Upscaling from grain-scale processes to alluviation in bedrock channels using a cellular automaton model

Authors


Abstract

[1] The presence of sediment cover in bedrock rivers inhibits saltation-driven incision, and so accurate predictions of the relationship between bedrock exposure (Fe) and relative sediment flux (sediment supply rate over capacity sediment transport rate, Qs/Qt) are necessary to model incision and hence landscape evolution. Theoretical predictions of a linear or negative exponential form for this relationship are not consistent with laboratory data that instead demonstrate a range of different relationships. Here we use a cellular automaton (CA) model to establish how the relationship between Fe and Qs/Qt evolves from the dynamics of, and interactions between, individual sediment grains moving through a bedrock channel. The key model parameter is the probability of grain entrainment, which is altered as a function of the number of neighboring grains in order to reproduce the enhanced mobility of isolated grains on bedrock surfaces. For each model run, an equilibrium sediment cover is attained for a specified sediment input, enabling the relationship between Fe and Qs/Qt to be established. As well as both linear and exponential relationships, model runs reproduce other relationships observed in laboratory experiments. These other relationships require isolated grains to be more easily entrained than grains in sediment clusters, which is consistent with field observations of grain mobility. There is therefore a continuum of relationships between Fe and Qs/Qt; the relationship that is most applicable to a particular reach will depend on the role of channel slope, roughness and shear stress in controlling the entrainment of grains from bedrock and alluvial surfaces.

1. Introduction

[2] Quantifying the role of sediment in bedrock rivers remains a significant research challenge, despite bedrock rivers being key features of upland environments and drivers of landscape evolution. Bedrock rivers control landscape evolution because their incision sets and disseminates base level changes [Howard et al., 1994]. Explaining and predicting bedrock river incision rates requires an understanding of the critical role of sediment in incision, a role that is mediated through a balance between erosive tools and protective cover effects [Gilbert, 1877; Sklar and Dietrich, 1998]. Developing physically based predictions of incision is complicated because sediment dynamics are characterized by nonlinear relationships between multiple interacting factors. In this paper we focus on one aspect of sediment dynamics in bedrock rivers, which is the spatial and temporal distribution of in-channel sediment cover in bedrock channels of low roughness.

[3] A key process driving incision is the impact of coarse (>2 mm) saltating grains on uncovered areas of the river bed. In order to model this process, the fractional exposure of the bed is typically predicted as a function of sediment flux. One of the simplest proposed forms of the relationship between sediment flux and sediment cover is [Kooi and Beaumont, 1994; Sklar and Dietrich, 2004]:

display math

where Fe is the fractional exposure of the bedrock bed, qs is the sediment supply per unit channel width (kg m−1 s−1), qt is the sediment transport capacity per unit width (kg m−1 s−1) for a fully alluvial bed and qs/qt is referred to as sediment flux. Sklar and Dietrich [2004] formulated equation (1) from the assumptions of no sediment cover when qs = 0, full sediment cover when qs = qt and a linear relationship between these two end-members (Figure 1). Turowski et al. [2007] derived an alternative formulation under the assumptions of random deposition of uniform sediment on a flat bed, no grain-grain interactions, no sediment redistribution, and no feedback between sediment and hydraulics:

display math

where φ is a dimensionless parameter. In the case that a grain is equally likely to be deposited anywhere on the bed, φ = 1 (seen in Figure 1). If sediment is preferentially deposited on bedrock areas of the bed, then sediment cover is greater than for a purely random distribution and φ > 1; this condition is likely in a bedrock river [Turowski, 2009]. φ could also vary both temporally and spatially across a bed.

Figure 1.

Relationships between fractional bedrock exposure (Fe), erosion rate (E), and relative sediment flux (qs/qt) as predicted by Sklar and Dietrich [2004] and Turowski et al. [2007]. For the latter curve, φ = 1. E is calculated as by Sklar and Dietrich [2004].

[4] Bedrock erosion rates predicted using equations (1) and (2) have different magnitude maxima that occur at different values of qs/qt (Figure 1). Furthermore, when φ = 1, Turowski et al.'s [2007] maximum erosion occurs where Sklar and Dietrich [2004] predict no erosion. To be able to use these models to understand landscape evolution, it is necessary to identify which is most applicable to bedrock channels, and under what circumstances.

[5] That the effect of sediment cover is important in determining bedrock incision has been demonstrated in field settings [Johnson et al., 2009, 2010; Yanites et al., 2011]. However, identifying the relevant relationship between sediment flux and sediment cover is harder because present-day sediment cover may not reflect long-term dynamics [Lague, 2010]. In one of the few existing studies, Cowie et al. [2008] presented field data that they claimed are consistent with Sklar and Dietrich's [2004] relationship. In another example, Hobley et al. [2011] used field data to constrain the relationship between sediment flux and erosion, with the resulting function incorporating the roles of both tools and cover effects, consistent with both equations (1) and (2).

[6] Laboratory data have also been used to assess these relationships. Finnegan et al. [2007] and Johnson and Whipple [2007, 2010] quantified the development of a bedrock channel under different sediment feed conditions. The former two identified both tools and cover effects; Johnson and Whipple [2010] measured a relationship between sediment flux and sediment cover, but did not distinguish between equations (1) and (2). In all these experiments, the channel was actively incising, complicating the relationships governing sediment cover. Finally, Turowski et al. [2007] demonstrated that the relationship between normalized erosion rate and sediment mass measured in a circular abrasion mill [Sklar and Dietrich, 2001] is better represented by equation (2) than equation (1).

[7] Here we focus on sediment cover development on planar surfaces, and so primarily consider a series of flume experiments in which the degree of sediment cover on a rough bed was directly quantified from images of the bed [Chatanantavet and Parker, 2008]. Several sets of experiments were conducted with different boundary conditions. Within each set, hydraulic properties were kept constant whereas each run had a different sediment feed rate and was allowed to evolve to a steady state sediment cover. Chatanantavet and Parker [2008] found that two variables, the initial extent of sediment cover and the channel slope, determined the resulting relationship between sediment flux and sediment cover, as outlined in Table 1.

Table 1. Alluviation Style as a Function of Channel Slope and Initial Sediment Cover [Chatanantavet and Parker, 2008]
Slope (S)aNo Initial Sediment CoverInitial Sediment Cover
<0.005(1-Fe) ∝ qs/qt(1-Fe) ∝ qs/qt
0.005 < S < 0.015Runaway alluviationWith initial sediment patches: (1-Fe) ∝ qs/qt
>0.015Runaway alluviationWith initial sediment cover > threshold depth and qs/qt > (qs/qt)cr: (1-Fe) ∝ qs/qt

[8] The style of alluviation in these experiments is a function of the relationships between sediment entrainment and deposition, flow hydraulics and surface roughness. The simplest behavior, gradual alluviation, occurred at low slopes (Table 1), in which Fe decreased gradually as qs/qt increased. This result is approximately described by both equations (1) and (2) [Turowski, 2009]. However, under different conditions nonlinear behaviors were displayed that are not described by either equation.

[9] These nonlinear behaviors can be described as: (1) runaway alluviation; (2) threshold effects depending on channel slope and/or Froude number; and (3) sensitivity to initial sediment cover. Runaway alluviation occurred in runs with channel slope ≥∼0.005 and no initial sediment cover [Chatanantavet and Parker, 2008]. In these runs, no alluviation occurred until sediment feed reached an ‘oversaturated’ capacity value, at which point full alluviation rapidly occurred, hence the term runaway alluviation.

[10] In one set of experiments with initial sediment cover, channel slope was increased between sets of runs from 0.015 to 0.053. In each set of runs, the equilibrium channel was free from sediment cover until a threshold value of qs/qt, known as (qs/qt)cr; when qs/qt was greater than (qs/qt)cr, Fe was approximately a linear function of qs/qt. As slope increased, (qs/qt)cr also increased such that the channel was free of sediment cover until a higher value of qs/qt (Figure 2a). Similar results were also reported by James et al. [2011], who identified the threshold as a function of dimensionless bed load, a dimensionless grain parameter and a movability number (shear velocity over settling velocity).

Figure 2.

Results from the flume experiments of Chatanantavet and Parker [2008]. (a) The impact of channel slope (given in %) on the relationship between Fe and qs/qt. (b) The time evolution of Fe in runs started with different sediment depths. Initial depth is given in cm with approximate number of grain diameters in parentheses. Gravel had a uniform diameter of 0.7 cm. (Redrawn from Figures 9b and 13 of Chatanantavet and Parker [2008].)

[11] Another set of experiments held hydraulic conditions and slope constant, but altered the initial sediment depth. Runs with an initial depth of 0 or ∼1 grain diameters stabilized with no alluviation, whereas runs with an initial sediment depth of ∼3 to ∼9 grain diameters stabilized at a Fe = ∼0.4 (Figure 2b). In the runs that developed sediment cover, sediment transport capacity was less than for a run with the same hydraulic conditions but that experienced runaway alluviation.

[12] Chatanantavet and Parker [2008] reported further flume experiments by Demeter et al. [2005], in which the development of sediment cover was observed on rough and smooth beds with no initial sediment cover. On the smooth bed, sediment cover only started to occur at relatively high values of qs/qt, and only limited sediment cover developed before runaway alluviation occurred. On the rough bed, partial alluviation occurred at a lower value of qs/qt than on the smooth bed, and runaway alluviation did not occur.

[13] Some of Chatanantavet and Parker's [2008] experiments produced results that are reproduced by equations (1) and (2), but many experimental results are quite dissimilar from the predictions of these equations. We suggest that these dissimilarities are because both equations (1) and (2) and the models of Turowski et al. [2007] and Turowski [2009] omitted interactions between grains. A recent field and theoretical study [Hodge et al., 2011] has demonstrated significant differences between the behaviors of grains on bedrock and alluvial surfaces within a bedrock channel. On a bedrock surface (compared to an alluvial surface within the same channel) grain entrainment occurs at a lower shear stress, grains travel farther in a single translation, and grain deposition is unlikely. Correspondingly, entrainment from an alluvial surface is less likely than from a bedrock surface, and deposition is more likely. These contrasts occur because of the differing effects of the bedrock and alluvial surfaces on grain sheltering, pivot angles and local flow profiles [Hodge et al., 2011]. Because of these differences the extent of sediment cover directly affects the dynamics of individual grains. We hypothesize that grain interaction explains aspects of Chatanantavet and Parker's [2008] experimental results, and that the absence of such interactions from Turowski's [2009] model accounts for the inability of this model to reproduce Chatanantavet and Parker's [2008] data.

[14] In this paper we demonstrate that the emergent, nonlinear behaviors reported by Chatanantavet and Parker [2008] (Figure 2 and Table 1) can be explained through consideration of the contrasting behaviors of grains on bedrock and alluvial surfaces. To achieve this, we use a cellular automaton model that reproduces the movement of individual grains in a bedrock channel. In this model, the contrasting effects of bedrock and alluvial surfaces are parameterized through the probability of grain entrainment. We quantify the effect of model parameterization on the development of sediment cover and demonstrate the sensitivity of sediment cover to different model parameter values. The model results are used to provide a framework that unites the different relationships between sediment flux and sediment cover observed by Chatanantavet and Parker [2008]. Finally, we consider the implications of these results for the application of relationships between sediment flux and sediment cover in landscape evolution models.

[15] We have developed the model for smooth bedrock river beds, noting that Chatanantavet and Parker [2008] used beds of both smooth and moderate roughness. Previous theoretical work (including Sklar and Dietrich [2004], Turowski et al. [2007], Stark et al. [2009], and Lague [2010]) has also assumed low roughness. Many bedrock river reaches are characterized by moderate to extreme roughness elements, and development of comprehensive models of incision will ultimately require explicit incorporation of bed heterogeneity. Our simplifying assumption of low bedrock roughness aims to address grain interactions independent of morphological effects.

[16] Another assumption of previous theoretical and experimental work is that sediment flux is in equilibrium with sediment cover over relatively short timescales, such as within a flood event. This assumption may not be true in orogenic settings such as Taiwan, where sediment cover fluctuations over short timescales are driven by the large temporal variability in discharge and sediment flux [Lague, 2010; Yanites et al., 2011]. Here we focus on conditions where short-term equilibrium can be expected.

2. Cellular Automaton Model

[17] Cellular automaton (CA) models use rules that are simple, local, often empirically based, and that operate over discrete time steps [Fonstad, 2006]. These models typically operate on regular computational grids and, as a result of the local nature of the rules, the evolution of any particular grid cell is a function of the properties of the surrounding cells. CA models fall along a continuum of levels of abstraction [Paola, 2001; Fonstad, 2006], the form of the rules depending on this level and how processes are represented in the model. The degree of abstraction represents a balance between reproducing system dynamics and simplification to make the model computationally efficient [Paola, 2001]. Depending on the aim of the model, CA model parameters may be properties that can be directly measured in the real system, or may be more conceptual lumped parameters that cannot be measured directly. In the latter case, model calibration has to be through comparison between measurable data and identifiable model outputs. The flexible formulation of CA models, combined with their ability to represent complex systems simply while retaining key dynamics, means that they have been widely used in geomorphology to model the upscaling of small-scale processes to produce large-scale emergent features including fluvial systems [Murray and Paola, 1994; Coulthard et al., 2002; Thomas and Nicholas, 2002; MacVicar et al., 2006], hillslopes [Chase, 1992; Tucker and Bradley, 2010], and bed forms [Werner and Fink, 1993; Favis-Mortlock, 1998; Nield and Baas, 2008; Narteau et al., 2009].

2.1. CA Model Outline

[18] Here we use a cellular automaton framework to simulate the movement of individual grains through a bedrock channel, and hence to investigate how the dynamics of single grains upscale to produce an emergent sediment cover. We demonstrate how incorporating the effect of bedrock and alluvial surfaces on grain entrainment within a simple framework can affect the development of sediment cover and qualitatively reproduce the behaviors observed by Chatanantavet and Parker [2008]. Given the complexity of interactions between sediment transport, overall surface roughness and hydraulics [e.g., McEwan et al., 1999; Recking et al., 2008], we use an intentionally reduced complexity approach; we do not aim to produce a fully scaled and parameterized model of Chatanantavet and Parker's [2008] flume experiments.

[19] The model domain is a regular grid, 1000 cells long × 100 wide for all runs reported here, representing a bedrock river reach (Figure 3). Discrete, uniform-sized, grains of diameter D are introduced at the upstream boundary and travel downstream through repeated processes of entrainment, transport and deposition (Figure 3a). Each square grid cell has length and width X, where X is equal to D, and can contain a vertical stack of grains with one grain occupying each layer (Figure 3a).

Figure 3.

(a) A schematic of the CA model, illustrating the different operations that are applied within each model time step. The full model domain is 100 cells wide by 1000 cells long. Note that grains may be entrained in multiple sequential time steps, producing a total translation distance that is a multiple of L. (b) A section of model domain with a binary parameterization of grain entrainment. Grains where five or more neighboring cells contain grains have entrainment probability Pc, whereas those with fewer neighboring grains have probability of entrainment Pi. (c) The evolution of the number of grains leaving the model during a run starting with no sediment cover. (d) Evolution of fractional bedrock exposure (Fe) in a downstream section of the model domain.

[20] In each model time step, a prescribed sediment supply (Qs, grains per time step across the entire model width) is introduced to the upstream boundary of the model domain and is deposited over a downstream introduction distance (Figure 3a). All grains in the model are assigned an entrainment probability P, and unbiased random numbers are used to determine which grains are entrained in each time step. The entrained grains are deposited a step length L downstream from their starting position. In most runs (specified below) a smoothing algorithm is applied to ensure that the number of grains in each cell is not more than one greater than the number in the adjacent downstream cell. Unstable grains (those grains >1 grain higher than the downstream neighbor) are identified and moved one further cell downstream. This process is iterated until the smoothing condition is achieved. This provides a first order approximation of local toppling of unstable grains.

[21] An equilibrium sediment cover develops as the model evolves to a stochastic steady state (Figures 3d and 4), in which the time-averaged inputs and outputs of sediment are equal but output remains variable (Figures 3c and 3d). The time series of sediment output was therefore smoothed using a running mean over 100 time steps to give Qs_out_sm (grains per time step across the entire model width). Model runs were terminated once the model had run in steady state for 500 time steps. From empirical analysis of model results, steady state was said to be achieved when Qs_out_sm/Qs was within 1 ± 0.015 for 96% of time steps. Median model run duration was 1000 time steps. For a subset of runs designed to identify the effect of initial sediment cover, the model was run for 10,000 time steps.

Figure 4.

The development of sediment cover in a model run with Pi = 0.95, Pc = 0.1 and Qs/Qt = 0.75 and no initial sediment cover, at 1000, 2000, 3000 and 4000 time steps (top to bottom). Sediment is transported from left to right.

2.1.1. CA Model Parameters

[22] Developing the CA model is a balance between including parameters that will affect the formation of sediment cover, while ensuring that there are not too many poorly defined parameters and that the model is not overly complex. Parameters that are not included in the CA model are discussed in section 4.5. The parameters of the CA model, and the ways in which they are given values and tested, are described below. Methods of quantifying sediment cover are also presented.

2.1.2. Probability of Entrainment (P)

[23] A key model parameter is the probability that each grain has of being entrained in a single time step. Physically, this is the probability that the hydraulic forces exerted on the grain overcome the forces resisting grain entrainment within the time step. For a patch of sediment in a river, both the hydraulic and resisting forces exhibit a distribution of values. Hydraulic forces vary because of turbulence [Schmeeckle et al., 2007; Dwivedi et al., 2011], and resisting forces vary due to variation in grain size, exposure, pivoting angles and packing [Kirchner et al., 1990; Johnston et al., 1998; Dancey et al., 2002; Papanicolaou et al., 2002]. These variations are not explicitly incorporated in the model, but are implicitly incorporated through the value of P and the stochastic nature of grain entrainment. Values of P near to zero represent a situation in which resisting forces are greater than the majority of hydraulic forces, whereas values of P near to one therefore represent a situation in which hydraulic forces are generally greater than resisting forces.

[24] P can be parameterized in many different ways. We use two approaches: uniform values of P where all grains have the same probability of entrainment, and binary values of P. In the latter case P is a function of the number of surrounding cells that contain grains. If five or more of the eight surrounding cells contain grains, then the grain is considered to be in a sediment cluster and P = Pc. If fewer than five of the surrounding cells contain grains, then the grain is considered to be isolated and has P = Pi, where Pi > Pc. An example of the spatial pattern of Pi and Pc is shown in Figure 3b. To avoid edge effects when calculating the number of neighboring cells with grains, a one cell thick border is added to the model domain prior to the neighborhood calculation. This border is probabilistically populated with grains according to the grain density in the proximal areas of the model domain. This border is removed following the neighborhood calculation.

[25] Using uniform values of P assumes that there is no spatial variation in sediment processes. The binary parameterization is used to provide an approximation of the variation in sediment processes that result from the contrast between bedrock and alluvial surfaces, with Pc and Pi respectively representing entrainment from alluvial and bedrock surfaces [Hodge et al., 2011]. Instigating the switch from Pi to Pc at five neighboring grains, rather than any other value, is essentially arbitrary. Five was selected because it is midway between full and zero exposure. More complex parameterizations in which P decreases as a function of the number of neighboring grains would also be feasible. However, more complex parameterizations require more detailed empirical data with which to justify them; such empirical data are currently not available. Values of P used in the CA model runs are selected to explore the resulting range of behaviors, rather than trying to identify real values of P for particular scenarios.

2.1.3. Sediment Cover (1 − Fe)

[26] The main aim of the CA model is to quantify the response of sediment cover to different model parameterizations, necessitating a definition of sediment cover. Fractional sediment cover could be considered to be the proportion of cells that contain grains. Calculating this at any one instant overestimates effective cover by including grains that are entrained in successive time steps and that only provide only very temporary cover. Instead, sediment cover is defined as the proportion of cells containing grains that are not entrained in that time step. This definition is more consistent both with the sediment cover that entrained grains experience and with other published work. Turowski [2009] used a similar definition and excludes grains undergoing transport when calculating cover. Chatanantavet and Parker [2008] measured sediment cover from images of the active flume. In some runs, they record no cover when Qs ∼ 0.5 the capacity sediment transport rate. Given that the flume must contain sediment under these conditions (albeit sediment undergoing transport), then their image analysis must only identify stationary sediment.

2.1.4. Initial Sediment Depth

[27] In most CA model runs, the model domain is initially free from sediment cover; sediment cover subsequently evolves from the deposition of sediment introduced at the upstream boundary. A small number of runs use an initial sediment cover to see whether the CA model shows the same sensitivity to initial sediment cover as Chatanantavet and Parker's [2008] flume experiments. In these runs the initial distributions of sediment depths are selected to reproduce those used by Chatanantavet and Parker [2008].

2.1.5. Step Length (L)

[28] The step length of a mobile grain L has a uniform value of L = 10X, where X is the cell size. L is equivalent to the grain saltation distance; if a grain moves in sequential time steps its total translation distance will be a multiple of L. Grains do not move laterally. The value of L is also used to determine the introduction distance over which grains input into the model domain are initially distributed; in the case of uniform L, the introduction distance is equal to L. As our results are independent of the value of L for reasons explained in the following section, we do not consider alternative constant values or distributions of L.

2.1.6. Capacity Sediment Transport (Qt)

[29] The model time step is arbitrary and unscaled, so comparison with empirical or theoretical results requires a means of scaling, for which we use a capacity sediment transport rate (Qt, grains per model width per time step) that is defined for each run. Sklar and Dietrich [2004] defined Qt using a bed load transport equation, selecting that of Fernandez Luque and Van Beek [1976], whereas Chatanantavet and Parker [2008] defined this capacity as the sediment feed rate that caused their flume bed to be fully covered by sediment, which was consistent with application of a revised version of the Meyer-Peter and Müller formula. The former definition is not applicable to the CA model because the model does not explicitly determine the hydraulic properties needed for computing bed load transport from hydraulic equations. Although the latter definition could be applied, uncertainties in exactly when full cover was achieved could lead to significant differences in the estimated value of Qt, given that the CA model often asymptotically approaches full sediment cover and that sediment cover is inherently stochastic.

[30] Herein, we adopt an alternative definition of the capacity sediment transport by incorporating an active layer into the CA model. In gravel bedded alluvial rivers, sediment is typically only active within the top layer of the bed; this active layer is assumed to have depth of the order of 2D90, where D90 is the 90th percentile of the sediment grain size distribution (GSD) [Parker, 1991; DeVries, 2002]. Although the CA model has a uniform GSD, sediment in bedrock rivers is typically poorly sorted, with D90 ∼ 2D50 [Ferguson and Paola, 1997]. We therefore assume an active layer depth of 4D. Hence, in the CA model only grains within a four grain thick active layer are allowed to move in a given time step. If a cell contains more than four grains, all grains deeper than the active layer have P = 0. A fully alluvial bed where every cell contains ≥4 grains is used to calculate a capacity sediment transport rate, as:

display math

where Qt is the total sediment transport rate (in grains per time step) from the model, LA is the active layer depth (in number of grains) and w is the width of the model domain (in number of cells).

[31] The definition of Qt means that when Qs is expressed as Qs/Qt, model results are insensitive to the value of L, with the same relationship between sediment cover and Qs/Qt being produced by model runs with different values of L. This is demonstrated in Table 2, in which model runs with different values of L, but the same value of Fe are demonstrated to have the same values of Qs/Qt. The same is true if L is drawn from a distribution.

Table 2. The Relationship Between Fe and Qs/Qt for Different Values of L
 Run ARun B
  • a

    inline image, mean value of P for all mobile grains; inline image, mean number of mobile grains per cell. Both runs have the same value of Fe, so inline image and inline image are the same in both.

  • b

    Under steady state, Qs = Qs_out.

Step length (L)L2L
Sediment cover (Fe)FeFe
Capacity sediment transport (Qt, from equation (3))LPcLAw2LPcLAw
Output flux (Qs_out)aL inline image inline imagew2L inline image inline imagew
Relative sediment flux (Qs/Qt)b inline image inline image/PcLA inline image inline image/PcLA

2.1.7. Time Step

[32] The time step for the CA model is not absolute, but is defined as the time taken for all entrained grains to move distance L. In using the CA model to explore the patterns and dynamics of sediment cover, the value of the time step is not important.

2.2. CA Model Applications

[33] We apply the CA model in three sets of experimental runs, summarized in Table 3. In each set the emergent property of interest is the steady state sediment cover, i.e., mean sediment cover during the last 500 model time steps. The first two sets of runs are designed to quantify how the form of the relationship between relative sediment flux (Qs/Qt) and sediment cover (Fe) varies as a function of the model parameter values (uniform or binary values of P). The third set is designed to identify whether the steady state sediment cover is sensitive to initial sediment cover.

Table 3. Parameter Values Used in Each Set of Model Runs
 Set 1Set 2Set 3
Smoothing algorithmNoYesYes
Active layerNoYesYes
Parameterization of PUniform, P = 0.05, 0.25, 0.5, 0.75, 0.95Binary, pairs of Pi and PcBinary, Pi = 0.95 and Pc = 0.05 and 0.1
Initial sediment coverNoNoYes
Number of time stepsRun to steady stateRun to steady state10,000

[34] Set 1 runs demonstrate model behavior under the simplest possible parameterization, and use uniform values of P, no smoothing algorithm and have no active layer (i.e., all grains within a cell can potentially be entrained in a time step regardless of the number of grains in that cell). Although very simplified, these assumptions are consistent with those made by Turowski [2009].

[35] Set 2 runs use physically more realistic settings, including binary parameterization for P, use of the active layer formulation and application of the smoothing algorithm. Multiple runs were performed to explore the Pi and Pc parameter space, subject to the constraint that PiPc. Model runs in which Pi = Pc are included for comparison with Set 1, in which this equality is implicit. Varying values of Pc and Pi can be considered to be a proxy for varying controlling parameters such as channel slope and surface roughness; this is discussed in detail in section 4.4. Consequently, the experiments of Chatanantavet and Parker [2008] can be approximately placed within this parameter space allowing comparison to the CA model results.

[36] Set 3 runs use the same model set up as Set 2, but each run has an initial sediment cover produced by assigning each cell in the model domain an integer number of grains drawn from an appropriate normal distribution truncated at zero. Chatanantavet and Parker [2008] report results from beds composed of 7 mm gravel that are 10, 20, 40 and 60 mm deep. Initial sediment beds for the CA model are therefore drawn from normal distributions with means of 1, 3, 6 and 9 grains and standard deviations of 0.5 grains. For comparison with the previous sets, runs with no initial sediment cover are also conducted.

3. Model Results

3.1. Set 1: Equal Entrainment Probability

[37] When all grains have probability of entrainment P, and the model is run without using the smoothing algorithm or the active layer, the relationship between Fe and Qs/Qt is negative exponential (Figure 5a). For a given value of Qs/Qt, a higher value of P results in a higher value of Fe. The relationships for P = 0.05, 0.25, 0.5, 0.75 and 0.95 are well described by equation (2), with φ of between 3.8 (P = 0.05) and 0.2 (P = 0.95). For comparison, Turowski et al. [2007] suggest that φ = 1. The different curves and values of φ in Figure 5a are caused by the differing mobility of grains in the different model runs. However, the total amount of sediment on the bed is the same for all values of P at any given value of Qs/Qt; this is seen in Figure 5b in which sediment cover is calculated as the proportion of cells that contain grains (regardless of whether they are stationary). Fitting equation (2) to the curve in Figure 5b gives φ = 4.0 (±≤0.016).

Figure 5.

Relationships between Fe and Qs/Qt when all grains have probability of entrainment P. (a) Results from model runs with no smoothing and no active layer, P = 0.05 to 0.95. Sediment cover (1 − Fe) is calculated as the proportion of cells containing one or more grains that are stationary for one or more time steps. Relationships are described by equation (2); for P = 0.05, 0.25, 0.5, 0.75 and 0.95, φ is respectively 3.8 (±0.014), 3.0 (±0.008), 2.0 (±0.002), 1.0 (±0.001) and 0.2 (±0.0002). Values in brackets are 95% confidence intervals. (b) The same relationships as in Figure 5a, only sediment cover is instead calculated as the proportion of cells that contain grains, regardless of whether or not they are static. The relationships are described by equation (2), with φ = 4.0 (±≤0.016).

[38] Another feature of the curves in Figure 5a is that at higher values of P, Fe is not zero when Qs/Qt = 1 i.e., the bed is not fully covered when sediment supply is equal to capacity transport rate. This occurs because the high probability of entrainment means that even with a layer of grains four grains deep (i.e., the depth of the active layer), there is a high probability that all grains in a cell will be entrained and that the cell will therefore not have any cover. But, if Qs > Qt, the model will not be able to transport all the supplied sediment. It is envisaged that ‘excess’ sediment (i.e., QsQt) will be deposited within the first length L of the model domain. This aggrading zone will supply sediment at rate Qt to the downstream model domain, but, because the smoothing algorithm is not applied, will not spread downstream.

3.2. Set 2: Binary Entrainment Probability

[39] In model runs in which Pi and Pc are both varied, the relationship between Fe and Qs/Qt is not always described by a negative exponential curve. Depending on the values of Pi and Pc, the relationship takes different forms (Figure 6), which were classified into six categories: negative exponential, convex exponential, steep sigmoidal, gentle sigmoidal, horizontal linear, and decreasing linear (Table 4). The functions in Table 4 were fitted to all relationships using a least squares fit, and the relationship was classified according to the fit with the smallest sum of squared errors.

Figure 6.

Forms of the relationship between Fe and Qs/Qt for different combinations of Pi and Pc. When PiPc < 0.7, the relationship is negative exponential as hypothesized by Turowski et al. [2007]. However, much of the parameter space is instead populated by relationships with a sigmoidal form, which is consistent with many of Chatanantavet and Parker's [2008] flume results. At high values of Pi, relationships with horizontal linear, decreasing linear and convex exponential forms also occur.

Table 4. Forms and Occurrence of the Relationships Between Fe and Qs/Qt (Figure 6)
Relationship NameEquationaDescriptionRange of Pi and PcOccurrence in Literature
  • a

    Here a and b are fitted parameter values.

Negative exponentialequation imageRate of decrease in Fe decreases as Qs/Qt increases.Pc ≤ 0.6; [PiPc] ≤ 0.15Turowski et al. [2007], Turowski [2009], and Chatanantavet and Parker [2008]
Sigmoidal (a > −40)Fe = inline imageFe ∼ 1 when Qs/Qt < 0.5. Gradual decrease in Fe as Qs/Qt increases.Pc ≤ 0.6; [PiPc] ≥ 0.05Chatanantavet and Parker [2008]
Sigmoidal (a < −40)Fe = inline imageRapid decrease between Fe = 1 and Fe = 0, at Qs/Qt ≥ 0.5.Pc ≤ 0.15; Pi ≥ 0.5 (If Pc = 0.05, Pi ≤ 0.75)Chatanantavet and Parker [2008]
Horizontal linearFe = 1 − aQs/QtFe = 1 for all Qs/Qt. Rapid decrease to Fe = 0 at some value of Qs/Qt > 1.Pi ≥ 0.8; Pc ∼ 0.05Runaway alluviation in the work of Chatanantavet and Parker [2008]
Decreasing linearFe = 1 − aQs/QtLinear decline in Fe with Qs/QtPc = 0.6 to 0.75Sklar and Dietrich [2004] and Chatanantavet and Parker [2008]
Convex exponentialequation imageInitial linear decline in Fe with Qs/Qt; sharp decrease in Fe when Qs/Qt > 0.8.Pi and Pc ≥ 0.85none

[40] Relationships best described by a linear equation were further subdivided by gradient (i.e., horizontal or not). Relationships best described by a sigmoidal equation were further subdivided according to whether the value of a was greater than, or less than, −40. The parameter a describes the steepness of the transition between Fe = 1 and Fe = 0, and the value of −40 was selected because it split the relationships into two contrasting populations.

[41] Table 4 outlines the values of Pi and Pc under which the different relationships occur. Negative exponential relationships that are consistent with Set 1 results only occur over a small proportion of the parameter space (Figure 6). Linear functions also have limited occurrence. Instead, most relationships between Fe and Qs/Qt are best described by a sigmoidal function (Figure 6).

[42] Transects through the parameter space (Figure 6) demonstrate the effect of systematically varying one or both parameter values. Results extracted from three types of transects are shown in Figure 7: different values of Pi = Pc (Figure 7a), constant Pc with varying Pi (Figures 7b and 7c), constant Pi with varying Pc (Figure 7d).

Figure 7.

Variations in relationships between Fe and Qs/Qt along different transects of the parameter space. (a) Pi = Pc; (b) Pc = 0.1, Pi = 0.1 to 0.99; (c) Pc = 0.25, Pi = 0.25 to 0.99; and (d) Pc = 0.05 to 0.95, Pi = 0.95. Symbols show the form of the relationship.

[43] A transect along the line Pi = Pc shows that when Pi and Pc are 0.6 or less, the relationship between Fe and Qs/Qt is negative exponential (Figure 7a). As Pi and Pc increase, the relationship first becomes decreasing linear, and then convex exponential. Transects with constant Pc (Figures 7b and 7c) show that as Pi increases the form of the relationship evolves away from the negative exponential or linear form that occurs when Pi = Pc toward a more sigmoidal form. Within the sigmoidal forms, as the value of Pi increases, the transition between Fe = 1 and Fe = 0 becomes steeper and occurs at higher values of Qs/Qt. At high values of Qs/Qt, all relationships tend to the same line regardless of the value of Pi.

[44] When Pi is constant (Figure 7d) patterns are similar to those on transects with constant Pc in that as Pc decreases the relationship changes from negative exponential or linear to sigmoidal. Within the sigmoidal forms, the steepness of the transition section increases as Pc decreases. Pc = 0.05 produces a horizontal linear relationship with no sediment cover, which is discussed in section 4.2. In contrast to the constant Pc case, the constant Pi transects do not all tend to the same line at high values of Qs/Qt. With constant Pc, the relationships tend to the curve described when Pi = Pc, the form of which varies according to the value of Pc. With constant Pi, each curve has a different value of Pc, and is therefore tending toward a different line at high values of Qs/Qt.

[45] In summary, the relationship between Fe and Qs/Qt takes different forms across the parameter space and is more complex than the relationships derived by Sklar and Dietrich [2004] and Turowski et al. [2007].

3.3. Set 3: The Effect of Initial Sediment Cover

[46] In the majority of CA model runs, the presence or absence of sediment cover on the bed at the beginning of the run did not affect the steady state sediment cover, although it does affect the number of time steps needed for the model to achieve steady state. For example, in runs with Pi = 0.95 and Pc = 0.1 for Qs/Qt = 0.25, 0.625 and 1, the time taken to achieve steady state is longer when there is a larger difference between the initial bed and the steady state sediment cover (Figures 8a8c).

Figure 8.

Evolution of sediment cover in CA model runs with initial sediment cover; legend shows mean depth of sediment cover in number of grains. (a–c) Models run with Pi = 0.95, Pc = 0.1 and Qs/Qt = 0.25, 0.625 and 1. (d–f) Models run with Pi = 0.95, Pc = 0.05 and Qs/Qt = 0.5, 0.75 and 1.

[47] However, for some combinations of parameter values, the presence or absence of sediment cover at the start of a run results in different steady states. Figures 8d8f show the evolution of sediment cover in model runs with Pi = 0.95 and Pc = 0.05 for Qs/Qt = 0.5, 0.75 and 1. When Qs/Qt is 0.5, all model runs eventually stabilize with full bedrock exposure, although this occurs more rapidly with less initial sediment cover. When Qs/Qt is 1, all runs with initial sediment cover converge to a steady state with full sediment cover, whereas the run without initial sediment cover maintains a steady state with no sediment cover. When Qs/Qt is 0.75, runs with initial sediment cover three grains or more thick stabilize with full sediment cover, whereas the run with no initial sediment cover stabilizes at no sediment cover. The run with initial cover one grain thick oscillates between zero and full sediment cover due to downstream movement of channel-wide sediment patches.

[48] The transition from an initial bed (with or without sediment) to the steady state condition typically occurs in a systematic way with a front separating the initial and new sediment covers advancing down the model domain (e.g., Figure 4). In some cases there can be two fronts, with an intermediate cover in between. Within a front-bounded region, sediment cover is spatially homogeneous and the number of grains per cell has a low standard deviation (∼1). However, some parameter combinations produce spatially heterogeneous sediment cover (e.g., Figure 9).

Figure 9.

The development of sediment cover in a CA model run with (a) Pi = 0.95, Pc = 0.1 and Qs/Qt = 0.625 and no initial sediment cover and (b) Pi = 0.5, Pc = 0.1 and Qs/Qt = 0.375 and no initial sediment cover. Figures shows model state at 1000 time step intervals for model run lengths of between 1000 and 10,000 time steps (top to bottom). Sediment is transported from left to right. In Figure 9a, patches of thicker sediment cover are initiated at the upstream end; these patches grow and merge downstream.

[49] The evolution of sediment cover is typically synchronous with the evolution of the number of grains leaving the model (e.g., Figures 10a and 10b). However, in some model runs, the output number of grains (Qs_out, grains per time step across the entire model width) equilibrates before Fe reaches a steady value. In Figures 10c and 10d, model runs with different initial sediment covers all reach Qs_out/Qs = 1 after 3000 time steps, whereas they only converge to the same value of Fe after 10,000 time steps. Between 3000 and 10,000 time steps, different model runs are at different quasi-stable values of Fe.

Figure 10.

The evolution of (a and c) Fe and (b and d) Qs_out/Qt for CA model runs with different initial sediment cover as indicated in the legend (expressed as mean depth in grains). For Figures 10a and 10b, Pi = 0.5, Pc = 0.1, Qs/Qt = 0.5. For Figures 10c and 10d, Pi = 0.5, Pc = 0.1, Qs/Qt = 0.375.

4. Discussion and Implications

4.1.

[50] The CA model produces several distinct forms of relationships between Fe and Qs/Qt, depending on the values of Pi and Pc. Furthermore, for some values of Pi and Pc, the steady state to which the model evolves depends on whether or not there is initial sediment cover. Turowski et al. [2007] predicted that, under the assumption of uniform grains distributed randomly on a flat bed, Fe should be a negative exponential function of Qs/Qt. This behavior occurred in two ways in the model: first, in all model runs with the restrictive assumptions of neither smoothing nor an active layer (Figure 5); second, in runs with both an active layer and smoothing, and when PiPc, and Pc < 0.6. In contrast, Chatanantavet and Parker [2008] observed different forms of the relationship between Fe and Qs/Qt, including negative exponential, decreasing linear, sigmoidal and horizontal linear, all of which are also reproduced by some parameterization of the CA model (Figure 6). The conditions under which these forms occurred in Chatanantavet and Parker's [2008] data and in the CA model are summarized in Table 5.

Table 5. The Occurrence of Different Relationships Between Fe and Qs/Qt in the Data of Chatanantavet and Parker [2008] and the CA Model Results
Relationship Between Fe and Qs/QtChatanantavet and Parker [2008]CA Model Results
Negative exponential or linearS < 0.015 and initial sediment coverPiPc
SigmoidalS > 0.015 and initial sediment cover (e.g., Figure 2a)Pi high relative to Pc (e.g., Figure 7b)
Horizontal linearNo initial sediment cover, named runaway alluviationLow Pc and high Pi

[51] The similarities in Table 5 between the CA model results and Chatanantavet and Parker's [2008] flume observations suggest that the CA model is an improvement over previous models of sediment cover formation [Sklar and Dietrich, 2004; Turowski et al., 2007; Turowski, 2009]. This improvement reflects the use of different probabilities of sediment entrainment from bedrock and alluvial surfaces. The CA model behavior indicates that the emergence of these different behaviors can be explained in the context of grain-scale dynamics.

[52] However, the initial sediment cover appears to have different effects in the CA model and in Chatanantavet and Parker's [2008] experiments. The latter only recorded sigmoidal relationships when the bed had initial sediment cover (Table 1). In the CA model, a sigmoidal relationship is produced both by runs without initial sediment cover (Figure 6) as well as by runs with initial sediment cover (e.g., the equilibrium values of Fe for runs with thick initial cover in Figure 8) but at differing values of Pc and Pi.

4.2. Interpreting Model Results via Grain Dynamics

[53] The steady state extent of sediment cover in the CA model is determined by the input sediment supply and the probabilities of grain entrainment. For the input sediment supply to be conveyed through the model, each section of the model domain with length L has to supply this flux to the next downstream section. The volume of sediment entrained from any area of the bed is the product of the volume of sediment in that area and the probability of grain entrainment (equation (3)). To reach steady state, the amount of sediment on the bed adjusts until this volume of sediment is in equilibrium with the sediment input. If there is too little sediment on the bed, then sediment is deposited; if there is too much, sediment is eroded.

[54] Where all grains are equally mobile, an increase in sediment supply requires a proportional increase in sediment cover. The random placement of sediment on the bed means that as Qs/Qt increases, Fe decreases in a negative exponential manner [Turowski et al., 2007]. For example, at low sediment cover, doubling the amount of sediment on the bed results in a proportionally larger decrease in bedrock exposure than doubling the amount of sediment from a higher initial sediment cover.

[55] When interaction between grains affects their mobility, implemented in the CA model as a function of the number of neighboring grains, the relationship between Fe and Qs/Qt is more complex. Increasing the input flux increases the number of grains on the bed, but also increases the mean number of neighboring grains that each grain will have and so reduces overall grain mobility. As grains accumulate, the switch from Pi to Pc produces the widely observed sigmoidal behavior. As more grains switch to Pc the flux entrained from a given volume of grains decreases, and therefore a disproportional increase in sediment volume is needed to convey the input sediment supply through the model. This is achieved by sediment deposition, resulting in an increase in sediment volume and a decrease in Fe. This behavior is consistent with the flume experiments of James et al. [2011], who observed that as sediment supply rate increased, sediment storage increased with a concurrent change from discrete to collective grain movement. Switching from Pi to Pc at a different number of neighboring grains would alter the value of Qs/Qt at which sediment cover starts to develop, but will generally not alter the form of the relationship.

[56] The rapidity of the decrease in Fe depends on the difference between Pc and Pi. If Pc is much smaller that Pi (e.g., Pi = 0.95, Pc = 0.1), then the decrease in Fe is very rapid because a large increase in sediment volume is needed to account for the rapid decrease in the probability of grain entrainment (e.g., Figure 6). However, if the difference between Pc and Pi is smaller, then the switch from Pi to Pc results in a smaller decrease in the probability of grain entrainment, the resulting increase in sediment volume is smaller and Fe decreases less rapidly. This effect of the difference between Pc and Pi can be seen in the distribution of sigmoidal relationships with a less than −40 (rapid decrease, large difference between Pc and Pi) and greater than −40 (gentle decrease, smaller difference between Pc and Pi).

[57] In horizontal linear relationships between Qs/Qt and Fe, Qt is relatively small. Large values of Qs/Qt are therefore transmitted with a relatively small volume of sediment on the bed and high Fe. Consequently, even when Qs/Qt = 1, grains on the bed typically have probability of entrainment Pi and are highly mobile. Sediment can therefore be transmitted through the CA model at the capacity sediment transport rate with minimal sediment cover, suggesting that alluvial definitions of transport capacity, e.g., as used by Sklar and Dietrich [2004], may not always be applicable to the bedrock case. Furthermore, transport capacity may vary as a function of Qs/Qt as a result of changing hydraulic roughness [Chatanantavet and Parker, 2008].

[58] The implementation of an active layer within the model also affects the form of the relationship between Qs/Qt and Fe. The active surface layer is four grains deep; any further grains in a cell are considered to be buried in the sub-surface and have an entrainment probability of zero. Hence, once the volume of sediment in the CA model is such that some cells contain more than four grains sediment cover increases disproportionately. This effect only occurs at high values of Qs/Qt and is only noticeable if Fe is not already close to zero. The resulting decreases in Fe can be seen in the convex exponential and decreasing linear relationships that occur at high values of Pc and Pi (Figure 6). Changing the active layer depth would primarily affect the scaling of the relationship along the Qs/Qt axis, but not the form of the relationship.

4.3. Relationships Between Grain Density and Sediment Flux

4.3.1. Predicting Sediment Flux From Grain Density

[59] The occurrence of runaway alluviation and the sensitivity of some model parameterizations to initial sediment cover can be further explained by considering the relationship between the density of grains on the bed (expressed as grains per cell) and the flux of sediment entrained from those grains. Whereas the previous section considered how the cover of sediment on the bed (1 − Fe) developed in response to Qs/Qt, here we consider the sediment flux that is produced by a given sediment volume. This additional analysis demonstrates how, under some circumstances, different sediment volumes can produce the same sediment flux; this explains why some parameterizations can have more than one equilibrium sediment cover and how spatially heterogeneous sediment cover occurs.

[60] For a given distribution of sediment on the bed, the rate of sediment leaving the model (Qs_out, grains per time step across the entire model width) can be predicted by

display math

where inline image is the mean probability of entrainment for all mobile grains (i.e., up to four grains in any given cell) and inline image is the mean number of mobile grains per cell:

display math
display math

where Nm is the number of mobile grains in each cell, c is the number of cells, and Ni and Nc are the number of grains with entrainment probabilities of Pi and Pc respectively. To use equations (4)(6) to identify how Qs_out varies with the volume of sediment on the bed, it is necessary to simulate the distribution of different volumes of sediment across the model domain, enabling Ni and Nc to be calculated. Simulated beds are produced by using values drawn from a normal distribution to determine the number of grains in each cell. All values are rounded to the nearest positive integer. The smoothing algorithm is then applied to the simulated bed. Parameters for the normal distribution were derived by fitting normal distributions to the distributions of the number of grains in each cell from steady state CA model runs.

[61] For each pair of Pi and Pc equilibrium values of Qs_out were predicted from 13 simulated beds with different sediment cover, to quantify how Qs_out/Qt varied with sediment cover. The mean number of grains per cell ( inline image, including immobile grains) in the simulated beds ranged from 0 to 6. For each pair of Pi and Pc the variation in Qs_out/Qt with inline image takes one of three forms, the distribution of which across the Pi and Pc parameter space is shown in Figure 11a.

Figure 11.

Relationships between the mean number of grains per cell, inline image, and sediment flux (Qs_out/Qt) for different combinations of Pi and Pc. (a) Summary of the form of all relationships for all combinations of Pi and Pc used. (b) Example forms of relationships between inline image and Qs_out/Qt: excess, in which the maximum value of Qs_out/Qt is greater than 1 (e.g., Pi = 0.95, Pc = 0.05); multiple, in which some values of Qs_out/Qt are produced by more than one value of inline image (e.g., Pi = 0.5, Pc = 0.1); and, unique, in which each value of inline imageproduces a unique sediment flux (e.g., Pi = 0.99, Pc = 0.5).

[62] Most combinations of Pi and Pc produce a unique relationship between inline imageand Qs_out/Qt (Figure 11b), and so every value of Qs/Qt has a unique steady state value of Fe. Model runs with these combinations of Pi and Pc will not be sensitive to the initial sediment cover; runs initiated with more or less sediment than the equilibrium value of inline image will undergo erosion or deposition until inline image is in equilibrium.

[63] The second form occurs with pairs of Pi and Pc with low Pc and high Pi. In this ‘multiple’ form, the same value of Qs_out/Qt is produced by multiple values of inline image. In these cases initial sediment cover will be important because some values of Qs/Qt could produce multiple steady state values of Fe. The resulting steady state value of Fe depends on the form of the inline image and Qs_out/Qt curve. inline image of the initial bed gives the value of Qs_out/Qt with which the bed is in equilibrium. If the input value of Qs/Qt < Qs_out/Qt, then erosion occurs, hence moving left along the inline image and Qs_out/Qt curve until inline image is in equilibrium with Qs /Qt. Conversely, if Qs/Qt > Qs_out/Qt, then inline image will evolve right along the inline image and Qs_out/Qt curve.

[64] Such evolution is demonstrated in Figures 8e and 8f, in which runs with and without initial sediment cover, but the same value of Qs/Qt, evolve to different steady state values of Fe. This is consistent with Figure 11b, which predicts multiple steady states for this parameterization when Qs/Qt is greater than ∼0.45. In the runs with initial sediment cover, steady state Fe ∼ 0, whereas in runs without initial sediment cover, Fe ∼ 1, both of which are consistent with evolution along the inline image and Qs_out/Qt curve. Inconsistent behavior occurs when inline image = 1; Figure 11b suggests that these runs should evolve to inline image < 1, but instead they evolve to inline image > 1, i.e., Fe ∼ 0. This may be because some steady state values of inline image are more stable than others, as discussed below.

[65] The final form is the excess form, in which some values of inline image produce a value of Qs_out that is greater than the theoretical capacity sediment transport rate, i.e., Qs_out/Qt > 1 (e.g., Figure 11b). This occurs when Pc = 0.05 and Pi ≥ 0.6. Under this form, model runs evolving from a bed without initial sediment cover could achieve steady state with Qs/Qt > 1 and Fe ∼ 0. This will only occur up to a maximum value of Qs/Qt (1.64 in Figure 11b); if Qs/Qt is greater than this maximum value, rapid alluviation will occur. This behavior of Fe ∼ 0 for increasing values of Qs/Qt, followed by runaway alluviation when Qs/Qt > 1 was observed by Chatanantavet and Parker [2008], who found that in some flume runs, the measured capacity sediment transport rate was higher when the run was initiated without a sediment bed than when it was initiated with sediment cover.

4.3.2. Sediment Cover Stability

[66] The above analysis demonstrates that under some parameter combinations, a single value of Qs/Qt can produce multiple steady state values of Fe. More than one steady state sediment cover can occur in the same model, for example the heterogeneous sediment covers in Figure 9. In Figure 9a, the sparser and thicker covers have respective values of inline image of 0.3 and 2.4. Interpolating steady state values of inline image for Qs_out/Qt = 0.625 from the relevant inline image and Qs_out/Qt curve gives inline image = 0.3, 0.8 and 2.5, suggesting that both values of inline image correspond to steady state conditions. Figure 9b also shows patches of sparser and thicker sediment, which have respective inline image of 0.35 and 1.4. This is again consistent with predictions of inline image = 0.35, 1.3 and 1.4 from the inline image and Qs_out/Qt curve.

[67] The contrast between Figures 9a and 9b is that in Figure 9a the thicker sediment patches coalesce downstream. As the model runs, small patches of thicker sediment cover spontaneously develop at the upstream end of the model. These patches grow as they move downstream, so that the downstream end of the model all has denser sediment cover. This change is because the thicker sediment cover is more stable than the sparser sediment cover. The conversion from sparse to thick sediment cover only requires a small number of grains to be clustered together and hence acquire probability of entrainment Pc. They are then less mobile and so accumulate on the bed, creating a thicker sediment layer. Grains subsequently deposited into and around this area are more likely to become Pc grains, increasing the size of the cluster until it covers the model domain. The converse transition from clustered grains to sparse grains is far less likely under constant Qs/Qt.

[68] That some values of Fe are more stable than others is also illustrated in Figures 10c and 10d. Model runs with different initial sediment covers initially equilibrate at different values of Fe (these runs are in equilibrium because Qs_out/Qs = 1), before all converging to the same, more stable, value of Fe. In another example, all runs in Figure 8d evolve to the same value of Fe, despite Figure 11b predicting three possible steady state sediment covers.

[69] Analysis of the relationships between grain density and sediment flux has demonstrated that under a limited range of parameter values (as outlined in Figure 11a), the CA model can evolve to different equilibrium values of Fe for the same value of Qs/Qt. Furthermore, these can occur within the same model domain. However, not all these values of Fe are equally stable, suggesting that the value of Fe that a model run will achieve cannot always be predicted from a relationship like those in Figure 11b.

4.4. Interpretation of the Parameter Space

[70] Analysis of the CA model results has demonstrated how the parameterization of Pi and Pc affects the form of the relationship between Qs/Qt and Fe, determines the possible number of equilibrium values of Fe and thus influences model sensitivity to initial sediment cover. The CA model results also reproduce the predictions of Turowski et al. [2007] and the findings of Chatanantavet and Parker [2008] (e.g., Figures 2a and 7b). Furthermore, the observed model behaviors have been explained in the context of the dynamics of and interactions between individual grains.

[71] However, the key model parameters, Pi and Pc, are typically unknown in both field and laboratory settings. Ancey et al. [2008] measured entrainment probabilities for grains from an alluvial bed in a grain-width flume, data which is not readily transferable to the bedrock case. Field measurements such as Hodge et al. [2011] only capture the probability of entrainment at the temporal scale of one or more flow events. To relate the model results to empirical data, it is necessary to consider how Pi and Pc are likely to relate to the parameters that are typically identified as controlling sediment transport, i.e., channel slope, roughness and shear stress.

[72] Values of Pi and Pc represent the balance between the entraining forces of the flow and the resisting forces of the grain. In a channel with both bedrock and alluvial patches, Pi and Pc respectively characterize grains on bedrock and alluvial surfaces. The difference between Pi and Pc is because of the effect of the different surfaces on grain exposure, pivot angles, packing, the local flow profile and turbulence. These properties are in turn described by parameters including slope, shear stress and channel roughness, which are the controlling parameters identified by Chatanantavet and Parker [2008].

[73] Based on the above reasoning, we consider systematic variation in parameter values in the flume experiments of Chatanantavet and Parker [2008] to be paths through the parameter space of Figure 6. In one of their experiments, channel slope (S) is increased between runs (Figure 2a). Increasing S will increase the shear stress (τ) applied to the channel bed. Theoretical analysis of grain entrainment from bedrock and alluvial surfaces [Hodge et al., 2011] demonstrated that critical entrainment shear stress (τc) is an order of magnitude lower on bedrock surfaces than on alluvial surfaces. As S and hence τ increase, both Pi and Pc will increase. However, Pi is likely to increase faster both because of the larger excess shear stress (ττc) on bedrock surfaces and because grains on a bedrock surface tend to have a smaller range of τc. The results in Figure 2a can therefore be considered to represent an approximately vertical line though the parameter space of Figure 6, explaining the similarity between Figures 2a and 7b.

[74] A potential caveat to this explanation is that Chatanantavet and Parker [2008] only observed sigmoidal relationships when there was initial sediment cover, otherwise runway alluviation occurred. Runaway alluviation is observed in the CA model, but only with Pc = 0.05 and high Pi. This could indicate that all Chatanantavet and Parker's [2008] runs fall within this narrow parameter range. Critical entrainment shear stress on bedrock surfaces was estimated to be ∼0.1 of those on alluvial surfaces [Hodge et al., 2011]; consequently such a large difference between Pi and Pc may readily occur. Alternatively, runs with and without initial sediment cover could have different values of Pi and Pc; the initial sediment cover would alter the bed roughness, changing the flow and hence the values of Pi and Pc, meaning that Chatanantavet and Parker's [2008] runs would fall within a wider area of the parameter space in Figure 6.

[75] Increasing bedrock roughness is likely to represent a vertical path through the parameter space in Figure 6. At low roughness, individual grains will be exposed to the flow and thus are more likely to be entrained, representing a high value of Pi. If roughness is increased, Pi will decrease as roughness elements shelter grains. The effect on Pc of changing roughness will be less because in both the rough and smooth cases the grain is sheltered by surrounding sediment grains. This interpretation is compatible with the results of Demeter et al. [2005], who observed a strongly sigmoidal relationship between Fe and Qs/Qt on a smooth bed, but a more gradual relationship between Fe and Qs/Qt on a rough bed.

4.5. Parameters Not Implemented in the CA Model

[76] The CA model is a reduced-complexity approximation of the transport of individual grains in a bedrock river. The model reproduces and provides explanations for the key findings of Chatanantavet and Parker's [2008] flume experiments, and therefore may contain sufficient process representation. However, properties currently not included in the CA model might also be expected to influence grain scale processes.

[77] The relationship between sediment flux and sediment cover can be considered a sediment continuity problem, which has implications for the nature of sediment transport in bedrock streams, including the degree of size-selectivity in this process [Hodge et al., 2011]. The model uses a single grain size and so no size-specificity of sediment processes is enabled. Implementation of different grain sizes and size-specific rules could affect the cover dynamics through the potential for size-selective entrainment, vertical sorting and downstream fining [Chatanantavet et al., 2010]. The extent to which sediment processes are size specific in bedrock rivers may also depend on the amount of sediment cover. Tracer studies [Hodge et al., 2011] demonstrate that in a bedrock river with 20% sediment cover, grain entrainment and transport are largely size-independent. At 80% sediment cover, transport distances are a significant inverse function of grain size. Bedrock topography can further complicate these relationships [Goode and Wohl, 2010].

[78] The CA model only includes spatial variation in the probability of grain entrainment caused by the effect of neighboring grains, and even then only in a binary parameterization. The model thus omits the effects of other spatially variable factors, including bedrock topography, immobile boulders and bed form development. In a bedrock river, local variations in roughness and shear stress will result in grains being more likely to be entrained from certain areas of the bed than from others. This will result in a spatial pattern of sediment cover, which is likely to be relatively static over time (i.e., not migrate downstream like the sediment cover in Figure 9a). Bedrock river beds will be likely to have sediment cover that consists of stable elements reflecting mesoscale roughness, and transient elements reflecting the interaction of sediment supply and transport processes.

[79] Extreme spatial variations in the probability of grain entrainment could further alter the relationship between Qs/Qt and Fe. For example, in the case of a bed with an incised inner channel, most sediment will be routed through the inner channel, resulting in approximately constant values of Fe for a range of values of Qs/Qt [Johnson and Whipple, 2007; Finnegan et al., 2007]. The results from the CA models, and indeed many flume experiments of Chatanantavet and Parker [2008] therefore may not be directly applicable to cases with extreme channel topography.

5. Implications for Modeling Bedrock River Incision

[80] Application of the CA model shows that different relationships between Qs/Qt and Fe are generated under different circumstances. Furthermore, in cases with high Pi and low Pc, multiple steady state values of Fe may exist; as a result, equilibrium sediment cover is sensitive to the initial sediment cover, in a sometimes unpredictable way. There is therefore no universally applicable relationship between Qs/Qt and Fe; instead the relationship between Qs/Qt and Fe changes in a complicated but structured way across the Pi and Pc parameter space.

[81] The linear and negative exponential relationships proposed by Sklar and Dietrich [2004], Turowski et al. [2007] and Turowski [2009] only occur within a small range of the parameter space in which PiPc (Figure 6). Furthermore, theoretical and field data [Hodge et al., 2011] indicate that in bedrock rivers, Pi is likely to be greater than Pc, suggesting that other forms of Fe and Qs/Qt relationships are more probable. A sigmoidal relationship between Qs/Qt and Fe is most commonly produced by the CA model and therefore may be the optimum choice in the absence of additional information. However, predictions of bedrock incision made using a sigmoidal relationship will be sensitive to the value of Qs/Qt at which Fe decreases from one to zero. This is because once Fe = 0, no bedrock incision will occur. As a result, different sigmoidal relationships may predict different incision rates and hence different patterns of landscape evolution.

[82] To determine the most applicable relationship between Fe and Qs/Qt, it is necessary to place real rivers within the parameter space of Figure 6, and doing this depends on factors including channel slope, shear stress and overall channel roughness. Furthermore, the CA model results indicate that the form of the relationship between Fe and Qs/Qt may vary spatially within a catchment as a function of these determining factors. The relationship may also vary temporally as a result of hydraulic and sediment supply changes. It is noted that in cases with extreme variability in sediment supply this variability can outweigh the influence of the relationship between Fe and Qs/Qt [Lague, 2010]. Understanding the variability in the relationship between Fe and Qs/Qt will benefit the development of landscape evolution models with improved process representation of bedrock incision [e.g., Gasparini et al., 2007].

6. Conclusions

[83] The relationship between bedrock exposure (Fe) and relative sediment flux (Qs/Qt) in a bedrock river is a key determinant of the rate of bedrock incision. However, the form of this relationship is poorly constrained, with conflicting theoretical predictions [Sklar and Dietrich, 2004; Turowski et al., 2007; Turowski, 2009] and laboratory data [Chatanantavet and Parker, 2008].

[84] We have used a cellular automaton (CA) model to quantify how this relationship evolves from the dynamics of and interactions between individual grains. The model behavior is governed by the probability of entrainment of individual grains. By altering this probability as a function of the number of neighboring grains (Pi and Pc), empirical observations of contrasting grain mobility on bedrock and alluvial surfaces [Hodge et al., 2011] are incorporated into the model.

[85] The CA model results show systematic variation in the relationship between Fe and Qs/Qt across the parameter space of Pi and Pc. Linear and negative exponential relationships between Fe and Qs/Qt are produced when PiPc. The sigmoidal relationships and runaway alluviation observed by Chatanantavet and Parker [2008] are only reproduced when Pi > Pc, suggesting that the difference in grain dynamics on bedrock and alluvial surfaces is a key factor in the development of sediment cover. Further research is needed to identify how channel-scale parameters including slope, roughness and shear stress map onto the parameter space of Pi and Pc.

[86] The development of transient landscapes in landscape evolution models is sensitive to the form of the bedrock incision model that is employed [Gasparini et al., 2007]. Furthermore, an important component of saltation-driven incision models is the relationship between Fe and Qs/Qt. We suggest that the simple relationships between Fe and Qs/Qt typically applied to bedrock incision are not universally applicable. There is a need to quantify the factors that control the form of this relationship, hence enabling improved representation of the relationship in landscape evolution models.

Acknowledgments

[87] Thanks to Phairot Chatanantavet, Jens Turowski, an anonymous reviewer and the AE Dimitri Lague for reviews that helped to improve an earlier version of this paper.