ESurfEarth Surface DynamicsESurfEarth Surf. Dynam.2196-632XCopernicus PublicationsGöttingen, Germany10.5194/esurf-6-1-2018Developing and exploring a theory for the lateral erosion of bedrock channels for use in landscape evolution modelsLangstonAbigail L.alangston@ksu.eduTuckerGregory E.https://orcid.org/0000-0003-0364-5800Department of Geography, Kansas State University, Manhattan, KS, USACooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado, Boulder, CO, USADepartment of Geological Sciences, University of Colorado, Boulder, CO, USAAbigail L. Langston (alangston@ksu.edu)8January2018611272May20178May20176November201714November2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://esurf.copernicus.org/articles/6/1/2018/esurf-6-1-2018.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/6/1/2018/esurf-6-1-2018.pdf
Understanding how a bedrock river erodes its banks laterally is a frontier in
geomorphology. Theories for the vertical incision of bedrock channels are widely
implemented in the current generation of landscape evolution models. However,
in general existing models do not seek to implement the lateral migration of
bedrock channel walls. This is problematic, as modeling geomorphic processes
such as terrace formation and hillslope–channel coupling depends on the accurate
simulation of valley widening. We have developed and implemented a theory for
the lateral migration of bedrock channel walls in a catchment-scale landscape
evolution model. Two model formulations are presented, one representing the
slow process of widening a bedrock canyon and the other representing
undercutting, slumping, and rapid downstream sediment transport that occurs
in softer bedrock. Model experiments were run with a range of values for
bedrock erodibility and tendency towards transport- or detachment-limited
behavior and varying magnitudes of sediment flux and water discharge in order
to determine the role that each plays in the development of wide bedrock valleys.
The results show that this simple, physics-based theory for the lateral erosion
of bedrock channels produces bedrock valleys that are many times wider than
the grid discretization scale. This theory for the lateral erosion of bedrock
channel walls and the numerical implementation of the theory in a
catchment-scale landscape evolution model is a significant first step towards
understanding the factors that control the rates and spatial extent of wide
bedrock valleys.
Introduction
Understanding the processes that control the lateral migration of bedrock
rivers is fundamental for understanding the genesis of landscapes in which
valley width is many times the channel width. Strath terraces are a clear
indication of a landscape that has experienced an interval during which lateral
erosion has outpaced vertical incision . Broad
strath terraces and wide bedrock valleys that are many times wider than the
channels that carved them are found in mountainous and hilly landscapes
throughout the world e.g., and provide clues about the nature of their evolution.
Wide bedrock valleys and their evolutionary descendants, strath terraces, are
erosional features in bedrock that are several times wider than the channels
that carved them and range in spatial scale from tens to thousands of meters
(Fig. ). Wide bedrock valleys created by incising rivers
provide the opportunity for sediment storage in the valley bottom, influence
hydraulic dynamics by allowing peak flows to spread out across the valley,
and decrease the average transport velocity of sediment grains
.
Changes in climate that drive changes in sediment flux, changes in discharge
magnitude, and/or changes in discharge frequency have been cited as causes of
periods of lateral erosion in bedrock rivers. The frequency of intense rain
is correlated with higher channel sinuosity and lateral erosion rates on
regional scales . Several studies demonstrate that
significant lateral erosion in rapidly incising rivers is accomplished by
large flood events
resulting from cover on the bed during extreme flood events
and exposure of the bedrock walls to
sediment and flow . Sediment cover on the bed that
suppresses vertical incision and allows lateral erosion to continue unimpeded
is a critical element for the development of wide bedrock valleys, as
determined from modeling, field, and experimental studies
.
Lateral erosion that outpaces vertical incision and creates wide bedrock
valleys and strath terraces has been linked to weak underlying lithology,
such as shale , although strath terraces certainly exist in stronger
lithologies, such as quartzite . The relationships
among river sediment flux, discharge, lithology, and rates of lateral bedrock
erosion are not well defined. Because we do not sufficiently understand the
processes of lateral erosion, landscape evolution models lack a physical
mechanism for allowing channels to migrate laterally and widen bedrock
valleys, in addition to incising bedrock valleys.
Existing landscape evolution models do not address the lateral erosion of
bedrock channel walls and the consequential migration of the channel, in no
small part because of the lack of a rigorous understanding of the processes
that control the lateral erosion of bedrock channel walls. If this theoretical
hurdle can be cleared, an algorithm for lateral erosion must be applied
within a framework of models that currently only erode and deposit
vertically. To our knowledge, this study is the first attempt at
incorporating a generalized physics-based algorithm for lateral bedrock
erosion and channel migration on a drainage basin scale to a two-dimensional
landscape evolution model.
Background
Theories for the vertical incision of bedrock channels have advanced
considerably since the first physics-based bedrock incision models were
presented in the early 1990s. For example, bedrock incision models now
include theories for the adjustment of channel width
e.g.,, the role of sediment size and bed cover
(e.g., ; ;
), and thresholds for incision
(e.g., ; ). Rivers may
respond to changing boundary conditions by adjusting both slope and channel
width e.g.,
and landscape evolution models must be able to capture both of these responses
if we are to fully describe the behavior and function of landscapes. Research
on bedrock channel width gives important insights into the larger-scale
problem of bedrock valley widening. In particular, sediment cover on the bed
plays an important role in the evolution of channel cross-sectional shape
because sediment cover on the bed can slow or halt vertical incision
while allowing lateral
erosion to continue. Models of channel cross-sectional evolution predict that
increasing sediment supply to a steady-state stream results in a wider,
steeper channel for a given rate of base-level fall
. While theories that account for dynamic
adjustment to bedrock channel width continue to be refined (for a review, see
), landscape evolution models that include a
relationship between sediment size and cover
and incision thresholds in bedrock channels are available and widely used
.
Numerical models for alluvial rivers have made considerable advances in
capturing the planform dynamics of both meandering and braided rivers, which
necessarily include lateral bank erosion.
developed the first numerical model that simulates lateral bank movement in
alluvial rivers and produces realistic patterns of river meandering. In this
study, bank erosion scales inversely with the radius of curvature such that
more rapid erosion occurs in tighter bends with a smaller radius of
curvature. A more recent treatment of radius of curvature as a control on
lateral erosion rates has been implemented in CAESAR, a cellular landscape
evolution model that calculates a two-dimensional flow field
. This model is appropriate for studying alluvial
river dynamics in meandering or braided streams at reach and small catchment
scales and timescales of up to thousands of years
, but it is not designed to model the evolution of
bedrock rivers. The EROS model is a morphodynamic–hydrodynamic model that also
allows for the lateral erosion of bank material . In EROS, the lateral erosion of bank
material is equal to the vertical erosion rate multiplied by the lateral
topographic slope and a coefficient of unknown value .
This treatment of lateral erosion allows for lateral channel mobility and the
development of realistic braided rivers, but it lacks a mechanistic process
specifically for the lateral erosion of bedrock channels.
As noted above, considerable advances have been made in developing theory and
models for the planform dynamics of single-thread meandering channels. As a
result, the scientific community has a good understanding of how meander
patterns form and evolve and how meander wavelength and migration rate scale
with properties such as water discharge, valley gradient, and sediment grain
size e.g.,.
This body of work addresses the planform pattern of river channels, but does
not deal with the broader drainage basin topography in which those channels
are embedded. The principal state variable in channel meander models is the
trace of the channel, x(λ), where λ represents
streamwise distance and x=(x,y,t) is the channel centerline
position. Some more recent models also incorporate a vertical channel
coordinate, so that x=(x,y,z,t)e.g.,, but the emphasis remains on the channel
trace rather than on the topography. For example, the slope of the channel
and/or valley is normally treated as a boundary condition rather as an
element of topography that evolves dynamically as it steers the flow of
water, sediment, and energy.
There is also a well-developed literature on process models of landscape
evolution, in particular the evolution of ridge–valley topography
sculpted around drainage networks. We refer to these models as landscape
evolution models, or LEMs e.g.,. With LEMs, the emphasis lies on computing the
topographic elevation field, η(x,y,t). Water and sediment cascade
passively downhill across this surface. In some of these models, channel
segments are assumed to exist as sub-grid-scale features that are free to
switch direction arbitrarily as the topography around them changes. Other
LEMs represent water movement as a two-dimensional flow field, whether
through multiple-direction routing algorithms
e.g., or with a simplified form of the shallow-water equations
. Regardless of the approach to
flow routing, LEMs differ from meander models in treating a self-forming,
two-dimensional flow network rather than a single channel reach and in
explicitly modeling the evolution of topography.
The lateral migration of bedrock channel walls has only been implemented into
landscape evolution models in a limited number of studies
. model
bedrock valley widening using a one-dimensional stream power model for vertical incision
and assume that valley widening rates depend on stream power. They note that
the width of the valley floor is related to the duration of steady state in
the river, as theorized by . This model is based on the
key observation that lateral erosion exceeds vertical incision when the
channel is carrying the maximum sediment load dictated by the transport
capacity. By varying sediment supply to the channel, their model predicts the
development of a series of strath terraces. Strath terrace sequences have
also been produced by coupling a meandering model with a river incision model
. modeled meandering
channels in a valley section using a two-dimensional landscape evolution model and an
adaptive grid approach. A vector-based approach to modeling the lateral migration
of meandering streams in heterogeneous bed material has been used to
reproduce a range of bedrock valley forms , but
this model is primarily a channel-scale model. While each of these studies
model the lateral migration of bedrock channel banks, they all operate with a
meandering model that is not applicable to lateral migration in low-sinuosity
channels or in a generalized landscape evolution model.
Approach and scope
Until now, landscape evolution models have lacked a generic mechanism for
allowing channels to migrate laterally and widen bedrock valleys, as well as
incise bedrock valleys. While advances in controls on bedrock valley width
have been made using meandering models, the representation of a sinuous
channel does not describe all rivers, and often such models are constructed on
a channel scale rather than on a drainage basin scale. In this study, we
develop a theory for the lateral migration of bedrock channel walls and
implement this theory in a two-dimensional landscape evolution model for the first time.
We seek to explore the parameters that exert primary control on the
morphology of bedrock valleys and the rate of bedrock valley widening using a
series of numerical experiments.
With a few exceptions noted below, most LEMs treat erosion and sedimentation
as purely vertical processes. When the flow of water and sediment collects in
a “digital valley”, the elevation of that location may rise or fall, but
lateral erosion by channel impingement against a valley wall is usually
neglected. Yet nature seems to be perfectly capable of forming erosional
river valleys much wider than the channels they contain (Fig. ). The question arises of how one might honor the process of
valley widening by lateral erosion (and narrowing by incision) within the
topographically oriented framework of an LEM. In other words, how might the
key features of LEMs and channel planform models be usefully combined?
In addressing this issue, it is useful to consider that the typical LEM
treatment of topography as a two-dimensional field η(x,y,t) is itself a
simplification, albeit a practical one. Consider an alternative framework in
which the boundary between solid material (rock, sediment, soil) and fluid
(air, water) is treated as a surface in three-dimensional space, σ(x,y,z,t). The surface possesses at each point a
surface-normal velocity, σ˙, which represents the combined
surface-normal rates of erosion, sedimentation, and tectonic motion. Such a
framework would lend itself to representing lateral erosion because any
movement of this surface where it is not flat implies a horizontal component
of motion. The cost of such an approach lies in computational complexity. For
practical reasons, it is desirable to find methods by which a lateral
component of erosion by stream channels could be represented within the much
simpler framework of a two-dimensional elevation field η(x,y,t).
Field examples of wide bedrock valleys cut by lateral erosion. All
cross sections are from north to south. (a) The Drôme River in the French
Alps is transport limited and meandering in reaches that carve wide bedrock
valleys. The bedrock valley at this location (44.69∘ N, 5.14∘ E)
is 500 m wide and the channel is ∼45m wide (indicated by light blue
shade of cross section line). (b) Gower Gulch (36.41∘ N,
116.83∘ W) in Death Valley, USA widened significantly in response to
increased discharge from a stream diversion in the 1940s
. The bedrock valley is 30 m wide and the channel
braids are ∼2m wide (indicated by light blue shade of cross section
line). (c) Left Hand Creek drains the Colorado Front Range (40.11∘ N,
105.25∘ W) and has undergone multiple cycles of lateral erosion that
produced flights of strath terraces, outlined in white on the image. The
cross section shows Table Mountain at ∼70m above the current stream
height on the north side of cross section and a lower terrace level at
10 m
above the current stream level on the south side of the cross section. (d) Arroyo
Seco in the California Coast Range (36.27∘ N, 121.33∘ W) carved a
600 m wide strath terrace during a period of lateral erosion that is
30 m
above the current stream level. The current bedrock valley is 125 m wide and
the channel is ∼15m wide (indicated by light blue shade of cross
section line). Images: Google Earth. Cross sections: NCALM and 30 m SRTM.
In this paper, our objective is to define and explore a theory for lateral
erosion that has the following characteristics: simple and sufficiently
general in nature to be applicable in landscape evolution models; containing
as few parameters as possible; requiring relatively few input variables, such
as channel gradient and water discharge plus gross channel planform
configuration. The aim of this theory is to model valley widening or
narrowing over timescales relevant to drainage basin evolution and across
multiple branches within a drainage network. The theory is not designed to
predict the movement of a particular channel segment over a period of a few
years, but rather is intended to provide a general basis for understanding
when and why valleys tend to narrow or widen during the course of their
long-term geomorphic evolution. Theoretical predictions about these trends
then serve as quantitative, mechanistically based hypotheses that can be
tested through experiments and observations. Through a set of numerical
experiments, we seek to answer the following set of questions.
How does this lateral erosion model compare with purely vertical erosion models?
How do two alternative formulations, which treat bank material differently, compare to each other?
What combinations of bedrock erodibility, sediment mobility, water flux, sediment flux, and model type result in wide bedrock valleys?
What are the predictions of the model that could be readily tested through experiment and/or observation?
In the following sections we outline our theory for lateral channel wall
migration and explain the two algorithms we have developed to apply this
theory to an existing model. We then present the results from our set of
numerical experiments and discuss how well the model describes the formation
of wide bedrock valleys. The approach presented here is intended to be a
starting point, but not an ending point. Our main goal is to draw attention
to the importance of lateral stream erosion within the context of
drainage basin evolution and to offer some ideas for how this might be
addressed in the framework of a conventional grid-based LEM.
Theory
We have deliberately chosen the most simple formulation
possible for deposition and erosion, while still capturing the role of
sediment. We do this in order to focus on developing the lateral erosion
component of our model. Evolution of the height of the landscape, η,
through time is described by the deposition rate, d, minus the erosion rate, e,
plus a constant rate of uplift relative to base level, U.
∂η∂t=-e+d+U
The deposition rate is assumed to depend on the concentration of sediment (Cs)
in active transport and its effective settling velocity, νs. Sediment
concentration is expressed as the ratio of volumetric sediment flux, Qs,
to water discharge, Q:
Cs=QsQ.
We treat water discharge as the product of runoff rate and drainage area
such that Q=RA. The deposition rate is therefore given by
d=νsd*QsRA,
where d* is a dimensionless number describing the vertical distribution of
sediment in the water column, which is equal to 1 if sediment is equally
distributed through the flow ; νs, d*, and R
are lumped into a single dimensionless parameter, α, that represents
the potential for deposition.
α=νsd*R
A large value for α implies more rapid deposition (all else being
equal) either because settling velocity, νs, is high and sediment is
quickly lost from the flow, or because runoff rate, R, is low and there is
little water in the channels to dilute the sediment. A small value for
α represents slower settling velocity, or more intuitively, greater
runoff; α can be thought of as a sediment mobility number. When
α<1, sediment is easily transported and the model tends towards
detachment-limited behavior. When α>1, sediment is less mobile and the
model tends towards transport-limited behavior.
Vertical erosion theory
In the model presented here, we use the stream power incision model
e.g., to calculate the vertical incision rate;
the stream power model is the simplest bedrock incision model that represents
fluvial erosion for steady-state topography. The vertical erosion rate is derived
from the rate of energy dissipation on the channel bed, which is given by
ωv=ρgQWS,
where ρ is the density of water, g is gravitational acceleration, Q
is water discharge, W is channel width, and S is channel slope. We assume
that the rate of vertical erosion scales as
Ev=Kv′ωvCe,
where Kv′ is a dimensionless vertical erosion coefficient and Ce is
cohesion of bed and bank material. We use bulk cohesion simply as a
convenient reference scale for rock resistance to erosion. This choice allows
us to express the erosion rate as a function of the hydraulic power applied
(ωv), a commonly used measure of material strength (Ce), and a
dimensionless efficiency factor (Kv′).
We assume that channel width is a function of discharge :
W=kwQ0.5,
where kw is a width coefficient. It is important to recognize that channel
width is not explicitly represented in the model we describe. Rather, it is
one element of the lumped parameters Kv and Kl (erosion coefficients
discussed below). The channel-width scaling parameter values we discuss
(kw) are used only in the estimation of reasonable ranges for these
parameters. The bank width coefficient, kw, is constant along the channel
length based on data sets from both alluvial and
bedrock rivers that show a relationship
between channel width and discharge. Substituting RA for Q and Eq. () for W in Eq. () and then combining
Eqs. () and () gives the following:
Ev=Kv′ρgR1/2kwCeA1/2S,Ev=KvA1/2S.
Lumping several parameters gives Kv, a dimensional vertical erosion
coefficient (with units of years-1) that consists of known or
measurable quantities, and one unknown dimensionless parameter, Kv′.
Although evidence indicates that sediment in the channel plays an important
role in inciting lateral erosion in bedrock channels
, the
model presented here uses the stream power incision model to represent
vertical erosion, which does not account for sediment-flux-dependent incision
e.g.,. The standard stream power model (Eq. 8)
has some limitations, especially in the lack of threshold effects and
the assumption of constant channel width . Despite these
limitations, the stream power model is a good approximation for long-term
vertical bedrock incision on large spatial scales
e.g., and is appropriate
here given that the goal of this work is to explore the dynamics of lateral bedrock
erosion as a function of channel curvature.
Lateral erosion theory
Lateral erosion requires hydraulic energy expenditure to damage the bank
material and/or dislodge previously weathered particles
.
Consistent with earlier meandering models
e.g.,, we hypothesize that the lateral erosion
rate is proportional to the rate of energy dissipation per unit area of the
channel wall created by centripetal acceleration around a bend. Erosion of
the channel wall is the result of the force of water acting on the channel
wall. We know from basic physics that the force of water acting on the wall
is equal to the force of the wall acting on the water, which is equal to
centripetal force. Centripetal force is Fc=mv2rc, where m
is mass, v is velocity, and rc is the radius of curvature. The centripetal
force of a unit of water can be found by replacing m with ρLHW, where
ρ is the density of water and L, H, and W are unit length, water
depth, and channel width, respectively. The centripetal force of water flowing
around a bend can be expressed in terms of centripetal shear stress, which is
analogous to bed shear stress, by dividing both sides by HL, giving
σc=ρWv2rc.
Centripetal shear stress can be turned into a rate of energy expenditure by multiplying by fluid velocity, giving
ωc=ρWv3rc.
To express this in terms of discharge, Q, instead of velocity, we employ
the Darcy–Weisbach equation, giving v3=gqS/F, where q is discharge per
unit width and F is a friction factor, which yields
ωc=ρgQSrcF.
Equation () describes a quantity that might be termed centripetal
unit stream power, as it represents the rate of energy dissipation per unit
bank area. The centripetal unit stream power is similar to the more familiar
quantity unit stream power, except that channel width is replaced by the
radius of curvature multiplied by a friction factor.
We hypothesize that lateral erosion rate scales with energy dissipation rate around a bend according to
El=Kl′ωcCe,
where Kl′ is a dimensionless lateral erosion coefficient. Combining Eqs. () and () gives
El=Kl′ρgRCeFASrc,El=KlASrc,
where Kl is a dimensional erosion coefficient for lateral erosion composed
of known or measurable quantities and one unknown dimensionless parameter,
Kl′. If Kl′ is equal to Kv′, we find a ratio between Kl
and Kv given by
KlKv=R1/2kwF,
which consists of runoff rate, R, bank width coefficient, kw, and
friction factor, F. We can measure or make reasonable estimates of each of
these parameters in order to determine what the ratio of lateral to vertical
erodibility should be. Mean annual runoff rate can vary widely, but a higher
peak runoff intensity will lead to a higher Kl/Kv ratio and more
lateral erosion.
A fixed kw is common in landscape evolution models that model long-term
landscape erosion e.g.,, but channel width can vary with incision rate in
models and natural systems ,
suggesting there are cases when dynamic width scaling is important
. In this model, kw is given a value of 10 m(m3s-1)-1/2,
which is reasonable for natural rivers
, but the value can range between 1 and 10 due to
differences in runoff variability, substrate properties, and sediment load
. The friction factor, F, is the Darcy–Weisbach
friction factor, which can range from 0.01–1.0 for natural rivers
. With a lower friction factor
(representing smooth channel walls), the lateral erosion ratio would be
higher due to less energy being dissipated on the channel walls, leaving more
energy available for lateral erosion.
Numerical implementation
One challenge in modeling both vertical and lateral erosion in a drainage
network lies in the representation of topography. Typically, landscape
evolution models use a numerical scheme in which the terrain is represented
by a grid of points whose horizontal positions are fixed and whose elevation
represents the primary state variable in the model. Such a framework does not
lend itself to the motion of near-vertical to vertical interfaces (such as
stream banks and cliffs), and for this reason, incorporating lateral stream
erosion in a conventional landscape evolution model requires a modification
to the basic numerical framework. A vertical rather than horizontal grid
can be used for near-vertical landforms in
isolation, but is inappropriate when one wishes to represent vertical
interfaces that are inset within a larger landscape. Grid-node movement
combined with adaptive re-gridding provides a possible solution, but is
computationally expensive and particularly difficult to implement when
multiple branches of a drainage network may undergo lateral motion. Here, we
adopt a simpler approach in which valley walls are viewed as sub-grid-scale
features that migrate through the fixed grid. Rather than tracking the
position of these vertical interfaces, we instead track the cumulative
sediment volume that has been removed from the cell surrounding a given grid
node as a result of lateral erosion. When that cumulative loss exceeds a
threshold volume, the elevation of the grid node is lowered.
More specifically, at each node in the model, we calculate a vertical
incision rate at the primary node and a lateral erosion rate at a neighboring
node (Fig. ). The lateral neighbor node for the primary
node is chosen on the outside bank of two stream segments that flow into and
out of the primary node. The stream segments used to identify the neighboring
node over which lateral erosion should occur are the incoming stream segment
to the primary node with the greatest drainage area and the stream segment
that connects the primary node to its downstream neighbor (Fig. ). If the two segments are straight, then a neighboring
node of the primary node is chosen at random and lateral erosion occurs at
this node until elevation changes at the node.
The calculation of the radius of curvature along two stream segments in a raster grid
with D8 flow routing presents a challenge, as the angle between segments is
discretized; the two segments may form a straight line, in which case the
angle is equal to 0∘, form a 45∘ angle, or form a 90∘
angle. In order to reduce the impact of this discretization, we assume that
each of these three cases represents a continuum of possible radii of
curvature. Cases of two straight segments are treated as if the actual angle
between them ranges anywhere between +22.5 and -22.5∘. If one
takes the average among these possible angles, the resulting inverse radius
of curvature is 0.23/dx, where dx is the cell size in the flow direction.
Similarly, we assume that a 45∘ bend represents a continuum of
possible angles between the two segments ranging from
22.5–63.5∘, resulting in an inverse radius of curvature of
0.67/dx. Following the same principle for a 90∘ bend gives a mean
inverse radius of curvature of 1.37/dx (see the Supplement).
The volumetric rate of material eroded laterally for each lateral node is
calculated by El×dx×H, where H is water depth given in
meters. Water depth at each node is calculated by H=0.4Q0.35, where Q is given in m3s-1. The volume of sediment
eroded laterally per time step is sent downstream along with any material
eroded from the primary cell. The volumetric erosion rate is multiplied by the
time step duration to get the volume eroded at the lateral nodes, and the
cumulative volume eroded from each lateral node is tracked throughout the
entire model run. When the cumulative volume eroded from the lateral node equals
or exceeds the volume needed to erode the node (see end-member model
descriptions below), the elevation of the lateral node is set to the
elevation of the downstream node (Fig. ). Flow is then
rerouted and water flows down the path of steepest descent. The model does
not distinguish between sediment and bedrock in the model grid and all
material that is eroded has the bedrock erodibility of the Kv or Kl
terms. When material is eroded vertically or laterally from bedrock nodes,
the volume of the eroded material is sent downstream as part of the Qs
term. If deposition occurs in the model, deposited material is added to the
topography of the node as bedrock. Thus, sediment is not “seen” in the
model as material that can be easily re-eroded after deposition; rather,
sediment works to increase the deposition term (Eq. ).
The lateral erosion rate presented here (Eq. ) relates lateral
erosion to the radius of curvature, but the application of this model is not
limited to meandering streams. Streams with fully developed meandering are
part of a relatively small subset of streams that are able to widen valleys
through lateral erosion; there are examples of streams that are classified as
single-thread or braided, yet which clearly show evidence of erosion and
lateral migration at locations where an outer bend in the channel impinges on
a valley wall or terrace .
Conceptually, therefore, this approach is not meant to exclusively represent
channels with fully developed meandering.
Conceptual illustration of model nodes showing the stream segments
(in light blue) from the upstream node, to the primary node (in green), to the
downstream node. Vertical erosion (Ev) occurs at the primary node. The
neighbor node (in pink) where lateral erosion (El) occurs is located on
the outside bend of the stream segments. The height over which lateral
erosion occurs, H, is shown by the dashed blue line. (a) For the total block
erosion model, the volume that must be laterally eroded before elevation is
changed is (Zn-Zd)dx2, the difference in elevation between the
neighbor node and the downstream node (indicated with double-sided black
arrow) times the surface area of the neighbor node. (b) The elevation of the
lateral node is changed after the entire block is eroded and flow can
potentially be rerouted. (c) In the undercutting-slump model, the volume that
must be laterally eroded (representing bank undercutting) before elevation is
changed is (H-Zd)dx2. H-Zd is the difference in elevation between
the water surface height and the elevation of the downstream node, indicated
with the double-sided black arrow. (d) When the neighbor node has been
undercut, elevation is changed, allowing water to be rerouted, while the
slumped material is transported downstream as wash load.
End-member model formulations
We have implemented two methods of determining whether enough lateral erosion
has occurred to lower the lateral node. The first method, the total block
erosion model, dictates that the entire volume of the lateral node above the
elevation of the downstream node must be eroded before its elevation is
changed (Fig. a, b). This formulation assumes that the
height of the valley walls is a controlling factor in the ultimate width a
valley can achieve, and thus valley width scales with valley wall height. In this
method, lateral migration depends on bank height so that taller banks
experience slower lateral migration, as all of the volume of the lateral node
must be eroded for the valley to widen. The second method, the
undercutting-slump model, dictates that only the volume of the water height
on the bank times the cell area must be eroded for the elevation to change
(Fig. c, d), while the remaining material slumps into the
channel and is transported away as wash load, i.e., not redeposited in the
model or included in Qs calculations. This model formulation represents
the migration of valley walls independent of valley wall height. With these
two end-member models, we address whether the lateral erosion rate should scale
with valley wall height. Valley wall or bank height is known to limit lateral
channel migration and valley width in transport-limited streams where
additional sediment from valley walls cannot be transported out of the
channel . However, whether valley wall height should limit
valley widening in detachment-limited bedrock channels is less clear
and likely depends on the bedrock lithology
(; ). The links between these
end-member model formulations and the natural processes they represent are
explored in the Discussion section.
Model runs and parameters discussed in this paper.
Model versionKl/KvKαNumber of runsTotal block1.0–1.51×10-40.2–2.010Total block1.0–1.55×10-5–2.5×10-40.810Undercutting-slump1.0–1.50.00010.2–2.010Undercutting-slump1.0–1.50.00005–0.000250.810TB water flux1.0–1.50.00005–0.00250.86UC water flux1.0–1.50.00005–0.00250.86TB sed flux1.0–1.50.00010.2–2.010UC sed flux1.0–1.50.00010.2–2.010Model experiments
In order to constrain the conditions that result in significant lateral
bedrock erosion and valley widening, we ran sets of models using a range of
values for bedrock erodibility, α (sediment mobility number), and the
Kl/Kv ratio using both the total block erosion model and the
undercutting-slump model (Table ). The model domain was
600 m
by 600 m with a 10 m cell size, three closed boundary edges, and an uplift rate
relative to base level of 0.0005 myr-1 imposed on the entire model domain.
Water flux was introduced at the top of the model by designating a node as an
inlet with an area of 20 000 m2 and sediment flux at carrying capacity.
This setup allowed each run to have a primary channel on which to measure
width and channel mobility. All models were spun up to an initial condition
of approximately uniform erosion rate with vertical incision only. The models
were then run for 100–200 kyr with the lateral erosion component. In order to
isolate the effect of bedrock erodibility, a set of model calculations was
run in which erodibility ranged from 5 × 10-5 to 2.5 × 10-4 while α was held constant at 0.8. In order
to isolate the effect of detachment-limited vs. transport-limited behavior,
another set of models was run in which erodibility was held constant at
1 × 10-4 and α values ranged from 0.1 to 2, which represents
a detachment-limited system when α<1 and a transport-limited system
when α>1 (Table ).
Kl/Kv ratios for all model runs were set to 1.0 or 1.5, resulting in a
runoff rate of 14 or 36 mmh-1 from Eq. (). These runoff
rates do not represent a yearly mean annual runoff, but rather peak event runoff
rates that are likely to result in appreciable lateral erosion due to the
scaling with the Kl/Kv ratio. found that
bedrock erosion rates in abrasion mill experiments are an order of magnitude
higher in samples from channel margins compared to the channel thalweg. This
suggests that Kl in this model should be at least equal to Kv and
could be much higher .
Understanding the model behavior in response to detachment- vs. transport-limited behavior (represented by α) and the Kl/Kv ratio is
complex and requires understanding how runoff plays into both parameters. The
value of α is calculated by vs, a proxy for grain size, and runoff
rate, R, although neither grain size nor runoff is explicitly set in the
model runs. Values of α that capture a range of detachment- or
transport-limited behavior are set instead (α= 0.2–2.0). When
the Kl/Kv ratio is set for a given model (either 1.0 or 1.5 in all model
runs), the runoff rate is calculated inside the model. Once a runoff rate for
a given Kl/Kv ratio is calculated, by extension a value of vs can be
calculated from runoff rate and the set α value. Therefore, in model
runs with the same Kl/Kv ratio and therefore the same runoff rate, a
transport-limited system (α greater than 1) has a larger grain size
(approximated by vs) compared to a detachment-limited system with a low
α.
Measures of lateral erosion in model landscapesChannel mobility
Channel mobility distinguishes models with lateral erosion from models with
only vertical incision. At steady state, channels in models with only
vertical bedrock incision do not migrate across the model domain. However, a
mobile channel is necessary to carve wide valleys and it is enticing to say
that the more mobile the channels, the wider the bedrock valley. In our
model, channel mobility is not controlled by sediment flux, as found in
alluvial channels , but by the
lateral erosion of bedrock. However, the term “channel mobility” is used
here in the same sense as in the alluvial literature; channel mobility describes
lateral channel planform changes along the length of the channel.
The effect of bedrock erodibility and α on channel migration through
time for both model versions is shown in Fig. . Channel
migration over 200 kyr is shown for six selected runs that span the range of
bedrock erodibility and α values for the two different model
formulations: the undercutting-slump model in which Kl/Kv=1.5 and the
total block erosion model in which Kl/Kv=1.5. In all runs, the total block
erosion model produced more confined channels compared to the
undercutting-slump model. The undercutting-slump model produces more dynamic
channel migration over the model domain, especially in the high K model. In
both model formulations, the high K and high α runs have the widest
extent of channel migration (recall that high α represents lower
sediment mobility) and the low K and low α runs have the most
restricted channel migration.
In order to describe channel mobility in our model runs in a single term, we
calculate a cumulative migration metric, λ, which is calculated
by first determining the migration distance of the channel between time steps
at all model cells the main channel occupies. Most often, the migration
distance between time steps at a single cell will be 0 or 10 m, indicating no
migration or migration to a neighboring cell. The mean of migration distances
between time steps is taken and summed over the duration of the model run to
give the cumulative migration metric, λ, which indicates how often the
channel has migrated during the model run. A model run can have the same
λ value if the channel marches across the entire model domain or if
the channel repeatedly switches between two nearby channel courses; λ
can also be used as an indicator for the maximum lateral extent occupied by
the channel during the model run. That is, the maximum possible extent of x
positions occupied by the channel is equal to λ, but the actual x
distance occupied by the main channel could be lower as the channel migrates
over the same area repeatedly.
Bedrock erodibility and the Kl/Kv ratio have the strongest control on
channel migration distance. Channel mobility increases as bedrock erodibility
increases in both the total block erosion model and the undercutting-slump
model (Fig. a, b). When K is low, representing strong
bedrock lithology, there is limited channel movement in the total block
erosion models with λ values between 15 and 35 m. This means that on
average during the model run the channel occupied 1–3 cells (Fig. c). With low values of K, the undercutting-slump model had
λ values around 200 m, but a lateral extent of only 5 model cells
(Fig. c). This indicates that in the undercutting-slump
model, the channel was actively migrating within a small area of the model
domain. In model runs with high K values representing weak bedrock, total
channel migration, λ, and the spatial extent of the
channel migration increase (Fig. a). With the total block model,
λ appears to be a good proxy for the total spatial extent of channels,
but for the undercutting-slump model, λ tends to overestimate the
lateral extent of channel occupation (Fig. ).
Increasing the Kl/Kv ratio from 1.0 to 1.5 results in 1.5–2 times
more channel mobility, with the largest relative increases in total block
erosion model runs with high erodibility and higher α values (Fig. a, b).
This is because the undercutting-slump models
already have high channel mobility with Kl/Kv equal to 1. Increasing
the Kl/Kv ratio to 1.5 increases channel mobility in UC models, but the
total block erosion models have a larger threshold for lateral erosion, so the
increased Kl/Kv ratio results in relatively more channel mobility in
the total block models.
For model runs with the same bedrock erodibility but different α
values (which represents sediment mobility), channel mobility is lower in
models with lower values of α (representing high sediment mobility)
and higher when α>1 (representing less mobile sediment; Fig. b). In the following presentation and discussion of
models with varying values of α, high α model runs will be
termed “transport limited” and low α model runs will be termed
“detachment limited”. This effect is most pronounced in the total block
erosion models in which channel mobility increases by a factor of 4 as
α increases. In the undercutting-slump models, channel mobility also
increases with α, especially when Kl/Kv=1.5. When Kl/Kv=1 in the undercutting-slump models, the trend in channel mobility vs. α is less well defined.
Channel positions over 200 kyr with different values for bedrock
erodibility and α in the undercutting-slump model (UC model, blue
lines) and total block erosion model (TB model, red lines). (a) High bedrock
erodibility (K=2.5×10-4), medium α value (α=0.8).
(b) High α value, indicating low sediment transport (α=2.0),
medium bedrock erodibility (K=10-4). (c) Low bedrock erodibility (K=5×10-5), medium α value (α=0.8). (d) Low α,
indicating high sediment transport (α=0.2), medium bedrock erodibility
(K=10-4).
Cumulative channel-averaged migration (a, b) and mean valley width
(c, d) averaged over 100 kyr for spin-up models with no lateral erosion (spin,
black triangles), total block erosion models (TB, red markers), and
undercutting-slump models (UC, blue markers) with Kl/Kv=1 (square
markers) and 1.5 (circle markers). (a) Cumulative channel-averaged migration
(λ) for model runs with α=0.8 plotted against bedrock
erodibility, K; (b)λ for model runs with K=10-4 plotted
against α. Mean valley width averaged over 100 kyr of the model runs.
(c) Mean valley width for model runs with α=0.8 plotted against bedrock
erodibility, K. (d) Mean valley width for model runs with K=10-4
plotted against α.
Valley width
Valley width is the primary indicator of lateral erosion; a wide bedrock
valley implies that significant lateral erosion has occurred relative to
vertical incision. Valleys can be defined in a few different ways and valley
width needs to be quantified in our model. Many studies use low gradient
areas of a DEM to determine valley width
e.g.,. This gives the width
for the valley bottom that has been shaped by channel processes, but excludes
areas that have been recently shaped by channel processes and then reworked
by hillslope processes. Another way to measure valley width is by determining
the width of the valley at a certain height above the channel. This simple
metric is often used for finding valley width in the field, for example using
eye height above the channel e.g.,. Using a certain height above the channel to
determine valley width in the models cannot distinguish between a fluvially
carved bedrock valley and low relief in a landscape with weak bedrock.
Instead we define valley width as the width of the area perpendicular to the
main channel where slope is characteristic of the fluvial channel rather than
hillslopes for a given bedrock erodibility and α value. The reference
slope for a fluvial channel is given by the slope–area relationship, assuming
that the height of the landscape and Qs are steady in time. When the
height of the landscape is in equilibrium, Eqs. () and
() are combined and rewritten as
U=e-νsd*QsRA.
At steady state, Qs is the total upstream eroded material given by Qs=AU. Substituting the steady-state equation for Qs and Eq. (8) into Eq. () gives
U=KvA1/2S-αU.
Solving the above equation for S gives the equation for reference slope
that determines whether a model cell is shaped by fluvial or hillslope
processes .
S=UKvA1/2(α+1)
Our models successfully produce bedrock valleys that are several model cells
wider than the channels that created them (Fig. ). Models
with only vertical incision have v-shaped valleys that are only 1 model cell
wide (10 m in our experiments) and the channels do not shift laterally
(Fig. a). Given the specifications of the total block and
undercutting-slump models, it is not surprising that the total block models
take longer to respond to the onset of lateral erosion and valleys are more
narrow than in the undercutting-slump models. The total block erosion models
take on the order of 10 kyr to produce an observable response to lateral
erosion and ultimately produce bedrock valleys that are up to 25 m wide,
while the undercutting-slump models take about 5 kyr to show a response to
lateral erosion and ultimately produce valleys that are up to 50 m wide.
Model topography and cross sections at y=500 showing examples of
valley widening. The black line indicates the position of the main channel on the
landscape. The red triangle shows the position of the main channel in the cross
section. (a) Model with vertical incision only. (b) Total block erosion model
after 70 kyr of lateral erosion. (c) Undercutting-slump model after 50 kyr of
lateral erosion.
Figure 6 shows slope maps of total block and
undercutting-slump models that show the width of the valley shaped by fluvial
processes. The blue areas have slopes that are characteristic of fluvial
channels and red areas have slopes that are characteristic of hillslopes. The
total block erosion model with a low α value shows very little bedrock
valley widening as evidenced by the thin band of blue along the main channel
1–2 model cells wide (Fig. a). Increasing α to
obtain transport-limited behavior in the model results in wider valleys that
have been shaped by the channel that are 2–3 model cells wide in the total
block erosion model (Fig. b). The landscape in the
undercutting-slump model has wider valleys that result from more extensive
carving by channels. The fluvially carved valleys in the detachment-limited
model are about 2–3 model cells wide and the valleys in the transport-limited
model are over 50 m wide in some places (Fig. c, d).
Slope maps showing fluvially carved valleys in total block erosion
and undercutting-slump models with high and low values of α. The white
and blue areas in the maps indicate slopes that are characteristic of
fluvial channels, i.e., lower than the reference slope value (Eq. ). (a) Total block erosion model, low α
(detachment limited). (b) Undercutting-slump model, low α
(detachment limited). (c) Total block erosion model, high α
(transport limited). (d) Undercutting-slump model, high α
(transport limited).
Figure c and d show valley width for the lower two-thirds
of the model channels averaged over the duration of the model runs in 54
model runs. To ensure that using characteristic fluvial slope as the
criterion for a valley in all model runs gives valley width resulting from
lateral erosion, and not valley width inherent in the model, we first use
this criterion to measure valley width for the spin-up models that include no
lateral erosion component. Valley width for the spin-up models is
consistently 10 m, the width of 1 model cell. Valley width does not change
significantly for any of the total block model runs in which K is varied
and α is held constant (Fig. c). When the
Kl/Kv ratio is increased from 1 to 1.5, valley width increase slightly
for all model runs, but wide valleys are not possible in the total block
erosion model with this value of α. Valley width in the
undercutting-slump model for changing bedrock erodibility shows a somewhat
counter-intuitive signal (Fig. c); the
undercutting-slump model results in wider valleys for lower values of bedrock
erodibility. The reasons for this signal are discussed in the section below.
When α is varied and K is constant, valley width increases with the
tendency towards transport-limited conditions (α>1) in all
undercutting-slump models, but only in total block erosion models when the
Kl/Kv ratio is equal to 1.5 (Fig. b). The widest
valleys for a given bedrock erodibility occur with high α values as a
result of higher slope. The models predict more channel mobility and wider
valleys under transport-limited streams (set by α) compared to
detachment-limited streams (Fig. b, d). As α
increases, the deposition term increases, and a steeper slope is needed to
maintain the landscape in steady state relative to uplift. Higher channel
slopes in transport-limited model runs also cause increased lateral erosion
according to Eq. ().
Linking channel mobility and valley width
We have shown that the greatest channel mobility occurs in the
undercutting-slump models and increases significantly with increasingly soft
bedrock (Fig. a). However, maximum channel mobility
does not translate into maximum valley width. In the undercutting-slump
models, the widest valleys occur in the low erodibility model runs that have
relatively low channel mobility. This reflects the fact that the areas visited by the
migrating channel in the low-relief, high K model runs are easily
overprinted by small-scale fluvial processes and lose the slope signature of
the larger channel. This prevents our algorithm from finding an area of
the model that has recently been shaped by the channel. The mismatch between
channel mobility and valley width also reflects the fact that hard bedrock valleys are
allowed to erode very easily in the undercutting-slump model and the surface
smoothed by the channel is persistent through time. The relationship between
hard bedrock and wide valleys reflects the use of the undercutting-slump
model, which is inappropriate for hard bedrock wall erosion in natural
systems. With the undercutting-slump model, only a small volume threshold
must be overcome for lateral erosion to occur, and the rest of the node
material is transported downstream as wash load. However, models that have
resistant bedrock (low K) are least suitable for the undercutting-slump
model. In order for this model to be a good description of how nature works,
the bed material would need to be able to break up into small pieces that are
easily transported away, which is conceivable for resistant clay banks.
However, the total block erosion model is generally more appropriate for
representing the erosion of resistant bedrock channels that erode into
material transported as bedload.
Valley width averaged over the model domain vs. model time for total
block erosion and undercutting-slump models with Kl/Kv= 1 and 1.5.
Increased water flux occurs from 100 to 150 kyr, indicated by light blue
shading.
Adding complexity: water flux, sediment fluxEffects of increased discharge on lateral channel migration
In order to investigate how transience in landscapes affects lateral erosion,
we introduce increased discharge at the inlet point in the upstream end of
the model. Using drainage area as a proxy for discharge, increasing water
flux in the model represents how a larger stream on the same landscape
influences valley width. Increasing drainage area also allows us to observe
the extent of landscape change and how rapidly the different model runs
respond to an event such as stream capture. The drainage area at this input
point is increased from 20 000 to 160 000 m2 and sediment load is
set to the carrying capacity of the new drainage area. For a typical model
run, the additional drainage area approximately doubles the magnitude of
drainage area at the outlet of the main channel in the model domain; i.e.,
maximum drainage area in model runs increases from ∼1×105 to
∼3×5m2. Models with increased water flux were run using both model
formulations, Kl/Kv=1.0 and 1.5, and erodibility values that ranged
from 5×10-5 to 2.5×10-4 with alpha held constant at 0.8
(Table ).
Recalling that lateral erosion scales with drainage area (Eq. ),
while vertical incision scales with the square root of
drainage area (Eq. 8), we therefore expect that increasing
drainage area will increase lateral erosion and valley width in every case
for the undercutting-slumping model in which the numerically imposed condition
for lateral erosion to occur is much smaller than in the total block erosion
model. In the total block erosion model, lateral erosion temporarily stalls
because of the volume threshold that must be exceeded before lateral erosion
occurs. There is no threshold for vertical incision, which speeds up when
additional water flux is added to the model.
Surface topography and cross section at y=420 during the period of
increased water flux for the total block erosion models. The red triangle on
cross sections indicates the channel position. (a) Total block erosion model
with low K and Kl/Kv=1.0 at 100 kyr before the increase in water
flux. Note that this model looks similar to the spin-up model runs with no
lateral erosion. (b) After 15 kyr of increased water flux, the cross section
shows vertical incision in the channel and increased relief between the
channel and the hillslopes. (c) At 30 kyr after water flux increased,
equilibrium is reached. Lateral erosion can begin and the valley widens to
20 m at y=420.
Surface topography and cross section at y=420 during the period of
increased water flux for the undercutting-slump models. The red triangle on cross
sections indicates the channel position. (a) Undercutting-slump model with low
K and Kl/Kv=1.5 at 100 kyr before the increase in water flux. Valley
is 30 m wide. (b) After 15 kyr of increased water flux, the channel has both
vertically incised and laterally widened the valley to a width of 40 m.
(c) After 30 kyr of increased water flux, the valley has a width of 60 m at y=420.
Longitudinal profile, cross sections, and slope maps from model run
TB1.5, with medium K after cessation of increased water flux. (a) Longitudinal channel
profiles show uplift and aggradation, which produces a convexity that
propagates upstream. (b) Cross sections across the model domain at y=400 show
channel aggradation and new lateral erosion of valley walls. (c, d, e) Slope
maps show valley narrowing following the passage of the knickpoint where
y=400 (dashed line) at 155, 159, and 163 kyr.
Total block erosion models
In all of the model runs, increased water flux resulted in increased lateral
erosion and wider valleys. Figure shows valley width
averaged over the model domain vs. model time for all of the water flux
models. The total block erosion model and undercutting-slump model respond
differently to a step change in water flux. The total block erosion models
first incise vertically to a new steady-state stream profile then erode
laterally as a result of the increased water flux (Fig. ), while the undercutting-slump model incises vertically
and erodes laterally simultaneously (Fig. ).
Total block erosion models in which the Kl/Kv ratio is equal to 1.5
(TB1.5) show an interesting pattern in valley widening after increased water
flux (Fig. c). All of the TB1.5 model runs show a
significant increase in valley width during the 50 kyr period of increased
water flux. After 6 kyr of increased water flux (model time = 106 kyr), the
high and medium erodibility model runs have greater valley widths, but the
low erodibility model shows a gradual increase in valley width over 14 kyr of
increased water flux (model time 100–114 kyr). For the first 14 kyr of the
increased water flux, the channel of the low K model run incises rapidly,
increasing the gradient between the channel and the adjacent cells and
preventing lateral erosion. After the channel profile comes into new
equilibrium, the increased water flux accelerates lateral erosion on the
valley walls and valleys widen by 10 m compared to before the increased water
flux in the total block erosion models.
After the increased water flux stops at 150 kyr, the wider valleys persist for
∼10–20 kyr in the low and medium erodibility models (Fig. c) for two reasons. First, after the cessation of
increased water flux, the channel returns to equilibrium through aggradation
and uplift. While aggradation occurs, lateral erosion can occur more easily
in the total block erosion models. In this case, the total volume that must
be eroded from any lateral node cell is reduced as the channel floor moves up
in vertical space. The second reason for persistent wide valleys is that in
the medium and low K model runs, the increase in water flux eroded wide
valleys into relatively resistant bedrock. These flat surfaces near the
channel persist in harder bedrock, even after water flux has decreased to
original levels. Following the end of the period of increased water flux,
valley width in the TB1.5 medium K model run remains elevated for
10 kyr
(model time 160 kyr) before channel narrowing that propagates upstream
(Fig. ). After cessation of the increased water flux at
150 kyr, the channel profile returns to equilibrium through uplift and
aggradation (Fig. a). Channel aggradation begins at the
bottom of the channel profile and results in a convexity that propagates
upstream (Fig. a). At model position y=400, from 150–158 kyr
the channel increases in elevation due to uplift (Fig. b). Wide valleys created during increased water flux are
maintained, and new lateral erosion of valley walls is seen (Fig. b). At 159 kyr, 9 kyr
after the cessation of increased water
flux, the aggradational knickpoint reaches y=400 and incision and valley
narrowing is observed (Fig. d, e).
Figure shows surface topography and cross sections across
the model domain for three times in the low erodibility model run using the
total block erosion model. This figure demonstrates the effect of valley
deepening and then widening in response to increased water flux. Before water
flux is increased, the channel is narrow and has steep valley walls (Fig. a). After 15 kyr
of increased water flux and increased
vertical incision, the topography reaches a new equilibrium and channel
elevation is stationary. Only after this period of re-equilibration can
lateral erosion begin to widen the valleys. After 30 kyr of increased water
flux, the entire channel has incised, especially in the upper valley. At
y=420, the position of the cross section, the channel has been incised by 3 m,
and the valley has widened to about 20 m (Fig. c).
This response of primarily vertical incision is expected when using the total
block erosion model, which sets a high threshold for lateral erosion.
Undercutting-slump models
In the undercutting-slump models, all of the model runs show a significant
increase in channel mobility with additional water flux (Fig. b, d).
The largest valley widths occur in the models with
low bedrock erodibility for reasons discussed above. Unlike the total block
erosion models, there is no discernible lag between the onset of water flux and
valley widening in the undercutting-slump models (Fig. ).
This is because erosion of the valley wall is independent of the height of
the valley wall for the undercutting-slump model formulation and the increase
in drainage area results in larger increases in lateral erosion rates faster
compared to vertical incision rates (Eqs. 8, ).
Figure shows topography and cross sections for three times
in the low erodibility model run using the undercutting-slump model. Before
water flux is increased, the channel is significantly wider than in the total
block erosion model. The cross section shows a 30 m wide valley with low
gradient areas next to the channel, indicating that these areas were shaped
by the lateral erosion (Fig. a). Following the increase
in water flux, the valley is much wider across the entire model domain,
especially at the upstream segments of the channel. After 15 kyr of increased
water flux, the channel has both vertically incised and widened the valley to
∼40m at y=420 (Fig. b). After 30 kyr of increased
water flux, the valley has widened further to ∼60m at y=420 (Fig. c).
The undercutting-slump model runs with medium and low
erodibility maintain increased valley width after water flux has decreased,
particularly in Kl/Kv=1.5 models (UC1.5) (Fig. d).
This indicates that wide valley floors can persist for
long periods of time after the conditions that created them have stopped.
Mean valley width for the upper half of the model domain over the duration
of additional sediment flux model run for total block erosion and
undercutting-slump models with Kl/Kv ratios of 1 and 1.5. Light blue
shading indicates the duration of increased sediment flux.
Effects of increased sediment flux on lateral erosion
In order to explore how the addition of sediment to a stream affects lateral
erosion and valley widening, we added sediment to the inlet point at the top
of the model. The sediment flux models were run for 100 with 50 kyr of
standard lateral erosion followed by 50 kyr of increased sediment flux. Before
additional sediment was added, the sediment flux at the inlet was equal to
the carrying capacity of the stream, which is equal to UA. Models with
increased sediment flux were run using both model formulations, Kl/Kv=1.0 and 1.5, and α values that ranged from 0.2–2.0, with bedrock
erodibility held constant at 1×10-4 (Table ).
During the 50 kyr periods of increased sediment flux, 5 times more sediment
flux was added, forcing all of the streams to aggrade initially. Adding
sediment increases the deposition term (Eq. ), which results
in aggradation if the model is initially in steady state, that is e-d=U.
Aggradation in the channels continues until the channel slopes become steep
enough to increase the vertical erosion term so that e-d=U again, and the
landscape is in a new equilibrium state. In this model, no distinction is
made between the erodibility of deposited material and bedrock; any deposited
material in the model has the properties of bedrock rather than sediment. The
model responds to changes in sediment flux by adjusting channel slope, rather
than both slope and channel width as observed in natural systems
, because of the fixed-width scaling in this
model.
Model topography and cross sections at y=420 during the period of
increased sediment flux for the total block erosion model with α=1.5
and Kl/Kv=1.5. The black line indicates the position of the main channel on the
landscape. The red triangle shows the position of the main channel in the cross
section. (a) Before increased sediment flux is introduced at the input point,
indicated with the arrow. (b) After 20 kyr of increased sediment flux, the
channel has aggraded by 4 m and has eroded the valley wall by 30 m.
Figure shows valley width averaged over the upper half
of the model domain (closest to the sediment source) plotted against model
time. After sediment is added to the models, all of the model runs show a
significant increase in valley width except the low α model runs,
which show little change in width. Valley width increases more and valleys
stay wide for longer with higher values of α. Valleys are the narrowest
and least persistent through time in the TB1 model group (Fig. a), and valleys are the widest and most persistent through
time in the UC1.5 model group (Fig. d). Valley widths
and the duration of wide valleys after the addition of sediment are similar
between the TB1.5 group and the UC1 group (Fig. b, c).
The addition of sediment to these models results in channel aggradation and
valley filling that accounts for a substantial fraction of measured increases
in valley width for all of these model runs. It is not possible to
distinguish between widening due to valley filling and widening due to
bedrock wall retreat from this spatially averaged value of valley width.
Lower values of α showed little or no increase in bedrock valley width
after the addition of sediment flux. This is because channels in the low
α runs (high sediment mobility) easily adapt to the increased sediment
flux without significant or far-reaching changes to the channel slope. It is
interesting to note that mean valley width increases at 50 kyr for all model
runs, then declines to close to pre-sediment values by about 80 kyr. Mean
valley width begins to decline as the models come into steady state with the
increased sediment flux, indicating that lateral erosion can most readily
occur when the channel is in a transient, aggradational state.
Figure shows an example of simultaneous valley filling
and significant bedrock erosion in the TB1.5 model group. Before the addition
of sediment flux (t=50kyr), the channel is 10 m wide. Other channels
shown in the cross section (at 80 and 250 m) are immobile and show little
evidence of lateral erosion. After the addition of sediment to the model, the
main channel aggrades by 4 m while also shifting 30 m to the right,
eroding a significant amount of bedrock valley wall over 20 kyr.
Figure shows the α=1.5 run from model group
UC1.5 before and after the added sediment flux that results in true bedrock
valley widening. At 50 kyr in the model run before the additional sediment is
added, the valley in the upper half of the model domain (y=240) is about
30 m
wide (Fig. a). Over 50 kyr, sediment is added to the
model and the channel aggrades for ∼20kyr before it comes into steady
state; i.e., its slope is steep enough to carry the additional sediment load
and aggradation stops. During the 20 kyr of aggradation, this model run shows
both retreat of the valley walls and channel aggradation. By 70 kyr in the
model run, the channel has aggraded by 5 m and the valley is 50 m wide
(Fig. b). During this 20 kyr period, the channel has
migrated 50 m to the right, eroding the hillslope and forming steep valley
walls.
Model topography and cross sections at y=420 during the period of
increased sediment flux for the undercutting-slump model with α=1.5
and Kl/Kv=1.5. The black line indicates the position of the main channel on the
landscape. The red triangle shows the position of the main channel in the cross
section. (a) Before increased sediment flux is introduced at the input point,
indicated with the arrow. (b) After 20 kyr of increased sediment flux, the
channel has aggraded by 5 m and has eroded the valley wall by 50 m.
Before the increase in sediment flux, all channels are in equilibrium by
definition. Adding sediment to the inlet point in the models causes the
channels to aggrade in all model runs, increasing the channel slope. This
increase in channel slope increases the lateral erosion term and the vertical
erosion term (Eqs. 8, ), but while the channel is
aggrading, vertical incision is effectively zero. Therefore, for the total
block erosion models, most new lateral erosion should occur while the channel
is aggrading because the threshold volume that must be eroded becomes
smaller when relief between the channel node and neighboring nodes decreases
(Fig. ). Figure shows that after
sediment flux is added, there is a persistent increase in valley width for
many model runs even after the channel profile has come into steady state
with respect to the added sediment flux. The permanent increase in slope
should result in higher lateral erosion rates, resulting in permanently wider
valleys because the increased vertical incision rates that result from the
higher slope are offset by increased deposition. This suggests the possibility
that if a channel experiences increased slope through aggradation, then more
lateral erosion occurs.
DiscussionComparison among purely vertical incision models and end-member lateral erosion models
The simple theory for lateral bedrock channel erosion presented here,
combined with a landscape evolution model, produces valleys that are several
times wider than the channels they hold. The development of wide valleys is
sensitive to the end-member model formulation selected, which is discussed
below. The widest valleys in this set of models occur in transport-limited
model runs (high α values) when using the undercutting-slump model
formulation, which represents lateral erosion that is independent of valley
wall height. Wider bedrock valleys under conditions of relatively immobile
sediment (high α value; Fig. ) reflect conditions
observed in natural systems where wide bedrock valleys are considered a
diagnostic feature of transport-limited streams .
The results presented here show that the lateral erosion component allows for
mobile channels in all model runs (Fig. a, b), even
when the model has reached steady state, unlike models with vertical incision
only that have stationary channels at steady state. The modeling experiments
show that landscapes with highly erodible bedrock have the most mobile
channels. In the total block erosion model formulation, weak bedrock allows
for greater channel mobility because the amount of lateral erosion that must
occur to erode valley walls is lower in low-relief landscapes with easily
eroded bedrock . The model also predicts more
channel mobility and wider valleys in models with high values of α
(low sediment mobility), especially in the total block erosion models.
Channel mobility is a critical factor in the development of wide bedrock
valleys because all of the erosion of the valley must be accomplished
through erosion by the channel (e.g., ). The
width of surfaces beveled by lateral erosion has been framed as a competition
between channel mobility and relative rock uplift rate
, with greater channel mobility resulting in more area
shaped by lateral erosion. The mobility of river channels increases with
increasing sediment flux , which emphasizes the
potential importance of high sediment load as a requirement for the
development of wide bedrock valleys. Landscapes in weaker bedrock are more
likely to have more channel mobility and wider valleys
e.g.,. Rivers flowing through soft bedrock are also more
likely to behave as transport-limited rivers as a result of the increased
sediment flux in the stream from the surrounding hillslopes and lower channel
slopes in easily eroded bedrock. Channel mobility as a parameter extracted
from the model is also important because measures of channel mobility during
periods of lateral planation e.g., can be used
to validate future lateral erosion models.
The two model formulations presented here describe end-member behavior for
how the lateral erosion of valley walls scales with wall height and can also be
considered in terms of the physical processes of valley widening found in
natural systems. The total block erosion model, in which the entire volume of
a neighboring node must be eroded before lateral erosion can occur, best
describes lateral erosion in resistant material and/or material that erodes into
blocks that are not easily transported by the stream. This approach is used
to represent, in a simple way, a system in which the undermining of a channel
bank leads to the gravitational collapse of resistant material that must itself
then be eroded in place . The dependence of
rates of valley widening on wall height has been demonstrated in alluvial
systems where sediment transport rates in the channel are low relative to the
sediment eroded from valley walls . One can imagine a similar limitation in bedrock
gorges where lateral valley wall movement is accomplished through rockfall
into the river . Valley widening may also be
limited when valley wall height exceeds the height of the flood stage;
note that the vertical erosion of flat surfaces next to
the channel can result in valley erosion rates that are
orders of magnitude greater than lateral erosion rates alone suggest.
The undercutting-slump algorithm represents the lateral erosion of valley walls
that is independent of bank height. This model represents lateral erosion on
a bank that has been laterally undercut and the remaining material slumps
into the channel and is transported away as wash load, i.e., not added to the
Qs term or redeposited in the model. The undercutting-slump model is
applicable in locations with an under-capacity stream and lithology that
slumps easily and rapidly breaks down into small grains that are easily
transported . Lateral
erosion that is independent of valley wall height allows for the development of
wider bedrock valleys (Fig. ); the mechanism described by
the undercutting-slump model more likely occurs in weak bedrock that breaks
down into easily transportable grain sizes as observed in many natural
systems e.g.,. The undercutting-slump model consistently produces
wider bedrock valleys and more mobile channels than the total block erosion
model because less lateral erosion is required to erode valley walls in the
undercutting-slump model algorithm. However, this undercutting-slump model is
not appropriate for landscapes with very hard bedrock (low erodibility), as
evidenced by overhanging cliffs along many rivers and persistent blocks of
collapsed material following slumping or delivery from adjacent hillslopes
. The behavior of the models varies significantly
based on which model is selected, although the same general trends are seen
in both models. In nature, the lateral erosion of valley walls likely follows
neither of these end-members perfectly, but will operate on a continuum
between the two .
presented two end-member relationships between
channel erosion and valley erosion that are similar to the models presented
in this study and found similar behavior between their two models.
Model limitations and future directions
While the model captures several important markers of lateral bedrock
erosion, such as mobile channels and bedrock valleys that are up to 5–6
times the channel width, the model did not develop broad, smooth valleys
that are up to 100 times the width of their channel (Fig. ) and that are sustained over many years, as observed in
flights of strath terraces in the Front Range of Colorado, for example
. Some important elements of reality have been
simplified or omitted in this model, and future versions of the model should
address (1) resolving the effects of grid resolution on total lateral erosion
and valley width, (2) setting runoff variability and magnitude separately from
grain size, (3) including tools-and-cover effects and thresholds in the
vertical incision model, and (4) treating sediment and bedrock erodibility
separately.
In LEMs that use single-direction flow-routing schemes, such as the model
presented here, it is possible in principle to have an “implied width”
(implied by the width–discharge relation embedded in K; Eq. 8)
that is larger than a grid-cell size. This issue is not unique
to our particular model; any non-hydrodynamic LEM with sufficient resolution
faces the same inconsistency. We explored the effects of a modification to
the model through which the lateral erosion rate is calculated to account for both the
position of the channel within the model cell and cases in which the implied channel
width is greater than the cell size (Figs. S6, S7 in the Supplement). Using a flow-routing
algorithm that allows flow to be distributed to two downstream pixels when
the implied width is greater than the pixel size is a justifiable adaptation
that would improve the hydrodynamic handing of water flow in this model,
particularly with smaller pixel sizes. However, the intent of developing this
new lateral erosion model within an LEM was to investigate how lateral erosion
might be implemented within the context of an otherwise fairly generic and
common model formulation without excessive complexity.
Sensitivity tests were conducted to explore the effect of grid size on total
lateral erosion and valley width during model runs with dx=10, 15, and 20 m (Figs. S8–S12).
Grid size effects on cumulative lateral erosion are
particularly pronounced in the total block erosion formulation (Fig. S8)
due to the increased volume that must be eroded for lateral erosion to occur
when grid size is increased. Using the total block erosion model in which
lateral erosion scales with valley height, larger grid size can result in
less lateral erosion, more narrow valleys, and longer response times for
lateral erosion to occur. Using the undercutting-slump model resulted in
valley widths estimated from cross sections and slope maps that are
reasonably similar among models with dx=10, 15, and 20 m. The finding
that grid size affects the magnitude of lateral erosion and valley widening
to varying degrees is a limitation of the model that must be overcome before
model parameters can be calibrated. Grid-scale effects have been previously
documented in LEMs, and achieving solutions that are grid-scale independent
remains an open challenge .
In the case of lateral erosion, we suggest that identifying and implementing
a sub-grid scaling factor so that valley width becomes independent of cell
size in all model realizations is needed in order to predict the absolute timing
and magnitude of lateral widening. There are several strategies that could
usefully be explored, including the use of multidirectional routing schemes to
represent flow dispersion and the use
of downscaling techniques to correct for resolution bias
.
In order to focus on implementing the equations for lateral erosion into the
model, the simplest possible erosion–deposition model was used. This
erosion-deposition model (Eq. ) has the advantage of not
requiring the calculation of transport capacity and prevents potential
problems with abrupt transitions from erosion to deposition, but does so at
the expense of losing some details of runoff rate and grain size, which are
lumped into the parameter α. In this model, detachment- or
transport-limited behavior is set through α, which works well for
general model exploration, but becomes problematic when exploring specific
model responses to spatial and temporal changes in runoff rate and multiple
grain sizes. Setting runoff and grain size explicitly is an important next
step for determining how these factors independently impact bedrock valley
width and channel mobility. Including a dynamic Kl/Kv that is
calculated with runoff from discrete events and channel widths is a target
for future models. Runoff rate can vary widely, but a higher runoff intensity
will lead to a higher Kl/Kv ratio and more lateral erosion, as
suggested by field observations of lateral erosion in bedrock channels during
large flood events and correlation of increased
sinuosity and storminess of climate . The model
presented here does not have the capability to represent changes in
Kl/Kv based on processes that cause increased lateral erodibility, such
as changes in the distribution of sediment during high flow
or increased mass wasting of hillslopes
. A more process-specific representation of the
Kl/Kv ratio is a target for future model development.
The model presented in this paper uses the stream power incision model, the
simplest reasonable vertical incision model, in order to focus on our goal of
exploring the novel application of lateral bedrock erosion in a landscape
evolution model. Using a tools-and-cover incision model
e.g., in a future lateral erosion bedrock
model would be closer to the way we conceptualize lateral erosion in natural
systems. The main impact of using a tools-and-cover incision model in a
lateral erosion model would be less efficient vertical incision as relative
sediment flux increases . Slowing vertical erosion so
that lateral erosion can catch up is an important part of the mechanism cited
by many studies for allowing lateral erosion in incising streams
. Slowing vertical incision may be a necessary
condition for significant lateral erosion and bedrock valley widening, but it
is not by itself a sufficient condition. A model that describes how sediment
tools carry out lateral erosion needs to be constructed
, but tools-and-cover incision models do not offer
any mechanism for changing the rate of lateral erosion, just decreasing the
efficiency of vertical incision.
Another limitation of the current model is that sediment is not treated
explicitly, but rather is tracked in the model through the Qs term. No
distinction in erodibility is made between sediment and bedrock. In the
current model, when the landscape is in steady state, vertical erosion minus
deposition is equal to the uplift rate. Increasing sediment flux, Qs, in
the deposition term immediately results in channel aggradation and increasing
channel slope. In natural systems, channels respond to increased sediment
flux by increasing both slope and width. Changes in channel width are not
captured in this model due to the fixed value of kw, which is appropriate
for landscapes in quasi-equilibrium . How bedrock
channel width responds to changes in boundary conditions, such as uplift rate
and sediment, is the subject of ongoing research
e.g., with important implications for driving the channel
incision of slump deposits and terrace generation .
In not differentiating between sediment and bedrock explicitly in this model,
the different erodibilities of sediment and bedrock are not accounted for. In
most cases, sediment in a channel should be much easier to erode than the
bedrock in a channel by allowing more rapid lateral migration through cells
that have been previously occupied and contain some amount of sediment
. But in some cases, sediment in a soft bedrock
channel can be composed of coarse-grained, resistant lithology sourced from
upstream. For example, the streams that drain the Colorado Front Range flow
from hard, crystalline bedrock onto soft, friable shale bedrock
. The granitic cobbles that cover the channel
bed in stream segments underlain by shale bedrock take much more energy to
move than it takes to transport the friable flakes of shale that line the
walls of the channel. Different erodibilities should also result in more
active channel migration once a wide valley is established because the
channel erodes laterally through sediment that is more easily eroded than
bedrock .
Comparison between models and field studies
Lateral erosion rates depend on the magnitude of shear stress and tools
applied to channel walls and the resistance of the bedrock to erosion. Our
model of lateral bedrock erosion proposes that channel curvature controls
the lateral erosion rate. showed that extremely efficient
bedrock wall erosion of up to ∼80m over 5 years occurred where the
river encountered sharp bends. They attribute this rapid lateral bedrock
erosion in river bends to abrasion from sediment particles that detach from
flow lines in the curve and impact the wall. also
suggest that the lateral erosion rate by bedrock abrasion depends on how often
sediment particles are deflected towards the channel walls, specifically by
channel roughness elements. There is an important distinction between this
study and the work of in that their conclusions are
based on observations of lateral erosion in a straight flume. Lateral erosion
that occurs in the absence of channel curvature highlights the point that
channel curvature is not the only control on lateral erosion, but it appears
to be an important one.
The total block erosion model demonstrates how landscapes with hard bedrock
and detachment-limited conditions respond to increased discharge by first
incising the channel bed and then widening after the channel has come into
equilibrium (Fig. ). This behavior is similar to
narrowing and incision of bedrock channels in response to increased uplift
or vertical incision followed by channel widening
in response to increased discharge . The model
predicts that not only will channels in easily eroded bedrock reach
equilibrium more quickly than channels in resistant bedrock, but valleys will
also begin to widen faster in easily eroded bedrock than in more resistant bedrock
.
One of the few studies that has been able to report bedrock valley widening
through time is from a unique case in Death Valley .
Stream capture increased the drainage area of a small basin by 75-fold in the
1940s, and channel response over the following 60 years was mapped by aerial
photos. found that mean valley width in a channel
segment with weak bedrock increased by 9 m in 60 years. In contrast, in
channel segments in hard bedrock, they found vertical channel incision and
the development of knickpoints. They attribute the difference in response to
lithological differences and suggest that the presence of sediment on the bed
in the weak bedrock channel segments protects the bed from incision, allowing
the valley walls to migrate laterally. This difference in response is similar
to the behavior of the end-member models presented here: the total block
erosion model shows rapid incision and narrowing in response to increased
water flux, whereas the undercutting-slump models show incision and valley
widening.
Lateral erosion in nature is often attributed to increased sediment delivery
to channels, which suppresses vertical bedrock incision and gives lateral
erosion a chance to become the dominant mode of bedrock erosion
e.g.,. If this is the case,
then we expect increased sediment flux to have the largest effect on the low
α, detachment-limited model runs. The same amount of new sediment was
added to each model run, but the sediment resulted in more aggradation in the
high α runs. In the high α, transport-limited runs, the channels
already behave as if they are loaded with sediment. In low α runs, the
model tends towards detachment-limited behavior, so additional sediment is
rapidly and easily transported out of the system. The slope needed to
transport the additional sediment is lower in the detachment-limited runs,
resulting in less aggradation in response to the increased sediment flux. The
addition of sediment in this model does not lead to increased sediment cover
on the bed, as bedrock and sediment are not differentiated in the model;
rather, increased sediment flux results in the “deposition” of bedrock material
that aggrades the channel. This channel aggradation in the model certainly
indicates that vertical incision has stopped, allowing lateral erosion to
become the primary erosive agent even in models in which the Kl/Kv ratio is
low or in the total block erosion models. This predicted increase in lateral
erosion (relative to vertical incision) during periods of aggradation occurs
in some of the model runs, especially those with high α values. When
the channel has reached a new equilibrium following increased sediment flux,
many model runs maintain wider valleys due to the higher slope and increased
lateral erosion rates.
A potential test of the lateral erosion model
One of the goals of developing landscape evolution models is to develop and
test hypotheses about how dynamics in natural systems work over spatial and
temporal scales that are not readily observable. A challenge remains in how
to test a newly developed numerical model with field data. In order to test
simply how well this model captures the development of wide bedrock valleys,
we would need a field location where channel curvature is identified as the
primary mechanism for lateral erosion, for example rivers in mudstone
bedrock where particle detachment from the bank is from fluid stresses alone
. A field data set to test
this lateral erosion model could conceivably be derived from experimental
data, a well-constrained “natural experiment” of wide bedrock valleys that
developed over geologic timescales , or from rapid
valley widening associated with an extraordinary event. To our knowledge,
experimental data sets that describe the effect of channel curvature on
lateral bedrock erosion do not exist, nor have we identified an appropriate
natural experiment to evaluate bedrock valley widening over geologic
timescales.
Researchers have only recently started to study the mechanistic processes of
lateral bedrock erosion e.g.,.
The model presented here does not include all of the processes the community
has identified as relevant to lateral erosion; rather, we formulated the
simplest reasonable model to test the hypothesis that stream power exerted on
channel walls is a primary control on lateral bedrock erosion. We do not
consider small-scale processes, such as the abrasion of channel walls by
sediment, but rather focus on reach-scale drivers of valley wall erosion. Because
of the simplicity of our model and the grid size effects on valley widths
produced by the model (Fig. S8), this model is not currently suitable for
predicting the absolute timing or magnitude of lateral widening in natural
systems.
In the broadest terms, the key prediction of this model that can be compared
to field sites is the relationship of increasing valley width with drainage
area. So far, no other landscape evolution models consider lateral bedrock
erosion in a catchment-scale model; therefore, most LEMs predict no
relationship between valley width and drainage area. Our model does predict
increasing valley width with drainage area (Figs. , S7)
and a scaling relationship between width and drainage area that can be
compared with data from natural systems. Figure shows
valley width vs. drainage area data from one undercutting-slump model run
with increased water flux for a period of 50 kyr. The data shown in the figure
are from six time slices when the model is in steady state and are time
averaged over 2500 model years. The time-averaged valley width data have some
scatter, vary by ∼30m for a given value of drainage area, and
cover a limited range of drainage areas. Log-binned averages of valley width
show a scaling prediction that can be tested against field measurements of
valley width and drainage area. The scaling relationship predicted by this
model has a Kv coefficient of 0.16 ± 0.052 and a c exponent of
0.46 ± 0.027.
Several studies have shown a power-law relationship between valley width and
drainage area in natural systems . The power-law equation describing
the relationship between valley width and drainage area takes the generic
form of W=KvAc, where Kv is a widening factor and c is an exponent that
ranges in value from ∼0.3–0.75. Comparing model data with a field data
set of valley width vs. drainage area could be used to determine how well
this model of lateral erosion driven by channel curvature captures valley
widening in natural systems. A key next step in this line of research is to
analyze in detail the predicted scaling relationship between width vs.
drainage area through a sensitivity analysis on grids much larger than those
used in this paper so as to cover several orders of magnitude in drainage
area and use this as a basic test of our model formulation.
Valley width vs. drainage area for six time slices in the
undercutting-slump increased water flux model with dx=10, K=10-4, and
α=0.8. All six time slices are from the model at steady state, with
three time slices taken from the period of normal water flux and three time
slices taken from the period of increased water flux. Each time slice
represents data averaged over 2500 years of model time, or 1.6 % of the
total length of the model run. The red dots show log-binned averages of valley
width. The black line shows a least squares power-law fit for the binned
data. The Kv coefficient has a value of 0.16 with a standard error of
0.052 and the c exponent has a value of 0.46 with a standard error of
0.027.
Conclusions
The most important finding of this work is that a simple, physics-based
theory for lateral bedrock channel migration, when combined with a landscape
evolution model, produces wide bedrock valleys that scale with drainage area,
as predicted in natural systems. So far, other landscape evolution models do
not address lateral bedrock erosion and therefore predict no relationship
between valley width and drainage area. Two end-member algorithms were
presented that describe how lateral erosion occurs on the model grid: the
total block erosion model requires that the entire volume of a node is
laterally eroded before elevation is changed, while the undercutting-slump
model requires that the node is laterally undercut and the overlying material
is transported away as wash load. These two algorithms represent end-members
of how lateral bedrock erosion can occur in natural systems and show
significant differences in the patterns and timing of lateral erosion and the
development of wide bedrock valleys. Significant bedrock valley widening, represented by valleys that are
several model cells wide, only occurs when using the undercutting-slump model. Differences in the transient
model response to
changes in boundary conditions (e.g., first vertical incision followed by
lateral erosion vs. simultaneous vertical and lateral erosion) can be used to
determine the appropriate application of the end-member models.
The model presented here also produces mobile channels in an eroding rather
than an aggrading landscape. Channel mobility is a fundamental factor for
developing and maintaining a bedrock valley that is several times wider than
the channel it holds . Increased channel
mobility and wider flat-bottomed valleys under transport-limited conditions
in the model suggest that slowing vertical incision amplifies the effect of
lateral erosion . However, this model lacks some
important elements that prevent it from predicting the absolute timing and
magnitude of lateral erosion, specifically lateral erosion that is
independent of grid size and the separate treatment of bedrock and sediment. Our
theory for the lateral erosion of bedrock channel walls and the numerical
implementation of the theory in a catchment-scale landscape evolution model
is a significant first step towards understanding the factors that control
the rates and spatial extent of wide bedrock valleys.
The lateral erosion models described in this text will be
made available as a Landlab component in winter 2018.
Data sets used for this paper can be accessed by emailing Abigail L. Langston and the
Landlab toolkit can be accessed from http://landlab.github.io.
The Supplement related to this article is available online at https://doi.org/10.5194/esurf-6-1-2018-supplement.
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported in part by NSF grants EAR-1331828 and ACI-1450409.
Thanks go to
Dimitri Lague, Aaron Bufe, and an anonymous reviewer for their insightful
comments that improved this paper.
Edited by: Jens Turowski
Reviewed by: Aaron Bufe and two anonymous referees
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