Geophysical Self-Organization by Differential Frost Heave
Recurrent freezing in water-rich soils can generate polygonal, striped, and circular patterns at the ground surface made prominent by stone borders, ordered vegetation, or regular topography. A mathematical model is advanced that demonstrates that one-dimensional
freezing can become unstable and evolve into differential frost heave, which can explain the
initiation of several patterned ground forms. This analysis establishes differential frost
heave as a new instability mechanism that can lead to geophysical self-organization. This is
being explored as a potential indicator of global climate change.
Cold climate regions subject to recurrent freezing and thawing can display prominent patterned
landforms such as hummocks (Fig 1a), frost boils (Fig 1b), and stone circles (Fig 1c). These are
examples of patterned ground, i.e., surface features made prominent by the segregation of stones,
ordered variations in ground cover, or regular topography1. Patterned ground is of interest since
it is a form of geophysical self-organization2. It is an example of a highly dissipative system that
must transfer significant heat to its surroundings to satisfy the Second Law of Thermodynamics.
Patterned ground forms such as frost boils are being studied as potential indicators of global climate change since their characteristic spacing and height, which can be measured via remote
sensing, change in response to environmental conditions3.
Washburn4 identified 19 mechanisms that could influence the formation of patterned ground
in recurrently frozen soils. Only the frost-cracking5, Rayleigh free convection6,7,8, and buoyancyinduced soil circulation9 mechanisms have been incorporated into predictive models. However, it
is doubtful whether these can explain the occurrence of patterned ground forms such as shown in
Fig. 1. The latter do not display frost cracking; moreover, it is unlikely that the small density differences developed during freezing or thawing are sufficient to promote buoyancy-induced free
convection in most soils. These models were developed based on observations of pattern formation in nonfreezing/thawing processes such as polygonal crack formation in biaxially stressed
solids and hexagonal convection cells in liquid layers heated from below. That is, the hexagonal
geometry associated with stress cracking or Rayleigh free convection suggested that patterned
ground formation might arise owing to the same mechanisms. However, such inferences have
possibly hindered advances in understanding the origins of diverse patterned ground forms. Note
that the hexagonal geometry is a natural consequence of the linear stability theory (LST) analysis
that was used in the frost-cracking and free convection models to determine whether onedimensional freezing or thawing has any propensity to evolve into multidimensional patterning.
In fact, any LST analysis will lead to a spatial dependence f  x, y  , where x and y denote lateral
spatial coordinates, dictated by solutions to the Helmholtz equation:
2 f 2 f

2 f
x 2 y 2
where  denotes the wave number. Superposition of the exponential solutions to this equation
can describe stripes, squares, rectangles, hexagons, and circles10,11. In this paper we present a
LST analysis of a novel mechanism for patterned ground formation for which there are no analogies in nonfreezing/thawing processes, namely differential frost heave. As such, this is an instability mechanism that has not been described heretofore, which also appears to explain the occurrence of patterned ground forms such as shown in Fig. 1.
The emphasis in this paper will be on the underlying physics of the differential frost-heave
mechanism and on its implications for patterned ground formation. The mathematical details,
which are straightforward but tedious, will be presented in a future publication12.
Physics of freezing in porous soils
Differential frost heave (DFH) refers to laterally nonuniform uplifting of the ground surface due
to freezing of water within the soil. Frost heave can greatly exceed that due merely to the expansion of water upon freezing (9%) owing to additional water being drawn upward via cryostatic
suction13,14. This arises because soils are wet preferentially by unfrozen water rather than ice.
Hence, the ice within a pore is separated from the soil by a submicron film of unfrozen water.
Owing to different nearest neighbor molecular interactions (London dispersion forces) in this
thin film of unfrozen water, the pressure parallel to the pore wall will be less than the normal
component. This anisotropic pressure phenomenon, referred to as “disjoining pressure”, in combination with capillary (curvature) effects causes a depression in the normal freezing point T0
described by the Gibbs-Thompson equation:
T  T0 
 pi  pw 
 H f
where    w  i is the mass density of liquid water, which is assumed to be the same as for
ice, and H f is its latent heat of fusion. The difference between the ice pressure pi and the unfrozen water pressure (parallel to the pore wall) pw is proportional to the unfrozen water content
and hence to the temperature. The latter relationship varies markedly between different soil types
as is referred to as the “soil characteristic function”. This implies that the pressure gradient in the
thin film of unfrozen water will be parallel to the temperature gradient, thereby providing a driving force (i.e., cryostatic suction) for upward water permeation during freezing.
Since soils have a pore-size distribution, ice penetration will progress from the larger to the
smaller pores as the temperature decreases. Hence, freezing in a water-saturated soil will not occur at a discrete plane, but rather within a relatively thin (0.11 cm) region that is referred to as
the “frozen fringe”. As freezing progresses, the pressure within the ice in the frozen fringe increases until it is sufficient to cause separation of the soil particles at some plane at which an “ice
lens”, i.e., a layer of nearly pure ice, will begin to grow owing to additional water being drawn
upward by cryostatic suction. This ice lens will become thicker until downward penetration of
the freezing front decreases the soil permeability sufficiently to cut off its subsurface water supply. Subsequent ice-lens formation will occur as the freezing front continues to penetrate downward. These ice lenses become progressively thicker and more widely spaced as the rate of freezing decreases with increasing depth. It is the formation of these ice lenses that can result in substantial heave of the ground surface.
In the absence of any model, it is unclear how DFH can occur since some mechanism must
be available to remove the latent heat of fusion from the excess water that is frozen under the
heaved (elevated) ground surface. This cannot occur owing to purely one-dimensional upward
heat transfer since the conduction path between the ambient air (or snow) and underlying freezing front is longer under elevations relative to depressions in the ground surface. The DFH model
developed in this paper will provide the explanation for this.
Previous frost-heave modeling studies
A LST analysis seeks to determine whether a particular solution for a physical process, referred
to as the “basic state”, is stable with respect to infinitesimal perturbations; i.e., will it evolve to a
solution that describes an alternate lower energy-state of the system if any of the dependent variables are minutely perturbed. Here we seek to determine whether one-dimensional freezing and
heave will spontaneously evolve into multidimensional differential frost heave. We will base our
LST analysis on the basic state solution for one-dimensional freezing and heave advanced by
Miller15, which will briefly be reviewed here.
Fig. 2 shows a cross-section of a water-saturated soil undergoing freezing downward from
the ground surface at zs. Here we show zb as a fixed plane defining the permafrost table that is
assumed to be at 0C, in which case the region between zs and zb that undergoes seasonal freezing and thawing is referred to as the “active layer”. Note that the DFH mechanism does not require that permafrost be present. However, the presence of permafrost can facilitate patterned
ground formation since it helps to keep the soil water-saturated by impeding drainage. The frozen fringe is bounded by zf, the plane at 0C defining the maximum instantaneous depth at which
freezing of pore water is occurring, and zl, the plane at which active ice-lens growth occurs. Between zl and zi, additional pore water is freezing although no additional ice-lens formation or
growth occur since the lowermost ice lens blocks any upward water permeation to this region.
The region between zi and zs, containing ice lenses and pore ice, undergoes essentially no additional freezing. The planes defined by zf, zl, and zi move downward whereas that defined by zs
moves upward in time.
The Miller model involves solving unsteady-state one-dimensional conservation equations
in each of the regions described above. These constitute an energy balance in the frozen region
between zl and zs; energy and unfrozen water and ice mass balances within the frozen fringe between zf and zl; and a water mass balance in the unfrozen isothermal region between zb and zf.
The overall conservation of water relates the ice and unfrozen water volume fractions to the soil
porosity. Darcy’s law relates the unfrozen water velocity to the ice and unfrozen water pressures
and introduces the hydraulic permeability, which is a function of the unfrozen water content. The
Gibbs-Thompson equation relates the ice and unfrozen water pressures to the temperature. The
soil characteristic function relates the unfrozen water content to the ice and unfrozen water pressures. A force balance at the bottom of the lowest (actively freezing) ice lens relates the pore water volume fraction to the ice and unfrozen water pressures and the weight of the overlying soil,
and determines when a new ice lens is initiated. A water mass balance at the lowest ice lens relates the ice velocity to the water permeation rate. The original Miller model invoked a rigid ice
approximation whereby the ice velocity was assumed to be constant throughout the frozen soil.
Fowler and Krantz16 recognized that this assumption does not permit DFH and incorporated a
regelation mechanism suggested by Gilpin14 for which the ice velocity is proportional to the local
temperature gradient. This results in a set of five coupled partial differential equations in two dependent variables, the temperature and unfrozen water pressure, in two independent variables,
time and the vertical spatial coordinate. The initial conditions correspond to unfrozen watersaturated soil at 0C. The boundary conditions are continuity of temperature at 0C, heat flux,
and pressure at zf, continuity of the temperature dictated by the Gibbs-Thompson equation and a
load pressure equal to the weight of the overlying soil at zl. Whereas Miller15 considered only a
constant subfreezing temperature boundary condition at zs, here we also allow for a heat flux that
is either constant or proportional to the difference between the instantaneous ground-surface and
constant ambient air temperatures (a Newton’s law of cooling condition), or continuity of tem-
perature and heat flux with an overlying conducting layer of snow. This one-dimensional frostheave model introduces the following soil properties, physical and transport properties, and environmental parameters: soil type characterized by its specific soil characteristic function and hydraulic permeability relationships, mass density, porosity, and thermal properties; heat of fusion,
ice and unfrozen water mass densities, thermal conductivities, and heat capacities; and specified
conditions at the ground surface constituting either its temperature, cooling flux, heat-transfer
coefficient, or snow depth; in the latter case, the thermal properties of the snow are also required.
Several investigators15,17,18 have solved the Miller model for one-dimensional freezing and
heave. The model as outlined above is cumbersome, subject to numerical convergence problems
owing to the frozen fringe, and not readily extended to multidimensional (differential) frost
heave. Fowler19 applied scaling analysis to the Miller model and showed that quasi-steady state
conditions apply within all three regions and that the frozen fringe can be collapsed to a plane
across which jump boundary conditions apply. Moreover, he demonstrated that a permeation
boundary layer existed below the lowest ice lens across which the entire pressure drop in the water phase essentially occurs. Fowler and Noon20 have shown that the solution to this simplified
model, which can be solved analytically, gives results in excellent agreement with a numerical
solution to the full Miller model. Krantz and Adams21 proved that this simplified model agrees
well with one-dimensional frost-heave measurements in laboratory core experiments. Subsequently, Fowler and Krantz16 generalized the equations for the simplified Miller model to multidimensional frost heave. However, they made no attempt to solve these multidimensional equations and to explore their implications for DFH and patterned ground formation. The latter is the
subject of this paper.
Linear stability analysis for differential frost heave
Here we will present a model for DFH based on a LST analysis of a basic state described by the
solution for one-dimensional frost heave described by the solution of Fowler and Krantz16 for the
simplified model of Miller15. This involves allowing spatial perturbations in all the dependent
variables appearing in the multidimensional frost-heave equations; these constitute the temperature, ice and unfrozen water pressures, unfrozen water volume fraction, water permeation velocity, and the moving planes defined by zf and zs. The solution for any dependent variable
X   x, y, z, t  is of the general form
i  x  y 
X   Xˆ  z, t  e 
where  and  are the wave numbers in the x and y directions, respectively, for which only the
real part of X  has physical significance. If the basic state is either time-independent or quasisteady-state (i.e., slowly changing in time), X   z, t  will have an exponential time-dependence
with a growth coefficient (exponential argument)  . If any perturbation characterized by wave
numbers  and  can be found for which   0 , the basic state is unstable. The basic state for
freezing is time-dependent but slowly varying; i.e., the planar freezing front and ground surface
move very slowly in time. Hence, we carried out both a quasi-steady-state (frozen time) as well
as a real time LST analysis. Because time enters the LST analysis only implicitly through the
freezing and frost-heave velocities and boundary conditions, the conclusions drawn from both
analyses were the same. Hence, for simplicity we will present only the results for the quasisteady-state analysis. We also considered only two-dimensional modes characterized by wave
number  . Since hummocks, frost boils, and stone polygons are three-dimensional patterns,
their geometrical characteristics can be related to  via Squire’s transformation22.
An additional consideration for DFH is the resistance to bending offered by the frozen soil.
We considered two models for this resistance: a purely elastic (Hookian) bending described by
thin plate and shell theory23 and, a purely viscous resistance model suggested by Fowler and
Noon20. Since laboratory experiments24 indicate that frozen soil behaves elastically on short time
scales and measurements25 are available for the elastic modulus for frozen soils (assumed to be 5
MPa unless stated otherwise), whereas only estimates are available for the effective viscosity, we
present results only for the elastic model.
Linear stability theory model predictions
This LST analysis indicates that indeed one-dimensional freezing and frost heave can be unstable
and thereby display a propensity to develop into DFH. This is illustrated in Fig. 3, which shows a
plot of the growth coefficient  (nondimensionalized with a conductive time scale) as a function
of the dimensionless wave number   2 d0  , where d 0 is the maximum depth of freezing
(chosen here to be 1 meter, a typical active layer depth) and  is the wavelength, for three representative soil types, sand, silt, and clay. Coarse-grained sand (i.e., large pores) is seen to be stable with respect to DFH (i.e.,   0 for all  ), whereas the medium-grained silt and fine-grained
clay are unstable (i.e.,   0 for some  ). Note however that the unstable modes for clay grow
considerably more slowly (i.e.,  is smaller) than for silt.
The LST model provides insight concerning the mechanism for DFH. The isotherms are
compressed in the trough relative to the crest regions. This is a consequence of the amplitude 
of the perturbed freezing front at z f always being less than the amplitude  of the perturbed
ground surface at z s . This implies that significant lateral temperature gradients exist to conduct
heat from the crest to the trough regions, which then conduct this heat upward to the ambient air.
This is confirmed by an energy balance over one wavelength, which indicates that additional
frost heave occurs in the crest regions because the latent heat of fusion is conducted laterally into
and upward within the trough regions.
Manifestation of this instability leading to DFH depends critically on the soil properties and
can be mitigated by the environmental conditions. Fig. 3 indicates that DFH will not occur in
coarse-grained soils such as sands that do not generate sufficient cryostatic suction. The model
also indicates that it may not be possible to observe DFH in very fine-grained soils such as clays
for which the unstable modes grow very slowly owing to a very small hydraulic permeability.
Bending of the frozen soil is a stabilizing influence that damps higher wave number modes as
can be seen in Fig. 3. The elastic modulus is zero at the freezing point and increases with decreasing temperature24. An increase in the elastic modulus of the frozen soil will decrease the
maximum wave number as well as the growth rate of the unstable modes. However, only an infinite value of the elastic modulus can stabilize all modes. Moreover, soils such as sand are stable
to DFH even for a zero elastic modulus. An increase in the instantaneous freezing depth h1 also
has a stabilizing effect associated with elastic bending. The predictions in Fig. 3 are for
h1  0.05d0 . As h1 increases, the growth rates of all modes decrease and higher wave numbers
are progressively stabilized until at some critical depth all modes are stabilized. As h1 increases,
DFH must elastically deform a thicker frozen layer, which favors longer wavelength modes.
The environmental parameters enter the DFH model through the ambient temperature, heattransfer coefficient, and snow depth. Their effect on this instability is of particular interest relative to using patterned ground forms associated with DFH as indicators of global climate change.
An increase in the ambient air temperature causes a decrease in the growth rate such that all
modes are stabilized when T = 0C. This effect is associated with the decreased ability to conduct away the latent heat of fusion at higher ambient temperatures. The effect of increasing the
heat-coefficient coefficient (HTC) is non-monotonic. The maximum growth coefficient and
range of unstable wave numbers are largest for a dimensionless HTC (nondimensionalized with
respect to ???) of order one. This interesting effect arises because for small values of the HTC
there is insufficient capability to remove the latent heat of fusion, whereas for large values of the
HTC, freezing occurs too rapidly to permit DFH to develop. The effect of increasing snow depth
is also non-monotonic. A small snow depth causes a relatively greater insulating effect above the
troughs and thereby favors more heave under the crests, whereas a large snow depth causes an
increase in the ground surface temperature and can result in complete stabilization.
Relating model predictions to observable patterned ground features
A LST analysis can indicate only whether one-dimensional freezing and frost heave is unstable.
It is limited in what it can indicate regarding observable pattern formation. Nonetheless, in many
analyses for which the nonlinearities are weakly manifest, the most highly amplified mode predicted by LST corresponds to observable characteristics of the developed (finite amplitude) state
of the system. For example, Fig. 3 indicates that silt has a most highly amplified wave number.
However, as discussed above, the most highly amplified wave number will decrease with increasing freezing-front penetration until an h1 is reached at which there are no unstable wave
numbers. Clearly there will be some wave number  max that will experience more growth during
the entire frost-penetration process. This is shown in Fig. 4 that plots the relative amplitude of
the heaved ground surface   0 , where  0 is some arbitrary initial perturbation amplitude (LST
analysis permits determining only the relative amplitude of the perturbations), as a function of
wave number for the conditions used in Fig. 3 and two different maximum freezing depths
d0  5 cm and d0  1 m. The ratio   0 is a maximum at  max  ??? for d0  5 cm and at
 max  2.8 for d0  1 m.
It is reasonable to expect that  max might be related to the spacing of patterned ground forms
associated with DFH. The only regular interconnected geometric forms that satisfy the Helmholtz equation for soils with laterally isotropic properties are hexagons and stripes, corresponding to patterns expected on horizontal and sloped terrain, respectively. Note that squares will not
display equidistant center-to-center spacings, whereas equilateral triangles are a subset of hexagons. Fig. 5 shows a computer-generated pattern based on interlocking hexagons; the heaved
crests and depressed troughs are shown by the light and dark areas, respectively. Clearly this
hexagonal packing can describe the geometry of patterns such as those shown in Fig. 1. To facilitate quantitative comparison between model predictions and data, it is useful to determine the
center-to-center distance for hexagons, which is given by the following:
Lh 
4 3 d 0
3 max
Field study validation of the model predictions
Prior field studies do not have sufficient information on the soil properties and environmental
conditions to permit any quantitative validation of these model predictions. Hence, the authors
carried out extensive studies of active earth hummocks at a field site 6825' N, 133°45' W in the
Mackenzie Delta in the Northwest Territories of Canada. This site was chosen both because it
had been studied by several previous researchers and was near the Aurora Research Institute in
Inuvik NWT, which provided logistical support30,31,32,33. The soil was a silt containing a small
amount of clay. Its soil characteristic function and hydraulic permeability were obtained from the
studies of Smith34. Values for the active layer depth (1 m) and average snow depth during freezing (??? cm) were obtained from the Aurora Research Institute. A heat-transfer coefficient of ???
W/m2K was used to characterize the prevailing environmental conditions. The average centerto-center distance for the 47 contiguous hummocks measured at this field site was ??? m ± ???
m. This agrees well with the spacing of ??? corresponding to  max  ??? inferred from the LST
predictions in Fig. 4. The LST model also indicates that the observed average ground-surface
heave of ??? cm would have approximately ??? years to develop, which is entirely reasonable for
these patterns.
Laboratory validation of the model predictions
Field data do not provide a discriminating test of these model predictions since they represent
patterns that have existed for hundreds of years and hence have been subjected to changing environmental conditions. A demanding test of any model is to validate its predictions via controlled
laboratory studies. We are not aware of any prior successful attempts to grow patterned ground
associated with DFH in the laboratory. We utilized the insight provided by the LST model for
DFH in order to design an apparatus for growing frost boils in the laboratory. Key considerations
were choosing a soil that readily promotes frost heave (silt); assuring one-dimensional freezing
and thawing; maintaining a water supply below the freezing front to sustain ice-lens growth; and
freezing sufficiently slowly to permit measurable frost heave. A commercial open top freezer
with overhead infrared panels was programmed for daily freeze/thaw cycles to permit facilitating
the growth of patterned ground on a relatively short time scale (1-2 weeks). Fig. 6 shows a photograph of four frost boils made prominent by soil surface heave and sorting of stones. The center-to-center spacing of ??? cm for these frost boils agrees well with the spacing of ??? corresponding to  max  ??? inferred from Fig. 4. Note that  max is larger for these laboratory experiments because the maximum freezing depth (5 cm) was less than that required to stabilize all
modes for silt under the specified environmental conditions (23 cm).
DFH-induced Patterned Ground as an Indicator of Global Climate Change
Fig. 7 is an aerial photograph showing the sharp demarcation line between moist nonacidic and
moist acidic tundra observed south of Prudhoe Bay in the arctic region of Alaska, which has
been suggested as a possible indicator of global climate change35,36,37. Oxidation and surface
leaching will cause acidification of tundra soils over time. However, nearly neutral soil conditions can be maintained if mineral-rich material is convected from depth to the soil surface. The
cryoturbation associated with frost boil activity provides this convective mechanism. Interestingly, active frost boils are ubiquitous in the moist nonacidic tundra regions, but nearly absent in the
moist acidic tundra. However, the occurrence of moist nonacidic tundra does not correlate with
mean annual temperature as might be expected. Indeed, active frost boils are nearly absent near
the arctic coast where the mean annual temperature is lowest and prevalent at some sites south of
the coast where warmer temperatures prevail. The LST model developed here not only explains
the occurrence of DFH-induced frost boils in the silt-rich soils characterizing this arctic region,
but accounts for the nonmonotonic behavior with respect to mean annual temperature. Indeed,
this is explained by the effect of snow depth on the occurrence of DFH and associated patterned
ground forms. The arctic coast is subject to high winds and therefore has a high heat-transfer coefficient and relatively little snow accumulation during the onset of freezing conditions. In contrast, interior arctic sites are more sheltered such that lower heat-transfer coefficients and greater
snow depths are observed. Moderate heat-transfer coefficients and snow depths can have a destabilizing effect on DFH and thereby promote frost boil activity. Field studies are currently underway to explore further the implications of DFH and frost boil activity as potential indicators
of global climate change.