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# Alluvial cover controlling the width, slope and sinuosity of bedrock channels

### Jens Martin Turowski

Bedrock channel slope and width are important parameters for setting bedload transport capacity and for stream-profile inversion to obtain tectonics information. Channel width and slope development are closely related to the problem of bedrock channel sinuosity. It is therefore likely that observations on bedrock channel meandering yields insights into the development of channel width and slope. Active meandering occurs when the bedrock channel walls are eroded, which also drives channel widening. Further, for a given drop in elevation, the more sinuous a channel is, the lower is its channel bed slope in comparison to a straight channel. It can thus be expected that studies of bedrock channel meandering give insights into width and slope adjustment and vice versa. The mechanisms by which bedrock channels actively meander have been debated since the beginning of modern geomorphic research in the 19th century, but a final consensus has not been reached. It has long been argued that whether a bedrock channel meanders actively or not is determined by the availability of sediment relative to transport capacity, a notion that has also been demonstrated in laboratory experiments. Here, this idea is taken up by postulating that the rate of change of both width and sinuosity over time is dependent on bed cover only. Based on the physics of erosion by bedload impacts, a scaling argument is developed to link bedrock channel width, slope and sinuosity to sediment supply, discharge and erodibility. This simple model built on sediment-flux-driven bedrock erosion concepts yields the observed scaling relationships of channel width and slope with discharge and erosion rate. Further, it explains why sinuosity evolves to a steady-state value and predicts the observed relations between sinuosity, erodibility and storm frequency, as has been observed for meandering bedrock rivers on Pacific Arc islands.

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Bedrock channels are the conveyer belts of mountain regions. Once sediment produced on hillslopes by mass wasting reaches a channel, it is evacuated along the stream network. In the process, the moving particles act as tools for bedrock erosion and the river adjusts until it reaches a steady state. Then, channel morphology, parameterized for example by the width and bed slope of the channel, stays constant over time, and the vertical erosion rate adjusts to match tectonic uplift. In turn, width and slope determine the transport capacity of the channel and thus the channel's efficiency in evacuating sediment. As a result, channel long profiles and width can be used as indicators for local tectonic process rate, and uplift rates and histories can in principle be calculated from morphologic characteristics of the channel network (e.g. Kirby and Whipple, 2001; Roberts and White, 2010; Wobus et al., 2006b). Conventionally, the inversion of channel morphology to obtain tectonic information has focussed on long profiles or slope, despite the observation that channel width also adjusts to tectonic forcing (e.g. Duvall et al., 2004; Lavé and Avouac, 2001; Yanites et al., 2010). For reliable inversion we thus need a model that can predict the effects of uplift both on channel width and slope (e.g. Turowski et al., 2009) and possibly other morphologic parameters.

Davis (1893) sparked a long-standing debate in geomorphology when he described the meanders of the Osage River as inherited from a prior alluvial state of the channel. Although this explanation is still frequently encountered to explain why bedrock channels are sinuous, even Davis' contemporaries argued that active meandering occurs in incised channels (e.g. Winslow, 1893). By now, numerous field observations of features such as cut-off meander loops and gentle slip-off slopes in inner meander bends have confirmed that actively meandering bedrock channels exist and are common (e.g. Barbour, 2008; Ikeda et al., 1981; Mahard, 1942; Moore, 1926; Seminara, 2006; Tinkler, 1971). However, the mechanics of bedrock river meandering are still debated and have recently attracted research interest (e.g. Johnson and Finnegan, 2015; Limaye and Lamb, 2014), since the meandering problem is closely related to the problems of terrace formation, lateral planation, gorge eradication and bedrock channel width (cf. Cook et al., 2014; Finnegan and Balco, 2013; Turowski et al., 2008a). In fact, active meandering is dependent on the lateral erosion of the channel walls and is therefore directly related to the adjustment of channel width. Similarly, meandering lengthens the channel over a given drop of height and thereby reduces bed slope. Thus, it seems likely that observations on bedrock channel sinuosity are informative also for the study of channel width and slope and vice versa. While the power-law scaling of channel width and slope with discharge with typical exponents of ∼ 1∕2, positive for width and negative for slope, is widely acknowledged (e.g. Lague, 2014; Snyder et al., 2003; Whipple, 2004; Whitbread et al., 2015; Wohl and David, 2008), observations of the scaling relationships of sinuosity are less commonly discussed. In a detailed study of Japan, Stark et al. (2010) demonstrated that lithology poses a first-order control on the sinuosity of actively incising bedrock channels, with weak sedimentary rocks displaying higher values of a regional measure of sinuosity than volcanic or crystalline lithologies. Once this influence was accounted for, a positive trend of sinuosity with the variability in precipitation emerged, quantified by typhoon-strike frequency or by the fraction of days with rainfall exceeding a threshold. This positive trend with storm frequency could generally be confirmed for other islands of the Pacific Arc, including Taiwan, Borneo, New Guinea and the Philippines (Stark et al., 2010). The prediction of the relationships observed by Stark et al. (2010) remains a benchmark for any theory of bedrock channel meandering, but an explanation is lacking so far. Further, in addition to observations on channel bed slope and width, the sinuosity scaling provides another line of evidence for the validation of general models of bedrock channel morphology.

Sinuosity increases when, within a channel bend, the bank at the outer bend erodes faster than at the inner bend. Alluvial meander theory relates this imbalance in lateral erosion to hydraulics (e.g. Edwards and Smith, 2002; Einstein, 1926; Ikeda et al., 1981). Within the bend, there are higher flow speeds in the outer bend than in the inner bend. Erosion rate and therefore the meander migration rate is assumed to be dependent on the velocity difference. In contrast, in many bedrock channels, erosion is driven by particle impacts in the two most common fluvial erosion processes: plucking and impact erosion (e.g. Beer et al., 2017; Chatanantavet and Parker, 2009; Cook et al., 2013; Sklar and Dietrich, 2004). Abrasion means the erosion due to impacts of moving bedload particles. Plucking means the removal of larger blocks of rock by hydraulic forces. In the latter process, impacts drive crack propagation and thus the production of pluckable blocks, which is also known as macro-abrasion (Chatanantavet and Parker, 2009). In environments where particle impacts drive erosion, the outer bends of meanders are particularly prone to erosion as particle trajectories detach from flow lines and can thus impact the walls (e.g. Cook et al., 2014). If bedrock channel sinuosity is indicative of past climate, as Stark et al. (2010) suggested, then bedrock channels need the ability to first adjust to the required sinuosity and second keep this sinuosity constant over long time periods, while continuing vertical incision. The latter feat can be achieved either by stopping lateral erosion once the required sinuosity is reached or by maintaining a balance of those processes that increase sinuosity and those that decrease it. The only known mechanism for decreasing sinuosity is meander cut-off. However, cut-off can only occur if the channel meanders actively, and it is only effective when sinuosity is high. The sinuosity of bedrock channels observed by Stark et al. (2010) span a wide range of values, including low sinuosities that cannot be kept steady with recurring cut-off. Thus, it seems unlikely that the cut-off mechanism can balance lateral erosion rates at low sinuosity to achieve a steady state. The argument suggests that channels cease or at least strongly decrease active meandering once they have reached the steady-state sinuosity, but why they do this is an unsolved problem. This raises the question as to when and why some bedrock channels actively meander while others do not. In general, two lines of argument have been proposed to answer this question.

The first line of argument asserts that the process of bedrock erosion controls lateral erosion rates, and local lithology determines this process and thus whether a channel actively meanders or not. Johnson and Finnegan (2015) compared two bedrock channels in the Santa Cruz Mountains, California, USA: one actively meandering in a mudstone sequence, the other one incising without meanders into a sandstone. While both lithologies showed similar strength when dry, the mudstone lost strength through slaking due to wetting–drying cycles and could thereafter be eroded by clear water flows. In this case, essentially, active meandering could be achieved by a similar hydraulic mechanism as has been described for alluvial streams (e.g. Edwards and Smith, 2002; Ikeda et al., 1981; Seminara, 2006). Moore (1926) likewise described an influence of lithology on the meanders of streams on the Colorado Plateau – there, meanders can be found in sandstone units, while in weaker shales, the valleys are wide and straight. However, Moore (1926) did not describe different erosion mechanisms (e.g. slaking, impact erosion) for the two lithologies, and it is unclear what causes the different channel behaviour in his study region. While the slaking mechanism should be more efficient in a variable climate due to more frequent wetting–drying cycles, in line with Stark et al.'s (2010) observations, it fails to explain why a stream can continue incising while maintaining a constant sinuosity. Further, Stark et al. (2010) described sinuous bedrock channels in a range of lithologies, including hard crystalline rock, where slaking erosion is likely not important.

The second line of argument builds on the relative availability of sediment in the channel. In resistant bedrock, erosion is driven by the impacts of moving particles in the two most common fluvial bedrock erosion processes, abrasion and plucking. The increasing erosion rate with increasing relative sediment supply is known as the tools effect (e.g. Cook et al., 2013; Sklar and Dietrich, 2004). Conversely, stationary sediment residing on the bed can protect the bedrock from impacts. This is known as the cover effect (e.g. Sklar and Dietrich, 2004; Turowski et al., 2007), which has been argued to play a key role in the partitioning of vertical to lateral erosion (e.g. Hancock and Anderson, 2002; Turowski et al., 2008a). Moore (1926) suggested that whether a bedrock river actively meanders or not depends on the relative availability of sediment, a notion that was later investigated experimentally by Shepherd (1972). In Shepherd's (1972) experiments, a sinuous channel was cut into artificial bedrock made of sand, kaolinite and silt, which was not erodible by clear water flow. Base level, water discharge and sediment supply were kept constant over the entire run time of 73 h. At first, all sediment could be entrained by the flow and the channel cut downwards, without changing the planform pattern. But as the channel bed slope declined due to erosion over the course of the experiment, patches of sediment formed on the inside bends and the channel started to widen and to meander actively. Shepherd (1972) suggested that lateral erosion rates stayed similar throughout the entire run, while vertical erosion rates declined due to the increasing importance of the cover effect. Thus, at first, lateral and vertical erosion were balanced such that channel width remained constant over time, while the later decrease in vertical incision led to channel widening and ultimately migration and active meandering.

Shepherd's (1972) experimental observations point to the fundamental importance of bed cover in setting bedrock channel width and meandering dynamics. In this paper, I develop a physics-based scaling argument to explain the observed scaling of bedrock channel width, slope, and sinuosity. The argument is motivated by the behaviour of the experimental channel of Shepherd (1972) and is built on the fundamental assumption that bed cover controls lateral erosion. It exploits general considerations and observations about bedload transport and process knowledge of fluvial bedrock erosion. Since channel morphology is set by the partitioning of erosion between bed and banks, the problem is approached by assessing under which conditions lateral erosion can occur and how these conditions relate to channel bed cover. The physical considerations lead to a model of incising channels with stable width, slope and sinuosity. Model predictions are compared to observed scaling relationships of bedrock channel width and slope with discharge, drainage area and erosion rate and to the sinuosity scaling observed by Stark et al. (2010).

Previous attempts at predicting bedrock channel morphology can be grouped into four classes (Shobe et al., 2017, also gave a recent overview):

- i.
1-D models using a shear stress or stream power formulation (e.g. Seidl et al., 1994; Whipple, 2004). These models capture the fundamental scaling of slope with discharge and, to an extent, of slope with erosion rate, but need to make assumptions on width–discharge scaling for closure (see Lague, 2014, for a review). Zhang et al. (2015) described a morpho-dynamic model that also captures alluvial dynamics and includes both tools and cover effects. However, this model is restricted to channels with macro-rough beds, i.e. topography with a relief that is a substantially larger than the dominant grain size.

- ii.
1-D models that treat channel width explicitly, but, instead of assuming a width–discharge scaling, make an alternative assumption to close the system of equations. Suggested assumption have been a constant width-to-depth ratio (Finnegan et al., 2005) or the optimization of energy expenditure (Turowski et al., 2007). These models have been proposed assuming a shear stress or stream power erosion law (Finnegan et al., 2005; Turowski et al., 2009), as well as sediment-flux-dependent erosion laws including either just the cover effect (Yanites and Tucker, 2010) or both tools and cover effects (Turowski et al., 2007). For the shear stress erosion model, the closing assumption has at least been partially validated against models treating the cross-sectional evolution of a channel (Turowski et al., 2009). Although these models can predict a range of observed scaling relations, especially if sediment flux effects are included in the erosion model (see Turowski et al., 2007; Yanites and Tucker, 2010), they suffer from a lack of physics-based arguments for connecting lateral erosion to channel morphology and from the essential arbitrariness of the closing assumption.

- iii.
2-D models that explicitly model some aspects of the width dynamics. For a shear stress erosion law, Stark (2006) used a slanted trapezoidal channel shape, while Wobus et al. (2006a) and Turowski et al. (2009) described models with a fully adjustable channel cross section. Lague (2010) used a trapezoidal cross section and included the cover effect in his formulation. The success of these models in predicting scaling relationships is similar to that of the models of class (ii), but none of the models published so far include all aspects of the current understanding of the process physics of fluvial bedrock erosion. Further, none of these models properly treat fully alluviated beds, where alluvial channel processes dominate, which can strongly affect long-term erosional dynamics and channel adjustment timescales (cf. Turowski et al., 2013).

- iv.
3-D models that, to some extent, resolve the interaction of hydraulics and sediment transport and their effect on bedrock erosion (e.g. Inoue et al., 2016; Nelson and Seminara, 2011, 2012). These models are generally numerically expensive and have not been used to investigate scaling relations on the reach to catchment scale.

As a result, we lack a model that is rooted in the current understanding of process physics and can predict channel width, slope and sinuosity on the catchment scale. Here, inspired by the experiments described by Shepherd (1972), I put forward the fundamental postulate that the partitioning between lateral and vertical erosion, and therefore width adjustment and sinuosity development, is controlled by a single variable: bed cover. Parameters such as sediment supply, river sediment transport capacity and bed topography directly control cover, but they only indirectly control the distribution of erosion by altering bed cover. Formalizing the observations made in Shepherd's (1972) experiments, we can make the following statements: (i) at low degrees of cover, channel width stays constant and the channel does not meander actively, and (ii) channel widening and active meandering commences when a threshold cover is exceeded. In Sect. 2.1, based on considerations based on the physics of erosion by particle impacts and of bedload transport, I develop a scaling argument for bedrock channel width. In Sect. 2.2, the slope of the channel is discussed. In Sect. 2.3, the argument is applied to the development of bedrock channel sinuosity.

## 2.1 Lateral erosion and bedrock channel width

Consider a straight bedrock channel with sub-vertical walls. The general
direction of water and particle discharge is parallel to the walls, although
we can expect some lateral motion due to secondary currents and turbulent
fluctuations. As bedrock erosion is achieved by particle impacts, the
requirement for lateral erosion is a sideward deflection of travelling
particles such that they (i) reach and impact the wall and, (ii) upon
impact, have enough energy to cause damage to the rock. Lateral motion of
sediment particles can be driven by secondary currents, turbulent
fluctuation and momentum diffusion (e.g. Diplas et al., 2008; Parker,
1978), cross-stream diffusion of particle paths (Seizilles et al., 2014),
gravitationally driven migration on cross-sloping beds (e.g. Parker et al.,
2003), or sideward deflection by obstacles on the bed (Beer et al., 2017;
Fuller et al., 2016). For given conditions – hydraulics, bed morphology,
sediment supply and grain characteristics – we can define a sideward
deflection length scale *d*_{xy} for every point on the bed, which depicts
the maximum distance a particle can be deflected sideward while still
causing bedrock wall erosion. This length scale should be a function of hydraulics or
transport capacity, channel bed slope, channel curvature, bed roughness,
sediment properties (size, shape, density), and possibly of the erodibility
of the bedrock via the threshold for erosion. Crucially, it can be expected
that *d*_{xy} can vary considerably over short distances both along and
across the channel, depending on bed topography and the local distribution
of roughness and alluvial cover. For the construction of a reach-scale model
of bedrock channel morphology, we need to first find the relevant point
within each cross section and the corresponding *d*_{xy} that determines
lateral erosion (which we call *d*_{x}) and then the relevant cross section
and the corresponding *d*_{x} that determines channel width in the reach
(which we call *d*). For a given channel, the propensity to lateral erosion
then depends on the ratio of the sideward deflection length scale *d* and the
channel width *W* (Fig. 1). In a channel with a width much larger than *d*, only
bedload moving close to the walls – or more precisely, within a distance *d* of the
walls – can contribute to lateral erosion. In contrast, in a channel with
*W* ∼ *d*, all bedload can contribute to lateral erosion.

In general, a bedrock channel widens only when bedload particles impact the
walls, i.e. in the framework proposed above that some bedload is moving
within a distance *d* from the walls. For the purpose of illustration, consider a
narrow, straight bedrock channel with *W* ∼ *d* (Fig. 1). Due to frequent particle
impacts on the walls, lateral erosion rates are high and the channel widens.
This leads to a decrease in the areal sediment concentration and thus a
decrease in the number of bedload particles that can cause lateral erosion.
At some point bedload impacts on the wall become so unlikely that widening
ceases. The channel has reached a steady-state width. However, this argument
does not capture the entire story, since we have neglected vertical
incision. Next, this aspect will be included in the consideration and the
ratio *d*∕*W* will be tied to one of the common observables in bedrock channel
morphology: the covered fraction of the bed *C* (e.g. Sklar and Dietrich,
2004; Turowski and Hodge, 2017).

The relative efficiency of lateral to vertical erosion has been tied to bed
cover in conceptual–theoretical arguments (e.g. Hancock and Anderson, 2002;
Moore, 1926), experimental observations (e.g. Finnegan et al., 2007;
Johnson and Whipple, 2010; Shepherd, 1972) and field studies (e.g. Beer et
al., 2016; Johnson et al., 2010; Turowski et al., 2008a). Using a
combination of experiments and modelling, it has been argued that the
fraction of covered bed area is an adequate proxy for the reduction in
erosion due to the shielding effect of sediment on the reach scale (Turowski
and Bloem, 2016). Consequently, cover *C* is commonly defined as the covered
bed area fraction, i.e. the bed area covered by sediment *A*_{cover} divided
by the total bed area of the considered reach *A*_{tot}. Normalizing by the
length of the considered reach *L*, we can write *C* also as a ratio between two
length scales, the relevant covered width *W*_{cover} (which could be a reach
average or the covered width for the cross section relevant for setting
lateral erosion rates) and the channel width *W*.

At low sediment supply, cover is low to non-existent. Sufficient tools for
incision are available only where the particle stream concentrates. There,
an inner channel is formed, and so the channel narrows (e.g. Finnegan et
al., 2007; Johnson and Whipple, 2010). To a similar effect, in wide
channels, several longitudinal grooves tend to form at low sediment supply
(Inoue et al., 2016; Wohl and Ikeda, 1997). One of these draws most sediment
and water and, after some time, develops into an inner channel that captures
the entire water and sediment supply. At high sediment supply, the bed is
covered by sediment, which reduces vertical erosion to zero. Lateral erosion
occurs in a strip just above the cover, where bedrock is exposed and tools
are abundant (Beer et al., 2016; Turowski et al., 2008a). The channel
widens. We can formalize the observations outlined above by relating the
rate of change of channel width, d*W*∕d*t*, to relative sediment supply
${Q}_{\mathrm{s}}^{\ast}$, which is the ratio of bedload supply *Q*_{s} to transport
capacity *Q*_{t} (Fig. 2). At ${Q}_{\mathrm{s}}^{\ast}$ = 0, lateral erosion and
therefore d*W*∕d*t* is also zero, due to the lack of erosive tools. For small
${Q}_{\mathrm{s}}^{\ast}$, the channel narrows and d*W*∕d*t* must be negative. For high
${Q}_{\mathrm{s}}^{\ast}$, the channel widens and d*W*∕d*t* must be positive. Since cover *C* is
generally related to ${Q}_{\mathrm{s}}^{\ast}$ (e.g. Sklar and Dietrich, 2004;
Turowski and Hodge, 2017; Turowski et al., 2007), a similar relationship
must arise between d*W*∕d*t* and cover. At a critical value, ${Q}_{\mathrm{c}}^{\ast}$ or
*C*_{c}, the channel behaviour switches from narrowing to widening and
d*W*∕d*t* = 0. This is the only point where the channel both has a steady width
and incises vertically with a finite erosion rate. At the critical cover,
the distance of bedload particles from the walls needs to be equal to the
sideward deflection length scale *d*. If *d* is larger than this typical distance,
frequent impacts will occur on the channel walls and the channel widens
(Fig. 1). If it is smaller, few bedload particles move in the vicinity of
the walls, leading to a lack of erosive tools, and the bed near the walls is
not eroded. An inner channel forms for which the above condition is true.

As can be seen from the following argument, the critical cover *C*_{c} must
depend on channel width and should indeed scale with *d*∕*W*. Chatanantavet and
Parker (2008) demonstrated with experiments that in wide straight channels
in the cover-dominated domain, alternating gravel bars formed. Inoue et al. (2016) modelled this situation and found that a meandering thread of
alluvial material between alternating submerged gravel bars migrates
downstream over uniformly eroding bedrock, leading to a channel with a
symmetric bedrock cross section. From studies on alluvial rivers it is known
that the main path of bedload particles in a straight channel with submerged
bars is offset from the main path of water and the thalweg (e.g. Bunte et
al., 2006; Dietrich and Smith, 1984; Julien and Anthony, 2002). Gravel
bedload moves across the bar, enters the thalweg at the bar centre,
traverses it and climbs the next downstream bar at its head (Fig. 3).
Similarly, it has been observed that in a partially alluviated bedrock
channel, sediment moves from patch to patch or from bar to bar (Ferguson et
al., 2017; Hodge et al., 2011). However, the precise bedload path over
partially covered bedrock has not yet been described. For the following
argument, I make two main assumptions. (i) The bedload path determined by
Bunte et al. (2006) for gravel bed channels with alternating submerged bars
applies also to bedrock channels (Fig. 3). This assumption is plausible and
is adopted since there is a lack of direct relevant data. (ii) The sideward
deflection length of bedload is largest at the edge of alluvial patches or
bars in the direction of the uncovered bedrock (Fig. 4). This assumption is
made for three reasons. First, the bedrock is typically smoother than the
alluviated section and provides less impediment to particle movement, in
particular to sideward deflection toward the uncovered part of the cross
section (cf. Chatanantavet and Parker, 2008; Ferguson et al., 2017; Hodge et
al., 2011, 2016). Second, at the edge of bars, the alluvium provides
roughness elements that can lead to sideward deflection (cf. Beer et al.,
2017; Fuller et al., 2016). Third, at this point the velocity vector of the
bedload particles has a large cross-stream component; in fact, it is at its
maximum (Fig. 3). In a channel with steady-state width, bedload particles at
this point should just fail to reach the wall, and we can assume that the
sideward deflection length scale *d* is approximately equal to the uncovered
width (Fig. 3). At steady state, we therefore expect that the following
relation holds:

Using the equation for critical cover (Eq. 2), we can relate channel width to vertical erosion rate using one of the established models for incision (e.g. Auel et al., 2017; Sklar and Dietrich, 2004). I assume a sediment-flux-dependent erosion law, including tools and cover effect, of the form

Here, *E* is the vertical erosion rate and *k* is a parameter that describes the
erodibility of the rock. As before, *Q*_{s} is the upstream bedload sediment
supply. Note that in the original saltation–abrasion model (Sklar and
Dietrich, 2004), *k* depends explicitly on hydraulics, but, consistently, in all
of the field and laboratory studies where all relevant parameters have been
measured, this dependency has not been found (Auel et al., 2017; Beer and
Turowski, 2015; Chatanantavet and Parker, 2009; Inoue et al., 2014; Johnson
and Whipple, 2010). At steady state, *C* = *C*_{c}. Substituting Eq. (2) into
Eq. (3) and solving for width, we obtain an equation for the steady-state
width of bedrock channels.

## 2.2 Channel bed slope

To extend the argument to channel bed slope, an additional equation is needed relating bed cover to sediment supply and transport capacity. Several equations have been suggested in the literature, including the linear decline model (Sklar and Dietrich, 2004) and the negative exponential (Turowski et al., 2007). Recently, Turowski and Hodge (2017) derived a model of the form

Here, *e* is the base of the natural logarithm, *U* is the average bedload
particle speed and *M*_{0} is the minimum mass per bed area necessary to
completely cover the bed, which is dependent on grain size (Turowski, 2009;
Turowski and Hodge, 2017). Note that Eq. (5) reduces to the linear decline
model at high sediment supply, i.e. for large *Q*_{s}.

We can write the bedload transport capacity per unit width as a power
function of both discharge *Q* and channel bed slope *S* (e.g. Rickenmann, 2001;
Smith and Bretherton, 1972):

Here, *K*_{bl} is a constant, and it has been argued that the exponents *m* and
*n* typically take values between 1 and 4 (Barry et al., 2004; Smith, 1974).
Note that in Eq. (6), the threshold of the motion of bedload has been neglected.
Such a threshold is generally accepted to be relevant for bedload motion
(e.g. Buffington and Montgomery, 1997) and will become important when
linking sinuosity to storm frequency. Assuming steady state at the critical
cover *C*_{c}, substituting Eqs. (2) and (6) into Eq. (5), and solving for *S*, we
get

## 2.3 Sinuosity

At a given location, lateral erosion and therefore the development of
curvature and sinuosity is of course dependent on local conditions such as
the channel width, bed slope and long-stream curvature (e.g. Cook et al.,
2014; Howard and Knutson, 1984; Inoue et al., 2016). But rather than trying
to predict the detailed evolution of the planform pattern, here I propose a
reach-scale view of sinuosity development. As is conventional, sinuosity
*σ* is defined as the ratio of the total channel length *L*_{C} to the
straight length *L*_{V} from end to end. Note that this is equivalent to the
ratio of valley slope *S*_{V} to channel bed slope *S*.

Sinuosity can only increase if the walls of the channel are eroded. Thus,
the rate of change of sinuosity d*σ*∕d*t* should be zero when d*W*∕d*t* is negative.
Sinuosity development commences at the same critical cover *C*_{c} that marks
the transition from channel narrowing to widening, and d*σ*∕d*t* should be
positive when d*W*∕d*t* is also positive (Fig. 2). However, we need to slightly
adjust the picture that has been advanced in Sect. 2.1, since instead of a
straight channel, we are now dealing with a curved channel. Further, channel
curvature is varying along the stream. As before, lateral erosion should
stop once the channel walls are outside the reach of particle impacts.
Due to curvature, particle trajectories detach from water flow lines and
wall erosion rates can be expected to be highest in regions with the highest
curvature (e.g. Cook et al., 2014; Howard and Knutson, 1984). Point bars
develop in the inside bends, providing roughness for sideward deflection
(Fig. 5). Substantial particle impacts can thus be expected in the outside
bends, probably a little downstream of the bend apex (cf. Fig. 5). The rest
of the argument can stay essentially the same: lateral erosion stops once
the bedrock wall is just outside the reach of the deflected particles.
The bedrock channel is driven to a steady state at which *C* = *C*_{c}. At
this point, sinuosity development ceases and the channel essentially stalls
itself in its active meandering. Treating valley slope as an independent
parameter, Eq. (8) can be substituted into Eq. (7) and solved for sinuosity
to obtain

In this section, I will compare the model to field and laboratory observations. First, I will interpret the experiments of Shepherd (1972) in light of the arguments that lead to the model equations. Then, I will compare field observations to the predictions by the equations. Since for most field sites many essential parameters are not known, I will focus on accepted scaling relations. Lague (2014) has summarized the available data for geometry and dynamics of bedrock channels and has identified six lines of evidence that any model needs to match. Two of these are related to transient channel dynamics and knick point migration. Since the model developed in the present paper is only concerned with steady-state channels, the remaining lines of evidence, namely slope–area scaling, slope–erosion rate scaling, width–area scaling, and width–erosion rate scaling, are discussed below. To these I add the two scaling relations for the sinuosity of channels – sinuosity–erodibility scaling and sinuosity–storm-frequency scaling – as observed by Stark et al. (2010).

For the comparison with field data, I use six data sets that include information on erosion rates, with scaling relationships as compiled by Lague (2014) (Table 1). Two of these data sets arise from studies of rivers crossing a fault: the Bakeya, Nepal (Lavé and Avouac, 2001), and the Peikang river, Taiwan (Yanites et al., 2010). The data for the Bagmati, Nepal (Lavé and Avouac, 2001), were not used, since a tributary joins the stream within the studied reached, supplying unknown amounts of both water and sediment and thereby altering boundary conditions (see Lague, 2014; Lavé and Avouac, 2001; Turowski et al., 2009). Four of the data sets arise from studies comparing different catchments that are thought to be in a topographic steady state along a gradient in uplift rate with otherwise comparable conditions. These are channels from the Siwalik Hills, Nepal (Kirby and Whipple, 2001; reanalysed by Wobus et al., 2006b), the Mendocino Triple Junction (Snyder et al., 2000), Eastern Tibet (Ouimet et al., 2009) and the San Gabriel Mountains (DiBiase et al., 2010). I did not use the data from the Santa Inez Mountains (Duvall et al., 2004), since a lack of coarse bedload in these mudstone channels has been reported (Whipple et al., 2013). There, impact erosion may not be the dominant erosion process, which could alter channel processes, morphology and dynamics. The channels studied by Tomkin et al. (2003) and Whittaker et al. (2007), draining catchments with strong long-stream gradients in uplift rate, are under-constrained for the purpose of model comparison, since the variation in erosion rates and therefore sediment supply along the stream is unknown.

For parts of the discussion it is useful to work with two approximations for
the cover equation, Eq. (5), both for the sake of algebraic simplicity and
ease of argument. First, in the tools-dominated domain, cover is scarce and
bedrock erosion rate is controlled by the availability of tools. Then,
*Q*_{s}∕*W* is small and the exponential term can be approximated with a
first-order Taylor expansion, reducing Eq. (5) to

Then, we can reduce the first term in the slope and the sinuosity equations, Eqs. (8) and (10), which yields

Second, in the cover-dominated domain, tools are abundant, but most of the
bed is covered. Then, the erosion rate is set by the fraction of the exposed
bedrock. Sediment supply per unit width *Q*_{s}∕*W* is large, the exponential
term vanishes, and we retrieve the linear model (Sklar and Dietrich, 2004).

Then, the first term in the slope and the sinuosity equations, Eqs. (8) and (10), reduces to 1.

The cover-dominated approximation (Eqs. 12 and 13) is likely most relevant
for the data discussed here. It is known that many actively incising bedrock
rivers exhibit substantial cover at least at low flow (Meshkova et al.,
2012; Tinkler and Wohl, 1998; Turowski et al., 2008b, 2013), and it seems
likely that for many rivers the sideward deflection length scale *d* is much
smaller than the channel width (formally, *W*≪*d*,
leading to $W-d\approx \phantom{\rule{0.125em}{0ex}}W$), implying substantial cover at steady state.
Therefore, it can be expected that the tools-dominated approximation (Eqs. 10 and 11) is only relevant for small headwater streams or for channels that
do not receive much coarse sediment, for example due to an upstream
reservoir.

## 3.1 Shepherd's (1972) experiment

Shepherd's observations have been described in detail in the introduction.
From a model perspective, consider a stream that re-incises its bed after a
base level drop. At constant sediment supply, as the stream incises, bed
slope and therefore transport capacity decreases. As a result, cover
increases (Eq. 5). At some point the critical cover *C*_{c} is exceeded and
the stream starts active meandering. Meandering lengthens the flow path and
therefore also decreases bed slope and transport capacity. The subsequent
increase in cover leads, at some point, to full cover stopping vertical
incision. Once the steady-state width is reached, lateral erosion drops to
zero. Then, the stream also stops active meandering. It essentially stalls
itself and reaches a steady state for sinuosity. The described scenario is
equivalent to the one observed by Shepherd (1972), although the stalling
phase was not reached in his experiments.

## 3.2 Channel width

A number of studies report the sensitivity of channel width to uplift rate (for summaries of the available data, see Lague, 2014; Turowski et al., 2009; Whipple, 2004; Yanites and Tucker, 2010). Several different behaviours have been observed (see also Table 1). In comparisons of channels in catchments that differ only by uplift rate, channel width was comparable at similar drainage areas, indicating that there was no response to uplift rate (Snyder et al., 2003; DiBiase and Whipple, 2011). In another study, Duvall et al. (2004) found narrower channels in catchments with higher uplift rates, but this could also be related to the lack of coarse bedload in the mudstone channels (Whipple et al., 2013). Similarly, some channels display a typical width–area scaling despite strong gradients in uplift rate (Tomkin et al., 2003; Whittaker et al., 2007). In contrast, channels crossing an uplifting fault block tend to narrow (Lavé and Avouac, 2001; Yanites et al., 2010).

According to the proposed model, steady-state channel width scales with the
square root of the product of bedload supply *Q*_{s}, erodibility *k* and
sideward deflection length scale *d* and inversely with the square root of the
vertical incision rate *E* (Eq. 4). The different response of channel width in
studies comparing different channels in areas with gradients in uplift rate
(no channel narrowing) and those that looked at single channels crossing an
uplifting fault block (channel narrowing) can be explained by the role of
sediment flux. I will discuss the latter case first.

When a channel crosses from a region without uplift into an uplifting fault
block, water discharge and sediment load stay the same, provided there are
no tributaries or major hillslopes sediment sources. Thus, in the width
equation (Eq. 4), bedload supply *Q*_{s} is constant and the channel
responds by increasing erosion rate *E* to match the increased uplift rate.
Provided that *k* and *d* are independent of erosion rate, the channel narrows and
channel width should scale with incision rate to the power of $-\mathrm{1}/\mathrm{2}$. Two of
the cases mentioned above allow a direct evaluation of this prediction. In
the Bakeya River (Lavé and Avouac, 2001), the width–erosion rate scaling
exponent is −0.63, and in the Peikang River (Yanites et al., 2010), the
scaling exponent is −0.42 (Table 1); both are close to the predicted value of
$-\mathrm{1}/\mathrm{2}$.

In catchments in a topographic steady state, the channel geometry adjusts
such that the long-term incision rate matches the long-term uplift rate or
base level lowering rate. Averaged over the catchment, bedload supply
can be written in terms of erosion rate *E* and catchment area *A*.

Here, *β* is the fraction of material that contributes to bedrock
erosion, i.e. the bedload fraction. The steady-state channel width Eq. (4) then becomes

As vertical incision rate *E* cancels out, steady-state channel width in this
case is independent of uplift rate. This is in agreement with field
observations (Table 1). Equation (15) also predicts the typical scaling of
channel width *W* with the square root of drainage area *A*. However, it is likely
that both the gravel bedload fraction *β* and the sideward deflection
length scale *d* vary in a systematic fashion with drainage area. The bedload
fraction tends to decrease with increasing drainage area (e.g. Turowski et
al., 2010), possibly even to the extent that bedload supply *Q*_{s} is
independent of drainage area (see Dingle et al., 2017). There are additional
complications that arise from non-linear averaging of sediment supply both
with varying floods and stochastically varying bedload supply. Further, the
bedload fraction *β* is likely dependent on erosion rate *E*, in a
currently unknown way. At the moment little is known about how *d* varies along
a stream. I will return to this point in the discussion.

## 3.3 Channel bed slope

A power-law scaling of slope with drainage area with an exponent of $-\mathrm{1}/\mathrm{2}$ is widely assumed to be indicative of steady-state bedrock channels.

This relationship is known as Flint's law (Flint, 1974), although it was studied by Hack (1957) at an earlier point. The pre-factor *k*_{s} is called the
steepness index, and the exponent *θ* is called the concavity index.
For the concavity index, a range of values of 0.4–0.6 is often reported
(Lague, 2014). Whipple (2004) gives a range of 0.4–0.7 for actively incising
bedrock channels in homogenous substrates with uniform uplift, while higher
concavities (0.7–1.0) are associated with decreasing uplift rates in the
downstream direction. Using data from catchments where erosion rate have
been constrained using cosmogenic nuclides, Harel et al. (2016) found a
median value of the concavity index of 0.52 ± 0.14, with a similar
range as reported by Whipple (2004). It seems, therefore, that the observed
variability in the value of the concavity index is higher than is generally
acknowledged, with observed values as low as 0.4 and as high as 1. In
comparisons of channels in steady-state landscapes, the steepness index
*k*_{s} has been observed to increase with incision rate according to a power
law, with an exponent ranging from 0.25 to 0.93 (Table 1), derived from four
data sets (Lague, 2014). The two channels crossing a fault block exhibit
different scaling. The Bakeya (Lavé and Avouac, 2001) shows a positive
relationship with an exponent of about 0.49, while for the Peikang (Yanites
et al., 2010), little to no slope changes in response to uplift have been
reported.

The brief summary of observations above implies that a model should be able to account for the following observations. (i) Slope should decrease with drainage area according to a power law with an exponent value varying between about 0.4 and 0.7. (ii) The exponent may be altered if there are gradients in uplift rate along the stream; in particular, a downstream decrease in uplift may drive the concavity index up to higher values of up to about 1. (iii) In channels draining catchments in a topographic steady state, the steepness index should increase with uplift rate according to a power law with an exponent value varying between about 0.25 and 1.0. (iv) In channels crossing a fault block, slope may or may not increase in response to uplift.

Often, the concavity index in the slope–area relationship is related to a slope–discharge scaling by assuming that discharge scales with drainage area following a relationship of the form

Here, *k*_{h} and c are catchment-specific values describing the hydrology. In
particular, the exponent c takes a value of 1 if the exchange of water with
groundwater storage and evapotranspiration are spatially uniform in the
catchment (e.g. Snyder et al., 2003). For natural data, the value of c is
dependent on the discharge chosen for the regression. For the long-term mean
annual discharge, various effects should average out and c should be close to
1 (Dunne and Leopold, 1978, as cited by Snyder et al., 2003). Leopold et al. (1964) reported values between 0.70 and 0.75 for bankfull discharge. When
transforming the observed values of the concavity index of the slope–area
scaling to an exponent of the slope–discharge relationship, we thus obtain a
range of values for the slope–discharge exponent of 0.4–1.0 for steady-state
channels in uniform conditions and 0.7–1.4 for channels with a downstream
decrease in uplift rate.

In the model equation (Eq. 7), slope scales with discharge to a power of
$-m/n$. Many bedload transport equations can be written in the form of Eq. (6) (Smith and Bretherton, 1972), and the theoretical values of *m* and *n* depend
on the chosen equation. For example, the Einstein (1950) bedload equation
yields $m\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}n\phantom{\rule{0.125em}{0ex}}=$ 2 (Smith and Bretherton, 1972), while Meyer-Peter and
Müller (1948) type equations yield *m*=1 and *n* = 1.5 (Rickenmann,
2001). However, in the latter case, the linear scaling arises only if the
threshold of bedload motion is neglected and is thus valid only for large
floods. Rickenmann (2001) argued that *n* = 2 gives a better fit for both
laboratory and field data at stream gradients larger than 3 %. However, he
also included relative roughness as a separate predictor, which is
implicitly dependent on slope. If written out explicitly, the dependence on
slope should be stronger, with values of *n* potentially much larger than 2
(see also Nitsche et al., 2011; Schneider et al., 2015). Measured *m* values
are usually much larger than those derived from models. For example, Bunte
et al. (2008) reported *m* values ranging from about 7.5 to 16, using data
obtained during the snowmelt period of American streams with portable
bedload traps. Analysing bedload data sampled with Helley–Smith pressure
difference samplers from a large number of streams, Barry et al. (2004)
found values of *m* in the range of about 1.5–4.0. They used drainage area
instead of slope in their transport equation, and the data given in their
paper do not allow a re-evaluation in terms of slope. Nevertheless, a
regression of channel bed slope of the sites against drainage area yields an
exponent of −0.48, giving an estimate of *n* ≈ 7.1. From the mentioned
cases, it is clear that depending on the choice of equation or data set, a
wide range of values for the *m* and *n* scaling exponents can be obtained.
Finally, it needs to be noted that most bedload data and bedload transport
equations in the literature have been derived for channels with a mobile
bed. Bedload equations specifically for natural bedrock channels are not
known to the author. In addition to the explicit relationship of slope and
discharge, slope is implicitly related to discharge via sediment supply,
channel width and the sideward deflection length scale, all of which could
depend on discharge or drainage area.

Of the discussed approaches, the field data evaluation by Rickenmann (2001) may be most appropriate for the purpose at hand, since the data were
derived from long-term monitoring of deposition in retention basins. The
timescale of the data is thus closer to the timescales of bedrock erosion
and channel adjustment than the near-instantaneous measurements used for
example by Barry et al. (2004). This would yield values of *m*= 1 and
*n* = 2 and a ratio *m*∕*n* = 0.5 (Rickenmann, 2001). For the remainder of the
discussion, I will use this case as standard, as well as a range of
*n* values of 1.5–7 for evaluating possible ranges of the values of scaling
exponents.

The equations and the discussion are considerably simplified in the tools- or cover-dominated approximations (see Eqs. 10–13). In the tools-dominated case, channel bed slope is given by

Here, we can recognize two different cases. First, consider narrow headwater
channels. There, the sideward deflection length scale *d* is of the order of
the channel width *W*. As a result, slope depends strongly on the actual values
of *d* and *W* and their scaling with other morphological parameters, e.g. bed
roughness. I will not consider this case further, as there are few relevant
data available. Second, consider a wide channel carrying little coarse
sediment, for instance due to an upstream reservoir. Then, *W*≪*d* and Eq. (18) reduces to

Since bedload particle speed *U* is dependent on hydraulics, there is an
implicit dependence of *U* on slope and discharge, which needs to be taken into
account. With standard assumptions on flow velocity and shear stress
(Appendix A), Eq. (19) becomes

Here, *k*_{tools} is assumed to be constant (see Eq. A9, Appendix A), and
*α* is a constant that typically takes a value of 0.6 (e.g. Nitsche
et al., 2012). In the case of a steady-state channel crossing an uplifting
fault block, *Q*_{s} and *Q* can be considered constant and only *E* varies. In
this case, the discharge exponent is equal to −0.5 as long as $m/n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}/\mathrm{2}$. For
*n* = 1.5, the dependence on erosion rate and erodibility yields an exponent
of 0.47, with decreasing values as *n* increases (it reaches 0.375 for
*n* = 2, 0.20 for *n* = 4 and 0.12 for *n* = 7). For a channel in a
steady-state landscape, we can substitute Eq. (14) to obtain

Now, the exponent on erosion rate varies between 0.27 and 1.05. As before,
slope–area scaling cannot be evaluated in a meaningful manner, since the
dependence of *β* and *d* on area is unknown.

In the cover-dominated case, Eq. (7) reduces to

Here, I also used the approximation *W*≪*d*, and
channel width was eliminated using Eq. (4). For rivers crossing an uplifting
fault block, where all parameters apart from erosion rate can be treated as constant, slope scales with the incision rate *E*^{1∕2n}, with the exponent
lying in the range of 0.07–0.33 and using a range of *n* values of 1.5–7, as
discussed above. For catchments in a topographic steady state, *Q*_{s} can be
expected to scale linearly with erosion rate (Eq. 14), yielding a slope
equation of the form

In this case, the exponent on erosion rate yields the range of values of
0.14–0.67. The dependence on *Q*_{s} introduces an additional dependence on
area, affecting the slope–area exponent. Assuming that *Q* is proportional to
drainage area (*c* = 1) and *m* = 1 and *n* = 2, the slope–area exponent
evaluates to −0.25. However, both bedload fraction and sideward deflection
distance can be expected to scale with drainage area in an unknown way,
which would alter the relationship. In addition, if *E* varies systematically
along the stream, the slope–area scaling will be affected. For example, if
*E* decreases in the downstream direction, it also decreases with increasing
drainage area, resulting in an increase in the concavity index. This is in
line with observations.

In summary, the values for the scaling exponents for the relationship between slope and erosion rates for the different cases that have been discussed encompass the range of observed values (Table 1). All four observations regarding channel bed slope, as outlined in the beginning of this chapter, can be obtained.

## 3.4 Sinuosity

Recapitulating the results of Stark et al. (2010), we expect sinuosity to
increase both with increasing erodibility *k* and increasing storm strike
frequency. After substituting Eq. (4) into Eq. (9) to eliminate channel
width and employing the approximation *W* ≪ *d*, the
tools-dominated case gives

As before, the bedload particle speed *U* is dependent on slope and discharge.
Accounting for this gives

Here, I have also replaced discharge *Q* with effective discharge
*Q*−*Q*_{c}, subtracting a critical discharge for the onset of bedload motion
*Q*_{c} (e.g. Buffington and Montgomery, 1997; Rickenmann, 2001), which is
important when considering discharge variability (e.g. Lague et al., 2005;
Molnar, 2001) and thus sinuosity dependence on storm frequency. In the
cover-dominated case, we get

For the following discussion, *S*_{V} is treated as a constant but could in
principle be a function of local tectonics and, therefore, implicitly
erosion rate. The expected scaling with erodibility is directly obvious from
both Eqs. (25) and (26); sinuosity scales with ${k}^{(\mathrm{3}+\mathit{\alpha})/(\mathrm{4}n+\mathit{\alpha}+\mathrm{1})}$ in the tools-dominated case and with
*k*^{1∕2n} in the cover-dominated case. Since there is currently no
accepted way of measuring *k*, no quantitative data exist and the comparison
cannot go further.

Next, we link sinuosity to the variability in precipitation. The variability in forcing parameters is important for threshold processes (e.g. Lague,
2010), and the only relevant threshold process that we have considered is
bedload transport. When considering variable forcing, mean discharge needs
to be replaced by the effective discharge *Q*_{eff} that determines bedload
transport and incision on long timescales (e.g. Lague et al., 2005;
Molnar, 2001). In general, if the threshold discharge is higher than the
mean discharge, a higher discharge variability results in a higher effective
discharge (Deal, 2017). In storm-driven catchments, such as the streams on
the Pacific Arc islands studied by Stark et al. (2010), geomorphically
active floods are generally rare (e.g. Molnar, 2001) and erosion is limited
to a few days per year and often less, making this assumption valid.
Variability in discharge *V*_{Q} scales with the frequency of large storms
*F*_{Storm} (cf. Deal, 2017; Rossi et al., 2016). We thus find a scaling that
agrees with the observations of Stark et al. (2010):

## 4.1 Comparison to previous models

The model proposed here connects channel width, bed slope and sinuosity to discharge, erosion rate and substrate erodibility, via the core variable of bed cover. It fills a gap within the available published models, as it is a 1-D reach-scale model constructed from considerations of the physics of bedload transport and fluvial erosion, without the need of arbitrary closing assumptions. I have used a fluvial bedrock erosion model (Eq. 3) that includes both tools and cover effects and that is consistent with current process understanding (e.g. Beer et al., 2017; Fuller et al., 2016; Johnson and Whipple, 2010; Sklar and Dietrich, 2004), as well as quantitative field and laboratory measurements (Auel et al., 2017; Beer and Turowski, 2015; Chatanantavet and Parker, 2009; Inoue et al., 2014; Johnson and Whipple, 2010). The model presented here thus improves upon existing 1-D reach-scale models both in the plausibility of the underlying assumptions and, as has been shown in Sect. 3, in the predictive power concerning the observed scaling relationships. In addition, the model is complete in the sense that it does not feature a lumped calibration parameter with obscure physical meaning. All model parameters have a direct physical interpretation and can, at least in principle, be measured in the laboratory or the field.

## 4.2 Sideward deflection of bedload

To further validate or refine the model, we need information on some of the
unconstrained parameters. In particular, we are missing observations on
bedload paths in partially alluviated beds and on sideward deflection of
bedload particles. While no data are available on the former, at least some
initial observations have been reported on the latter. From laboratory
observations, Fuller et al. (2016) argued that roughness dominantly controls
sideward deflection of bedload in bedrock channels and therefore lateral
erosion. This interpretation is supported by field data of Beer et al. (2017). For a full quantification of the model, the sideward deflection
length scale *d* would need to be measured for a realistic range of boundary
conditions, varying hydraulics, bed roughness, particle size and
characteristics. To upscale the model to the reach scale, we would need
scaling relationships of bed roughness with drainage area or other
morphological parameters that vary along a stream. A comprehensive
investigation of the controls on the scaling of bed roughness of bedrock
channels is not known to the author. An additional complication arises from
the role of alluvium. An alluviated bed is typically rougher than bedrock
(e.g. Chatanantavet and Parker, 2008; Ferguson et al., 2017; Hodge et al.,
2011, 2016), and the effect of stationary sediment on a bedrock bed on
sideward deflection of moving particles has not yet been investigated.

We can obtain some tentative constraints on these scaling relationships by
considering catchments in a topographic steady state. I assume that, in the
cover-dominated domain, sideward deflection length scale *d* and bedload
fraction *β* are dependent on drainage area *A* according to a power law,
with exponents *a* and *b*, respectively. The slope–area scaling can be written as

Here, I used the hydraulic scaling (Eq. 17) to replace discharge with area.
If we assume that the concavity index, which includes both the explicit and
implicit dependence on drainage area in Eq. (28), is equal to 1∕2 and use
*m* = 1, *n* = 2 and *c* = 1 (see Sect. 3.2), we obtain *b*−*a* = 1. Similarly,
assuming that the width–area scaling in Eq. (15) should have an exponent of
1∕2, from the width Eq. (15), we obtain

This yields *a*+*b* = 0. Solving, we obtain $a\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}/\mathrm{2}$ and *b* = $-\mathrm{1}/\mathrm{2}$. This means
that the sideward deflection length *d* increases when moving downstream while
the bedload fraction *β* decreases, both with the square root of
drainage area. At least for the bedload fraction, this seems to be a
plausible value (see Turowski et al., 2010). For *d*, at first glance, an
increase with drainage area seems somewhat surprising, since it is often
assumed that roughness decreases in the downstream direction (e.g.
Ferguson, 2007; Nitsche et al., 2012). However, this assumption is made for
alluvial channels and is related to downstream fining that is observed in
many alluvial streams (e.g. Parker, 1991). In a bedrock channel, it seems
plausible that a progressive increase in cover leads to an overall increase
in roughness when moving downstream.

## 4.3 Implications for stream-profile inversion

The theoretical framework of the stream power model has been frequently used to obtain information about tectonic uplift or fluvial erosion rates by stream-profile inversion (e.g. Kirby and Whipple, 2001; Wobus et al., 2006b). Within the stream power framework, the steady-state profile of bedrock channels is given by

Here, *k*_{e} is a lumped calibration parameter that is commonly interpreted
to reflect bedrock erodibility. For the analysis, it is usually assumed that
*m*^{′} = 0.5, *n*^{′} = 1 and *c* = 1 (see Lague, 2014), to obtain a concavity index
equal to 1∕2, although evidence points to *n*^{′} typically being larger than
1 (DiBiase and Whipple, 2011; Harel et al., 2016; Lague, 2014). Then, slope is
fitted with a power law against area and a value for *E*∕*k*_{e} can be derived.
More sophisticated inversions exploit the transient dynamics of models that
can resolve erosion histories and find separate fit solutions for both *E* and
*k*_{e} (e.g. Roberts and White, 2010). Comparing Eq. (30) to the four slope
equations obtained by the model (Eqs. 20–23), the steady-state equations
show the same power-law dependence of slope *S* on drainage area *A* and erosion
rate *E*, although, depending on the domain (cover- vs. tools-dominated) and the
type of forcing (crossing a fault or topographic steady state), the scaling
exponents differ. In particular, the relationship between the scaling
exponent of slope with erosion rate and the scaling exponent on drainage
area (the concavity index) may be different to the one inferred from Eq. (30).
For example, for steady-state catchments in the cover-dominated domain, the
scaling exponent on erosion rate evaluates to 1∕*n* (Eq. 23), while the
concavity index evaluates to ($\mathrm{1}+cm)/n$ (Eq. 28; using *b* – *a* = 1). Further,
the physical interpretation of *m* and *n* is different from the interpretation of
*m*^{′} and *n*^{′}. While in the stream power model, *m*^{′} and *n*^{′} are directly related to the
mechanics of fluvial bedrock erosion, in the model proposed here, *m* and *n* are
related to the mechanics of bedload transport. Clearly, a wrong choice for
the value of *n*^{′}, in particular, leads to incorrect estimates of erosion rates.
If *m*^{′} and *n*^{′} are determined by bedload transport, as suggested here, *n*^{′} may fall
in the plausible range between 1.5 and 7 (see Sect. 3.2) and could be
very different from *n*^{′} = 1 that is typically used when deriving tectonic
information from stream-profile inversion.

## 4.4 The role of cover for sinuous bedrock channels

Here, I have argued that in streams where impact erosion is the dominant fluvial erosion process, cover is the central variable that needs to be considered. Nevertheless, it can be expected that bed cover modulates sinuosity development also in streams where other erosion processes are dominant. As has been argued by Johnson and Finnegan (2015), the dominant erosion process – slaking or impact erosion – determines whether a particular stream actively meanders or not in their study region. However, even weak rock that can be worn away by clear water flow will not erode if it is covered by a thick layer of sediment. Moreover, arguably, wetting–drying cycles are both less frequent and less efficient when water needs to flow through the pores of a gravel or sand layer. Although the erosion mechanism may likely make certain channels more prone to active meandering than others, I suggest here that bed cover plays a role in all of them.

Based on the idea that relative sediment supply controls bedrock channel meandering (Moore, 1926; Shepherd, 1972) and by making links to lateral erosion and channel width evolution, a physics-based 1-D model of bedrock channel morphology was constructed. The model correctly predicts the observed scaling relations between channel width and slope, on the one hand, and discharge and erosion rate, on the other, as well as those between sinuosity, on the one hand, and erodibility and storm strike frequency, on the other. In addition, it yields plausible ranges of values of the exponent values and can explain why a channel should develop to a steady-state sinuosity. The model is rooted in process physics, is fully parameterized and does not include lumped calibration parameters. It therefore describes bedrock channel morphology more completely than previously proposed models.

By predicting steady-state long profiles of bedrock channels similar to the stream power model, the model proposed here explains the success of the stream power model in describing steady-state channel bed slope and its failure to account for the scaling of width. In addition, it reconnects channel long-profile analysis with the insights that have been obtained on the physics of fluvial bedrock erosion over the last 2 decades. If the physical argument proposed here is correct, methods of stream-profile inversion to obtain data on erosion rate or tectonic history using the stream power model are based on incorrect assumptions. The results obtained with these methods are likely incorrect, especially if they were used to derive uplift histories. A further interesting point is that, if the model is correct, the scaling of width with erosion rate (Eq. 4) seems less complicated than that of slope (Eq. 7), since its dependence on the details of the hydraulics and hydrology is less pronounced. This may indicate that tectonic information can be more robustly obtained from channel width than from slope.

The model proposed here opens a new view to reach-scale bedrock channel morphology. Although the assumptions that have been made are physically plausible, many of them are as yet untested and few data are available to constrain the values of and the controls on some of the key parameters, such as the sideward deflection length scale. Nevertheless, the strong rooting of the model in process physics and its success in correctly predicting scaling relationships of slope, width and sinuosity is encouraging and warrants further investigation. For a comprehensive evaluation of the model and the underlying assumptions, we need detailed investigations of the sediment dynamics in partially alluviated bedrock channels. In particular, this includes bedload transport equations for particles moving over a bare bedrock bed, maps of bedload particle concentrations in the channel for various bed morphologies and flow conditions, and research into the controls on sideward deflection of moving particles.

No data sets were used in this article.

In the tools-dominated domain, the channel bed slope is given by Eq. (19):

Here, the bedload particle speed *U* depends on shear stress and therefore
slope and discharge. Based on laboratory flume measurements, Auel et al. (2017) gave an equation (their Eq. 19) for particle speed as a function
of shear stress *τ*, including various previous measurements, both over bedrock
and alluvial beds.

Here, *t**a**u*_{c} is the critical shear stress for the onset of bedload motion.
To eliminate shear stress, I use the DuBoys equation and the water
continuity equation:

Here, *H* is the water depth. Water flow velocity *V* can be computed by the variable power flow resistance
equation, which can be expressed as a function of slope, discharge and width
(Ferguson, 2007; Nitsche et al., 2012):

Here, *R* is a measure of bed roughness with dimensions of length, for example
the standard deviation of the bed surface (e.g. Nitsche et al., 2012), and
*α* ≈ 0.6 is a constant. Shear stress can then be written as

For substitution into Eq. (A1), I neglect the threshold (i.e. $\mathit{\tau}/{\mathit{\tau}}_{\mathrm{c}}-\mathrm{1}\approx \mathit{\tau}/{\mathit{\tau}}_{\mathrm{c}})$ to obtain

Here, *k*_{tools} is assumed to be constant:

Substituting the width equation (Eq. 4),

A |
Drainage area (m^{2}). |

A_{cover} |
Covered bed area (m^{2}). |

A_{tot} |
Total bed area (m^{2}). |

a |
Scaling exponent, d−A. |

b |
Scaling exponent, β−A. |

C |
Fraction of covered bed. |

C_{c} |
Critical cover. |

c |
Scaling exponent, Q−A. |

d |
Sideward deflection length scale, reach (m). |

d_{x} |
Sideward deflection length scale, cross section (m). |

d_{xy} |
Sideward deflection length scale, at a point (m). |

E |
Vertical erosion rate (m s^{−1}). |

e |
Base of the natural logarithm. |

F_{Storm} |
Storm strike frequency (s^{−1}). |

g |
Acceleration due to gravity (m s^{−2}). |

H |
Water depth (m). |

K_{bl} |
Bedload transport efficiency (kg m${}^{-\mathrm{3}m-\mathrm{1}}$ s^{m−1}). |

k |
Erodibility (m^{2} kg^{−1}). |

k_{e} |
Erodibility in stream power model (m^{1−3m} s^{m−1}). |

k_{h} |
Hydrology coefficient (m^{3−2c} s^{−1}). |

k_{s} |
Steepness index (m^{2θ}). |

k_{tools} |
Lumped constant, tools-dominated channel slope. |

k_{V} |
Velocity coefficient. |

L |
Reach length (m). |

L_{V} |
Straight length from reach start to end (m). |

M_{0} |
Minimum mass per area necessary to cover the bed (kg m^{−2}). |

m |
Discharge exponent in bedload equation. |

m^{′} |
Discharge exponent in the stream power model. |

n |
Slope exponent in bedload equation. |

n^{′} |
Slope exponent in the stream power model. |

Q |
Water discharge (m^{3} s^{−1}). |

Q_{c} |
Critical discharge for the onset of bedload motion (m^{3} s^{−1}). |

${Q}_{\mathrm{c}}^{\ast}$ | Relative sediment supply at the critical cover. |

Q_{eff} |
Effective discharge (m^{3} s^{−1}). |

Q_{s} |
Upstream sediment mass supply (kg s^{−1}). |

${Q}_{\mathrm{s}}^{\ast}$ | Relative sediment supply; sediment transport rate over transport capacity. |

Q_{t} |
Mass sediment transport capacity (kg s^{−1}). |

R |
Bed roughness length scale (m). |

S |
Channel bed slope. |

S_{cover} |
Channel bed slope predicted in the cover-dominated approximation. |

S_{tools} |
Channel bed slope predicted in the tools-dominated approximation. |

S_{V} |
Valley slope. |

U |
Bedload speed (m s^{−1}). |

V |
Water flow velocity (m s^{−1}). |

V_{Q} |
Discharge variability parameter. |

W |
Channel width (m). |

W_{cover} |
Covered length within the channel width (m). |

α |
Scaling exponent, V−Q. |

β |
Fraction of sediment transported as bedload. |

θ |
Concavity index; scaling exponent S−A. |

ρ |
Density of water (kg m^{−3}). |

σ |
Sinuosity. |

σ_{cover} |
Sinuosity predicted in the cover-dominated approximation. |

σ_{tools} |
Sinuosity predicted in the tools-dominated approximation. |

τ |
Bed shear stress (N m^{−2}). |

τ_{c} |
Critical bed shear stress at the onset of bedload motion (N m^{−2}). |

The author declares that he has no conflict of interest.

I thank Joel Scheingross and Eric Deal for insightful discussions. The model
presented here is rooted in discussions with Colin Stark and Jonathan Barbour about a
decade ago. The author was prompted to take up the problem of bedrock
channel sinuosity again during a field visit to the Chinese Tien Shan in
2016, studying rivers affected by active folding with Kristen Cook. Mitch D'Arcy, Joel Scheingross and Aaron Bufe as well as two anonymous reviewers looked through a
previous version of this paper, and their helpful comments led to tremendous
improvements.

The article processing charges for this open-access

publication were covered by a Research

Centre of the Helmholtz Association.

Edited by: Daniel Parsons

Reviewed by: two anonymous referees

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