EARTH SURFACE PROCESSES AND LANDFORMS
Earth Surf. Process. Landforms (2012)
Copyright © 2012 John Wiley & Sons, Ltd.
Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/esp.3330
Rock damage and regolith transport by frost: an
example of climate modulation of the
geomorphology of the critical zone
Robert S. Anderson,1,2* Suzanne P. Anderson1,3 and Gregory E. Tucker2,4
Institute of Arctic and Alpine Research (INSTAAR), University of Colorado, Boulder, CO. USA
2
Department of Geological Sciences, University of Colorado, Boulder, CO. USA
3
Department of Geography, University of Colorado, Boulder, CO. USA
4
Cooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado, Boulder, CO. USA
1
Received 7 December 2011; Revised 11 September 2012; Accepted 13 September 2012
*Correspondence to: Robert S. Anderson Institute of Arctic and Alpine Research (INSTAAR), University of Colorado, Boulder, CO. USA. E-mail: [email protected]
ABSTRACT: In this article we craft process-specific algorithms that capture climate control of hillslope evolution in order to elucidate
the legacy of past climate on present critical zone architecture and topography. Models of hillslope evolution traditionally comprise rock
detachment into the mobile layer, mobile regolith transport, and a channel incision or aggradation boundary condition. We extend this
system into the deep critical zone by considering a weathering damage zone below the mobile regolith in which rock strength is
diminished; the degree of damage conditions the rate of mobile regolith production. We first discuss generic damage profiles in which
appropriate length and damage scales govern profile shapes, and examine their dependence upon exhumation rate. We then introduce
climate control through the example of rock damage by frost-generated crack growth. We augment existing frost cracking models by
incorporating damage rate limitations for long transport distances for water to the freezing front. Finally we link the frost cracking damage
model, a mobile regolith production rule in which rock entrainment is conditioned by the damage state of the rock, and a frost creep
transport model, to examine the evolution of an interfluve under oscillating climate. Aspect-related differences in mean annual surface
temperatures result in differences in bedrock damage rate and mobile regolith transport efficiency, which in turn lead to asymmetries
in critical zone architecture and hillslope form (divide migration). In a quasi-steady state hillslope, the lowering rate is uniform, and
the damage profile is better developed on north-facing slopes where the frost damage process is most intense. Because the residence
times of mobile regolith and weathered bedrock in such landscapes are on the order of 10 to 100 ka, climate cycles over similar
timescales result in modulation of transport and damage efficiencies. These lead to temporal variation in mobile regolith thickness,
and to corresponding changes in sediment delivery to bounding streams. Copyright © 2012 John Wiley & Sons, Ltd.
KEYWORDS: critical zone; frost processes; hillslope evolution; weathering; mobile regolith production; climate change
Introduction
The geomorphologist views the critical zone as a threedimensional (3D) structure formed by weathering and transport
processes, which are themselves powered by tectonics, climate,
biota, and gravity. Understanding the complex interactions
that govern the evolution of this structure is among the ‘grand
challenges’ in Earth surface processes (National Research
Council, 2010). A slice through the critical zone (Figure 1) reveals
three interfaces of significance: (1) the boundary between
atmosphere and the ground surface, (2) the boundary between
disaggregated and displaced material that comprises mobile
regolith, and the weathered but in-place bedrock beneath it,
and (3) the boundary between weathered and fresh bedrock. To
understand the evolution of critical zone structure requires
process models that describe transformations and mass transport.
Together these processes produce the structure we see and
control the relative movement of these three interfaces. At a
minimum, models of hillslope evolution require rules for the rate
of detachment of rock into the mobile regolith layer, for the rate of
mobile regolith transport, and for channel incision or aggradation
rates that serve as boundary conditions for the hillslopes. A
more complete model would also describe the evolution of
material properties (e.g. mineralogy, grain size and derived
quantities such as porosity, hydraulic conductivity and
thermal diffusivity) within the weathered bedrock and on the
conveyor belt of the mobile regolith layer. Here we present
an approach to understanding hillslope evolution and critical
zone structure in which the rules are based upon specific
processes, rather than on generic descriptors of the system
such as slope angle and soil thickness.
A note on terminology
Because standard terminology varies somewhat between the
geomorphology, geochemistry and soils communities, it is useful
to start by defining several key terms. We use the word ‘bedrock’
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
HIL
LS
L
O
Regolith
transport
Entrainment
probability
Damage
crack
density
time-average
parcel
trajectory
Weathering as Damage
PE
1 at m
os
ph
ere
mo
bil
er
2
e
we
go
lit h
at h
ere
dr
oc
k
C
H
A
N
N
E
L
3
fres
h ro
ck
Figure 1. Schematic diagram of the critical zone in a hillslope crosssection. Three interfaces are identified with numbers: 1, the ground
surface, or atmosphere-mobile regolith interface, 2, the bottom of soil,
or the mobile regolith-weathered bedrock interface, and 3, the weathered-fresh bedrock interface. A parcel of rock over time follows the
trajectories shown with large arrows. In the rock layers, material moves
toward the surface as it is exhumed. At interface 2, particles are released
into the mobile regolith, which may be defined as a granular material in
which most grains have moved by more than one grain diameter relative
to their immediate neighbors. The time-averaged trajectory of material
in the mobile regolith is downslope, although individual displacement
events may move material in any direction. Two profiles at the left
edge of the diagram illustrate the important processes driving evolution of the critical zone. The damage profile in the weathered bedrock
shows the accumulated effect of damage processes as material moves
toward the surface. Damage transforms fresh bedrock to weathered
bedrock by altering its physical and hydrologic properties through
growth of cracks. Increasing damage renders weathered bedrock more
susceptible to entrainment into the mobile regolith layer, shown with a
change in entrainment probability from low (white) to high (black).
Creep processes are responsible for the transport profile in the mobile
regolith layer. Mobile regolith is ultimately delivered to the stream
channel, whose elevation and transport capacity serve as boundary
conditions for the hillslope. This figure is available in colour online
at wileyonlinelibrary.com/journal/espl
to mean fresh, unweathered bedrock and often use the qualifier ‘fresh’ to further reduce ambiguity. This is the geochemists’
protolith. To the geomorphologist, weathered material can be
subdivided into material that has been physically displaced
by transport or mixing processes and material that has not
been transported or disrupted by mixing processes. We refer
to the mobile material, what often is called soil, as ‘mobile
regolith’, honoring the recent use of that term (e.g. Lebedeva
et al., 2010; Riggins et al., 2011). Mobile regolith is distinguished by the loss of rock structure due to transport and/or
mixing processes. The regolith that is not mobile is here
termed weathered bedrock: bedrock that has been weathered
or damaged by surface-related processes, but has not
been transported by slope processes or mixing processes.
Weathered bedrock encompasses rock weathered to any
degree, including saprock and saprolite, so long as it retains
original rock structure and has not been physically displaced.
Although weathered bedrock is immobile, it may well be
capable of being moved by surface processes; the presence
of completely weathered saprolite for instance, implies that
mixing and transport processes have not operated with
sufficient vigor or to sufficient depth to entrain it into mobile
regolith. We refer to the lowering of the mobile regolith–
weathered bedrock interface (labeled 2 in Figure 1) as
‘production of mobile regolith’ or ‘release of saprolite into
mobile regolith’. This process of conversion of weathered
bedrock to mobile regolith is the geomorphologist’s ‘regolith
production’, sometimes referred to simply as ‘weathering’, a
usage we will avoid.
Copyright © 2012 John Wiley & Sons, Ltd.
Weathering describes everything from solute flux from watersheds (e.g. Gaillardet et al., 1999), to changes in soil mineralogy
(White et al., 1996), to generation of fractures (Hall, 1999;
Fletcher et al., 2006), to production of mobile regolith (e.g.
Anderson and Humphrey, 1989). Here we address weathering
as a suite of processes that damage rock. We focus specifically
on rock damage that accrues through initiation and growth of
fractures. In other words, we will focus on physical processes
rather than chemical processes, although we acknowledge that
physical damage can promote chemical processes (e.g. Fletcher
and Brantley, 2010; Jin et al., 2011), while chemistry rather than
physics leads the way in many settings. Our focus on rock
damage through fracturing has parallels in the analysis of the
effect of stress on rock, for instance in landsliding, rock cliffs,
and faults (e.g. Amitrano and Helmstetter, 2006). Weathering
processes that fracture rock include thermal stress (Hall, 1999),
biotite hydration (e.g. Fletcher et al., 2006; Fletcher and Brantley,
2010), and frost cracking (Matsuoka and Murton, 2008). Growth
of microcracks increases rock porosity and permeability (Sousa
et al., 2005; Graham et al., 2010; Jin et al., 2011); greater fluid
circulation leads to a cascade of processes that further alter and/
or damage the rock. Indeed, the full spectrum of cracks is important. Larger cracks can govern rock mass strength, and will strongly
influence bulk permeability because they are likely to be more
interconnected and have larger apertures than microcracks.
The damage that rock accrues on its way toward the surface
can be thought of as a reduction in the rock’s ability to resist
shear, and ultimately to resist disintegration. This resistance is
likely related to the density of flaws in the rock parcel. In this
view, the rate at which damage accumulates in a rock mass
may be cast as the rate at which new crack surface area is
generated per unit volume of rock [=] L2/L3T = 1/LT. (Note that
we use [=] to mean ‘has dimensions of’ throughout this paper,
with M, L and T representing mass, length, and time, respectively.) Damage, D, which is the time integral of damage rate,
therefore has dimensions of 1/L. Its inverse is a length scale, and
may be thought of as the mean spacing between microcracks.
We argue that the susceptibility of weathered bedrock into the
mobile regolith ought to increase with increasing damage; this
is equivalent to assuming that the smaller the pieces are, or the
weaker the mass is, the more easily the weathered bedrock will
become part of the mobile regolith. Note that the concept of
damage as used here is similar in spirit to Kirkby’s (1977, 1985)
‘soil deficit,’ which represents the amount of parent rock remaining at a particular level within a soil profile. It is also consistent
with the concept of weathering-related damage used by Hoke
and Turcotte (2002) in an analysis of surface disintegration
through mineral dissolution.
Damage therefore results in reduction of physical strength
(ability to resist deformation under stress), which depends upon
the numbers and strengths of the bonds within the material.
One may then argue that (1) flaws matter, (2) the density of flaws
affects rock porosity and permeability, as well as strength
(Graham et al., 2010; Luque et al., 2011), (3) strength is modified
by a rock’s tectonic history (Molnar et al., 2007; Koons et al.,
2012), which can produce flaws, (4) strength (and damage)
can be quantified with tensile or shear strength (e.g. Aydin and
Basu, 2006), as measured by tools such as splitters and shear
vanes, (5) the huge range in strength among geomaterials, and
notably between fresh bedrock and saprolite, raises a practical
challenge in quantifying strength and damage across this range,
and (6) strength may be related to mineralogy, cement, bulk
density (in the sense that it reflects grain proximity and bond
strength), degree of chemical alteration, connectedness of the
fracture network, and so on. We note that the properties of the
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
rock mass that matter to entrainment in mobile regolith will likely
differ from the strength that matters to the disintegration or failure
of a cliff (e.g. Moore et al., 2009).
In the remainder of the paper we will describe a generic view
of the damage zone, which we follow by working a specific
example in which damage occurs by frost cracking of the rock.
Armed with this model of damage, we link damage, entrainment,
and transport processes to produce a model of hillslope evolution
relevant to landscapes in which frost-related processes dominate.
Reactor on a Hillslope
We first describe the critical zone in its hillslope context, focusing on the concept of a reactor on a hillslope that is emerging as
an appropriate analog. Consider the path taken by a parcel of
rock as the landscape surface erodes down toward it (Figure 1).
From the point of view of the rock, it moves toward the surface
at a rate governed by the rate of exhumation (lowering) of the
landscape. In effect, the rock advects through the weathering
zone en route to the base of the mobile regolith. As the rock
advects vertically through the weathering zone, it will begin to
be altered by surface-related critical zone processes. These
processes are physical, chemical, and biological, and the
relative roles of these processes in altering the rock mass differ
by climate, rock type, and the density of flaws in the rock that
reflects a legacy of its tectonic past. The rock accumulates
damage as it passes through the critical zone. If the dominant
changes are chemical, the analogy with a ‘reactor’ is direct. If
they are dominantly physical, perhaps the better term would
be ‘mill’ or ‘grinder’. However, because the comminution of
the material by any physical processes will in turn alter its
hydrologic properties, with attendant feedbacks on the rates of
chemical alteration, the term reactor perhaps remains appropriate. Upon emerging at the top of the weathered bedrock, the
rock parcel is available to be released into the mobile regolith.
The probability of release may depend on the degree of
damage. For example, release into mobile regolith may require
the parcel to be smaller than some critical size, or to possess a
sufficient density of cracks to allow plant roots to pry it from
the subjacent rock. In any case, the state of the rock parcel upon
entrainment on the mobile regolith conveyor belt governs the
initial mineralogical and grain size distribution of the mobile
regolith. Subsequent motion of the parcel down the hillslope
is governed by the processes that both translate and stir the
mobile regolith. Here the chemical and biological reactor
intensifies as hydrologic, thermal, and biological activity are
all amplified in the near surface. During translation, other
parcels of rock are added to the base of the mobile regolith, their
own character upon release being dependent upon the particular damage profile through which they were advected within the
base of the critical zone. We expect the mineralogy and grain
size distribution of the mobile regolith to evolve as it translates
downslope. The rate of motion of the mobile regolith conveyor
on which these reactions take place is governed by the local
slope, and by climatically modulated chemical, physical and
biological processes.
The soil ultimately delivered to the adjacent stream therefore
reflects the evolution of rock as it moved through the intact rock
portion of the critical zone – the weathering zone - and further
digestion during downhill transport on the mobile regolith
conveyor. The timescale over which this evolutionary play takes
place is thousands to hundreds of thousands of years (Yoo et al.,
2011). However, the chemistry of the water delivered to the
stream reflects the operation of present-day hydrologic, chemical
and biological processes integrated over timescales of minutes to
years. The rocks and minerals through which the water passes,
Copyright © 2012 John Wiley & Sons, Ltd.
and the reactions that these allow, will of course condition the
chemistry, allowing examination of the rates and types of
weathering reactions taking place.
The timescales over which the rock parcels transit this critical
zone system are long. This requires that we acknowledge deep
time in studies of the critical zone. One timescale may be derived
from steady state landscape analysis, in which the horizontal
length scale of the hillslope, L, and the effective diffusivity of
the landscape, k, combine to yield a hillslope timescale T = L2/k
(e.g. Koons, 1989). Given hillslope lengths of order hundreds of
meters, and landscape diffusivities of order 10–2 m2/yr (e.g.
Fernandes and Dietrich, 1997), this suggests that the time to
develop steady hillslope forms is on the order of a million years.
Another timescale that is more relevant to the critical zone that
forms the carapace of such hillslopes can be constructed from
the residence time of mobile regolith on a hillslope. This is scaled
by T = H/w, where H is the thickness of mobile regolith, and w the
rate of mobile regolith production (rate of lowering of the mobile
regolith–weathered bedrock interface). Given mobile regolith
thickness of order 1 m, and mobile regolith production rates
constrained by concentrations of cosmogenic radionuclides
of order 10–4–10–5 m/yr, we expect residence times of order
10–100 thousand years. If the thickness of the zone of chemical
and/or physical weathering in the rock beneath the soil is several
times the thickness of the soil, the residence time within the
weathering zone will be commensurately longer. It is therefore
clear from even this simple scaling that the critical zone evolves
on timescales that are similar to those of the major swings in
climate that characterize the Plio-Pleistocene glacial ages, with
accompanying major changes in physical, chemical and biological processes and process rates. It is therefore inevitable that the
architecture of the critical zone, and the mineralogical, grain size,
and hydrologic properties of the materials presently residing in
the critical zone, reflect a legacy of past climates, reaching back
at least one glaciation. The present is therefore not a proper or
complete key to the past. Any study of the critical zone must
honor this legacy.
Damage Profile – A Generic Case
Consider the changes in a parcel of initially unweathered bedrock
as it is brought toward the weathered bedrock–mobile regolith interface by exhumation (Figure 1). We expect that many processes
responsible for altering the rock mass strength will decline in intensity with distance below the surface. For now, suppose for simplic_ falls off exponentially with depth and is
ity that damage rate, D,
independent of the damage state of the rock mass:
_ ¼D
_ 0 e z=zdamage
D
(1)
where zdamage is a length scale characterizing the rate of decay of
damage rate with depth. This would take the place of the length
scale L in our discussion of timescales earlier. In later sections of
the paper, we acknowledge specific processes that exert depth
profiles of damage rate that are not necessarily exponential.
The rock parcel moves through this damage zone at a rate
governed by the steady landscape lowering rate, e_ . The accumulated damage as the bedrock arrives at the base of the mobile
regolith is therefore simply the integral of the damage rate, and is
Z0
D¼
1
_ 0 zdamage
D
þ Do
D_ 0 e _et=zdamage dt ¼
e_
(2)
Note that the rock will arrive in the near-surface zone with some
initial flaw density and hence initial state of damage, D0, setting
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
_ ¼D
_ o e ðe_ t=zdamage Þ 1 D
D
Dcrit
(3)
Here we have assumed the simplest linear dependence on
damage, although one could consider more complex dependence if the physics warrants it. [Note that a formulation analogous to this has been used to model the rate of damage to
vegetation by erosion (Collins et al., 2004; Tucker et al., 2006).]
Integrating Equation 3 from great depths to the surface yields
the state of damage in the rock at the base of mobile regolith
_ z
D
0 damage
D ¼ Dcrit 1 e e_Dcrit
(4)
Exhumation rates high enough to assure that the ratio in the
argument of the exponential is much less than unity, in other
_ z
D
words, e_ > 0Ddamage
, result in the dependence captured in
crit
Equation 2 (because in this case D/Dcrit < < 1 in Equation 3).
Lower exhumation rates [or thicker zone (zdamage) or greater
_ 0)] allow sufficient damage to accumumaximum damage rate (D
late that the limiting damage Dcrit begins to play a role. Numerical
integration of Equation 3 for a number of exhumation rates results
in steady state damage profiles (Figure 2) that illustrate this
dependence. Under high exhumation rates, the time spent in
the damage zone is insufficient to achieve the limiting damage
Dcrit. When exhumation rates are low, the limiting damage is
achieved; further lowering of the exhumation rate results in
profiles that differ only in the depth at which this limiting damage
is attained.
Copyright © 2012 John Wiley & Sons, Ltd.
0
.
ε=1 cm/yr
damage-limited cases
1
Normalized depth (z/zdamage)
the ‘intact rock strength’ of the fresh bedrock. In general, this will
vary with rock type and with the tectonic history to which the
rock has been subjected prior to interaction with the surface
(Molnar et al., 2007; Clarke and Burbank, 2010, 2011), and therefore will vary spatially. The additional accumulated damage upon
reaching the base of the mobile regolith (at t = 0) depends directly
on the damage process rate at the mobile regolith–weathered
_ 0 , and the characteristic length scale over
bedrock interface, D
which it declines with depth, zdamage, and depends inversely on
the lowering rate of the mobile regolith–weathered bedrock
interface, e_ . This formalism is perfectly analogous to that used to
constrain local erosion rates from cosmogenic radionuclide
concentrations in rock at the base of the regolith (e.g. Small
et al., 1999). Just as more rapid erosion yields lower
beryllium-10 (10Be) concentrations in the rock that emerges at the
base of mobile regolith, the more rapidly the rock moves through
the damage zone, the less damage it accumulates. The framework
is also similar in spirit to the model of granite weathering by
albite dissolution explored by Lebedeva et al. (2010).
Accumulated damage could reflect more than one factor. For
example, if pyroxene grains in a gabbro dissolve, both the bulk
tensile strength and the bulk density would decrease. They
might do so at different rates and with different length scales,
zdamage. In another case, root stresses may weaken intergranular bonds without producing much change in bulk
density. Thus, one might expect variations in the damage rate
profiles with landscape position.
We now acknowledge the potential role of a limiting damage,
Dcrit, beyond which it becomes much more difficult to generate
additional damage. For example, Dcrit might represent a state of
complete transformation of a reactive mineral species, such as
albite to kaolinite (Lebedeva et al., 2010), or a case in which all
or most intergranular bonds have been broken by physical
stresses from root growth, ice lens formation, or similar processes.
The rate of damage then becomes
.
ε=1 mm/yr
2
e
ted
imi
cas
-l
ent
m
n
rai .
ent
ε=0.1 mm/yr
3
4
5
6
0
0.2
0.4
0.6
0.8
1
Normalized damage (D/Dcrit)
Figure 2. Profiles of damage expected for three very different exhumation
rates. Depth axis normalized by the depth scale zdamage. Damage axis
normalized by the limiting damage, Dcrit. Fast exhumation (and more
generally, any large value of the ratio e Dcrit / D0 zdamage) leads to less
damage throughout the profile. Given the choices for other parameters
_ 0 and zdamage, for rates less than roughly 09 mm/yr, damage reaches
D
its asymptotic value governed by the limiting value Dcrit. Damage-limited
and entrainment-limited cases are described in the Discussion section. This
figure is available in colour online at wileyonlinelibrary.com/journal/espl
Consider now the top and bottom of the damage profile. One
may ask at what depth one would encounter ‘fresh’ rock. Operationally, one might assign a particular threshold damage below
which the rock is considered intact, or fresh, and we would call
that the base of the weathered bedrock. One might document this
depth in rock cores. Less expensively, the depth to intact rock
may be remotely sensed using shallow seismic methods (e.g.
Clarke and Burbank, 2011) in which velocity gradients may be
used to estimate fracture density profiles. If however profiles exist
in the form of cores or pits, one may document the depth scale,
zdamage, for decay of some strength measure of damage, and
define fresh rock as some multiple of zdamage at which depth the
damage falls below some level.
At the top end of the damage profile, the weathered bedrock–
mobile regolith interface, a potential coupling exists between the
weathering of rock in the subsurface and the likelihood that a
parcel of rock will be detached from its neighbors to become
entrained in the mobile regolith. This might occur, for example,
if the detachment were accomplished by the burrowing of
rodents, the growth of tree roots, or growth of ice lenses, and
these processes could only loft rocks below a given size. This
critical size will depend upon the available entrainment
mechanisms, and will therefore be site-specific and dependent
upon the climate.
The damage profiles shown in Figure 2 are analogous in
many ways to the one-dimensional (1D) profiles developed
by geochemists to capture the essence of weathering. As
recently summarized by Brantley and Lebedeva (2011), chemical depletion profiles display dependence on both the rate of
lowering of the land surface (erosion rate) and the depthdependence of the process involved in altering the rock and
the minerals within it. The damage-limited case could also
be analogous to the geochemist’s kinetically limited case, in
which the shape of the weathering front is governed by the
reaction constants in the mineral transformations.
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
Damage by Frost Cracking
Theory of frost cracking
We now consider the case in which the damage imparted to
the rock mass as it passes through the weathering zone is due
solely to frost cracking (e.g. Murton et al., 2006). Ice lens growth
in pre-existing flaws in rock can expand openings and extend
fractures, leading to shattering of rock (Matsuoka, 1990). Studies
of freezing soils (e.g. Taber, 1930; Williams, 1967) have shown
that the transformation of water into ice in a porous medium
proceeds somewhat differently from the phase transformation in
bulk water. In porous material such as soil or rock, ice forms over
a range of temperatures, so that unfrozen water and ice are both
found at sub-zero temperatures (Dash et al., 2006). Ice growth
begins in the largest pores, and continues in progressively smaller
pores as temperatures decline. Unfrozen water migrates in thin
films through partially frozen porous media to feed the growth
of discrete ice lenses in the larger voids of soil or rock at sub-zero
temperatures. Frost cracking is important at temperatures just
below freezing, a temperature range called the ‘frost cracking
window’; although it varies with rock properties and freezing
rate, we take this range to be –3 to –8 C (Walder and Hallet,
1985). Experiments confirm that cracking continues in rock
within the frost-cracking window as long as liquid water is
available to move to the freezing front (Walder and Hallet 1985;
Walder and Hallet, 1986; Hallet et al., 1991; see also Murton,
1996; Murton et al., 2006; Matsuoka and Murton, 2008). Finally,
we follow Hales and Roering (2007) in supposing that the rate of
frost cracking by growth of segregation ice is proportional to the
temperature gradient (see also Worster and Wettlaufer, 1999).
We develop a model of the depth profile of frost cracking in
several steps. We begin with a consideration of temperature
variations in the subsurface, as the duration of sub-freezing
temperatures and the rate of freezing are important parameters in frost cracking. We then present several models of
increasing complexity that quantify the effect of thermal state
on frost cracking.
20
20
Temperature (°C)
Similarly, recent models of corestone development in weathering profiles (Fletcher and Brantley, 2010) acknowledge the roles
of initial flaw densities, and the rate of propagation of reaction
fronts into the blocks between these fractures as setting the sizes
of the corestones as they encounter the earth’s surface. They
differentiate between incompletely developed profiles, in which
corestones remain in the profile, and completely developed
profiles in which no remnants of the original blocks remain in
the top of the profile.
While the approach outlined in this section is simple, it is
impoverished in that it fails to address any particular process.
Next we develop models based on a specific process, namely
frost cracking, by which rock can be damaged during its passage
through the weathered bedrock zone toward the surface. We
argue that frost cracking is relevant to the evolution of many
regions, in that while modern temperatures may not reach deeply
into frozen conditions, peri-glacial conditions were widespread
in the last glacial maximum (LGM) when mean annual temperatures were depressed by at least 6 C and in places by 12 C.
15
T (0-3 cm)
T (16-19 cm)
T (32-35 cm)
T (48-51 cm)
15
snow last
covers snow
surface melts
10
10
5
5
0
0
-50
0
50
100
150
200
DOY 2009
Figure 3. Shallow ground temperature histories, measured hourly over
most of an annual cycle, from the Gordon Gulch site in the Boulder Creek
Critical Zone Observatory, Colorado, USA. Snow events are noted; when
significant snow covers the ground, surface temperatures approach 0 C,
and lose any significant daily oscillation. This figure is available in colour
online at wileyonlinelibrary.com/journal/espl
dramatically damps the amplitude of surface temperature oscillation as long as snow cover exists. And the freezing and thawing of
moisture in the ground releases and consumes latent heat that
must be accounted for in any calculation that conserves heat.
In Figure 3, the shallow ground temperatures are depicted from
the Gordon Gulch site of the Boulder Creek Critical Zone
Observatory (CZO), Colorado, USA, from four depths. When
snow covers the ground, the surface temperature approaches
0 C, and is held there while the snow persists. Temperatures at
all other depths more slowly approach 0 C as this serves as the
new boundary condition.
Inspired by such temperature records, and in the hope of
generating a thermal model of the near subsurface that permits
exploration of the dependence of surface and subsurface
processes on temperature, we take the temperature of the surface
to be sinusoidal at both daily and annual timescales. We
prescribe the mean annual temperature, and amplitudes of both
daily and annual variations around the mean to yield
2pt
2pt
Ts ¼ T s þ ΔTa sin
þ ΔTd sin
365
1
(5)
where ΔTa and ΔTd are the amplitudes of annual and daily
temperature oscillations, respectively, and t is time in days.
The temperatures at all depths and times are then calculated by
conserving heat, assuming that conduction dominates (no heat
transport by moving water), and acknowledging the role of latent
heat at the phase boundary. In one dimension, the rate of change
of temperature T with time t reflects conservation of heat:
@T
1 @Qh
¼
@t
rc @z
(6)
where z is the depth into the ground, r is local material density, c
its heat capacity, and Qh the vertical heat flux. Ignoring the role of
motion of water as a heat transport mechanism, heat transport is
governed by conduction through Fourier’s Law:
Qh ¼ k
@T
@z
(7)
Temperature variations in the critical zone
Measured temperature time series such as those illustrated in
Figure 3 reveal several features that we must capture in any
proper model of the system. Temperature varies on both
annual and daily cycles, with comparable amplitudes. Snowfall
Copyright © 2012 John Wiley & Sons, Ltd.
where k is the local (vertical) conductivity of the substrate. We
allow thermal properties to vary with water and ice content. We
account for the role of phase change of water using a strategy
employed by Ling and Zhang (2003), and mimicked by West
and Plug (2008) (see also Matell et al., 2011). The thermal
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
conductivity and specific heat capacity are updated in each time
step according to whether the material is thawed (T > 0 C) or
frozen. For temperatures within a ‘phase change envelope’
(defined here as the 1 C envelope between –1 C and 0 C),
the thermal constants are calculated differently (Ling and Zhang,
2003; see also West and Plug, 2008) to incorporate the effects of
latent heat, as well as to acknowledge the fact that the freezing of
water in soil and rock does not take place entirely at the freezing
temperature. Thermal conductivity and volumetric specific heat
capacity are calculated as
8
>
< kf
k kf
k ¼ kf þ u
½T ðTf DT Þ
>
DT
:
ku
T < Tf DT
Tf -DT < T < Tf
T > Tf
converted into specific heat capacity c by dividing by the density
of the ice/water/soil combination.
As shown from the data depicted in Figure 3, the presence of
snow can greatly influence the subsurface temperature histories
(e.g. Sturm et al., 2001; Bartlett et al., 2004). We do not perform
a full energy balance model to compute the temperatures within
the snowpack. Instead, we employ the surface air temperature to
melt the snow surface when air temperature rises above 0 C, and
redeposit the latent heat associated with that melt within the
snowpack. If air temperature is below 0 C, heat is transported
purely by conduction within the snowpack with a thermal
conductivity appropriate for snow. We emphasize that this is
not a full snowpack thermal model, but wish through this exercise
to illustrate the essence of the role of snow in altering subsurface
temperature histories.
In the example calculation shown in Figure 4, the top
40 cm is taken to be mobile regolith with 40% porosity,
whereas the underlying bedrock is taken to have 3% porosity. Both are assumed to be 90% saturated. While water
content of rock is very difficult to measure directly, recent
work in alpine rock masses has shown that water content
in rock is quite high and relatively steady at depth (Sass,
2005; Graham et al., 2010; Langston et al., 2011). At shallow
depths, the assumption of 90% saturation certainly overestimates the role of water in stalling the penetration of heat
into the subsurface. We employ this value as a means of
highlighting the effects here. To illustrate the role of snow
cover, we have prescribed a 40 cm snowfall event that occurs
at night on day 60.
That the snow persists for two months allows this case to
serve as an illustration of the expected effects of snow cover.
We assumed that snow cover of 40 cm fell at midnight, when
the surface temperature was at its lowest. Given the high
thickness of the snowpack, the amplitudes of the temperature
variations at all depths are subsequently highly damped. When
the daily oscillations of air temperatures begin to go above 0 C,
on day 80, melting of the snow begins to occur. Re-deposition
of that heat in the snowpack allows it to warm, which is reflected
in the rise of ground surface temperatures (Figure 4, red line). The
snowpack becomes isothermal (at 0 C; not shown) on day 90,
after which the temperatures at all depths asymptotically
approach 0 C; the problem has now become an instantaneous
warming problem. The snowpack then remains isothermal until
the snowpack disappears on day 120. At each depth the temperature stays within the –1 C to 0 C phase change envelope, DT,
until sufficient heat to melt all ice in the pores has been supplied.
The stalling of subsurface temperatures during phase change,
called the ‘zero curtain’ effect, reflects the latent heat required
9
>
=
>
;
(8)
and
8
C
>
T < Tf DT
< f
W Wu
Tf -DT < T < Tf
C ¼ Cf þ Lrb
>
DT
:
Cu
T > Tf
9
>
=
>
;
(9)
where Tf is the freezing temperature (in kelvin), DT the width of
the phase change envelope (in kelvin), L the latent heat of freezing for water (in J/kg), W the fractional total water content of
the soil by mass, Wu the unfrozen water content that remains at
Tf – DT (here taken to be 5% of W), and the subscripts ‘f’ and
‘u’ refer to the frozen and unfrozen values of the thermal conductivity and specific heat capacity, both of which are calculated as
inputs to the model based on soil properties and water/ice
content (Lunardini, 1981). Thermal conductivities are calculated
separately for frozen and unfrozen soils as geometrically
weighted values
k ¼ k i υi k w υw k s υs
(10)
where ki, kw, and ks are the thermal conductivities of ice, water,
and mineral or rock, and where υi, υw, and υs are the fractions
of ice, water, and mineral in the substrate (Ling and Zhang,
2004). Specific heat capacity c (in J/kg/K) is calculated as
c ¼ ci υi þ cw υw þ cs υs
(11)
where ci, cw, and cs are the specific heat capacities of ice, water
and mineral (Ling and Zhang, 2004). While υs does not vary in
time, the fractions of ice and water evolve as the substrate freezes
and thaws. Volumetric specific heat capacity C (in J/m3/K) is
10 regolith = 0.4m, 40% porosity, 90% saturated
Temperature (°C)
rock 3% porosity, 90% saturated
5
0
-5
time series at these depths:
0, 10, 20, 30, 40 cm
-10
snow cover
-15
0
50
100
150
200
250
300
350
Time (days)
Figure 4. Modeled shallow ground temperature histories at four depths over an annual cycle, starting January 1, driven by the surface temperature
history (depth = 0 cm) shown. Model incorporates a two-month period of snow cover, initially with 04 m thickness, and temperature of –12 C at the
time of deposition. The zero-curtain is clearly illustrated, during which time the latent heat associated with freezing and thaw of the water stalls
propagation of the thermal wave into the ground.
Copyright © 2012 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
to thaw the ice. The zero curtain persists for roughly 10, 20 and
30 days for depths of 20, 30 and 40 cm, respectively. The zero
curtain occurs again when the ground freezes in the fall, such that
near-surface temperatures drop well below freezing while the temperatures at depth are stalled by the release of latent heat in the
freezing process. The duration of this effect is dependent upon the
total water content of the soil and subjacent rock. Our example,
using 90% saturation maximizes the duration of the zero curtain.
Another way to view temperature histories is shown in Figure 5,
where we depict temperature profiles at many different times
within an annual cycle. Note the asymmetry of the envelope withinwhich temperature varies. In the absence of latent heat effects,
this envelope would be a symmetrical exponential funnel with
pffiffiffiffiffiffiffiffiffiffiffi
length scale kP=p (e.g. Gold and Lachenbruch, 1973), where
k is the thermal diffusivity (¼ k = cr ), and P is the period of the
temperature oscillation. For a typical diffusivity k of 10–6 m2/s,
the length scale for daily fluctuations would be 017 m, while that
for the annual cycle would be 32 m. In the case illustrated, with
mean annual temperature of –2 C and daily and annual oscillations of 6 C, temperatures never rise above 0 C at depths below
about 2 m. The area above this depth, where temperatures do rise
above 0 C, is known as the active layer. Rock spends some
amount of time in the cracking window for all depths down to
5 m. But given that the easily available water in the larger pores
is permanently frozen below the active layer, access to water is a
problem for these greater depths. As water must come from the
surface in such permafrost situations, it becomes less and less likely
that cracking occurs for greater depths. Inspection of specific
temperature profiles suggests that the primary times at which water
is readily available to drive ice growth and hence cracking are in
the spring, when daily surface temperature rises above 0 C.
Frost cracking models
We present several models of frost damage (Figure 6) with
increasing levels of complexity and realism. For the first exercise,
meant to allow comparison with past models, we omit daily
0
regolith
1
2
active layer depth
Depth (m)
3
4
5
frost
cracking
window
6
rock
7
8
9
-10
-5
0
5
10
Temperature (°C)
Figure 5. Modeled temperature profiles through the critical zone
at weekly intervals that span one year, with mean annual temperature of
–2 C, and annual and daily amplitudes of 6 C, and mobile regolith
thickness, porosity, and water saturation the same as for Figure 4. The
temperature range from –3 to –8 C (shaded), optimal for frost cracking,
is called the ‘frost cracking window’. Phase change of water, most of
which resides in the 04 m thick mobile regolith, governs the asymmetry
of the envelope within which temperatures vary. Tilt of the entire profile
reflects geothermal heat flux of 40 mW/m2. This figure is available in
colour online at wileyonlinelibrary.com/journal/espl
Copyright © 2012 John Wiley & Sons, Ltd.
temperature oscillations, we employ a uniformly low porosity
material (with no distinction between mobile regolith and
bedrock), which minimizes the effects of latent heat discussed
earlier, and we ignore effects of snowpack. These simplifications
allow us to compare with the models of Anderson (1998) and of
Hales and Roering (2007), and to focus on the first order controls
on frost cracking.
In the simplest model, put forth by Anderson (1998), the time
rock spends in the frost cracking window (middle panels of
Figure 6) serves as a proxy for frost damage. Hales and Roering
(2005, 2007) improved this model by acknowledging that this
simple proxy does not capture the role of water as a limiting factor
in frost lens growth in rock. In their model, two conditions must
be met for frost cracking to occur: the rock is in the frost-cracking
window, and a source of liquid water (T > 0 C) is available in a
pathway along which temperatures monotonically increase
(unfrozen water at the surface and downward decreasing
temperatures; or water at depth and upward decreasing temperatures). Reasoning that cracking intensity is proportional to temperature gradient (as is growth rate of segregation ice; Worster and
Wettlaufer, 1999), Hales and Roering calculate crack intensity
as the product of time in the frost cracking window and temperature gradient, which should yield a measure with units of temperature-time/length. The calculation is carried out for each time
step, and summed for one year. Hales and Roering (2007) report
crack intensity in units of C/cm, however, leaving the time
increment implicit. We can reproduce their results by recalling
that they employ a one day time step and noting a factor of 100
difference that arises from temperature gradients in C/m rather
than C/cm. We track cracking intensity in units of C-days/m,
although on several of the plots we have normalized against
the maximum value to allow comparison of the patterns
between models that generate very different magnitudes of
cracking intensity.
Comparison of these models is illustrated in Figure 6. We show
two cases, one in which the mean annual temperature is well
above 0 C, and one in which it is well below 0 C. On the left
we show temperature profiles at many times of the year, as in
Figure 4; in these cases, however, the profiles define a symmetric
exponential funnel, because latent heat effects are ignored. In the
middle panels we show the expected time spent in the cracking
window [here taken to be (–8 C to –3 C)]. Following Anderson
(1998), these patterns are well predicted with an analytic solution
(dashed black line) as long as the time series of surface temperatures is sinusoidal, and the material properties governing thermal
diffusivity are uniform with depth and constant in time. The
numerical tracking of time spent in the frost-cracking window
(solid red line) reproduces well the expected analytic solution in
both cases.
In the right panels, we show in gray the temporal sum of
thermal gradient [the proxy for cracking intensity proposed by
Hales and Roering (2007)], here normalized against its maximum
value. We show in the left panel with heavy gray lines the
segments of the instantaneous temperature profiles with the
appropriate combination of temperature gradient and presence
of unfrozen water elsewhere in the profile such that frost cracking
can occur. In Figure 6a, with T s = 3 C, the predicted pattern of
annual cracking intensity is clearly governed by time in the cracking window. But given that the only times when the temperatures
dip into the frost cracking window correspond with times when
the surface temperatures are < 0 C, the (unfrozen) water available for driving cracking must always come from depths that
are well below the site of potential frost cracking. In the sub-zero
mean annual temperature case (Figure 6b), there are times in
summer when water is available from the surface, and times
in winter when it is available at depth (two sets of heavy gray lines
in the left panel, one dipping to the left, one dipping to the right).
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
pattern due to time in
frost-cracking window
pattern due to temperature
gradient and access to water
0
0
0
1
1
1
2
2
2
3
3
3
4
4
4
a Ts = 3°C, ΔTa = 12°C
5
6
5
7
8
8
9
9
-5
0
5
10
10
15
analytical
solution
6
7
10
Depth (m)
Depth (m)
Depth (m)
not penalized
penalized
5
6
7
numerical
solution
8
9
0
Temperature (°C)
0.1
0.2
0.3
10
0.4
Time (fraction of year)
0
0.5
1
Normalized cracking intensity
0
0
1
1
1
2
2
2
3
3
3
4
4
4
5
6
Depth (m)
0
Depth (m)
Depth (m)
b Ts = -3°C, ΔTa = 12°C
5
6
6
7
7
8
8
8
9
9
9
-10
-5
0
Temperature (°C)
5
10
0
0.2
0.4
Time (fraction of year)
0.6
not penalized
5
7
-15
penalized
10
0
0.5
1
Normalized cracking intensity
Figure 6. Comparison of models for frost cracking intensity. Profiles of temperature (left), time in the frost cracking window (middle), and frost cracking
intensity (right) for (a) T s = +3 C, and (b) T s = –3 C. In both cases, the annual temperature amplitude (ΔTa) is 12 C, and both daily oscillations and latent
heat effects are ignored. Left panels: Calculated temperature profiles at weekly intervals, with frost cracking window shaded gray. Parts of profiles that lie in
the frost cracking window are red; of these, cases where some portion of the profile is above 0 C are shown with heavy gray lines. Middle panels: Analytical
solution (dashed line), and numerical solution (solid line) for time spent in the frost cracking window. Right panels: Frost cracking intensity as calculated from
the magnitude of the thermal gradient (as suggested by Hales and Roering, 2007) and water availability in the direction of heat flow (bold gray lines). This
pattern is further modified when the rate of cracking is taken to be dependent on the distance to the unfrozen water (‘penalized’ cases, blue lines). This figure
is available in colour online at wileyonlinelibrary.com/journal/espl
The resulting crack intensity profile is significantly different from
the simpler but less defensible proxy of time in the cracking
window (shown for comparison in the middle panel, Figure 6b).
We now add a further element of realism to the Hales and
Roering frost cracking intensity model in the water limited or
‘penalized’ case shown in cyan on the right panels of Figure 6.
Here we acknowledge that the rate of growth of cracks should
be limited not only by the presence of liquid water, as in the
Hales and Roering algorithm, but also by the distance that
water must be moved through cold materials to get to the site
of potential frost cracking. This effect is analogous to the
process by which a growing ice lens in soil is abandoned in
favor of a new ice lens when the water supply to it becomes
limited (Rempel, 2007). This ‘penalty function’, here represented
by a simple exponential of the form e-l/l*, where l* is a specified
length scale (we take for this example l* = 05 m), significantly
Copyright © 2012 John Wiley & Sons, Ltd.
alters the predicted pattern of cracking intensity. In general, this
modification moves the locus of predicted cracking damage to
greater depth in the non-permafrost case (Figure 6a), and to a
single maximum in the permafrost case (Figure 6b).
In the case of negative T s (in Figure 6b, –3 C), deep sites
have access to unfrozen water in the spring, as the surface
temperature rises (left-dipping gray profiles on the left panel),
while shallow sites have access to water in the fall, as surface
temperature drops (right-dipping profiles). These give rise to the
two peaks in the predicted Hales and Roering cracking rate
profile. However, when the system is penalized for water transport distance, the resulting pattern displays a single peak. In the
fall, water must be drawn from depths of 2 to 3 m to near-surface
sites, while in the spring water must be drawn from the surface to
sites of increasing depth. But early in the spring (uppermost leftdipping gray lines), sites at ~1 m depth begin to have access to
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
effects are maximized by our choice of high water content. In
addition, the daily oscillations in temperature, which are felt
in only the top few tens of cm of the surface, allow additional
cracking in these shallow depths.
The predicted profiles differ not only in the shapes, but also in
their integrals, and hence in the predicted total crack damage
accomplished as a rock passes through the cracking rate profile.
We plot this integral metric of frost damage for a wide range of
mean annual temperatures (T s ) in Figure 8 for models with and
without a daily temperature cycle, and with and without the
penalty for water travel distance. Consider first the models that
omit daily temperature oscillations. Total frost damage increases
monotonically as T s drops from +1 to –8 C (the lower edge of
the frost cracking window) in the penalized case, while it displays
two peaks in the unpenalized case. The principal reason for the
reduction of predicted cracking as T s rises is that during times
in which temperatures are in the frost cracking window, water
is available only at depths that are far below all sites of potential
cracking. The penalty on frost cracking for inefficient water
transfer is very high, an impact well documented in field studies
(Hall et al., 2002; Sass, 2005). This is reflected in the bias toward
cracking at the base of the frost cracking window in Figure 6. Also
note that these profiles of damage are normalized to emphasize
the difference in the spatial pattern. The non-normalized values
are tremendously different owing to the penalty; this accounts
for the 35-fold difference in y-axis scales in Figure 8.
Adding in the daily temperature cycles produces an even
greater increase in the accumulated damage sustained by
weathered bedrock when it reaches the base of the mobile
regolith as the mean annual temperature declines, in the most
realistic case (including daily cycles and a transport-distance
penalty) (Figure 8). Not much damage is sustained for mean
annual temperatures greater than 0 C, whereas significant
damage has occurred for bedrock exhumed in conditions with
surface melt that therefore does not have much distance to travel
to the frost-cracking window. In the water-limited model, the
greatest rates of cracking will occur at these times, giving rise to
the broad peak at ~1 m depths.
Additional elements of reality
We now relax the assumptions that allowed us to compare
directly with past models of Anderson (2002) and Hales and
Roering (2005, 2007), so that we may incorporate elements of
reality we consider important in the hillslope problem. In the
following models, we (1) include a daily temperature cycle;
(2) use a non-uniform porosity structure (and hence k and c
structure) in the subsurface to represent a typical critical zone
architecture of high porosity mobile regolith overlying low
porosity bedrock; (3) incorporate latent heat, which stalls the
freezing front; (4) acknowledge and attempt to illustrate the role
of snowcover.
As in the case depicted in Figure 4, we assume that 04 m thick
mobile regolith, with 40% porosity, overlies rock with 3% porosity,
and, we assume that both bedrock and mobile regolith are 90%
saturated. We perform a no-snow case, and a case in which the
snow is 05 m deep and remains on the landscape for two months.
The daily surface temperature amplitude is taken to be 4 C.
In Figure 7, we show the resulting cracking intensity profiles
when these elements of reality are included, for cases both with
and without the penalty for distance that water travels to drive
frost cracking. The profiles for T s of +3 and –3 C (dashed) can
be compared with the right panels of Figure 6 to visualize
the effects of all elements of reality listed earlier. In general the
patterns are all compressed toward the surface, reflecting the
stalling of the freezing front by latent heat effects (as illustrated
in Figure 4) and by the high porosity of the mobile regolith. These
a
b
0
0
0
0
1
1
2
regolith = 0.4 m
40% porosity
90% saturated
3
Mean Annual Temperature
-8
-6
-4
-3
-2
0
2
4
6
8
3
4
4
Depth (m)
Depth (m)
-8
2
-8
rock 3% porosity
90% saturated
5
ΔTa = 12°C
no snow
6
5
6
7
7
8
8
no penalty
9
10
-4
water distance penalty
l* = 0.5m
9
0
500
1000
1500
Cracking intensity (°C-day/m)
2000
10
0
5
10
15
Cracking intensity (°C-day/m)
Figure 7. Profiles of cracking intensity for more realistic cases for unpenalized (a) and penalized (b) cases, for a range of T s (mean annual temperature) at intervals of 2 C. In both cases, we include a daily cycle of temperatures, different porosities and hence thermal properties of the mobile
regolith and bedrock, and latent heat associated with phase changes. We have ignored the role of snow, and assume that the bedrock and mobile
regolith are 90% saturated at all times of the year. Profiles for mean annual temperatures of +3 C (green-blue) and –3 C (brown) are heavy and
dashed. Note the large difference in the calculated magnitudes of cracking intensity between the two cases.
Copyright © 2012 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
a
no penalty
summed crack with daily
summed crack no daily
Frost damage [°C day/m]
700
frost
cracking
window
600
500
400
300
200
100
0
-15
-10
-5
0
5
10
Mean annual temperature (°C)
b
with penalty
summed crack with daily
summed crack no daily
Frost damage [°C day/m]
20
frost
cracking
window
15
10
5
0
-15
-10
-5
0
5
10
Mean annual temperature (°C)
Figure 8. Integrated frost cracking intensity as a function of T s for both
(a) unpenalized and (b) penalized cases and for cases with (black) and
without (red) daily temperature oscillations. Mobile regolith and bedrock
are assumed to be 90% saturated in all cases. The maximum expected
cracking intensity in positive T s is effectively eliminated when we restrict
the distance that water can travel to reach sites in the frost cracking
window (‘with penalty’ cases).
mean annual temperatures between –2 and –12 C. The damage
peaks at T s = –12 C because time spent in the frost cracking
window is greatest at the outer edges of the cycle; when the mean
is –12 C, the temperatures in the summer turn around within the
frost-cracking window, and therefore spend significant time in the
window. The expected total damage again drops off with further
decline in T s , as the times when the bedrock is both in the frostcracking window and has access to unfrozen water declines. At
these very low temperatures, unfrozen water is available only in
the active layer near the surface in the mid-summer, and less
and less time is spent in the unfrozen conditions. The presence
of a daily cycle allows more expected frost damage for T s > 0
for two reasons: the daily cycle allows more near-surface
excursions into the frost cracking window, and it provides
more times of available liquid water during mid-days when
the daily mean temperature < 0 C.
To return to the Gordon Gulch site in the Boulder Creek CZO,
mean annual ground surface temperature is presently about
+45 C. Modern cracking rates are therefore predicted to be very
low. However, it is likely that Colorado alpine sites saw ~ 7 C
cooling in LGM conditions (e.g. Brugger, 2010), meaning that
Gordon Gulch surface temperatures would have dropped into a
much higher damage-rate regime during glacial times. More
locally, as discussed in Dühnforth and Anderson (2011), a mean
annual temperature depression of ~6 C can best explain the
LGM extent in the headwaters of north Boulder Creek only a
Copyright © 2012 John Wiley & Sons, Ltd.
few kilometers to the west of the Gordon Gulch site. A greater
than four-fold increase in cracking rates in the subsurface would
therefore be expected.
In summary, we expect the intensity of damage associated with
frost cracking to be strongly dependent upon the T s , and in most
cases it will be limited to the top few meters of the rock. The
strong T s dependence translates into a strong modulation of
cracking damage as climate oscillates. Given the typical many-
C amplitude of T s changes associated with glacial–inter-glacial
cycles, the descent from inter-glacial to glacial climate can
take alpine sites from conditions in which little to no damage is
occurring into conditions in which the rate at which damage is
occurring is intense (or vice-versa at very high altitudes).
The revised model contrasts with Hales and Roering model
predictions in that it predicts that rates peak only when T s < 0.
This is largely a consequence of the reduction of expected cracking with the distance water must travel through frozen materials
(compare Figures 8a and 8b). This behavior is consistent with
field data suggesting much higher mobile regolith production
rates under T s < 0, in both the western Alps (Delunel et al.,
2010) and central Apennines (Tucker et al., 2011). The effects of
latent heat associated with freezing, which serve to stall the freezing front and to reduce the time the subsurface spends in freezing
conditions, will be minimized in rocky landscapes in which high
porosity mobile regolith is thin or absent, and in which the degree
of saturation is small.
The presence of snow cover can reduce the intensity of frost
cracking by buffering daily temperature swings, as illustrated by
the difference in intensity with and without daily cycles in
Figure 8. Nonetheless, even for cases in which the ground is fully
insulated from daily oscillations, cracking intensity is still predicted
to be substantially higher when the mean annual temperature
falls within or slightly below the frost-cracking window.
While we have focused earlier on the impact of the thermal state
of the landscape on the damage process, other processes will
clearly be affected strongly by the thermal regime. In particular,
when the mean annual temperature drops significantly below
0 C, permafrost develops, and the hydrologic depth scale
shortens to the active layer; the permafrost table becomes an aquiclude. This will alter hydrologic flow paths to become essentially
surface-parallel. Deep flow paths that dominate many alpine
fracture-flow systems will be eliminated, with consequences for
chemical weathering and for solute chemistry of the rivers.
Decreased near-surface permeability on frozen ground may also
lead to increased surface runoff, with potential impacts on sediment transport and landscape evolution (e.g. Bogaart et al., 2003).
In addition, the hillslope conveyor system into which
weathered bedrock is entrained upon reaching the mobile regolith interface may be changed significantly in a cooler regime.
One might imagine that the active layer may more efficiently pass
mobile regolith downhill by solifluction and frost creep (e.g.
Matsuoka and Murton, 2008). The sign and magnitude of the
change will likely also reflect any change in vegetative cloak on
the landscape. Loss of trees may for example release mobile
regolith to be more efficiently transported due to frost processes.
However, in some cases loss of trees may result in lower transport
rates if the transport is governed by tree throw and root-related
transport (e.g. Hughes et al., 2009). In the following section we
attempt to link damage processes with transport processes in
order to explore their co-dependence in a model setting.
Numerical Hillslope Model
We now explore the emergent behavior of an interfluve in the
face of climate-controlled variations in both the damage and
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
transport processes. In Gordon Gulch of the Boulder Creek CZO,
opposing slopes that face dominantly north (N) and south (S)
receive very different radiation and snow cover, and are cloaked
with contrasting vegetation. We observe a broad asymmetry
in weathering on these opposing slopes. The depth to fresh
rock, as measured with shallow seismic refraction, is greater on
N-facing slopes than on S-facing slopes (Befus et al., 2011). Thin
mobile regolith of order 05 m depth is found throughout
the catchment.
Our modeling strategy is to link the processes responsible for
damaging bedrock, and for producing and transporting mobile
regolith, with the meteorological characteristics that govern these
processes. In the present model we simplify the process set to one
in which damage in the subsurface, production of mobile regolith, and transport of mobile regolith are all driven by frost-related
processes. The required meteorological forcing therefore reduces
to the thermal state and moisture balance of the hillslope, which
will importantly vary in both time (climate change) and space
(radiative setting and hence slope aspect). We acknowledge that
other processes we have discussed earlier (such as the roles of
trees in all three of these hillslope processes) will also vary with
climate and with aspect. It remains a target for future research
to abstract the roles of trees, and the linkage between the vegetative landscape and the frost-related processes, so that they can be
incorporated in critical zone evolution models. We describe first
the climate forcing, followed by how this is incorporated in the
hillslope processes rule set.
Climate
We impose an oscillation in climate with 40 ka period. As the
LGM glacier lengths in the headwaters of the Boulder Creek
watershed may best be modeled by imposing a 6 C drop in T s
from present conditions (Dühnforth and Anderson, 2011), we
impose a 40 ka oscillation in mean annual surface temperature
at all locations with an amplitude of 3 C (6 C full range). Dependence on aspect is incorporated by aligning our 1D hillslope
model to be N–S oriented, such that left is S and right is N. We
have not incorporated an energy balance model to evaluate the
surface temperatures as a function of position on the slope
transect, although this is an excellent target for future studies.
Such a model would need to incorporate the roles of trees of
different species and density, and how they affect the snow cover,
as well as the more straight-forward radiative problem. In the
absence of a full energy balance model for our landscape, we
simply assert a slope dependence of T s that is reasonable. This
is accomplished for northern hemisphere settings by allowing
the T s to depend linearly on hillslope gradient in the model,
which is positive on S-facing slopes (left side of the simulations)
and negative on N-facing slopes. The spatial and temporal
distribution of mean annual temperature is therefore taken to be
@z
T s ðx; t Þ ¼ T s0 þ ΔT sinð2pt=P Þ þ a
@x
(12)
where the second term on the right hand side represents the
temporal variation, with ΔT the amplitude of the oscillation, and
the third term represents spatial variation. The parameter a, which
has units of temperature per unit slope gradient, controls the sensitivity of mean annual temperature (MAT) on the slope of the landscape. In our current model we employ a = 2 C per unit slope
gradient, which results in slope-related differences of about 2 C.
The distribution of T s across the landscape serves to modulate
both the depth and intensity of the frost cracking in the weathered
bedrock, and the number and depths of the freeze–thaw events in
Copyright © 2012 John Wiley & Sons, Ltd.
the mobile regolith. Hence it governs both the weathering and
transport components of the hillslope evolution. We assume that
sufficient moisture is available in the unfrozen parts of the
subsurface throughout the year.
As in many hillslope evolution models (e.g. Dietrich et al.,
1995; Anderson, 2002), the basic statement of mobile regolith
conservation in one dimension is
@H
@Q
¼w
@t
@x
(13)
where H is thickness of mobile regolith, w the rate of release of
weathered bedrock into the mobile regolith layer, Q the volumetric discharge of mobile regolith per contour length [=] L3/LT, and x
the distance from the hillcrest. In order to proceed, we require
rules for the rate of conversion of weathered bedrock to mobile
regolith, w, and for transport of mobile regolith downslope, Q.
We add to this list the rule discussed earlier for the progressive
damage of rock within the weathering zone, and propose one
possible linkage between damage and mobile regolith release
or production rate.
Mobile regolith transport
In keeping with our argument that frost-related processes
dominate in this landscape, the transport rule for mobile regolith
is based upon earlier models of mobile regolith motion by frost
heave (Anderson, 2002). As illustrated in the work of Matsuoka
and Moriwaki (1992), the volumetric downslope transport by an
individual frost heave event may be taken to be
1
@z
q ¼ bz2
2
@x
(14)
where b is the slope-normal strain induced by frost and z is the
depth of frost penetration [L]. The dimensions of q are therefore
L2, the cross-sectional area of material displaced per event. This
may be extended to include the frequency of frost events and to
embrace a distribution of frost penetration depths to become
1
H @z
Q ¼ f bz 2 1 e H=z 1 þ
2
z @x
(15)
for the total volumetric discharge [=] volume/LT. Here we have
integrated over an exponential probability density function of
frost depths characterized by a depth scale z* (see Anderson,
2002). The parameter f represents the frequency of freeze–thaw
(frost heave) events [=] number/year. The strain induced by frost,
b is in turn related to the water content of the soil and the frostsusceptibility of the soil. The complexity of this equation largely
reflects the fact that it accounts for the role of mobile regolith
thickness, H: when H = 0, Q = 0, and as H > > z*, Q approaches
the limit
1
@z
Q ¼ f bz 2
2
@x
(16)
Climate therefore comes into the mobile regolith discharge rule
through its governance of f, b, and z*. High values of each
increase the mobile regolith discharge on a given slope; given
the square dependence on frost depth, this exerts the strongest
control. The equation is of the common form Q = –k(dz/dx) that
lies behind the diffusive nature of hillslopes (e.g. Culling, 1965).
Here we may identify the transport efficiency, k, as
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
1
H
k ¼ f bz 2 1 e H=z 1 þ
2
z
(17)
This formulation captures the dependence of transport efficiency
on both climate and mobile regolith thickness for this particular
frost-driven surface transport process.
In Figure 9 we show the results of thermal models in which we
have tracked the number of freeze–thaw events in an annual
cycle, f, and their depths, z, so that the total mobile regolith
discharge may be assessed as a function of the mean annual
temperature. Here rather than impose a probability density function for freeze depths, we simply tally those that occur in the
model, and sum over all events, so that the efficiency becomes
N
1 X
k¼ b
fn z 2
2 n¼1 n
(18)
Here n indexes depth into the mobile regolith, and there are N
such depth intervals. A frost event is said to have occurred if the
temperature at that depth passes both downward and upward
through 0 C (i.e. we count the zero-crossings and divide by
two). In the frost-creep algorithm, therefore, no account is made
for how much time is spent in these excursions into subfreezing
temperatures; in this sense it differs from the algorithm used for
frost damage of rock, in which time spent in the frost cracking
window matters. This acknowledges the fact that frost heave
associated with ice lens growth in soil can occur much more
rapidly than frost cracking, which entails transport of water
through much less permeable material. In the models depicted,
the mobile regolith thickness is taken to be 04 m, and the amplitudes of the annual and daily oscillations are taken to be 8 C and
3 C, respectively, in accord with temperature records in Gordon
Gulch. Two sets of models were run, one with no snow, and the
other with an imposed snowfall of 045 m that persists for 60 days.
The values of the calculated hillslope transport efficiency, or
landscape diffusivity, k, range from 0005 to 004 m2/yr, which
indeed spans the range of efficiencies reported from many
700
0.05
transport efficiency limited by soil no snow
600
damage (integrated crack intensity)
500
0.03
400
300
0.02
200
0.01
100
0
-8
-4
0
4
8
Damage, Integrated crack intensity
Transport efficiency, κ (m2/yr)
transport efficiency with 2 months snow
0.04
0
Mean annual temperature (°C)
Figure 9. Integrated frost cracking intensity (damage shown as bold
line) and mobile regolith transport efficiency (k), as functions of mean
annual temperature. Both frost damage and frost-related mobile regolith
transport are much more efficient for moderately negative T s . Presence
of snow (dashed line) diminishes the transport efficiency for cases in
which T s is near 0 C.
Copyright © 2012 John Wiley & Sons, Ltd.
landscapes [roughly 4 10–2 to 4 10–4 m2/yr in temperate
to arid landscapes (Fernandes and Dietrich, 1997), and
2–6 10–2 in peri-glacial landscapes (Small et al., 1999, in
the Wind River Range, USA; and Riggins et al., 2011, in Bodmin
Moor, UK)]. The pattern displays two maxima at high and low
temperatures, at which mobile regolith transport is greatest.
(For mean annual temperatures higher and lower than these,
transport rates decline precipitously, as the 0 C crossings vanish.)
These peaks correspond to T s at which the annual oscillations
take the surface temperature to around 0 C, allowing the daily
oscillations that ride atop the annual cycle to cross 0 C
frequently. The presence of snow reduces the number and depths
of frost-penetration events over a broad range of mean annual
temperatures. The effect is most pronounced for T s near 0 C. In
the no-snow case, the efficiency changes little over the range
+4 to –2 C, below which the efficiency climbs rapidly. In the
snow-on case, the flat pattern is transformed into a broad lull in
efficiency; this results in a more pronounced increase in
efficiency as the mean annual temperature is lowered (e.g. k
increases by ~ four-fold from +1 to –3 C).
In our hillslope models, we parameterize this dependence of k
on the local mean annual surface temperature. This allows us to
honor how the efficiency with which mobile regolith is moved
depends both upon climate and upon position on the hillslope,
while avoiding the need to run the thermal model on sub-daily
time steps throughout the million year hillslope simulation.
Mobile regolith entrainment
Here we craft a rule for mobile regolith production in which
we capture the fact that it should depend upon the state of
the weathered bedrock at the mobile regolith–weathered
bedrock interface (its susceptibility to entrainment), and the
processes available to entrain it. In this sense the problem
becomes similar to sediment transport mechanics, in that
the theory builds on a quantification of resisting and driving
stresses. As discussed earlier, the susceptibility to entrainment
is quantified by its initial state (e.g. how cracked the original
rock is, or how bedded it is) and the damage it has sustained
en route to the surface.
We take the mobile regolith production rate to be the product
of two terms, one capturing the dependence on distance from
the surface to the mobile regolith interface (i.e. the soil depth),
the other representing the damage-dependent susceptibility of
bedrock at that interface. We propose that the susceptibility to
entrainment depends upon the damage in a fuzzy thresholded
way (Figure 10). We employ a hyperbolic tangent to switch
smoothly from very low probability to very high probability of
entrainment over a specified range:
pðw Þ ¼
1
D D
1 þ tanh
2
sD
(19)
where D* is the damage scale at which the probability of
entrainment is 50%, and sD is a damage scale over which the
entrainment probability changes significantly.
The processes responsible for entrainment are taken to be
dependent upon depth of mobile regolith, which can be
conceived as a proxy for either root dislodgement or frost-lofting
of rock. In the sprit of the present emphasis on thermal processes,
we choose an exponential function to capture the essence of the
decay of the number of frost events with depth. Hence, the
mobile regolith production function takes the form:
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
D*
1000
1
S
1 Ma
Elevation (m)
Susceptibility to entrainment
990
σD
980
higher T
S-facing slopes
960
er
ow e
l
l
g
a
sh ama
d
940
-100
Figure 10. Rule for susceptibility to entrainment of weathered bedrock
into mobile regolith. The more damaged the parcel of bedrock, the more
likely it is to be entrained in the overlying mobile regolith. The functional
dependence is here captured using a damage scale D* and a sensitivity
scale sD.
1
D D
1 þ tanh
w_ ¼ w_ 0 e H=H
2
sD
(20)
Here the first factor represents the dependence of the efficiency of
the entrainment process on mobile regolith thickness, and the
second factor represents the susceptibility of the weathered
bedrock immediately below the mobile regolith to entrainment
(Equation 18). Note that for damage D > > D*, we have chosen
this form such that it reduces to the oft-used exponential dependence of mobile regolith production rate on mobile regolith
thickness (e.g. Heimsath et al., 1997). We could also entertain
other forcing functions in which the dependence on mobile
regolith thickness takes on a humped rather than exponential form
(e.g. Riggins et al., 2011; see also discussion in Humphreys and
Wilkinson, 2007, and an example from tree-related processes
in Gabet and Mudd, 2010).
Initial and boundary conditions
We initiate all models with a triangular hillcrest with small roughness (001 m amplitude random perturbations on grid with 1 m
spacing) and uniform mobile regolith thickness of 01 m. The
bounding channels are lowered at a prescribed steady rate, ėch.
The reported steady state results are not sensitive to the choice
of initial conditions, although the transient route to steady state
form is. The rock mass may be assigned any initial state of
damage, D0, reflecting the initial concentration of flaws in the
rock that in turn reflect both the rock type (the density of bedding
planes, for example) and the tectonic history that has produced
additional flaws prior to interaction with surface processes (e.g.
Molnar et al., 2007; Clarke and Burbank, 2011; Koons et al.,
2012). In the present model we assume zero initial damage in
the rock mass.
Results
In Figure 11 we show an example of the hillslope model after
1 Ma model time. The straight-sloped initial conditions have
been effectively ‘forgotten’, and the hilltop form has evolved
toward a convex shape with nearly uniform mobile regolith
thickness. This is in accord with many earlier models (e.g. those
reported in Anderson, 2002). We focus here on the more novel
aspects of the model results.
Copyright © 2012 John Wiley & Sons, Ltd.
Depth to
10% damage (m)
D, Damage of rock at base of regolith
N
colder T
N-facing slopes
970
950
0
shift
in
divide
-50
de
da eper
ma
ge
0
Distance (m)
50
100
0
50
100
10
5
0
-100
-50
Distance (m)
Figure 11. Example result of hillslope evolution model. Simulated
critical zone architecture after 1 Ma, with initial conditions of a triangular
symmetrical interfluve centered at x = 0. Resulting morphology is
asymmetric, showing divide migration toward the warmer S-facing slope.
More efficient transport and damage on colder N-facing slope requires
lower slopes to accomplish mobile regolith transport than on warmer less
efficient S-facing slopes. Mobile regolith thickness (pink layer) is nearly
uniform, although it varies in time through imposed 100 ka climate cycles
(see Figure 12). Degree of damage in weathered rock depicted with color
shading. Damage occurs to greater depths on the N-facing slopes, as
depicted by the depth to weathered bedrock with 10% of the damage at
the mobile regolith–weathered bedrock interface (bottom plot).
Lower mean annual temperatures on N-facing slopes (by a
maximum of ~2 C at all times; Equation 12 with a = 2 C/unit
slope) result in deeper and more intense frost damage profiles
on N-facing slopes (bottom plot in Figure 11). In addition, and
more importantly to the form of the hillslope, the efficiency of
mobile regolith transport is much greater on the N-facing
slopes. This results in an asymmetry of the ridge profile, with
steeper slopes facing south. A corollary to this asymmetry in
efficiency is that in order to achieve a steady form, divide
migration is required: the slopes must everywhere be steeper
on the S-facing slope to compensate for the inefficiency of
transport there. We discuss this asymmetry further later.
The imposed 100 ka oscillation in thermal forcing (3 C
amplitude; 6 C range) results in oscillation of mobile regolith
thickness (shown in Figure 12 for one slope position). We find
that the amplitudes of change are greatest nearest the bounding
streams.
Given that the rule for mobile regolith production entails a
damage threshold, albeit a fuzzy one (Equation 18), the damage
of the weathered bedrock just below the mobile regolith becomes
essentially uniform at the damage threshold value. What differs
between N- and S-facing slopes is the depth to which weathering
damage has proceeded (Figure 12). This length scale is greater on
the N-facing slopes.
Discussion
We have attempted to craft a model in which each of the
components of the system is linked to the climate. Not surprisingly, we find that both the damage and transport processes are
strongly modulated by mean annual temperature when frostrelated processes dominate. We hope that this effort can serve
to show the way forward when other geomorphic and climatic
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
MAT (°C), Q (m2/ky)
15
of saprolite to mobile regolith, are thought to be accomplished
by reactive transport of water and oxygen to sites of mineral
dissolution within the shale (e.g. Jin et al., 2011). The density
and efficiency of the flowpaths through which these reactants
are transported are in turn governed both by tectonically generated fractures, and by fractures generated by near-surface
processes, including the peri-glacially generated frost cracks on
which we have focused in this article.
Q: N-facing slope
Q: S-facing slope
Temperature
10
5
0
Damage-limited versus entrainment-limited cases
Regolith (m)
-5
1.5
1
glacial
time
0.5
0
0
200
400
600
800
1000
Time (ky)
Figure 12. Time series of model input of T s (red line in top plot), and
model output for (top) mobile regolith discharge on N- and S-facing
slopes, and (bottom) mean mobile regolith thickness. During glacial times
(low temperatures; example shaded), mobile regolith discharge is high.
This results in draining of mobile regolith from the landscape, as weathered bedrock entrainment into mobile regolith cannot keep pace with
more efficient frost-related transport. This figure is available in colour
online at wileyonlinelibrary.com/journal/espl
settings are approached. This model predicts several features of
the landscape and its critical zone that can be compared with real
landscapes. Novelties of the model include its prediction of (1)
the thickness of mobile regolith and how this varies in space
and in time on opposing slopes, (2) the profile of damage and
hence depth to ‘fresh’ rock (defined here as a threshold of
damage), (3) the history of sediment production from the landscape. The first two may be compared with information from pits
or boreholes or geophysical surveys (e.g. Befus et al., 2011); the
latter may be compared with the sedimentary record in associated sediment sinks, or with sediment delivery rates inferred from
terrace sequences downstream of the landscape.
In the damage models we have developed, both the generic and
frost-crack specific cases, the damage profiles result from combination of an initial condition (related to tectonic and geologic history
of the rock package) and a damage process that is ‘attached’ to the
instantaneous surface. This results in damage profiles that reflect
the intensity of the damage process, here modulated by the mean
annual temperature, and the rate of lowering of the surface which
governs the time spent in the damage zone. In steady topography,
the rate of lowering is everywhere uniform, meaning that differences in the degree of development of the damage zone will reflect
solely the spatial variation in surface-related damage. Hence we
see deeper damage profiles on colder N-facing slopes in the quasisteady landscape depicted in Figure 11. To first order, this reflects
observations in the Gordon Gulch landscape of the Boulder Creek
CZO, in which shallow geophysical surveys reveal deeper
weathered rock zones on N-facing slopes (Befus et al., 2011).
A strong analogy may be made between this system and the
development of chemical weathering profiles in an eroding
system (e.g. Brantley and Lebedeva, 2011, their figure 3d; see
also Brantley and White, 2009). The magnitude of the depletion
of particular minerals is governed by the reaction constants, and
by the time the parcel of rock spends within the weathering zone.
As the latter is governed by the rate of landscape lowering, these
minerals are far less depleted in rapidly eroding landscapes.
The coupling of the cracking problem with hydrology is
indeed important in many settings. In the Shale Hills CZO, the
conversion of bedrock to saprolite, and the further breakdown
Copyright © 2012 John Wiley & Sons, Ltd.
The pace of lowering of the mobile regolith–weathered bedrock
interface may be controlled by either the degree of damage of
the rock, or by the efficiency of the available entrainment
processes. Consider the case in which the rock mass quality is
so poor that even before reaching the near-surface it has sufficient
density of flaws to be entrained by the processes available at the
base of the mobile regolith. In this case, the rate of release and
hence lowering of the interface will be governed entirely by the
rates of processes active at the base of the mobile regolith [e.g.
upfreezing of clasts (Anderson, 1988a, 1988b) or tree-root
prying]. This we may call the entrainment-limited case. However,
if the rock mass possesses so few flaws to begin with, or is so
tough that it is difficult for new cracks to form as it approaches
the surface, the rate of release of pieces of bedrock into the
mobile regolith will be governed entirely by the rate at which
new damage occurs. We may call this case the damage-limited
case. In a steady landscape the rate of lowering of the mobile
regolith–weathered bedrock interface must be uniform; in this
damage-limited case, the degree of rock damage in the
weathered bedrock at the interface must therefore be uniform,
as damage controls the rate of lowering. In the entrainmentlimited case, the bedrock damage at the interface may vary
considerably, but everywhere exceeds that required to release
it into mobile regolith. In this case, whatever attribute of the
weathered rock is responsible for governing the entrainment
process must be uniform. Most earlier models of hilltop evolution
have appealed to mobile regolith thickness as the proxy for
the process intensity; the likelihood of upfreezing (progressive
upward motion of rocks in soil due to frost heave; e.g. Anderson,
1988a, 1988b), the density of roots, and the availability of water
to drive chemical alteration of the substrate indeed all depend
upon thickness of mobile regolith.
Divide migration
The relief, R, of a hilltop of steady form above its bounding stream
is a function of the lowering rate of the landscape (both the
hillslope and the stream), the mobile regolith transport efficiency
(here k), and the length of the hillslope from the divide to the
stream, L (e.g. Anderson, 2002):
R¼
rr w_ 2
L
rb 2k
(21)
On a steady asymmetric hilltop, with bounding streams at the same
elevation, the relief calculated from both streams is identical
R ¼ R1 ¼
rr w_
rr w_
L 1 2 ¼ R2 ¼
L2 2
rb 2k1
rb 2k2
(22)
so that we may deduce the required shift in the divide due to
differences in the landscape diffusivity on one versus the
other slope:
Earth Surf. Process. Landforms, (2012)
ROCK DAMAGE AND REGOLITH TRANSPORT BY FROST
L1
¼
L2
rffiffiffiffiffi
k2
k1
(23)
In the case illustrated in Figure 11, the mean diffusivities on the
two slopes differ by 32% at the end of the simulation (maximum
difference 71%), while the divide has migrated so that the lengths
of the two slopes differ by ~22%. The calculated divide migration
pffiffiffiffiffiffiffiffiffiffi
should result in a length ratio of 1:32 = 115; in other words, one
slope is 15% longer than the other. The present position of the
divide at least crudely follows the expected scaling. Reasons for
departure from the expected scaling include the fact that
landscape diffusivity is non-uniform on both slopes (following
the spatial pattern of mean annual surface temperature), and
varies through time as climate dictates.
The asymmetry of peri-glacial hillslopes has been studied by
a number of workers, as summarized by French (2007; see
section 13.4.2). The steepest slope is most commonly west to
southwest in northern hemisphere sites. These will be the warmest slopes, as in our simulation. Two explanations have been
advanced. One conforms with what we suggest, that the steeper
slope corresponds to the one on which the mobile regolith
transport process is less efficient. In our simulations, this reflects
higher mean annual temperatures that result in fewer and less
deep frost-heave events. A second explanation involves the
lateral movement of the stream that bounds the slopes; in other
words, it has to do with the boundary condition at the base of
the slope. The steep slope then corresponds to that toward which
the stream is shoved, presumably by rapid mobile regolith
deposition from the more efficient slope. The stream is not
allowed to migrate laterally in our model, and hence this
mechanism cannot explain the asymmetry produced. While
slope asymmetry is a prominent outcome in our simulation, we
do not see marked slope asymmetry in the Gordon Gulch
watershed.
Information transfer across the landscape
While the models here are most pertinent to the Gordon Gulch
section of the Boulder Creek CZO, we have made use of information gleaned from other portions of the Front Range landscape.
The glacial headwaters to Boulder Creek lie only 5 km to the west
of Gordon Gulch. The 6 C amplitude reduction in mean annual
temperature required to explain ice extent in the Front Range
alpine zone (Dühnforth and Anderson, 2011) likely describes
the thermal cycle to which Gordon Gulch was subjected;
these temperature swings take the landscape into and out of
peri-glacial conditions, and strongly modulate the importance
of frost-related mobile regolith transport and bedrock damage of
the substrate. Outboard of the crystalline core of the Front Range
lie scraps of alluviated surfaces that appear to have widened and
aggraded during glacial times (Dühnforth et al., 2012). Formation
and subsequent abandonment of these surfaces has been interpreted to reflect significant swings in sediment supplied to the
streams from their non-glaciated headwaters; high sediment supply leads to widening and alluviation, and low sediment supply
leads to abandonment of the surfaces. Our models of thermally
modulated, frost-heave mobile regolith transport operate with
the proper sign, with low discharge of mobile regolith during inter-glacial times, and many times higher discharge of mobile regolith during cold glacial times.
Future Work
Despite these successes, the present model has a long way to go
before we can consider it a viable model of landscapes such as
Copyright © 2012 John Wiley & Sons, Ltd.
the one that we target in the Boulder Creek CZO. There is
much room for improvement. We have ignored the following
complexities and feedbacks:
• In crystalline rocks, the density of flaws in the rock mass will
govern the permeability of the rock. As it is the access of water
to the rock mass that in turn governs the chemical attack of the
rock, which in turn weakens it to further cracking, we expect a
strong feedback here. To the degree that permeability structure
controls the efficiency and the pathways available for water in
the landscape, this feedback will also exert control on the
chemistry and the biological activity in the subsurface. Indeed,
in most 1D geochemical models of weathering front form and
advance, hydrologic feedbacks in the system are either
ignored or simply parameterized. For example, the evolution
of the surface area on which chemical reactions play out,
and which strongly governs the advection velocity (and the
role of lateral flow) of fluids within the profile, is rarely incorporated (see Brantley and Lebedeva, 2011). We have similarly
ignored this potentially important feedback in that our model
is not coupled to a model of the hydrology of the hillslope. As
the problem of hillslope evolution is not 1D but (at least) two
dimensional (2D), full treatment requires a 2D transient
hydrologic model that is capable of addressing the evolving
spatial distribution of crack-induced porosity and permeability.
• In order to simplify the problem and demonstrate how rich
even this simple system can be, we have focused solely on
the physical processes of frost cracking and frost heave. But
there remains a parallel effort that could and should be taken
to explore the roles of other damage processes, and other
transport processes. This reflects a broader effort of the
geomorphic community in the last decade toward development of geomorphic transport rules (e.g. Heimsath et al.,
2001; Dietrich et al., 2003). We hope that the attention to
the role of climate in modulating the processes we have tried
to capture will inspire similar attention to the roles of climate
change in governing the roles of trees in the landscape. For
example, in the Boulder Creek CZO we know that trees differ
across the current landscape in terms of both species and
density, and the elevation of the sites suggests that the vegetative cloak on the landscape will have shifted dramatically from
a potentially treeless tundra to the present montane forest as
the climate swings from glacial to inter-glacial conditions.
• The frost-related damage that we have attempted to capture in
our model provides only one of the next steps in moving
toward models that address both the climatic and geologic
reality of any particular site. We have broken the problem of
cracking into two pieces: the initial condition associated with
rock structure and tectonically-generated cracks; and additional frost-cracking that is always strongly related to distance
from the instantaneous earth surface. We have ignored the
influence of topography on the state of stress in the nearsurface rocks, which may either enhance or dampen the rate
of crack growth. This is addressed in another paper in this
volume (Slim et al., in press) in which the superposition of topographic and farfield stresses in the Shale Hills CZO, Pennsylvania, is hypothesized to enhance crack growth in valley
bottoms and reduce it on ridgetops.
• We have simplified the climate to a sinusoidal history of mean
annual temperature. In fact, the climate has likely spent much
more time in the middle range of temperatures than such an
algorithm produces (for example, the d18O record from deep
sea suggest that the global ice volume is normally distributed
about the mean; see Anderson et al., 2006).
• Our thermal model is also inadequate in its characterization
of aspect-related dependence of mean annual temperature.
We have employed a linear dependence of mean annual
Earth Surf. Process. Landforms, (2012)
R. S. ANDERSON, S. P. ANDERSON AND G. E. TUCKER
•
•
•
•
•
temperature on S-directed slope angle (last term in Equation
12). In reality, we require closer attention to the radiation field
that will govern the thermal differences between the thermal
states of N- versus S-directed slopes. While the mean annual
direct radiation is easily calculated on any arbitrary slope,
the additional roles of non-uniform tree cover and of nonuniform snow cover make calculation of the anticipated
ground surface temperature histories quite complex. We have
resorted to a simple algorithm that can be constrained empirically in the present day, and that at least has the correct sign.
We have greatly simplified the role of snowfall, and for that
matter of water in the landscape. In the present model we have
at least attempted to capture the fact that water is required to
drive frost cracking, and have done so by enacting a penalty
on the cracking rate that increases with the distance the water
would have to be pulled. But there is no true assessment of the
availability of water. The wetness in the mobile regolith and in
the bedrock mass does not evolve with time. As we have used
high values for both mobile regolith and weathered bedrock in
the majority of our calculations (90% saturation), our calculations of frost cracking should therefore be considered overestimates, although the present algorithms are far more defensible
than the simple tallying of time in the frost-cracking window
used in some earlier models.
There is great need for documentation of the damage profile in
weathered bedrock. While we have suggested that the crack
surface area per unit volume is an appropriate metric, this is
a difficult quantity to measure, and has not often been documented in the real world. We seek proxies for the degree of
damage in rock specimens, which may include the tensile
strength of rock (e.g. Kelly et al., 2011). On the landscape
scale, Clarke and Burbank (2011) demonstrate the utility of
shallow seismic methods, from which the velocity structure
can be inferred and the effective fracture density can be
calculated. Documentation of cracking frequencies in
boreholes (e.g. Slim et al., in press) is indeed helping to inform
models of cracking in other critical zone observatories.
Again for simplicity, and to isolate the role of bedrock damage
evolution in this hillslope system, we have assumed that
mobile regolith production proceeds in a continuous fashion.
In reality, discrete blocks are released whose size is governed
by the degree of damage (the inverse of which is a length scale,
or distance between flaws). While this element of reality is
captured in a model presented by Riggins et al. (2011), there
is utility in merging these approaches.
The initial rock mass is assumed in the present model to be
uniform in all of its properties. In reality, rock mass quality, or
‘intact rock strength,’ will vary due to stratigraphic architecture
(bedding planes) and in its susceptibility to cracking by both
tectonic and surface processes (Molnar et al., 2007; Clarke
and Burbank, 2011). This variability will in turn exert considerable control on real hillslopes by governing variations in
the initial condition of a rock as it begins to feel the surface
(D0 in Equation 2) and the rate of damage that subsequently
accrues.
1D versus 2D. In the present model, mobile regolith must
move in one direction only; there can be no component of
lateral motion. As discussed in Anderson (2002), this will
eliminate the potential for formation of bedrock outcrops
on midslopes; these are effectively killed by mobile regolith
being forced to move over bare outcrops. This is particularly
the case if the rate of rock damage is minor on bare rock
outcrops (e.g. the humped rule; see Anderson, 2002), or if
there are large spatial variations in the initial state of damage
of the rock. Preliminary 2D models of mobile regolith
motion indeed demonstrate a more complex motion of mobile
regolith as it interacts with bedrock knobs.
Copyright © 2012 John Wiley & Sons, Ltd.
• Finally, the boundary conditions in the present model are
held steady: the bounding streams are made to lower at a
prescribed steady pace. Yet in many of our stream systems,
there is ample evidence in the form of small terraces that the
streams have experienced episodes of aggradation. Where
dated, these imply aggradation during glacial times. This is a
failing of 1D calculations: stream dynamics depend not only
on the local delivery of sediment to the stream, but to the
delivery at all points along the channel, and the ability of the
channel to transport the sediment delivered to it. The latter will
clearly vary with climate. In a very real sense the hillslopes are
slaves to the history of the stream. But it is a two-way coupling.
The streams respond to climate change by not only handling
the water from rain and snowmelt, but by attempting to pass
on the sediment delivered to them by the hillslopes.
Conclusions
We espouse the view of the critical zone as one in which rock is
attacked by damage processes en route to the mobile regolith
interface, and is subsequently released into the hillslope
conveyor system. Progress in understanding this system occurs
when we can construct self-consistent models that acknowledge
specific processes that damage the bedrock of a specified character, release it into mobile regolith, and transport it under specific
climatic conditions. The time scales involved in the exhumation
and the transport down typical hillslopes suggest that the legacy
of past climates will be strong, emphasizing the need to characterize the modulation of these processes, and in some cases the
switching of one suite of processes with another, by climate.
We have worked a specific example in which frost-related
processes both exert damage on the rock in the subsurface by
frost-cracking, and transport it by frost creep once released into
mobile regolith. Results suggest that persistent aspect-related
differences in microclimate translate into differences in hillslope
transport efficiency that can lead to asymmetries in critical zone
architecture and hillslope form (divide migration). Modulation
of climate between glacial and inter-glacial conditions can result
in significant variation in the mobile regolith cloak on the landscape. A direct corollary to this is a history of mobile regolith
delivery to the streams that fluctuates significantly, with peaks in
delivery during glacial maxima. This can affect both aggradation
events (fill terraces) within the mountain front, and lateral planation to produce broad strath surfaces on the adjacent plains.
There remains much to do in crafting process-based rules for
bedrock damage, weathered bedrock release into mobile regolith, and mobile regolith transport in different climates and different rock types. Advances are needed to better our treatment of
frost-related processes, specifically the explicit linkage to hydrologic processes that govern the degree of saturation of the soils
through annual cycles. In other settings, other suites of processes,
some of them biological, will require a parallel treatment.
Acknowledgments—This research was funded by a grant from the
National Science Foundation (Boulder Creek CZO: NSF-0724960).
The authors thank Taylor Perron, Sue Brantley, and two anonymous
reviewers for insightful comments that greatly clarified the manuscript.
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