Quaternary Research 75 (2011) 325–333

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Quaternary Research

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Assessing the role of differential frost heave in the origin of non-sorted circles

Rorik A. Peterson ⁎University of Alaska, Fairbanks, AK, USA

⁎ PO Box 755905, University of Alaska, Fairbanks, FaFax: +1 907 474 1519.

E-mail address: rorik@alaska.edu.

0033-5894/$ – see front matter © 2010 University of Wdoi:10.1016/j.yqres.2010.08.003

a b s t r a c t

a r t i c l e i n f o

Available online 19 September 2010

Keywords:Patterned groundDifferential frost heave

A. L. Washburn famously proposed and reviewed 19 hypotheses for the origin of patterned ground inperiglacial environments over 50 years ago (Washburn, 1956). Of these 19 mechanisms, only a few havebeen analyzed starting from a fundamental description of the physics to assess their potential contribution tothe initiation of patterned ground. Here, differential frost heave (DFH) is investigated as the origin of non-sorted circles in periglacial landscapes. Model results indicating that DFH can spontaneously lead topatterned ground are compared to measurements of patterned ground in the Canadian Arctic Archipelago.The characteristic size of the predicted emerging pattern depends strongly on the maximum depth offreezing but is only weakly dependent on the soil composition. The predicted emerging patterns may also bedramatically different in size with a small change in active layer when a relatively thin (~10 cm) insulatingsnow covers the ground during freezing. The model predicted trends agree with field observations of patternsize and active layer depth at two distinct sites. Although two data points cannot conclusively indicate atrend, this correlation gives support for the possibility of determining some subsurface properties usingremote sensing images of surface patterned ground.

© 2010 University of Washington. Published by Elsevier Inc. All rights reserved.

Introduction

Washburn's seminal paper on the origins of patterned ground(Washburn, 1956) looked in detail at many possible causes for theorigins of patterned ground. He found some faults or limitations witheach mechanism and concluded that a single mechanism could not beresponsible for the range of different patterned-ground types. Theoverall conclusion was that in all likelihood many concomitantmechanisms were responsible for each classification of patternedground (polygenetic). Currently, there is some consensus in agree-ment with this conclusion, but identifying the specific mechanismsresponsible for a particular type of patterned ground is still debated(Mann, 2003).

In the 50 yr since Washburn's publication, there has beensignificant work on attempting to clarify the exact mechanismsresponsible for the different types of patterned ground, but a unifyingset of theories has proved elusive and remains a subject of debate. Thispaper suggests that one of the 19mechanisms, differential frost heave,is likely the original cause of one type of patterned ground inparticular: non-sorted circles. In support, it will be demonstrated howanalytical and numerical modeling of the process corroboratesobservations in the field for non-sorted circles at field study sites in

irbanks, AK 99775-5905, USA.

ashington. Published by Elsevier In

the North American Arctic. As examples, two sites with non-sortedcircles with distinct dimensions will be analyzed in greater detail. Andfinally, this paper will make the case that finding the origin of non-sorted circles has important implications beyond purely satisfyingscientific curiosity. Existing non-sorted circle cryoturbation plays asignificant role in storing organic carbon in arctic soils (Horwath et al.,2008; Ping et al., 2008). Because climate plays a major role in whethercircles initially form or not, future changes in climate may cause newnon-sorted circle systems to form where they were previously sparseor absent. Therefore, the possibility of new circles forming should beconsidered when attempting to quantify accurately the potentialeffects of climate change on arctic landscapes because they play such asignificant role in the sequestration of organic carbon.

Non-sorted circles are characterized by the absence of vegetationin the center with areas of vegetation and peat in between, as shownin Figure 1. Horizontally non-uniform (differential) frost heave (DFH)has been observed and measured (Romanovsky et al., 2008) wherethe circle center heaves more than the perimeter and inter-pattern,vegetated region. Mature, patterned ground features clearly owe theircurrent dynamic state of equilibrium to a balance of factors includingclimate, vegetation, chemistry, and hydrology (Van Vliet-Lanoë, 1991;Boike et al., 2008), and there has been some success with modelingthese interrelated and complex processes (Nicolsky et al., 2008;Daanen et al., 2008). When discussing the question of patternedground genesis as Washburn (1956) first did, these spatially varyingprocesses had not yet developed into the state that they are observedtoday.

c. All rights reserved.

Figure 1. Two forms of non-sorted patterned ground observed in the Canadian Arctic Archipelago. (a) Non-sorted polygons of about 20 cm with sparse vegetation at Isachsen, and(b) non-sorted circles of about 70–100 cm diameter with vegetated inter-circle regions.

326 R.A. Peterson / Quaternary Research 75 (2011) 325–333

This paper addresses the question of origin only, and therefore theseprocesses will not be addressed in detail; it discusses how a uniform,non-spatially differentiated ground surface can spontaneously developa surface pattern, and how pattern dimensions are largely dictated bysoil thermal characteristics and conditions. Once this pattern initiallydevelops, the remaining ecological, biological, and hydrological pro-cesses may act to further differentiate the surface into the mature formthat is observed today. However, the initial pattern size and spacing arefixed by the dominant DFH pattern, and these subsequent processesdevelop around this initial template. The dominant pattern is thatwhichgrowsmore thanany other sizedpatterndue to recurrent freezingof theactive layer, and therefore becomes most expressed over time. Thisinitial pattern size is fundamentally important because it is assumed inthis analysis that the size does not evolve or change significantly with

time, and is therefore observed in some mature patterned-groundfeatures.

The argument for DFH will be initially developed in terms ofattempting to explain what led to the differences between the twodistinct examples of non-sorted patterned ground shown in Figure 1.The broader implications of the role of DFH in patterned ground willbe addressed in the latter discussion. Figure 1a shows small, non-sorted patterns observed at Isachsen, located at about 78°30′N on EllefRingnes Island in the Canadian Arctic Archipelago. The patternsshown are on the order of 20 cm in diameter, with sparse vegetationin between and smaller-scale desiccation cracks within the pattern.Figure 1b shows a larger form of non-sorted patterns from GreenCabin, located at about 73°10′N on Banks Island in the Canadian ArcticArchipelago. These circular features are on the order of 70–100 cm in

10−2 10−1 100 1010

20

40

60

80

100

subfreezing temperature [°C]

unfr

ozen

wat

er c

onte

nt [%

vol

ume]

claysilty−claysandy−silt

Figure 2. Volumetric unfrozen water content as a function of subfreezing temperaturebelow 0 °C for three representative soil types encountered in arctic regions.

327R.A. Peterson / Quaternary Research 75 (2011) 325–333

diameter and have more developed vegetation in between. Both ofthese sites were components of a larger study of the biocomplexity ofpatterned ground along a transect that extended from the foothills inNorthern Alaska to Isachsen (Walker et al., 2008).

The active layer in arctic regions refers the uppermost layer of soilthat undergoes perennial freezing and thawing. It may or may notoverlie permafrost, but permafrost is pervasive at the sites discussedhere. The smaller patterns in the more northern region have ashallower active layer around 30 cm, the mean annual temperature islower, freeze-up occurs more rapidly, snow cover is generally thinner,and the soil more clay-like. The soil composition at Isachsen is 55%clay, 32% silt, and 13% sand (Michaelson et al. 2008). The largerpattern at Green Cabin has a deeper active layer around 70 cm, slowerfreeze-up, slightly deeper snow, and higher silt content. The soilcomposition is 11% clay, 35% silt and 55% sand (Michaelson et al.,2008). The smaller pattern in Figure 1a has distinct cracking bothwithin and in between the polygons that has been suggested aspossibly the original cause of the pattern. However, a cracking theorysimilar to that for much larger ice-wedge polygons (Lachenbruch,1961) has not been developed that predicts a patterning on thislength scale. There are distinct cracks of a smaller scale (several cm)within a single polygon that are likely due to desiccation, but the non-sorted polygon has a much larger dimension (tens of centimeters).Therefore there must be some process other than the cracking that isleading to a pattern of a larger scale.

It is being suggested here that DFH initially forms both the largerpatterns at Green Cabin and smaller patterns at Isachsen, and isotherwise independent of the cracking phenomenon.Althoughcrackingstill occurs, it does not lead to the non-sorted circles observed at theselocations. In support of this, it will be demonstrated that a predictivemodel for DFH indicates that the observed pattern size is predicted atboth sites.

Frost heave

Different soil types affect frost heave through their different un-frozen water content behaviors. Frost heave refers to the uplifting ofthe ground surface owing to the freezing of water within the soil. Itstypical magnitude exceeds themere expansion of water upon freezingdue to freezing of additional water drawn upward from the unfrozensoil below the freezing front. This is referred to as thermally inducedwater migration, and it is also responsible for some fractured andbrecciated near-surface rock (Murton et al. 2006). Unfrozen waterexists in the thin region between soil and ice because of bothcurvature and intermolecular forces that combine to lower the freeenergy of a soil/water/ice system below that of a soil/ice system attemperatures below 0 °C (Dash et al, 2006). The presence of saltlowers the bulk freezing temperature and can modify the influence ofintermolecular forces, but an unfrozen liquid film remains (Wet-ettlaufer, 1999). The effects of salt are not investigated here.

Unfrozen water as a function of subfreezing temperature can befairly accuratelymeasured in the lab andalso reasonablywell in thefield(Romanovsky and Osterkamp, 2000). The data are then fit to anempirical model, traditionally using two free parameters. Here we use atwo-parameter relationship (p, q) shown below that captures non-linear behavior at both warm and cold temperatures. Utilizing thegeneralized Clapeyron relationship (Rempel, 2007), the unfrozenwatercontent volume fractionW as a function of the subfreezing temperatureT takes the form

T =P0T0ρL

1−Wð ÞpWq

where P0 and T0 are bulk water freezing pressure and temperature, ρis density, and L is latent heat of fusion. Frost heave with no significantoverburden, which is the case where nearly all patterned ground is

observed, is controlled by the high-water content parameter p, so q isheld constant for all soils in this analysis. The analysis methodologyremains the same for high-overburden situations where q is thecontrolling parameter, although the results may be different. Figure 2shows the volumetric unfrozen water content as a function oftemperature below 0 °C for three soil types that represent the rangefound in many arctic regions. Unfrozen water is present below thebulk freezing temperature due to curvature and surface forces, andthe particular behavior is characteristic of each soil type due to theirdifferent chemical composition, particle shape and size distribution.

The general shape of unfrozen water curves when plotted with asemi-log scale is a backwards “S”, and there is often observed hys-teresis between cooling and heating curves, with higher unfrozenwater on the descent. This is primarily the result of capillary effectsthat preclude the intrusion of ice into small pore conduits that mayconnect relatively larger pore spaces, although nucleation kinetics canalso play a role (Kozlowski, 2004). On heating, the larger pore spacesretain their ice content as determined by surface effects only, andcapillary forces do not interfere. Although the heating and coolingcurves do not differ greatly, we will use parameters that are fit towarming data, and therefore somewhat lower unfrozen watercontents. This has been called the freezing point obtained uponmelting (Kozlowski, 2004). The rationale for this choice is that thetemperature fluctuations that occur during natural soil freezing resultin pores being repeatedly frozen and thawed to some degree beforeeventually completely freezing.

Figure 2 shows three curves of unfrozen water content that rep-resent clay, silty clay, and sandy silt soils. These names, or classifications,are approximate only sincemany soils termed silty clay canhave a rangeof freezing behaviors. In order to investigate trends and not restrict theanalysis to exact soils from a particular region, the curves shown haverepresentative p-values of 2.5 (sandy silt), 1.5 (silty clay) and 0.5 (clay).The behavior of a particular soil could be determined approximately byits behavior relative to these benchmarks.

The amount of frost heave that occurs depends onmany factors, butsome of the most important ones are the unfrozen water content, thepermeability (or hydraulic conductivity), the ground thermal con-ditions, and the degree of saturation. The water content andpermeability are independent properties fromamodeling perspective,but are interconnected because a particular soil type determines bothof them. The analysis that follows will examine the behavior of thethree soil types shown in Figure 2with their respective permeabilities.Two different ground thermal conditions will be investigated:constant subzero ground surface temperature, and snow cover witha constant snow surface temperature since the latter has markedly

0 20 40 60 80−80

−60

−40

−20

0

20

Isachsen Green Cabin

days

elev

atio

n [c

m]

Figure 3. Frost heave and frost depth model predictions (solid curves) and themaximum heave at two sites (circles) measured after complete freeze-up of the activelayer.

328 R.A. Peterson / Quaternary Research 75 (2011) 325–333

different behavior than the former. The degree of saturation will beassumed constant at 100% since most arctic soils are near saturationduring autumn freeze, particularly below the top few centimeters(Hinzman et al, 1991). Lower water saturation levels would simplylead to a lower effective permeability, however, those effects on thefrost heave model used here have not been fully examined.

The amount of heave that occurs during downward groundfreezing can be calculated using various frost heave models. Popularmodels include the segregation potential (SP) model (Konrad andMorgenstern, 1981), and coupled heat/mass balance models like thatof O'Neill and Miller (1985) or Gilpin (1980). The SP model iscomputationally easier to solve and often used in engineering analysis,but requires knowledge of the SP parameter which is a function of soiltype, porosity, overburden pressure, pore fluid, freezing rate, andpossibly others (Konrad, 2005). The other models are more difficult tosolve due to discontinuities whenever a new ice lens is formed. Fowlerand Krantz (1994) have developed a continuum version of the O'Neilland Miller model that is relatively straight-forward computationallyyet captures the important behavior, and only requires knowledge offundamental soil properties and thermal conditions that are easilymeasured in either the lab or field. The analysis in this paper is basedon this continuum model for frost heave.

The fundamental mass and energy balances are

∂S∂t + ∇⋅U = −m

ρ

∂ 1−Sð Þ∂t + ∇⋅V =

mρ

−Lm + ρCpdTdt

= k∇2T

where S is volumetric ice content, t is time, U and V are liquid and icevelocity vectors,m is mass freezing rate, and k is thermal conductivity.The liquid velocity is described using Darcy's Law with a permeabilitythat is dependent on ice content

U = − khρWg

� �Wϕ

� �γ∇pW

where g is gravity, kh is unfrozen permeability, ϕ is void fraction, andpw is the liquid water pressure. It is a gradient in the liquid waterpressure that leads to upward water percolation and subsequent ice-lens formation during frost heave.

Figure 3 shows the amount of heave and the frost depth as afunction of time for a silty clay exposed to a constant ground surfacetemperature of−10 °C. Initial conditions for the active layer include asaturated, unfrozen soil at 0 °C with homogenous composition.Freeze-up takes about 3 months for an active layer of 1 m, and lessthan a month for a relatively shallow active layer like that at Isachsen.Total frost heave at both Isachsen and Green Cabin was measured in2005 and reported by Romanovsky et al. (2008). The maximum heaverecorded at Isachsen and Green Cabin is shown by the circles at a timethat corresponds to a freezing depth equal to the active layer recordedat each specific site. Because only maximum heave was manuallyrecorded much later after the autumn freeze, the time required forfreeze-up of the active layer is not available from their data. Thecurves represent the predicted heave with complete saturation andunlimited supply of ground water for lens formation. Less heave mayoccur in practice because of under-saturation before freeze.

The agreement between the model and these data points is decentconsidering the range of unaccounted factors that can influence thetotal amount of heave. The ability of this model to accurately predictthe time-evolution of one-dimensional heave has also been demon-

strated before (Krantz and Adams, 1996) for laboratory frost heaveexperiments.

Pattern formation by DFH

Regularly spaced patterns can arise spontaneously due to in-stabilities in steady-state situations, or one-dimensional transientsituations (Ball, 1999). Because there is no steady-state during soilfreezing of the active layer, the patterning arises due to perturbations inthe otherwise one-dimensional downward freezing situation. Thepropensity for this spontaneous pattern formation that could lead topatterned groundhas beendemonstrated theoretically before (PetersonandKrantz, 2003; Fowler, 2003) and is only briefly reviewedhere. In thecase of differential frost heave, it is the concomitant occurrence ofseveral factors that leads to positive feedback. It is helpful to firstconsider an unstable systemwith fewer parameters to understand howinstability leads to pattern formation.

Consider a liquid water film on the underside of a rigid body with aless-dense fluid below, like air. This is the common situation of watercondensation on the ceiling, and is known as a Rayleigh-Taylorinstability (Rayleigh, 1883). This situation is shown in the top ofFigure 4a. As long as the film remains planar and perpendicular to theforce of gravity, it will remain fixed to the rigid support. However, inresponse to any system perturbations such as vibrations or non-uniform condensation, the liquid surface can become non-planar onthe bottom boundary. A force balance there leads to liquid flowing tothe already slightly thicker regions from thinner regions as shown bythe arrows, which has a positive feedback effect. This is an unstablesituation. Furthermore, it is unconditionally unstable because theperturbation can be infinitesimally small and the situation is stillunstable. This is often called linearly unstable because the mathe-matical equations that describe the system are linearized prior tosolution.

The other important aspect of this system is the fact that there is apreferred size for the two-dimensional waves (or three dimensionaldroplets) to form due to surface tension. Surface tension provides aforce that is inversely proportional to the radius of curvature.Therefore, there is a particular droplet size at which the force due tosurface tension balances the downward force of the growing wave.This is the droplet size that is most often observed, for example, whencondensation occurs on a ceiling.

A somewhat analogous situation is occurring during differentialfrost heave; however, a few more processes are occurring simulta-neously. Figure 4b is a schematic of what occurs during downwardfreezing of the active layer. Frozen soil overlies unfrozen soil, and the

(a)

(b)

Figure 4. Diagram of two linearly unstable systems. (a) A liquid film above a less densefluid attached to a rigid support, such as condensation on the ceiling, and (b) soilundergoing downward freezing and frost heave.

2 1 0.5 0.3 0.15 0.1

wavelength [m]

0 10 20 30 40 50 60 70−10

0

10

20

30

40

50

60

wavenumber [m−1]

initi

al g

row

th r

ate

claysilty−claysandy−silt

Figure 5. Linear stability analysis of downward freezing and frost heave for threedifferent soil types. The growth rate is normalized by the rate of freezing; thecorresponding wavelength, or pattern spacing, is shown on the top axis.

329R.A. Peterson / Quaternary Research 75 (2011) 325–333

interface is moving downward. A slight perturbation to either theground surface or the interface between frozen and unfrozen soilresults in an unstable situation. Consider a slight perturbation to theground surface as shown in Figure 4b. This results in a wavy surfacewith an increased surface area by which heat can be removed,analogous to adding fins to a heat exchanger. The extra heat lossresults in either increased ice lensing at the freezing front andtherefore heave, or an increased depth of freezing. The latter reducesthe thermal gradient and concomitantly the local freezing rate,providing a negative feedback (i.e., stabilizing mechanism), and iswhy the freezing of pure water is not unstable. One-sided freezing ofpure water does not lead to dendrites, but rather progresses as aplane.

It is at this point that the frost heave process is critically differentfrom most solidification processes. Because cryosuction bringsunfrozen water to the freezing front to form ice lenses, the effect ofthis initial perturbation on the suction determines whether more lensice forms, or alternatively the frost depth increases (Fowler andKrantz, 1994). Without cryosuction, only the frost depth increaseswith increased heat flux. The suction is a function of the soil type, andsome soils will react to this increase in heat flux by bringing morewater to the freezing front to form lenses, instead of substantiallyincreasing the frost depth. It is these soils that are unstable becausethe increase in lensing results in further heaving of the groundsurface, and a positive feedback situation arises. Furthermore, theincreased heat flux occurs directly below a peak, so more lens ice isformed below peaks than below troughs, leading to differentiallensing. This differential lensing further perturbs the ground surface,leading to further increases in heat flux.

The soils that display this type of behavior are only a relativelysmall subset of all frost-susceptible silty clays because the character-istics of both unfrozen water content and permeability must fallwithin a specific range of values (Peterson, 2008). Soils without thecorrect characteristics are stable to DFH because small perturbationslead to increased soil freezing instead of differential lensing.Specifically, both the permeability and unfrozen water content ofpartially frozen unstable soils are less sensitive to changes in icecontent than stable soils. The propensity for DFH can be furtherdiminished by the elastic force required to bend the overlying frozen

layer, and small wavelength perturbations are mitigated by a morerigid frozen soil.

There is also a mechanism that determines the pattern size,analogous to the surface tension in droplet formation. The perturba-tion in heat fluxwill be diffused at deeper depths resulting in a smallerreaction at the freezing front to ground-surface perturbations. Just likemore fins on a heat exchanger result in more heat flux, smallerwavelength perturbations result in greater differential frost heave.However, when the wavelength of the perturbation becomes muchsmaller than the depth of freezing, the effect becomes too diffused atdepth and no differential frost heave results. Therefore, there is acritical depth below which differential frost heave will not spontane-ously form. This critical depth is a function of the soil thermal prop-erties but not its cryosuction properties, because there must first be amechanism for differential freezing at the interface to occur. A soilwith DFH-susceptible cryosuction properties will not differentiallyheave without also having a differential thermal gradient. Forexample, the silty clay in Figure 2 is susceptible to spontaneous DFHuntil the frost depth is 10 cm, while the clay is susceptible until adepth of nearly 30 cm (Peterson, 2008). Because of this depthdependence, a larger perturbation in heat flux is required to sustaindifferential heave in silty clay than in clay soil.

The linear stability of frost heave using the continuum modeldiscussed above has been investigated under various environmentalconditions (Peterson and Krantz, 2008) and soil types (Peterson 2008);a characteristic result of this type of analysis is shown in Figure 5 wherethe initial growth rate of a perturbation is plotted as a function of itswavenumber at an average frost depth of 10 cm. The correspondingwavelength is shown on the top axis in units of meters. The mostimportant qualitative aspect of Figure 5 is whether the growth rate ispositive or negative. The perturbation growth rate is normalized usingthe time scale for freezing of the entire active layer, and its exactnumerical value is of little significance when simply determining thestability of a system. The initially predicted growth rates would changequickly in an unstable situation when non-linear effects would rapidlybecome significant as the amplitude increases. These predictions areanalytical rather than numerical, and therefore the stability of aninfinitesimally small perturbation is being determined.

The significant aspect of these results is that some types of soil thatare susceptible to one-dimensional heave are not susceptible tospontaneous initiation of differential frost heave. In Figure 5, thesandy silt is not susceptible to differential heave, and this is a con-sequence of how both the permeability and the unfrozen watercontent behave as a function of temperature. Specifically, the soil does

330 R.A. Peterson / Quaternary Research 75 (2011) 325–333

not generate sufficient additional suction when a spatial perturbationoccurs. Therefore, the perturbation leads to additional downwardfreezing instead of lens ice formation, which is not unstable, muchlike one-sided freezing of pure water discussed earlier. Any initial,random perturbation will decay and the system will return to one-dimensional frost heave.

This type of analysis can provide considerable insight but haslimitations as well. Its major limitation is that it only indicateswhether DFHwill spontaneously initiate but can not indicate how thatperturbationwill grow and eventually manifest itself as a pattern. Thisanalysis identifies the size (i.e., wavelength) of patterns that caninitially form, and it is assumed here that this size does not evolve orchange significantly as the pattern develops. For example, Figure 5indicates that the clay and silty clay are susceptible to DFH. Fur-thermore, 30-cm patterns in clay would initially grow faster thanthose in silty clay, but the opposite is true for patterns smaller thanabout 12 cm. The exact numerical values (i.e., 30 and 12 cm) dependon factors such as the freezing rate, but the relative trend (i.e., largerpatterns growing faster in clay initially) remains the same. However,this analysis only provides information about the initial growth ofinfinitesimally small perturbations. Once a tiny perturbation begins togrow, the linear stability results become less indicative of how thesystem will evolve, which is a general characteristic of all linearstability analyses (Drazin and Reid, 1981).

The next step is therefore to solve the multi-dimensional problemnumerically. A small perturbation of a single wavelength is added tothe flat top surface. The initial amplitude of these small perturbationsare 1 mm in the results discussed here. It was verified that the sametrends are observed with smaller (0.1 mm) initial perturbations andso the initial amplitude is small enough that non-linear effects are notyet significant at the beginning. The evolution of the pattern can thenbe calculated as the freezing process progresses. The results of thistype of non-linear numerical analysis are shown in Figure 6, wherethe amplitude of the perturbation is plotted as a function of thefreezing depth. The calculations were carried out for freezing depthsup to 1.0 m for a silty clay, starting with an initial 1-mm perturbation.Each curve represents an initial perturbation of a different wave-length. The growth rate varies as the depth of freezing progresses. Forsmall patterns such as 30 cm, the growth rate decreases and even-tually becomes negative around a depth of 70 cm, at which point thepattern begins to shrink in amplitude. In practice, these patternswould never be observed at the end of a freezing season. In general,smaller patterns grow faster initially, but then slow down before the

0 0.2 0.4 0.6 0.8 11

2

3

4

frozen depth [m]

diffe

rent

ial h

eave

[mm

]

30 cm50 cm75 cm100 cm200 cm300 cm

Figure 6. Amount of differential frost heave as a function of the depth of freezing forpatterns of different sizes. The most likely pattern observed at a given site will be thatwhich grows the most when the frost depth is equal to the active layer depth at thatsite.

larger patterns. The 3-m pattern is still growing at its fastest rate at anactive layer depth of 1 m.

These type of predictions need to be comparedwith differential frostheave observed in the field, or in controlled laboratory experiments.Specifically, the amount of heave (one-dimensional and differential)would be measured continuously (e.g., daily) throughout freeze-up.This ismore feasible in a laboratory than in the field because extraneousfactors can be controlled or mitigated. However, a natural area wherepatterned ground exists could be homogenized by tilling and reconso-lidated, and then the surface heavemeasured in a gridded network overseveral seasons to observe any pattern initiation and formation.Continuous data at the sites discussed here are not currently availabledue the challenges of field measurements in these remote regions, andonly net heave for an entire year has been recorded.

Several of these curves in Figure 6 intersect at different freezingdepths, which indicates that the pattern that grows the most is afunction of the active layer depth. The implication of this is that givena spectrum of white noise perturbations, one wavelength, or narrowrange of wavelengths, will emerge over time, and that wavelength is afunction of the active layer depth. This constitutes one mechanismthat could initiate the regularity in the spacing of patterned ground.Figure 7 shows the pattern size that grows most significantly as afunction of the active layer depth for both the clay and the silty clay.There is no curve for the silty sand because perturbations in those soilsdo not result in spontaneous differential frost heave.

Implications and discussion

The curves shown in Figure 7 indicate the pattern size that has thelargest amplitude after complete freeze of the active layer. Thesevalues are often significantly different than the size predicted by thelinear analysis because the full two dimensional equations have beensolved and non-linear terms are taken into account. This often occursin both hydrodynamic systems (Drazin and Reid, 1981) and biologicalsystems (Murray, 1989). Although the curves in Figure 7 are distinct,there is not a substantial difference in the predicted pattern sizebetween the two soil types. This means that the pattern size observedat a site is only a weak function of the soil type if DFH is the sole causeof patterning. It would be very difficult to predict the type of soil basedsolely on the pattern size (for a given active layer depth). Thesignificantly higher clay content at Isachsen (55%) compared to GreenCabin (11%) does not appear to be responsible for the substantiallysmaller pattern size. Although the soil type strongly determineswhether patterning will occur, it does not significantly affect the sizeof the pattern that will ultimately emerge.

0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

1.2

active layer depth [m]

patte

rn s

ize

[m]

claysilty−clay

Figure 7. Themost likely pattern size observed as a function of the active layer depth forthe two soil types that can spontaneously form patterned ground by differential frostheave.

331R.A. Peterson / Quaternary Research 75 (2011) 325–333

The depth of the active layer has a more significant effect on thepattern size; in general, the pattern width on the surface scales withthe local active layer depth. This prediction is observed at both thesites shown in Figure 1. The active layer at Isachsen (Fig. 1a) isreported in the range of 30 cm (Michaelson et al. 2008; Walker et al.2008), while the active layer at Green Cabin is reported in the range of70–80 cm. The patterns at each site are comparable to these activelayer depths. Exact numerical agreement is not expected since anaccurate unfrozen water curve is not available for these exactlocations. Additionally, there are other factors remaining to beaccounted for, such as the ground surface thermal conditions thatare discussed next. However, the predicted trend does agree with thefield observations.

The predictions of Figure 7 were generated assuming a constanttemperature at the ground surface of−10 °C. While this is a reasonableorder-of-magnitude estimate during active layer freeze, it may bemoreaccurate to model the system with a layer of snow and match the heatflux between the snow and the soil. The significantly different thermalconductivities between soil and snow have a significant effect on thestability of frost heave. The depth of snow primarily affects the rate offreezing, and it also has a strong influence on pattern size because thesnow acts to further diffuse the spatially-variable temperature gradient.Figure 8 shows quantitatively the predicted amount of differential frostheave for several pattern sizes as a function of the active layerdepth. Theoverall trend is similar to the constant temperature case, with smallerpatterns growing faster initiallywhile also reaching amaximum sooner.The depth of snow is 10 cmfor all curves shown,which is a typical depthat both Isachsen and Green Cabin. Total snow depthmeasured in springat both locations has been recorded at about 20 cm. Unfortunately, thelogistical difficulty in obtaining continuous or even a few early wintermeasurements at these remote locations is such that these data are notavailable. Clearly it is the snow depth during early winter freeze that isnecessary in this analysis, however somereasonable assumptions canbemade while recognizing that they should be re-examined when suchdata are available. Therefore, although the exact snow depth duringfreeze-up is not known, it is reasonable to assume it is somewherebetween zero and 20 cm; 10 cm is used here.

A trend is shown in Figure 8 that could possibly have significantimplications. For the entire range of pattern sizes investigated from10 cm to 3 m, the smallest pattern growsmost significantly for all activelayers depths up to 60 cm. This occurs when a snow cover is presentbecause the insulating effects mitigate otherwise enhanced heat fluxeffects at all depths except at relatively shallow ones. This is in contrastto thepredicted trendwhen the ground surface temperature is constant,and the pattern size increases monotonically with active layer depth

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

frozen depth [m]

Diff

eren

tial H

eave

[mm

]

10 cm30 cm50 cm100 cm200 cm300 cm

Figure 8. Amount of differential frost heave as a function of the depth of freezing forpatterns on different sizes. The thermal ground conditions include a 10-cm depth ofsnow.

(Fig. 7). The vertical arrow in Figure 8 indicates the depth at which thesmallest pattern no longer grows more than larger patterns. At thispoint, the smallest pattern is overtaken by the largest pattern. At nopoint do intermediate pattern sizes demonstrate the most overalldifferential heave.

This discontinuity is clearly evident when plotting the pattern sizethat grows the most as a function of the active layer as shown inFigure 9. The solid curve shows the behavior with a 10-cm snowdepth, and there is an obvious discontinuity around 60 cm. As theaverage snow depth is increased to 15 cm, there is only a small rangeof active layer depths around 50 cm with small patterns. As the snowdepth is further increased to 20 cm, the discontinuity disappears and amore continuous trend emerges, similar to the constant temperaturepredictions. This sharp discontinuity is striking and deserves someattempt at explanation. The behavior is somewhat analogous to a footrace involving sprinters, mid- and long-distance runners. Wheneveryone is running fast (fast freezing due to shallow snow), sprinters(small patterns) win at short race distances (shallow active layer), butcannot maintain the speed at longer distances where endurancerunners excel. The mid-distance racers do not win until the averagespeed of everyone is slower (thick snow cover) and the racers aremore spread out, providing them with a middle distance at whichthey excel. This analogy can be carried over and observed in Figure 8where the sprinters begin to lose to distance runners at the arrowaround 0.6 m, while mid-distance runners never win overall.

These results appear to indicate that a relatively shallow (~10 cm)but finite snow depth during freeze-up can play a significant role indetermining the pattern size that is observed at a particular location,and the maximum depth of freeze further determines whetherrelatively small or larger patterns emerge over time. This leads to apossible explanation for the markedly different pattern size betweenIsachsen and Green Cabin. A somewhat more shallow depth of freezecombined with a slightly thinner snow cover compound to yield amarkedly smaller pattern at Isachsen. The behavior demonstrated inFigure 9 can be partially explained in terms of the heat flux effectsdiscussed earlier. Surface perturbations (peaks) are like heat finsincreasing the overall heat transfer. A snow cover reduces overall heattransfer and therefore more heat fins would be required to maintainthe positive feedback effect that results in DFH. When a snow cover ispresent, the smallest pattern maintains an advantage in terms ofgrowth because it has the greatest perturbation in overall heat flux.However, as the depth of freeze increases further, this advantage isdiminished because the closely spaced heat fins lose their discrete

0.1 0.3 0.5 0.7 0.9

1

2

3

4

active layer depth [m]

patte

rn s

ize

[m]

10 cm15 cm20 cm

Figure 9. Themost likely pattern size observed as a function of the active layer depth forthree different depths of snow. The discontinuity at 60-cm depth with 10 cm of snowindicates that themost likely pattern observed can change dramatically when the activelayer changes only a small amount.

332 R.A. Peterson / Quaternary Research 75 (2011) 325–333

contributions due to thermal diffusion. For the silty clay soil shown inFigure 9, this occurs around 0.5–0.6 m.

The potential broader implication of these predictions can bespeculated by considering the following scenario. Due to changingclimatic conditions, an area that previously had no patterned groundmay become susceptible to differential frost heave. This might occurdue to increasing water availability, changes in the time and durationof freeze-up, or changes in vegetation that indirectly affect thethermal conditions on the ground. For an area that has relativelyshallow snow during this period, the pattern size is drasticallyaffected by the active layer depth, and a slight difference between say50 and 70 cm can result in a pattern that is 10–30 times different insize.

It should be made clear that the DFH process modeled here onlypartially describes the entire yearly freeze/thaw cycle. Most impor-tantly, the processes involved during thaw and consolidation are notdescribed. The lateral movement of soil due to peak/trough formationduring DFH is only completely reversible uponmelt of the ice lenses inthe absence of any other physical processes. However, there are otherprocesses that lead tomass displacement of soil, although they are notcompletely understood and predictable. Corte (1972) observedsurface perturbations in an initially uniform frost-susceptible soilexposed to repeated freeze/thaw events in the laboratory. Mackay(1980) discusses the potential for both buoyancy-driven movementand grain-size driven circulation of soil during thaw of ice-rich soils inhis “equilibrium model” of hummocks. According to Van Vliet-Lanoë(1991), desiccation and hardening of the upper-most layer into a“carapace” during thaw can occur when binding agents like organicsand amorphous clay are present, aiding subsequent fine-soil accu-mulation below the carapace with subterranean water movement.Peterson and Krantz (2008) presented laboratory experiments of soilheaving due to cyclic freeze/thaw resulting in equally sized surfaceexpressions. Therefore, there is evidence that thaw-related processesexist that can provide the lateral soil movement necessary foreventual pattern formation.

The cryoturbation associated with patterned ground and differ-ential frost heave has been shown to be a major factor in sequestering

Figure 10. Panchromatic remote sensing image from Quickbird of Howe Island, Alaska, showand 2-m spacing. The image is viewed using GoogleEarth.

organic carbon in arctic soils (Horwath et al., 2008; Ping, et al., 2008).Organic carbon that originates on the surface is brought to depthwhere it can be frozen into an intermediate, or transition zone directlyabove the permafrost (Shur et al., 2005). Cryoturbation associatedwith larger patterns would likely lead to deeper sequestration andexhumation than smaller patterns. Patterned ground regions can be acarbon sink by this mechanism, but also a source when climaticwarming occurs and the transition region thaws. Because theexistence of patterned ground plays a factor in the amount of carbonsequestration that can occur, it should perhaps be considered in somelonger term climatic change models and assessments. Althoughoverall changes in soil freezing depth and snow cover are now oftenincluded, the spontaneous occurrence of patterned ground where itdid not occur before is not taken into account.

There has been some effort toward investigating what possibleclimatic, biological, and hydrological information might possibly bededuced from simple surface-pattern size measurements (Walkeret al., 2008). A major motivation for this effort is that ground-surfacepattern formations in remote regions such as the Arctic are relativelyeasily determined by remote-sensing techniques. An excellentexample of how readily available and accessible these data can be isshown in Figure 10, which is a panchromatic visual image from theQuickbird remote-sensing platform viewed using GoogleEarth. Theimage shows a plan view of Howe Island just off the northern Alaskacoast. Ice-wedge polygons about 25 m across are easily visible, as arethe non-sorted circles that appear as white dots. Because thesepatterns are fairly clear, an estimate of their size, number density, anddistribution is easily determined by automated pattern-recognitionsoftware. Here, the average pattern spacing is approximately 2 m.

Conclusions

Because much of the Arctic is remote and difficult to access, it isobviously desirable to use remote-sensing data when possible todetermine aspects of the subsurface thermal regime and soilcharacteristics that influence frost heaving. There is perhaps evengreater impetus for extraplanetary investigations, and features on

ing both ice wedge polygons of about 20-m size, and non-sorted circles of about 1 m size

333R.A. Peterson / Quaternary Research 75 (2011) 325–333

Mars are providing some information about the soil conditions there(Levy et al., 2008; Soare et al., 2008). Three possibilities that have beeninvestigated here are determining soil composition, active layerdepth, and ground thermal conditions from the observed pattern size.With regards to soil composition, results indicate it would be fairlydifficult to discriminate between a clay and an silty clay based on theobserved pattern, as shown by the proximity of the two curves inFigure 7. The best conclusion that may be drawn is that the absence ofpatterning may indicate a more sandy material that is not susceptibleto differential frost heave.

There is a fairly robust trend between active layer depth andpattern size in Figure 7. To a first-order approximation, the patternsize is on the order of the active layer depth. This prediction issupported by comparing the average pattern size at Isachsen andGreen Cabin shown in Figure 1. When a thin snow cover exists duringfreeze-up, there is the possibility that the pattern size is greatlyaffected by the active layer depth, and the resulting pattern can varyby a factor of 10 or more. This effect diminishes with increasing snow.This may also have played a role in the significantly differentpatterned ground at the two sites. This prediction also indicates thatpattern size is not always a direct indicator of the active layer depth,and ground surface thermal conditions must be considered. Theresults discussed here do not disprove alternative theories, such asinitial contraction or desiccation cracking, and further developmentsin that area may prove that they are also viable explanations. DFH canalso result after other phenomenon have already initiated a pattern,and determining which was first can probably only be conclusivelydetermined by experiment. Controlled field and laboratory experi-ments will ultimately help support one or more of these theories.

The broader implication of the fact that differential frost heave candevelop spontaneously is the possibility that new areas of patternedground may appear due to a subtle change in climate, either a modestwarming or an increase in snow cover. The cryoturbation associatedwith patterned ground such as non-sorted circles will then have asignificant effect on sequestration and exhumation of carbon andother nutrients. The depth at which this occurs is closely related to thesize of the pattern, which is, as has been presented here, determinedby differential frost heave. The effect that patterned ground has onsequestering organic carbon near the permafrost table should betaken into account when attempting to accurately quantify the effectsof climate change.

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