Introduction
Hillslopes take on a rich variety of forms. Their profile shapes may be
convex-upward, concave-upward, planar, or some combination of these. Some
slopes are completely mantled with soil, whereas others are bare rock, and
still others draped in a discontinuous layer of mobile regolith. The
processes understood to be responsible for shaping them are equally varied,
ranging from disturbance-driven creep to dissolution to large-scale mass
movement events.
Considerable research has been devoted to understanding the evolution of
soil-mantled slopes that are primarily governed by disturbance-driven creep,
such as downslope soil transport by biotic and abiotic soil-mixing
processes. As a result, the geomorphology community has mathematical models
that account well for observed slope forms and patterns of regolith thickness
e.g.,. Furthermore, stochastic-transport theory
provides a mechanistic link between the statistics of particle motion, the
resultant average rates of downslope transport, and the emergence of
convex-upward, soil-mantled slope forms
.
One gap that remains, however, lies in understanding steep, rocky slopes
(Fig. ). “Rocky” implies slopes that lack a continuous
soil cover e.g.,and references therein; here,
a transport law that assumes the existence of such a cover no longer applies.
“Steep” implies angles approaching or exceeding the effective angle of
repose for loose, granular material, so that ravel may be an important
transport mode
e.g.,
and particles have the potential to fall as soon as they are released from
bedrock. This type of relatively fast, long-distance transport does not fit
comfortably in the framework of standard diffusion-based models of hillslope
soil transport, which derive from an underlying assumption that the
characteristic length scale of motion is short relative to the length of the
slope.
Examples of rocky hillslopes, sometimes referred to as Richter
slopes. (a) Chalk Cliffs, Colorado, USA. (b) Canadian Rockies. (c) Grand
Canyon, Arizona, USA. (d) Rocky Mountain National Park, Colorado, USA.
(e) Guadeloupe Mountains, Texas, USA. (f) Waterton Lakes National Park, Canada
(photos by Gregory E. Tucker).
Rocky slopes are rarely completely barren. More commonly, they have a patchy
cover of loose material, which may either retard rock weathering by shielding
the rock surface from moisture or temperature fluctuations, or enhance it by
trapping water and allowing limited plant growth. A discontinuous cover does
not fit easily within the popular exponential-decay regolith-production
models e.g.,, which assume an
essentially continuous soil mantle.
An additional issue, which pertains to both rocky and soil-mantled slopes, is
the connection between sediment movement at the scale of individual “motion
events,” and the resulting longer-term average sediment flux, which forms
the basis for continuum models of hillslope evolution. Recent theoretical and
experiment work has begun to forge a mechanistic connection between these
scales
.
However, the community's resources for computational analysis of
particle-level dynamics remain limited, lagging behind developments in
understanding sediment transport in coastal environments
and rivers
.
To further our understanding of how grain-level weathering and transport
processes translate into hillslope evolution, both for hillslopes in general
and rocky slopes in particular, it would be useful to have a computational
framework with which to conduct experiments. Ideally, such a framework should
be sophisticated enough to capture the essence of weathering and granular
mechanics, while remaining simple enough to involve only a small number of
parameters and provide reasonable computational efficiency.
Our aim in this paper is to describe one such computational framework, test
whether it is capable of reproducing commonly observed hillslope-profile
forms, and examine how its parameters relate to the bulk-behavior parameters
used in conventional continuum models of soil creep and regolith production.
The model uses a pairwise, continuous-time stochastic (CTS) approach to
combine a lattice grain model with rules for stochastic bedrock-to-regolith
conversion (“weathering”) and disturbance of surface regolith particles.
One goal of this event-based approach is to study how bulk behavior, such as
the diffusion-like net downslope transport of soil, can emerge from a large
ensemble of stochastic events. In this paper, we present the “Grain Hill”
model and examine its ability to reproduce three common types of slope
profile: (1) convex-upward, soil-mantled slopes
(Fig. a, b), (2) quasi-planar rocky slopes
(Fig. c, d), and (3) cliff-rampart morphology in layered
strata (Fig. e, f).
We begin with a description of the modeling technique. We then present
results that illustrate the macroscopic behavior of the model under a variety
of boundary conditions, and define the relationship between the cellular
model's parameters and the parameters of conventional continuum mechanics
models for hillslope evolution.
Examples of three characteristic types of hillslope profile. Red
line in map view depicts hillslope profile location. (a, b) Soil-mantled,
convex-upward slope (Gabilan Mesa, California, USA). (c, d) Quasi-planar,
thinly mantled slope (Yucaipa Ridge, California, USA). (e, f) Cliff formed in
resistant Tertiary laccolithic intrusive rocks overlying Jurassic sedimentary
rocks (Cedar Mountain, Utah, USA).
Hypothetical time sequence of transition events (a–h), illustrating several of the states and transitions in the Grain
Hill model. Note that although this example shows a single particle in
motion, it is possible for multiple cells to exist in a state of motion at
any given time.
Model description
The model combines a cellular automaton representation of granular mechanics
with rules for weathering of rock to regolith and for episodic disturbance of
regolith. Cellular automata are widely used in the granular mechanics
community, because they can represent the essential physics of granular
materials at a reasonably low computational cost. Because the principles are
often similar to those of lattice-gas automata in fluid dynamics
e.g.,, cellular automata for granular mechanics
are sometimes referred to as lattice grain models (LGrMs)
.
CTS lattice grain model
Our approach starts with a two-dimensional (2-D) CTS lattice grain cellular automaton. A cellular automaton can be
broadly defined as a computational model that consists of a lattice of cells,
with each cell taking on one of N discrete states (represented by an
integer value). These states evolve over time according to a set of rules
that describe transitions from one state to another as a function of a
particular cell's immediate neighborhood. A continuous-time stochastic model
is one in which the timing of transitions is probabilistic rather than
deterministic. Whereas transitions in traditional cellular automata occur in
discrete time steps, in a CTS model, they are both stochastic and
asynchronous. A CTS model can be viewed as a type of Boolean delay equation
, though the number of possible states is not
necessarily limited to just two.
The method we present here, which we will refer to as the Grain Hill model,
is implemented in the Landlab modeling framework .
The lattice grain component, on which Grain Hill builds, is described in
detail by . Here, we present only a brief overview of
the lattice grain model's rules and behavior. The framework is based on the
pairwise (“doublet”) method developed by Narteau and colleagues
, which has been applied to problems as diverse as
eolian dune dynamics
and
the core-mantle interface .
In the basic CTS lattice grain model, the domain consists of a lattice of
hexagonal cells. Each cell is assigned one of eight states
(Table , states 0–7). These states represent the nature and
motion status of the material: state 0 represents fluid (an “empty” cell
into which a solid particle can move), states 1–6 represent a grain moving
in one of the six lattice directions, and state 7 indicates a stationary
grain (or aggregate of grains, as discussed below). For purposes of modeling
hillslope evolution, we add an additional state (8) to represent rock, which
is immobile until converted to granular material, representing regolith. An
optional additional state (9) is used to model
large blocks, as described below. Figure shows several of these
states in the form of a time sequence of transition events. Note that the
timing of transition events is purely stochastic; there are no time steps in
the usual sense.
States in the Grain Hill model.
State
Description
0
Fluid
1
Grain moving upward
2
Grain moving up and right
3
Grain moving down and right
4
Grain moving down
5
Grain moving down and left
6
Grain moving up and left
7
Resting grain
8
Rock
(9)
Block (optional)
Like other lattice grain models, the CTS lattice grain model is designed to
represent, in a simple way, the motion and interaction of an ensemble of
grains in a gravitational field. The physics of the material are represented
by a set of transition rules, in which a given adjacent pair of states is
assigned a certain probability per unit of time of undergoing a transition to a
different pair. For example, consider a vertically aligned pair of cells in
which the top cell has state 4 (moving downward) and the bottom cell state 0
(empty/fluid) (Fig. f). Downward motion (falling) is
represented by a transition in which the two states switch places
(Fig. g).
The stochastic pairwise transitions in the CTS lattice grain model are
treated as Poisson processes. The probability density function for the
waiting time, t, to the next transition event at a particular pair is given
by an exponential function with a rate parameter r, which has dimensions of
inverse time:
p(t)=re-rt.
Each transition type is associated with a rate parameter that represents the
speed of whichever process the transition is designed to represent. To
implement these transitions, the CTS lattice grain model steps from one
transition to the next, rather than iterating through time steps of fixed
duration. Whenever the state of one or both cells in a particular pair
changes, if the new pair is subject to a transition, the time at which the
transition is scheduled to occur is added to a queue of pending events. The
soonest among all pending events is chosen for processing, and the process
repeats until either the desired run time has completed or there are no
further events in the queue. Further details on the implementation and
algorithms are provided in .
Grain motion through fluid is represented by a transition involving a moving
grain and an adjacent fluid cell in the direction of the grain's motion: the
two cells exchange states, representing the motion of the grain into the
fluid-filled cell, and the replacement of the grain's former location with
fluid (Fig. c, d and f, g). During this transition, the
grain's motion direction remains unchanged (Fig. , top
left). Note that the lattice itself never moves; rather, material motion is
represented simply by an exchange of grain and fluid states between an
adjacent pair of cells.
Gravity is represented by transitions in which a rising grain decelerates to
become stationary, a stationary grain accelerates downward to become a
falling particle, and a grain moving upward at an angle accelerates downward
to move downward at an angle (Fig. ). An additional rule
allows for acceleration of a particle resting on a slope: a stationary
particle adjacent to a fluid cell below it and to one side may transition to
a moving particle (Fig. , bottom row). Importantly for our
purposes, this latter rule effectively imposes an angle of repose at
30∘.
For gravitational transitions, the rate parameter, rg, is determined by
considering the time it would take for an initially stationary object to fall
a distance of one cell width under gravitational acceleration without fluid
drag. This works out to be
rg=2δ/g,
where δ is cell width and g is gravitational acceleration. This rate
parameter is used for all of the gravitational transitions illustrated in
Fig. .
Because of the stochastic treatment of all transitions – including
gravitational ones – it is possible for grains in the model to hover in
mid-air for a brief period of time before plunging downward
e.g.,. For purposes of modeling hillslope
evolution, this is fine; what matters most is that there is a distinct
timescale gap between “fast” (large rate constant) processes associated
with grain motion and “slow” (small rate constant) processes associated
with weathering and disturbance, which are described below. First, however,
we must consider frictional interactions among moving particles.
Rules for motion and frictional (inelastic) collisions, illustrated
here for one of the six lattice directions.
Illustration of gravitational rules. The bottom row shows the
“falling on a slope” rule, which effectively imposes a 30∘ angle of
repose. Modified from .
We assume that biophysical disturbance events such as the growth of roots and
burrowing by animals, and the settling motions that follow, tend to impart
low kinetic energy, with “low” defined as ballistic displacement lengths
that are short relative to hillslope length and comparable to or less than
the characteristic disturbance-zone thickness. We consider such motions to be
dominated by frictional dissipation rather than by transfer of kinetic energy
by elastic impacts. This view is similar to the reasoning of
that the mean free path of mobile grains will
typically be short relative to hillslope length, scaling with the grain
radius and particle concentration. For this reason, unlike the original
lattice grain model of , the present formulation
includes only inelastic collisions (Fig. ). These inelastic
(frictional) collisions are represented by a set of rules in which one or
both colliding particles become stationary, representing loss of momentum and
kinetic energy as a result of the collision. The particular choices for
frictional interaction are motivated simply by the geometry of the problem.
They are non-unique in the sense that one could imagine reasonable
alternatives to the rules illustrated in Fig. ; however,
the details of frictional interactions have little influence on the outcomes
of the Grain Hill model. In the general lattice grain CTS model, the rate
parameter for frictional transitions is set equal to the product of the
gravitational parameter and a dimensionless friction factor, f∈[0,1]
(there is also a corresponding elastic factor equal to 1-f). In the Grain
Hill implementation explored in this paper, f=1, such that particle
collisions are purely frictional.
One limitation of the CTS lattice grain model is that falling grains do not
accelerate through time; instead, they have a fixed transition probability
that implies a statistically uniform downward fall velocity. This treatment
is obviously unrealistic for particles falling in a vacuum, though it is
consistent with a terminal settling velocity for grains immersed in fluid.
Consistent with the above reasoning, the relatively short ballistic
displacement lengths asserted for the modeled hillslopes also reduce the
importance of this assumption, as a particle would typically have little time
to accelerate before impacting another particle.
Tests of the CTS lattice grain model show that it reproduces several basic
aspects of granular behavior . For example, when
gravity and friction are deactivated, the model conserves kinetic energy.
When gravity and friction are active, the model reproduces some of the common
behaviors observed with granular materials. For example, Fig.
illustrates a simulation of the emptying of a silo to form an angle-of-repose
grain pile. For our purposes, what matters most is simply that the model
captures, in a reasonable way, the response of particles on a slope to
episodic disturbance events.
Lattice grain simulation of emptying of a silo. Light-shaded grains
are stationary; darker-shaded ones are in motion. Black cells are walls
(rock). Time units are indicated in seconds. From .
Weathering and soil creep
Weathering of rock to form mobile regolith is modeled with a transition rule:
when a rock cell lies adjacent to a fluid cell (which here is assumed to be
air), there is a specified probability per unit of time, w (1/T), of
transition to a grain–air pair (Figs. a–b, and , top). In other words, w is the Poisson rate constant
for the weathering transition process. This treatment means that the
effective maximum expected weathering rate, in terms of the propagation of a
weathering front, is cell diameter, δ, multiplied by w. An indirect
consequence of this approach is that the weathering rate declines with
increasing regolith thickness. As average regolith thickness increases, the
fraction of the surface where rock is in contact with air diminishes, and
consequently so does the average transition rate. A limitation of the
approach is that when the rock is completely mantled, no further weathering
can take place. We explore the consequences of this rule below, and compare
it with the behavior of continuum regolith-production models.
Transitions representing rock-to-regolith transformation by
weathering (a) and regolith disturbance (b), in which a stationary
particle becomes mobile and switches position with a air cell. The illustration
represents one of the six possible orientations.
Soil creep is modeled by a transition rule that mimics the process of
episodic disturbance of the mobile regolith (which we use here as a generic
term that includes various forms of unconsolidated granular material, such as
soil, colluvium, and scree). For each resting grain that is adjacent to an
air cell, there is a specified probability per unit of time, d (1/T), that the
regolith and air will exchange places, representing movement
(Fig. b, c). The regolith cell is also converted from a
stationary state to a state of motion (Fig. b). An
advantage of this approach is that it mimics, in a general way, the
effectively stochastic disturbance processes that are understood to drive
soil creep.
Our definition of d is closely related to the activation rate, Na, in
the probabilistic theory for soil creep developed by
. When combined with the lattice grain
gravitational rules, the resulting cellular model captures both the
scattering (disturbance) and settling (gravitational) behavior articulated by
. In the Grain Hill cellular model, as in their
theory, downslope regolith flux arises because, on average, scattering occurs
perpendicular to the local surface while setting is vertical. The Grain Hill model
includes an additional element not present in the
theory: an increase in (downward) scattering
distance for particles on slopes steeper than 30∘. This behavior, as
illustrated below, promotes a non-linear relationship between gradient and
flux, and leads to the possibility of threshold slopes.
Note that the weathering and disturbance rate constants (w and d,
respectively) are understood to be considerably smaller than the
gravitational rate constant, rg. As noted above, a key concept here is
that there are two distinct timescales: a short timescale associated with
grain motion, and a much longer scale associated with weathering and
disturbance frequency.
Cells as grain aggregates
Natural regolith disturbance events usually impact many grains at once.
Raindrop impacts on bare sediment typically dislodge several grains at once
. Excavation of an animal burrow disturbs a volume of
grains equal to the volume of the burrow, and the fall of a tree mobilizes a
volume of regolith similar to the volume of the tree's root mound.
Observations of such processes suggest that there may be a characteristic
volume of disturbance that in some cases may be much larger than the volume
of a single grain. For this reason, we envision regolith cells as being grain
aggregates, with a length scale (width of a cell) δ and a volume scale
δ3.
Initial and boundary conditions
The 2-D model domain represents the cross section of a hypothetical hillslope,
on which particles move within the cross-sectional plane. Any regolith cells
that reach the model's side or top boundaries disappear. This treatment is
meant to represent the presence of a stream channel at the base of each side
of the model hillslope; particles reaching these channels are assumed to be
eroded. Progressive lowering of the base level at the two model boundaries is
treated by moving the interior cells upward away from the lower boundary, and
adding a new row of rock or regolith cells along the bottom row. A new row of
cells is added at time intervals of τ.
Cells around the lattice perimeter retain their initial states. If, for
example, a transition occurs in which a grain “moves” into a fluid cell on
the lattice perimeter, its former location will correctly transition to a
fluid cell, but the perimeter cell itself will retain its status as a fluid
cell. Effectively, this treatment means that grains or blocks reaching either
of the two vertical boundaries are instantly eroded.
The initial condition for most runs presented here has the bottom two rows
filled with regolith grains. The lower left and lower right cells are
assigned to be rock, which represents the base-level (and incidentally helps
keep a consistent color scheme among different model configurations, because
the rock state is always present). The rest of the domain is initialized as
air cells.
Scaling and non-dimensionalization
The basic model has four parameters: the disturbance rate, d (cells / T),
weathering rate, w (cells / T), base-level lowering interval, τ (T), and
width of domain, λ (cells). The base-level lowering timescale τ
represents the time interval between episodes of relative uplift in which the
interior domain is lifted by one cell relative to its side boundaries. The
domain width might properly be considered a boundary condition rather than a
parameter, but we include it here with an eye toward examining how slope
width impacts hillslope properties such as mean height. Once we define the
width of a cell, δ (L), we can define versions of these four
parameters that explicitly incorporate this length scale:
D=dδ,W=wδ,U=δ/τ,L=λδ.
Consider the case of dynamic equilibrium, in which the rate of base-level
lowering is balanced by the hillslope's rate of erosion. The mean height of
this steady state hillslope, H, is a function of the above four parameters
plus the characteristic length scale δ, such that we end up with a
total of six variables:
H=f(D,W,U,L,δ).
Buckingham's Pi theorem dictates that these six variables, which collectively
include dimensions of length and time, may be grouped into four dimensionless
quantities:
Hδ=fDU,WU,Lδ.
The ratio d′=dτ=D/U is a dimensionless disturbance rate. Similarly,
w′=wτ=W/U is a dimensionless weathering rate. Noting the definitions
above, Eq. () is equivalent to
h=fd′,w′,λ,
where h=H/δ is dimensionless hillslope height. Hence, we have a
dimensionless property of the hillslope, h, that depends uniquely on three
other non-dimensional variables, representing disturbance rate, weathering
rate, and length.
One can similarly define a dimensionless regolith thickness, r=R/δ,
where R is the dimensional equivalent; it too should depend on the three
dimensionless parameters that represent disturbance rate d′, weathering
rate, w′, and hillslope length, λ, respectively. For a hillslope
composed entirely of regolith, r and h depend solely on d′ and
λ. Finally, we define a fractional regolith cover Fr. In the Grain
Hill model, Fr is calculated as the number of air–regolith cell pairs
divided by the total number of cell pairs that juxtapose air with either
regolith or rock.
Blocks
The foregoing model is designed to represent regolith as grain aggregates
composed of gravel-sized and finer grains: material fine enough that it is
susceptible to being moved by processes such as animal burrowing, frost
heave, tree throw, and so on. Some hillslopes, however, are adorned with
grains that are simply too large to be displaced significantly by such
processes. For example, presented a case study and
model of slopes formed beneath a resistant rock unit that periodically sheds
meter-scale or larger blocks. On at least some of these types of slope, the
distance between surface blocks and their source unit is considerably greater
than the distance that they could roll during an initial release event
. This observation implies that the blocks are
transported downslope by a process of repeated undermining.
hypothesized that erosion of soil beneath and
immediately downhill can cause a block to topple and hence move a distance
comparable to its own diameter in each such event.
We wish to capture this form of “too big to disturb” behavior in the Grain
Hill model. The CTS approach, at least as it is defined here, does not lend
itself to variations in grain size or geometry. Instead, we introduce an
additional type of particle that represents the behavior of blocks rather
than treating their difference in size explicitly. In a sense, the approach
can be viewed as treating blocks as having greater density, rather than
greater size, than other grains. A block particle differs from a normal
regolith cell in that it cannot be scattered upward by disturbance. Motion of
a block particle can only occur under two circumstances: when it lies
directly above an air cell (in which case it falls vertically, trading places
with the air cell) and when it lies above and to the side of an air cell (in
which case it falls downslope at a 30∘ angle, with probability per
time d). These rules mimic the undermining process discussed by
.
As in the model, block particles can also undergo
weathering. Here, weathering is again treated in a probabilistic fashion:
blocks form from weathering of bedrock, at probability per time w. Once
created, a block can undergo a conversion to normal regolith with probability
w when it sits adjacent to an air cell. This treatment of blocks captures,
in a simple way, the weathering of blocks as they move downslope. For
purposes of this paper, the block component is included simply to test
whether a cellular automaton treatment produces results that are
qualitatively consistent with observations, and also consistent with the
hybrid continuum–discrete model of and
.
Results
Fully soil-mantled hillslope
We start by considering the case of fully soil-mantled hillslopes, in which
the supply of mobile regolith is effectively unlimited
(Fig. a, b). Under this condition, the Grain Hill model
represents a testable mechanistic hypothesis: that a transport limited,
soil-mantled hillslope behaves essentially as a granular medium subject to
periodic, quasi-random disturbance events. This concept was also the essence
of the acoustic-disturbance experiments by . To
test the hypothesis, we run the Grain Hill model with a constant rate of
material uplift relative to base-level until the system reaches quasi-steady
state, to determine whether its steady form is smoothly convex upward (when
the gradient is below the failure threshold) to planar (when the gradient
lies at or near the failure threshold). Model runs were performed using a
251-row by 580-column lattice. Disturbance rates were varied from 0.001 to 0.1 yr-1 and intervals between relative-uplift events from
100 to 10 000 years.
Results show that the Grain Hill model produces parabolic to planar hillslope
forms, depending on the ratio of disturbance to uplift rates, which is
encapsulated in the dimensionless parameter d′ (Fig. ). At high
d′ (frequent disturbance and/or slow base-level fall), hillslope relief is
low and the form is smoothly convex upward (Fig. , lower right
panels). At somewhat lower d′, the lower part of the slope approaches a
threshold angle while the upper part remains smoothly convex
(Fig. , middle diagonal panels). At low d′, the form becomes
predominantly planar and achieves a threshold relief that is insensitive to
further increases in d′ (Fig. , upper left panels).
Equilibrium topographic cross sections using only regolith particles
(no rock) and a variety of disturbance frequencies (d) and time interval
between base-level fall events (τ). Fast basal incision and/or infrequent
disturbance lead to planar threshold hillslopes; slow basal incision and/or
frequent disturbance lead to parabolic hillslopes.
Scaling of mean height as a function of d′ is shown in
Fig. . The figure shows results for 125 model runs
spanning 2 orders of magnitude in each parameter (d, τ, and
λ) in half-decade intervals. The 125 runs represent a 5×5×5 grid of experiments, in which each grid point represents a
particular combination of the three parameters (d, τ, and λ).
For any given hillslope length, there are three regimes of behavior. Low d′
(upper left of graph) leads to threshold hillslopes, in which relief depends
only on hillslope length. Under moderate d′, mean height scales inversely
with d′, as expected from linear diffusion theory. At high d′, we have a
finite-size regime in which dimensionless hillslope mean height is comparable
to the disturbance scale, δ (cell size in the model); in other words,
the hill is only one or a few cells high.
Dimensionless mean hillslope height, h, as a function of
dimensionless disturbance rate d′ for a range of hillslope lengths. Data
points include 125 sensitivity analysis runs in which d∈[10-3,10-2.5,10-2,10-1.5,10-1], τ∈[102,102.5,103,103.5,104], and λ as shown in the legend.
The behavior of the Grain Hill model in its simple, transport-limited
configuration can be compared to diffusion theory, which relates volumetric
sediment flux per unit contour length, qs, to topographic gradient:
qs=-Ds∂η∂x,
where η is land-surface height, x is horizontal distance, and Ds is
an effective transport coefficient. The
probabilistic theory for transport due to particle scattering and settling
formulates Ds as
Ds=krpRaNa1-ccm2‾cos2θ,
where k is a dimensionless coefficient, rp is particle radius, Ra is
active regolith thickness, Na is the activation rate, θ is slope
angle, c is particle concentration, and cm is a maximum concentration.
The over-bar denotes an average over the active regolith thickness. For the
Grain Hill model, Ra scales with the characteristic disturbance depth,
δ. Further, because we treat grain aggregates, we may also assume
rp∼δ. Therefore, we have the prediction that
Ds=aδ2Nacos2θ,
where a is a dimensionless proportionality constant.
The mean expected activation rate, Na, is closely related to the Grain
Hill model's disturbance frequency parameter, d. To relate the two
quantitatively, one needs to make a trivial lattice-geometry correction. A
straight-as-possible cut through the hex lattice exposes on average two faces
per cell, both of which are susceptible to a disturbance event. Because d
is the expected disturbance frequency per cell face, and because independent
Poisson events are additive, the resultant disturbance frequency for each
cell exposed along a quasi-horizontal surface is Na=2d.
A more important difference is that whereas Na is defined as activation
rate per unit horizontal area, d represents the rate per unit surface area
regardless of orientation. For a given d, Na will increase with surface
roughness (because there is more exposed area of regolith–air contact), and
with gradient (because the slope length increases).
An additional effect arises from the model's effective 30∘ angle of
repose. On slopes steeper than this, the expected disturbance rate increases
substantially because gravitational dislodgement becomes activated
(Fig. , bottom row). Thus, the Grain Hill model incorporates
an additional non-linear relationship between flux and gradient inasmuch as
Na depends on gradient.
We can derive an effective diffusivity, De, from the modeled topography by
applying the expected relationship between mean elevation and diffusivity.
Here, De is defined as that value which, if it were spatially uniform,
would yield the same mean steady-state elevation as that produced by the
particle model. Framing it this way allows us to interrogate how the
effective transport coefficient varies as a function of mean slope gradient.
At steady state, mass balance implies that
qs=Ex,
where E is the rate of erosion – equal to the rate of material uplift
relative to base level – and x is horizontal distance from the ridge top.
Substituting Eq. () and rearranging gives
dηdx=-EDs(x)x≈-EDex.
Integrating and then averaging over x, we can solve for the average
elevation, η‾:
η‾=E3DeLh2,
where Lh=L/2 is the length of the slope from ridge top to base (in other
words, half the total length of the domain). We can then rearrange this to
find De:
De=E3η‾Lh2.
To examine how De scales, we can define a dimensionless form, normalizing
by the disturbance frequency, d, and the square of active regolith
thickness (equal to particle diameter), δ2:
De′=Dedδ2=ELh23η‾dδ2.
Noting that E=δ/τ, L=2Lh, and L/δ=λ, this is equivalent to
De′=λ212h‾dτ,
where h‾ is the mean hillslope height in particle diameters.
As expected, De′ increases with hillslope gradient
(Fig. ). The effective diffusivity approaches an asymptote at
30∘ (mean gradient ≈ 0.6), representing an angle of repose. The
pattern resembles the family of non-linear flux–gradient curves introduced by
and explored further by
and . At low
gradients, De′ approaches a value of about 60. (This method of estimating
De′ is similar to fitting the standard theoretical parabolic curve to the
experimental profiles, except that here we use the integral of the profiles.)
The link between De and d provides a way to scale the Grain Hill model
to field-derived estimates of Ds and Ra. Here, we equate the theoretical
effective diffusivity, De, with the definition of the transport
coefficient Ds of . Noting that, at low
gradients, cos2θ in Eq. () approaches unity, and using
the prior relation Na=2d, we may write Ds for low slope angle as
Ds(θ→0)=2aδ2d.
In the Grain Hill model, the fact that low-angle De′≈60 implies
that the dimensional equivalent De(θ→0)≈60δ2d. Equating Ds (the transport coefficient derived by
) and De (the effective transport
coefficient derived from the Grain Hill model),
Ds(θ→0)≈60δ2d.
This relation can be used to scale the parameters in the Grain Hill model
with field data. For example, if one were to assume an active regolith
thickness of 0.4 m and a low-gradient transport coefficient of Ds=0.01 m2 yr-1, and set δ to the active regolith thickness, then
d=Ds60δ2≈0.001yr-1.
Here, d represents the frequency with which a given exposed patch of
regolith of width and depth δ is disturbed upward. With the above
values, the simulated hills in Fig. would be 232 m long
(valley to valley), with height ranging from 1.6 to 57.6 m and base-level
lowering rate from 0.04 to 4 mm yr-1.
Relationship between dimensionless diffusivity and mean gradient,
from the series of 125 model runs of which a subset is shown in
Fig. .
Hillslope with regolith production from rock
Having established that the Grain Hill model reproduces classic soil-mantled
hillslope forms and has parameters that can be related to the parameters in
commonly used continuum hillslope transport theories, we turn now to the case
in which regolith is generated from bedrock with a production rate that may
(or may not) limit the rate of erosion. We explore the role of regolith
production with a series of model runs in which w′ varies from 0.4 to 40.
The upper end of this range represents a condition in which the potential
maximum rate of regolith production greatly exceeds the rate of base-level
lowering. The lower end, 0.4, is less than the rate of base-level fall, and
would seem to be insufficient to allow for equilibrium to occur, and yet
nonetheless it does. Examples of equilibrium hillslope forms found in this
parameter space are shown in Fig. .
Final equilibrium profiles from Grain Hill runs with rock and
weathering. Domain size is 222 rows by 257 columns, and uplift interval
ranges from 100 to 10 000 years.
Relationships among mean gradient, fractional regolith cover, dimensionless
disturbance rate d′, and dimensionless weathering rate w′ are illustrated
in Fig. . For w′>1, the gradient–d′ relation
(Fig. a) has the same shape as in the purely regolith models:
a threshold regime at lower d′ transitioning to an inverse gradient–d′
relation at higher d′. This indicates that when the maximum weathering rate
(for a flat surface) is substantially greater than the rate of base-level
fall, we recapture transport-limited conditions. With w′<1, however, the
hillslope achieves an equilibrium gradient that is greater than that for the
transport-limited case, and at lower d′, is greater than the threshold
angle (Fig. a, b).
We can also examine the fractional regolith cover, which is defined here as
the number of rock–air cell pairs divided by the total number of cell pairs
at which air meets either regolith or rock (Fig. c, d). The
fractional regolith cover shows relatively little sensitivity to d′
(Fig. c). The cover hovers around unity for high w′ and
d′ but systematically declines with w′ when w′ is below about 10.
(Note that the data points representing d′=1000 and w′>1 have
hillslope heights of only a few particles and are therefore sensitive to
finite-size effects.)
Mean equilibrium gradient and regolith thickness for models with
rock and weathering, as a function of d′ and w′. Data represent 125 runs
with d∈[10-3,10-2.5,10-2,10-1.5,10-1] yr-1, w′∈[100,100.5,101,101.5,102], and τ∈[102,102.5,103,103.5,104] yr. Horizontal dashed lines show model's angle of repose for regolith.
The models with w′<1 present a seeming paradox: how is it possible to
achieve an equilibrium form when the maximum weathering rate appears to be
lower than the rate of uplift relative to base level? The solution to the
paradox lies in surface area. The surface area of rock that is exposed to
weathering is not fixed but rather depends on the overall slope length, the
terrain roughness, and the fractional regolith cover. To appreciate the first
effect, consider a planar slope at angle θ with no regolith cover. If
wδ represents the maximum slope-normal bedrock weathering rate, then
the vertical rate is simply wδ/cosθ. All else equal,
increasing gradient will increase vertical weathering rate, thereby providing
a feedback between gradient and rock lowering rate. A second feedback relates
to topographic roughness: all else equal, a rougher surface will experience a
greater weathering rate because it provides more surface area. The third
feedback, which is embedded in the depth-dependent regolith production
hypothesis lies in regolith cover: the greater the
exposure of rock (or the thinner the cover), the faster the average rate of
rock-to-regolith conversion. In the Grain Hill model, this third feedback is
represented by fractional bedrock exposure (since weathering only occurs when
rock cells are juxtaposed with air cells).
To test whether these are indeed the feedbacks responsible for equilibrium
topography in the Grain Hill model, we can compare the rate of material
influx (uplift relative to base-level) with the expected rate of
rock-to-regolith conversion. In the Grain Hill model, the expected rate of
regolith production, P, in cross-sectional area per time, is the product of
weathering rate per cell face, w, the cross-sectional area of a cell, A,
and the number of rock–air cell faces, nra,
P=wAnra.
The rate of material addition due to uplift relative to base-level, U, again
in cross-sectional area per time, is the area of a cell, A, multiplied by the
horizontal width of the domain in cells, nH, divided by the interval
between uplift events, τ:
U=nHA/τ.
Equality between rock uplift and weathering can be expressed as
1τ=wnranH.
The ratio on the right side represents the surface-area effect, in the form
of surface area exposed to weathering per unit horizontal area. The balance
is illustrated in Fig. , which compares the left-hand and
right-hand terms for each of the 125 model runs with weathering. Each data
point represents a single snapshot in time, and so scatter is to be expected.
To help diagnose the scatter around the 1:1 line, the data are divided into
quintiles by fractional regolith cover, Fr (note that some of the points
in the lower quintiles are obscured by being over-plotted along the 1:1
line). Many of the points that fall off the 1:1 line, especially at the high
end (higher 1/τ), come from runs with Fr>80%; with very few exposed
rock–air pairs, a small fluctuation in the nra can produce a relatively
large change in predicted weathering rate. At the low end, many of the points
above the 1:1 line come from runs with a maximum height of only a few cells,
which are subject to finite-size effects.
The main message of Fig. is that the Grain Hill model
demonstrates an equilibrium adjustment between rock uplift and rock
weathering. The weathering rate does not have a fixed upper “speed limit,”
but rather is set by the exposed surface area, which in turn is a function of
gradient, roughness, and regolith cover. Solutions with a discontinuous
regolith cover are indicative of this adjustment. Slopes can grow arbitrarily
steep, with weathering and erosion increasingly attacking from the sides as
the gradient rises.
Comparison between rate of material input, 1/τ (cells / year),
with effective rate of weathering, wnra/nH, from 125 model runs (see
text).
Comparison between weathering rule and inverse-exponential model
The most popular function to describe regolith production from bedrock is the
decaying exponential formula proposed by , which has
proved consistent with estimates of production rate obtained using cosmogenic
radionuclides . The production
rate is given by
P=P0exp-R/R*,
where P0 is the maximum (bare bedrock) production rate, R is regolith
thickness, and R* is a depth-decay scale on the order of decimeters. On a
flat surface, assuming no erosion or deposition, the expected rate of change
of R over time is
dRdt=ρrρs(1-ω)P0exp-R/R*,
where ρr and ρs are the bulk densities of
parent material and regolith, respectively, and ω is the fraction of
parent material removed in solution upon weathering. Starting from a bare
surface, and assuming isovolumetric weathering (in which case
ρs=(1-ω)ρr), the expected regolith
thickness as a function of time can be found by integrating Eq. ():
RR*=lnP0R*t+1.
We can compare this with the behavior of the cellular weathering rule by
running the case of a flat, initially bare-rock surface from which weathered
material may neither enter nor leave (Fig. , case dw-1=0).
When the disturbance rate is zero, the cellular weathering model
asymptotically approaches a steady regolith thickness of exactly one cell
(thickness equal to δ). This is so because the model allows weathering to
occur only when rock cells are exposed to air cells, and there is no
disturbance process that would juxtapose rock and air once the initial
weathered layer has formed. When disturbance rate is non-zero, however,
regolith continues to form even after the mean thickness r exceeds unity
(representing one characteristic disturbance depth). Continuation of regolith
production occurs because the disturbance process intermittently exposes
rock, at which point it becomes subject to weathering. The greater the
disturbance rate, the more frequent the exposure and hence the more rapid
weathering (Fig. ). For any ratio dw-1, the model's
weathering behavior clearly differs from the logarithmic growth in thickness
predicted by exponential theory. This represents both a strength and a
weakness in the Grain Hill model. On the one hand, the model under its
present configuration cannot account for rock-to-regolith conversion
resulting from processes that penetrate more than one characteristic
disturbance depth δ into the subsurface. For example, the model
neglects the possibility that some plant roots may penetrate deeply and
contribute to disaggregation, or that an unusually deep freezing front in a
cold winter might cause rock fracture and displacement of the resulting
fragments e.g.,. On the other hand, the model
honors the likelihood that soil disturbance and regolith production are
closely linked processes, rather than independent: all else equal, a
greater disturbance rate will tend to produce faster rates of both regolith
production and downslope soil movement.
Regolith thickness versus time, as predicted by inverse-exponential
theory (log growth; solid cyan curve) and the Grain Hill model with a range
of ratios of disturbance rate (d) to weathering rate (w). Time
(horizontal axis) is non-dimensionalized by multiplying by w.
Rock collapse and vertical cliffs
Some rock slopes display a cliff-and-rampart morphology in which a vertical
or near-vertical rock face stands above an inclined, often sediment-mantled
buttress (Figs. and ). Although common
in sedimentary rocks where a resistant unit forms the cliff and a weaker unit
the buttress, the same morphology is sometimes found in apparently
homogeneous lithology (Fig. b). The cliff portion of such
slopes suggests a process of undermining and collapse, with the cliff-forming
material being cohesive enough to maintain a vertical face but too weak to
support overhangs.
Two examples of cliff-and-rampart morphology. (a) Near Palisade,
Colorado, USA, after recent rock-fall event (photo courtesy of D. Nathan Bradley and
Dylan Ward). (b) Colorado Plateau, Utah, USA. Note that contact between lower
rampart and subvertical slopes, both of which have formed in a gray shale
unit, occurs without any apparent break in lithology.
To explore the origins of ramp-and-cliff morphology, we consider a version of
the Grain Hill model that adds an extra rule to represent collapse: any rock
particle that directly overlies air has the possibility to transition to a
falling regolith particle, with the same rate as gravitational transition
from resting to falling; in other words, as soon as a rock particle has been
undermined, it behaves like cohesionless material.
Under dynamic equilibrium, this rule produces a morphology with slopes that
are roughly planar, with alternating vertical and sloping sections and patchy
regolith cover (Fig. ). With w′≤1, gradient and
regolith cover depend strongly on w′ and show little or no sensitivity to
d′. When w′≪1, the hillslope forms resemble pinnacles. These examples
demonstrate two combined feedbacks between weathering and base-level fall: the
surface area susceptible to weathering, and the frequency and magnitude of
material collapse through undermining.
The case of transient evolution under a stable base level leads to the
formation of a regolith-mantled, angle-of-repose ramp (Fig. ).
The slope break remains relatively sharp as it retreats headward. The ramp
forms as a transport slope. The angle of repose is an attractor state: if the
angle were steeper, weathered material would be rapidly removed as a result
of gravitational instability; if it were substantially lower, material would
accumulate, because transport would be limited to the (much lower) rate of
disturbance-driven creep motions. Hence, the Grain Hill model predicts that
formation of a sediment-mantled ramp beneath a steeper, actively weathering
rock slope is an expected outcome for a steep rock slope under stable
base level.
Quasi-steady model hillslope profiles created using a collapse rule,
under four different combinations of d′ and w′. Insets show magnified
views of a portion of each hillslope.
Time series showing transient erosion of a steep rock slope under a
stable base-level, highlighting formation of ramp-and-cliff morphology.
Simulation shows 20 000 years of slope evolution under
d=w=10-3 yr-1. Nominal width, assuming δ=0.1 m, is 12 m.
Blocks
Weathering and erosion in landscapes underlain by relatively massive,
fracture- or joint-bounded rock can sometimes produce large “blocks” of
rock, defined here as clasts that are too large to be displaced upward by
normal hillslope processes. The release of blocks from dipping sedimentary or
volcanic strata can alter both the shape and relief of hillslopes
. When blocks are delivered to streams, they can
influence the channel's roughness, gradient, erosion rate, and longitudinal
profile shape .
As discussed in Sect. , the Grain Hill model can be
modified to honor blocks by defining an additional cell type that represents
blocks. The weathering process is modified such that a rock cell now weathers
into a block, and the block in turn may weather to form regolith. When a
block is undermined directly from below, it will fall just as a normal
regolith particle would. When a block particle lies adjacent to and above an
air cell, a disturbance event may occur that causes the block to shift
downward on the slope. By these means, blocks in the model may move downward,
or downward and laterally, but never upward. An implicit assumption in this
treatment is that blocks do not roll long distances (further than their own
diameter) upon release.
We examine model runs in which a resistant rock layer is embedded in a weak
sedimentary material that is soft enough to be treated as regolith
(Fig. ). The modeled hillslopes are qualitatively consistent
both with field observations and with the mixed continuum–discrete model of
and in that block-mantled
slopes are generally concave upward, reflecting a downslope decrease in the
flux of blocks as weathering progressively transmutes them into regolith.
Examples of models that include blocks. Rock (black) weathers to
blocks (dark red), which can only move downward or downward plus laterally.
Blocks in turn weather to regolith (light brown) (GIF-format animations of
similar runs are available as an online resource; see Tucker, 2018a).
Comparison to field sites
We perform a basic validation of the Grain Hill model by comparing its output
to real field sites, testing whether the model is capable of reproducing
realistic hillslope forms at the correct spatial scale under known boundary
conditions. Field sites were chosen such that model boundary conditions could
be derived from independent field estimates of rate parameters such as Ds
and the rate of base-level fall. To perform this test, we consider two
examples: a convex-upward, soil-mantled hillslope in Gabilan Mesa,
California, USA (Fig. a, b), and a steep, quasi-planar,
discontinuously mantled hillslope in the Yucaipa Ridge, California, USA
(Fig. c, d). For each of these two case studies, the
hillslopes appear to be approximately at steady state, and independent
estimates exist for the rate of base-level fall, U
. We estimated
the effective transport coefficient, Ds, for the profiles shown in
Fig. a, c by measuring the second derivative of the
one-dimensional hillslope elevation profiles, ∂2η∂x2, and solving for Ds using
Ds=-U∂2η∂x2.
For the Gabilan Mesa profile, we estimated the profile-averaged effective
transport coefficient as 0.0345 m2yr-1. The effective rate of
base-level lowering has been estimated at U≈1.47×10-4 m yr-1 . To construct a Grain Hill model for the
Gabilan profile, we begin by assuming a characteristic disturbance depth of
δ=1 m. This value was chosen to be consistent with measured soil
depths that typically range between 0.2 and 1.2 m .
We treat the system as transport-limited, consisting of mobile material, so
that weathering is not explicitly modeled. The disturbance parameter, d, is
then calculated from the independently estimated value of Ds using
Eq. (). The interval between uplift events is τ=δ/U≈6800 years. The resulting modeled equilibrium profile provides a
reasonably good match to the observed Gabilan profile, with a convex-upward
shape and a hilltop height of about 45 m above the slope base
(Fig. a).
For Yucaipa Ridge, we estimated the transport coefficient at
Ds∼0.028 m2 yr-1 on the basis of hilltop curvature and an
estimated effective rate of base-level lowering of
≈ 0.0027 m yr-1 . Using Eq. (), this equates to a
disturbance-rate parameter d=0.00468 yr-1 and an uplift interval of
370 years in the Grain Hill model. Bedrock outcrops are common on the Yucaipa
Ridge hillslopes, implying a thin, discontinuous regolith cover. We therefore
treat the system as consisting of bedrock that must be weathered before it
can become mobile. Because we do not have independent information on the
effective maximum rock weathering rate, the Yucaipa case is a somewhat weaker
test: we can only ask whether there exists a geologically reasonable value of
w such that the model reproduces the observed relief and shape of the slope
profile. Through trial and error, we find that with a weathering rate
parameter w=0.002 yr-1 (which corresponds to a maximum regolith
production rate of 2 mm yr-1), the model does a credible job of capturing
the shape and size of the Yucaipa profile (compare Fig. c
with b). Although this particular value was obtained
through a simple calibration process, it is at least both geologically
reasonable and, as one might expect, somewhat lower than the rate of
base-level lowering.
To test the sensitivity of the Yucaipa example to the assumed characteristic
disturbance depth, we ran a second experiment in which δ was reduced
to 0.75 m, and the weathering, disturbance rate, and uplift parameters were
rescaled accordingly. The relief and mean gradients of the two cases are
nearly identical, with planar slopes and a relief in both cases of
∼100 m.
Steady state models using parameters estimated from observed
hillslope profiles. (a) Parameters are based on Gabilan Mesa, with the profile
shown in Fig. a, b for comparison. (b) Parameters based on
Yucaipa Ridge, with the profile shown in Fig. c, d for
comparison.
These two examples demonstrate that the Grain Hill model parameters are not
arbitrary but instead can be linked through straightforward reasoning to
field estimates of transport efficiency and base-level lowering. When one does
so, the model successfully reproduces both the shape and scale of observed
slopes.
Discussion
With just three parameters – disturbance frequency (d), characteristic
disturbance depth (δ), and base-level fall frequency (u) – the Grain
Hill algorithm can reproduce the convex-upward to quasi-planar forms
associated with soil-mantled hillslopes (Fig. ). With the
addition of a parameter that represents rock-to-regolith conversion rate, the
algorithm accommodates partly mantled, rocky hillslopes
(Figs. , , ). By adding a rule
for detachment of blocks from resistant rock, the model reproduces hillslope
forms associated with hogbacks and ledge-forming escarpments
(Fig. ).
A common criticism of cellular automaton models is that they involve
arbitrary rules and/or parameters that can neither be measured nor verified
in the real world. That is not the case for the Grain Hill model, for which
the parameters are tied to measurable physical quantities. For example, the
disturbance frequency d is directly related to the frequency parameter
Na in statistical theory of soil transport developed by
, and through that theory to the diffusion-like
transport coefficient Ds that is commonly estimated in field studies. This
connection between model parameters and field measurements is illustrated by
the model's ability to reproduce the correct shape and scale of observed
hillslope forms when estimates of Ds and U are available
(Figs. , ). In the transport-limited
case, there are no tunable parameters: given independent estimates of Ds
and U, the correct morphology is recovered (Figs. a,
a). In the case where rock weathering appears to play a
role, and an independent estimate of P0 is not available, the model
requires an estimation of maximum weathering rate w. Nonetheless, a
plausible value of w (0.002 m yr-1), somewhat smaller than the rate of
base-level fall (0.0027 m yr-1), reproduces the observed shape and relief
in the Yucaipa Ridge case study.
The transport dynamics predicted by the Grain Hill model are consistent with
continuum soil-transport theory, which treats soil as a fluid with a
downslope flow rate that depends on slope gradient. Like the popular
Andrews–Bucknam non-linear transport law
e.g.,,
the transport-limited form of the Grain Hill model predicts diffusion-like
behavior in which the effective diffusivity increases with slope gradient,
with an asymptote at a threshold angle (Fig. ). In one sense,
the Grain Hill model is actually closer to the process level than fluid-like
continuum models, because net downslope mass flux arises from a sequence of
stochastic disturbance events rather than being dictated by a macroscopic
transport law.
One limitation of the Grain Hill model is that its threshold-like behavior
arises from the lattice geometry: regolith cells perched at a 30∘
angle above and to one side of an air cell are treated as unconditionally
unstable. Whereas the timing of motion is treated as a stochastic process,
the occurrence of motion is inevitable (unless some other event occurs
first). This treatment neglects the possibility of frictional locking among
noncohesive grains at angles somewhat above 30∘, as well as the
possibility of cohesion. This limitation could be overcome by introducing a
probabilistic treatment of grain stability: a grain aggregate will be stable
with a given probability p, and unstable with probability 1-p. Such a
treatment would introduce an additional parameter, but this parameter could
in principle be estimated from physical experiments. The addition of a
“sticking rule” like this might also make it possible for models with
alternative lattice geometries to manifest the same dynamics, thereby
decoupling the basic model framework from the geometry of the lattice on
which it is implemented.
The inclusion of rock-to-regolith conversion enables the Grain Hill model to
predict a continuum of slope forms from fully soil mantled to intermittently
covered to bare. However, there are several limitations in the treatment of
regolith production that could be improved on. The weathering rule assumes
that regolith production can only occur when rock is exposed to air, which
obviously neglects the role of shallow subsurface processes such as root or
frost wedging. The effective weathering depth scale is the same as the
disturbance scale, and equal to the cell size. This assumption is probably
reasonable if the processes responsible for weathering and disturbance were
one and the same, but not if they are distinct processes with different
length scales. The Grain Hill model also does not account explicitly for
chemical weathering, which in some cases can extend well below the surface.
Finally, the model's effective regolith-production behavior does not follow
the log-growth curve predicted by inverse-exponential theory for a stable
surface (Fig. ). With these caveats in mind, one advantage
of the stochastic model of regolith production is that it effectively treats
the disturbance and regolith-production processes as being closely linked:
all else equal, the production rate is higher when disturbance is more
frequent.
The popular inverse-exponential model for regolith production implies the
existence of a speed limit to landscape evolution: in the absence of rock
landsliding, erosion rate cannot exceed the maximum rate of rock-to-regolith
conversion. Moreover, the model implies the existence of a bare landscape
once the rate of erosion exceeds the maximum rate of regolith production.
found evidence, however, that in fact there are
additional stabilizing mechanisms, and that these manifest in landscapes with
thin, patchy soils. The Grain Hill model is consistent with these
observations in that it predicts the natural emergence of a discontinuous
regolith cover, with the fractional cover exerting an influence on the
average rate of weathering and erosion. Furthermore, the model behavior
highlights the importance of slope length and roughness in modulating the
regolith production rate: all else equal, steeper or rougher slopes allow
higher production rates, leading to an additional feedback between relief and
erosion rate for rocky hillslopes. The possibility of rock collapse upon
undermining by weathering provides another feedback mechanism that may allow
rates of erosion to exceed the flat-surface maximum regolith production rate
(Fig. ).
The Grain Hill model also provides insight into transient evolution of rocky
slopes. Experiments on the relaxation of rocky slopes that are steeper than
the threshold angle predict the formation of a regolith-mantled pediment at
the angle of repose, which extends upslope as the steep upper slope gradually
recedes (Fig. ). This scarp–pediment morphology emerges without
any variation in material strength, requiring only a period of base-level
stability.
As a computational framework for exploring hillslope forms, the Grain Hill
model has the advantage that it provides a mechanistic link between events
(disturbance and weathering) and long-term morphologic evolution, without the
need to specify a flux law. The model has the further advantage of being
fully two dimensional, allowing disturbance and weathering events to initiate
from the side as well as vertically. A further key element is that the model
can mix timescales: a short timescale associated with grain motion, an
intermediate timescale associated with disturbance events, and a much longer
timescale for slope evolution. Mixing these disparate timescales in a single
computer model is made possible by the fact that most of the time grains are
stationary: the algorithm operates on small (stochastic) time steps during
those moments when grains are moving, and on much longer steps when no grains
are in motion (for further information on the discrete-event algorithm behind
the model, see ).
The Grain Hill framework has several important limitations. It is not
practical to simulate motion of individual grains unless the spatial scale is
quite limited (e.g., Fig. ) or the grains are unusually large
(Fig. ). If one wished to model individual grains (of order, say,
10-3 m) at the scale of a hillslope (of order 102 m), a much more
efficient solution algorithm would be needed. Furthermore, the nature of a
cellular automaton is such that physical interactions are limited to adjacent
cells only; long-distance effects such as stress transmission cannot easily
be represented. In one sense, the restriction to short-range influence could
be seen as an advantage, in that it forces one to think about how it is that
mass or energy is actually transmitted in a granular medium. But the
restriction means that well-known principles such as solid-state stress
cannot easily be represented. On the other hand, the model does capture
non-local transport, in which particles set in motion can travel a distance
comparable to the slope length
. Non-local
transport emerges in the Grain Hill model when the slope angle is near or
above 30∘, such that there is a high probability that a disturbed
particle will land in an unstable location and continue moving without the
need for a second disturbance event.
A further limitation concerns the fixed cell size. Because the model is
restricted to a fixed cell size, the Grain Hill framework does not lend
itself to treatment of multiple grain sizes (apart from the simple
“aggregates and blocks” approach illustrated in Fig. ).
Despite these limitations, the Grain Hill model provides a useful framework
for exploring hillslope process and form in the context of stochastic events.