gravel and 195-mm cobbles (Figures 1a and 1b). However, compared with the 195-mm cobbles, a lower discharge is required for E a to exceed zero and the maximum E a within the undercut zone to peak for 10-mm gravel (Figures 1a and 1b). This occurs because finer sediment can be transported at lower shear stresses and hence lower discharge for a given channel slope. Overall, the advection-abrasion model predicts similar patterns of erosion on channel banks for 10-mm gravel and 195-mm cobbles (Figures 1a and 1b). The advection-abrasion model predicts a negligible erosion rate within the bedload layer, E cb , compared with the suspended load layer, E cs , for both 10-mm gravel and 195-mm cobbles (Figures 1c and 1d) because the bedload layer is much thinner than the suspended layer. The total load erosion, E ca , is almost the same as the erosion rate within the suspended load layer. For the gravel, E ca increases with increasing discharge once the transport stage is above the threshold of motion (Figure 1c) due to the increase in impact velocity and the erosional height on channel banks. However, E ca starts to decline at discharges higher than 2 × 10 5 m 3 /s (Figure 1c). This occurs because increases in impact velocity and the erosional height are more than offset by the decrease in near-bed sediment concentration with increasing discharge. For the cobbles, the increase in transport stage is much lower than for the gravel with increasing discharge, resulting in a slower decrease in near-bed sediment concentration. Therefore, E ca is more influenced by the increase in impact velocity and the erosional height on channel banks than the decrease in near-bed sediment concentration for the cobbles, as compared to the gravel, which leads to a near monotonic increase in E ca with increasing discharge (Figure 1d).
The deflection-abrasion model predicts an increase in lateral erosion rate, E cd , at low discharge due to the increase in impact energy with increasing shear stress, but starts to decline at high discharge due to the decrease in the extent of alluvial cover with increasing transport capacity for given sediment supply (Figures 1c and 1d; Li et al., 2021).
The deflection-abrasion model predicts negligible lateral erosion at discharge higher than ∼2.5 × 10 2 m 3 /s for the gravel and ∼3.9 × 10 3 m 3 /s for the cobbles because there is not enough alluvium to deflect bedload particles (Figures 1c and 1d). The advection-abrasion, in contrast, predicts continued lateral erosion at high discharge. The erosion rate predicted by the advection-abrasion model is higher than the deflection-abrasion model except at low discharge when the bed is near fully covered by alluvium. Therefore, the erosion rate predicted by the combined-abrasion model shows similar patterns with the deflection-abrasion model at low discharges where the deflection-abrasion mechanism dominates, but becomes fully controlled by the advection-abrasion model at intermediate and high discharges where the advection-abrasion mechanism dominates (Figures 1c and 1d).

Influence of Slope
Predictions by the advection-abrasion model for 10-mm gravel and 195-mm cobbles are qualitatively similar when the slope is varied, with all other parameters held constant (Figures 2a and 2b). The distribution of the lateral erosion rate on channel banks predicted by the advection-abrasion model forms an undercut erosional pattern. The maximum E a within the undercut zone increases with increasing slope due to the increase in transport stage and hence impact velocity. The advection-abrasion model predicts a decrease in the height of the undercut zone on channel banks with increasing slope as a result of the decrease in water depth. In contrast, the height of maximum E a within the undercut zone increases with increasing slope due to the increase in bedload layer height with increasing transport stage. For the same channel slope, the lateral erosion rate for the gravel is larger than for the cobbles (Figures 2a and 2b) because, for the same shear stress, smaller grains have a higher transport stage and hence higher impact energy.
The advection-abrasion model predicts a monotonic increase in the erosion rate with a slope within the bedload layer, E cb , due to the increase in bedload layer height and the impact velocity (Figures 2c and 2d). Compared with E cb , the erosion rate within the suspended load layer, E cs , increases with increasing slope due to the increase in impact energy at low slope values, but starts to decline at S > 0.8 for the gravel and S > 0.06 for the cobbles due to the decrease in height of suspended load layer as bedload layer height approaches the water surface at steep slopes (Figures 2c and 2d). The total load erosion rate, E ca , monotonically increases with increasing slope for both the gravel and the cobbles, even at steep slopes where E cs declines because E cs declines at a lower rate than the growth of E cb (Figures 2c and 2d). E cb is smaller than E cs at S < 0.3 for gravel and S < 0.04 for cobbles, but becomes larger than E cs at very steep slopes where the bedload layer grows to encompass most of the flow depth (Figures 2c  and 2d). Therefore, E cs dominates the total load erosion at low slopes and E cb dominates at steep slopes, while both bedload and suspended load layers are important in eroding channel banks at intermediate slopes.
The erosion rate predicted by the deflection-abrasion model, E cd , increases with increasing slope once the transport stage is above the threshold of motion, but starts to decline at S ≈ 0.00008 for 10-mm gravel ( Figure 2c) and S ≈ 0.0012 for 195-mm cobbles (Figure 2d) due to the decrease in the extent of alluvial cover as the transport stage increases for given sediment supply. Compared with the advection-abrasion model, the deflection-abrasion model predicts negligible erosion at S > 0.00021 for the gravel and S > 0.0017 for the cobbles (Figures 2c  and 2d). The erosion rate predicted by the deflection-abrasion model is lower than the advection-abrasion model 12 of 25 for the whole range of slope variation, except at 0.000075 < S < 0.00008 for gravel and 0.00115 < S < 0.0012 for cobbles (Figures 2c and 2d). Therefore, the erosion rate predicted by the combined-abrasion model E c follows the advection-abrasion model, except at a narrow range of small gradients where the erosion rate predicted by the deflection-abrasion model dominates.

Influence of Grain Size
The advection-abrasion model, E a , also shows an undercut erosional pattern when grain size is varied with all other parameters held constant at values for the reference site (Figure 3a). Within the undercut zone, E a initially increases with increasing grain size due to the increase in near-bed volumetric sediment concentration for larger sediment and the increase in impact velocity. However, the impact velocity starts to decrease when the increase in grain size is more than offset by the decrease in transport stage, resulting in a decrease in E a for large grain size.
The advection-abrasion model predicts an increase in the erosion rate within both the bedload layer, E cb , and the suspended load layer, E cs , for grain sizes smaller than ∼4 and ∼15 mm, respectively, and a decline in these erosion rates for larger grain sizes (Figure 3b). E cs is higher than E cb for the full range of grain sizes that can cause erosion (Figure 3b), so the total load erosion E ca is almost fully controlled by E cs . Compared with the advection-abrasion model, the deflection-abrasion model predicts a negligible erosion rate at grain size smaller than ∼280 mm, an increase in erosion rate for larger grain size due to the increase in the extent of alluvial cover, and then a Areal erosion rate predicted by advection-abrasion, deflection-abrasion, and combined-abrasion models as a function of discharge for (c) 10-mm gravel and (d) 195-mm cobbles. Orange lines are the conditions for the representative field case of the Black Canyon. The low transport stages for the 10-mm gravel and the large transport stages for the 195-mm cobbles correspond to unrealistic low slopes (<0.0001) and large slopes (>0.1) in natural bedrock rivers, respectively, but are shown here for comparison of the influence of slope on 10-mm gravel and 195-mm cobbles.
13 of 25 decline in the erosion rate at grain size larger than ∼305 mm due to the decrease in transport capacity (Figure 3b). The erosion rate predicted by the deflection-abrasion model is only larger than the advection-abrasion model near the threshold of motion at D ≈ 300 mm (Figure 3b). Therefore, the combined erosion rate follows a similar pattern with the advection-abrasion model, except at D ≈ 300 mm where the deflection-abrasion model dominates.

Influence of Sediment Supply
Increasing sediment supply in the advection-abrasion model, with all other variables held constant at the reference site values, increases the sediment concentration and hence the erosion rate until the bedload sediment supply exceeds the transport capacity (Figures 4a and 4b). The erosion rate within the suspended load layer E cs is much higher than the erosion rate within the bedload layer E cb for the full range of sediment supply rate (Figures 4c and 4d). Therefore, the total load erosion rate E ca is nearly the same with E cs (Figures 4c and 4d).
Compared with the advection-abrasion model, the deflection-abrasion model predicts zero erosion rates for 10-mm gravel because it is transported in suspension under hydraulic conditions at the reference site ( Figure 4c). Therefore, the combined erosion rate E c is equal to E ca for the gravel (Figure 4c). The deflection-abrasion model for the cobbles predicts negligible erosion at q < 0.0009 m 2 /s ( Figure 4d) due to the lack of alluvial cover and an increase in erosion rate E cd at larger sediment supply as the number of deflections increases with the extent of alluvial cover. E cd for the cobbles starts to decline at q ≈ 0.02 m 2 /s ( Figure 4d) due to the shift of deflection locations toward the top of alluvium surface and hence the decrease in the efficiency of deflecting bedload particles when deflectors become densely packed at a high sediment supply rate (Li et al., 2021). Compared with the deflection-abrasion model, the advection-abrasion model predicts a higher erosion rate at a low supply rate (q < 0.004 m 2 /s) but is outpaced by E cd at a high supply rate (q < 0.004 m 2 /s) for the cobbles (Figure 4d). Therefore, the combined erosion rate E c for the cobbles is mainly due to the advection-abrasion mechanism at a low supply rate but becomes dominated by the deflection-abrasion mechanism at a high supply rate (Figure 4d).

Nondimensional Erosion Rate
We explored the advection-abrasion model, the deflection-abrasion model, and the combined-abrasion model behaviors over a wide range of parameter space defined by transport stage and relative sediment supply (Figures 5-8). The transport stage is varied in two ways: varying discharge (Figures 5 and 6) and varying slope (Figures 7 and 8), because of the dependency of * on flow depth.
For the constant slope case, the advection-abrasion model predicts an increase in non-dimensional lateral erosion rate with increasing transport stage because the impact velocity increases with increasing discharge at the same relative sediment supply for 10-mm gravel ( Figure 5a) and 195-mm cobbles (Figure 6a). The nondimensional lateral erosion rate predicted by the advection-abrasion model also increases with increasing relative sediment supply due to the increase in impact rate with increasing sediment supply until the bedload supply rate reaches bedload transport capacity (Figures 5a and 6a). The advection-abrasion model predicts an increase in the nondimensional lateral erosion rate where the total sediment supply exceeds the bedload transport capacity because some of the sediment is transported as suspended load.
Compared with the advection-abrasion model, the deflection-abrasion model predicts a peak in lateral erosion rate at intermediate transport stage ( * ∕ * ≈ 16 for the gravel and * ∕ * ≈ 5.5 for the cobbles; Figures 5b and 6b) when the increase in impact velocity and the decrease in impact rate with increasing flow depth are well balanced. The deflection-abrasion model predicts zero erosion at * ∕ * ≥ 29 because of the onset of suspension (Figure 5b) for the 10-mm gravel. The erosion rate for the 195-mm gravel predicted by the deflection-abrasion model remains above zero even at extreme, unrealistic discharges (10 7 m 3 /s) at our reference site (Figure 6b) because the cobbles are transported at a much lower stage than the gravel for a given discharge and remain bedload at extreme discharge events (10 7 m 3 /s).
The deflection-abrasion model predicts a peak in the lateral erosion rate at high relative sediment supply (Figures 5b  and 6b) when the increase in the number of deflections is balanced with the decrease in the deflection efficiency, when the deflection location shifts toward the top of deflectors with increasing sediment supply (Li et al., 2021). The erosion rate predicted by the deflection-abrasion model is higher than the advection-abrasion model at low transport stage ( * ∕ * < 4.5 ) and high relative sediment supply (q/q t > 0.5), but is outpaced by the advection-abrasion model at lower relative sediment supply (q/q t < 0.5) or high transport stage ( * ∕ * > 4.5 ) (Figures 5c and 6c). The combined-abrasion model predicts an increase in erosion rate with increasing transport stage and increasing relative sediment supply, except at low transport stage where the erosion rate peaks at high relative sediment supply due to the dominance of the deflection-abrasion model here (Figures 5 and 6d).

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In the parameter space defined by relative sediment supply and transport stage, the non-dimensional erosion rate predicted by the advection-abrasion model for the constant discharge cases (Figures 7a and 8a) is qualitatively similar to the constant slope case (Figures 5a and 6a). This occurs because the increasing slope decreases the flow depth, resulting in lower impact height on the banks (impact area) but higher near-bed sediment concentration (impact rate), while increasing discharge increases the flow depth, resulting in higher impact height on the banks (impact area) but lower near-bed sediment concentration (impact rate). These opposite effects on impact area and impact rate cause similar patterns of erosion rate for the constant discharge and constant slope cases.
When the transport stage is varied by varying slope, the erosion rate for 10-mm gravel is generally higher than the constant slope case (Figures 5a and 7a), but the opposite relation occurs for 195-mm cobbles at the same transport stage and relative sediment supply (Figures 6a and 8a). This occurs because the cobbles have a higher bedload layer height than the gravel and hence the increase in impact area with increasing discharge has a larger effect on erosion rate for cobbles. Compared to the constant slope case (Figure 6b), the transport stage can go beyond the threshold of suspension for cobbles when the slope is varied, resulting in a peak erosion rate at * ∕ * = 16 and zero erosion rate at * ∕ * > 29 (Figure 8b). The advection-abrasion model predicts a higher erosion rate than the deflection-abrasion model in the full range of transport and sediment supply conditions for the gravel, except within a small range where 1.2 < * ∕ * < 3.5 and 0.75 < q/q bc < 1.5 (Figure 7c). For cobbles, the deflection-abrasion model predicts higher erosion rate than the advection-abrasion model at low-to-intermediate transport stage (1 < * ∕ * < 10.5 ) and high relative sediment supply (q/q bc > 0.5) but becomes negligible beyond the threshold of suspension (Figure 8c). The combined-abrasion model for 10-mm gravel and 195-mm cobbles predicts a similar erosional pattern with the advection-abrasion model, where the erosion rate increases with increasing transport stage and relative sediment supply except at a small range of conditions where the deflection-abrasion model dominates (Figures 7 and 8d).

Comparison Between Advection-Abrasion and Deflection-Abrasion Models
The advection-abrasion process is important to consider in sediment-starved or detachment-limited bedrock rivers, where there is not enough alluvial cover to deflect particles to erode the banks. For example, Black Canyon in the Fraser River receives relatively little coarse sediment supply compared to the transport capacity (q/q bc = 0.034 for 195-mm cobbles) during a characteristic discharge event (Q w = 7,000 m 3 /s). The low sediment supply rate results in negligible bed coverage and hence a relatively small erosion rate predicted by the deflection-abrasion model. However, the advection-abrasion model relies on the turbulence intensity and predicts a lateral erosion rate of ∼60 mm 2 /yr for 195-mm cobbles.
The deflection-abrasion model assumes that the saltation hop length is infinite and predicts a zero lateral erosion rate for the suspended sediment. In contrast, the advection-abrasion model predicts a higher lateral erosion rate at larger transport stages, especially beyond the threshold of suspension. This condition mostly occurs for finer grain sizes, in narrow slot canyons that experience flash floods, in coarse-grained bedrock rivers during large flood events, or in steep bedrock rivers or knickzones. For example, Wire Pass, a slot canyon in Utah, is characterized by undulating sidewalls with a wavelength of 5-10 m (Carter & Anderson, 2006). These undulating sidewalls have been suggested to be created by the abrasion of sediment particles during flash floods, where the majority of the sediment (sand) can be easily transported in suspension (Carter & Anderson, 2006). The deflection-abrasion model would predict zero erosion, but the advection-abrasion model can predict the sidewall erosion caused by the suspended sediment impacts on these slot canyons. In bedrock rivers, large sediment (e.g., gravel and cobbles) can also be transported in suspension during extreme flood events or in steep reaches. For example, the typhoon induced extreme floods in Taiwan are capable of suspending large sediment with grain size ∼10 cm in the Liwu River (Hartshorn et al., 2002) and ∼57 cm in the narrow knickpoint of the Da'an River (Cook et al., 2009(Cook et al., , 2013, causing rapid lateral erosion of bedrock banks above the bedload layer height or near the water surface, which indicates the importance of lateral erosion within the suspension regime. Compared with the advection-abrasion model, the deflection-abrasion model dominates in a small parameter space where the transport stage is near the threshold of motion and the relative sediment supply is close to the threshold of full bed coverage (Figures 5-8). Low transport stage leads to relatively low turbulence energy and hence low impact energy and impact rate from the advection-abrasion model, but the large extent of alluvial cover on the bed is beneficial for deflecting the bedload particles to erode the channel banks. For example, the bedrock beds of the South Fork Eel River (Sklar & Dietrich, 2004) and the downstream reach of the Boulder Creek (Finnegan et al., 2017), California, USA, are covered by a nearly continuous alluvial cover with bedrock exposed only in isolated patches. Both of these channels have a transport stage below 2 for the median grain size during annual floods. Without considering the advection-abrasion model, application of the deflection-abrasion model in the downstream reach of the Boulder Creek has successfully captured the channel widening and steepening dynamics (Li et al., 2021), supporting the inference that the deflection-abrasion dominates lateral erosion in channels with low transport stage and high bed coverage.

Implications for Natural Bedrock Rivers
Our combined-abrasion model has implications for the relative importance of sediment size in eroding bedrock channel banks. Natural bedrock rivers transport a wide range of grain sizes, where finer sediment can be transported easily in suspension within almost the full distribution of discharge events but coarser sediment can only be mobilized over a limited duration. Previous bedrock lateral erosion models only consider bedload impacts, assuming that the long-term lateral erosion rate is controlled by the coarser sediment transported over a limited duration and that the influence of finer sediment is negligible in eroding bedrock cannel banks (Li et al., 2020(Li et al., , 2021Turowski, 2018Turowski, , 2020. However, our combined-abrasion model suggests that the finer sediment transported in suspension can be advected toward the banks to cause lateral erosion over a longer period and hence might dominate lateral erosion. To explore the competition between finer sediment and coarser sediment over the full distribution of discharge and sediment supply events, we use our reference site (Black Canyon) as an example. We calculate the probability density function (PDF) of the discharge at our reference site (Figures 9a and 9b) using the daily discharge data for the full 1912-2019 period of record from Water Survey of Canada at the Hope Gauge station (08MF005). Mixed grain sizes would complicate the entrainment of each grain size and the interaction amongst them (e.g., Parker, 1990;Wilcock & Crowe, 2003;Wilcock et al., 2001), and extending our model to mixed grain sizes would

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19 of 25 be more realistic, but would require reevaluation of several formulas, such as the critical shear stress for motion, the boundary roughness, and the transport capacity. For simplicity, we calculate the total erosion on bedrock channel banks from the sum of erosion by 10-mm gravel and 195-mm cobbles.
To incorporate the variability of sediment supply, we assume that the sediment supply Q s (t) follows a power law relation with water discharge, Q w (t), where k sw is a scaling factor which can be obtained from the annual sediment flux at our reference site, and m can be viewed as a rating exponent which generally varies between 1 and 3 based on sediment transport measurements in several bedrock rivers (Lague, 2010). For simplicity, we choose m = 2. We consider two sediment supply conditions: low sediment supply and high sediment supply. We use the annual coarse mass flux of 0.35 Mt/yr for the low sediment supply case where the bed is near fully exposed (reference site condition) and 35.0 Mt/yr for the higher sediment supply case where the bed is near fully covered. In each sediment supply scenario, the 10-mm gravel and the 195-mm cobbles are assumed to have the same mass flux.
For the lower sediment supply scenario, the combined-abrasion model predicts a much higher lateral erosion rate for the finer sediment (10-mm gravel) than the coarser sediment (195-mm cobbles) along the whole bedrock channel bank at our reference site (Figure 9c). The peak erosion rate for 10-mm gravel (∼0.1 mm/yr) is 9 times higher than 195-mm cobbles (∼0.01 mm/yr) and occurs at a lower elevation above the bed (∼0.08 m) than 195-mm The dominance of lateral erosion by the finer sediment causes the total erosion rate to follow a pattern similar to the 10-mm gravel, with 85% of the total erosion caused by the 10-mm gravel. In contrast, at high sediment supply, the bed is near fully covered by alluvium. Therefore, the coarser sediment can be deflected by the alluvium and can cause higher erosion rates (Figure 9d). The lateral erosion rate for the coarser sediment is larger than the finer sediment at an elevation of z < 0.4 m above the bed. The maximum erosion rate for the coarser sediment (∼13 mm/yr) is slightly higher than the finer sediment (∼10 mm/yr) (Figure 9d). At high sediment supply, the coarser sediment accounts for nearly the same percentage of the total erosion rates as the finer sediment (∼50%). Therefore, our combined-abrasion model implies that the finer sediment controls the lateral erosion in bedrock rivers with low sediment supply, and the importance of coarser sediment in eroding bedrock channel banks increases with increasing sediment supply.
An important implication of our advection-abrasion model is that the most frequent, small magnitude events might be more effective in eroding bedrock channel banks than the least frequent, extreme magnitude events (Figures 9a and 9b). For example, the small magnitude events (Q w < 2,000 m 3 /s) at our reference site are effective in transporting the finer sediment (10-mm gravel) and eroding bedrock channel banks. The small magnitude events are also >100 times more frequent than the extreme magnitude events (Q w > 12,000 m 3 /s). The net effect of the magnitude and frequency of discharge events reveals that the smallest events (Q w < 2,000 m 3 /s) are >10 times more effective in eroding bedrock channel banks than the extreme events (Q w > 12,000 m 3 /s) at our reference site (Figure 9). Like classic analyses of geomorphic work (Leopold et al., 1964), moderate magnitude and frequency might be the most effective in eroding bedrock channel banks. This is because of the net effect of the reduced impact velocity for small magnitude events and the reduced frequency of the extreme events. At our reference site, the majority of lateral erosion is caused by the discharge with a moderate magnitude that is between 6,000 and 8,000 m 3 /s.
Our combined-abrasion model includes two lateral erosion mechanisms: lateral erosion by bedload particle impacts that are deflected by the alluvium (deflection-abrasion mechanism) and lateral erosion by bedload and suspended load particle impacts that are advected by turbulence eddies (advection-abrasion mechanism). The model uses the law of the wall and the Rouse-Vanoni equation to calculate the vertical profile of sediment concentration, which does not incorporate the lateral variation of sediment concentration. More sediment can concentrate near the channel center due to the higher flow velocity here (e.g., Chatanantavet & Parker, 2008;Finnegan et al., 2007;Inoue et al., 2014;P. A. Nelson & Seminara, 2012), but more sediment can also be steered near the channel walls by the strong secondary flow found in natural bedrock rivers (Venditti et al., 2014). The future development of our model will need to more accurately estimate the sediment concentration near the walls. A quantitative test of our model performance is not possible due to a lack of direct field and laboratory observations of erosion by suspended particles, but a qualitative comparison of our model and field observations in a natural bedrock gorge (Beer et al., 2016(Beer et al., , 2017 has successfully reproduced several key patterns of lateral erosion, including the concentration of lateral erosion near the bottom of the channel walls, the occurrence of wall erosion near the water surface by impacts of suspended load, and the increased erosion of channel walls adjacent to the bed roughness due to the combined effect of advection-abrasion and deflection-abrasion mechanisms. Our model does not consider other lateral erosion mechanisms, such as plucking (Beer et al., 2017) that may dominate in rivers with weak or well-jointed bedrock channel banks. In meandering bedrock rivers, channel curvature can also enhance the sediment transport perpendicular to the bedrock channel banks and hence accelerate the bank abrasion (Cook et al., 2014;Mishra et al., 2018). To build a complete lateral erosion model, future work is needed to develop lateral erosion models by plucking and to incorporate the influence of channel curvature.
Our advection-abrasion model for lateral erosion does not incorporate turbulent fluctuations influenced by bank roughness, which might underestimate the lateral erosion rate. Future field and experimental work are needed to develop a relation of turbulent fluctuations caused by bank roughness, which can be notably greater than bed roughness (Li, Venditti, Rennie, & Nelson, 2022). Furthermore, current lateral erosion models calculate the hydraulic conditions based on the assumption of steady, uniform flow. However, 3D complex flow structure has been observed in laterally constricted bedrock rivers characterized by flow plunging toward the bed, with a high velocity core near the bed, and flow upwelling along the banks, causing a counter-rotating secondary flow structure (Hunt et al., 2018;Venditti et al., 2014). The presence of the complex flow structure causes higher and lower local flow velocities, turbulent intensities, shear stresses and erosion rates (Cao et al., 2022;Hunt et al., 2018;Li, Venditti, Rennie, & Nelson, 2022;Venditti et al., 2014), and hence needs to be coupled with erosion models to predict the morphodynamics of bedrock rivers. Nevertheless, our combined-abrasion model considers both LI ET AL.
10.1029/2022JF006806 21 of 25 bedload and suspended load, which is a more realistic representation of lateral erosion processes in bedrock rivers than the bedload deflection-only models. Our combined-abrasion model highlights the importance of tiny but energetic particles in eroding bedrock channel banks. Our model also predicts erosion rate explicitly as a function of sediment supply, shear stress, grain size and rock strength. These variables are rarely well constrained in bedrock rivers, but our model indicates the quantities that need to be constrained to enable mechanistic prediction of lateral bedrock erosion in the field.

Application to a Natural Bedrock Gorge
We applied our model to the natural bedrock gorge of the Gornera catchment, near Zermatt, Switzerland. The bedrock gorge is 25 m long and 5-6 m wide, located roughly 300 m downstream of a water intake (Beer et al., 2016). Rock tensile strength is 2.3 MPa and the median bedload grain size is 40 mm (Beer & Lamb, 2021;Beer et al., 2016). There are discharge and fine and coarse sediment flux time series from 2012 to 2013 (Beer et al., 2016), but there are no observations of suspended or bedload grain size, which are necessary for model inputs. Therefore, we cannot predict the erosion rate in the bedrock gorge, but we can explore the range of possible erosion rates for physically reasonable combinations of fine sediment grain size and sediment flux. We set the upper limit of fine sediment grain size as the largest grain size (5 mm) that can be suspended by the largest flood and the lower limit as the smallest grain size (0.1 mm) that is not viscously damped and hence can cause erosion. Delaney et al. (2018) measured the sediment flux of fine and coarse sediment within the Gornera catchment in 2016 and 2017 and found that the total sediment flux is ∼10 5 m 3 over 2 years and the fine sediment flux is 10 times of the coarse sediment flux. Therefore, the upper limit of the total sediment supply in the bedrock gorge is set to 10 5 m 3 and the lower limit is set as 1% total sediment supply (10 3 m 3 ). The fine sediment flux time series is predicted using an empirical relation presented by Delaney et al. (2018) that relates sediment flux to water discharge and time. Coarse sediment flux is estimated to be 1/10 of fine sediment flux (Delaney et al., 2018). Beer et al. (2016) measured the lateral erosion rate on three different channel banks, including banks with no flow obstacle, low flow obstacle, and high flow obstacle in the nearby streambed. Here, we select the field case that has no major obstacles near the channel banks so that we can solely focus on lateral erosion caused by impacts of particles deflected by alluvium and advected by turbulent eddies. We compared the erosion rate averaged over the channel bank and the distribution of erosion rate on channel banks (Figures 10a-10c). Compared with the mean observed erosion rate (∼0.9 mm/2 yr; Beer et al., 2016), the model predicts a lower erosion rate for particles finer than 0.3 mm and for sediment supply lower than 5 × 10 4 m 3 , and a higher erosion rate for coarser grain size and higher sediment supply (Figure 10a). The model prediction matches field observations for limited sediment supply and grain size conditions where grain size falls between 0.3 and 5 mm, and sediment supply falls between 5 × 10 4 m 3 and 6 × 10 4 m 3 (Figure 10a). The model also shows that the majority of erosion is caused by fine sediment except for very fine grain sizes (<0.22 mm) that are highly influenced by the viscously damping effect (Figure 10b). To further compare erosional patterns on the channel banks, we select a combination of grain size (2.5 mm) and sediment supply (5 × 10 4 m 3 ) that has the same erosion rate as the measurement. Our model overpredicts the erosion rate near the bed but underpredicts the erosion rate near the water surface ( Figure 10c). This is likely due to our use of a simple expression that shows the lateral flow turbulence intensity declines exponentially from the bed to the water surface (Nezu & Nakagawa, 1993). The predicted erosion rate is nearly fully caused by the impacts of fine sediment, because the erosion rate by the impacts of coarse sediment is more than one order of magnitude smaller than that of fine sediment (Figure 10c).

Conclusion
We developed an advection-abrasion model for lateral erosion by bedload and suspended load particle impacts that are advected by turbulence eddies. The model calculates the lateral erosion rate as a function of sediment concentration and impact velocity that are controlled by sediment supply, discharge, slope and grain size. The model predicts an undercut erosional bank shape, where the lateral erosion rate concentrates on the lower part of the banks and decreases progressively up to the water surface. The maximum erosion rate within the undercut zone peaks at intermediate discharge due to the reduction of near-bed sediment concentration as water depth increases, but increases with increasing slope due to the decline in water depth and hence the increase in nearbed sediment concentration, for given sediment supply. The maximum erosion rate within the undercut zone peaks at intermediate grain size due to the reduction in impact energy with increasing grain size for given shear 22 of 25 stress, but increases with increasing sediment supply until the bedload supply approaches the transport capacity due to the increase in sediment concentration. The erosion rate within the suspended layer is larger than the bedload layer for all supply and transport conditions explored, except at steep slopes where bedload layer height approaches water depth.
We combined the advection-abrasion model with the deflection-abrasion model. Both the advection-abrasion and deflection-abrasion models can be nondimensionalized as a function of transport stage and relative sediment supply for a given grain size. The deflection-abrasion model predicts a lower erosion rate than the advection-abrasion model for all supply and transport conditions explored, except within a limited condition where sediment is transported near the threshold of motion and the bedload sediment supply is close or higher than transport capacity. Therefore, the combined-abrasion model follows a pattern similar to the advection-abrasion model, where the erosion rate increases with increasing transport stage and increasing relative sediment supply until the bedload sediment supply approaches the transport capacity.
Application of the combined-abrasion model in a natural bedrock river with the wide distribution of discharge and sediment supply events and mixed grain size (finer and coarser sediments) indicates that the finer sediment causes more lateral erosion than coarser sediment in a low sediment supply environment, but coarser sediment Figure 10. Contour plots of (a) the ratio of predicted to measured erosion rate and (b) the ratio of predicted erosion rate by fine to coarse sediment as a function of volumetric sediment supply and grain size, and (c) an example of the comparison of the vertical profile of measured erosion rate and modeled erosion rate by fine sediment, coarse sediment and them combined, where the grain size of fine sediment is 2.5 mm and the total sediment flux is 5 × 10 4 m 3 .
LI ET AL.
10.1029/2022JF006806 23 of 25 becomes as important as finer sediment for eroding bedrock channel banks in rivers with high sediment supply rates.

Data Availability Statement
The code for Figure 1 was written in Python 3.7 by Tingan Li and can be adjusted to generate all other figures by changing variables in the code. The daily discharge data at Hope Gauging Station is available at https://wateroffice. ec.gc.ca/ for station WSC 08MF005. Both the code and data are archived at https://doi.org/10.20383/103.0638 in the Federated Research Data Repository for publication.