ESurfEarth Surface DynamicsESurfEarth Surf. Dynam.2196-632XCopernicus PublicationsGöttingen, Germany10.5194/esurf-7-67-2019Short communication: flow as distributed lines within the landscapeFlow as distributed linesArmitageJohn J.armitage@ipgp.frhttps://orcid.org/0000-0003-2806-8181Dynamique des Fluides Géologiques,
Institute de Physique du Globe de Paris, Paris, FranceJohn J. Armitage (armitage@ipgp.fr)17January201971677513June201827June201820December201821December2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://esurf.copernicus.org/articles/7/67/2019/esurf-7-67-2019.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/7/67/2019/esurf-7-67-2019.pdf
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.
Introduction
It is known that resolution impacts landscape evolution models (LEMs)
. The resolution dependence of LEMs is caused by how
run-off is routed down the model surface. It has been demonstrated that
either distributing flow down all slopes (multiple flow direction, MFD) or
simply allowing flow to descend down the steepest slope (single flow
direction, SFD), gives different outcomes for landscape evolution models
. It has been noted that landscape
potentially has a characteristic wavelength for the spacing of valleys
. Therefore, a landscape evolution model should be
able to reproduce such regular topographic features independently of the
model resolution. For a model of channelized flow, it was, however, found that
the routing of run-off led to a resolution dependence in the valley spacing,
which could be overcome by the addition of a parameterized flow width that
was less than the numerical grid spacing .
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
. This means that the model response time
becomes likewise dependent on the chosen flow width. Ideally, the LEM would be
independent of grid resolution without introducing a predefined length scale
that impacts the model response.
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 . The very wide rivers, greater than
1 km, are, however, outliers within this global data set, with the median of
the distribution of mean river width being 124 m, with the upper quartile at
432 m (Fig. ). In LEMs developed for understanding long-term
landscape evolution, the large timescales necessitate large spatial scales,
where a single grid cell can be 1 km wide or more
. A spatial resolution of cells larger than a few
metres becomes necessary when modelling at the scale of a continent
e.g.. This means that flow has a width at a
subgrid level.
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. ), where water collects along these lines? To
explore this idea and understand LEM sensitivity to resolution, I wish to
explore how a simple LEM evolves under four scenarios (Fig. ):
(1) simple SFD from cell area to cell area, (2) an MFD version of this cell-to-cell algorithm, (3) a
node-to-node SFD, and (4) a node-to-node MFD.
A landscape evolution model
In this study I will assume landscape evolution can be effectively simulated
with the classic set of diffusive equations described in :
∂z∂t=∇κ+cqwn∇z+U,
where κ is a linear diffusion coefficient, c is the fluvial
diffusion coefficient, qw is the water flux, n is the water
flux exponent, and U is uplift. This heuristic concentrative–diffusive
equation is capable of generating realistic landscape morphology, with the
slope–area relationships commonly observed
. Strictly, it assumes that there
is always a layer of material to be transported by surface run-off, and as
such it can be classed as a transport-limited model. It accounts for both
erosion and deposition and is, therefore, appropriate for modelling landscape
evolution beyond mountain ranges and into the depositional setting (see
models such as DIONISOS; ). It differs from mixed
erosion and deposition models such as and
because those models split the divergence of the sediment flux into two
terms: a rate of erosion and a rate of deposition. Here, instead I assume that
the sediment flux is a function of water flux and slope.
Equation () is solved with a finite-element scheme written using
Python and the FEniCS libraries (I will call the code “fLEM”; see the “Code
availability” section). The equations are solved on a Delaunay mesh, where the mesh is
made up of predominantly equilateral triangles with an opening angle of
60∘. Model boundary conditions are initially of fixed elevation on
the sides normal to the x axis and a zero gradient on the sides normal to
the y axis. The model aspect ratio is 4 to 1. Uplift is fixed at
U=10-4 m yr-1, the linear diffusion coefficient is κ=1 m2 yr-1, the fluvial diffusion coefficient is
c=10-4 (m2 yr-1)n-1, and the water flux exponent is
n=1.5.
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
qw[cell]=αals,
where α is precipitation rate, a is the cell area, and
ls is the length from cell centre to cell centre down the
steepest slope (Fig. a and b). This gives a water discharge per
unit length, which has the advantage of not having to explicitly state the
sub-grid width of the flow . However, implicitly this
implies that the flow is over the width of a cell. An alternative is to route
water from node to node along cell edges and for it to accumulate. I assume
that along the length of each cell edge water can be added to the flow line,
assuming that the input is linearly related to the length of the flow line,
qw[node]=αl,
where l is the length of the edge that joins the upslope node to the
downslope node (Fig. c and d). This means that the cell area is
ignored and instead water enters the flow path uniformly along its length and
accumulates downslope.
Equation () makes the assumption that water accumulates as a
function of length. Water flux is observed to be related to catchment area:
Qw∝A0.8. The catchment length,
l, is then related to area by l∝A1/p, where 1.4<p<2.0. At the lower end of the range, this gives
Qw∝l1.12, suggesting that accumulating water as a
linear function of flow length is a reasonable simplification. A knock-on
effect of this assumption is that the magnitude of the water flux predicted
for the node-to-node routing is less than that of the cell-to-cell, as in the latter
water is accumulated over cell areas which is naturally larger than the
cells' edges.
Both Eqs. () and () do not attempt to capture the
interaction between water flux and river width; rather, these are two methods
to approximate run-off within a coarse numerical grid. For both the
cell-to-cell and node-to-node methods the flow can then be routed down a
SFD or routed down MFDs weighted by the relative gradient, as in, for
example, . I run the numerical model with a uniform
precipitation rate of α=1 m yr-1.
Equation () is made dimensionless following
using the linear diffusion timescale and the model length in the
x direction, L. This means that Eq. () can be rewritten as
∂z̃∂t̃=∇1+Dq̃wn∇̃z̃+U
and
∇̃⋅∇̃z̃|∇̃z̃|q̃w=-1,
where x=x̃L, y=ỹL, z=z̃L, t=t̃L2/κ, q=q̃αL, and
D=cαnLnκ.
The effect of model resolution
At a low model resolution, 512×128 cells, all four methods of flow
routing give a similar landscape morphology after 5 Myr of model evolution
(Figs. and ). However, elevations are significantly
lower for the cell-to-cell flow routing model as the water flux term is lower
for the node-to-node routing algorithm (Figs. and ).
As the resolution is increased to 2048×512 cells, the landscape
morphology starts to diverge. For the cell-to-cell SFD algorithm, the
landscape shows more small-scale branching, as previously discussed by
(Fig. b and c). For the SFD algorithm it can be
seen that the high-resolution model has multiple peaks along the ridges
(Fig. b). This roughness to the topography is removed if the flow
is distributed downslope from cell to cell (MFD; Fig. d).
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 1.563×10-6 (5 Myr),
with an aspect ratio of 4×1. (a) Cell-to-cell single flow
direction (SFD) algorithm with a resolution of 512×128 cells.
(b) The same model but with a resolution of 2048×512 cells.
Panels (c) and (d): cell-to-cell multiple flow direction (MFD)
algorithm.
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 1.563×10-6 (5 Myr),
with an aspect ratio of 4×1. (a) Node-to-node single flow
direction (SFD) algorithm with a resolution of 512×128 cells.
(b) The same model but with a resolution of 2048×512 cells. Panels (c) and (d): node-to-node multiple flow direction (MFD)
algorithm.
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. a and b). For the node-to-node MFD
algorithm, the morphology and distribution of water flux are similar for both
the low and high resolution (Fig. c and d); yet as with the
cell-to-cell algorithm, increased resolution
leads to increased branching of the network. The two MFD models give a
smoother topography, as by distributing flow, local carving of the landscape
is reduced.
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 y axis is
shown. (a) Sediment flux and (b) valley-to-valley
wavelength for the cell-to-cell SFD algorithm. (c) Sediment flux and
(d) valley-to-valley wavelength for the cell-to-cell MFD algorithm.
The dashed line in panels (a), (c), and (e) marks
the time at which erosion balances uplift, given by t≥3Hr/U, where Hr is the relief height and U is the uplift rate
.
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. and
). To calculate the valley spacing I take horizontal swaths of the
spatial distribution of water flux. For each swath profile a peak finding
algorithm is used to find the distance from peak to peak
in water flux. This distance is then averaged over the 100 swath profiles
and over 10 model runs to give the minimum, lower quartile, median, upper
quartile, and maximum valley wavelength (Figs. and ).
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 y axis (Fig. a). Furthermore, the
mean valley spacing reduces with increasing resolution (Fig. b).
This behaviour is not ideal, as it means that model behaviour to perturbations
in forcing might become resolution dependent. For the MFD wind-up times
remain resolution dependent, while the mean valley spacing is similar for the
four different resolutions (Fig. c and d).
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 y axis is
shown. (a) Sediment flux and (b) the node-to-node SFD
algorithm. (c) Sediment flux and (d) valley-to-valley
wavelength for the node-to-node MFD algorithm. The dashed line in panels (a) and (c) marks the time at which erosion balances
uplift, given by t≥3Hr/U, where Hr is the relief
height and U is the uplift rate .
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. a and b). For the
node-to-node SFD, at a resolution of 256 cells or less along the y axis, there is an instability in the sediment flux output. This is due to the flow
tipping between adjacent nodes due to small differences in relative elevation
after each time iteration. This unstable behaviour disappears for the higher
resolution of 512 cells along the y axis (Fig. a).
It is only when node-to-node MFD is used that the LEM becomes significantly
less resolution dependent (Fig. c and d). For the node-to-node MFD
the time evolution of sediment flux is similar for all resolutions, and the
valley spacing is similar as resolution is increased. The steady-state sediment flux is, however, not completely stable (Fig. c). This is
due to the migration of the flow across the valley floors created within the
model topography (Fig. ). Even once a balance has been achieved
between erosion and uplift, small lateral changes in elevation can be seen to
create a negative to positive change in elevation of a few metres between
time iterations, where the time step is 100 years (Fig. b). This is
associated with an equivalent change in water flux (Fig. c).
Final steady state of an example model run for the node-to-node MFD
algorithm. (a) Final model elevation where the domain is 800 km
long by 100 km wide and uplift is fixed at U=10-4 m yr-1, the
linear diffusion coefficient is κ=1 m2 yr-1, the fluvial
diffusion coefficient is c=10-4 (m2 yr-1)n-1, and the
water flux exponent is n=1.5. (b) Difference in elevation between
the last two model time steps, where the time step duration is 100 years.
(c) Difference in water flux between the last two model time steps.
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 n. Therefore, if the drainage network forms
rapidly, as is the case for cell-to-cell routing, then the model wind-up is
more rapid. For the node-to-node routing, it takes longer for the network to
grow (Fig. ). Furthermore, the MFD model is the slowest to evolve
to a steady state, where the total sediment flux is balanced by the uplift
(Fig. ). I have chosen to focus on n=1.5 as this value
previously gave more realistic slope–area relationships at steady state
. However, it is interesting to note that
growth of the network is a function of both the routing algorithm and the
value of n.
Sub-grid-scale processes
The model that has the least resolution dependence is the node-to-node MFD
(Figs. c and d and c and d). The difference between
this model and the other three is that this version has the maximum possible
flow directions available within my set-up. By treating flow paths as lines
within the numerical grid, from any node there are six paths, which is twice as
many as in the cell-to-cell MFD. This means that there is greater
distribution of the flow and a reduced localizing of flow paths within the
node-to-node distributed model. For SFD, increasing resolution, however, leads
to multiple branches (Figs. b and b).
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. ). In order to generate such valleys, a detachment-limited
model, such as the stream power law, would be more appropriate. However, many
stream power law models also suffer resolution dependence, as they typically
use an SFD to route water e.g..
looked at using MFD routing for the stream power law
and found that there remained some spatial resolution dependence. The model
of used a rectangular grid and removed resolution
issues by using a predictor–corrector algorithm to adjust for resolution
effects. However, for the transport-limited model used here, I find that with
a triangular grid the MFD routing is resolution independent without
additional corrections. This is likely related to the fact that the length of each cell
face is equal, while for rectangular cells the diagonal flow direction is
longer than the cell faces. The implication is that for LEMs, a mesh that has
cells with node-to-node spacing of equal length is preferable to a
rectangular grid; however, this hypothesis will require further exploration.
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 . DAC, therefore,
uses a variant of a stream power law model; yet like the transport-limited
model I present, DAC uses a triangular grid. However, DAC routes flow down
the steepest route of descent (SFD). By exploring how model resolution
impacts the main drainage divide, it was demonstrated that the inclusion of a
sub-grid level calculation for water divides is crucial to remove otherwise
spurious results .
By using the same set-up of a domain of 4 to 1 aspect ratio, uplift at
0.1 mm yr-1, and a precipitation rate of 1 m yr-1, I have
explored how valley spacing varies as a function of resolution in the DAC
model. DAC uses an adaptive mesh; therefore, the settings on how the
re-meshing occurs needed to be altered to achieve an increase in the number
of cells. By comparing two models at a different resolution (23 172 cells
compared to 93 734), it can be seen that the median wavelength is very
similar (Fig. ).
Comparison of two model results using Divide And Capture (DAC;
) at different resolutions. (a) Model
steady state for an initial resolution of 51 by 204 cells, which after
adaptive re-meshing increases to 23 172 cells. (b) Model steady
state for an initial resolution of 101 by 404 cells, which after adaptive
re-meshing increases to 93 734 cells. (c) Comparison of the
wavelength of valleys for the two models, taken from 20 swaths 1.25 km
wide from the left-hand boundary (see code availability for python scripts
and DAC input files).
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 , which has a
rectangular grid, has been tested for different resolutions and is found to
converge to the same solution at sufficiently high resolution.
CAESAR-LISFLOOD uses a
version of the shallow-water equations to solve for river flow, where water
flows in four directions (Manhattan neighbours) and, therefore, uses an MFD
rather than an SFD algorithm. Furthermore CAESAR-LISFLOOD operates on a
resolution that is smaller than the width of an individual channel. This
suggests that at a small spatial scales, where water depth is captured, a
rectangular grid combined with an MFD algorithm is appropriate. Such a
high-resolution model, however, cannot be run over periods greater than
several millennia e.g.. Therefore, to explore how
landscape evolves over millions of years, I suggest we must distribute flow
across the model domain and use meshes of equal node-to-node spacing to avoid
resolution dependence.
Steady state but not steady topography
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 . It has been previously suggested
that this behaviour can be replicated within an LEM by introducing a
distributed routing algorithm . This modelling result
has, however, been challenged by, for example, , where it
has been suggested that distributive flow routing algorithms in fact create a
fixed topography at steady state. My model, however, is in agreement with the
initial findings of . It has been previously noted that
an MFD algorithm will give more diffuse valley bottoms compared to an SFD
algorithm . If landscapes are indeed never steady, then
perhaps this unsteady nature is due to the diffuse sediment transport across
wide flood plains, which feeds up into the drainage basins. It is, after all,
within the valley floor that the distributed flow routing is the most
unsteady (Fig. c).
Application of the cell-to-cell SFD and node-node MFD algorithms to
a palaeo-DEM (digital elevation model). (a) Palaeo-DEM created from
ASTER data of the Ebro region of Spain. (b) Water flux after
20 kyr of model evolution assuming an SFD with a model resolution of
1024×1024 cells. Uplift is assumed to be very small at
10-5 m yr-1, with a precipitation rate held constant at
0.1 m yr-1. (c) Water flux for after 20 kyr for a model
assuming the node-to-node MFD routing. The white box in the top right
highlights a region of the Riu Bergantes catchment where the river is known to
have shifted course during the Holocene.
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. a). The river valleys have been
filled and the landscape has been smoothed in an attempt to approximate
this landscape in the late Pleistocene. This landscape is then allowed to
evolve, assuming a uniform uplift of 10-5 m yr-1 and a
precipitation rate held constant at 0.1 m yr-1. I assume that
c=10-5 (m2 yr-1)n-1,
κ=10-1 m2 yr-1, and n=1.5. Under these conditions the
landscape is left to evolve for 20 kyr (Fig. ) with zero gradient
boundaries on the east, west, and southern sides and fixed elevation on the
northern boundary.
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. b and c). This problem of too much deposition within regions
of low slope, such that the water flux does not reach the model boundaries,
can be overcome with the application of a “carving” algorithm. As for
example applied within the TopoToolbox Landscape Evolution Model (TTLEM), a minima imposition can be
used to make sure rivers keep on flowing down through regions of low slope
. Such an additional algorithm will, however, affect
how the network grows within the model, so for this example, I have left the
routing algorithm to drain internally.
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. b). After the 20 kyr duration, it is observed that
high water flux is concentrated within the deep valleys. The node-to-node MFD
algorithm creates multiple flow paths that exit the mountain valleys and
migrate onto the flood plains (Fig. c). Field studies of the Riu Bergantes have found that this catchment has experienced periods of
significant sediment reworking, potentially related to climatic change
. The region outlined with the white box in
Fig. c shows evidence of terrace formation related to lateral
movement of the Riu Bergantes during the Holocene
. In particular, where the flow paths create a
small island (see Fig. c, centre of the white box), there is
evidence from terrace deposits that the course of the Riu Bergantes has
flipped from the eastern to the western side of this island. The cell-to-cell
SFD cannot create this observed behaviour. Therefore, as well as creating
landscape evolution that is not resolution dependent, the MFD algorithm
creates landscape evolution that is, relative to the SFD, closer to that
observed in nature.
Conclusions
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 source of this resolution dependence is the
numerical methods that we employ to route surface water. Unless we model
landscape evolution at a spatial scale that is smaller than an individual
river, we must somehow approximate this flow. By treating flow from node to
node within the model mesh and by distributing flow down these lines, the LEM
developed here is no longer resolution dependent. Furthermore the model
evolution is closer to what we observe. Therefore, I would strongly suggest
that for LEMs that operate at a scale larger than the resolution of a river,
we must use MFDs.
The code fLEM is available from the following repository:
https://bitbucket.org/johnjarmitage/flem/. The
valley wavelength Python script and DAC input files are available from the
following repository: https://bitbucket.org/johnjarmitage/dac-scripts/. DAC was developed by Liran Goren; see
https://gitlab.ethz.ch/esd_public/DAC_release/wikis/home (last access:
16 January 2019).
The author declares that there is no conflict of
interest.
Acknowledgements
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
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