Landscapes form by the interplay of bedrock erosion and the entrainment, transport and deposition of sediment, as determined by various drivers such as climate, tectonics, and biological activity. Each erosion process has a minimum of two phases: the breakdown of rock mass by chemical or physical weathering and the entrainment and evacuation of loose pieces of rock that are produced in this way (Gilbert, 1877). From this, it is clear that a minimum description of any eroding landscape needs to include a mass balance equation each for bedrock and for loose sediment. Consider a control volume within a river (Fig. 1), with width W, length L, and a height ranging from the surface, i.e. the interface between bedrock or sediment and the water, to a fixed reference level somewhere in the bedrock below. The loose material, sediment, overlays the bedrock. Uplift pushes new bedrock into the control volume at a rate TU, while incision converts it into sediment at a rate I. We assume that the erosion products are small enough so that they are subsequently transported in suspension. Then the rate of change of bedrock mass per unit area Mb is given by 5∂Mb∂t=ρrTU-I. Here, ρr represents the density of the bedrock. Dividing Eq. (5) by ρr, and realizing that hb=Mb/ρr, we retrieve the usual form of the bedrock mass balance (Eq., 4). Details of the derivation of the mass balance for sediment have been given by Turowski and Hodge (2017). Note that working with mass instead of a deposit thickness is advantageous for bedrock channels, because sediment may not be equally distributed on the bed. The entrainment–deposition framework is preferable, because it makes it possible to distinguish between moving and stationary sediment, which is necessary to treat the cover and the tools effects. This is not possible when using the Exner approach (Appendix A). The mass balance for the mobile sediment per unit area Mm is given by Eq. (2): 6∂Mm∂t=-∂qs∂x+E-D. The mass balance for the stationary sediment per unit area Ms is given by 7∂Ms∂t=D-E. Finally, sediment flux per unit width qs and the mobile sediment mass per unit area Mm are connected via the downstream particle speed U: 8qs=UMm.

Considering impact erosion to be the dominant erosion process, the lateral erosion rate EL of bedrock channels is driven by particle impacts. It can therefore, similarly to the formulation of the saltation–abrasion model (Sklar and Dietrich, 2004), be written as the product of two terms: (i) the average volume eroded by a single impact, Vi, and (ii) the impact rate per area and time Ir. The latter term can be subdivided into two terms. The first of these quantifies the number of available particles per unit time and area, FT, which describes the tools effect. The second term, FC, describes the effect of bed cover, which captures the effects of the distribution of sediment in the channel on lateral erosion. The need for this term arises because bedload particles generally travel parallel to the channel walls. Sideward deflection is controlled by the interaction of moving particles with the bed (Beer et al., 2017; Fuller et al., 2016) and specifically with stationary sediment, i.e. bed cover (cf. Turowski, 2018). 9EL=ViIr=ViFTFC The volume eroded per impact for lateral erosion should be the same as for vertical erosion and has been quantified by Sklar and Dietrich (2004) as the energy of the impact divided by a material constant. It can be evaluated by 10Vi=2YkvσT2Mpwi22. Here, the first term is related to material properties, where Y and σT are Young's modulus of the bedrock and its tensile strength, respectively, and kv is the rock resistance coefficient. The second term gives the kinetic energy of the impacting grain. Here, Mp is the mass of a single particle and wi is the impact speed normal to the wall.

As in vertical bedrock erosion (Beer and Turowski, 2015; Inoue et al., 2014; Sklar and Dietrich, 2004), the tools effect can be modelled as a linear function of bedload supply Qs (Mishra et al., 2018) multiplied by a dimensionless factor κT with values between 0 and 1 that describes the fraction of bedload available for lateral erosion. To obtain the number of impacting particles per unit area, this product needs to be divided by the mass of a single particle and the total area of the wall, Aw, that is eroded to give 11FT=κTQsAwMp. Substituting Eqs. (10) and (11) into Eq. (9), the lateral erosion rate of a bedrock channel can thus be written as 12EL=κTYkvσT2Qswi2AwFCD. In Eq. (12), there are three parameters that require further discussion: the impact speed wi, the eroded area Aw, and the cover-dependent term FCD. In a previous paper (Turowski, 2018), I argued that lateral erosion and channel width development are intimately related to bed cover. The quantification of all three parameters springs from the physical–conceptual model developed in this previous paper. For this reason the cover-dependent term, FCD, will be discussed first, leading to a quantification of the other two terms, wi and Aw.

In a straight bedrock channel the motion of water and sediment is generally parallel to the walls. Lateral erosion occurs when sediment particles are deflected sideways such that they impact the walls with sufficient force to cause damage. For a given reach, we can define a sideward deflection length scale d, which is relevant for reach-scale lateral erosion (Turowski, 2018). The relevant cross section for setting reach-scale channel width is assumed to be located where the sinuous bedload particle stream crosses from the gravel bar onto the smooth bedrock at the apex of the bar (Fig. 2). Only there, several conditions come together that are favourable to achieve the maximal sideward deflection distances (Turowski, 2018). These are (i) the high particle concentration, (ii) a vector of motion of the particle stream that is already pointing towards the walls, (iii) the existence of roughness necessary for sideward deflection provided by the alluvium, and (iv) the smooth bedrock that does not hinder sideward motion. We expect that the wall is eroded if the uncovered width Wuncovered in the cross section is smaller than d (Fig. 3). As a result, we can quantify the cover-dependent term FCD as 13FCD=1ifd>Wuncovered0otherwise. We can write the eroded area on the wall Aw as the product of a length scale and a height. From the argument above, the same particle can attack the wall once when passing each gravel bar. Therefore, the relevant length scale for lateral erosion is the distance between bars on a given side of the channel, i.e. the wavelength of bar spacing, λ. Fuller et al. (2016) observed that for sideward-deflected particles, the erosion height on the wall is larger than the typical saltation hop height. Beer et al. (2017) observed a similar increase in wall erosion rates near boulder obstacles in the channel. When the roughness elements that cause deflection are related to stationary alluvium, we can expect that the height scale is the maximum saltation height of bedload particles at the wall, Hw. 14Aw=λHw Note that the entire area is not eroded at the same time. Rather, particles are deflected towards the wall near the apex of the bars (cf. Turowski, 2018). Consequently, only a small area is eroded at a given time, and the locus of erosion slowly moves downstream as the bars migrate. Likewise, the impact speed wi and the sideward deflection distance d are related to saltation properties. It is, of course, possible that a particle undergoes several saltation cycles until it impacts the wall. However, in this case, in each additional saltation hop, the sideward component of motion would reduce due to downstream hydraulic forces and frictional loss of momentum. Here, I assume that only during the first hop do particles have sufficient lateral momentum to cause erosion upon impact on the wall. This assumption needs to be verified experimentally.

Since, within the model, sideward deflection is caused by stationary alluvium, particle trajectories should follow those observed for saltation over alluvium (e.g. Abbot and Francis, 1977; Niño et al., 1994), rather than those over bedrock (e.g. Chatanantavet et al., 2013; Auel et al., 2017a). Because the wall-normal component of the motion is relevant for impact erosion (e.g. Sklar and Dietrich, 2004), the particle trajectory needs to be corrected for its angle γ of motion with respect to the wall. Then, the sideward deflection distance d is related to the saltation hop length Ls by 15d=Lssin⁡(γ). Likewise, the impact speed wi is related to the particle speed U by 16wi=Usin⁡(γ). Auel et al. (2017a) proposed empirical equations to describe saltation properties over a sediment bed as a function of hydraulics, based on their own experiments and a data compilation from various sources. They give the saltation hop length Ls by 17LsD=1.17θθc-1. Here, D is grain diameter, θ is the Shields stress, and θc is the critical Shields stress for the onset of bedload motion. Similarly, hop height Hs is given by 18HsD=0.025θθc-1+24 and downstream particle speed U is given by 19U=1.46ρsρ-1gD0.5θθc-10.5. Here, ρ and ρs are the densities of the water and sediment, respectively, and g is the acceleration due to gravity. Finally, to close the system of equations, we need some relations describing the geometry of the gravel bars. Alternating bars in bedrock channels have been little studied (e.g. Nelson and Seminara, 2012), and the necessary relations are not available. From a large data compilation of bar width and length in braided channels, Kelly (2006) found that bar length Lbar is related to bar width Wbar by 20Lbar=4.95Wbar0.97. Based on this observation, I assume that in bedrock channels the wavelength of the bars scales with their width, such that 21λ=κbarWbar=κbarWcovered. Here, the bar width has been identified with the covered width Wcovered (Figs. 2 and 3), and κbar is a dimensionless constant with a value of 2–10, by analogy with bar shapes in alluvial rivers (e.g. Kelly, 2006). Bed cover C is the ratio of covered bed area Ac to total bed area Atot, which can be related to the covered width Wcovered as follows: 22C=AcAtot=WcoveredW=WbarW. Here, W is the channel width. As a result, the bar length can be written as 23λ=κbarWC. Assuming that the maximum saltation height at the wall corresponds to the maximum saltation hop height, Hw=Hs, and substituting Eqs. (13) to (16) and Eqs. (18) to (23) into Eq. (12), we obtain 24EL=κYgkvσT2ρsρ-1sin⁡2(γ)QsWCθθc-1θθc-1+24ifd>Wuncoveredandθ>θc0otherwise. Here, κ=85κT/κbar is a dimensionless constant. The sideward deflection length scale d can be estimated by the hop length Ls (Eq. 24): 25d=1.17Dsin⁡(γ)θθc-1. Finally, the uncovered width can be related to bed cover using Eq. (22). 26Wuncovered=W-Wcovered=W(1-C) The rate of change of channel width, in the case of a widening channel, should be twice the lateral erosion rate given in Eq. (24), since both sides are eroded at the same time. 27dWdt=2EL Note that, when d=Wuncovered, the model gives a steady-state channel width consistent with the model of Turowski (2018), with the sideward deflection distance given by Eq. (25).

I will now derive analytical expressions for the response time of the channel to perturbations in the boundary conditions, such as changes in discharge, sediment supply, or uplift rate. This will be done for three key parameters: channel bed slope, channel width, and cover. For the derivation, it is necessary to assume that, on the timescale of adjustment of one variable, the other variables stay essentially constant. This assumption is reasonable if a particular variable adjusts much slower than another variable. For example, slope adjustment takes much longer time than the adjustment of bed cover.

The sideward deflection length scale d is an important parameter setting channel morphology in steady state, in particular the channel width, which depends on the square root of d (Turowski, 2018). 48W=kQsdI=1.17Dsin⁡(γ)θθc-1kQsI Here, d is estimated using saltation hop length of bedload particles over bare bedrock (Eq. 25). Saltation hop length is dependent on the Shields stress, and the new formulation can consequently alter steady-state scaling of channel width and slope. Unfortunately, Eq. (48) cannot be solved analytically, since Shields stress θ is non-linearly dependent on channel width and slope (see Eq. D6), and a numerical solution is necessary (Fig. 4). As in the model by Turowski (2018), channel width is independent of discharge (Fig. 4a), and the observed scaling between width and discharge arises from a co-dependence of discharge and sediment supply (see Fig. 4b).

For the calculation of adjustment timescales, the dependence of width and slope on discharge, sediment supply, and uplift rate, and on each other, needs to be explicitly taken into account. From the derivation (Appendix C), the relevant width and slope in the timescale equations are those of the steady-state morphology corresponding to the relevant control variables. As such, they are not independent of sediment supply, discharge, or other control variables. Within the model, steady-state channel width and slope cannot be evaluated analytically or written in a closed-form equation. Thus, a numerical solution is necessary. Adjustment timescales of width are generally longer than those for slope and for cover (Fig. 5), at least for the parameter values used in the example calculations (Table 1).

Equation (24) is a mechanistic description of lateral fluvial bedrock erosion by impacting particles. Field and laboratory data that can be used to test the model are scarce, and the few data sets that exist do not include information on all necessary parameters to test it (e.g. Cook et al., 2014; Fuller et al., 2016; Suzuki, 1982; Mishra et al., 2018). The minimum parameters needed for a meaningful test are the lateral erosion rates measured in parallel with relevant driving variables, including water discharge and bedload transport rate, in a channel with self-formed sediment cover and alternating gravel bars. Nevertheless, the model provides a starting point for future investigations, providing a clear mechanistic description and a host of testable assumptions and predictions.

Due to a lack of direct relevant data and to keep the complexity of the model reasonable, it was necessary to make some simplifications and assumptions on relevant processes and geomorphic response. For example, bedrock channels at high slopes tend to adjust their bed into a step-pool morphology (Duckson and Duckson, 1995; Scheingross et al., 2019). The feedbacks necessary to develop these bedforms, and how they may affect the flow hydraulics and erosion rates, have not been considered in the present model (e.g. Scheingross and Lamb, 2017; Yager et al., 2012). In addition, it was necessary to quantify the wavelength of alternating bars. For the considerations on timescales presented here, the assumption of steady-state cover had to be made, implying fully developed bars and ignoring a potential braiding instability at large channel widths. Nelson and Seminara (2012) provided a linear stability analysis of bar formation over an initially bare bed. They stated explicitly that their considerations do not apply to the geometry of fully formed bars. However, their results and numerical model predictions by Inoue et al. (2016) could be interpreted to suggest that during the transient adjustment to fully formed bars from an initially empty bed under constant forcing conditions, bar wavelength varies little over time. Experimental evidence is rare. Some circumstantial observations can be found in the paper of Chatanantavet and Parker (2008), but these authors do not provide a systematic investigation or conclusive evidence for any type of scaling. In summary, none of the available studies was set up to investigate the controls of fully formed alternating bars, and a full understanding of the controls of their geometry is currently lacking. In absence of a full theory of alternating bars in bedrock channels, I have chosen to keep bar aspect ratio constant (Eq. 21), by analogy with observations in alluvial channels (e.g. Kelly, 2006). Yet, due to the coupling with bed cover (Eq. 23), this assumption leads to unphysical behaviour in the limit of small degrees of cover. In this case, the bar wavelength is small, implying small bar width in comparison to channel width. As a consequence, the meandering bedload path has a large amplitude in comparison to its wavelength, and the deflection angle γ approaches 90∘. The assumption about bar wavelength is a minor piece in the model, affecting only the response time channel widening, which is linearly dependent on bar aspect ratio. For a full treatment of bar wavelength, we can speculate on the behaviour at two limits. First, at low values of cover, bar wavelength should be independent of cover, and it is likely controlled by channel width or depth. Second, at high cover values, neighbouring bars start to overlap and the relationship to cover likely becomes more complicated. For a fully covered bed, bar dynamics should resemble those in alluvial channels. Further theoretical and experimental investigations are necessary to resolve this issue.

The lateral erosion equation (Eq. 24) generally aligns with expected relations. Lateral erosion rates increase with increasing shear stress, sediment supply, and erodibility. However, they are inversely proportional to bed cover. This negative relationship arises because gravel bars increase their length as cover increases, due to their constant aspect ratio (Eq. 23). This leads to less frequent impacts on the wall by travelling bedload. Fuller et al. (2016) observed that bedrock wall erosion is positively correlated with bed roughness in laboratory experiments. Similarly, Beer et al. (2017) observed higher wall erosion rates next roughness elements in a field study. The data from both of these papers are not sufficient for constraining a functional relationship between roughness and lateral erosion rates. In the model, lateral erosion rate (Eq. 24) depends implicitly on roughness, with a positive relationship, via the dependence on shear stress (see Eq. D6). A similar implicit dependence can be found for the sideward deflection distance d (Eq. 25). Nevertheless, dedicated data on sideward deflection distances are needed to test the current equations and to guide future theoretical developments. Another aspect that is lacking in the current formulation is the dependence of lateral erosion rate on channel curvature. Recent work has attempted to address this within the stream-power framework of bedrock erosion (e.g. Langston and Tucker, 2018; Limaye and Lamb, 2014). Including channel curvature into the present model needs further work on bar deposition and bedload paths within curved channels (cf. Bunte et al., 2006; Fernández et al., 2019; Mishra et al., 2018; Turowski, 2018).

In comparison to the model by Turowski (2018), the sideward deflection length scale d has been explicitly quantified in terms of hydraulics (Eq. 25), which may alter steady-state relationships in comparison to the previously published model. In general, the updated model's predictions align with the results of Turowski (2018). It is somewhat surprising that channel width, like in the model by Turowski (2018), is explicitly independent of discharge (Fig. 4a), and, instead, it is set by sediment supply (Fig. 4b). This implies that channel bed slope adjusts to changes in discharge without an effect on channel width, as long as sediment supply stays constant. The results arise because slope and discharge only feature in the same two equations, in that for Shields stress (Eq. D6) and that for the bedload transport capacity (Eq. C7). Using common parameter values for the relevant exponents m, n, and α, the relationship between slope and discharge is the same in these two equations, allowing the two parameters to co-vary without affecting other parameters. Considering all other parameters constant, the first of these (Eq. D6) gives the relation 49S∼Q2(α-1)α+1, while the second one (Eq. C7) gives the relation 50S∼Q-mn. With the common parameter choice of α=0.6 (see Nitsche et al., 2012), m=1, and n=2 (see Turowski, 2018), we find that the two exponents are equal: 512(α-1)α+1=-mn=-12. Thus, a change in discharge can be offset by a change in slope, without the need to vary any of the other parameters. Mathematically, this means that by substitution the number of parameters and equations can be reduced by one, and slope can be eliminated. A different choice of the m/n ratio or of α would yield a direct dependence of width on discharge, and a dynamic co-evolution of slope and width.

The condition of grade can be stated as what I call an order principle, which is a principle after which a dynamic system adjusts state variables to comply with forcing variables. Considering a stream without tributaries or hillslope sediment supply, the order principle for the condition of grade can be stated as follows: a river adjusts such that sediment flux is constant along the stream. The order principle is a direct consequence of the description of the sediment mass balance of the stream (see Sect. 1).

Unlike alluvial channels, which feature a single type of material (the alluvium), in bedrock channels we need to also consider bedrock. This necessitates a second mass balance equation for bedrock (Eq. 4), in addition to that for alluvium (see Sect. 2.1). Accepting that a sediment mass balance cannot be neglected for a mechanistic description of bedrock channel dynamics, a bedrock river thus adjusts to two order principles, rather than one. The first of these is related to the mass balance of sediment (Sect. 2.1) and leads to a condition of grade, as discussed above. The second of these is related to the mass balance of bedrock (Eq. 4) and can be stated as follows: the river adjusts such that the vertical erosion rate is equal to the uplift or base-level lowering rate.

When control variables change, the river responds by adjusting its morphology – slope, width, and bed cover – to comply with both of the order principles. However, due to the different adjustment timescales, the path to a new steady-state morphology may be complex. As an example, consider a river at steady state when sediment supply increases. The river responds by depositing sediment and increasing stationary sediment mass (Eqs. 6 and 7). The increase in available stationary sediment increases entrainment rates (cf. Turowski and Hodge, 2017). Deposition continues until the river reaches a graded state in which sediment outflux from the considered reach is equal to sediment supply (Eqs. 6 and 7). At the same time, any change in stationary sediment directly affects bed cover (Eq. 32), and the immediate response of the stationary sediment mass is reflected in the short response times of bed cover (Eq. 42; Fig. 5). Changes in cover, in turn, affect both vertical and lateral incision rates, initiating slope and width adjustment. These adjust much more slowly than bed cover (Fig. 5) until the vertical erosion rate matches the uplift rate. Yet, adjustments in width and slope feed back into the sediment dynamics, e.g. by affecting transport capacity. Again, the river responds by depositing or entraining material to maintain grade. The mutual feedback continues until both order principles – grade and the erosional balance with matching incision and uplift rates – are satisfied.

With two order principles controlling bedrock channel adjustment, the river may be in a steady state with respect to one of them but not with the other. Because the adjustment timescale for cover is shortest (Fig. 5), with values that range from minutes to days, it can be expected that bedrock rivers are close to a graded state most of the times (cf. Phillips and Jerolmack, 2016). Given the long adjustment times for width and slope, this does not necessarily mean, however, that they are in a steady state with respect to bedrock elevation, where incision rate matches uplift rate.

The considerations and arguments presented in this paper affect the conceptual view of a bedrock channel, and the use of relevant terminology. We can distinguish detachment-limited and transport-limited channels, which are identified with the two endmember descriptions focusing on the mass balance description of bedrock (detachment limited) and sediment (transport limited), respectively (cf. Shobe et al., 2017). For detachment-limited channels, we assume that the transport of sediment (Eq. 6) can be neglected; i.e. sediment transport does not significantly impact channel dynamics and morphology. Formally, this assumption is valid if sediment supply is very much smaller than transport capacity, implying that stationary sediment mass Ms∼0 or that sediment transport does not affect bedrock erosion. For transport-limited channels, we assume that bedrock incision can be neglected (Eq. 4). Formally, this assumption is valid if deposition or erosion has a negligible effect on the stationary sediment mass, in the mathematical limit as Ms goes to infinity. The latter point implies that entrainment or deposition of sediment does not significantly affect stationary sediment mass.

A formal definition of bedrock channels should fulfil a number of criteria (cf. Turowski et al., 2008b). First, the definition should comply with the intuition of field workers. Alluvial and bedrock channels are endmembers on a continuum of channel types, and, therefore, there will always be debated cases. But generally, most geomorphologists would agree on whether the particular river should be classified as an alluvial or bedrock river when seeing it in the field. Second, it should not rely on observations of field parameters that can change quickly, e.g. over a single flood. Third, a useful definition should not rely on parameters that cannot be measured. Fourth, it should not rely on theoretical concepts that are untested, untestable, or debated. Fifth, a definition rooted in the understanding of relevant processes or dynamics is preferable to one that relies solely on descriptions of morphology.

Bedrock channels, in general, have often been classified as detachment-limited channels, in which long-term sediment-supply is (much) smaller than long-term sediment transport capacity (e.g. Whipple, 2004; Whipple et al., 2013). Further, this condition is generally assumed to result in partial sediment cover and exposed bedrock on channel bed and banks. Bedrock exposure in the channel can easily be observed in the field, and it is therefore often used for channel classification (e.g. Montgomery et al., 1996; Tinkler and Wohl, 1998). A number of formal definitions of bedrock channels have been put forward based on these considerations. Exemplary, I will quote and discuss the most recent definition by Whipple et al. (2013):

Bedrock rivers may satisfy either or both of the following conditions: (1) the long-term capacity of the river to transport bedload (Qc) exceeds the long-term supply of bedload (Qs), resulting in generally sediment-starved conditions, significant rock exposure in bed and banks, and only thin, patchy, and temporary alluvial cover; or (2) the river is, over the long term (millennial to geologic timescales), actively incising through in-place rock.