The cover effect in fluvial bedrock erosion is a major control on bedrock channel morphology and long-term channel dynamics. Here, we suggest a probabilistic framework for the description of the cover effect that can be applied to field, laboratory, and modelling data and thus allows the comparison of results from different sources. The framework describes the formation of sediment cover as a function of the probability of sediment being deposited on already alluviated areas of the bed. We define benchmark cases and suggest physical interpretations of deviations from these benchmarks. Furthermore, we develop a reach-scale model for sediment transfer in a bedrock channel and use it to clarify the relations between the sediment mass residing on the bed, the exposed bedrock fraction, and the transport stage. We derive system timescales and investigate cover response to cyclic perturbations. The model predicts that bedrock channels can achieve grade in steady state by adjusting bed cover. Thus, bedrock channels have at least two characteristic timescales of response. Over short timescales, the degree of bed cover is adjusted such that the supplied sediment load can just be transported, while over long timescales, channel morphology evolves such that the bedrock incision rate matches the tectonic uplift or base-level lowering rate.

Bedrock channels are shaped by erosion caused by countless impacts of the sediment particles they carry along their bed (Beer and Turowski, 2015; Cook et al., 2013; Sklar and Dietrich, 2004). There are feedbacks between the evolving channel morphology, the bedload transport, and the hydraulics (e.g. Finnegan et al., 2007; Johnson and Whipple, 2007; Wohl and Ikeda, 1997). Impacting bedload particles driven forward by the fluid forces erode and therefore shape the bedrock bed. In turn, the morphology of the channel determines the pathways of both sediment and water, and the forces which the latter exerts on the former, and thus sets the stage for the entrainment and deposition of the sediment (Hodge and Hoey, 2016). Sediment particles play a key role in this erosion process; they provide the tools for erosion and also determine where bedrock is exposed such that it can be worn away by impacting particles (Gilbert, 1877; Sklar and Dietrich, 2004).

The importance of the cover effect – that a stationary layer of gravel can
shield the bedrock from bedload impacts – has by now been firmly established
in a number of field and laboratory studies (e.g. Chatanantavet and Parker,
2008; Finnegan et al., 2007; Hobley et al., 2011; Johnson and Whipple, 2007;
Turowski and Rickenmann, 2009; Turowski et al., 2008; Yanites et al., 2011).
Sediment cover is generally modelled with generic relationships that predict
the decrease in the fraction of exposed bedrock area

The most commonly used function to describe the cover effect is the linear
decline (Sklar and Dietrich, 1998), which is the simplest function connecting
the steady-state endmembers of an empty bed when relative sediment supply

In contrast, the exponential cover function arises under the assumption that
particle deposition is equally likely for each part of the bed, whether it
is covered or not (Turowski et al., 2007).

Hodge and Hoey (2012) obtained both the linear and the exponential functions using a cellular automaton (CA) model that modulated grain entrainment probabilities by the number of neighbouring grains. However, consistent with laboratory flume data, the same model also produced other behaviours under different parameterizations. One alternative behaviour is runaway alluviation, which was attributed by Chatanantavet and Parker (2008) to the differing roughness of bedrock and alluvial patches. Due to a decrease in flow velocity, an increase in surface roughness, and differing grain geometry, the likelihood of deposition is higher over bed sections covered by alluvium compared to smooth, bare bedrock sections (Hodge et al., 2011). This can lead to rapid alluviation of the entire bed once a minimum fraction has been covered. The relationship between sediment flux and cover is also affected by the bedrock morphology; flume experiments have demonstrated that on a non-planar bed, the location of sediment cover is driven by bed topography and hydraulics (e.g. Finnegan et al., 2007; Inoue et al., 2014). Johnson and Whipple (2007) observed that stable patches of alluvium tend to form in topographic lows such as potholes and at the bottom of slot canyons, whereas Hodge and Hoey (2016) found that local flow velocity also controls sediment cover location.

The relationship between roughness, bed cover, and incision was explored in a number of recent numerical modelling studies. Nelson and Seminara (2011, 2012) were one of the first to model the impact that the differing roughness of bedrock and alluvial areas has on sediment patch stability. Zhang et al. (2015) formulated a macro-roughness cover model, in which sediment cover is related to the ratio of sediment thickness to bedrock macro-roughness. Aubert et al. (2016) directly simulated the dynamics of particles in a turbulent flow and obtained both linear and exponential cover functions. Johnson (2014) linked sediment transport and cover to bed roughness in a reach-scale model. Using a model formulation similar to that of Nelson and Seminara (2011), Inoue et al. (2016) reproduced bar formation and sediment dynamics in bedrock channels. All of these studies used slightly different approaches and mathematical formulations to describe alluvial cover, making a direct comparison difficult.

Over timescales including multiple floods, the variability in sediment supply is also important (e.g. Turowski et al., 2013). Lague (2010) used a model formulation in which cover was written as a function of the average sediment depth to upscale daily incision processes to long timescales. He found that over the long term, cover dynamics are largely independent of the precise formulation on the process scale and are rather controlled by the magnitude–frequency distribution of discharge and sediment supply. Using the CA model of Hodge and Hoey (2012), Hodge (2017) found that, when sediment supply was very variable (alternating large pulses with no sediment supply), the amount of sediment cover was primarily determined by the recent supply history, rather than by the relationships identified under constant sediment supply.

So far, it has been somewhat difficult to compare and discuss the different cover functions obtained from theoretical considerations, numerical models, and experiments, since a unifying framework and clear benchmark cases have been missing. Here, we propose such a framework and develop type cases linked to physical considerations of the flow hydraulics and sediment erosion and deposition. We show how this framework can be applied to data from a published model (Hodge and Hoey, 2012). Furthermore, we develop a reach-scale erosion–deposition model that allows the dynamic modelling of cover and prediction of steady states. Thus, we clarify the relationship between cover, deposited mass, and relative sediment supply. As part of this model framework, we investigate the response time of a channel to a change in sediment input, which we illustrate using data from a natural channel.

Here we build on the arguments put forward by Turowski et al. (2007) and
Turowski (2009). Consider a bedrock bed on which sediment particles are
distributed. We can view the deposition of each particle as a random process,
and each area element on the bed surface can be assigned a probability for
the deposition of a particle. When assuming that a given number of particles
are distributed on the bed, the mean behaviour of the exposed area

Cartoon illustration of a bed partially covered by sediment. For
purpose of illustration, the bed is divided into a square raster, with each
pixel of the size of a single grain. For a given number of particles in the
area of the bed of interest, the exposed area fraction of the bed is
dependent on the distribution of particles. Grains that sit on top of other
grains do not contribute to cover. The probability that a new grain is
deposited on uncovered bed is given by

We can make some general statements about

For purpose of illustration, we will next discuss two simple forms of the
probability function

Integrating, we obtain

Similarly to above, if we set

It is clear that the assumption of

The argument used here to obtain the exponential cover effect in Eq. (8)
essentially corresponds to the one given by Turowski et al. (2007). Since
this case presents the simplest plausible assumption, we will use it as a
benchmark case, to which we will compare other possible functional forms of

In principle, the probability function

A simple functional form that can be used to take into account either one of
these two effects is a power law dependence of

Here, the probability of deposition on uncovered ground is increased in
comparison to the benchmark exponential case if

A convenient and flexible way to parameterize

Here,

Here,

In principle, a suitable function

Examples for the use of the regularized incomplete Beta function
(Eq. 11) to parameterize

To illustrate how the framework can be used, we apply it to data obtained
from the CA model developed by Hodge and Hoey (2012). The CA model reproduces
the transport of individual sediment grains over a smooth bedrock surface. In
each time step, the probability of a grain being entrained is a function of
the number of neighbouring grains. If five or more of the eight neighbouring
cells contain grains, then the grain has a probability of entrainment

The model is run with a domain that is 100 cells wide by 1000 cells long, with each cell having the same area as a grain. Up to four grains can potentially be entrained from each cell in a time step, limiting the maximum sediment flux. In each time step random numbers and the probabilities are used to select the grains that are entrained, which are then moved a step length of 10 cells downstream and deposited. Model results are insensitive to the step length. A fixed number of grains are also supplied to the upstream end of the model domain. A smoothing algorithm is applied to prevent unrealistically tall piles of grains developing in cells if there are far fewer grains in adjacent cells. After around 500 time steps the model typically reaches a steady-state condition in which the number of grains supplied to and leaving the model domain are equal. Sediment cover is measured in a downstream area of the model domain and is defined as grains that are not entrained in a given time step. Consequently grains that are deposited in one time step and entrained in the following one do not contribute to the sediment cover, and so the model implicitly incorporates the effect of local sediment cover on grain deposition.

Model runs were completed with six different combinations of

Cover bed fraction and total mass on the bed produced by the model were
converted using Eq. (3) into the new probabilistic framework (Fig. 4). The
derivative was approximated by simple linear finite differences, which, in
the case of run-away alluviation, resulted in a non-continuous curve due to
large gradients. The exponential benchmark (Eq. 8) is also shown for
comparison. The different model parameterizations produce results in which
the probability of deposition on bedrock is both more and less likely than in
the baseline case, with some runs showing both behaviours. Cases where the
probability is more than the baseline case (i.e. grains are more likely to
fall on uncovered areas) are associated with runs in which grains in clusters
are relatively immobile. These runs are likely to be particularly affected by
the smoothing algorithm that acts to move sediment from alluviated to bedrock
areas. All model parameterizations predict greater bed exposure for a given
normalized mass than is predicted by a linear cover relationship (Fig. 3b).
Runs with relatively more immobile cluster grains have a lower exposed
fraction for the same normalized mass. Runs with low values of

Probability functions

Previous descriptions of the cover effect relate the exposed fraction of the
bed to the relative sediment supply

Sediment dynamics at the bed are modelled by two reservoirs for
stationary and mobile mass, which can exchange material by entrainment
(

The difference form of the mass balance for the mobile sediment is then given
by (cf. Fig. 5)

Here,

Here,

It is useful to work with dimensionless variables by defining

Here,

The dimensionless entrainment and deposition rates,

We also need sediment entrainment and deposition functions. The entrainment
rate needs to be modulated by the availability of sediment on the bed. If

Here,

Integrating, we obtain

Here, we used the condition

When

Note that

Finally, the sediment transport rate needs to be proportional to the mobile
sediment mass times the downstream sediment speed

After incorporating the original equation between

In this chapter, we discuss the steady solution to the system of equations
and thus clarify the relationship between cover, stationary sediment mass,
sediment supply, and transport capacity. Setting the time derivatives to 0,
we obtain a time-independent solution, which links the exposed area directly
to the ratio of sediment transport rate to transport capacity. From Eq. (23)
it follows that in this case, the entrainment rate is equal to the deposition
rate, and we obtain

Here, the bar over the variables denotes their steady-state value.
Substituting Eq. (24) to eliminate

Note that we assume here that sediment cover is only dependent on the
stationary sediment mass on the bed, and we thus neglect grain–grain
interactions known as the dynamic cover (Turowski et al., 2007). In analogy
to Eq. (24), we can write

Here,

If we use the exponential cover function (Eq. 8) with Eqs. (27), (28), and
(29), we obtain

Similarly, equations can be found for the other analytical solutions of the
cover function. For the linear case (Eq. 6), we obtain

For the power law case (Eq. 9), we obtain

The exponential cover function essentially leads to a combined linear and
exponential relation between

Analytical solutions at steady state for the exposed fraction of the
bed (

Steady-state solutions using the Beta distribution to parameterize

The previous analysis shows that steady-state cover is controlled by the
characteristic dimensionless mass

The minimum mass necessary to completely cover the bed per unit area,

Here,

Here,

Here, the Shields stress

The characteristic dimensionless mass

Variation of the exposed bed fraction as a function of transport capacity, assuming that particle speed scales with transport capacity to the power of one-third.

To calculate the temporal evolution of cover on the bed within a single reach, we solved Eqs. (3), (22), (23), and (24) numerically for a section of the bed with homogenous conditions using a simple linear finite difference scheme. In this case sediment input is a boundary condition, while sediment output, mobile and stationary sediment mass, and the fraction of the exposed bed are output variables. In general, a change in sediment supply leads to a gradual adjustment of the output variables towards a new steady state (Fig. 10). It is desirable to obtain expressions for the response time of the system to external perturbation, such as a change in sediment supply or hydraulic conditions. Such a response time could then be compared to the timescales of changes in boundary conditions. For example, during a flood event, both transport capacity and sediment supply change over time. If these changes are slow in comparison to the response time of cover, the bed cover state can essentially keep up with the imposed changes at all times, and therefore steady-state equations (Sect. 3.2) can be used to calculate its evolution. In contrast, if the imposed change is rapid in comparison to the response time, cover may lag behind, and an approach that resolves cover as a dynamic variable is necessary. This may, for example, be important when studying the erosional behaviour of channels in response to floods (see Lague, 2010; Turowski et al., 2013). Unfortunately, a general analytical solution is not possible, but results can be obtained for special cases. We first derive analytical solutions for the response time for a reach without upstream sediment supply and for a system responding to small perturbations in sediment supply or transport capacity (Sect. 3.3.1) and discuss the system behaviour (Sect. 3.3.2). Finally, we apply the concepts to data from a flood in a natural river and demonstrate that, for this specific case, because of the response times, the steady-state relations do not capture cover behaviour.

First, consider a reach without upstream sediment supply; i.e.

Using the exponential cover model (Eq. 8), we obtain

Equation (39) is separable and can be integrated to obtain

Letting

By making the parameters in the exponent on the right-hand side of Eq. (42)
dimensional, we get

which allows a characteristic system timescale

Since this timescale is dependent on the transport capacity

We can make some further progress and define a more general system timescale
by performing a perturbation analysis (Appendix A1). For small perturbations
in either

Here, exp denotes the natural exponential function. The characteristic
system timescale can then be written as

Since bed cover is more easily measurable than the mass on the bed, Eq. (46)
can help to estimate system timescales in the field. Further,

To illustrate these additional dependencies, we have used numerical solutions
of Eqs. (3), (22), (23), and (24) to calculate the time needed to reach
99.9 % of total adjustment after a step change in transport stage (chosen
due to the asymptotic behaviour of the system), analysed across a plausible
range of particle speeds

Temporal evolution of cover for the simple case of a control box
with sediment through-flux, based on Eqs. (3), (22), (23), and (24). Relative
sediment supply (supply normalized by transport capacity) was specified to
0.25 and increased to 1 at

Evolution of the exposed bed fraction (removal of sediment cover)
over time starting with different initial values of bed exposure, for the
special case of no sediment supply; i.e.

Dimensionless time to reach 99.9 % of the total adjustment in
the exposed area as a function of

The perturbation analysis (Appendix A) gives some insight into the response
of cover to cyclic sinusoidal perturbations. Let sediment supply be perturbed
in a cyclic way described by an equation of the form

Here, the overbar denotes the temporal average,

Here,

The gain can be written as

Here,

Phase shift (Eq. 49) and gain (Eq. 50) as a function of the ratio of
the period of perturbation

To illustrate the magnitude of the timescales using real data, we use a flood data set from the Erlenbach, a sediment transport observatory in the Swiss Prealps (e.g. Beer et al., 2015). There, near a discharge gauge, bedload transport rates are measured at 1 min resolution using the Swiss Plate Geophone System, a highly developed and fully calibrated surrogate bedload measuring system (e.g. Rickenmann et al., 2012; Wyss et al., 2016). We use data from a flood on 20 June 2007 (Turowski et al., 2009) with the highest peak discharge that has so far been observed at the Erlenbach. The meteorological conditions that triggered this flood and its geomorphic effects have been described in detail elsewhere (Molnar et al., 2010; Turowski et al., 2009, 2013). The Erlenbach does not have a bedrock bed in the sense that bedrock is exposed in the channel bed; however, the data provide a realistic natural time series of discharge and bedload transport over the course of a single event. Rather than predicting bed cover evolution for a natural system, for which we do not currently have data for validation, we use the Erlenbach data to illustrate possible cover behaviour during a fictitious event with different initial sediment cover extents, using natural data to provide realistic boundary conditions.

Using a median grain size of 80 mm, a sediment density of
2650 kg m

Setting

Assuming that all change in the response time is due to changes in the period
(i.e. assuming a constant amplitude,

In the exemplary event, the evolution and final value of bed cover depends
strongly on its initial value (Fig. 14), indicating that the adjustment is
incomplete. The system timescale is generally larger than 1000 s and is
inversely related to discharge via the dependence on transport capacity. The

Calculated evolution of cover during the largest event observed at the Erlenbach on 20 June 2007 (Turowski et al., 2009). Bedload transport rates were measured with the Swiss Plate geophone sensors calibrated with direct bedload samples (Rickenmann et al., 2012). The final fraction of exposed bedrock is strongly dependent on its initial value.

In principle, the framework for the cover effect presented here allows the
formulation of a general model for bedrock channel morphodynamics without the
restrictions of previous models (e.g. Nelson and Seminara, 2011; Zhang et
al., 2015). To achieve this, the dependency of

The dynamic model put forward here is a minimum first-order formulation, and
there are some obvious future alterations. We only take account of the
static cover effect caused by immobile sediment on the bed. The dynamic
cover effect, which arises when moving grains interact at high sediment
concentration and thus reduce the number of impacts on the bed (Turowski et
al., 2007), could in principle be included in the formulation but would
necessitate a second probability function specifically to describe this
dynamic cover. It would also be possible to use different

We will briefly outline in this section the main differences to previous formulations of cover dynamics in bedrock channels. Thus, the novel aspects of our formulation and the respective advantages and disadvantages will become clear.

Aubert et al. (2015) coupled the movement of spherical particles to the simulation of a turbulent fluid and investigated how cover depends on transport capacity and supply. Similar to what is predicted by our analytical formulation, they found a range of cover function for various model set-ups, including linear and convex-up relationships (compare the results in Fig. 6 to their Fig. 15). Aubert et al. (2015) presented the most detailed physical simulations of bed cover formation so far, and the correspondence between the predictions is encouraging.

Nelson and Seminara (2011, 2012) formulated a morphodynamic model for bedrock channels. They based their formulation on sediment concentration, which is in principle similar to our formulation based on mass. However, Nelson and Seminara (2011, 2012) did not distinguish between mobile and stationary sediment and linked local transport directly to sediment concentration. Further, Nelson and Seminara (2011, 2012) assumed a direct correspondence between sediment concentration and degree of cover, which is equivalent to the linear cover function (Eq. 6). In this case, it is assumed that grains are always deposited on uncovered bed, and the different possible distributions of particles within a grid node are not taken into account. Practically, this implies that the grid size needs to be of the order of the grain size because, strictly, the assumption is only valid if a single grain can cover an entire grid node (cf. Fig. 1). Although different in various details, Inoue et al. (2016) have used essentially the same approach as Nelson and Seminar (2011, 2012) to link bedload concentration, transport, and bed cover. Both of these models allow the 2-D modelling of bedrock channel morphology. Although we have not fully developed such a model in the present paper, our model framework could easily be extended to 2-D problems.

Inoue et al. (2014) formulated a 1-D model for cover dynamics and bedrock
erosion. There, they distinguish between stationary and mobile sediment using
an Exner equation to capture sediment mass conservation. The degree of bed
cover is related to transport rates and sediment mass via a saturation
volume, which is related to our characteristic mass

Zhang et al. (2015) formulated a bed cover model specifically for beds with
macro-roughness. There, deposited sediment always fills topographic lows from
their deepest positions, such that there is a reach-uniform sediment level.
While the model provides a fundamentally different approach to what is
suggested here, its applicability is limited to very rough beds, and the
assumption of a sediment elevation that is independent of the position on the
bed seems physically unrealistic. In principle, the probabilistic framework
presented here should be able to deal with macro-rough beds, by making the

In this paper, we focused on the dynamics of bed cover rather than on
the modelling of the dynamics of entire channels. The probabilistic
formulation using the parameter

Based on field data interpretation, Phillips and Jerolmack (2016) argued that bedrock rivers adjust such that, similar to alluvial channels, medium-sized floods are most effective in transporting sediment and that channel geometry therefore can quickly adjust their transport capacity to the applied load and therefore achieve grade (cf. Mackin, 1948). They conclude that bedrock channels can adjust their morphologic parameters (channel width and shape) quickly in response to changing boundary conditions. In contrast, our model suggests that instead bed cover can be adjusted to achieve grade. In steady state, time derivatives need to be equal to 0. Thus, entrainment equals deposition (Eq. 14), implying that the downstream gradient in sediment transport rate is equal to 0 (Eq. 13). When sediment supply or transport capacity change, the exposed bedrock fraction can adjust to achieve a new steady state, and a change in the channel geometry is unnecessary. These changes in sediment cover can occur far more rapidly than changes in width and cross-sectional shape (compare to Eq. 46). Whether a steady state is achieved depends on the relative magnitude of the timescales of perturbation and cover adjustment (see Sect. 3). Our results imply that bedrock channels have two distinct timescales to adjust to changing boundary conditions to achieve grade. Over short times, bed cover is adjusted. This can occur rapidly. Over long timescales, channel width, cross-sectional shape, and slope are adjusted.

The probabilistic view put forward in this paper offers a framework into
which diverse data on bed cover, whether obtained from field studies,
laboratory experiments, or numerical modelling, can be easily converted to be
meaningfully compared. The conversion requires knowledge of the mass of
sediment on the bed and the evolution of exposed fraction of the bed. Within
the framework, individual data sets can be compared to the exponential
benchmark and linear limit cases, enabling physical interpretation.
Furthermore, the formulation allows the general dynamic sub-grid modelling
of bed cover. Depending on the choice of

It needs to be noted here that the precise formulation of the entrainment
and deposition functions also affects steady-state cover relations. When
calibrating

The system timescale for cover adjustment is inversely related to transport capacity. This timescale can be long and in many realistic situations, cover cannot instantaneously adjust to changes in the forcing conditions. Thus, dynamic cover adjustment needs to be taken into account when modelling the long-term evolution of bedrock channels.

Our model formulation implies that bedrock channels adjust bed cover to achieve grade. Therefore, bedrock channel evolution is driven by two optimization principles. On short timescales, bed cover adjusts to match the sediment output of a reach to its input. Over long timescales, width and slope of the channel evolve to match long-term incision rate to tectonic uplift or base-level lowering rates.

The model data used in Sect. 2.2 (Fig. 3) can be obtained by contacting the authors. The Erlenbach data used in Sect. 3.3.3 (Fig. 14) belong to the Mountain Hydrology and Mass Movements Group at the Swiss Federal Research Institute for Forest Snow and Landscape Research WSL and have been used with permission.

Here, we derive the effect of a small sinusoidal perturbation of the driving
variables, namely sediment supply

Here, the bar denotes the average of the quantity at steady state, while

Steady-state cover is directly related to the mass on the bed

Substituting Eq. (A3) and a similar equation for

Here, the averaged terms drop out as they are independent of time. If

Here, since the

It is therefore sufficient to derive the perturbation solution for

First, let us look at a perturbation of sediment supply

Again, since the

A similar approximation applies for the exponential in

The perturbation is assumed to be sinusoidal

Here,

and the phase shift by

The perturbation of the transport capacity

Both

Here,

Using the system timescale

Here,

The authors declare that they have no conflict of interest.

We thank J. Scheingross and J. Braun for insightful discussions and two anonymous reviewers and associate editor D. Egholm for their comments on the paper. The data from the Erlenbach is owned by and was used with permission of the Mountain Hydrology and Mass Movements Group at the Swiss Federal Research Institute for Forest Snow and Landscape Research WSL.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: David Lundbek Egholm Reviewed by: two anonymous referees