The 1-D saltation–abrasion model of channel bedrock incision of Sklar and Dietrich (2004), in which the erosion rate is buffered by the surface area fraction of bedrock covered by alluvium, was a major advance over models that treat river erosion as a function of bed slope and drainage area. Their model is, however, limited because it calculates bed cover in terms of bedload sediment supply rather than local bedload transport. It implicitly assumes that as sediment supply from upstream changes, the transport rate adjusts instantaneously everywhere downstream to match. This assumption is not valid in general, and thus can give rise to unphysical consequences. Here we present a unified morphodynamic formulation of both channel incision and alluviation that specifically tracks the spatiotemporal variation in both bedload transport and alluvial thickness. It does so by relating the bedrock cover fraction to the ratio of alluvium thickness to bedrock macro-roughness, rather than to the ratio of bedload supply rate to capacity bedload transport. The new formulation (MRSAA) predicts waves of alluviation and rarification, in addition to bedrock erosion. Embedded in it are three physical processes: alluvial diffusion, fast downstream advection of alluvial disturbances, and slow upstream migration of incisional disturbances. Solutions of this formulation over a fixed bed are used to demonstrate the stripping of an initial alluvial cover, the emplacement of alluvial cover over an initially bare bed and the advection–diffusion of a sediment pulse over an alluvial bed. A solution for alluvial–incisional interaction in a channel with a basement undergoing net rock uplift shows how an impulsive increase in sediment supply can quickly and completely bury the bedrock under thick alluvium, thus blocking bedrock erosion. As the river responds to rock uplift or base level fall, the transition point separating an alluvial reach upstream from an alluvial–bedrock reach downstream migrates upstream in the form of a “hidden knickpoint”. A tectonically more complex case of rock uplift subject to a localized zone of subsidence (graben) yields a steady-state solution that is not attainable with the original saltation–abrasion model. A solution for the case of bedrock–alluvial coevolution upstream of an alluviated river mouth illustrates how the bedrock surface can be progressively buried not far below the alluvium. Because the model tracks the spatiotemporal variation in both bedload transport and alluvial thickness, it is applicable to the study of the incisional response of a river subject to temporally varying sediment supply. It thus has the potential to capture the response of an alluvial–bedrock river to massive impulsive sediment inputs associated with landslides or debris flows.

The pace of river-dominated landscape evolution is set by the rate of downcutting into bedrock across the channel network. The coupled process of river incision and hillslope response is both self-promoting and self-limiting (Gilbert, 1877). Although there are multiple processes that can lead to erosion into bedrock, we here focus on incision driven by abrasion of a bedrock surface as moving particles collide with it. Low rates of incision entail some sediment supply from upstream hillslopes, which provides a modicum of abrasive material in river flows that further facilitates bedrock channel erosion. Faster downcutting leads to higher rates of hillslope sediment supply, boosting the concentration of erosion “tools” and bedrock wear rates, but also leading to greater cover of the bedrock bed with sediment (Sklar and Dietrich, 2001, 2004, 2006; Turowski et al., 2007; Lamb et al., 2008; Turowski, 2009). Too much sediment supply leads to choking of the channels by alluvial cover and the retardation of further channel erosion (e.g., Stark et al., 2009). This competition between incision and sedimentation leads long-term eroding channels to typically take a mixed bedrock–alluvial form in which the pattern and depth of sediment cover fluctuate over time in apposition to the pattern of bedrock wear.

Theoretical approaches to treating the erosion of bedrock rivers have shifted over recent decades (see Turowski, 2012, for a recent review). The pioneering work of Howard and Kerby (1983) focused on bedrock channels with little sediment cover; it led to the detachment-limited model of Howard et al. (1994), in which channel erosion is treated as a power function of river slope and characteristic discharge, and the “stream-power-law” approach, in which the power-law scaling of channel slope with upstream area underpins the way in which landscapes are thought to evolve (Whipple and Tucker, 1999; Whipple, 2004; Howard, 1971, foreshadows this approach). At the other extreme, sediment flux came into play in the transport-limited treatment of mass removal from channels of, for example, Smith and Bretherton (1972), in which no bedrock is present in the channel and where the divergence of sediment flux determines the rate of lowering. Whipple and Tucker (2002) blended these approaches, and imagined a transition from detachment limitation upstream to transport-limited behavior downstream. They also discussed, in the context of the stream-power-law approach, the idea emerging at that time (Sklar and Dietrich, 1998) of a “parabolic” form of the rate of bedrock wear as a function of sediment flux normalized by transport capacity. Laboratory experiments conducted by Sklar and Dietrich (2001) corroborated this idea, and they led to the first true sediment flux-dependent model of channel erosion of Sklar and Dietrich (2004, 2006). This saltation–abrasion model was subsequently extended by Lamb et al. (2008) and Chatanantavet and Parker (2009). It was explored experimentally by Chatanantavet and Parker (2008) and Chatanantavet et al. (2013); evaluated in a field context by Johnson et al. (2009), Chatanantavet and Parker (2009), Hobley et al. (2011) and Turowski et al. (2013); adapted to treat alluvial intermittency by Lague (2010); and given a stochastic treatment by Turowski et al. (2007), Turowski (2009) and Lague (2010), the latter of whom introduced several new elements. Howard (1998) presents an alternative formulation for incision that relates bedrock wear to the thickness of alluvial cover rather than sediment supply, in a form that can be thought to be a predecessor of the present work.

At the heart of their saltation–abrasion model lies the idea of a cover
factor

The saltation–abrasion model is considerably more sophisticated and flexible
(Sklar and Dietrich, 2004, 2006) than this sketch explanation can encompass.
It does, however, have three major restrictions. First, it is formulated in
terms of sediment supply rather than local sediment transport. The model is
thus unable to capture the interaction between processes that drive evolution
of an alluvial bed and those that drive the evolution of an incising of
bedrock–alluvial bed. Second, for related reasons, it cannot account for
bedrock topography significant enough to affect the pattern of sediment
storage and rock exposure. Such a topography is illustrated in
Fig.

Here we address all three of these issues in a model that allows both alluvial and incisional processes to interact and coevolve. We do this by relating the cover factor geometrically to a measure of the vertical scale of elevation fluctuations of the bedrock topography, here called macro-roughness, rather than to the ratio of sediment supply rate to capacity sediment transport rate. Our model encompasses downstream-advecting alluvial behavior (e.g., waves of alluvium), diffusive alluvial behavior and upstream-advecting incisional behavior (e.g., knickpoint migration). In order to distinguish between the model of Sklar and Dietrich (2004, 2006) and the present model, we refer to the former as the CSA (Capacity-based Saltation-Abrasion) model, and the latter as the MRSAA (Macro-Roughness-based Saltation-Abrasion-Alluviation) model. We point out here that the first and third issues indicated above have also been addressed by Lague (2010), although in a substantially different way than presented here. The notation used in this paper is defined in Table A1.

Views of the Shimanto River, a mixed alluvial–bedrock river in
Shikoku, Japan.

Sklar and Dietrich (2004, 2006) present the following model, referred to here
as the Capacity-based Saltation-Abrasion (CSA) model, for bedrock incision in
mixed bedrock–alluvial rivers transporting gravel. Defining

In the above formulation, it is assumed that the gravel transport rate

Before introducing the relation of Sklar and Dietrich (2006) for abrasion
coefficient

The relations of Sklar and Dietrich (2004, 2006) to compute

The relations above define a 0-D formulation. It must be augmented with other parameters and relations, including channel width, relations for hydraulics, quantification of flow discharge or flow duration curve, etc., to allow application at the river reach scale.

It is useful to cast Eq. (

A relation for the evolution of bedrock surface elevation

The MRSAA model (introduced below) has several new features as compared to
CSA. These are best illustrated by first characterizing the mathematical
nature of CSA in the context of Eq. (

Any solution of Eqs. (

The CSA model (Sklar and Dietrich, 2004, 2006) was a major advance in the
analysis of bedrock incision due to abrasion because it (a) accounts for the
effect of alluvial cover and tool availability on the incision rate through
the term

The model does, however, have a significant limitation in that it
specifically does not include either alluvial morphodynamics or the
morphodynamics of transitions between bedrock and alluvial zones. Here we
study this limitation, and how to overcome it, in terms of the highly
simplified configuration of a reach (HSR, highly simplified reach) with
constant width; fixed, non-erodible banks; constant water discharge; and
sediment input only from the upstream end. For simplicity, we also neglect
abrasion of the gravel itself, so that grain size

In the CSA model, the bedload transport rate

We illustrate this behavior in Fig.

In a more realistic model, the effect of a change in bedload feed rate

Schematic diagram illustrating downstream modification of
a sedimentograph. At the upstream feed point (

A second limitation concerns alluviation of the bedrock surface. Consider
a wave of sediment moving over this surface, as shown in Fig.

Bed elevation

The goal of this paper is the development and implementation of a model that overcomes these limitations by (a) capturing the spatiotemporal coevolution of the sediment transport rate, alluvial cover thickness and bedrock incision rate, and (b) explicitly enabling spatiotemporally evolving transitions between bedrock–alluvial morphodynamics and purely alluvial morphodynamics. The form of the model presented here is simplified in terms of the HSR outlined above, including a constant-width channel and a single sediment source upstream.

Schematic diagram illustrating the propagation of a wave of sediment
over bedrock (orange line shifting to dashed orange line over time). Here

The geomorphic incision law of the MRSAA model is identical to that of CSA,
i.e., Eq. (

The specific case we consider here is one for which (a) the bedrock surface
is rough in a hydraulic sense (as opposed to a hydraulically smooth or
transitional surface; see Schlichting, 1979), and (b) the characteristic
vertical scale of bedrock elevation fluctuation about a mean value based on
an appropriately defined window, here denoted as the macro-roughness

Illustration of the statistical structure or local hypsometry of the
bedrock surface topography (dark-grey line). Here

We formulate the problem by considering a conservation equation for the alluvium, in standard Exner form, appropriately adapted to include below-capacity transport over a non-erodible surface. The first model of this kind is due to Struiksma (1999), and further progress has been made by Parker et al. (2009, 2013), Izumi and Yokokawa (2011), Izumi et al. (2012), Tanaka and Izumi (2013) and Zhang et al. (2013). These models are expressed in continuous form; Lague (2010) presents a discrete version based on a series of reaches of finite length that allows for generalization to a continuous form.

None of the above models is specifically designed to handle the clast-rough
case, in particular that shown in Fig.

In such a statistical formulation, bedrock relief has neither a precise
“bottom” nor a precise “top”. Rather, the “bottom” and “top” of the
bedrock topography, as well as the macro-roughness

Let

The problem can now be rephrased in terms of a vertical coordinate

Schematic diagram for derivation of the Exner equation of sediment
continuity over a bedrock surface (dark-grey line). As in Figs. (

The alluvial sediment is taken to have constant porosity

For the case of sediment of constant density, the Exner equation for mass
balance of alluvial sediment can be expressed as

The combination of Eqs. (

In the present formulation, the cover fraction

Illustration of the MRSAA model relation between areal fraction of
alluvial cover of bedrock

Note that the cover relation of Fig.

In applying the MRSAA model to general cases, it is useful to delineate the
simplest functional form for the closure relation for cover fraction that
satisfies the constraints of Eq. (

The form for the derivative of Eq. (

The formulation presented here has an obvious limitation. Since it is a 1-D
expression of sediment conservation over a bedrock surface, it cannot capture
2-D variation, which will result in a more complex pattern than that shown in
Fig.

Sections

Equation (

It is important to realize that alluvial wave speed

It is of interest to inquire as to how the model would behave if the
clast-rough condition, i.e.,

The form of Eq. (

The full MRSAA model consists of the kinematic wave equation with a source
term Eq. (

In MRSAA, then, the spatiotemporal variation in the cover fraction

The flow model, and in particular Eqs. (

In the numerical analysis below, the actual equations used to solve for
morphodynamic evolution are not those of Sects.

In the restricted case of the highly simplified reach (HSR) configuration
constrained by (a) temporally constant, below-capacity sediment feed
(supply) rate

The steady-state form of Eq. (

Equation (

In solving for this steady state, and in subsequent calculations, we use the
bedload transport relation of Wong and Parker (2006a), a development and
correction of the semi-empirical relation of Meyer-Peter and Müller (1948),
rather than the similar formulation of Fernandez Luque and van Beek (1976);
in the case of the former,

In the case of a specified constant abrasion coefficient

We performed calculations for conditions loosely based on (a) field
estimates for a reach of the bedrock Shimanto River near Tokawa, Japan
(Fig.

Two sediment feed rates were considered. The high feed rate was set at
3.5

The value

Variation at steady state (black curves) of

For the high feed, predicted relations for a steady-state abrasion
coefficient

These results require interpretation. It can be seen from
Eqs. (

The results for the low feed rate are very similar. The values for variable
steady-state abrasion coefficient

Variation at steady state (black curves) of

The lack of dependence of steady-state bedrock slope

In order to compare the steady-state predictions of the slope–area
relation in Eq. (

The issue as to the values of

In their Table 1, Whipple and Tucker (2002) quote a range of values of

Normalized steady-state bedrock slope

One more difference between the CSA and slope–area formulations is worth
noting. If the slope–area relation is installed into Eq. (

Having conducted a fairly thorough analysis of the steady state common to the
CSA and MRSAA models, it is now appropriate to move on to examples of
behavior that can be captured by the MRSAA model, but are not captured
by models that assume a relation for cover based on the ratio of sediment
supply to capacity transport rate, i.e., Eq. (

Let

In order to illustrate the essential features of the new formulation of the
MRSAA model for the morphodynamics of mixed bedrock–alluvial rivers, it is
useful to consider the most simplified case that illustrates its expanded
capabilities compared to the CSA model. Here we implement the HSR
simplification. In addition, based on the results of the previous section, we
approximate

In the numerical solution of the differential Eqs. (

Three numerical solutions of the MRSAA model are studied here:
(a) stripping of an alluvial cover to bare bed, (b) emplacement of an
alluvial cover over a bare bed and (c) advection–diffusion of an alluvial
pulse over a bare bed. Reach length

None of these three cases can be treated using models that assume a relation
for cover based on the ratio of sediment supply to capacity transport rate,
i.e., Eq. (

The case of stripping of an initial alluvial layer to bare bedrock is
considered here. In this simulation, the bedload feed rate

Of interest in Fig.

In this simulation, the initial thickness of alluvium

MRSAA model solutions for

In this example the initial bed is bare of sediment. The sediment feed rate
is set equal to 0.0012 m

Here we consider three cases of channel profile evolution to steady state
that include both rock uplift and incision. In the first case, the initial
bedrock slope is set to a value below the steady-state value, and the
sediment feed rate is set to a value that is well above the steady-state
value for the initial bedrock slope, causing early-stage massive alluviation.
The configuration for the second case is a simplified version of a graben
with a horst upstream and a horst downstream. The configuration for the third
case is such that there is an alluviated river mouth downstream and
a bedrock–alluvial transition upstream. In all cases, MRSAA predicts
evolution that cannot be predicted by models that assume a relation for cover
based on the ratio of sediment supply to capacity transport rate, i.e., Eq. (

Here we set

Progression to steady state after an impulsive increase in sediment
supply:

Evolution predicted by the MRSAA model for localized subsidence at a
narrow graben superimposed on broader uplift. Note the bedrock–alluvial and
alluvial–bedrock transitions at the margins of the graben. By 15 kyr,
the bed top has reached steady state, even though the bedrock surface in the
graben continues to subside. The regional rock uplift rate and graben
subsidence rate are assumed constant for simplicity. Here

The results for the CSA model are shown in Fig.

Figure

In this example,

This case cannot be implemented in models that assume a relation for cover
based on the ratio of sediment supply to capacity transport rate,
i.e., Eq. (

In this example

The result of CSA for this case, with base level

CSA model evolution of an initial bedrock profile towards a
steady-state profile. Compare with the MRSAA model behavior in
Fig.

MRSAA is implemented with somewhat different initial and downstream boundary
conditions in order to model the case of a bed that remains alluviated at
the downstream end. This condition thus corresponds to an alluviated river
mouth. The initial bedrock slope is again 0.004, and the downstream bedrock
elevation

Figure

Figure

MRSAA model evolution of bed top and bedrock profiles with an
imposed alluvial river mouth at the downstream end and an upstream-migrating
bedrock–alluvial transition. The results are for

The MRSAA model is a direct descendant of the model of Sklar and Dietrich (2004) in terms of the formulation for bedrock incision, and the model of Struiksma (1999) in terms of the formulation of the conservation alluvium over a partly covered bedrock surface. In terms of its capabilities, however, it shares much in common with the previous work of Lague (2010), and in particular with his SSTRIM model. These include (a) the melding of incision and alluviation into a single model, (b) the inclusion of a cover relation that is based on geometric bed structure, and (c) the ability to track simultaneously the spatiotemporal variation in both incision rate and alluvial cover. Priority should accrue to Lague (2010) in regard to these features. The present model has the following advantages: (a) the Exner equation of sediment conservation is specifically based on a formulation of the statistics of partial and complete cover over a rough bedrock surface; (b) the formulation yields a specific relation for alluvial wave velocity as a function of cover, ranging to a maximum value for minimum cover to 0 for complete alluviation; and (c) it allows for explicit description of the nonlinear advective–diffusive physics of the problem in terms of an alluvial diffusivity and two wave celerities, one directed upstream and associated with bedrock incision, and one directed downstream and associated with alluviation.

The form of the MRSAA model presented here has been simplified as much as
possible, i.e., to treat a HSR (highly simplified reach) with constant grain
size

The MRSAA model presented here is applied to several 1-D cases with
spatiotemporal variation. The model can easily be generalized to 2-D simply
by expressing Eq. (

The MRSAA model in the form presented here has a weakness in that the flow
resistance coefficient

Because MRSAA tracks the spatiotemporal variation in both bedload transport and alluvial thickness, it is applicable to the study of the incisional response of a river subject to temporally varying sediment supply. It thus has the potential to capture the response of an alluvial–bedrock river to massive impulsive sediment inputs associated with landslides or debris flows. A preliminary example of such an extension is given in Zhang et al. (2013). When extended to multiple sediment sources, it can encompass both the short- and long-term responses of a bedrock–alluvial river to intermittent massive sediment supply due to landslides and debris flows. As such, it has the potential to be integrated into a framework for managing sediment disturbance in mountain rivers systems such as those affected by the 2008 Wenchuan earthquake in Sichuan, China. Over 200 landslide dams formed during that event (Xu et al., 2009; Fu et al., 2011). A similar potential application is the case of drastic sediment supply to, and evacuation from, rivers in Taiwan due to typhoon-induced or earthquake-induced landsliding (e.g., Yanites et al., 2010).

We present a 1-D model of alluvial transport and bedrock erosion in a river
channel whose bed may be purely alluvial, or mixed bedrock–alluvial, or may
transition freely between the two morphologies. Our model, which we call the
Macro-Roughness-based Saltation-Abrasion-Alluviation (MRSAA) model,
specifically tracks not only large-scale bedrock morphodynamics but also the
morphodynamics of the alluvium over it. The key results are as follows:

The transport of alluvium over a bedrock surface cannot in general be described simply by a supply rate that instantaneously affects the entire river reach downstream as it is varied in time. Here we track the alluvium in terms of a spatiotemporally varying alluvial thickness.

The area fraction of cover

The MRSAA model captures three processes: downstream alluvial advection at a fast timescale, alluvial diffusion, and upstream incisional advection at a slow timescale. Only the third of these processes is captured by models that assume a relation for cover based on the ratio of sediment supply to capacity transport rate rather than a measure of the thickness of alluvial cover itself. The CSA model can be thought of as a 0-D model that applies locally. The MRSAA model lends itself more directly to application to long 1-D reaches because it embeds the elements necessary to route sediment down the reach.

The MRSAA model reduces to the CSA model under the conditions of steady-state incision in balance with rock uplift and below-capacity cover. The steady-state bedrock slope predicted by both models is insensitive to the rock uplift rate over a wide range of conditions. This insensitivity is in marked contrast to the commonly used incision model in which the incision rate is a power function of bedrock slope and drainage area upstream. The two models can differ substantially under transient conditions, particularly under those that include migrating transitions between the bedrock–alluvial and purely alluvial state.

In the MRSAA model, inclusion of alluvial advection and diffusion lead to the following phenomena: (a) a wave-like stripping of antecedent alluvium over a bedrock surface in response to cessation of sediment supply, (b) advection–diffusional emplacement of a sediment cover over initially bare bedrock and (c) the propagation and deformation of a sediment pulse over a bedrock surface.

In the case of transient imbalance between rock uplift and
incision with a massive increase in sediment feed, MRSAA captures an
upstream-migrating transition between a purely alluvial reach upstream and
a bedrock–alluvial reach downstream (here abbreviated as a alluvial–bedrock
transition). The bedrock profile shows an upstream-migrating knickpoint, but
this knickpoint is hidden under alluvium. Models that assume a relation for
cover based on the ratio of sediment supply to capacity transport rate,
i.e., Eq. (

MRSAA captures the mixed incisional–alluvial evolution for the case of a simplified 1-D subsiding graben bounded by two uplifting horsts. It captures alluvial filling of the graben, and thus converges to a steady-state top-bed profile with a bedrock–alluvial transition at the upstream end of the graben and an alluvial–bedrock transition at the downstream end.

In the case studied here of an uplifting bedrock profile with an alluviated bed at the downstream end modeling a river mouth, MRSAA predicts an upstream-migrating bedrock–alluvial transition at which the bedrock undergoes a sharp transition from a higher to a lower slope. MRSAA further predicts a bedrock long profile under the alluvium that has the same slope as the top bed. It also predicts that the cover is thin, so that the purely alluvial reach is only barely so. The steady state for this case is purely alluvial.

The new MRSAA model provides an entry point for the study of how bedrock–alluvial rivers respond to occasional large, impulsive supplies of sediment from landslides and debris flows. It thus can provide a tool for forecasting river-sedimentation disasters associated with such events. An example application would be treatment of the aftereffects of the 2008 Wenchuan earthquake, which triggered massive alluviation and the formation of over 200 landslide dams.

Consider a clast or grain of size

The rate at which a grain strikes the bed per unit distance moved is

The incision rate of the bedrock

Notation.

Continued.

The participation of L. Zhang and X. Fu in this work was made possible by the National Natural Science Foundation of China (grant nos. 51379100 and 51039003). The participation of G. Parker was made possible in part by a grant from the US National Science Foundation (grant no. EAR-1124482) The participation of C. P. Stark was made possible in part by grants from the US National Science Foundation (grant nos. EAR-1148176, EAR-1124114 and CMMI-1331499). The participation of T. Inoue was made possible by support from the Hokkaido Regional Development Bureau. Edited by: T. Coulthard