Landscape evolution models (LEMs) aim to capture an aggregation of the processes of erosion and deposition within the earth's surface and predict the evolving topography. Over long timescales, i.e. greater than 1 million years, the computational cost is such that numerical resolution is coarse and all small-scale properties of the transport of material cannot be captured. A key aspect, therefore, of such a long timescale LEM is the algorithm chosen to route water down the surface. I explore the consequences of two end-member assumptions of how water flows over the surface of an LEM – either down a single flow direction (SFD) or down multiple flow directions (MFDs) – on model sediment flux and valley spacing. I find that by distributing flow along the edges of the mesh cells, node to node, the resolution dependence of the evolution of an LEM is significantly reduced. Furthermore, the flow paths of water predicted by this node-to-node MFD algorithm are significantly closer to those observed in nature. This reflects the observation that river channels are not necessarily fixed in space, and a distributive flow captures the sub-grid-scale processes that create non-steady flow paths. Likewise, drainage divides are not fixed in time. By comparing results between the distributive transport-limited LEM and the stream power model “Divide And Capture”, which was developed to capture the sub-grid migration of drainage divides, I find that in both cases the approximation for sub-grid-scale processes leads to resolution-independent valley spacing. I would, therefore, suggest that LEMs need to capture processes at a sub-grid-scale to accurately model the earth's surface over long timescales.

It is known that resolution impacts landscape evolution models (LEMs)

There is a potential problem with parameterizing the flow width to be fixed
at a sub grid level. The response time of LEMs to a change in external
forcing is strongly dependent on the surface run-off

Water is the primary agent of landscape erosion. There are multiple pathways
within the hydrological cycle from evaporation, transpiration, and groundwater flow; however, for many landscapes the river network is the primary
route through which water flows downslope. Mean river width varies from
5 km to a few metres

Distribution of mean river width taken from the Global River Widths
from Landsat (GRWL) Database

If the width of the flow path for run-off is narrower than can be reasonably
modelled, then can the flow paths be treated as lines, from model
node to node (Fig.

In this study I will assume landscape evolution can be effectively simulated
with the classic set of diffusive equations described in

Equation (

Diagram of flow routing from cell to cell and node to node for either a single flow direction (SFD) or a multiple flow direction (MFD) algorithm weighted by the relative gradient.

Water can be routed from cell to cell, where precipitation is collected over
the area of each cell, sent downwards, and accumulates. In this cell-to-cell
configuration the water flux has units of length squared per unit time and is
given by

Equation (

Both Eqs. (

Equation (

At a low model resolution,

Dimensionless elevation from the cell-to-cell flow routing landscape
evolution model with different flow routing algorithms at different numerical
resolutions after a dimensionless runtime of

Dimensionless elevation from the node-to-node flow routing landscape
evolution model with different flow routing algorithms at different numerical
resolutions after a dimensionless runtime of

For the node-to-node SFD algorithm, the increase in resolution has led to
significant branching of the valleys, which is clearly visible when the water
flux is plotted (Fig.

Dimensional sediment flux that exits the model domain and box–whisker plots of the dimensionless valley-to-valley wavelength for each model
for different resolutions, where the number of cells along the

To understand better how increasing resolution impacts the model evolution
the total sediment flux eroded from the model domain is plotted against time,
and the final valley spacing is calculated (Figs.

For the cell-to-cell SFD it can be seen that the evolution of the model is
resolution dependent, as the wind-up time reduces as resolution is increased
from 64 to 512 cells along the

Dimensional sediment flux that exits the model domain and box–whisker plots of the dimensionless valley-to-valley wavelength for each model
for different resolutions, where the number of cells along the

The node-to-node SFD algorithm is no better than the cell-to-cell SFD. In
this case wind-up time is resolution dependent, and the valley spacing
increases with increasing resolution (Fig.

It is only when node-to-node MFD is used that the LEM becomes significantly
less resolution dependent (Fig.

Final steady state of an example model run for the node-to-node MFD
algorithm.

Changing the flow routing algorithm changes the model wind-up time. This is
because the rate at which the network grows and the magnitude of the water
flux are affected by the choice of flow routing. The response time of the
model is proportional to the water flux raised to the power

The model that has the least resolution dependence is the node-to-node MFD
(Figs.

The grid cells in the models presented are large. At the highest resolution (2048 by 512 cells), the width of each triangle is of the order of 200 m if I was modelling a landscape 100 km wide. The model is, therefore, some approximation of local processes that give rise to the large-scale landscape. By distributing flow in multiple directions the model is in a sense approximating the hydrological processes that operate on a sub-grid-scale that give rise to the river network. The assumption of SFD is, however, too strong, and the sub-grid-scale processes are ignored.

The transport-limited model that I explore has certain limitations. In
particular the valleys floors are wide and not representative of V-shaped
valleys that would be expected from fluvial incision into bedrock
(Fig.

MFD routing might approximate local processes that distribute flow. Another
key sub-grid-scale process is the migration of drainage divides. A drainage
divide is the opposite of the flow path, as it separates the valleys. The
numerical model Divide And Capture (DAC) was developed to explore whether by using
an analytical solution to the stream power law, the sub-grid-scale migration
of drainage divides could be captured

By using the same set-up of a domain of 4 to 1 aspect ratio, uplift at

Comparison of two model results using Divide And Capture (DAC;

The implication of the results I present here, and from the development of
DAC, is that processes at a sub-grid level are of a crucial importance to
model stability, and hence great care must be taken in generating
reduced-complexity LEMs. At a small spatial and temporal scale, the landscape
evolution model CAESAR-LISFLOOD

In experiments of sediment transport it has been noted that when the
catchment outlet is fixed in time, the landscape does not achieve a steady
fixed topography

Application of the cell-to-cell SFD and node-node MFD algorithms to
a palaeo-DEM (digital elevation model).

In nature we observe that river networks are not fixed in space and time; rather, various processes lead to changing flow directions. To further explore
how realistic the cell-to-cell SFD and node-to-node MFD algorithms are, I
compare how the flow of water is predicted to evolve after a 20 kyr
interval. The initial condition is a palaeo-DEM generated from ASTER data
from the Ebro Basin, Spain (Fig.

The initial condition is derived from a real landscape, and as the model
allows for deposition in regions of low slope, both model routing algorithms
do not create drainage patterns that fully connect to the boundaries
(Fig.

Despite this imperfection, the internal drainage patterns still prove to be
insightful. The cell-to-cell SFD algorithm creates single paths for the flow
of water (Fig.

In the study of the evolution of the earth's surface we are increasingly
turning to models that attempt to capture the complexities of surface
processes. It is, however, clear that many LEMs are resolution dependent

The code fLEM is available from the following repository:

The author declares that there is no conflict of interest.

This work was inspired by a series of meetings organized by the Facsimile working group and by a visit to the Riu Bergantes catchment in Spain in October 2017. John Armitage is funded through the French Agence National de la Recherche, Accueil de Chercheurs de Haut Niveau call, grant “InterRift”. I would like to thank Kosuke Ueda and Liran Goren for help in running DAC. I would also like to thank Liran Goren and Andrew Wickert for their reviews. Edited by: Jean Braun Reviewed by: Liran Goren, Andrew Wickert, and one anonymous referee