Geophysical Research Letters

Modeling the evolution of bedrock channel shape with erosion from saltating bed load



[1] The cross-sectional shape of a bedrock channel reflects the channel's history of incision. Although bedrock erosion due to impacts from saltating bed load particles is an important mechanism shaping bedrock channels, models of channel shape have thus far neglected the effects of sediment. Here, we present a model in which the cross-sectional shape of a bedrock channel evolves in response to abrasion from saltating bed load particles. We calculate the distribution of velocity and boundary shear stress with a two-dimensional hydrodynamic model, and we assume bed load is transported at capacity at the lowest point of the channel. Localized areas of alluviation emerge depending upon channel geometry and the imposed sediment supply, and bedrock erosion occurs in nonalluviated areas where there is bed load transport. The model captures the so-called “tools” and “cover” effects, and it simulates the dynamics between sediment supply and channel shape previously observed in experiments.

1. Introduction

[2] Bedrock rivers link climate, tectonics, and topography and drive landscape evolution. The shape of a bedrock channel provides a first-order control on its ability to convey water and sediment and implicitly controls incision rates, which provide boundary conditions for landscapes in tectonically-active environments. An understanding of channel shape is therefore critical to understanding the dynamics of bedrock channels and ultimately landscape evolution.

[3] Scaling arguments have traditionally been used to parameterize channel dimensions in the context of landscape evolution models. Typically this is done through an empirical hydraulic geometry law expressing channel width as a power-law function of river discharge, wQwb, where the exponent b generally falls in the range of 0.3–0.5 [Whipple and Tucker, 1999; Whipple, 2004; DeLong et al., 2007]. Although most channels tend to follow this simple width-discharge scaling, there are many exceptions [e.g., Finnegan et al., 2005, 2008; Whittaker et al., 2007] and there has been considerable discussion on the relationship between width and channel slope, uplift rate, and sediment transport [e.g., Finnegan et al., 2005; Turowski et al., 2007; Yanites and Tucker, 2010]. There also has been some debate on what controls the width-to-depth ratio of bedrock channels, with some theoretical and modeling work suggesting the ratio should be constant [e.g., Finnegan et al., 2005; Wobus et al., 2006], while other studies indicate that most bedrock channels should not exhibit a constant ratio [e.g., Turowski et al., 2009; Yanites and Tucker, 2010].

[4] The uncertainty surrounding the controls on bedrock channel dimensions suggest that explicitly modeling channel shape can inform these scaling relationships and provide additional insight on bedrock channel evolution. Notable efforts by Stark [2006], Wobus et al. [2006], and Turowski et al. [2009] have provided insight on how discharge and slope variations and tectonic effects can influence width adjustment in bedrock channels. These models, however, use some form of a shear stress/stream power erosion rule (i.e., E = k(τbτc)α, where E is erosion rate, τb is boundary shear stress, τc is a critical shear stress for erosion, and k and α are constants), effectively lumping all relevant erosional processes into a single hydraulic parameter, thereby ignoring the effects of sediment transport in the channel.

[5] In many bedrock river systems, the abrasion of bedrock due to impacts of saltating bed load particles is a critical mechanism for erosion [Sklar and Dietrich, 2004, 2006, 2008]. In these cases, sediment supply can be the primary control on channel erosion and the evolution of channel shape. Recent flume experiments in artificial bedrock channels [e.g., Finnegan et al., 2007; Johnson and Whipple, 2007, 2010] observed feedbacks between sediment transport, erosion, and the flow field in which the width of channel incision depended primarily on the sediment supply rate, with erosion focused at topographic low points when the channel was free of sediment cover, and focused at higher points when cover was present.

[6] In this study, we present a model of the evolution of bedrock channel shape that builds upon previous efforts by explicitly modeling erosion due to saltating bed load particles. Simulations of the Finnegan et al. [2007] experiments show that the model captures the dynamic relationship between channel shape and sediment supply, and the model recovers the so-called “tools” and “cover” effects wherein width-averaged erosion rates reach a maximum value at an intermediate sediment supply rate.

2. Model Description

2.1. Hydrodynamics

[7] We calculate flow and boundary shear stress with the ray-isovel model, described fully by Kean and Smith [2004] and outlined below. This model incorporates the effect of the wall shape on the flow field and thus should be appropriate for low width-to-depth ratio channels, and it has been supported by flume data [Kean et al., 2009]. It does not, however, model secondary flows, so we restrict ourselves at present to relatively straight reaches.

[8] The momentum equation for steady, uniform flow, averaged over turbulence, through a channel cross section is:

equation image

where a Boussinesq-type closure for Reynolds stresses has been employed. Here, ρw is the density of water, g is gravitational acceleration, S is the channel slope, y and z are the cross-stream and vertical coordinates, u is the downstream velocity, and K is an isotropic kinematic eddy viscosity. While the velocity field is defined in a Cartesian coordinate system, the fluid stress and eddy viscosity fields are more effectively defined in a coordinate system composed of isovels, contours of constant velocity, which are orthogonal everywhere to rays, which are streamwise surfaces of zero cross-ray shear (Figure 1b). The local boundary shear stress, τb, is equal to the downstream weight of the water between adjacent rays divided by the wetted perimeter separating them δpb; i.e.:

equation image

where L is the length of the ray. Similarly, the longitudinal shear stress acting on each element of isovel surface along a ray τlx can be defined as

equation image

where l is the distance along the ray from the boundary and δpl is the length along an isovel between two adjacent rays. The basic idea of the model is that turbulent mixing along a ray may be treated as in a plane open channel flow. More precisely, shear stress along a ray is related to the velocity gradient along the ray through an eddy viscosity K:

equation image

Near the channel boundary, the eddy viscosity is described by a generalization of the algebraic form of Rattray and Mitsuda [1974]:

equation image

until it reaches the channel-scale eddy viscosity K0, given by:

equation image

Here, u*equation image is the shear velocity, κ is von Karman's constant (≈0.41), and β is 6.24 [Shimizu, 1989].

Figure 1.

(a) Schematic showing model set up. Bed load transport occurs between yLsyyRs, with a maximum elevation of zbl. Local alluviation occurs where zblz(y) > t, where t is an empirical threshold value. In this example, the lowest portion of the channel is alluviated. (b) Schematic for the ray-isovel flow model. Velocity contours are shown every 0.5 m/s. The zone of bed load transport is outlined in red. (c) Distribution of boundary shear stress computed with the ray-isovel model (RIM). The depth-slope product is shown for comparison.

[9] We specify boundary conditions of ∂u/∂z = 0 at the water surface and u = 0 at l = l0, where l0 is the roughness height. To reduce the number of grid points needed to obtain a solution, the velocity a short distance above the boundary is computed using the law of the wall: u(l) = (u*/κ) ln (l/l0).

[10] Figure 1b shows the computed rays and isovels for a particular cross section and water discharge, and Figure 1c shows the computed boundary shear stress distribution. We also plot the depth-slope product (τb(y) = ρwgH(y)S), which in this example dramatically overpredicts boundary shear stress in the deepest part of this relatively narrow channel. As pointed out by Wobus et al. [2008], use of the depth-slope product here could lead to runaway deepening when combined with a stream power-type erosion rule.

2.2. Sediment Transport

[11] Having calculated the distribution of boundary shear stress at every point in the channel, we must now determine where in the channel sediment transport is occurring. Our fundamental assumption here is that bed load is concentrated in the lowest portions of the channel where it is transported at the local sediment transport capacity, qt(y), which can be calculated with a sediment transport equation such as that of Fernandez Luque and van Beek [1976]:

equation image

where Rb = ρs/ρw − 1, τ* is the Shields stress (τ* = τb/[(ρsρw)gD]), τc* is the critical Shields stress for incipient motion, ρs is the density of the sediment, and D is sediment diameter. In (7), qt(y) is expressed as a mass flux per unit width.

[12] The bed load layer width then is found by constraining the total calculated sediment flux to be equal to the imposed sediment supply Qs:

equation image

where yLs and yRs are the locations of the left and right edges of the bed load layer, which are iteratively determined in the model by starting at the lowest point in the channel and moving outward along the boundary until equation (8) is satisfied. This formulation assumes that bed load behaves as a longitudinal strip over which sediment is transported. Although it is not clear how universal this behavior may be, preferential transport paths of sediment transport have been observed in experimental bedrock channels [e.g., Chatanantavet and Parker, 2008], and over the long term, gravitational effects should force bed load to occupy the lowest part of the channel.

[13] If the lowest parts of the channel are unable to accommodate the supplied sediment, part of the channel will alluviate. We assume that a portion of the channel becomes alluviated if zblz(y) > t, where zbl is the bedrock elevation at yLs and yRs, z(y) is the bedrock elevation at coordinate y, and t is an alluviation threshold (see Figure 1a). We assume that this threshold scales with grain diameter. Although we lack systematic studies of the local conditions under which alluviation initiates, observations of erosion under varying sediment supply of Finnegan et al. [2007, Figure 10] and Johnson and Whipple [2010, Figure 4] suggest that t ≈ 5D. If it is determined that part of the bed is alluviated, the hydrodynamics are recalculated with the alluvial area “filled in” below the horizontal alluvial surface, and the model iterates until it converges on a velocity distribution and alluvial layer.

2.3. Bedrock Erosion

[14] Bedrock erosion is assumed to be entirely the consequence of abrasion by saltating bed load particles. Sklar and Dietrich [2004] have developed a mechanistic saltation-abrasion model, where the erosion rate E is the product of three terms: the volume of bedrock eroded per particle impact (Vi), the number of impacts per unit time (Ir), and the fraction of the bed area exposed (Fe):

equation image

[15] We use Sklar and Dietrich's [2004] formulations for Vi and Ir:

equation image
equation image

Here, Y is Young's modulus of elasticity, kv is a non-dimensional coefficient that depends partly on the properties of the bed load material, σT is the bedrock tensile strength, wf is the settling velocity of particles (computed here with the algorithm of Dietrich [1982]), and qs is the mass flux per unit width of bed load transport.

[16] The Fe term has received some attention in the literature; Sklar and Dietrich [2004] assume Fe decreases linearly as the ratio of sediment supply to sediment transport capacity increases, while Turowski et al. [2007] derived an exponential formula for Fe, and subsequent experimental work [e.g., Chatanantavet and Parker, 2008; Johnson and Whipple, 2010] have found greater variability in sediment flux - cover relations than predicted in either model, with Fe sensitive to local bed topography. Because our model explicitly calculates where sediment transport occurs, we do not need to include the Fe term, and we compute the local erosion rate as

equation image

Here, qs in equation (11) is evaluated as qt(y) (equation (7)). This formulation ensures that bedrock erosion occurs only in locations where bed load transport is occurring (zblz(y) ≥ 0) but not in topographically low areas that have been alluviated (zblz(y) > t). To remain consistent with the assumptions implicit in the Sklar and Dietrich [2004] model, the erosion rate calculated in equation (12) is directed vertically; proper evaluation of lateral erosion would require modeling particle saltation over sloping surfaces, which would significantly increase the model's complexity and would likely have a weak effect on channel shape.

3. Example Calculations

[17] Table 1 lists the parameters we use in a simulation of the Finnegan et al. [2007] experiments; water discharge Qw was held constant throughout the experiment while sediment supply Qs was varied in a stepwise manner. The Finnegan et al. [2007] experiments provide an ideal test case for our model since they varied the sediment supplied to their experimental bedrock channel and allowed it to adapt its geometry to the conditions they imposed. Figure 2 shows the results of the simulation compared to observations from the experiment.

Figure 2.

Bed topography from (a) the model simulation and (c) the Finnegan et al. [2007] experiment at the end of each period of constant sediment feed (Qs = 59 g/s from 0–12 hr, 25 g/s from 12–24 hr, 59 g/s from 24–42 hr, and 11 g/s from 42–60 hr). (b and d) The distribution of erosion in during each feed period by differencing the curves in Figures 2a and 2c.

Table 1. Parameters Used in the Simulation of the Finnegan et al. [2007] Experimenta
Water discharge Qw0.04 m3/sb
Grain size D6 mm b
Channel slope S0.02b
Bedrock roughness height l00.2 mmd
Critical Shields stress τc*0.045d
Alluvial thickness t30 mmd
Water density ρw1000 kg/m3d
Sediment density ρs2650 kg/m3d
Rock elastic modulus Y5 × 109 Pad
Dimensionless rock resistance parameter kv106c
Rock tensile strength σT0.5 × 106 Pab
Settling velocity wf0.351 m/se

[18] Although in some places the width and depth of erosion is overpredicted by the model, it captures the essential dynamics that were observed in the experiment. An erosional slot forms and becomes narrow when the sediment supply is reduced (from 12–24 hr). When the sediment supply is subsequently increased (from 24–42 hr), the low part of the channel alluviates and erosion is concentrated on the higher, lateral portions of the cavity.

[19] A significant contribution of the Sklar and Dietrich [2004] model is its incorporation of the so-called “tools” and “cover” effects, wherein the bedrock erosion rate is maximized at an intermediate sediment supply. This is achieved through their model's Fe = 1 − Qs/Qt term, which causes erosion rates to be small when Qs is very low (no tools for erosion) or very high (the bed becomes covered with sediment and protected from erosion). When we view our model from a width-averaged point of view, we find that it also captures the tools and cover effects. Figure 3 shows the width-averaged erosion rate (computed as equation image = 1/(yRwimage E(y)dy, where yRw and yLw are the right and left edges of the water surface, respectively) computed as a function of sediment supplied to a parabolic channel under constant average boundary shear stress. The curves for both the Sklar and Dietrich [2004] model and our model feature rising (tools-dominated) and falling (cover-dominated) limbs. In our model, the falling limb begins at the sediment supply at which alluviation starts, and because yLs, yRs, and zbl are strongly dependent upon channel shape, the shape of this curve will be different for different channel shapes.

Figure 3.

Influence of sediment supply Qs on width-averaged erosion rate for constant average boundary shear stress (equation image*/τc* = 5) for the parabolic cross-section shown in the inset figure. The erosion rate computed with Sklar and Dietrich [2004] is shown for comparison. The circled point corresponds to the water surface and alluviated layer shown on the inset figure. Parameters are the same as those in Table 1 with the exceptions of S = 0.015, D = 50 mm, l0 = 5 mm, t = 25 cm, and wf = 0.85 m/s.

4. Discussion and Conclusions

[20] Our model explicitly accounts for bedrock erosion from saltating bed load, an observable erosion mechanism based on measurable physical parameters, and it captures tools and cover effects. Of course, bed load abrasion is just one of many potential mechanisms of bedrock erosion; we currently ignore erosion due to plucking [e.g., Chatanantavet and Parker, 2009], lateral abrasion of the walls [e.g. Fuller et al., 2010], abrasion from suspended sediment [e.g., Lamb et al., 2008], or weathering from wet-dry cycles [e.g., Montgomery, 2004]. Nevertheless, our model does provide a basis for simulating the dynamic interaction between sediment transport, channel incision, and channel shape.

[21] In zero-dimensional saltation-abrasion models [e.g., Sklar and Dietrich, 2004; Turowski et al., 2007], the shape of the curve relating erosion rate to sediment supply (Figure 3) depends upon Fe, a function of Qs/Qt. In our model, the local value of qs/qt is always either 1 or 0 because we assume that bed load is always being transported at capacity. Yet, our model produces a equation imageQs relationship functionally similar to those in the zero-dimensional models. Because the distribution of bed load, alluviation, and erosion are controlled by channel shape, this suggests that the Fe term might better be thought of as a parameter that incorporates the effects of channel shape on the width of bed load transport and on patterns of local alluviation [e.g., Johnson and Whipple, 2010].

[22] As in the Finnegan et al. [2007] and Johnson and Whipple [2010] experiments, a distinct relationship between the shape of a channel and the record of sediment supplied to it emerges in our simulations. This suggests that for bedrock channels dominated by vertical incision, the shape (and therefore width) of channel is strongly, if not entirely controlled by the history of sediment supplied to it.


[23] This work has benefitted from the first author's discussions with Bill Dietrich, Leonard Sklar, Michael Lamb, Noah Finnegan, and Joel Johnson. Jason Kean helpfully provided code which became the basis for the hydrodynamic model implemented here. Reviews from Joel Johnson and an anonymous reviewer improved the clarity of the manuscript. This work was funded by a National Science Foundation Graduate Fellowship, the National Center for Earth-Surface Dynamics, and a National Science Foundation International Research Fellowship (grant 0965064).

[24] The Editor thanks Alan Howard and Joel Johnson for their assistance in evaluating this paper.