Bedrock rivers are characterized by conditions of insufficient sediment supply, wherein the amount of sediment supplied to the channel is less than the channel's sediment transport capacity. Consequently, bedrock channels generally exhibit discontinuous sediment cover, which current approaches to the morphodynamics of alluvial rivers are unable to handle. To fill this gap, we present a theoretical framework in which local sediment transport rates are proportional to the areal concentration of sediment available for transport on the bed and the sediment continuity equation is reformulated to account for temporal changes in areal concentration of sediment on nonalluviated surfaces. We then perform a linear stability analysis that shows that an initially uniform distribution of the areal concentration of sediment on the bed is unstable to small perturbations. This suggests that the sediment cover in bedrock channels tends to concentrate into regions which may eventually be alluviated, in general agreement with experimental observations.
 Progress in bedrock morphodynamics requires an understanding of the dynamics of sediment redistribution over the channel bed. Recent experiments in artificial bedrock channels [Finnegan et al., 2007; Johnson and Whipple, 2007, 2010] and numerical simulations of the evolution of a bedrock channel cross section due to erosion from saltating bedload particles [Nelson and Seminara, 2011] have demonstrated explicit feedbacks between the spatial distribution of bed load sediment, local erosion rates, and channel shape. Extended to the longitudinal direction, the spatial distribution of sediment on the bed may influence the planform evolution of bedrock channels.
 Observations of sediment transport in artificial bedrock flumes under varying bed roughness, sediment supply, and hydraulic conditions [e.g., Finnegan et al., 2007; Chatanantavet and Parker, 2008] suggest that bed load sediment tends to become spatially concentrated and form preferential transport paths. Additionally, recently-collected multibeam data from the Mississippi River suggest that even large lowland rivers can behave as mixed bedrock-alluvial channels, with local exposures of consolidated substratum (bedrock) and a discontinuous layer of alluvial sediment [Nittrouer et al., 2011].
 Currently, we lack a theoretical formulation that is able to deal with discontinuous sediment cover of the bed surface. The present work aims precisely at filling this gap: an analytical framework is developed to address the morphodynamics of bedrock channels in which the assumption of sufficient sediment supply has been relaxed.
 Consider a partially-alluviated bedrock channel that has sediment distributed over its bed with an areal concentrationC∗(x∗, y∗, t∗) (we use asterisk superscripts to denote dimensional variables). Here, x∗ and y∗ are coordinates in the downstream and cross stream direction, respectively, and t∗ is time. C∗ has units of length, since it is defined as the ratio of a volume of a monolayer of sediment over the area of the bed it occupies. This areal concentration may vary between zero and a maximum Cm∗, such that the bed is completely covered: if we assume the sediment to consist of spheres of a uniform diameter ds∗, then Cm∗ = πds∗/6. It is then appropriate to define a dimensionless areal concentration C(x∗, y∗, t∗) = C∗(x∗, y∗, t∗)/Cm∗ such that a continuum of bed configurations may be distinguished, varying from a fully exposed (C = 0) or partially exposed (0 < C < 1) bedrock surface to an alluviated bed (C = 1).
 Unlike problems in alluvial morphodynamics where bed load transport occurs at transport capacity, bedrock morphodynamics requires modeling of bed load transport on partially-exposed bedrock surfaces. Recalling that bed load particles are entrained in response to local turbulent fluctuations [e.g.,Drake et al., 1988], we assume that the intensity of the local bed load transport rate will be proportional to the areal concentration of sediment available on the bed [e.g., Sklar and Dietrich, 2004]. Thus, we can express the modulus of the local volumetric bed load transport rate per unit width, |qs∗|, as
where qsc∗ is the transport rate at capacity (i.e., when C = 1). The direction of bed load flux deviates from the direction of the local average bottom stress, as particles feel the effect of the lateral slope [e.g., Seminara, 1998]:
where is the unit vector aligned with the local boundary shear stress vector, ∇h∗ is the gradient operator defined on the plane tangent to the bottom, η∗ is the local and instantaneous bed elevation, and G is a tensor whose entries are coefficients that indicate transport sensitivity to longitudinal and transverse bed slopes, and can be estimated from experimental results. Note that η∗ is the sum of the elevation of the bedrock surface ηR∗ and the thickness of the alluvial cover e∗.
 Since in bedrock channels three distinct environments (fully exposed bedrock surface, partially covered bedrock surface, and alluviated bed) may coexist, the classical bed evolution equation for alluvial rivers (the Exner equation) must be replaced here by two distinct statements.
 First, we need an appropriate statement of sediment conservation able to account for both sediment redistribution on a partially covered bedrock surface (no variation of bed elevation) and aggradation and degradation of the alluviated portions of the bed. A modified Exner equation able to meet these requirements takes the following form:
Here, p is the porosity of the alluvial cover. In our framework, e∗ will not vary unless there is local alluviation; that is, if C = 1. Thus, ∂C∗/∂t∗ = 0 whenever there is alluvial cover (e∗ changes in response to the divergence of sediment transport and changes of e∗ have a direct feedback on the hydrodynamics), and ∂e∗/∂t∗ = 0 whenever there is not (C∗ changes in response to the divergence of sediment transport and changes of C∗ have an indirect feedback on the hydrodynamics through modifications of the local roughness).
 Secondly, the elevation of the bedrock surface ηR∗ will evolve in response to tectonic uplift and bedrock incision:
where is the uplift rate and is the erosion rate.
 The time scales of uplift and incision governed by equation (4)are much larger than the time scale of the redistribution and aggradation-degradation processes governed byequation (3). This implies that the rate of production of sediments by the incision process is so small that its contribution to mass conservation may be neglected. Moreover, it allows decoupling of short term morphodynamics from long term incisional phenomena. Below we concentrate on the former class of processes. The latter class has been given some attention by Nelson and Seminara .
3. Short Term Morphodynamics of Mixed Bedrock-Alluvial Channels
 We now examine the observed short term tendency of sediment to concentrate into regions where distinct mixed bedrock-alluvial patterns form. Various patterns have been reported in the literature: Barchan dunes, i.e., three-dimensional isolated sand patches migrating downstream over a rocky river bed [e.g.,Hersen, 2005]; alternate bars, i.e., alternate sequences of alluvial regions separated by exposed bed surface, which also migrate downstream [Chatanantavet and Parker, 2008]; and point bars, i.e., steady discontinuous alluvial regions observed in inner bends of meandering rivers where the consolidated substratum (bedrock) is locally exposed [Nittrouer et al., 2011]. Full understanding of each of these classes of pattern will require a variety of theoretical techniques developed in the neighboring field of alluvial morphodynamics. Here we limit ourselves to an attempt to answer the following question: is an initial, uniform distribution of sediment cover on a flat bed stable?
 We then consider a straight rectangular channel with constant width 2B∗ (Figure 1) and perform a linear stability analysis to explore how areal sediment concentration responds to small perturbations in space and time. Since we are interested in the initial instability of sediment in a non-alluviated bedrock channel, our analysis assumes that both ∂e∗/∂t∗ in (3) and ∂ηR∗/∂t∗ in (4)vanish. Moreover, we restrict ourselves to perturbations scaling with channel width, hence we assume that flow perturbations may be described by the shallow-water equations. Since the process of sediment redistribution over the bed is much slower than the process whereby the flow adjusts to it, the quasi-steady form of St. Venant equations will be sufficient. They read:
where u∗ and v∗ are the components of the velocity vector in the x∗ and y∗ directions, g is the gravitational acceleration, h∗ is the free surface elevation, and D∗ is the water depth. Cf is a friction coefficient taking the logarithmic form [Einstein, 1950]:
where kt∗ is a roughness parameter which consists of contributions from both bedrock and sediment. Note that kt∗ provides the only mechanism whereby the hydrodynamics feels the sediment redistribution process. As a first approximation, we weight the contribution of sediment roughness to total roughness by the local areal sediment concentration:
where ks∗ = 2.5ds∗ [Engelund and Hansen, 1967] is the absolute sediment roughness, kr∗ is the absolute roughness of the rock surface, and kr = kr∗/ks∗.
 At the channel sides, we specify no lateral flow of either fluid or sediments: v∗ = qsy∗ = 0 at y∗ = ±B∗. Moreover, we specify the sediment flux per unit width qsup∗ supplied to the channel, and stipulate that perturbations must not alter the total sediment flux through the channel.
 It is convenient to make the variables dimensionless, for which we use the following scaling:
Here, D0∗ and U0∗ are the average depth and speed for the unperturbed flow, s = ρs/ρw is the ratio of sediment density ρs to water density ρw, is the Froude number for the unperturbed flow, and T∗ is a characteristic timescale of perturbation growth.
The local sediment transport capacity qsc is calculated with an equation of the form
where τ∗ = Cf(u∗2 + v∗2)/[(s − 1)gds∗] is the Shields stress, τ∗c is the critical Shields stress for sediment entrainment and Ψ is an empirical coefficient.
Cf in (11) and (12) is calculated by substituting the dimensionless depth into (8):
where ξs = 6 + 2.5 ln (D0∗/ks∗).
 In nondimensional form, the lateral boundary conditions become v = qsy = 0 at y = ±1.
 We now perform a normal mode analysis of small perturbations of the basic state, which consists of a uniform flow with a uniform distribution of areal sediment concentration C0. We then write:
where ϵ ≪ 1 is a small parameter and c.c. denotes the complex conjugate. Note that we have considered the first of a discrete set of lateral modes of the perturbation and (corresponding to alternate bars in the alluvial analogy). The longitudinal structure of the perturbation, described by the function exp i(λx − ωt) in (18) and (19), defines a harmonic wave with dimensionless longitudinal wavenumber λ = 2π/(L∗/B∗), where L∗ is the wavelength. This wave migrates in the downstream (x) direction with wavespeed (ωr/λ) and grows with rate (ωi), ωr and ωi being the real and imaginary parts of the complex number ω.Figure 1aprovides a sketch of the spatially-periodic pattern of initial perturbations this formulation produces. Because we are not considering the case whereC = 1 and the bed alluviates, d1 = h1 in (18). The perturbed variables in (18) and (19) are substituted into the dimensionless equations (11)–(17) and any terms of O(ϵ2) or smaller are neglected. After a considerable amount of algebra, equations (11)–(14) at O(ϵ1) provide the following linear homogeneous algebraic system for the coefficients of the expansion u1, v1, h1, and C1:
Here G0 and Cf0 are
In order for the above algebraic system to have a solution, the determinant of the matrix on the left hand side of (20) must vanish. This constraint leads to a relationship for ω as a function of the perturbation wavenumber λ for given values of the dimensionless parameters C0, τ∗0, Cf0, kr and β. This is called a dispersion relationship and is reported in the auxiliary material. Once ωr and ωi have been calculated, one readily ascertains conditions under which ωi > 0, implying that perturbations are unstable, i.e., that sediment tends to organize itself into a bar-like pattern.
4. Example Calculations
 We now explore the stability of the areal concentration of sediment on the bed by using the dispersion relationship to calculate the angular frequency and growth rate of perturbations, using parameters describing experiment 1A-2 inChatanantavet and Parker . Figure 2a shows the nondimensional growth rate ωi as a function of the dimensionless wavenumber λ, for relative bedrock roughnesses kr between 0.01 and 0.9, and Figure 2b plots the dimensionless growth rate for unperturbed dimensionless areal sediment concentration C0 between 0.01 and 0.9. In all cases, the growth rate is positive for λ up to about 1.4, indicating that the system is unstable to perturbations with wavelengths longer than about 2 m. The maximum growth rate occurs between λ = 0.35 and 0.49, corresponding to wavelengths L∗ between 5.8 and 8.05 m. Photographs from this experimental run [Chatanantavet and Parker, 2008, Figure 8] show that the sediment formed a slightly sinuous longitudinal strip with a wavelength of approximately 6.5–8 m. For these conditions, the wave speed of the perturbation with the maximum growth rate is between 0.014 m/s and 0.025 m/s, suggesting that the “bar-like” pattern migrates downstream.
 Marginal stability curves, characterized by vanishing growth rate, are plotted in Figure 3 for several values of τ∗0/τ∗c. These curves are nearly vertical lines, suggesting that, unlike the alternate bar instability in alluvial channels, the instability of areal sediment concentration in bedrock channels does not depend on the aspect ratio β and only weakly depends on the excess shear stress ratio τ∗0/τ∗c.
5. Discussion and Conclusions
 We have presented a framework in which the morphodynamics of bedrock channels, where the rate of sediment supply is less than the channel's sediment transport capacity, may be explored analytically. The basic assumption we have made is that the local sediment transport rate is proportional to the local areal concentration of sediment on the bed. Divergences of sediment flux on nonalluviated areas of the bed then are balanced by temporal changes in areal sediment concentration, as expressed in our reformulated sediment continuity equation (3).
 Our linear stability analysis suggests that the difference in roughness between sediment and bare bedrock, as expressed in equation (9), forces the system to organize such that sediment on the bed will tend to become locally concentrated. This is in general agreement with experimental observations [e.g., Chatanantavet and Parker, 2008]. Our calculations suggest that as the bedrock roughness approaches the sediment roughness (i.e., as kr approaches 1), the instability vanishes (Figure 2).
 It is important to emphasize that the linear analysis presented here only addresses the initial instability which generates bar like patterns. Predicting the actual pattern emerging from this process will require a fully nonlinear analysis possibly able to treat regions where the areal sediment concentration C locally reaches 1 and local alluviation occurs. Moreover, the present hydrodynamic formulation may be inadequate to reveal the existence of finer scale patterns, like barchan dunes, which will require more refined turbulent closures. Finally, fully coupling this framework to a local mechanistic erosion model with the use of equation (3) [e.g., Nelson and Seminara, 2011] has the potential to lead to significant new insights on the behavior of bedrock river systems.
 The paper benefited from constructive reviews from Joel Johnson and Alan Howard. Funding for this work was provided by a National Science Foundation International Research Fellowship to P.A.N. (grant 0965064).
 The Editor thanks Joel Johnson and an anonymous reviewer for their assistance in evaluating this paper.