## 1. Introduction

[2] When debris flows sweep down steep canyons they often entrain most sediment in their paths and scour to underlying bedrock. Inspection of the bedrock after such a flow event indicates that the bedrock is also worn and cut by the flow [*Wieczorek et al.*, 2000; *Stock and Dietrich*, 2006].

[3] Considerable progress has been made toward understanding bedrock incision by fluvial processes [e.g., *Whipple and Tucker*, 2004; *Sklar and Dietrich*, 2004, 2008; *Turowski et al.*, 2008; *Johnson et al.*, 2009; *Johnson and Whipple*, 2010; *Lague*, 2010]. However, there is significant evidence that the process by which bedrock is incised in the steepest hillslopes is fundamentally different than the processes associated with fluvial bedrock incision. For example, the common practice of plotting local channel slope as a function of cumulative drainage area along a river longitudinal profile [e.g., *Whipple and Tucker*, 1999; *Sklar and Dietrich*, 1998; *Whipple and Tucker*, 2004] typically reveals a distinct scaling break at slopes below which mass flows typically do not entrain sediment or travel [*Stock and Dietrich*, 2003, 2006]. Experiments using flowing mixtures of gravel, sand, fine particles, and mud, here called *granular-fluid flows*, have provided further evidence for the ability of debris flows comprised of these materials to incise bedrock in a manner distinctly different from that by fluvial processes [*Hsu et al.*, 2007, 2008; *Hsu*, 2010]. Relatively little work has been done on granular-fluid flow incision mechanics and theory, leaving a significant gap in the understanding of landscape evolution in steep upland regions [e.g., *Stock and Dietrich*, 2003; *Stock et al.*, 2005; *Stock and Dietrich*, 2006].

[4] To develop a foundation for modeling bedrock incision by debris flows, *Stock and Dietrich* [2006] reasoned that the incision process is driven largely by the collisional forces associated with the relatively fluid-poor coarse front, or “snout” of such flows. They suggested that an expression for collisional stresses first proposed by *Bagnold* [1954] in simple sheared granular-fluid flows

could be used to estimate the relevant collisional bed stresses resulting from debris flows. In this equation, *ρ* is the density of the material comprising the particles, *d* is particle size, and is the shear rate. The parameter *λ* is what Bagnold called the linear concentration (of solids), where he showed that this parameter could be expressed as:

Here, *ν*_{s} is the volumetric solids concentration of the granular material, and *ν*_{s,max} is the maximum solids fraction, well-known for uniform spheres to be approximately 0.74. However, while some work has been done to understand packing limits for certain mixtures (e.g., theoretical work by *Rodine and Johnson* [1976] suggests *ν*_{s,max} ≈ 1 for multisize mixtures typical of natural debris flows), there is no general expression predicting *ν*_{s,max} for a mixture of different-sized particles. Based in part on equation (1), *Stock and Dietrich* [2006] proposed an expression for longterm bedrock incision rate *dz*/*dt* due to debris flows (where *z* is measured normal and outward from the free surface):

which depends on scaling of the excursion stresses with the average inertial stress (*K*_{0}), the relationship of rock resistance to incision rate (*K*_{1}), the inverse of the resistance of the bedrock to erosion (*R*), the frequency of occurrence of debris flows (*f*), the length of the snout (*L*), and a modified form of Bagnold's expression for collisional stress (bracketed term) raised to an empirically determined power, *n.* The modifications to Bagnold's collisional stress are as follows: the cos*ϕ* term is introduced to account for bed slope effects, *ν*_{s} (the volumetric solids concentration) replaces *λ*^{2}, *D*_{e} is a representative grain scale in the coarse front of a debris flow, and *w* is an empirical exponent. *Stock and Dietrich* [2006] derived expressions for the coefficients of equation (3) in terms of physical parameters such as the mechanical property of the bedrock and crack spacing to develop a geomorphic transport law for debris flow incision.

[5] In developing equation (3), *Stock and Dietrich* [2006] recognized challenges to applying equation (1) to natural flows composed of particle mixtures in water or muddy fluids. Most notably, debris flows are composed of particles of a wide size distribution for which the appropriate choice of *d* is not clear. This choice is complicated by the fact that particles tend to self-segregate by particle size [e.g., *Oyama*, 1939; *Gray and Thornton*, 2005], as well as by other particle properties such as density and shape [e.g., *Kakhakhar et al.*, 1997; *Plantard et al.*, 2006; *Hill et al.*, 2010], and there is substantial evidence that the constitutive behavior of even dry particle flows (*granular flows*) depends nonlinearly on the components in the mixture [e.g., *Rognon et al.*, 2007; *Hill and Zhang*, 2008; *Yohannes and Hill*, 2010; *Hill and Yohannes*, 2011]. Some have suggested that for predicting the dynamics of a granular-fluid flow in general, or a debris flow in particular, a particle size representative of a debris flow or of a region of interest be used [*Iverson*, 1997; *Stock and Dietrich*, 2006; *Hsu*, 2010]. For mixtures of different sized particles, the expression for *λ* (equation (2)) is also ambiguous, as *ν*_{s,max} is extremely sensitive to the particle size distribution [e.g., *Rodine and Johnson*, 1976; *Yohannes and Hill*, 2010; *Hill and Yohannes*, 2011]. Dimensional analysis by *Iverson* [1997] suggests it is appropriate to replace *λ* with *ν*_{s}/(1 − *ν*_{s}). Others [e.g., *Stock and Dietrich*, 2006; *Takahashi*, 2007] have used *ν*_{s} as a substitute for *λ*^{2}.

[6] An additional uncertainty in the application of equation (1) to debris flows involves the functional dependence of the normal stress (*σ*_{Bag} in equation (1)) on , in the presence of interstitial liquids. Many [e.g., *Hanes and Inman*, 1985; *Hsiau and Jang*, 1998], including *Bagnold* [1954] himself, suggested that for particle-fluid mixtures the power law dependence of stresses on , is somewhat less than quadratic. *Hsiau and Jang* [1998] suggested that the dependence of stress on varies if the granular temperature (or the variance of the particle velocities in a granular flow or granular-fluid flow) is non-constant, which is true for segregating mixtures [*Hill and Zhang*, 2008]. *Stock and Dietrich* [2006] addressed this uncertainty in the dependence of stress on by allowing *dz*/*dt* to vary as , where *w* must be determined by the nature of the flow.

[7] Data from physical experiments using small and large drums [*Hsu et al.*, 2008; *Hsu*, 2010] support the bedrock incision relationship proposed by *Stock and Dietrich* [2006]. In particular, *Hsu et al.* [2008] showed that incision rates into erodible basal panels in a 0.56 m diameter drum depended on grain size and bed material tensile strength in a manner similar to that predicted by *Stock and Dietrich* [2006]. In a 4 m diameter instrumented drum, *Hsu* [2010] simultaneously monitored the mean and dynamic loads on the bed and bedrock incision, from which she concluded that impact wear dominates incision in both dry granular flows and liquid-poor granular-fluid flows. Further, *Hsu* [2010] showed this wear is associated with large excursion forces and that the local variance of the force plate outputs increased roughly as the square of *D*_{84}. In the presence of moderate-to-high concentrations of interstitial fluids such as water and mud, *Hsu* [2010] found that sliding wear also becomes important in the erosion of relatively low tensile-strength bedrock substrates. In this paper, we focus on the dynamics of liquid-poor granular-fluid flows, typical of debris flow fronts, that we consider well-represented by dynamics in granular flows. We consider primarily those dynamics relevant to the incision of high tensile-strength bedrock found in nature and for which equation (3) is most relevant.

[8] To summarize, two key assumptions of the *Stock and Dietrich* [2006] debris flow incision model (equation (3)) supported by experimental investigations [*Hsu et al.*, 2008; *Hsu*, 2010] are 1) excursions in collisional stresses at the coarse front drive bedrock incision for the majority of rock strengths found in nature, and 2) these collisional stress excursions scale with the square of a representative grain size in the coarse front. Although the drum experiments by *Hsu* [2010] support these two assumptions, the data do not reveal how, theoretically, to calculate excursion stresses or the grain size dependency of these stresses. We propose that an important next step toward building a mechanistic and predictive model for bedrock incision by rocky debris flows, debris flow snouts, and other fluid-starved granular flows involves determining how the excursion stresses and the collisional forces scale with grain size distribution and flow dynamics.

[9] To address this, we use Discrete Element Method (DEM) simulations (first proposed by *Cundall and Strack* [1979]) to explore controls on boundary forces generated by granular flows. This approach is well-suited for dry granular flows, which we suggest represents salient features of the coarse front of even muddy debris flows, and has the distinct advantage over continuum models in explicitly predicting the dynamics of individual particles. There are several benefits to using this type of simulation. Segregation dynamics that concentrate large particles toward the flow front emerge spontaneously in the simulations, so there is no need to impose an empirical model for segregation. Further, the collective and individual impact processes of the particles on the boundaries can be directly monitored and related to other variables such as local and system-wide measures of particle size distribution [e.g., *Alam and Luding*, 2003; *Rognon et al.*, 2007; *Hill and Zhang*, 2008; *Yohannes and Hill*, 2010; *Hill and Yohannes*, 2011].