Rivers are dynamical systems that are thought to evolve towards a steady-state configuration. Then, geomorphic parameters, such as channel width and slope, are constant over time. In the mathematical description of the system, the steady state corresponds to a fixed point in the dynamic equations in which all time derivatives are equal to zero. In alluvial rivers, steady state is characterized by grade. This can be expressed as a so-called order principle: an alluvial river evolves to achieve a state in which sediment transport is constant along the river channel and is equal to transport capacity everywhere. In bedrock rivers, steady state is thought to be achieved with a balance between channel incision and uplift. The corresponding order principle is the following: a bedrock river evolves to achieve a vertical bedrock incision rate that is equal to the uplift rate or base-level lowering rate. In the present work, considerations of process physics and of the mass balance of a bedrock channel are used to argue that bedrock rivers evolve to achieve both grade and a balance between channel incision and uplift. As such, bedrock channels are governed by two order principles. As a consequence, the recognition of a steady state with respect to one of them does not necessarily imply an overall steady state. For further discussion of the bedrock channel evolution towards a steady state, expressions for adjustment timescales are sought. For this, a mechanistic model for lateral erosion of bedrock channels is developed, which allows one to obtain analytical solutions for the adjustment timescales for the morphological variables of channel width, channel bed slope, and alluvial bed cover. The adjustment timescale to achieve steady cover is of the order of minutes to days, while the adjustment timescales for width and slope are of the order of thousands of years. Thus, cover is adjusted quickly in response to a change in boundary conditions to achieve a graded state. The resulting change in vertical and lateral incision rates triggers a slow adjustment of width and slope, which in turn affects bed cover. As a result of these feedbacks, it can be expected that a bedrock channel is close to a graded state most of the time, even when it is transiently adjusting its bedrock channel morphology.
Bedrock rivers are important geomorphic landforms in mountain regions. They set the base level for hillslope response and evacuate the products of erosion, weathering, and hillslope mass wasting (e.g. Hovius and Stark, 2006). As such, they integrate the upstream erosional signal of the landscape, and the material transported in rivers can be used to estimate catchment-averaged denudation rates on various timescales (e.g. Turowski and Cook, 2017). Further, their morphology is thought to be indicative of past climate and tectonic conditions (e.g. Stark et al., 2010; Wobus et al., 2006). Consequently, they provide archives that can be exploited to unravel the Earth's history.
River channels are dynamical systems. Their state variables – for example, slope, cross-sectional shape, and bed roughness – evolve over time under the influence of externally imposed driving variables including water discharge, sediment supply, and tectonic uplift (e.g. Heimann et al., 2015; Lague, 2010; Parker, 1979; Wickert and Schildgen, 2019). Like in many other dynamical systems, there exists a fixed point in the descriptions of river dynamics at which all state variables are constant over time. In an alluvial river, at this fixed point, entrainment and deposition of sediment are in balance along the river profile, implying that sediment transport rate is constant and that sediment transport capacity matches sediment supply. A river that exhibits these features is said to be “in grade” or “graded”, because it is neither aggrading nor degrading (Mackin, 1948). Since its introduction, the graded stream concept has become a central paradigm in river morphodynamics (e.g. Blom et al., 2017; Church, 2006). There are several reasons for this importance. Chiefly, rivers are physically complicated systems, and the description of their steady-state forms is a problem that is considerably simpler than the full description of their dynamics. Further, many variables of natural rivers are challenging to measure. Yet, comparatively simple scaling relations have been observed between variables such as discharge or drainage area, on the one hand, and channel width or channel slope, on the other hand (e.g. Gleason, 2015; Leopold and Maddock Jr., 1953; Whitbread et al., 2015). These scaling relationships are thought to be explainable using steady-state models (e.g. Eaton and Church, 2004; Smith, 1974; Turowski, 2018; Wobus et al., 2006).
The condition of grade in a stream is tightly connected to the description
of its sediment mass balance. For alluvial rivers, this mass balance is
typically described by one of two approaches: the Exner equation or the
entrainment–deposition framework (e.g. An et al., 2018). In the Exner
equation (e.g. Chen et al., 2014; Paola and Voller, 2005), the rate of
change of the sediment bed elevation
In bedrock channels, the concept of grade has not been widely applied. One
of the main reasons for this is that bedrock channels are usually viewed as
detachment-limited systems, where sediment supply is much smaller than
transport capacity (e.g. Tinkler and Wohl, 1998; Whipple et al., 2013),
which is in direct contrast to the assumption of grade. As a result, the
system is assumed to be driven by its potency for erosion (e.g. Whipple,
2004). The evolution of bedrock channel bed elevation
Over the last 2 decades, evidence has been mounting that fluvial bedrock erosion is driven by the impacts of sediment particles in many settings (e.g. Cook et al., 2013; Johnson et al., 2010; Sklar and Dietrich, 2004). The amount of sediment in the channel affects erosion rates by two main effects. First, an increase in the number of moving particles leads to an increase in the number of impacts on the bed, increasing erosion rates. This is known as the tools effect. Second, sediment residing on the bed may protect the rock surface from impacts, reducing erosion rates. This is known as the cover effect. Evidence for both tools and cover effects has been described in laboratory and field studies (e.g. Beer et al., 2016; Cook et al., 2013; Johnson and Whipple, 2010; Sklar and Dietrich, 2001; Turowski et al., 2008a). In addition, large sediment bodies are common in many mountain regions, and they can reside in and around stream channels for potentially a long time (e.g. Korup et al., 2006; Schoch et al., 2018). All of these observations imply that a description of the mass balance of sediment should be an essential part of any theoretical description of bedrock channels. In addition, recent observations have been interpreted such that bedrock channels are in a graded state, similarly to alluvial channels (Phillips and Jerolmack, 2016). Thus, it seems that the view that bedrock channels are in a detachment-limited state, in which long-term sediment supply is smaller than transport capacity (e.g. Whipple et al., 2013), is insufficient to account for all observations made in natural streams.
In this paper, I have three separate, yet related, aims. First, I develop a description of the mass balance of bedrock channels, based on previous work by Turowski and Hodge (2017) and Turowski (2018). The mass balance is used to derive and discuss the concept of the graded stream for bedrock channels. Second, I derive expressions for response timescales for bedrock channels to adjust to a graded state. Third, for this it is necessary to develop a description of bedrock channel wall erosion by impacting particles. The concepts are used to discuss the current notion of bedrock channels, their possible routes to a graded state, and the relevant response timescales.
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
Schematic side view of a control volume within a bedrock channel. The bedrock (bottom) is overlain by stationary sediment (centre), which exchanges particles via entrainment
Considering impact erosion to be the dominant erosion process, the lateral
erosion rate
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
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
Schematic top view of a straight bedrock channel, with alternating submerged gravel bars (dark grey) on a bedrock bed (white). The sinuous thalweg (light grey) and bedload path (transparent dark grey) are indicated. The dashed black line indicates the cross section that is ideal for sideward deflection of particles; here, the bedload particle stream crosses the boundary between gravel and smooth bedload. The wavelength of the alternating bars and therefore of the bedload path should scale with channel width. Adapted from Turowski (2018).
The sideward deflection length scale
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
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.
Taking the spatial derivative of Eq. (4) and assuming spatially constant
uplift rate
For the adjustment of channel width, it is necessary to distinguish between
narrowing and widening channels. While channel widening is controlled by the
lateral erosion of bedrock walls (see Sect. 2.2, Eq. 24), a bedrock channel can only narrow when incising vertically. Therefore, the response timescale of narrowing is related to the vertical incision rate. The timescale of narrowing can be estimated by the time necessary to incise the flow depth
The response time for the adjustment of bed cover
The dynamics of the channel during adjustment is to some extent determined
by the relative magnitude of the response times. For example, if the response time for the adjustment of bed slope is always much longer than the response time for bed cover, on the timescale of slope adjustment, it can be assumed that bed cover is always at a steady state. The ratio of the response time for slope and width (widening channel) is given by
To illustrate the dependence of channel morphology and of the adjustment timescales on control and channel morphology parameters, I used parameter values for Lushui at the Liwu River, Taiwan (Table 1; see Turowski et al., 2007). The values of reach parameters were either measured in the field or estimated using literature data. The value for discharge is representative of bedload-carrying flows, using the partitioning method proposed by Sklar and Dietrich (2006). The values of the exponent and prefactor of the flow velocity equation (Eq. D4) were selected using data by Nitsche et al. (2012).
Parameter values used for the example calculations, following estimates by Turowski et al. (2007) for the Liwu River, at Lushui, Taiwan.
The sideward deflection length scale
Steady-state channel width (solid line), channel bed slope (dashed line), and bed cover (dotted line) against forcing variables discharge
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).
Timescales
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
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
In comparison to the model by Turowski (2018), the sideward deflection length scale
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
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 (
Bedrock channel dynamics are controlled by two dominant order principles. They adjust their morphology both to achieve grade, in which the sediment transport rate is constant along the stream, and to match incision rate to uplift or base-level lowering. The recognition of a steady state corresponding to one of these principles does not necessarily imply that the other has also been achieved. With minutes to days, the adjustment timescale for bed cover is short relative to the timescales for channel width and slope, and cover may be adjusted by changing supply conditions even over the duration of a single flood event. Thus, it can be expected that bedrock channels are close to a graded state most of the time. In the example calculations (Fig. 5), adjustment timescales for slope and width are of the order of thousands of years. This is shorter than the major cyclic variations of Earth's climate (e.g. Roe, 2006) or the typical timescales of mountain building. The results therefore suggest that many bedrock channels are also close to an erosional steady state, in which erosion rate is equal to uplift rate.
Substituting Eq. (7) into Eq. (6) to eliminate entrainment and deposition
rates, we obtain
Assume that the bedload particle path through the channel follows a sinusoidal path with a wavelength equal to the gravel bar spacing and an
amplitude
For the following analysis we assume that all parameters are kept constant
apart from sediment supply, which varies sinusoidally over time. This choice
allows one to obtain an analytical solution for the problem, and it does not affect
the result for the timescale of transient adjustment. Sediment supply can
then be written as the sum of the average supply
The development here is equivalent to that given by Turowski (2018).
The reach-averaged Shields stress
No data sets were used in this article.
The author declares that there is no conflict of interest.
I thank Claire Masteller, Aaron Bufe, and Joel Scheingross for discussions. Ron Nativ provided detailed comments on an earlier version of the article. Two anonymous reviewers provided constructive comments that helped to improve the paper.
The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.
This paper was edited by Eric Lajeunesse and reviewed by two anonymous referees.