This paper describes and explores a new continuous-time stochastic cellular automaton model of hillslope evolution. The Grain Hill model provides a computational framework with which to study slope forms that arise from stochastic disturbance and rock weathering events. The model operates on a hexagonal lattice, with cell states representing fluid, rock, and grain aggregates that are either stationary or in a state of motion in one of the six cardinal lattice directions. Cells representing near-surface soil material undergo stochastic disturbance events, in which initially stationary material is put into motion. Net downslope transport emerges from the greater likelihood for disturbed material to move downhill than to move uphill. Cells representing rock undergo stochastic weathering events in which the rock is converted into regolith. The model can reproduce a range of common slope forms, from fully soil mantled to rocky or partially mantled, and from convex-upward to planar shapes. An optional additional state represents large blocks that cannot be displaced upward by disturbance events. With the addition of this state, the model captures the morphology of hogbacks, scarps, and similar features. In its simplest form, the model has only three process parameters, which represent disturbance frequency, characteristic disturbance depth, and base-level lowering rate, respectively. Incorporating physical weathering of rock adds one additional parameter, representing the characteristic rock weathering rate. These parameters are not arbitrary but rather have a direct link with corresponding parameters in continuum theory. Comparison between observed and modeled slope forms demonstrates that the model can reproduce both the shape and scale of real hillslope profiles. Model experiments highlight the importance of regolith cover fraction in governing both the downslope mass transport rate and the rate of physical weathering. Equilibrium rocky hillslope profiles are possible even when the rate of base-level lowering exceeds the nominal bare-rock weathering rate, because increases in both slope gradient and roughness can allow for rock weathering rates that are greater than the flat-surface maximum. Examples of transient relaxation of steep, rocky slopes predict the formation of a regolith-mantled pediment that migrates headward through time while maintaining a sharp slope break.

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

One gap that remains, however, lies in understanding steep, rocky slopes
(Fig.

Examples of rocky hillslopes, sometimes referred to as Richter
slopes.

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

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

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.

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.

Hypothetical time sequence of transition events

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

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

The method we present here, which we will refer to as the Grain Hill model,
is implemented in the Landlab modeling framework

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 in the Grain Hill model.

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.

The stochastic pairwise transitions in the CTS lattice grain model are
treated as Poisson processes. The probability density function for the
waiting time,

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.

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.

For gravitational transitions, the rate parameter,

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

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

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

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

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 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,

Transitions representing rock-to-regolith transformation by
weathering

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,

Our definition of

Note that the weathering and disturbance rate constants (

Natural regolith disturbance events usually impact many grains at once.
Raindrop impacts on bare sediment typically dislodge several grains at once

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

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.

The basic model has four parameters: the disturbance rate,

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,

Buckingham's Pi theorem dictates that these six variables, which collectively
include dimensions of length and time, may be grouped into four dimensionless
quantities:

The ratio

One can similarly define a dimensionless regolith thickness,

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,

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

As in the

We start by considering the case of fully soil-mantled hillslopes, in which
the supply of mobile regolith is effectively unlimited
(Fig.

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

Equilibrium topographic cross sections using only regolith particles
(no rock) and a variety of disturbance frequencies (

Scaling of mean height as a function of

For any given hillslope length, there are three regimes of behavior. Low

Dimensionless mean hillslope height,

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,

The mean expected activation rate,

A more important difference is that whereas

An additional effect arises from the model's effective 30

We can derive an effective diffusivity,

Integrating and then averaging over

To examine how

Noting that

As expected,

The link between

In the Grain Hill model, the fact that low-angle

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

Here,

Relationship between dimensionless diffusivity and mean gradient,
from the series of 125 model runs of which a subset is shown in
Fig.

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

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

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.

Mean equilibrium gradient and regolith thickness for models with
rock and weathering, as a function of

The models with

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,

The rate of material addition due to uplift relative to base-level,

Equality between rock uplift and weathering can be expressed as

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.

The main message of Fig.

Comparison between rate of material input,

The most popular function to describe regolith production from bedrock is the
decaying exponential formula proposed by

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.

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 (

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.

Two examples of cliff-and-rampart morphology.

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.

The case of transient evolution under a stable base level leads to the
formation of a regolith-mantled, angle-of-repose ramp (Fig.

Quasi-steady model hillslope profiles created using a collapse rule,
under four different combinations of

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

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

As discussed in Sect.

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.

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).

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

For the Gabilan Mesa profile, we estimated the profile-averaged effective
transport coefficient as 0.0345 m

For Yucaipa Ridge, we estimated the transport coefficient at

To test the sensitivity of the Yucaipa example to the assumed characteristic
disturbance depth, we ran a second experiment in which

Steady state models using parameters estimated from observed
hillslope profiles.

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.

With just three parameters – disturbance frequency (

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

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

One limitation of the Grain Hill model is that its threshold-like behavior
arises from the lattice geometry: regolith cells perched at a 30

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.

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.

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.

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.

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.

A continuous-time stochastic cellular automaton model known as the Grain Hill model allows for computational simulation of two-dimensional slope forms that arise from stochastic disturbance and (possibly) weathering events. The model operates on a hexagonal lattice, with cell states representing fluid, rock, and grain aggregates that are either stationary or in a state of motion in one of the six cardinal lattice directions. An optional additional state represents unusually large grains (“blocks”) that cannot be displaced upward by disturbance events.

The Grain Hill model is able to reproduce a range of common slope forms, from
fully soil mantled to rocky and partially mantled. The bestiary of forms that
the model can produce includes convex-upward soil mantled slopes, planar
slopes (bare, soil mantled, or in between), and cliffs with basal ramparts.
When the model is configured to include a resistant rock layer that
decomposes into blocks, the model reproduces observed hogback-like slope
forms and qualitatively matches the behavior predicted by a recent
continuum–discrete model

In its simplest guise, the model has only three process parameters, which represent disturbance frequency, characteristic disturbance depth, and base-level lowering rate, respectively. Incorporating physical weathering of rock adds one additional parameter, representing the characteristic rock weathering rate. These parameters are not arbitrary but rather have a direct link with corresponding parameters in continuum theory. Comparison between observed and modeled slope forms demonstrates that the model can reproduce both the shape and scale of real hillslope profiles.

Experiments with the Grain Hill model highlight the importance of regolith cover fraction in governing both the downslope mass transport rate and the rate of physical weathering. Equilibrium rocky hillslope profiles are possible even when the rate of base-level lowering exceeds the nominal bare-rock weathering rate, because increases in both slope gradient and roughness can allow for rock weathering rates that are greater than the flat-surface maximum. Finally, experiments in transient relaxation of steep, rocky slopes predict the formation of a regolith-mantled pediment that migrates headward through time while maintaining a sharp slope break.

The Grain Hill model is freely available in an open-source repository (Tucker, 2018b), as is the Landlab toolkit (Hobley et al., 2017; Hutton et al., 2016). Videos showing simulations of mesa and hogback evolution are also available as an online resource (Tucker, 2018a).

The supplement related to this article is available online at:

The idea to develop a 2-D cellular rock-slope model arose from conversations among all three authors. The model code was written in Landlab by GT. Both GT and DEJH contributed to the underlying grid data structures and Python code. SWM extracted the hillslope profiles and estimated the parameters for the two field sites. GT performed the computational experiments and wrote the paper, with input and editing from SWM and DEJH.

The authors declare that they have no conflict of interest.

This research was supported by the US National Science Foundation
(EAR-1349390 and ACI-1450409 to Gregory E. Tucker and Daniel E. J. Hobley, and EAR-1349229 to Scott W. McCoy). Daniel E. J. Hobley's
participation was also supported in part by the National Center for Earth
Surface Dynamics (EAR-1246761). Support for high-performance computing and
software development was provided by the Community Surface Dynamics Modeling
System (CSDMS) (EAR-1226297). High-resolution topographic data were downloaded
from Open Topography (