Introduction
Most of the Earth's mountain ranges show clear signs of glacial
erosion, with distinct glacial landforms such as broad U-shaped
valleys, hanging valleys, and cirque basins. Many of these mountain
ranges have likely been thoroughly reshaped during the late Cenozoic
period, where the cooling climate allowed glaciers to form or expand
and erode into new parts of the landscape. Global cooling has also
expanded the realm of other cold-climate processes, but their
long-term influence on the shape of mountain ranges is not well
understood compared to the influence of glaciers.
The steep headwall and sides of glacial valleys are examples of
weathering-limited landforms where bedrock erosion may be controlled by frost
activity. Previous studies have modelled the relations between temperature
and frost-cracking processes in order to investigate the climatic control on
frost-related debris production on steep slopes and on the distribution of
threshold slopes . These studies show a good correlation between the
modelled zone of maximum frost-cracking intensity and the source area of
scree production in the Southern Alps , and in the Khumbu Himalaya, the base of the steepest
headwalls correlates with zones of peak frost-cracking intensity, indicating
frost-driven undercutting of these slopes . Such
findings together indicate that frost could be an important agent of erosion
on steep mountain slopes.
It has been suggested that periglacial processes are also involved in
the formation of the high and relatively smooth summit flats that
characterise the alpine landscapes of, for example, the Laramide
ranges in the western USA and the mountains of Scandinavia, Greenland,
and eastern Canada (Fig. ). These low-relief
surfaces, which are typically covered by a thin mantle of regolith,
are the focus of mass wasting and diffusive processes during ice-free
periods. Furthermore, ice masses are commonly observed to be
cold-based in such high-elevation settings and so tend to
preserve rather than erode the underlying terrain
.
Low-relief summit area at Ranastongi, southern Norway,
1900 ma.s.l. Widespread sorting patterns in the
debris surrounding the summit suggest periglacial processes are
either actively redistributing sediment today or
have done so in the past. Photo: Christian Brædstrup.
Using existing models to determine the periglacial influence on the
summit flats is, however, not straightforward because the
low-angle slopes allow sediment to accumulate. Depending on their
thickness, sediments are expected to influence frost cracking of
underlying bedrock in two ways: (1) sediments may retard frost
cracking by dampening the propagation of surface temperature
variations into the bedrock and (2) they promote frost cracking by
functioning as a high-porosity water reservoir for the ice-segregation
process. In order to quantify the
efficiency of periglacial processes on these low-relief landforms, it
is therefore vital to take a closer look at the sediment cover
shielding bedrock from direct contact with the atmosphere
. In this study, we quantify for the first
time mechanical bedrock weathering through frost cracking for
a range of sediment thicknesses and mean annual air
temperatures. This gives us new insights into how sediment and
temperature combine to scale frost-cracking intensity.
It has long been recognised that in soil-mantled landscapes, the denudation
rate is governed by the rate at which soil can be transported downslope
rather than by the weathering rate of the underlying bedrock
. When modelling periglacial processes in
soil-mantled landscapes, it is therefore essential to quantify the efficiency
of the active processes of sediment transport. We include an analysis of the
efficiency of frost heave as a transport mechanism across a range of mean
annual air temperature and sediment thickness. The quantification of both
frost weathering and frost-driven transport opens the possibility of
incorporating these relations into a large-scale landscape evolution model.
This enables us to explore the long-term feedbacks among climate,
frost-weathering intensity, sediment mobility, and the evolution of mountain
topography, as shown in an accompanying paper .
Background
The term “periglacial” encompasses a range of processes that involve
freezing and thawing in the cold but non-glaciated realm of polar and
alpine environments . There is still much
to be learned about how these mechanisms work and on what scales
Seefor a review, but here we focus on
two particular processes: frost cracking and frost creep, which are
both relatively well-studied via physical experiments. Frost cracking
was originally attributed to volume expansion (∼9%) during
the freezing of water trapped in cracks and pores in the bedrock. However,
more recent research has found that this mechanism requires very high
levels of saturation and confinement, and the experimental
focus has shifted to another process, known as ice segregation
. The ice-segregation
process has long been known from studies of frost heave and creep of
soils e.g. but has also been proposed as
a mechanism for breakdown of rock . Ice segregation causes ice lenses to grow by the freezing of water that is drawn from the surrounding rock or sediment
during periods of sustained subfreezing temperatures. The water
migrates to the ice lenses through films of water at the
grain–pore interface .
Information on frost cracking and frost creep comes primarily from
cold-room laboratory experiments where small amounts of sediment or
rock are subjected to multiple cycles of freezing and thawing
e.g.. Furthermore, field monitoring has established
empirical relationships between the frost-cracking process and natural
temperature variations e.g.. Several theoretical
models have simulated freezing of water in porous media
, but only a few studies
have attempted to model the spatial extent of periglacial processes in
relation to climate. developed the first
model relating frost cracking to climatic parameters. He introduced
the concept of the frost-cracking window (FCW) (-8 to
-3 ∘C), based on results by
, and quantified frost-cracking intensity
(FCI) as the length of time spent in this window during an annual
temperature cycle. extended
this model by quantifying frost cracking in bedrock as a function of
the temperature gradient when temperature falls within the FCW. This
model, furthermore, requires that water is available in the direction
of warming, which allows for continuous transport of water to the zone
of frost cracking (since ice segregation causes water to flow
from warm to cold areas).
quantified FCI as a function of mean annual
air temperature by depth-integrating frost cracking during an annual
cycle. extended this model by including
diurnal temperature oscillations, the effect of latent heat on
temperatures, a transient snow cover, a regolith layer with adjustable
porosity and constant thickness (0.4 m), as well as
limitations on water transport through frozen bedrock. The work
presented here builds on these previous models and extends them by
systematically delineating the effect of the thickness of the sediment
cover on frost-cracking and frost-creep processes.
Approach
We present a mechanistic model that quantifies the influence of
temperature and sediment cover on the two key periglacial processes:
(1) the production of mobile regolith from bedrock via frost cracking and
(2) the transport of sediment by frost creep. In order to provide
a sensitivity analysis, we compute the intensity of FCI and the frost-heave-induced transport diffusivity (κ) for
combinations of the mean annual air temperature (MAT) and thickness of
sediment overlying the bedrock (S). Our approach to computing FCI
and κ is based on , with some important
modifications.
The heat equation
Because frost cracking and frost creep take place within a limited
temperature range, quantification of these processes requires
knowledge of how changes in atmospheric temperature are propagated
into the ground. We therefore solve the one-dimensional transient
heat equation for the conduction of heat within the upper 20 m of
the subsurface:
C∂T∂t=∂∂zk∂T∂z+HL,
where T is temperature, C is bulk volumetric heat capacity, k is bulk
conductivity, and t is time. HL is the latent heat release or
consumption by phase change of water. We assume that the only heat exchanged
in the system is caused by conductive heat flow, and we ignore the thermal
effect of fluid advection. The heat equation is solved by the finite-element
method, which allows for irregular discretisation and for accurately
incorporating discontinuous variations in thermal properties, such as that
between water-saturated sediment and bedrock. We use the possibility of
irregular discretisation to increase the model resolution close to the
surface where the temperature gradient can be large (Fig. ).
Schematic of the model set-up. The water volume available for
frost cracking in an element at depth z in the frost-cracking
window is found by multiplying the porosity, ϕ, the water
fraction, wf, and the integrated flow resistance,
Γ, of all elements along the path ℓ. The integrated flow
resistance from an element containing water (z′) to z
is found by integrating the element flow resistance (γ)
along the path between
z and z′. The magnitude of γ depends on the
thermal state and porosity of each element. Bc: cold bedrock;
bw: warm bedrock; sc: cold sediment; sw: warm
sediment.
Panel (a): the temperature curve used as the surface
boundary condition for the heat equation. The curve is a sinusoidal
annual temperature oscillation of amplitude dTa=8 ∘C. Superimposed on this curve are diurnal
sinusoidal variations of random amplitude between 0 and
4 ∘C. The inset shows details of the two
superimposed variations. Panel (b): modelled temperature profiles
shown at weekly intervals throughout a year. The diurnal
oscillations are in this case omitted for clarity. Latent heat
exchange stalls the propagation of temperature variations, and this
leads to a kink in the temperature profiles around
0 ∘C. Panel (c): the corresponding weekly
evolution of water fraction along the profile. A water fraction of 0
corresponds to frozen conditions and 1 to completely thawed
sediment or bedrock. The partially thawed part of the profile (water
content between 0 and 1) extends deeper than the
0 ∘C isotherm because ice starts to thaw at
-1 ∘C. Panels (d and e): as for
(b) and (c) but with a sediment cover (yellow
fill). Note that the effects of the latent heat are more pronounced
because the sediment has higher porosity than bedrock. The latent heat
dampens the propagation of temperature variations into the
subsurface and the freeze–thaw events are shallow in the case of
thick sediment cover.
A constant heat flux from the Earth's interior, qb, is
used as a boundary condition at the bottom of the profile
(z=20 m). At the surface, the temperature is forced to vary
annually as a sinusoidal oscillation around a given MAT with
a superimposed diurnal sinusoidal variation (Fig. );
the amplitude of the annual variation is
dTa, while the diurnal variation has
a random amplitude between 0 and dTd
(Table ). The MAT is translated directly into ground
surface temperature.
Model parameters. These values apply to all model results unless otherwise specified.
Symbol
Description
Value
dTa
Amplitude of annual surface temperature variation
8 ∘C
dTd
Maximum amplitude of diurnal temperature variation
4 ∘C
γsw
Flow restriction in warm sediment
1.0 m-1
γsc
Flow restriction in cold sediment
2.0 m-1
γbw
Flow restriction in warm bedrock
2.0 m-1
γbc
Flow restriction in cold bedrock
4.0 m-1
Vcw
Critical water volume
0.04 m
ϕs
Porosity sediment
0.30
ϕbr
Porosity bedrock
0.02
qb
Basal heat flow
0.05 Wm-2
kw
Conductivity water
0.56 Wm-1K-1
ki
Conductivity ice
2.14 Wm-1K-1
kr
Conductivity bedrock
3.0 Wm-1K-1
Cw
Volumetric heat capacity water
4210 kJm-3K-1
Ci
Volumetric heat capacity ice
1879 kJm-3K-1
Cr
Volumetric heat capacity bedrock
2094 kJm-3K-1
β
Frost-heave expansion coefficient
0.05
L
Specific latent heat of water
333.6 kJkg-1
ρw
Density of water
1000 kgm-3
The presence of water and ice influences the thermal properties of
rock and sediment, and the parameters C and k hence
depend on porosity, ϕ, and the unfrozen fraction of the pore space, the water fraction, wf. Based
on porosity and the properties of bedrock, ice, and water, we first
apply standard mixing rules
to compute the conductivity and volumetric heat capacity for the frozen
(kf and Cf) and the unfrozen situation (ku
and Cu):
ku=kwϕkr1-ϕ,kf=kiϕkr1-ϕ,Cu=ϕCw+(1-ϕ)Cr,Cf=ϕCi+(1-ϕ)Cr,
where kw, ki, and kr are
the conductivities of water, ice, and bedrock,
respectively. Cw, Ci, and Cr are
the volumetric heat capacities for the same three materials. We then
mix these parameters according to water fraction (wf) in
order to obtain the bulk parameters:
k=kuwfkf1-wf,C=wfCu+(1-wf)Cf.
The effects of latent heat, HL, can be simulated by
increasing the volumetric heat capacity if the pore water is in the
process of changing phase
. Equation () is then replaced by
the simplification:
C̃∂T∂t=∂∂zk∂T∂z,
with
C̃=C+ϕρwdTLL for 0<wf<1C else ,
where L is the specific latent heat of water and dTL
is the width of the temperature window in which phase changes occur. We
follow this strategy and increase the heat capacity in elements where
freezing or thawing occur based on the following criteria:
freezing:T<0∘C and wf>0;thawing:T>-1∘C and wf<1.
Note that the two criteria are overlapping if -1∘C<T<0 ∘C and 0<wf<1. In this case, both
freezing and thawing may occur depending on the sign of
HL. We compute HL by combining
Eqs. () and (),
HL=(C-C̃)∂T∂t,
and relate this to water fraction by considering the heat involved in
freezing and thawing:
∂wf∂t=-HLϕρwL=1dTL∂T∂t.
The water fraction of each element can now be updated by the integration
of Eq. ().
The effect of latent heat is clear from the evolution of the thermal
profiles, mainly because latent heat exchange stalls the propagation
of temperature changes around 0 ∘C. The effect is strongest
when the porosity is high, for example in the case where bedrock is
covered by a thick layer of porous sediment (Fig. ).
Quantifying frost-cracking intensity
Our model for frost cracking is based on ice-segregation theory
see for reviews. Ice-segregation growth has been shown to operate
primarily in the temperature interval from -8 to
-3 ∘C . Furthermore, in order for ice lenses to grow,
water must be available and able to flow towards the ice lenses. This
requires a continuous positive thermal gradient (increasing
temperature) from the ice lenses to the water. Because the thermal gradient influences the flow rate of water,
suggested that frost-cracking intensity
scales with the thermal gradient in situations where water is
available along a path following the positive thermal
gradient. supplemented this approach with
a penalty function that reduces frost-cracking efficiency dependent
upon the distance that water must migrate through low-permeable frozen
bedrock to reach the ice lenses. We continue this line of
thought by scaling the frost-cracking intensity with the magnitude of
the thermal gradient multiplied by the volume of water available to
the ice-segregation growth process:
FCI(z,t)=dTdzVw(z)if-8∘C<T<-3∘C0 else .
Here FCI(z,t) is the cracking intensity at time t and
depth z below the surface.
The water volume available for ice-segregation growth, Vw,
is calculated by integrating the occurrence of water along the path,
ℓ, that starts at depth z and follows the positive thermal
gradient:
Vw(z)=∫ℓϕ(z′)wf(z′)e-Γ(z′)dz′.
Note that ℓ is either up or down along the vertical profile,
depending on the sign of the thermal gradient. ℓ extends from
depth z to either (i) the surface, (ii) the profile bottom, or (iii)
the point where the thermal gradient changes sign. Following
, we apply a penalty function
e-Γ(z′) that depends on the distance the water must
migrate to arrive at the point of ice segregation. This function
depends on the properties of the material located between the water
(at z′) and the point of ice segregation (z). We therefore
compute Γ(z′) by integrating the flow resistance,
γ, from z to z′:
Γ(z′)=∫zz′γ(z′′)dz′′.
Applying a constant value for γ (2 m-1)
results in a penalty function identical to that used by
. However, because ice segregation occurs at
temperatures below -3 ∘C and water is available only
for higher temperatures, water must always flow through a mixture of
frozen or unfrozen bedrock or sediment. To characterise the range of permeabilities, we apply four different γ values, representing (i)
unfrozen sediment, (ii) frozen sediment, (iii) unfrozen bedrock, and
(iv) frozen bedrock (Table ). Note that
e-Γ(z′)→0 for z′→D ensures that the existence of the profile bottom does not limit
the water volume available for the cracking process.
By incorporating a measure for the actual amount of water present in
the pore spaces of bedrock and sediment, we capture an important
effect of the frost-cracking process: that sediment overlying bedrock
acts as a water reservoir. The porosity of sediments is significantly
higher than for bedrock, and Vw therefore increases
rapidly if an unfrozen sediment layer exists in proximity to frozen
bedrock. Yet, we suggest that the positive effect of water
availability only increases up to a critical water volume,
Vw(z)≤Vcw (Table ), which
defines the situation when the frost-cracking process is no longer
limited by the availability of water. In Sect. ,
we explore the sensitivity of the model to the assumptions made
regarding water availability and discuss the implications for our results.
Finally, the annually averaged frost-cracking intensity,
FCI‾, is computed by depth-integrating the cracking
intensity and averaging this depth-integrated value throughout a year:
FCI‾=1Ta∫0Ta∫0DFCI(z,t)dzdt,
where Ta is 1 year and D=20 m is the thickness of the profile.
Quantifying frost creep
Frost heave results from the expansion of fine-grained sediment when
subjected to freezing and associated ice-lens growth e.g.. The expansion
due to freezing is perpendicular to the slope of the surface while the
thaw-conditioned contraction is vertical owing to gravity. The
surface-parallel displacement of sediment arises from this angle
discrepancy between directions of expansion and contraction
(Fig. ) , a process noted
long ago .
Sketch of the frost-creep mechanism. The downslope movement
of sediment (grey) occurs due to the angle discrepancy between frost
heave and thaw settling. The white area marks a zone of thickness, dz,
that expands and contracts (by βdz). The downslope
sediment flux caused by this freeze–thaw event is represented by
the hatched area. Note that the expansion has been exaggerated for
illustration purposes; in the model β is 0.05.
The contribution to horizontal sediment transport is
Q=-Nfβzdz∇h,
where a sediment layer of thickness dz located at depth z
experiences a number of freeze–thaw events, Nf. Here ∇h=[∂h∂x,∂h∂y] is the bed
topographic gradient and β is the relative expansion of the
layer when frozen (Fig. ). The unit of Q
is square metres as it records the volume of sediments advected
per unit width of the hillslope. We note that
the dependency on z arises from the fact that expansion of a buried
sediment layer causes passive transport of all sediments above this
layer.
We compute the number of freeze–thaw events in terms of the water fraction:
Nf=12∫t∂wf∂tdt.
Nf=1 if the integral covers the full period of a single
frost-heave event because a full frost-heave cycle involves shifting the
water fraction from 1 to 0 and back to 1 again. This formulation of
Nf also captures the effect of partial frost-heave events, which
may be important for the frost-creep rate if the MAT is close to the freezing
point. Note that ∂wf∂t is governed by
the transient heat equation as demonstrated in Sect.
(Eq. )
and is therefore rate-limited by thermal diffusion and the exchange of latent heat.
Integrating contributions from all sediment elements
along the vertical profile and averaging the total frost-heave
activity per year, yields the following annually averaged sediment
flux (m2a-1):
q=-β2Ta∫0Ta∫0S∂wf∂tzdzdt∇h.
We can now formulate the sediment frost-creep process as a standard
hillslope transport law:
∂S∂t=-∇⋅q,
where
q=-κ∇h
and
κ=-β2Ta∫0Ta∫0S∂wf∂tzdzdt
is the transport efficiency (m2a-1), and S is the local
sediment thickness.
This formulation for the transport efficiency (Eq. ) fulfils
the basic requirement that sediment discharge approaches 0 where there is
no sediment to transport. Note also that κ is limited by sediment
thickness until the thickness exceeds the annual penetration depth of the
phase-change window (-1 to 0∘C). The parameter β depends on the sediment grain-size
distribution, as this influences the susceptibility to frost
heave. Silty sediments are the most frost susceptible (high β)
, while coarser
sediments are likely associated with smaller values for β. We
do not vary β in this study (β = 0.05, adopted from
). However, we note here that the
influence of
β is to linearly scale the transport efficiency (κ), and variation in
frost susceptibility therefore has an highly predictable influence on
rates of frost creep.
Results
Rates of frost cracking
The modelled FCI shows that frost cracking can occur under very
diverse climatic conditions depending on sediment thickness and that
the effect of sediment cover on frost-cracking rate depends on
climate (Figs. and ). The model results point to peaks in frost-cracking activity in three
distinct environments.
The integrated frost-cracking intensity,
FCI‾, as a function of mean annual temperature,
MAT, and sediment thickness, S. Each point in the contour plot
corresponds to the annually integrated FCI‾ for
a particular combination of MAT and S. The contour plot was
constructed from 8100 such combinations. The porosity of both
sediment and bedrock is assumed to be 100 % saturated by
water. The dashed lines with numbers correspond to transects that
are shown in Fig. . The inset shows the
FCI‾ for positive MATs in detail. Here
FCI‾ is strongly limited by sediment thickness
and decays rapidly when S increases to more than a few centimetres.
The integrated frost-cracking intensity along profiles shown
in Fig. . Panel (a): profiles along the S axis
illustrate how FCI‾ depends on sediment thickness
for three different MATs. Note the exponential decay function for
MAT=5 ∘C and the humped function for
the two colder settings. Panel (b): inset highlighting the change
in FCI‾ for thin sediment cover. Panel (c): profiles of FCI‾ parallel to the MAT axis showing
how the effects of climate depend on sediment thickness. Areas with
no or very little sediment cover (red line) have the highest
FCI‾ in warm regions, where rapid freezing events
in winter promote intense cracking. In regions with permafrost
(MAT <0 ∘C), frost cracking is most intense under
thicker sediments.
Three examples of temperature profiles under different field
scenarios. The circles mark the position of finite-element nodes:
blue–violet colours indicate zones that contribute water; red–yellow
colours indicate zones with active frost cracking. The grey
temperature interval marks the frost-cracking window
(FCW). Panel (a): a warm, bare-bedrock setting with MAT =8 ∘C and therefore no permafrost. Frost cracking
occurs very close to the surface when temperatures drop into the
FCW in winter. Panel (b): a colder situation with permafrost and
MAT within the FCW (MAT =-4.5 ∘C). Frost
cracking takes place a few metres below the surface at the bottom of
the active layer. By providing water the sediment cover (yellow
fill) promotes frost cracking until the sediment thickness exceeds
the active-layer thickness. Panel (c): a situation with extensive
permafrost and MAT below the FCW (MAT =-8.5 ∘C). Frost cracking is now most active close
to the surface in summer. Again, because water must come from above
the FCW, a thin sediment layer amplifies frost cracking. The white
circles are nodes that are outside the FCW and that do not
contribute water.
Firstly, frost cracking can occur in a relatively warm climate
(MAT >0 ∘C), where the temperature drops below 0 ∘C
only during the coldest nights in winter (Fig. b). The
rapid diurnal temperature variations result in cold conditions and a
steep temperature gradient close to the surface. At the same time, positive temperatures,
and hence water, exist at only 20–30 cm depth below the
surface (Fig. a). Since there is no water present at
the surface, the water necessary for frost cracking must come from within
the bedrock and the resulting frost-cracking intensity thus depends
critically on the water migration capacity of the bedrock (parameters
γbc and γbw in
Table ). As the freezing front does not penetrate
far into the ground, the FCI is dampened by any sediment, which
prevents the bedrock from reaching temperatures in the frost-cracking
window. This effect is further enhanced by the higher water content of
the sediment that stalls the freezing front because of latent
heat. Frost cracking in this environment can therefore occur only
where the sediment cover is very thin (≤20 cm) or absent
(Fig. b and line 3 in Fig. a).
Secondly, frost cracking is promoted in an environment where the deep
subsurface is frozen (MAT <0 ∘C), but where surface
temperatures rise above the melting point during summer and on warm
days in spring and autumn (Figs. and b). The
water driving frost cracking in this setting comes from the surface
because the deeper ground is permanently frozen and because the annual
temperature oscillation ensures a negative thermal gradient
(temperatures decrease with depth) during summer periods, driving
water downwards from the surface. Because the water comes from above,
the presence of a sediment layer may accelerate cracking due to its
water-holding capacity. On the other hand, the relatively large
penetration depth (∼2 m) of the annual temperature
oscillation results in a rather gentle temperature gradient, and the
distance between the positive temperatures at the surface and the
bedrock within the FCW is a few metres (Fig. b). The
most efficient frost cracking therefore takes place at the
sediment–bedrock interface. Optimal conditions for frost cracking
occur where the lower limit of the sediment coincides with the depth
to the FCW, and this causes the FCI in cold environments to peak under
a sediment cover 1–2 m thick.
Thirdly, frost cracking also occurs in very cold climates (MAT ≤-8 ∘C), where only the warmest days in summer
experience surface temperatures above 0 ∘C. This is reflected
in the tail of the FCI distribution towards relatively thin sediment
covers for very cold temperatures (Fig. a). The low
temperatures limit the amounts of water generated at the surface, and
the transport of water is restricted through the frozen sediment and
bedrock below. Frost cracking thus occurs only due to the large but
shallow thermal gradient generated by the diurnal temperature
variation (Fig. c). As a result, sediment thickness is
critical: a thin sediment layer (≤10–20 cm) promotes
cracking owing to its water content, whereas thicker sediment (≥20 cm) limits cracking because the transport path of water
down to the top of the bedrock becomes too long. The colder the
environment is, the shallower the sediment cover must be to
accommodate frost cracking at the top of the bedrock.
Frost cracking in our model is very limited for MATs just below 0 ∘C, an outcome common to previous models . This result may seem counter-intuitive, but at
just below 0 ∘C MAT the mean annual ground temperature is
higher than the FCW. Temperatures required for frost cracking are therefore
restricted to near-surface zones in winter; a time when the deeper subsurface
is frozen and there is no water available. Conversely, when water does become
available in summer, subsurface temperatures are too warm for frost cracking.
Hence, frost-cracking activity is suppressed although temperatures oscillate
around 0 ∘C. The temperature variation does, however, lead to
efficient sediment transport by frost creep as demonstrated in the next
section. The model results thus point to an offset between the MATs that
promote efficient frost cracking and those that promote transport by frost
creep.
Rates of frost creep
The modelled frost-transport efficiency (Fig. ) shows
the combined result of frequent, but shallow, mass movement caused by
the diurnal temperature variations and the larger, but less frequent,
movement caused by the deep penetration of the annual temperature
oscillation. The contribution of the diurnal oscillations is minor
and the overall pattern is largely controlled by two properties
related to the annual temperature oscillation.
Firstly, the maximum penetration depth of the freezing and thawing front
depends on how close the MAT is to the phase-change interval (-1 to
0 ∘C). For thermal profiles with MATs close to the
phase-change interval, the front penetrates to greater depths in the
subsurface. Because sediment is passively transported on top of the
thawing layer, a deep-seated phase-change event leads to greater mass
movement than a shallower phase-change event. This effect is most
clearly seen at sediment thicknesses above a few metres
(Fig. ). For a constant sediment thickness, the
transport decreases with increasing difference between MAT and the phase-change interval
(Fig. c).
Panel (a): contours of the integrated frost-creep
efficiency, κ, as a function of MAT and sediment thickness,
S. Locations of transects plotted in (b) and
(c) are shown with grey, dashed lines and numbers.
Secondly, the maximum transport rate for any particular MAT (along
lines parallel to the S axis in Fig. ) occurs when
the sediment thickness corresponds to the maximum penetration depth of
the freezing and thawing front. At sediment thicknesses less than optimal
thickness, freezing or thawing continues into the underlying bedrock,
which does not increase the transport. The transport efficiency in
this interval is limited by the sediment thickness and thus increases
until the optimum thickness is reached (Fig. b).
For sediment thicknesses exceeding optimal thickness, the
frost-transport efficiency is limited by the penetration depth of the
freezing and thawing front and a plateau is attained with no further
increase in transport efficiency. The optimal sediment thickness is
highest for MATs close to the phase-change interval where the annual temperature variation
causes the deepest penetration of the freezing and thawing front.
The diurnal temperature oscillations extend the MAT interval for which
transport occurs to more extreme temperature values. For these extreme
temperatures, transport occurs in shallow sediment packages only
because the surface temperatures on warm summer days or cold winter
nights support freezing or thawing.
Model sensitivity
The modelled rates of frost cracking and frost creep reflect the basic
assumptions and chosen parameters in the model. A different set of
parameters may lead to different results. In this section, we hence explore and discuss the sensitivity of the model results to
variations in the most important parameters and assumptions.
Thermal properties
The frost-cracking intensity and the efficiency of frost-driven
sediment transport are both highly dependent on the modelled
temperature profiles. The parametrisation of the thermal model
therefore has a significant influence on the predicted rates of
sediment transport and bedrock weathering.
While variations in thermal conductivity and volumetric heat capacity
have only a minor influence on the rates of frost cracking and frost
creep, the amplitude of the annual temperature oscillation,
dTa, has a significant
impact. dTa is highly variable in natural
environments and varies over time due to changes in climate. In
general, dTa increases with distance from the
coast and from the equator . Increasing dTa in the model leads to higher
values of both FCI and sediment transport efficiency, κ
(Fig. ). It also leads to frost cracking and transport
in a wider range of MATs, as well as to frost cracking for thicker
sediment cover. For example, the FCI for MAT =-10 ∘C increases almost 2 orders of magnitude when
dTa is raised from 6 to 12 ∘C. It
is therefore important to incorporate knowledge of
dTa when estimating frost-cracking rates in areas
of unusually high or low annual temperature variations.
The effect of the amplitude of the annual temperature
variation, dTa, on frost cracking (left) and
frost creep (right). Note that the colour scale for
FCI‾ varies between situations with different
dTa.
Panel (a): the amplitude of the diurnal temperature
variations at the surface is dampened, simulating an insulating
snow cover on the ground during winter (indicated by arrows). Panel (b): the snow cover limits the integrated
frost cracking for MATs around and above 0 ∘C. The
lowest contour level of Fig. is indicated with
a black line for easy comparison. Panel (c): the snow cover
reduces the frost-creep efficiency at positive MATs because the
sediment is insulated from the cold air temperatures on cold nights
in winter. Again, the black line represents the lowest contour
level of Fig. .
Effect of a snow cover
We simulate the effect of snow cover by reducing the
amplitude of the diurnal temperature oscillations in winter
(Fig. a). The motivation for this approach
is that thick snow
insulates the ground below and dampens the downward propagation of rapid
variations in air temperature. This representation of snow
cover does not fully capture the
complex interactions between snow and ground temperatures that occur
in natural systems see, e.g.,. However, the
aim here is to illustrate first-order effects on the rates of both
frost cracking and sediment transport.
Because the frost-cracking intensity in positive MAT environments
depends on cold temperatures at night in winter, the FCI is lowered
substantially when diurnal variations are dampened. With a snow cover
present, efficient frost cracking occurs only for cold
temperatures (MAT<-5 ∘C) under sediment
covers up to 3 m thick (Fig. b). The frost-creep efficiency (κ) is less affected, but for high,
positive MATs (>5 ∘C) sediment transport is
reduced when the diurnal winter oscillations are dampened by snow
cover. Diurnal temperature variations that cause short-lived freezing
events during winter nights drive the sediment transport for high
MATs. Even without snow cover, the induced transport is modest
because the frost propagates only a few centimetres into the ground,
but with snow cover present the transport mechanism becomes negligible
for positive MATs.
Water availability
Water is essential for driving frost cracking at sub-zero temperatures,
but the implementation of the water dependency differs among existing
models. In the following, we briefly review the different approaches
and discuss the sensitivity of our results to how water availability
is quantified.
To account for the influence of water on frost-cracking rates,
included the condition that
positive temperatures should be present somewhere in the temperature profile
for frost cracking to take place. They also scaled the FCI with the
temperature gradient, which is thought to determine the flow rate of water.
p. 306 argued that the growth rate of cracks
should be limited not only by the flow rate but also by “the distance water
must be moved through cold materials to get to the site of potential frost
cracking”. Based on this idea, they introduced a penalty function that makes
the FCI decay exponentially over the distance between the zone of frost
cracking and positive temperatures. This implementation is in accordance with
experimental results, which indicate that “an external moisture source close
to a freezing rock contributes to ice segregation” p.
304. In line with these results, we suggest
that FCI should scale with not only migration distance but also the amount
of accessible water in the profile. Furthermore, we suggest that water should
be limited to a greater degree when it flows through cold bedrock than when
it flows through warm sediment, corresponding to differences in the permeability
of these materials. We have therefore integrated the available water and
included varying flow restrictions based on the thermal state and porosity of
the material. Yet, since the true water dependency is poorly constrained and
since the flow restriction parameters are chosen based on an entirely
qualitative estimate, we document the influence of our parametrisation by
exploring the FCI patterns that arise from a range of alternative choices
(Fig. ). The sensitivity of the model to variations alone in
the flow restriction parameters is shown in a supplementary figure. Overall,
an increase in bedrock flow restriction parameters leads to reduced frost
cracking at positive MATs, whereas an increased penalty for negative
temperatures shifts the maximum FCI towards thinner sediment. Below, we focus
on how variations in the fundamental assumption influence the general pattern
of FCI when compared to our standard model (Fig. a).
The integrated frost-cracking intensity as a function of
sediment thickness and MAT, for varying water-penalty
implementations. Panel (a): our standard model (same as Fig. ).
Panel (b): the distance to the
water source and the amount of water available influences the
penalty function, but in contrast to our standard model, the
flow restriction parameters, γ, are constant
(2 m-1) and independent of the porosity or thermal
state. Panel (c): flow restriction parameters are constant
(2 m-1), and the distance between an element in the FCW
and the nearest water source scales the FCI. Compared to our
standard model, the FCI is unaffected by the amount of water
available at the source (the porosity is not accounted for). This
model implementation is similar to that of
. Panel (d): frost cracking occurs whenever water is
present along the vertical profile in the direction of warming
temperature, independent of the distance to, or amount of water
available at, the source. This model implementation is similar to the
model by .
In the first case, the flow restriction of water is constant
(2 m-1) and hence independent of the porosity or thermal state
of the subsurface (Fig. b). Compared to the standard
model (Fig. a), this leads to a shift in the peak at negative MAT towards
smaller sediment thicknesses; an effect of more restricted water flow in
warm sediment, which diminishes the contribution of porous
water-filled sediment close to the frost-cracking zone. On the other
hand, the less restricted flow in cold bedrock leads to increased
frost cracking in zones with MAT ≥0∘C and S≤1 m. The
consequence of less restrictive flow in bedrock suggests that crack propagation in bedrock with
a high fracture density can be very efficient, leading to a positive
feedback between the degree of rock damage and frost-cracking
susceptibility.
In the second case (Fig. c), which resembles the
model by , FCI still depends on the
distance from bedrock in the FCW to the closest water, whereas the amount of water
available now has no effect. The flow restriction of
water is again constant (2 m-1) and independent of the
porosity or thermal state. Compared to our standard model, FCI is now
reduced for negative MAT and thick sediment. The humped relationship
between frost cracking and sediment thickness essentially disappears because frost-cracking is
not accelerated by water supplied from the porous sediment layer.
In the third case, following , frost cracking is
allowed to take place whenever water is present along a continuous thermal
gradient (Fig. d); there is no dependency on migration
distance or amount of water. Like in Fig. c, the lack of
sensitivity to water volume removes the humped FCI–sediment relation for
negative MATs. This implementation furthermore gives rise to a very sharp
jump in FCI at 0 ∘C, as is also seen in the original model by
. Frost cracking increases around 0 ∘C MAT
because water becomes available at great depth where the temperature reflects
the MAT. The increase is sudden because frost cracking is not restricted by
the great distance between cold bedrock and water (where
T ≥0 ∘C).
Frost-cracking intensity in our model scales with the volume of water
available (Eq. ), but only up to a fixed limit,
Vcw (Table ). We chose this
implementation because we believe that water availability can limit
frost cracking only up to a certain threshold. Although the value for
Vcw (0.04 m) is not supported by empirical data, our
experiments reveal that rates of frost cracking and sediment transport
are rather insensitive to variations in Vcw. Only when
Vcw is significantly reduced does it affect the computed
FCI values, first as a reduction in FCI for positive MATs
(Vcw=0.005 m) and then by also lowering frost
cracking for negative MATs (Vcw=0.001 m).
To summarise: the choice of flow restriction parameters influences the
conditions for which FCI is predicted to be highest, but the general humped
relationship between FCI and sediment thickness is preserved for negative
MATs (Fig. a, b). On the other hand, the humped relationship
disappears if the amount of water is not a rate-limiting factor as we assume
in our standard model. This means that FCI peaks with little or no sediment
cover for both negative and positive MATs (Fig. c, d).
As this analysis has demonstrated, water availability is clearly an important
issue for the predictive power of any frost-cracking model. We therefore
return to this topic in the discussion (Sect. ), and we
follow up on effects for long-term periglacial landscape evolution in the
companion paper .
Temperature thresholds for frost cracking
Finally, we explore the influence of the FCW.
summarises experiments and reports frost
cracking between -5 and 0 ∘C for high-porosity rocks (tuff,
shale, chalk) and between -6 and -3 ∘C for
medium-porosity rocks (limestone, sandstones). The model by
predicted frost cracking between -15 and
-4 ∘C for a marble and a granite. Thus, theory and
experiments indicate that the thresholds for frost cracking are more gradual
and lithology-dependent than what is captured by a general FCW from -8 to
-3 ∘C. To explore the consequence of varying the FCW, we
applied a range of FCW thresholds (some are shown in Fig. )
and found that the lower boundary of the FCW does not affect the
frost-cracking pattern appreciably unless the total width of the window
becomes very narrow (<1 ∘C). It also shows (predictably)
that raising the upper threshold for frost cracking leads to frost cracking
at higher MAT.
Integrated frost-cracking intensity as a function of sediment
thickness and MAT, for frost-cracking windows of (a)
[-8;-1], (b) [-8;-2], and
(c) [-15;-4] ∘C. Note the different colour
scales.
Discussion
The rates of frost cracking
The inclusion of the effects of variable sediment thickness in numerical
models of frost cracking generates new insights into how climate and
sediment cover combine to control bedrock weathering rates. First,
when studying the modelled values for FCI along the MAT dimension
(Fig. c), it is possible to compare our results to
those of previous models. In the following, we thus compare the FCI
results presented in this paper with the model introduced by
and that by
.
While the model includes two distinct zones of
efficient frost cracking, one for negative MATs and one for positive MATs,
suggested that only the cold-region cracking could
survive penalties to water transport in cold bedrock. Our model generally
confirms the findings of , although we suggest that
frost cracking can still be active in moderately warm climates, provided that
the sediment cover is sufficiently thin (<10 cm) and the surface
temperature is occasionally lowered into the FCW (Fig. c).
Our model thus corroborates results suggesting the most intense
scree production from bare bedrock at positive MATs .
Frost cracking under warm conditions is, however, very sensitive to bedrock
water saturation because water must come from within the bedrock when the
surface is frozen in winter. Frost cracking in warm regions is also
potentially limited by snow covers that insulates the bedrock from the cold
atmosphere in winter (Fig. ). Given that hillslope angle
governs the likelihood of retaining snow cover as well as a sediment layer,
our model indicates that frost cracking in warm regions mainly occurs in
winter along steep and wet bedrock surfaces, such as valley sides and
headwalls.
It is worth noting, however, that strong variations in bedrock
strength may potentially surpass the effects of water availability, as the
rate of ice segregation growth is likely to also depend on the size and
geometry of bedrock cracks .
Soil production function
It is generally accepted that the conversion of bedrock to sediment is
related to the thickness of the sediment mantle, and the production of new
sediment eventually stalls with increasing accumulation .
However, the exact nature of this relation
(soil production function) has been debated seefor
a review. Some have argued for an exponential
decline in sediment production rate with thickening sediment cover
, while others favour
a relation in which soil production rate peaks (the “humped”
function) under a finite sediment thickness . The strongest argument for
the latter model is that a thin sediment layer is more likely to
retain moisture, which is a vital ingredient for physical and chemical
weathering, whereas bare bedrock promotes run-off and remains dry much
of the time.
argued for a humped soil production model to
explain the presence of tors (bedrock knobs) at hill crests in the Laramide
ranges of the western USA – an area experiencing pronounced frost action.
According to , the tors protrude because (1)
divergent sediment transport away from the crests leaves the summits bare
and (2) sediment production from bare bedrock is slower than sediment
production in the surrounding soil-mantled slopes.
Assuming that sediment production scales with frost-cracking
intensity, our results suggest that the shape of the soil production
function depends on temperature
(Fig. ). For positive MATs, sediment production
decays exponentially with thickening sediment cover (line 3 in
Fig. a). We note that this result relies on the
assumption that moisture is present at all times. A variable moisture
content in the outer decimetres of bare bedrock, e.g. due to seasonal
variations as suggested by , could potentially
limit frost cracking on bare bedrock during dry periods.
The present model does not include such effects, since it assumes
saturated conditions at all times.
For negative MATs, sediment production first increases
with sediment thickness (up to 1–2 m) and then decays
exponentially with further sediment mantle thickening (lines 1 and 2
in Fig. a). The optimal sediment thickness (for
which sediment production is maximised) decreases with decreasing
MAT. Our model thus suggests that both the exponential decline and the
humped soil production
functions may be viable in the periglacial environment. Moreover, long-term
temperature fluctuations are likely to result in switching between the two.
Frost creep and depth-dependent transport
High-frequency diurnal temperature variations represent the main driver of
frost creep for thin sediment covers. The diurnal temperature variations do
not penetrate far into the sediment and the freeze–thaw events that they
cause are shallow and frequent. For sediment covers less than 1 m
thick, the most efficient transport (highest κ) occurs for MATs around
-6 and +6 ∘C (Fig. a and c) because
diurnal variations generate the highest number of freeze–thaw events for
these thermal settings . However, the annual
temperature variation is more dominant for thicker mantles (>1 m),
and its influence on creep is maximised for MATs around 0 ∘C
(Fig. a and b). This is because the seasonal temperature
variation is slow enough to penetrate much deeper into the sediment.
Freeze–thaw in a deep sediment layer contributes far more to the sediment
flux than shallow events because the overlying pile of sediment is passively
transported when the deep layer expands and contracts.
The efficiency of sediment transport by frost creep (κ) furthermore
depends on sediment thickness because the frequency of freeze–thaw events
decreases with depth into the sediment (Fig. ). Although
sediment flux initially increases with sediment thickness for all MATs, it
saturates at a temperature-dependent limiting thickness, beyond which it
becomes constant, akin to linear hillslope diffusion models
(Fig. b). This type of sediment-thickness-dependent function
is in close agreement with the hypothesised transport function by
although it depends on climate. For MATs around
0 ∘C, the annual temperature variation causes freezing and
thawing up to 3 m into the sediment, whereas the freeze–thaw
penetration depth is more limited under warmer and colder conditions,
respectively. We therefore emphasise that our frost-creep model supports
a transport function rooted in sediment thickness
e.g. up to
a limit of ∼3 m for MAT ∼0 ∘C and <1 m
for colder and warmer settings.
Finally, we emphasise that grain size is an important determinant of frost
susceptibility , and our
results are applicable mainly to fine-grained (silt-sized) sediment.
Furthermore, the creep mechanism quantified by our model is likely restricted
to gentle slopes of less than 15–20∘. Other processes, such as
gelifluction and debris flows, may take over at steeper slopes, resulting in
more non-linear relations between slope and sediment flux
.
Limits to water availability?
Whether it is necessary to penalise frost-cracking
intensity according to water availability remains an open question. On the one
hand, some water is always available in a porous media even at sub-zero
temperatures. The water is drawn to the zone of potential frost cracking
provided that a hydraulic connection exists via a film of unfrozen water
along grain boundaries (pre-melted films). The temperature gradient
determines the flow rate of water because it scales the suction force.
However, the thickness of pre-melted films gradually decreases when
temperature is lowered.
On the other hand, experimental results suggest that a nearby water source
(T>0 ∘C) increases frost-cracking intensity significantly
, which indicates that the distance
between the FCW and a water source is of importance. A possible reason is
that the hydraulic connection needed to maintain the suction force is less
likely to be preserved over great distances if permeability is low. From this
perspective, the flow-resistance parameter of our model can be seen as a
measure of the number of hydraulic connections between the site of ice
segregation and the site of available water. Low-permeability materials
should therefore be assigned higher flow-resistance values (γ) if
following this reasoning.
However, as the discussion above highlights, this
element of our model for frost cracking is unfortunately not well
constrained. We have therefore striven to explore this issue as openly as
possible, but further research is required to fully elucidate the influence
of permeability on frost cracking.