## 1. Motivation

[2] At the spatial scale of a bedrock river reach, the topography of the channel bed is a first-order control on spatial patterns of both basal shear stress and local sediment flux. Feedbacks between channel incision rate, bed roughness, shear stress and sediment flux are poorly understood yet fundamental to how fast channels will respond to changes in boundary conditions, including hillslope processes and discharge variability at short timescales and climate and tectonics over longer timescales.

[3] A variety of empirical and theoretical models have been proposed to predict bedrock incision rates and patterns along river networks. The intuitive assumption that bedrock incision should increase with increasing fluid shear stress, supported by some field data [*Howard and Kerby*, 1983], forms the basis of the shear stress (or stream power) family of erosion models [e.g., *Howard et al.*, 1994; *Stock and Montgomery*, 1999; *Whipple and Tucker*, 1999, 2002; *Whipple et al.*, 2000]. In this approach the combined effects of multiple erosion processes can be empirically calibrated to coefficients based on field data. However, lumping processes, and their different sensitivities to discharge and sediment flux, limits the predictive ability of these models, particularly when applied to spatial scales as small as channel reaches and timescales as short as flood events. In the saltation-abrasion model [*Sklar and Dietrich*, 1998, 2001, 2004, 2006], sediment flux rather than fluid discharge is the dominant driver of erosion, and incision rates have both positive and negative dependencies on sediment flux: the *tools effect*, in which higher sediment flux increases erosion rate due to more bed impacts, and the *cover effect*, in which higher sediment flux leads to greater local deposition, mantling the bed and shielding it from impact wear.

[4] Recent field studies have found evidence to support tools and cover effects on incision rates, through both short-term monitoring of sediment transport and bedrock incision [*Turowski et al.*, 2007a; *Johnson et al.*, 2010; *Turowski and Rickenmann*, 2008] and by interpreting patterns of bedrock river downcutting over long timescales [*Cowie et al.*, 2008; *Johnson et al.*, 2009]. Sediment flux–dependent models that include positive and negative feedbacks on incision rate are among the models that can simulate fluvial landscape forms including hanging valleys, knickpoints, and slope variability between adjacent channels [e.g., *Gasparini et al.*, 2006; *Wobus et al.*, 2006a, 2006b; *Brocard and van der Beek*, 2006; *Johnson et al.*, 2009]. However, the relationship between landscape form and a particular surface process model is typically nonunique [e.g., *Whipple and Tucker*, 2002], making morphological similarity necessary but not sufficient for validating a particular process model. In addition, process-based incision models often require better temporal and spatial constraints on variables (e.g., discharge, sediment flux, channel morphology) than can feasibly be measured in most field settings over a large parameter space, making model validation difficult.

[5] In this paper, we attempt to evaluate the form of proposed bedrock incision models by using controlled laboratory experiments to explore feedbacks between incision rate, basal shear stress, sediment flux, alluvial cover and evolving bed roughness for the case of river incision by abrasion. These independent and dependent variables are explicitly required, parameterized as a function of the other variables, or implicit in many bedrock erosion models. The flume experiments are idealizations of nature, but allow us to independently control variables that tend to be coupled in nature, letting us explore a range of parameter space.

[6] We use our experiments to evaluate the terms of a generic equation for sediment flux–dependent bedrock channel incision, based on the saltation-abrasion model [*Sklar and Dietrich*, 2004]:

where *E* is erosion rate, *Q*_{s} is sediment flux, *Fe* is the spatial fraction of bedrock exposed on the channel bed (i.e., not covered by sediment), *τ* is basal shear stress, and the shear stress term is raised to exponent *a*. While the right-hand side terms of (1) are not independent in that *Q*_{s} and *Fe* should also depend on *τ*, this formulation is useful to mechanistically isolate controls on *E*. Following *Sklar and Dietrich* [2004], we use *f*(*τ*) = *τ*/*τ*_{cr} − 1, where *τ*_{cr} is the critical shear stress necessary to initiate sediment motion. The quantity *τ*/*τ*_{cr} − 1 is referred to as the excess shear stress, and is zero at *τ* = *τ*_{cr}. Equation (1) is a parsimonious description of bedrock erosion rate that incorporates tools, cover and sediment transport dynamics in a manner consistent with the saltation-abrasion model [*Sklar and Dietrich*, 2004], although (1) neglects a range of dimensional coefficients as well as a possible reduction in erosion rate at high shear stress as sediment becomes suspended.

### 1.1. Erosional Dependence on Shear Stress

[7] In the theoretical case that all other parameters are held constant, it is unknown whether bedrock incision rates increase, remain unchanged or decrease with increasing fluid basal shear stress. Each of these dependencies has been proposed in recent fluvial bedrock incision models, corresponding to a range of proposed values for exponent *a*: *a* ≥ 1, *a* = 0, *a* = −0.5. In the saltation-abrasion model [*Sklar and Dietrich*, 2004], the exponent on excess shear stress is negative (*a* = −0.5), predicting that erosion rate will decrease with increasing shear stress, all else (e.g., sediment flux, alluvial cover) held equal. This scaling arises from an explicit parameterization of bed load saltation trajectories and their dependence on *τ* assuming a smooth, planar bed: saltation hop length increases with *τ*, reducing the number of particle impacts per unit bed area per time on a planar bed and thus counteracting a smaller increase in the vertical (normal to the bed) impact velocity with *τ*.

[8] *Chatanantavet and Parker* [2009] proposed a variation on equation (1) for bed load erosion in which *a* = 0, i.e., *E* ∝ *Q*_{s} · *Fe*. *Parker* [1991] studied downstream fining of clasts due to wear, and found no direct dependence on shear stress. *Chatanantavet and Parker* [2009] reasoned that impact wear on the bed should be proportional to impact wear on bed load clasts, hence *a* = 0.

[9] A range of positive values for shear stress exponent *a* have also been proposed, both with and without explicit sediment load considerations. *Howard and Kerby* [1983] found that channel incision rate scaled as *τ*^{a} with *a* ≈ 1 in rapidly eroding badlands. Based on scaling relations between discharge, fluid stresses and sediment transport, *Whipple and Tucker* [1999] and *Whipple et al.* [2000] argued that *a* should depend on erosion process, and propose 1 ≤ *a* ≤ 5/2 depending on whether plucking, bed load impact wear, or suspended load wear is the dominant erosion mechanism.

### 1.2. Erosional Dependence on Alluvial Cover

[10] The saltation-abrasion model [*Sklar and Dietrich*, 2004] assumes that the spatial extent of static alluvial patches increases as sediment flux increases, and zero bedrock is exposed where total sediment flux *Q*_{s} equals or exceeds total sediment transport capacity *Q*_{t}. Averaged over spatial and temporal variations along a channel reach, bedrock exposure *Fe* is predicted to vary as a linear function of *Q*_{s} and *Q*_{t}:

Flume experiments of *Chatanantavet and Parker* [2008] support the general form of equation (2).

[11] *Sklar and Dietrich* [2004] assume that the presence of sediment in active transport does not inhibit erosion, up to the point where the local bedrock bed is covered by a static alluvial deposit and local erosion rate drops to zero. In contrast, *Turowski et al.* [2007b] hypothesized that erosion may be inhibited not only by static alluvium but also by “dynamic” cover effects, in which sediment is mobile but the local erosion rate is nonetheless reduced (e.g., high concentrations of grains near the bed may reduce impact intensities). On theoretical grounds and consistent with the general form of some experimental results [*Sklar and Dietrich*, 2001], *Turowski et al.* [2007b] proposed that dynamic cover should vary exponentially with *Q*_{s}/*Q*_{t}, rather than linearly:

where ϕ should depend in some way on bed topography and is assumed to be 1 in the absence of additional constraints. When combined with a linear tools effect term (i.e., *E* ∝ *Q _{s}Fe*), the exponential cover model predicts a maximum erosion rate at

*Q*

_{s}=

*Q*

_{t}.

[12] One reason *Turowski et al.* [2007b] emphasized dynamic rather than static cover effects is their interpretation that when reach-averaged sediment flux (averaged both spatially and temporally) is less than reach-averaged transport capacity, static alluviation cannot occur anywhere along a channel because a positive feedback would occur where the excess transport capacity would entrain any static sediment. However, flume experiments [*Johnson and Whipple*, 2007; *Chatanantavet and Parker*, 2008] have demonstrated that even when overall *Q*_{s} < *Q*_{t}, local deposition can occur because local sediment flux and transport capacity depend on local bed topography and need not match the averaged or total values in a channel reach (by “local” we mean over a given small subset of bed area, such as a short distance along the bottom of an inner channel).

[13] Based on the flume experiments of *Chatanantavet and Parker* [2008], we hypothesize that the dependence of alluvial cover on *Q*_{s}/*Q*_{t} follows equation (2) rather than (3). Combining (1) and (2), the generic erosion equation that includes sediment and shear stress sensitivities becomes

Equation (4) represents a set of testable hypotheses: (1) erosion rate is linear in *Q*_{s} (all else held equal, including *Fe*), (2) *Fe* is linearly related to *Q*_{s}/*Q*_{t}, (3) erosion rate scales linearly with *Fe*, (4) erosion scales with excess shear stress to power *a* which can be determined from laboratory data, and (5) the combination of these factors can accurately predict reach-averaged experimental erosion rates. If dynamic rather than static cover effects are dominant, then equation (3) should provide a better fit to experimental data than (2). Another prediction of dynamic cover effects is that erosion should be inhibited uniformly and broadly in zones of concentrated sediment transport, not exclusively under patches of static cover. We later interpret the relative importance of static and dynamic cover effects from our experiments.