Corresponding author: R. M. Iverson, U.S. Geological Survey, 1300 SE Cardinal Ct., Vancouver, WA 98683, USA.
 Analyses of mass and momentum exchange between a debris flow or avalanche and an underlying sediment layer aid interpretations and predictions of bed-sediment entrainment rates. A preliminary analysis assesses the behavior of a Coulomb slide block that entrains bed material as it descends a uniform slope. The analysis demonstrates that the block's momentum can grow unstably, even in the presence of limited entrainment efficiency. A more-detailed, depth-integrated continuum analysis of interacting, deformable bodies identifies mechanical controls on entrainment efficiency, and shows that entrainment rates satisfy a jump condition that involves shear-traction and velocity discontinuities at the flow-bed boundary. Explicit predictions of the entrainment rateEresult from making reasonable assumptions about flow velocity profiles and boundary shear tractions. For Coulomb-friction tractions, predicted entrainment rates are sensitive to pore fluid pressures that develop in bed sediment as it is overridden. In the simplest scenario the bed sediment liquefies completely, and the entrainment-rate equation reduces toE = 2μ1gh1 cos θ(1 − λ1)/ , where θ is the slope angle, μ1 is the flow's Coulomb friction coefficient, h1 is its thickness, λ1 is its degree of liquefaction, and is its depth-averaged velocity. For values ofλ1ranging from 0.5 to 0.8, this equation predicts entrainment rates consistent with rates of 0.05 to 0.1 m/s measured in large-scale debris-flow experiments in which wet sediment beds liquefied almost completely. The propensity for bed liquefaction depends on several factors, including sediment porosity, permeability, and thickness, and rates of compression and shear deformation that occur when beds are overridden.
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 Here, to aid mechanistic prediction and interpretation of bed-sediment entrainment, I analyze the phenomenon in two ways. First I present a heuristic slide-block model aimed at clarifying the role that bed-sediment inertia and entrainment efficiency play when slide-block mass and momentum evolve simultaneously. I then formulate a three-layer, depth-integrated, continuum-mechanical model of mass and momentum exchange between a flow, an erodible bed, and a stable substrate. A depth-integrated approach has inherent limitations, but it holds some advantages over more-detailed approaches because it combines the effects of boundary conditions, mass fluxes, and momentum fluxes into conservation equations that encapsulate the mechanics of individual layers as well as the three-layer system as a whole. A depth-integrated analysis also facilitates evaluation of jump discontinuities, which play a crucial role at the interfaces between adjacent layers.
 New analyses of the entrainment process are warranted in view of results of recent field and laboratory studies, which have demonstrated that entrainment beneath debris flows and avalanches generally occurs by progressive scour rather than mass failure of bed material [Mangeney et al., 2010; Berger et al., 2011; Iverson et al., 2011; Reid et al., 2011; McCoy et al., submitted manuscript, 2012]. The latter three studies have additionally shown that entrainment rates are sensitive to pore fluid pressures that develop in bed sediment as it is overridden. Thus, a primary goal of the analyses here is to provide a theoretical basis for interpreting and predicting factors that determine entrainment rates and their dependence on pore pressures and effective-stress states in bed sediment. A secondary goal, which cannot be separated from the first, is to assess coupling between entrainment and flow dynamics.
2. Entrainment Velocity
 Neither of the analyses presented here considers detailed, grain-scale mechanics, but both analyses include constraints imposed by momentum conservation as the bed exchanges arbitrarily small elements of mass with the overriding flow. By definition, these elements of bed material must attain the basal flow velocity in order to be entrained [cf.Gray, 2001]. In this context “erosion,” “scour,” and “entrainment” are synonymous, because the analyses assume that material along the flow-bed interface has little or no velocity before it is entrained, and has the basal flow velocity after it is entrained. Intermediate velocities necessarily occur as small elements of bed material accelerate to merge with the flow, but they cannot be resolved by slide-block or depth-integrated continuum analyses.
 The presence of an erodible bed can make basal slip velocities difficult to assess because the location of the boundary between the flow and bed may be indistinct – as documented for idealized flows of nearly identical grains [Komatsu et al., 2001; Armanini et al., 2005; Larcher et al., 2007; Crosta et al., 2009b]. For flows of poorly sorted geological debris, however, distinctions between infinitesimal and finite slip velocities can hinge largely on the scale at which the flow-bed interface is observed (Figure 1). For example, continuum-scale velocity fields constructed using measured trajectories of large particles in experimental debris flows indicate that the flows have finite basal slip velocities [Johnson et al., 2012]. From this perspective entrainment analyses that consider the role of finite basal slip velocities appear appropriate. Such analyses also demonstrate that entrainment-rate data, interpreted in the context of momentum conservation and bed stress states, can help constrain basal slip velocities when they are not directly observable.
3. Slide-Block Analysis
 This analysis considers behavior of a single body (i.e., a slide block) that encompasses all mass in motion as the block descends a slope covered with entrainable bed material. By definition, the evolving block mass includes material actively undergoing entrainment, but does not include bed material prior to the onset of its entrainment (Figure 2). This definition precludes analysis of the behavior of slide blocks that lose mass through deposition, as clarified below.
 Newton's second law states that the evolving slide-block momentum obeys
where t is time, mis the slide-block mass,v is its velocity, and F is the sum of external forces acting on the block. If the block descends a plane inclined at a uniform angle θ, and its descent is resisted by basal Coulomb friction characterized by a friction coefficient μ, then F is given by
summarizes the net effect of gravitational acceleration, g. The net force F can be represented as F = mg′ because, like the gravitational driving force, the Coulomb friction force is proportional to mg.
 Use of a Coulomb friction rule for bed sediment undergoing entrainment assumes that the sediment immediately beneath the slide block attains a state of limiting equilibrium before entrainment begins. Gravity suffices to produce this limiting equilibrium state if θ equals the sediment's angle of repose. If θis less than the angle of repose, a limit-equilibrium state can result from reduction of frictional resistance that occurs as the slide block overrides the bed [cf.Mangeney et al., 2007]. Such a reduction might be caused by bed deformation that increases pore fluid pressure and thereby reduces the effect of mg μ cos θ, for example [e.g., Hungr and Evans, 2004]. Modeling pore pressure effects on bed-sediment resistance is beyond the scope of the slide-block analysis, but is emphasized in the continuum-mechanical analysis that follows.
 Coulomb friction coefficients for landslides, debris flows, and bed sediment are commonly assumed to be constant, and constant friction represents an important baseline case. On the other hand, evidence from diverse experiments with dense grain flows and grain-fluid suspensions indicates that Coulomb friction increases systematically as the shear rate increases [e.g.,MiDi GDR Group, 2004; Forterre and Pouliquen, 2008; Boyer et al., 2011]. This shear-rate effect cannot be fully incorporated in a slide-block model, but it can be represented in a rudimentary way by expressing the basal friction coefficient as the sum of a constant componentμ0and a velocity-dependent componentμV(v/V),
where μV and V are constants. More specifically, V is a velocity scale quantified in the analysis that follows.
 Substituting (2), (3), and (4) into (1)yields a slide-block momentum equation that can be written as
where the forcing terms on the right–hand side include constants defined as
The term −v(dm/dt) arises in (5) as a consequence of momentum conservation, but it can be interpreted more precisely as the reaction force experienced by the slide block as it exerts force to overcome the inertia of static bed material while accelerating that material to the ambient block velocity v.This effect of inertia constitutes an internal force because the slide-block definition includes all material in motion. By contrast, the action of an additional external force (i.e., a resistive force exerted by stationary bed material) would be required to cause deposition of basal slide-block material (i.e.,dm/dt < 0). Such externally driven mass loss differs fundamentally from internally driven mass loss that occurs when a rocket forcibly ejects matter to propel itself [Hungr, 1990; Erlichson, 1991]. As a result, “rocket equations” similar to (5) are inappropriate for analyzing motion of debris flows or avalanches that lose mass through deposition [cf. Cannon and Savage, 1988; Van Gassen and Cruden, 1989].
 The physical implications of (5) can be investigated by first considering some special cases in which constraints are placed on m and/or v. For example, if dm/dt = 0, then the entire net driving force m(G − Γv) causes slide-block acceleration, and(5) has solutions that describe maximum velocities attainable by slide blocks at any particular time. For the initial condition v = v0, one of these solutions applies if friction is exclusively rate-independent (i.e., Γ = 0):
Another solution applies if rate-independent and rate-dependent friction coexist:
Whereas (8)indicates that the slide-block velocity increases constantly,(9)shows that rate-dependent friction causes the velocity to asymptotically approach the steady statev = G/Γ. A fixed fraction of G/Γ is therefore a logical choice for the velocity scale V in (4) and (7).
 In more-general cases withdm/dt> 0, the slide-block behavior described by(5) can be more complex. The implications of dm/dt can be identified most readily by first rewriting (5) as
If the block velocity is constant (v = v0 and dv/dt = 0) and the initial condition is m = m0, then the solution of (10) shows that m grows exponentially:
This unbounded growth of m occurs in the presence of a constant v because the entire net force F = m(G − Γv0) acts to cause entrainment, while the net force itself grows in proportion to m.
 The behavior described by (11) reveals a fundamental asymmetry in the dynamics of Coulomb slide blocks entraining mass: a constant m implies stable growth of v, as described by (8) and (9), whereas a constant v implies unstable growth of m. (The mathematically stable case with Γ > G/v0 in (11) lacks physical relevance because it implies that mass loss occurs, violating the assumptions used to derive (5)). Equation (11)also demonstrates that a constant slide-block velocity cannot be accompanied by a constantdm/dt. Thus, no physically valid entrainment rule can have the seemingly plausible form dm/dt ∝ v.
 On the other hand, equations (10) and (11) provide a basis for proposing an entrainment formula in which dm/dt ∝ m/v. The basis is established by first considering an ideal case in which frictional resistance is zero (i.e., G = G0 = g sin θ, Γ = 0) and the driving force F = mG0 is allocated entirely to entrainment, implying that v is constant. Then (10) reduces to
where [dm/dt]max is the maximum entrainment rate that can occur without causing deceleration and eventual stoppage of the slide block. Because (12)describes an upper-bound condition for sustainable entrainment, it is reasonable to postulate that a more-general entrainment formula can have the analogous form
where αis an entrainment-efficiency parameter. The value ofα must satisfy 0 ≤ α ≤ 1, but it need not be constant. Indeed, if the value of α evolves so that it always obeys α = 1 − [(dv/dt + Γv)/G], then (13) simply restates (5), resulting in indeterminate behavior.
 Although a constant αis not mandatory, the behavior of slide blocks with constant entrainment efficiency is instructive. This behavior can be clarified by finding solutions of the two-equation system,(5) and (13), for cases in which αis constant. One solution applies if friction has no rate-dependence (i.e., Γ = 0):
In this case αregulates slide-block velocity and mass growth in a straightforward way. Ifα = 1/2, for example, normalized values of v and m grow at identical linear rates: v/v0 = m/m0 = 1 + (Gt/2v0). In this special case the net driving force is allocated equally between the two growth phenomena. In general, however, equal force allocation is unnecessary, and balanced growth of v and m does not occur. For α < 1/2, linear growth of v outpaces nonlinear growth of m, whereas the opposite is true for α > 1/2. If α → 1, then m → ∞, and this singular behavior reinforces the evidence of instability expressed by (11).
 A different solution of (5) and (13) applies if αis constant and rate-dependent friction exists (Γ > 0):
This solution indicates that exponential growth of m is inevitable for any nonzero value of α. Physically, this behavior results from v approaching the steady state v = (1 − α)G/Γ, implying that an increasing percentage of the net driving force is allocated to mass growth. Therefore, somewhat paradoxically, velocity stabilization by rate-dependent friction exerts a destabilizing effect on mass and momentum growth. For small values ofαand Γ, however, a seemingly stable near-balance between the slide-block velocity and entrainment rate can persist for a considerable time before conspicuous acceleration of mass growth begins (Figure 3). This behavior temporarily mimics that of an equilibrium relationship between v and dm/dt, although no true equilibrium is possible.
 Despite its simplicity, the slide-block analysis illustrates several important principles that apply to motion and mass entrainment resisted by rate-independent and/or rate-dependent Coulomb friction. Perhaps the most fundamental principle is that bed-sediment inertia produces an inverse relationship between the slide-block velocity and mass-entrainment rate, if all other factors are equal. Subsequent sections of the paper present a continuum mechanical analysis that reinforces this principle and also demonstrates how boundary shear tractions influence the entrainment efficiency parameterized above byα.
4. Depth-Integrated Continuum Analysis
 A more-thorough analysis considers mass and momentum exchange between three continuous layers, one representing a flow with a free upper surface, one representing an entrainable bed-sediment layer, and one representing a deeper substrate that cannot be entrained owing to its high strength (Figure 4). This configuration is similar to that of large-scale entrainment experiments involving tabular layers of nearly homogeneous sediment on a uniform, concrete slope [Iverson et al., 2011; Reid et al., 2011]. The analysis employs a Cartesian coordinate system in which x points in the downslope direction, ypoints in the cross-slope direction, andz points upward, normal to the slope. To emphasize essential mechanical principles and minimize mathematical complexity, it assumes that variations of all quantities in the y direction are negligible, and that temporal and spatial variations of the bulk density ρof the flow and bed material are negligible. A derivation of the depth-integrated balance equations used in the analysis shows how the effects of these variations can be included, however (Appendix A).
 Like most depth-integrated theories, the analysis assumes that thez-momentum balance everywhere reduces to an equation describing one-dimensional static equilibrium,
where σzz is the z-direction total normal stress (reckoned positive in compression),g is the magnitude of gravitational acceleration, and θ is the slope angle. Treatment of the z-momentum balance in such a simple way may be inappropriate in some instances, particularly if flows traverse irregular terrain [Denlinger and Iverson, 2004]. Here, however, this simplification sharpens the focus of the analysis by isolating the effects of the x momentum component on entrainment.
4.1. Balance Equations for Individual Layers
 Under the assumptions summarized above, the depth-integrated mass balance andx-momentum balance for each layer reduce to (Appendix A):
where h is the layer thickness in the z direction, is the depth-averaged velocity of the layer, ΣF is the sum of external forces per unit volume acting on the layer, and the subscripts top and bot denote the upper and lower boundaries of the layer, respectively. In (19)Etop and Ebot describe z-direction boundary-migration rates caused by transfer of mass through the upper or lower boundary. At each locationEis reckoned positive downward in order to emphasize boundary migration that occurs during bed-sediment entrainment by an overriding layer. In(20)the boundary momentum-flux terms − ρvtopEtop and ρvbotEbotarise mathematically from use of the mass-conservation equation during depth integration of the momentum-conservation equation (Appendix A). Physically, these momentum-flux terms account for the possibility that mass entering a layer through its boundary has a nonzerox-velocity component (vtop or vbot), and thereby transfers some x-momentum with it. Of course, momentum must be conserved during such a transfer – a matter addressed below by considering interactions between adjacent layers.
Equation (20)also contains a Boussinesq momentum-distribution coefficient,β, which accounts for the effects of nonuniform advection of x-momentum as a function ofz within each layer. The coefficient is defined mathematically by
where v is the local x-velocity component. Values ofβ determined from (21) range from β = 1 for a flow with all motion concentrated as basal slip to β= 4/3 for a flow with a linear velocity profile and no basal slip. Depth-averaged single-layer models of debris flows and avalanches commonly assume thatβ = 1, with little consequent error, but nonuniform velocity distributions can play an important role in momentum exchange that accompanies mass exchange between layers.
 An explicit expression for the force per unit area ΣFdz in (20) can be obtained in a standard way by considering the driving effect of gravity and resisting effect of Cauchy stresses [e.g., Malvern, 1969]. Application of Leibniz' theorem [Abramowitz and Stegun, 1964, pp. 11] during depth integration of the resulting stress-gradient terms ∂τxx/∂x and ∂τzx/∂z then yields
In this equation the longitudinal normal stress τxx, its depth-averaged value , and its boundary values τxxtop and τxxbotare reckoned positive in tension, a convention that is necessary in order to establish the correct signs of the boundary shear-traction terms,τzxtop and τzxbot. To reconcile this tension-positive sign convention with the compression-positive convention of the normal stressσzz in (18), I define
Taken in order, the six terms on the right-hand side of(24)account for (a) the downslope gravitational driving force, (b) the depth-averaged longitudinal force due to longitudinal thickness and normal-stress gradients, (c) the downslope force due to the compressive longitudinal boundary traction acting on upslope-facing facets of the layer's upper surface, (d) the upslope force due to the compressive longitudinal boundary traction acting on upslope-facing facets of the layer's lower surface, (e) the boundary shear traction acting on the upper surface of a layer, and (f) the boundary shear traction acting on the lower surface of a layer.
where κis a proportionality factor that can play the role of a Rankine earth-pressure coefficient or elastic Poisson's ratio. If the value ofκ is constant within any particular layer, as I assume here, then a derivation in Appendix B shows that (24) reduces to
where His the total thickness of the three-layer system (Figure 5), and the subscript zx has been omitted from τtop and τbotin order to simplify the notation. Collapse of the three longitudinal stress-gradient terms in(24) into a single term − κρgh cos θ(∂H/∂x) in (26) implies that the effect of ∂H/∂x is the same in all layers, except insofar as differing layers have differing values of κ and h.
 The simplification of (24) that leads to (26) indicates that geometric details of the boundaries between adjacent layers play no role in the theory, but this simplification is contingent on the applicability of (18) and (25). Indeed, these two equations represent the central assumptions that distinguish the elementary theory developed here from a more comprehensive theory that would include the effects of multidimensional, heterogeneous stress fields and thereby resolve the effects of stress variations in the vicinity of boundary irregularities. Such a theory would necessarily involve its own set of constitutive assumptions, however.
 Substitution of (26) in (20) yields the x-momentum equation I use to assess the relationship between flow dynamics and entrainment,
Except for the presence of τtop and ρvtopEtop, (27)has a form like that of a single-layer, depth-averaged flow momentum equation that accounts for possible mass and momentum fluxes through the basal boundary.
 A clear relationship between (27)and the slide-block momentumequation (5) can be established if β = 1, implying that v(z) is constant. For this special case, expansion of the derivatives on the left-hand side of(27)produces some terms that are identical to those on the left-hand side of(19), and use of (19) to cancel these terms introduces a new term, −ρEtop + ρEbot, in (27). Combining this term with the boundary momentum-flux terms −ρvtopEtop and +ρvtopEtopon the right-hand side of(27) then converts (27) to
where d/dt = ∂/∂t + (∂/∂x) denotes a Lagrangian total time derivative in a frame of reference that translates with the material velocity . This Lagrangian form of the momentum equation is correct mathematically but is “non-conservative” physically because its left-hand side describes evolution of rather than the conserved quantity ρh. The importance of this distinction can be clarified by considering a special case in which the upper boundary is a free surface with τtop = 0 and Etop = 0, and the lower boundary is an eroding surface with Ebot > 0 and vbot = 0. In this case the boundary momentum-flux terms vanish from the Eulerian momentumequation (27), but the Lagrangian equation (28) retains the term −ρEtop. The resulting form of (28),
is closely analogous to the momentum equation used by McDougall and Hungr to analyze the effects of entrainment, and is also closely analogous to the slide-block momentumequation (5). Indeed, except for the term −κρgh cos θ(∂H/∂x), (29) is identical to (5) if τbot is treated as a Coulomb stress and (5) is divided by the basal area ΔxΔy to find that m(dv/dt)/(ΔxΔy) = ρh(dv/dt) and v(dm/dt)/(ΔxΔy) = ρvEbot. The close correspondence of (5) and (29) clarifies that, in a Lagrangian reference frame, the term −ρEbot accounts for the inertia of bed material that is accelerated from rest to reach the ambient flow velocity . Use of the term −ρEbot to evaluate the effects of deposition in a Lagrangian frame is inappropriate, however–for reasons described in conjunction with (5).
 No term analogous to −ρEbot appears in the Eulerian momentum equation (27)because all inertial effects of entrainment or deposition are represented by the left-hand side of(27) unless mass passing through the upper or lower boundaries carries some momentum with it [cf. Chen et al., 2006; Sovilla et al., 2006]. A key advantage of (27) is that it can represent inertial effects more precisely than (28) or (29) can, because it does not require the assumption of a uniform velocity profile (i.e., β = 1). As shown below, the role of nonuniform velocity profiles is critical when evaluating interactions between adjacent layers.
4.2. Interactions Between Adjacent Layers
 Various special forms of (27) apply to the flow, bed, and substrate layers (Table 1). To distinguish these special forms, I use the subscript 1 to designate variables in the overriding flow layer, the subscript 2 for the entrainable bed-sediment layer, and the subscript 3 for the high-strength substrate layer (Figure 5). The equations of Table 1 assume that layer 1 experiences a negligible boundary shear traction on its upper surface. Thus, the only boundary shear traction affecting layer 1 is the resisting shear traction exerted by the top of the underlying sediment bed, −τ2top. The magnitude of −τ2top may differ from that of the traction the base of the flow exerts on the bed, +τ1bot, owing to a stress-field discontinuity at the flow-bed interface. (Such discontinuities occur commonly in deforming Coulomb materials, for example [Jaeger, 1969]). The sediment bed (layer 2) experiences the shear traction +τ1bot applied at its top surface by the overriding flow and also a resisting basal shear traction applied by the strong substrate, −τ3top. The substrate, in turn, experiences the traction +τ2bot applied by the overlying bed sediment and the basal resisting traction −τ3bot at some arbitrarily great depth where z = 0 and static stress equilibrium applies.
ρgh2 sin θ − κ2ρgh2 cos θ + τ1bot − τ3top − ρv1 botE
3 (strong substrate)
ρgh3 sin θ − κ3ρgh3 cos θ + τ2bot − τ3bot
 The only nonzero entrainment rate included in the equations of Table 1 is Eat the flow-bed boundary, which is the focus of this analysis. Recall thatEis reckoned positive if the boundary migrates downward as mass is transferred from the bed-sediment layer to the flow layer. Thus, if the mass involved in this transfer has nonzero momentumρv2top per unit volume, the momentum flux into the flow per unit area is ρv2topE, as shown in Table 1. Similarly, if mass is transferred from the flow to the bed during deposition (E < 0), the bed layer gains momentum at the rate − ρv1botE.
 The flow and bed layers can exchange momentum only if a velocity contrast exists along their interface, and momentum must be conserved during such an exchange. This requirement has ramifications that can be assessed by first adding the equations for layers 1 and 2 to obtain a momentum-conservation equation for the two-layer system as a whole. The right-hand side of the two-layer equation contains several forcing terms that must cancel one another because −τ3topaccounts for the only external force (aside from gravity) that acts on the two-layer system. This cancellation of terms yields −τ2top + τ1bot + ρv2topE − ρv1botE = 0, and the two-layer momentum equation thereby reduces to
The terms canceled from (30) can be rearranged to obtain
Importantly, the expressions in brackets in (31)can describe jumps (i.e., finite changes) in shear tractions and velocities that may exist at the flow-bed interface. If such jumps do not exist, then there is little reason to treat layers 1 and 2 as mechanically distinct, andequation (30)contains all the pertinent depth-integrated information about momentum conservation in the two-layer system. If jumps do exist, however, thenequation (31)contains crucial information that supplements the flow- and bed-layer equations ofTable 1.
 A result similar to (31)has been obtained previously by applying general Rankine-Hugoniot jump relationships to bed-erosion problems [Fraccarollo and Capart, 2002], but here (31)is derived directly from depth-integrated conservation laws. In the present context(31)is the only relevant jump condition, because the normal-stress distribution described by(18) is continuous across the boundary, and the mass distribution is continuous across the boundary by virtue of the constant bulk density ρ assumed for the flow and bed material [cf. Chadwick, 1976, pp. 114–120].
5. Entrainment-Rate Equations and Predictions
 Rearrangement of (31)shows that momentum conservation demands that the bed-sediment entrainment rate obeys
The numerator of this equation can be interpreted as an excess boundary shear stress, which expresses the difference between the basal shear traction exerted by the flow (τ1bot) and the boundary shear resistance exerted by the bed (τ2top). The proportionality of E to τ1bot − τ2top is consistent with some longstanding ideas about threshold conditions for entrainment of bed sediment [e.g., Vanoni, 1975]. On the other hand, the denominator of (32) indicates that E decreases as the basal slip velocity v1botincreases – a result that contradicts conventional wisdom but corroborates the results of the slide-block analysis insection 3.
 Two other implications of (32) warrant mention. First, the numerator of (32) implies that deposition rather than entrainment occurs if τ1bot < τ2top, provided that v1bot > v2top. Second, the denominator of (32) indicates that E → ∞ if τ1bot > τ2top and v1bot = v2top, consistent with the view that layers 1 and 2 behave as a single layer if there is no velocity contrast at the flow-bed interface. The conditionE → ∞ implies that downslope motion of the bed occurs en masse in response to quasistatic forcing, as envisaged by Takahashi . Indeed, this is the only style of bed motion that is feasible in the absence of basal boundary slip that transfers downslope flow momentum to the bed. In this case τ2top instantaneously jumps to τ3top as bed acceleration penetrates to a horizon at z = H − (h1 + h2), where a stronger substrate prevents motion. The remainder of this section emphasizes scenarios in which motion of the bed is localized at the flow-bed interface andE remains finite, but the next section revisits the possibility of deeper bed deformation.
 To further utilize (32) in assessments of finite entrainment rates, it is helpful to relate v1bot to a more readily observed quantity such as . The relationship between v1bot and depends on the flow velocity profile, v1(z), but for flows of opaque geological debris, direct measurements of v1(z) are at best problematic–and are perhaps impossible. Distinctions between alternative velocity profiles that involve various combinations of homogeneous simple shear and basal slip have been made on the basis of particle tracking studies, however [Johnson et al., 2012]. This family of profiles can be represented mathematically by
where s1 is a fitting parameter that ranges from s1 = 0 if there is no simple shear to s1 = 1 if there is no basal slip (Figure 6). Equation (33) implies that v1bot = (1 − s1) . As a result, for a broad range of plausible velocity profiles, (32) can be expressed as
Johnson et al. showed that for large-scale experimental debris flows, a good fit to particle-tracking data was attained withs1 = 1/2, implying that the basal slip velocity was half the mean flow velocity. The momentum distribution coefficient defined by (21) can also be expressed in terms of s1,
indicating that s1 = 1/2 corresponds to β1 = 13/12 (Figure 6). Thus, the entrainment-rate predictions of(34) can be very sensitive to small deviations of β1 from unity.
 An important special form of (34)applies for boundary conditions analogous to those considered in the slide-block model ofsection 3. In such cases the top of the bed-sediment layer is static prior to entrainment, and the boundary shear tractionsτ1bot and τ2top each obey a Coulomb friction rule τ = μσ′zz that incorporates a standard definition of effective normal stress,
where p is pore fluid pressure. Provided that the total normal stress σzz satisfies (18), boundary shear tractions that satisfy τ = μσ′zz can be expressed as
where μ1 and μ2 are Coulomb friction coefficients for layers 1 and 2, respectively, and p1bot and p2top are boundary pore fluid pressures for layers 1 and 2. By incorporating (37) and (38), (34) becomes
This equation indicates that E > 0 requires a contrast in Coulomb friction coefficients (μ1 > μ2), a contrast in pore fluid pressures (p2top > p1bot), or some combination of the two.
 A contrast that leads to E > 0 in (39)can result from rate-dependent friction that causesμ1 > μ2 to arise from shearing of basal flow material, analogous to the behavior represented in equation (4). Although no experiments have been conducted to isolate this effect in realistic debris-flow materials, recent ring-shear experiments with concentrated mixtures of solid spheres and liquid show that rate-dependent friction becomes significant if the value of a key dimensionless parameter,Iv, exceeds about 10−4 [Boyer et al., 2011]. Re-expressed here using the notation
this parameter includes the pore fluid viscosity η (which has values ranging from about 10−2 to 10−1Pa-s in many debris-flow mixtures), and the bulk shear rate (which probably has values exceeding 10 s−1 at the base of many rapid debris flows) [Iverson, 1997]. These values imply that Iv > 10−4 may commonly apply in flows with basal effective normal stresses that satisfy σ′zz1bot ≤ 1 kPa . Partial debris-flow liquefaction can produce basal effective normal stresses this small [Iverson et al., 2010], providing the requisite conditions to produce μ1 > μ2 in (39).
 Contrasts in boundary pore fluid pressure also can promote entrainment. For example, if the top of the bed sediment becomes completely liquefied by high pore pressures (i.e., p2top = ρgh1 cos θ) and the value s1 = 1/2 is adopted to describe the flow velocity profile, then (39) reduces to
is a pore pressure ratio that indicates the degree of flow liquefaction (0 ≤ λ1 ≤ 1) [cf. Iverson and Denlinger, 2001]. Strength reduction due to growth of pore pressure in overridden bed sediment also has implications for evolution of flow momentum. The flow momentum equation of Table 1 contains only one resistance term, −τ2top, which describes the shear traction exerted by the underlying bed sediment. With a nearly liquefied Coulomb bed (τ2top ≈ 0), there is little resistance to flow acceleration other than the inertial reaction force that develops during entrainment of bed sediment. Thus, the mass and momentum of a flow that liquefies its bed can grow explosively [Iverson et al., 2011].
 Predictions of (41)can be compared with entrainment rates of 0.05 to 0.1 m/s measured in large-scale experiments in which wet sediment beds liquefied almost completely as they were overridden by debris flows [Iverson et al., 2011; Reid et al., 2011]. Applicable parameter values in these experiments were μ1 = 0.84, g = 9.8 m/s2, h1 ≈ 0.2 m, θ = 31°, and ≈ 12 m/s, and use of these values in (41) yields a graph of E predicted as a function of λ1 (Figure 7). The graph shows that entrainment rates in the observed range of 0.05 to 0.1 m/s result if λ1 values range from about 0.5 to 0.8, consistent with λ1 values typically measured in experimental debris flows [Iverson et al., 2010]. Furthermore, Figure 7 shows that the full range of plausible entrainment rates extends upward only to ∼0.23 m/s for the conditions in the experiments. This upper bound on sustainable entrainment rates results from constraints imposed by momentum conservation with = 12 m/s.
 Predictions of (41)also can be compared with entrainment-rate measurements at the Chalk Cliffs field site in Colorado [Coe et al., 2008; McCoy et al., 2010]. There, relevant parameter values when debris flows overrode wet bed sediment at the upper monitoring station on June 12, 2010 were μ1 = 0.97, h1 ≈ 0.88 m, θ = 13°, and ≈ 3.9 m/s (McCoy et al., submitted manuscript, 2012). Use of these values in (41) yields entrainment rates ranging from E = 0 for λ1 = 1 to E = 4.2 m/s for λ1 = 0. Independent constraints on λ1did not exist for this Chalk Cliffs event, but adoption of typical debris-flow values 0.5 ≤λ1 ≤ 0.8 leads to predicted E values an order of magnitude larger than the measured values of 0.14 m/s. This discrepancy is consistent with the fact that full bed liquefaction did not occur at the Chalk Cliffs site (McCoy et al., submitted manuscript, 2012). Under such conditions, (39)may be a more-appropriate entrainment-rate equation than(41), but (39) includes additional quantities unconstrained by measurements at Chalk Cliffs.
 Further understanding of entrainment predictions can be gained by establishing a precise relationship between equation (39)and the slide-block entrainment-rateequation (13). Making the substitutions m = ρh1ΔxΔy, dm/dt = ρEΔxΔy, and μ0 = μ2 in (13)converts it to the equivalent continuum-mechanical form
After setting s1 = 0 and v = in (39)to mimic block-style motion,equation (39) matches (43)if the entrainment-efficiency parameterα is defined as
Equation (44) demonstrates that α depends on the ratio of the excess boundary shear traction ρgh1(μ1 − μ2)cos θ − (μ1p1bot − μ2p2top) to the net driving force per unit area at the base of the slide block, ρgh1 sin θ − μ2ρgh1 cos θ.At steady state (i.e., with no slide-block acceleration) these two quantities are equal, such thatα= 1 and the slide-block mass grows at the maximum sustainable rate indicated by(14)–(17).
 The same interpretation of the numerator and denominator of (44) applies if pore pressure effects are absent, in which case (44) reduces to the simple form,
This equation demonstrates that the relationship tan θ ≥ μ1 ≥ μ2 must be satisfied to ensure that 0 ≤ α ≤ 1. Thus, as stated in section 3, the slide-block model implies that sustained entrainment of dry, cohesionless Coulomb material can occur only if the bed slopes steeply enough to fail under static gravitational loading. On lesser slopes, entrainment can be a transient process accompanied by flow deceleration (as observed byMangeney et al. ), or it can result from transient growth of pore fluid pressure that weakens bed sediment as it is overridden.
6. Bed Deformation and Pore Pressure Generation
 The potential for bed deformation that leads to pore pressure growth depends partly on the weight of the overriding flow but also on downslope momentum transferred to the bed by the flow. The effects of this transfer are evident in the bed-layer momentum equation ofTable 1, and are clearest if longitudinal gradients are neglected in the equation (i.e., ∂/∂x = 0), reducing it to
In this case the kinematic boundary condition (A5) of Appendix A reduces to E = − ∂h2/∂t, and use of this relationship permits to be substituted into (46). The resulting bed momentum equation can then be rearranged to obtain
Consistent with (32), (47) reduces to E = [τ1bot − τ2top]/ρv1bot if the bed material is entirely static, because the values = 0 and τ3top = τ2top + ρgh2 sin θ then apply. Otherwise, (47) indicates that the net external force per unit area acting on the bed layer, τ1bot − τ3top + ρgh2 sin θ, may accelerate the layer at the depth-averaged rate ∂ 2/∂t and also confer the velocity v1bot to boundary material entrained by the overriding flow at rate E. Both phenomena can occur simultaneously, but (47) shows that increased values of ∂ 2/∂t imply diminished values of E, and vice versa. In the limit h2 → 0, (47) reduces to the jump condition (31) because the identities τ3top ≡ τ2top and apply. In this case entrainment and bed-layer acceleration are indistinguishable.
 The details of bed deformation that may occur prior to entrainment cannot be resolved by a depth-integrated analysis, but the potential for pore pressure growth during bed deformation can be assessed by utilizing a constitutive equation that specifies how changes in porosity,n, are related to shear deformation and changes in the mean effective normal stress, [cf. Iverson, 2009; George and Iverson, 2011]:
Here is the bed-sediment shear rate associated with > 0, ψ is a dilatancy angle that describes the tendency of the sediment to dilate (ψ > 0) or contract (ψ < 0) during irreversible shearing, Cis the drained compressibility of the sediment-fluid mixture (here assumed to greatly exceed the compressibility of the solid and fluid constituents), andd/dt is a total time derivative in a frame of reference that moves with the deforming sediment. If shearing were to occur in a closed container that prevents volume change, then (48) would reduce to d/dt = 2 tan ψ/C, implying that would increase with time if shearing proceeded with ψ > 0. This specious prediction demonstrates that ψ cannot be a material constant. Rather, ψ must evolve and must become zero during steady state shearing with no volume change – a state known in soil mechanics as a critical state [Schofield and Wroth, 1968]. With ψ = 0 or 2 = 0, (48) reduces to the standard constitutive equation used to define compressibility in soil consolidation theories, [Bear, 1972, pp. 206], and with C = 0 or d/dt = 0, (48) reduces to a kinematic description of porosity change due to diltatant shearing.
 Changes in porosity described by (48) imply corresponding changes in ρ, because ρ = ρfn + ρs(1 − n), where ρf and ρsare the mass densities of the fluid and solid constituents of the bed sediment. For water-saturated sediment with realistic ranges ofρf, ρs, and n values, however, implied changes in ρ are less than about 10 percent. Thus, use of a constant ρis unlikely to produce significant errors in order-of-magnitude assessments of pore pressure responses. Use of a constantρ also enables the mean effective normal stress in (48) to be defined more explicitly. By combining (18), (25), and (36), can be expressed as
An unambiguous relationship between the porosity change described by (48)and attendant pore pressure evolution can be obtained if the sediment bed behaves as effectively water-saturated. Experimental evidence indicates that such behavior does not require complete saturation, but also indicates that distinctly different behavior occurs when beds are relatively dry [Iverson et al., 2011; Reid et al., 2011]. For a saturated bed, mass-conservation equations for the bed's solid and fluid constituents may be combined with Darcy's law for pore fluid flow to obtain
where k is the Darcian permeability, and η is the pore fluid viscosity. A generalized form of (50) accounts for the effects of variations in k and η [Iverson, 2005], but here both parameters are treated as constants in order to highlight essential aspects of pore pressure evolution.
 Combination of (48), (49) and (50) yields a pore pressure diffusion equation,
which shows how evolution of p in a Lagrangian frame of reference depends on the diffusivity k/Cη and on two types of forcing. One type of forcing arises from increases in flow surface height H or from downward motion of the Lagrangian z coordinate (d(H − z)/dt > 0), both of which lead to gravity-driven pressurization of the pore fluid. The other type of forcing involves pore pressure modification due to shear-induced bed dilation at the rate tan ψ. If ψ > 0, then pore pressure tends to decline as a consequence of shearing, whereas pore pressure tends to increase if ψ < 0.
 In the limiting case with k/Cη → 0, equation (51) reduces to a description of undrained bed behavior. In this case pchanges instantaneously and uniformly in response to compressional loading or shearing that causes pore-volume change.Hutchinson and Bhandari first highlighted the role of undrained compression that may occur when mass movements override water-laden sediment, andSassa  extended the concept to include undrained shearing of sediments beneath debris flows. Pore pressure responses during undrained deformation can be muted by the influence of the compressibilities of the individual solid and fluid constituents [Skempton, 1954; Rice and Cleary, 1976], but this influence is generally negligible in water-saturated, soil-like materials at low confining stresses typical of Earth's surface [Lade and De Boer, 1997].
 If k/Cη → 0 does not apply, then the deforming bed layer behaves as variably drained, and the effects of compression and shearing on p depend on their rates relative to the rate of diffusive dissipation of pore pressure [cf. Rudnicki, 1984; Iverson and LaHusen, 1989; Iverson et al., 1997]. The magnitude of these rate effects can be quantified by identifying the diffusion timescale as T = Cηh22/k and the pressure scale as P = ρgh2 cos θ, and then multiplying all terms of (51) by T/P to obtain a normalized equation
Here p* = p/P, t* = t/T, ∇2* = ∇2h22, H* = H/h2, z* = z/h2, and is a dimensionless parameter that involves the timescale ratio 2T and normalized characteristic pressure CP.
Equation (52) indicates that the compression effects expressed by d(H* − z*)/dt* > 0 are of the same order as the diffusion effects expressed by ∇2*p*, whereas the magnitude of shearing effects depends on the value of I2. Algebraic substitutions show that this parameter can be defined in terms of its primitive components as
Values of some of the components of I2are readily estimated for water-saturated sediment beds (η ≈ 10−3Pa-s,ρ ≈ 2000 kg/m3, g ≈ 9.8 m/s2), and use of these values and a uniform simple-shear approximation ≈ 2 /h2 in (53) reduces the expression for I2 to
Because values of k can range widely, from about 10−7 m2 for gravel to 10−17 m2 for silty clay [Freeze and Cherry, 1979], (54) implies that the value of I2 can also range widely.
 A graph of (54) indicates that if sand or finer sediment is present in a bed that is overridden by a debris flow or avalanche, slight shear displacements of the bed can play a dominant role in generating pore fluid pressure. Indeed, with sufficient fine sediment, I2≫ 1 applies even for depth-averaged displacement rates as small as 2 = 10−6 m/s (Figure 8). In this case behavior is effectively undrained. Pore pressure generation can then produce very strong feedback (either positive or negative, depending on the sign of ψ) that influences further bed deformation and acceleration of the overriding flow. By contrast, shearing of beds composed entirely of sediment that is gravel-sized or coarser is likely to satisfyI2 ≪ 1 (Figure 8). In this case shear deformation is almost perfectly drained and is apt to generate little pore fluid pressure. Of course, if I2≈ 1, then intermediate responses that are not well-characterized as either drained or undrained are possible.
 Contractive shear displacements probably played an important role in causing liquefaction and rapid entrainment of wet bed sediment and explosive growth of debris-flow momentum in the experiments ofIverson et al.  and Reid et al. . In these experiments loosely packed sediment beds consisting of gravel and loamy sand had k values in the range from 4 × 10−11 to 4 × 10−12 m2 [Iverson et al., 2010]. According to Figure 8, these values imply that depth-averaged bed displacement rates of roughly 10−3 m/s sufficed to produce I2 ≥ 10.
 The preceding analyses show that, if all other factors are equal, conservation of downslope momentum demands that bed sediment entrainment rates decline as basal flow velocities increase. Although conventional wisdom holds that increasing a flow's velocity generally enhances its capacity for sediment entrainment, much of this wisdom derives from experience with near-equilibrium fluvial systems. Understanding of these systems has a context that differs fundamentally from that considered here, because it typically focuses on flows with macroscopic momentum that does not respond to entrainment, rather than on flows with fully coupled evolution of flow and bed momentum. The sharp contrast between fixed-momentum and evolving-momentum perspectives makes it worthwhile to recapitulate the predictionE ∼ 1/v1bot by using an analogy that appeals to kinesthetic intuition. In this analogy a human plays the role of a moving flow with evolving momentum.
 Consider a man descending a long version of a children's playground slide with a constant slope and constant friction coefficient. If the man is unencumbered as he descends, he has a constant acceleration and a basal slip velocity, v1bot, that increases linearly with time. However, the man is tasked with entraining static bricks that are positioned alongside the slide. Each brick has the same fixed mass and volume. Therefore, the entrainment rate is the local volumetric rate (per unit basal area) at which the man acquires bricks and accelerates them to his speed. The man's momentum is unaffected by his entrainment of static bricks, which have no momentum to contribute, but momentum conservation dictates that his downslope acceleration decreases in response to the force he exerts to perform the entrainment.
 Next consider an ideal case in which the man and the bricks are perfectly lubricated. Assume that the man has assistants (who may also be lubricated) positioned adjacent to the slide. The assistants push bricks laterally onto the man's lap as he passes by, allowing him to acquire bricks without exerting force or doing work. Nevertheless, to complete the task of entrainment, the man must exert a force to accelerate each brick and bring it up to his speed, counteracting the force that each brick exerts on him. The time integral of this force (i.e., the impulse) equals the mass of the brick multiplied by v1bot, which is the change in the brick's velocity as it comes up to speed. The man's downslope acceleration, integrated over the duration of the impulse, decreases in proportion to the magnitude of the impulse, and if the brick's mass or the man's speed is large enough, the impulse may produce a net negative acceleration that slows the man down. Whether or not slowing occurs, there is a tradeoff between the man's speed and the impulsive force he must exert to accelerate each brick, and this tradeoff dictates that E ∝ 1/v1bot. The tradeoff is a purely inertial phenomenon that is independent of the work done by the man to acquire bricks.
 In more-complicated circumstances the man or the bricks may be imperfectly lubricated, or his assistants may be absent, requiring the man to do additional work to pick up bricks before accelerating them. The added force required to do this work can be characterized, albeit imprecisely, as friction. The effects of friction are analogous to those of incomplete liquefaction of bed sediments, and they reduce the man's maximum rate of brick entrainment if all other factors remain unchanged. A further complication exists if the man is able to reduce his basal friction as he slides, analogous to a debris flow or avalanche that can liquefy its bed as it overrides it. Reduction of basal friction is the only means by which the momentum of the man or an analogous flow can grow more rapidly than it would in the absence of entrainment. In such circumstances flow momentum can grow unstably, however, because the net force available to cause entrainment grows in proportion to the mass of the entraining body [e.g.,Iverson et al., 2011].
 Although elementary mechanics establishes some important relationships between momentum conservation and bed-sediment entrainment, additional details remain unresolved, and some cannot be assessed using a depth-integrated continuum model. Like many geomechanical models, the depth-integrated continuum model presented here assumes that bed-normal total stresses (σzz) are lithostatic and that longitudinal normal stresses are proportional to σzz. These assumptions lead to useful mathematical simplifications and physical inferences, but they preclude assessment of the effects of boundary irregularities–except insofar as these effects can be parameterized by a boundary friction coefficient. If elements of bed material protrude a significant distance into an overriding flow, for example, shear tractions along the boundary vary spatially. Furthermore, as the flow impinges against such protuberances, it can transfer x-momentum to the bed by a mechanism that involves not only variable shear tractions but also variable normal tractions–an effect commonly parameterized as form drag. An inability to assess these effects of boundary geometry represents a fundamental limitation of the depth-averaged entrainment model presented here.
 From a practical viewpoint, perhaps a more significant limitation results from the model's focus on bed-sediment entrainment and neglect of entrainment of bank material. Rigorous modeling of bank entrainment, while very desirable, would be very challenging. It would entail formulation of a multidimensional boundary-traction model (in which depth averaging would likely be inappropriate), and would also entail coupling the traction model with a three-dimensional bank-failure model (because bank failures with infinite lengths or widths would contribute infinite mass to a passing flow). Moreover, a model of bank-material entrainment by debris flows or avalanches would necessarily include fully coupled evolution of flow momentum, distinguishing it from models of morphological evolution of stream banks that occurs under macroscopically steady flow conditions.
 Conservation of momentum constrains interpretations and predictions of bed-sediment entrainment rates by debris flows and avalanches that are free to accelerate as they descend steep slopes. Both a Coulomb slide-block analysis and a depth-integrated continuum analysis predict that, if all other factors are constant, the entrainment rateE decreases as the basal flow velocity v1bot increases. The largest sustainable entrainment rates occur in conjunction with steady flow, because the entire net driving force is then allocated to entrainment rather than acceleration. These findings help explain some key results of entrainment experiments in which dry granular avalanches moved across sloping beds covered with the same material [Mangeney et al., 2010]: entrainment was more pronounced during relatively slow, steady motion of avalanches than during rapidly accelerating motion.
 Predicted entrainment rates depend not only on flow velocities but also on excess boundary shear tractions. Although no universal model of shear tractions exists, a Coulomb model accounts for the effects of friction as well as modifications of bed friction due to pore fluid pressure. Entrainment-rate predictions are simplest if pore pressures in bed sediment locally grow large enough to liquefy it when it is overridden. Then the resisting Coulomb shear traction is essentially zero, and entrainment can occur at the maximum sustainable rate, expressed byE = [μ1gh1 cos θ(1 − λ1)]/v1bot or by E = [2μ1gh1 cos θ(1 − λ1)]/ if the approximation v1bot ≈ /2 is adopted. The latter equation predicts entrainment rates similar to those measured in large-scale debris-flow experiments in which wet bed sediments liquefied almost completely when they were overridden [Iverson et al., 2011; Reid et al., 2011].
 Growth of pore fluid pressure in wet bed sediment can occur as a result of both compressional loading by an overriding flow and shear deformation of the bed in response to transfer of downslope flow momentum. Loosely packed beds that exhibit contractive shear behavior are especially susceptible to this type of pore pressure growth, which may lead to liquefaction. An analysis of pore pressure generation and diffusion shows that the propensity for liquefaction may be very large, for example, if a water-saturated bed that contains significant sand or finer sediment begins to creep downslope at a depth-averaged rate greater than about 10−3 m/s as it is overridden. Bed liquefaction also promotes flow momentum growth, because it minimizes basal resisting forces that impede downslope acceleration.
 Evolution of flow momentum during entrainment illustrates a fundamental asymmetry in the relationships governing growth of mass, velocity and momentum for Coulomb bodies. Whereas a Coulomb body of constant mass undergoes stable velocity growth while descending a steeply inclined plane, a Coulomb body with constant velocity can undergo exponential mass growth in the same circumstances – provided that the frictional resistance of the bed is low enough to allow entrainment. This unbounded mass growth in turn implies unstable momentum growth. Therefore, the maximum momentum attainable by Coulomb bodies that entrain basal material is limited only by the volume of steeply sloping, erodible material they encounter.
Appendix A:: Depth Integration of Conservation Equations
 The general differential equations describing conservation of mass and the x component of momentum in a continuous material with variable bulk density ρ can be written as
where vx, vy, and vz, are, respectively, the x, y, and z components of velocity, ρgx is the driving force per unit volume due to the x component of the material's weight, and τxx, τyx and τzx are the components of the Cauchy stress tensor that resist motion in the x direction. For a discrete layer of material, depth averages of the dependent variables in (A1) and (A2) are defined as
where zbot(x, y, t) is the z coordinate of the layer's basal surface, ztop(x, y, t) is the z coordinate of the layer's upper surface, and h(x, y, t) = ztop − zbot is the layer's thickness. Subsequent equations assume that ρ varies only as a function of x, y, and t, such that ρ = , and also assume that of material potentially added to the layer locally equals of the layer itself.
 Integration of (A1) and (A2) through the layer thickness from its base at z = zbot to its upper surface at z = ztop involves use of kinematic boundary conditions that relate vz(zbot) and vz(ztop) to the other velocity components at z = zbot and z = ztop and to variations in the boundary positions:
Here Ebot and Etop are the z-direction boundary-migration velocities (reckoned positive downward) due to passage of material through the basal surface or upper surface of the layer, respectively.Gray  has shown that for (A4) and (A5) to describe mass fluxes through these surfaces exactly, Ebot and Etop must be multiplied by correction factors that account for local differences between the slopes of zbot and ztop and the slopes of the global coordinates, x and y. These correction factors are negligible, however, if zbot(x, y, t) and ztop(x, y, t) vary gradually enough to satisfy the conditions [∂zbot/∂x]2, [∂zbot/∂y]2, [∂ztop/∂x]2, and [∂ztop/∂y]2 ≪ 1, and here I assume that this is the case. Then (A5) shows that if a static layer with vx = vy = vz = 0 is subject to upward surface accretion (Etop < 0), the deposit's surface height increases at the rate ∂ztop/∂t = − Etop. The situation is more complicated at the base of a moving layer, where either erosion or sedimentation can occur and all of the terms in (A4) can evolve simultaneously. Despite such complications, Ebot < 0 always describes upward migration of the layer's base due to sedimentation, and Ebot > 0 describes basal lowering due to erosion.
 Provided that ρ = , depth integration of the mass-conservation (equation A1) from z = b to z = η yields
The third and fourth lines of (A6) illustrate the result of using Leibniz' rule [Abramowitz and Stegun, 1964, pp. 11] for interchanging the order of integration and differentiation during evaluation of and dz. The final line of (A6) results from substituting the kinematic boundary conditions (A4) and (A5) into the sixth and seventh lines of (A6) in place of vz(ztop) and vz(zbot), and then cancelling terms that sum to zero and making the identification ∂ (ztop − zbot)/∂t = ∂h/∂t. If is constant and Etop = Ebot = 0, then (A6)reduces to the standard depth-averaged mass-conservation division used in many shallow-flow theories [e.g.,Vreugdenhil, 1994; Pudasaini and Hutter, 2007].
 Depth integration of the momentum-conservation (A2) equation proceeds in a manner similar to that of the mass-conservation equation, and it yields
The momentum-exchange terms vx(ztop)Etop and − vx(zbot)Ebot arise in the seventh line of (A7) as a result of using the kinematic boundary conditions (A4) and (A5) to combine several terms in the second through fifth lines of (A7), which express contributions to the x-momentum fluxes at the flow boundaries. The integrals in the sixth and seventh lines of(A7) result from use of the identities
which show how the depth integrals of products are related to the products of depth integrals. Physically, the integrands (vx − )2 and (vx − )(vy − ) describe the effects of differential advection of momentum due to variations of vx and vy with z [Vreugdenhil, 1994]. Depth-averaged flow theories commonly account for these effects by introducing momentum-distribution coefficients that multiply hx2 and h, as exemplified by equation (21)in the main text. The one-dimensional depth-averaged conservation laws(19) and (20) in the main text are obtained from (A6) and (A7) by utilizing the momentum distribution coefficient β = (1/hx2)∫zbotztopvx2dz, neglecting all terms involving ∂/∂y, and assuming is constant.
 If the position of the upper surface of the three-layer system shown inFigure 5 is denoted by z = H, then the total normal stress σzz at any depth H − z below the surface is found by integrating equation (18) to obtain σzz = ρg(H − z)cos θ. Furthermore, if at all depths the lateral normal stress σxx is related to σzz by a simple proportionality rule, σxx = κσzz (where κcan be interpreted as a lateral earth-pressure coefficient or elastic Poisson's ratio), then
In the equations that follow, κ is assumed constant within any particular layer, although its value may differ between layers.
 The thickness of any layer is given by
and (B1)implies that the depth-averaged lateral normal stress within any layer is given by
Use of (B2) and (B3) and some algebraic rearrangement then enables the product xxh, which appears in (24), to be written as
Differentiation of (B4) with respect to x, followed by some further algebraic rearrangement, then yields a useful form of one of the terms in (24):
Two other terms in (24) can be evaluated directly through use of (B1) to obtain
Addition of (B5), (B6) and (B7), followed by some algebraic cancellations and use of the substitution ztop − zbot = h, then shows that this combination of terms reduces to
 I thank Christophe Ancey, Mark Schmeeckle, Nico Gray, Jason Kean, and Mark Reid for useful discussions and comments on a draft manuscript. Scott McCoy and his coauthors cordially provided access to their results from field monitoring at the Chalk Cliffs site in Colorado. I especially thank numerous colleagues who participated in USGS debris-flow flume experiments that helped motivate this work.