Over thousands to millions of years, the landscape evolution is predicted by
models based on fluxes of eroded, transported and deposited material. The
laws describing these fluxes, corresponding to averages over many years, are
difficult to prove with the available data. On the other hand, sediment
dynamics are often tackled by studying the distribution of certain grain
properties in the field (e.g. heavy metals, detrital zircons,

Numerical models of landscape evolution have significantly improved our
understanding of relief dynamics by recasting competing theories within a
general framework

At the same time, there are many techniques to trace the provenance and
transport of rock fragments (clasts) and minerals. For example, detrital
zircons, heavy minerals or trace elements in sedimentary rocks and river
streams are routinely used to determine sedimentary provenance and/or
constrain the exhumation history of orogenic highlands

This type of model would be key to quantitatively link the statistics for
provenance tracers with erosion rates in a catchment. For example,

In order to develop such a model, we couple a landscape evolution model with
a clast dispersion model. The landscape evolution model is a modified version
of Cidre

After briefly reporting previous modelling approaches based on flux-particle duality, we present Cidre and the probabilities used to move clasts. Then we analyse clast movement in the restricted cases of hillslope diffusion and river transport. Finally, we discuss the potential applications of this model. They include the 3-D tracing of weathered material which initially motivated this modelling approach.

Models coupling fluxes and particles have been developed in other scientific
fields, in particular in fluid mechanics, and are known as smoothed particle
hydrodynamics (SPH) models

The modelling approach presented in this paper is different from these
published works in the sense that (1) particles are not used to estimate the
water or erosion fields, (2) the topography evolves over time in our model,
and (3) our modelling is 3-D instead of 2-D. Nevertheless, our model is
inspired by the coupling of the landscape evolution model Eros and sediment
particles

Cidre is a C++ code modelling the topography dynamics on a regular grid of
square cells. At the beginning of a time step, a specified volume of rain
falls. Cells are sorted by decreasing elevations. The propagation of water
and sediment is proceeded in cascade starting from the highest cell to ensure
mass conservation. A multiple-flow algorithm propagates the water flux

The elevation

The length

Illustration of erosion–deposition processes in Cidre.

In the following we establish equations for

Here we use a different approach where the elevation variation results from
the difference between a local detachment rate and a deposition rate using
Eqs. (

The detachment rate is proportional to the local gradient. However,
the deposition rate (

Despite these conceptual differences, Eqs. (

Comparison between the profile evolutions of the hill predicted by
Eqs. (

It would be difficult to experimentally verify Eqs. (

In the case of river processes, we describe here a simplified version of
material detachment (sediment or bedrock), although the detachment threshold
and the explicit expression of bed shear stress in particular can be included

For river processes, the flow width

Erosion for sediment is different from that of bedrock (Eqs.

Flowing water can erode lateral cells (Fig.

The sediment leaving a cell is spread in the same way as water (Fig.

Compared to previous published versions of Cidre

Once

A clast has a specified radius

A clast is detached (eroded) if its depth is shallower than the erosion
calculated over the time step on that cell (if

For a moving clast entering a cell, the probability that it will be deposited
is simply the ratio between the volumetric deposition flux and the volumetric
incoming flux

A clast may be detached but not leave the cell. This may occur if the clast
was at depth. Removing material above the clasts takes part of the time step,
so that the remaining time may prevent the clast from leaving the cell.
Furthermore, a big clast should have a lower probability to leave the cell
than a small one. In order to take these realities into account, a
probability to leave the cell is determined by

Note that the probability of deposition could also depend on clast size. This is not implemented here but we will return to this point in the discussion.

The erosion–deposition–transfer fluxes calculated from the deterministic
rules of Cidre can be viewed as the mean values for the distribution of the
clast radius

The initial setup of the 3-D landscape model is an initial elevation grid, a grid or uniform value for the uplift-subsidence rates, a grid or uniform value for the precipitation rates, a geological model, and the boundary conditions. The geological model consists of different erodible layers. Their thickness is specified on each pixel. Values for the “erodibility” parameters (Eqs. 5, 6 and 7) are attributed to these different materials. Sediment erodibility (resulting from either the deposition of physically detached material or in situ regolith production by bedrock weathering) can be set differently from that of the bedrock. Nevertheless, in the experiments presented in this manuscript, there is only one bedrock type.

In addition, the clasts are initially listed in an input file specifying their initial location in the grid, as well as their depth, radius and mineralogy. There is no limitation to their number except for the one imposed by the computational times. Their initial localisation can be chosen according to a specific goal. For example, they can be grouped within one pixel to follow their transport from a specified source, or spread randomly in the model grid to study the mean transport rate of the sediment particles at the catchment scale. The distribution of the initial clast size can be freely chosen to trace one particular grain size, or a distribution of the grain sizes.

One way to validate the above model is to demonstrate that the displacement
of a clast population follows predictions in simple cases. We begin with the
case of linear diffusion. In Cidre, linear diffusion is obtained by using
Eqs. (

We consider an inclined plane of slope

These predictions are tested on a bedrock plan of

Figure

As predicted by the diffusion theory, the scattering of the clasts

Model test in the linear diffusion case (

In all cases,

In this version, we prefer probability law Eq. (

Here we illustrate the scattering of the clasts in the case of “non-linear”
diffusion. We use Eqs. (

Figure

River transport usually implies the formation of incisions, local depositions
and lateral movements of sediment by bank erosion. The movement of clasts
associated with this dynamic is an active research field

The mean travel velocity of a clast

Model test in the non-linear diffusion case (

We use larger clasts with a radius of 0.05 m in order to have a transport
rate of the same order of magnitude as in the diffusion case. Figure

We present here an illustration of clasts moving in a mountain–foreland context. Our goal is not to precisely analyse the clast dynamics in that case but to qualitatively describe a possible situation in a real-case scenario.

Model test in the river case (

Example of clasts exhumation from two locations (intrusive body)
located initially at a depth of 0.5 km (green clasts) and 0.4 km (red
clasts). The final maximum elevation for the mountain is 3000 m. The domain
size is 100 km

The mountain–foreland system consists of an uplifting block (the mountain)
and a stable domain where sediment eroded from the block is deposited or in
transit. The grid is 200

The experiment begins from a flat topography with small random elevations
(

If these grains were detrital zircons of different ages, the analysis of their age distribution at different places would allow the mixed zone to be mapped and, thus, the dynamics of the lateral alluvial fan to be reconstructed.

The modelling approach described in this paper and its developments may have different applications, which we propose in the following.

River dynamics involve processes acting on a large range of time periods,
from hours in the case of catastrophic flooding to thousands of years to
transport huge volumes of glacier sediment, for example. Determining
simplified laws to predict this complexity remains a challenge

Fining of the grain size by attrition in very steep catchments and by
selective deposition in most of the other catchments is a well-observed
phenomenon.

In the real world, material erosion and transport depend on the clast size
and the grain size evolves spatially. The feedback of grain size on the
landscape dynamics has been little explored

Provenance studies on detrital grains help constrain the chronology of the
exhumation of the sediment source

Placer-type deposits are secondary ores that can contain free particles with
very fine gold and other native metals (e.g. platinum-group elements).
The occurrence of gold grains in supergene environments, such as soil,
sediments and placers in rivers, is controlled by physical (as well as
bio-geochemical) processes of redistribution from a distant gold–quartz vein

The analysis and modelling of cosmogenic nuclide concentration in individual
clasts give quantitative information about the erosion–transport processes at
the landscape scale

In the 1990s,

The algorithm predicts a consistent clast velocity and surface erosion rate
in simple cases. The mean travel distance of the clasts does not depend on
the model cell size or time step. The scattering of the clasts depends on the
cell size and is overestimated. Nevertheless, decreasing the cell size
decreases the overestimation. This model has numerous potential applications
allowing field data on distinct grains to be linked to a large-scale
landscape evolution. The differences between the simple river and hillslope
cases illustrated here (e.g. Figures

We thank the two anonymous reviewers for constructive reviews. This paper is a contribution to the LMI COPEDIM (funding from IRD). S. Carretier thanks the Departamento de Geología of the Universidad de Chile for its welcome. M. Reich acknowledges funding by the MSI grant “Millennium Nucleus for Metal Tracing Along Subduction” (NC130065). Cidre sources are available upon request. The initial Cidre core was developed by B. Poisson at BRGM. This paper is dedicated to her memory. Edited by: S. Castelltort