Bedload sediment transport is one of the main processes that contribute to
bedrock incision in a river and is therefore one of the key control
parameters in the evolution of mountainous landscapes. In recent years, many
studies have addressed this issue through experimental setups, direct
measurements in the field, or various analytical models. In this article, we
present a new direct numerical approach: using the classical methods of
discrete-element simulations applied to granular materials, we explicitly compute
the trajectories of a number of pebbles entrained by a turbulent
water stream over a rough solid surface. This method allows us to extract
quantitatively the amount of energy that successive impacts of pebbles
deliver to the bedrock, as a function of both the amount of sediment
available and the Shields number. We show that we reproduce qualitatively the
behaviour observed experimentally by

The incision of bedrock channels is one of the key processes that govern the
formation and evolution of mountain ranges

In a bedrock mountain river, various processes contribute to incision:
chemical dissolution, cavitation, abrasion (or wear) by both bedload and
suspended load, plucking, and macroabrasion

The first direct measurement of the effect of sediment transport on abrasion
was performed by

In this article, we propose a new numerical approach of abrasion, based on
the discrete-element method. This method allows us to model the individual
trajectories of all particles within the bedload layer and therefore to
obtain a physically based value of the amount of energy transmitted to the
bedrock by impacts. The article in organized as follows. In Sect. 2 we
present our numerical setup and the physical laws implemented in our
simulations. In Sect. 3 we expose the numerical results regarding the
sediment transport rate, the energy delivered to the bedrock and the
influence of bedrock roughness. Finally, in Sect. 4 we discuss the
implications of our results on the influence of both the Shields number and
the sediment supply on the incision rate. We propose a new definition for the
cover function

We use the discrete-element method to simulate the individual dynamics of pebbles entrained by a turbulent water flow over a fixed bedrock. The same method would allow for modelling of non-spherical particles by considering composite particles made of two or more “glued” spheres, but for the sake of simplicity and to limit the number of control parameters, we restrict our study to the dynamics of spherical particles.

The computational domain is a parallelepipedic box of length

Snapshot of the numerical simulation. The colour scale codes the horizontal velocity of each pebble. The grey plane and (immobile) grey spheres constitute the rough bedrock.

List of the physical parameters used in the model.

The bedload consists of

A horizontal turbulent water flow in the

When two pebbles are in contact, the exact deformation of each solid particle
is not explicitly computed but spheres are instead allowed to overlap
slightly (see, for instance,

Two pebbles in contact, located respectively at

The tangential force

Finally, each collision between a pebble and the horizontal surface of
altitude

A stationary turbulent flow over a rough bedrock follows the
average velocity profile:

If the bedrock is covered with a layer of mobile pebbles, as in our
simulations, the turbulent velocity profile is modified. Recently,

The turbulent flow exerts on each mobile pebble a drag force given by

Pebbles are initially disposed on a regular lattice at a height

The time step used in the simulation is

Within the alluvial cover, some pebbles are almost immobile, either because
they got trapped by the bedrock roughness or because they belong to bottom
layers of the cover and are therefore not entrained by the water flow. These
pebbles constitute a static cover that contributes to protecting the bedrock
from rapid impacts by saltating pebbles. We quantify the cover fraction in
the following way: the bedrock surface is divided into square cells of side

As are collisions between mobile particles, each impact of a pebble on the
bedrock is inelastic: the impacting pebble loses a fraction of its incipient
kinetic energy during the collision (due to the dissipative term in
Eq.

Let us first investigate the structure and dynamics of the
bedload layer. Figure

Flux of sediment

Transport of sediment only occurs if the fluid drag force on a pebble is
large enough to overcome solid friction, that is, if the Shields number
exceeds a critical value

Average flux of sediment

In Fig.

Solid volume fraction and velocity of the fluid and the pebbles as a
function of height for a Shields number

Let us now investigate the transport law by varying the Shields number

Flux of sediment as a function of the relative excess shear stress

Let us now focus on the effect of the sediment supply on the bedload flux. In
Fig.

Figure

Saturated value

In Fig.

Static cover fraction as a function of the sediment supply, for
different values of the Shields number. The dashed line represents the
function

The flux of energy that is delivered to the bedrock by the impacts is given
by the work of the dissipative normal force during each collision between a
mobile pebble and the bedrock (whether it is the flat surface or one of the
glued spheres). In Fig.

Let us now investigate more precisely the effect of the sediment supply on
the energy transfer, which is plotted in Fig.

The inset in Fig.

The roughness of the bedrock can be modified by varying two
parameters: the surface density

In order to study the variation in energy transfer with the bedrock
roughness, we plot the flux of energy delivered to the bedrock with respect
to the sediment supply (

Flux of energy delivered to the bedrock as a function of the
dimensionless sediment supply. The energy transfer increases when the
sediment supply increases in the range

Two different cases of bedrock roughness: the positions of the glued
spheres are plotted as seen from above. Left:

Flux of energy delivered to the bedrock as a function of the
sediment supply for

We plot on the same graph our numerical prediction for the flux of
energy delivered to the bedrock (full circles), erosion rates measured
experimentally by

Empty symbols: cover function

Although the model that we adopt for the interaction between the water flow
and the pebbles is rather simple, the dynamics of the bedload layer appears
to be consistent with experimental observations: as shown by
Fig.

As shown in Fig.

We also quantified the influence of the Shields number on the abrasion
process and showed a power-law dependency of

Our results also show that abrasion only occurs beyond a given threshold
which is higher than the threshold of motion of pebbles, which can be
explained by the fact that rolling or sliding pebbles do not contribute
significantly to the erosion of bedrock. This is inconsistent, however, with
observations by

Our simulations indicate that the roughness of the bedrock does not affect
the general evolution of the energy delivered to the bed with respect to the
cover fraction. However, incision appears to be enhanced if the surface
density of asperities is low and if they are not too high. These two effects
can be related to the geometrical explanation of the cover effect: if the
roughness is denser or higher, mobile pebbles are more likely to get trapped
and immobilized along the bedrock, therefore protecting it from further
impacts by rapid pebbles. This enhanced cover effect will disappear if the
roughness density

Finally, let us remark that the influence of the coefficient of restitution
on the results of our simulations should be of importance and will be the
object of further investigation. Increasing the coefficient of restitution
would certainly facilitate the saltating motion of pebbles, whereas they only
roll along the bedrock at low

By analogy with both the linear

Coefficients used in the empirical fit of the flux of energy as a
function of sediment supply (see Eq.

Following the approach for incision rate by

Cover factor

We can estimate the rate of incision induced by the impacts on the bedrock,
based on the flux of energy delivered. Following

We have presented the results of a new model for incision of a river bedrock based on the direct simulation of physically based trajectories of pebbles in a stream. In this model we solved the equations of motion for a large number of pebbles entrained by a turbulent water flow, with a simplified retroaction of the presence of the pebbles on the flow. This allowed us to explicitly compute the trajectories of pebbles transported by the flow, and therefore to quantify the energy dissipated during collisions between the bedload and the bedrock, which is directly responsible for the incision of the bedrock. We found that the sediment transport rate can be fitted by a power law of the Shields number, similar to most classical transport laws at saturation. However, we also evidenced the influence of the sediment supply: the exponent of the transport law increases with the quantity of available pebbles. For a given Shields number, we showed that the bedload flux increases with the sediment supply until it reaches its saturated value. This allowed us to compute the sediment mass that the flow is able to transport. However, extracting a unique general expression for the flux of sediment as a function of both the Shields number and the sediment supply remains non-trivial.

The amount of energy that impacts of saltating pebbles deliver to the bedrock
can be directly computed from the simulation data. This flux of energy, which
is expected to be proportional to the incision rate, shows the same
qualitative variations with sediment supply as observed in experiments by

Though our results are qualitatively consistent with experimental
observations

Finally, let us note that in the prediction of the long-term evolution of a
river bed (see, for example,

Data supporting the results of this article can be obtained by contacting Vincent Langlois (vincent.langlois@univ-lyon1.fr).

The authors would like to acknowledge the insightful input of four reviewers, amongst whom P. Chatanantavet, on an earlier version of this article. Edited by: E. Lajeunesse