Journal of Glaciology , Vol. 49, No.164, 2003

Amechanism fordifferential hea ve and itsimplications forpatterned-ground formation

Rorik A.PETERSON, 1* William B. KRANTZ2{ 1Department of Chemical Engineering,University of Colorado,Boulder,Colorado 80309-0424,U.S.A. E-mail:[email protected] 2Institute of and Alpine Research, University of Colorado,CB-450,Boulder,Colorado 80309-0450,U.S.A.

ABSTRACT.Thegenesis ofsome types ofpatterned ground,including hummocks, frost boilsand sorted stone circles, hasbeen attributed todifferential frost heave(DFH) . However,atheoreticalmodel that adequatelydescribes DFH hasyetto be developedand validated.In this paper,wepresent amathematicalmodel for the initiationof DFH, and discuss howvariations in physical (i.e. /vegetationproperties) andenvironmental (i.e. ground/airtemperatures) properties affectits occurrence andlength scale. Using the Fowlerand Krantz multidimensionalfrost-heave equations, a linearstability analysis andaquasi-steady-state real-time analysisare performed. Results indicatethat the follow- ingconditions positively affect the spontaneousinitiation of DFH: silty soil,small Young’s modulus,small non-uniformsurface heattransfer orcolduniform surface temperatures, andsmall freezingdepths. Theinitiating mechanism forDFH ismultidimensionalheat transfer withinthe freezingsoil. N umericalintegration of the lineargrowth rates indi- cates that expressionof surface patterns canbecome evident on the10^100yeartime-scale.

1. NOTATION p Exponentin the soilcharacteristic function P Totalpressure Cp Heat capacity Pb Pressure dueto bending CT² Constantin the perturbed temperature-gradient p Pressure inthe farfield (e.g .basalplane) 1 solution q Exponentin the soilcharacteristic function CT± Constantin the perturbed temperature-gradient t Time solution T Temperature d0 Lengthscale corresponding to the maximumdepth T0 Dimensionless reference temperature inN ewton’s of freezing law-of-coolingboundary condition E Young’smodulus Tair Dimensionless airtemperature e1;2 Groupof constants inthe definitionof N 0 Tb Temperature atthe basalreference plane, zb p 1 À Tf;u Temperature ofthe frozen/unfrozensoil fl … ¡ † Àq Ts Temperature atthe groundsurface g Gravitationalconstant ¢T Temperature scaletypical for the freezingprocess V Gf;i Temperature gradientat zf f Freezingfront velocity h Dimensionalheat-transfer coefficient vi Icevelocity H Dimensionless heat-transfer coefficient w Deflection ofthe platemid-plane in thin-plate theory h Thicknessof the frozenregion z z Wl Unfrozenwater fraction of the totalsoil at 1 … s ¡ f † k Overallthermal conductivity the lowestice lens W k0 Unfrozensoil hydraulic conductivity l0 Derivativeof the unfrozenwater content inthe z kf;u Thermalconductivity of frozen/unfrozensoil downwarddirection at l ® x; y Horizontalspace coordinates Wl k k z Vertical spacecoordinate h 0 ¿ ³ ´ zb Basalplane, tablein the field L Heat offusionfor water zf Locationof the freezingfront P p z Locationof the lowestice lens N ¡ 1 l ¼ zs Locationof the groundsurface ¬ Dimensionless wavenumberin the x direction ¬ ­~ 1 ¯ · ¿ W 1 ­~ fk … ‡ † ‡ … ¡ l† ‡ f N * ~ l Present address: Institute forArctic Biology,Universityof ­ ­ l ¡ ¡ ¢ W f ¯® f N AlaskaF airbanks,F airbanks,Alaska 99709- 7000,U .S.A. µ¡ l l0 ¡ … l ¡ †¶ ®» L2k { i h Present address:DepartmentofChemical Engineering ,Uni- ­ l gkT versityof Cincinnati, Cincinna ti, Ohio4522 1-0171,U.S.A. 0 69 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Journal of Glaciology

»i amongothers. refers tosurface features made ¯ 1 prominentby the segregationof stones, orderedvariations ¡ »w ³ ´ inground cover or color ,orregulartopography .Sincepat- ® Exponentfor hydraulic-conductivity function terned groundis amanifestationof self-organizationin nat- ¡ Exponentialgrowth rate accordingto linear theory ure, its formationis aquestionof fundamentalsignificance. ² Perturbationin the freezingfront location Inthis paper,wepresent amathematicalmodel for a mech- ¶ Wavelengthof the perturbations anismby which DFH canoccur and may cause the initia- ¶ Constantin the Gilpinthermal regelationtheory G tionof some types ofpatterned ground.Using this model,a ¶ L» · G w limited parameter spaceof environmental (i.e. tempera- ku ture, snowcover) and physical (i.e. soilporosity ,hydraulic ¸ Poisson’sratio conductivity)conditions is definedwithin which DFH can »f Frozensoil density initiatespontaneously . »i;w Ice/watermass density ¼ Pressure scaleof 1bar ¿ Unfrozensoil porosity 3.PRIOR STUDIES À Unfrozen-waterfraction of the voidspace W =¿ ˆ l Á Ratio of ²=± Semi-empirical modelscorrelate observable characteristics ± Perturbationin the ground-surfacelocation ofthe SFHprocess withthe relevantsoil properties and climatic variables.The more prevalentmodels include the 2.INTR ODUCTION segregation potential (SP)model ofK onradand Morgenstern (1980,1981,1982)and that ofChen and Wang( 1991).Typically, Frost heave refers tothe upliftingof the groundsurface owing semi-empirical modelsare calibrated at a limited number of tofreezing of water within the soil.I tstypicalmagnitude sites underunique conditions. Since these uniqueconditions exceeds that owingto the mere expansionof water upon areaccounted for by using empirical constants, it canbe dif- freezing ( 9%)becauseof freezing additional water drawn ficult togeneralize the modelsfor use withvarying environ- ¹ upwardfrom the unfrozensoil below the freezingfront. The mentalconditions and field sites. Semi-empirical modelsare process ofdrawing water through a soilmatrix towards a easyto use, computationallysimple, andprovide fairly freezingdomain is knownas cryostatic suction . Primary frost accuratepredictions of SFH (Hayhoe and Balchin, 1 990; heave ischaracterizedby a sharpinterface betweena frozen Konradand others, 1998).However,theyprovide limited regionand an unfrozen region. In contrast, secondary frost insightinto the underlyingphysics and cannot predict DFH. heave (SFH) is characterizedby a thin,partially frozen Fullypredictive models arederived from the fundamen- regionseparating the completelyfrozen region from the taltransport andthermodynamic equations. In theory ,they unfrozenregion. Within this region,termed the frozen fringe , require onlysoil properties andboundary conditions in discrete ice lenses canform if there issufficient ice present to orderto be solved.Based on ideasdue to Gilpin ( 1980)and supportthe overburdenpressure. Thetheory of primaryfrost Hopke( 1980),Miller (1977,1978)proposed a detailedmodel heavecannot account for the formationof multiple, discrete forSFH. This model was later simplified byF owler( 1989), ice lenses. Miller (1980)states thatice-lens formationprob- extendedto include DFH byF owlerand Krantz (1994)and ablyonly occurs bythe primaryheave mechanism inhighly extendedfurther toinclude the effects ofsolutes andcom- colloidalsoils withnegligible soil particle-to-particle contact. pressible byN oon( 1996).Nakano( 1990,1 999)intro- Frost heavethat islaterallynon-uniform is referred toas duceda modelcalled M1 forfrost heaveusing a similar differentialfrost heave .Differential frost heave(DFH) requires approach.F ullypredictive models donot require calibra- that the freezingbe both significant and slow .Adequate tionand can be appliedto anywhere the relevantsoil prop- upwardwater flow by cryostatic suction requires either erties andthermal conditionsare known. Thesemodels also saturatedsoil conditions or averyhigh water table because provideinsight into the underlyingphysics involved. These unsaturatedsoil conditions can suppress the heavingpro- attributes facilitatemodel improvements andextensions. cess. Suppressionoccurs dueto insufficient interstitial ice, Due tothe complexityof some models,they can be compu- whichis required tosupport the overburdenpressure and tationallydifficult and inconvenient to use. allowfor ice-lens formation(Miller ,1977). Exceptunder asymptotic conditions, these models must TheDFH process is notfully understood. The complex besolved numerically .One-dimensional,numerical solu- interactionsthat occurin freezing soil have not been fully tions tothe modelby Miller (1977,1978)havebeen presented determined ordescribed andthere areas yet no predictive byO’ Neill andMiller (1982,1 985),Blackand Miller (1985), modelsfor this phenomenon.H owever,secondaryfrost heave, Fowlerand N oon( 1993),Black( 1995)andKrantz and whichis oftenassociated with DFH, hasreceived a certain Adams( 1996).Inseveral instances, numericalresults agree amountof attention from the scientific community.Several favorablywith laboratory results such asthose ofKonrad mathematicalmodels and empirical correlations have been (1989).Thepredictive capability of the M1 modelhas also developedto describe one-dimensionalSFH. Most ofthe beendemonstrated (Nakanoand T akeda,1991,1994). mathematicalmodels represent variationsand refinements Lewis (1993)and Lewis andothers (1993)were the first ofthe originalone-dimensional SFH model of O’Neilland touse alinearstability analysis (LSA) in an attempt topre- Miller (1985).Aset ofdescribing equations for three-dimen- dict the occurrence ofDFH. Usingthe modelequations de- sionalSFH based on the O’Neilland Miller modelhas been velopedby F owlerand Krantz (1994),theyfound that DFH developedby F owlerand Krantz (1994).Here these equa- occurs spontaneouslyunder a widerange of environmental tionsare used asabasisfor investigating the DFH process. conditions.Lewis (1993)includedthe necessary rheologyby DFH hasbeencited asapossiblecause for some forms of modelingthe frozenregion as anelastic materialand using patterned groundby T aber( 1929)and W ashburn( 1980) thin-platetheory .However,this ``elastic condition’’was 70 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Peterson and Krantz: Differentialfrost heave and patterned-ground formation

soil.The thickness ofthe frozenfringe in seasonally frozen groundis onthe orderof 1^1 0mm. Thefrozen region is characterizedby regular bands of pureice calledice lenses. Itisthe existence ofice lenses that givesrise tosurface heave greaterthan the 9%associatedwith the volumechange of waterupon freezing (W ashburn,1980).Intypical field situ- ationswhere frost heaveoccurs, permafrost underlies the unfrozenregion (Williams andSmith, 1989). Becausethe frozenfringe is relativelythin, it canbe reducedmathematically to a planeacross whichjump bound- aryconditions are applied (F owlerand Krantz, 1994).This planeis locatedat zf inFigure 1 .Freezingof saturated soils is fairlyslow ,ascanbe inferred fromthe largeStefan number 1 1 (St LC¡ ¢T ¡ )associatedwith this process. Becausethe ² p time-scale forfreezing is muchlonger than that forheat con- Fig.1.Diagram ofsaturated soil undergoingtop-downfreezing. duction,the temperature distributions inthe unfrozenand The region between zl and zf is the frozen fringe.Ice lenses frozenregions are quasi-steady state. Thethermal gradients occur in the frozen region, giving rise to frost heave. inthe unfrozenand frozen region are obtained by solving for the temperature distributions ineach region. Becauseice lenses occurin a discrete manner,the frost- incorrectlyintegrated into the analysis,giving unconven- heaveprocess is inherentlya discontinuousprocess. How- tionalresults that predicted asingleunstable wavenumber . ever,Fowlerand Krantz (1994)demonstrate usingscaling Independently,Noon(1996)andF owlerand N oon( 1997) argumentsthat the time-scales forequilibration of the pres- reported results fora LSAusing the Fowlerand Krantz sure andtemperature profileswithin the frozenfringe are modelwith a purelyviscous rheology .Noon’sresults indi- much shorter thanthe time-scale forthe freezingprocess. catedthat one-dimensionalheave is linearlyunstable using Thus,the entire frost-heaveprocess canbe modeled as twodifferent ground-surfacetemperature conditions:con- semi-continuous wheresome parameters areevaluated at stant temperature andsnow cover .However,the constant- the baseof the warmest (lowest) , zl. temperature results arenot physical due to unrealistic Forone-dimensionalfrost heave,the surfaces zs and zf parameter choices.The remaining condition necessary for areparallel planes. The frozen region including ice lenses instability,the presence ofsnow cover ,appearstoo restric- movesupwards relative to zb butdoes not deform. In this tive.Patterned ground is observedboth in the presence and instance,the rheologyof the frozenmaterial is irrelevant. inthe absenceof snow cover (Williams andSmith, 1989). InDFH, howeverthe frozenregion deforms asit moves, Furthermore, since onlya singlesoil was investigated, the andthe rheologyof the materialmust bespecified sothe influenceof soiltype on the predictionsis difficultto evalu- mechanicalproblem can be solvedin conjunction with the ate.In light of these points,further investigationof the linear thermal andfrost-heave problems. stabilityof DFH (this study) waswarranted. 4.2.Thermalproblem 4.THEDFH MODEL Due tothe largeStefan number forfreezing soil, the Themodel for SFH that weuse inthis paperis that dueto thermal problemsin boththe frozenand unfrozen regions Fowlerand Krantz (1994),whichis amodifiedversion of the aboveand belowthe frozenfringe, respectively ,aregreatly O’Neilland Miller model.Wechoseto use this modelbecause simplified.The two-dimensional, steady-state temperature it containsall the necessaryphysics to describe DFH while distributions inboth regions are obtained from the solution maintaininga numericallytractable form (F owlerand N oon, tothe Laplaceequation: 1993).Most importantly,this model properly takesinto account 2T 0 z < z < z 1 thermal regelationin the partiallyfrozen region termed the r u ˆ b f … † frozen fringe .Thermalregelation within the fringeallows for 2T 0 z < z < z : 2 r f ˆ f s … † differentialheaving .Theoriginal O’ Neilland Miller (1982, Theboundary condition at the groundsurface is Newton’s 1985)modelused arigid-iceapproximation that assumes all lawof cooling .Forboth problems, the boundarycondition the ice withinthe frozenfringe and lowest ice lens movesas at z specifies the temperature asthe standardtemperature auniformbody with a constantvelocity .Inorder for differ- f andpressure (STP)freezingpoint of water.Forthis paper, entialheave to occur ,this cannotbe the case.F owlerand the temperature at z willalso be specified. Krantz (1994)demonstrate that thermal regelationwithin b the fringeallows for DFH. Readers shouldrefer toF owler @Tf H Tf Tair at z zs 3 andKrantz (1994)and Krantz andAdams ( 1996)fordetails. @n ˆ ¡ … ¡ † ˆ … † T 0 at z z 4 f ˆ ˆ f … † 4.1.Physical description T 0 at z z 5 u ˆ ˆ f … † T T at z z ; 6 Thephysical system inwhich frost heavingoccurs isshown u ˆ b ˆ b … † inFigure 1 .Top-downfreezing occurs overa time-scale of where H is aheat-transfer coefficient, Tair is the dimensionless weeksto months. Theupper ,frozenregion is separatedfrom airtemperature atthe groundsurface and Tb isthe dimension- the lower,unfrozenregion by the frozenfringe, a partially less temperature ofthe underlyingpermafrost. Inthe absence frozentransition region containing ice, liquidwater and ofsignificant solutes, T 0,correspondingto 0³ C. b ˆ 71 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Journal of Glaciology

4.3.Mechanical problem

Thechoice of an elastic rheologycan introduce several con- ceptualand modeling difficulties. Toaddress these difficul- ties, weadopt several assumptions aboutthe geometryof the frozenregion as wellas its mechanicalproperties. The frozenfringe is, mathematically,reducedto a planeacross whichjump boundary conditions are applied (F owlerand Krantz,1994).Thisplane is the lowerboundary of the frozen region, zf .However,becauseit willbe perturbed anddis- torted, albeitinfinitesimally ,andwill no longer be a plane Fig.2.Diagram of the frozen fringe showing the initiation of ina strict mathematicalcontext, we refer toit asaninterface. DFH.Relative dimensions are not to scale.The dashed line in Figure2 showsa schematic ofthe frozenfringe before the left figure shows the location of zf which is defined where andafter inceptionof DFH dueto initiation of a newice the ice content iszero.Anew ice lens forms within the water- lens. Infact, the locationof the newlyforming ice lens is pressure boundary layer below the lowest ice lens.After initi- veryclose to the baseof the lowestice lens relativeto the ation of anew ice lens,the location of zf willmove either thickness ofthe frozenfringe. There is aboundarylayer in upwards or downwards relative to its previous position.The

the frozenfringe near the lowestice lens whereinmost ofthe twopossibilities areshown by zf0 and zf00.The icecrystals labeled pressure dropcausing upward permeation occurs. Thenew ``possible ice’’willexist for zf0 and not for zf00.The newlocation of ice lens willform within this boundarylayer (F owlerand zf is predicted by the LSAand is not prescribed in the model. Krantz,1994). Thedashed line labeled z is the bottomof the frozen f approachis convenientbecause it avoidssome inherent com- fringe,defined as wherethe ice contentis zero.At the incep- plexity,andthere is considerablejustification for its use. tionof DFH ,the locationof z canmove either upor down. f Experimentaldata for Y oung’smodulusof frozen soil as a Becauseof the quasi-steady-state assumption,any movement functionof temperature showthat the modulusapproaches of z isrelativeto the stationary,one-dimensionalboundary . f zeroas the sub-freezingtemperature approacheszero Although z isalwaysmoving downwards on the time-scale f (Tsytovich,1975,p. 185).Largeamounts of unfrozenwater at offreezing-front penetration, quasi-steady state must be 0³C arethe causeof this ``weakness’’.Thus,we would expect assumed inorder to conduct the LSA.While this is an the newlyaccreted materialto have a verylow Y oung’s abstractionfrom reality ,it isanexpedientand useful prac- modulus,so that neglectingit completelyintroduces only a tice forLSA of systems withunsteady basic states (Fowler, small error.Infact, as afirst approximationthe entire frozen 1997).Twopossible locations for zf areshown in Figure 2 regionis assumed tohave a constantY oung’smodulus.This (right side):onefor upward movement and one for down- approximationis crude since there isanon-zerotemperature wardmovement. Which ofthese twopossible outcomes actu- gradientwithin the frozenregion. In reality ,the Young’s allyoccurs isnotprescribed inthe modelbut is aresult ofthe modulusmonotonically increases withdecreasing tempera- LSA.However ,lineartheory does prescribe that z is either f ture andhas a maximumvalue at the groundsurface where inphase or anti-phasewith the ground-surfaceperturbation, it iscoldest. z .Anyrelative displacement otherthan 0 or º is not s Due tothe simplificationsdiscussed above,the mechan- allowed.The location of z inFigure 2 willbe inphase with l icalmodel reduces tothe bendingof athin,elastic platewith the groundsurface at z since the lowestice lens precludes s aconstantY oung’smodulus.If the deformationis small watermigration above it. Thus,Figure 2 demonstrates how relativeto the thickness ofthe plate,the deflectionof the z and z canbe either inphase or anti-phasedue to the for- s f plate’smid-planefrom equilibrium, w,canbe approximated mationof a newice lens. as(Brush andAlmroth, 1975): Mechanically,whennon-uniform ice-lens formation begins,the frozenregion will begin to deform and bend. P D 4 w 7 b ˆ rxy … † Wemodelthis asthe bendingof a thin,elastic plate,though E h3 aviscous,viscoelastic or visco-plasticmodel might also be D 1 8 ² 12 1 ¸2 … † used. Laboratoryexperiments byT sytovich( 1975)demon- … ¡ † 4 w w w w 9 strate that frozensoil does behave elastically on short time- rxy ² ;xxxx ‡ ;xxyy ‡ ;yyyy … † scales.Thenecessity forincluding the rheologyof the frozen where Pb is the pressure, E is Young’smodulus, h1 is the materialin the modelis toprovide a ``resistive force’’to frozen-regionthickness and ¸ is Poisson’sratio.In this deformationof the frozenmaterial. As willbe demonstrated analysis, ¸ 0.5. ˆ later,whenthe rheologyis neglected,a LSApredicts that Boundaryconditions for Equation ( 7)must bespecified the most unstablemode has a wavelengthapproaching zero, in the x and y directions. Becausethe problemis unbounded whichis intuitivelyincorrect. inthese directions, periodicboundary conditions are used. Aconceptualdifficulty is encounteredwhen assuming an For the x direction, elastic rheologyin this problemsince frozenmaterial is being w x 0 0 10 accreted.Thisproblem can be avoidedif we neglect the addi- … ˆ † ˆ … † tionof newly frozen material in the geometryof the elastic w x 0 0 no torque 11 ;xx… ˆ † ˆ ¡! … † materialbeing bent. What this means physicallyis that when w x 0 w x nº=2 periodicity 12 anon-uniformice lens beginsto form, only the materialspan- … ˆ † ˆ … ˆ † ¡! … † w;xx x 0 w;xx x nº=2 periodicity : 13 ningfrom the groundsurface, zs tothe lowestice lens, zl, is … ˆ † ˆ … ˆ † ¡! … † actuallybeing bent. In the sense ofthe elastic model,the new- Thereis acorrespondingset forthe y direction(how- lyaccreted materialhas an elastic modulusof zero. This ever,onlythe two-dimensionalproblem will be considered 72 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Peterson and Krantz: Differentialfrost heave and patterned-ground formation

here).Themechanical problem is coupledto the frost-heave the x and y directions, respectively.Becauseone-dimen- problemthrough the loadpressure at zl.Inthe one-dimen- sionalfrost heaveoccurs inthe z direction,perturbations sionalbasic state, the frozenregion is planarand there isno in both the x and y directions causea three-dimensional bending (P 0).Inthe two-dimensionalstate, the non- shapeto evolve. Initially ,however,onlythe propensityfor b ˆ zero Pb is afunctionof x.Theload pressure isthe sum of atwo-dimensionalinstability will be investigatedby setting the weightof the overlyingsoil and the pressure dueto bend- ­ 0.The functional form of X inEquation ( 18)depends ˆ ing. onwhether the problemis steady-state ortransient. b P »f gh1 Pb 14 Althoughthe frost-heaveprocess isinherentlytransient, it ˆ ‡ … † willbe treated asquasi-steady-state. If the instabilities grow veryrapidly relative to the basicstate, this assumptionis jus- 4.4.Non-dimensionalized frost-heave equations tified. Asuccessful exampleusing this techniquefor alloy TheF owlerand Krantz frost-heaveequations are two solidificationis presented inF owler( 1997).Inthis instance, coupledvector equations that describe the velocityof both assumingquasi-steady state yieldsresults somewhatclose to experimentalobservations. Thus, the solutionto the per- the upward-movingice atthe topof the lowestice lens, vi, turbed variablestakes the generalform: andthe downward-movingfreezing front, Vf .Specifically, the equationsare: i¬x ¡t X0 X0 z e e ; 19 k ˆ … † … † V 1 ¯ ¿ W · f G 1 ­~ G f ˆ ‡ ¡ ¡ l k i ¡ ‡ f where ¡ is anexponentialgrowth coefficient. » ³ u´ £ ¡ ¢ ¤ ¡ ¢ . Theperturbed temperature distributionis determined ¯W ¿ ­~ ¿ W · ¿ W W 15 interms ofthe basic-state temperature distribution.First, l ‡ ‡ ¡ l ¡ ¡ l l … † ¼ the solutionto Equation ( 2)subject toboundary conditions ¡ ¢ ¡ ¢ kf (3)and( 4)yields the unperturbedtemperature inthe frozen v ¬ G W V ; 16 i ˆ ¡ fk k i ¡ l f … † region: µ³ u´ ¶ where Gi and Gf arethe thermal gradientsin the unfrozen z zf Tf H Tair ¡ : 20 andfrozen regions, respectively , ¿ is the soilporosity , Wl is ˆ 1 H zs zf … † the unfrozen-watervolume fraction at the topof the frozen ‡ … ¡ † Solutionof Equation( 1)subject toboundary conditions ( 5) fringe(base ofthe lowestice lens),and kf and ku are the and( 6)is trivialwhen T 0. frozenand unfrozen thermal conductivities,respectively . b ˆ Vectors aredenoted in bold, and the coordinatesystem is T 0 21 orientedwith z pointingup as shownin Figure 1 .Thevalue u ˆ … † of Wl is determined usingthe lens-initiation criterion (Fowler Thefrost-heave equations ( 15)and( 16)require the andKrantz, 1994)that involvesthe characteristic function. thermal gradientat zf.Steadystate isalsoassumed inthe p 1 Wl ‡ perturbed regionin anticipation that the time-scale forper- N W 1 ¿ 1 l f ¡ : 17 turbationgrowth is greaterthan the time-scale forheat con- l ± ²q ¿ ˆ ¡ ¿ ˆ Wl … † duction.Starting with the two-dimensionalform of ³ ´ ¿ ± ² Equation( 2),the temperature is perturbed andthe differen- Thesolution to the one-dimensionalproblem is fairly tiationcarried out. Boundary conditions are applied at per- straightforward.Thesteady-state, one-dimensionalLaplace turbed locations,where the Taylorseries istruncated after equationfor temperature is solvedto determine the relevant the linearterm. Theperturbed temperature fieldis solved temperature gradients Gi and Gf.Workby F owlerand andthen differentiatedagain to yield: Noon( 1993)and Krantz andAdams ( 1996)hasshown that this modelpredicts the one-dimensionalfrost-heave beha- @Tf0 G0 C ² C ± ; 22 viorobserved in the laboratoryquite well. @z ² i ˆ T² ‡ T± … † ­z zf ­ˆ ­ where ­ 5.LINEAR STABILITYANAL YSIS e2¬zf e2¬zs CT² ¬Gi … ‡ † 23 LSAis performed here todetermine underwhat circum- ˆ ¡ e2¬zf e2¬zs … † ¬¡zf zs stances the one-dimensionalsolution to the SFHequations 2¬GiH e … ‡ † (15)and( 16)is unstableto infinitesimal perturbations. The CT± : 24 ˆ ¬ H e2¬zf e2¬zs … † conditionsunder which the one-dimensionalsolution is un- ‡ ¡ stable willprovide insight into the mechanisms responsible Thethermal gradientbelow the freezingfront is forDFH. Thisin turn willhelp explain the evolutionof assumed tobe zeroin this analysis.This condition is repre- some types ofpatterned groundthat aredue to DFH. sentativeof active-layer freezing above permafrost. Thus, G G0 0. TheLSA performed here followsstandard linear sta- f ˆ f ˆ bilitytheory (Drazin andR eid,1 981).Acomplete detailed Theperturbed mechanicalproblem begins by perturb- analysisis availablein Peterson (1999).Equations( 15)and ingthe pressure condition,Equation ( 14).Thepressure is (16)areperturbed aroundthe one-dimensional basic state non-dimensionalizedwith the pressure scale ¼. solutionand linearized. Because the equationsare linear in P z P 0 z; x »f gh1 Pb the perturbed variables,the solutionto these perturbed N N0 … † ‡ … † ‡ 25 variableswill take the generalform: ‡ ˆ ¼ ˆ ¼ … † i¬x i­ y X0 X z; t e ‡ ; 18 Thelinearized form of N 0 is: ˆ … † … † where ¬ and ­ arethe wavenumbersof the disturbancein N 0 e1² e2± ; 26 b ˆ ¡ … † 73 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Journal of Glaciology

where Table 1.Parameter values for frost-susceptible soils

»fgh1 e1 ¡ 27 ˆ ¼ … † p q ¿ k0 ® »s E 3 4 ^2 ^3 »f gh1 9 h1 ¬ m s kg m e ¡ ‡ : 28 2 ˆ ± ¼ ² … † ChenaSilt 1.480. 660.48 1 .22 610^8 6.4 1378 ^9 h1 is the thickness ofthe frozenregion and E is Young’s IlliteClay 2.44 7 .270. 671 .69 610 10.0 875 modulus. CalgarySilt 2.006.58 0.45 2.50 610^9 8.9 1529 Thefrost-heave equations ( 16)and( 15)arecomplicated functionsof the temperature gradientsand the pressure at the topof the frozenfringe. Each potentially varying param- 6.LINEARIZED MODEL PREDICTIONS eter must beperturbed (e.g.replacing ­~ with ­~ ­~0). Thus, ‡ the followingparameters areperturbed: ­ , f , G , G , N, V , l i f f Thissection presents the predictionsobtained from the LSA. v , W , z and z . i l f s Itwasshown in section 4that there aremany parameters Theperturbed variablesare substituted intoEquations that arise fromthe frost-heavemodel. Theseare divided into (15)and( 16),andthe equationslinearized. T woequations three groups:soil parameters, environmentalconditions and result thatdescribe the perturbed ice andfreezing-front physicalconstants. Wedefinea reference set that comprises a velocitiesin terms ofperturbed quantities: set ofenvironmental conditions about which we will vary kf some parameters andanalyze their effect onthe stabilityof G0 1 ¯ ¿ W · 1 ­~ G0 ­~0G i k ‡ ¡ ¡ l ¡ ‡ f ¡ f one-dimensionalfrost heave.These conditions were chosen ³ u´ £ ¡ ¢ ¤ ± ² tobe representative ofareas where DFH isobserved.There V 0 ¯W ¿ ­~ ¿ W · ¿ W W ˆ f l ‡ ‡ ¡ l ¡ ¡ l l arean additional six parameters that describe the particular h ~ ¡ ~ ¢ ¡ ¢ i soilbeing analyzed: p, q, ¿, k0, ® and »s.Thefirst twocome V f ¯W 0 ­ W 0 ­ 0 ¿ W l · ¿ W l W 0 ·W lW 0 ‡ l ¡ l ‡ ¡ ¡ ¡ l ‡ l from the characteristic function that relates the differencein ice h ¡ ¢ ¡ ¢ 29 i andwater pressures tothe amountof unfrozen water .The … † p q and values of and aredetermined empiricallyby fitting the characteristic functionto experimental data. In this paper, ~ ~ kf three different types offrost-susceptible soils areanalyzed: 1 ­ vi0 ­ 0vi ·¿ Gi0 W lVf0 Wl0V f ‡ ‡ ˆ ¡ ku ¡ ¡ ChenaSilt, Illite Clayand Calgary Silt. These soils were ± ² µ³ ´ ¶ kf kf 2 chosenbecause values for all six parameters couldbe obtained · Wl0Gi WlGi0 W l Vf0 2W lWl0V f ‡ ku ‡ ku ¡ ¡ frompreviously published data (Horiguchi and Miller ,1983) µ³ ´ ³ ´ ¶ withcurve fitting when necessary .Table1 lists the valuesfor kf kf 1 ¯ G0­~ G ­~0 ­~W V 0 allsix soil parameters forthe three soils investigated. ¡ … ‡ † k i ‡ k i ¡ l f µ³ u´ ³ u´ Theenvironmental conditions include the temperature z ­~W 0V ­~0W V : 30 boundaryconditions at the groundsurface, s, and at the ¡ l f ¡ l f … † ¶ permafrost table, zb.Thegeneral N ewton’slaw-of-cooling Theperturbed temperature gradientsand pressure were boundarycondition is appliedat the groundsurface. The determined inthe previoussections. Theremaining per- effects ofvarying degrees ofsnow cover (including none) anddiffering vegetative cover can be explored using this turbed variablescan be cast interms of N0 byuse oftheir definitions.Then using Equation ( 26),the twoequations boundarycondition. F orsimplicity ,the gradientin the areonly functions of ² and ±.Thisprocess isstraightforward lower,unfrozenregion is zerofor this analysis. butalgebraically complicated (Peterson, 1999). Thefinal environmental condition is the freezingdepth, Theperturbed velocitiesare determined usingthe kine- h1.Thisparameter hasa rangein dimensional form of 0 d , where d isthe depthof the activelayer .Itshouldbe matic condition.The kinematic condition at the freezing ! 0 0 frontis slightlycomplicated because the fluxof material apparentthat the freezingdepth is afunctionof time. As freezingprogresses, h1 increases.Thus,by specifying a value throughthe boundary zf must beaccountedfor .However, becausethe accreted materialis neglectedin the elastic for h1 inthe reference set ofparameters, the stabilityof one- model,both conditions are straightforward: dimensionalfrost heaveat a certain,frozen instant oftime is beinganalyzed. The value of h1 forthe reference set was @± chosento correspond to a time earlyin the freezingprocess, vi0 ¡± 31 ˆ @t ˆ … † whenit is believedDFH is most likelyto be initiated.I twill @² beshownin section 6.7that the mechanism forinstability is Vf0 ¡± : 32 ˆ @t ˆ … † relatedto differential heat transfer .Thetemperature gradi- Solutionof the perturbed frost-heaveequation is now ent inthe frozenregion is greatest earlyin the ground-freez- possible.Equations ( 29)and( 30)can be expressed interms ingprocess, soDFH ismore likelyto initiate at early times of ¡, ¬, ², ± andparameters that areconstant for the specific inthe freezingprocess. Thequasi-steady-state assumption case beinganalyzed. An eigenvalueproblem for the growth allowsfor specification of h1. coefficient ¡ asafunctionof ¬ results. Thephysical constants that arise inthis problemare pri- marilyconcerned with the properties ofice andliquid water . A ¬ ¡2 B ¬ ¡ C ¬ 0 33 … † ‡ … † ‡ … † ˆ … † Thesevalues are summarized inTable2. A constant-tempera- Thealgebraically cumbersome expressions for A, B and ture boundarycondition of ^1 0³C waschosen for the topsur- C canbe seen inPeterson (1999).Anexplicitexpression for face(Smith, 1985).Thisis accomplishedin the thermal the tworoots of ¡ iseasilyobtained. problemby allowing H .ChenaSilt waschosen as the ! 1 74 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Peterson and Krantz: Differentialfrost heave and patterned-ground formation

Table 2.Physical constants for the LSAmodel

Parameter Symbol Value

3 ^3 Ice density »i 0.9610 kg m 3 ^3 Waterdensity »w 1.0610 kg m Heatof fusion L 3.06105 J kg^1 ^1 ^1 Icethermal conductivity ki 2.2 W m K ^1 ^1 Waterthermal conductivity kw 0.56 W m K ^1 ^1 Heatcapacity Cp 2.0 kJ kg K Gravitationalacceleration g 9.8 m s^2

Fig.3.Plotof the positive root of the linearstability relation as soiltype because, as willbe demonstrated, it hasthe greatest afunction of the wavenumber for the reference set of parameter rangeof unstablebehavior .Adepthof freezingof 1 0cm, or values.Thesoil is Chena Silt.Wavenumbers greater than ¬ntl about1 0%of the maximumdepth of freezing, was also are stable and ¬max is the wavenumber with the maximum chosen.There is nosurface load because in field situations growth rate under these conditions. the onlyoverburden is the weightof the frozensoil. A Young’smodulusof 5 MPais used forthe elastic modulus (Anderslandand Ladanyi, 1994). growthrate asafunctionof wavenumber for the three soil types specified inTable1 . 6.1.Effects of parametervariations Allthree soils areconsidered frost-suscepti ble.However , Inthe followingsubsections, the effects ofvarying different forthe reference set ofparameter values, only Chena Silt has apropensityfor D FH(i.e. positivevalue of ¡ ).Itshouldbe parameters fromtheir reference-set valuesare investigated . ‡ Theresults arepresented byplotting the dimensionless growth notedthat these results donot indicate tha tCalgarySilt and rate, ¡,asa functionof the dimensionless wavenumber, ¬ (see Illite Claywill never heavedifferentially .Other parameters can Equation( 19)).Thescaling factors for these parameters are havea significanteffect onthe stabilitypredictions, and under 2 the correct conditionsthe soils mayheave differentially . 1 d0»wL 1 1 ¡ 350¡ days¡ 34 Theutility of a modelsuch asthis onehinges on whether it s ˆ t ˆ k ¢T º … † ‰ Š u canpredict observationsmade in the field.U nfortunately, 2º 6 1 there is alimited number ofsoil types toanalyze due to a ¬s meter¡ : 35 ˆ ¶d0 º ¶ … † dearthof adequate characterization data including hydraulic Forcomparison purposes, the results willbe presented in conductivityand su b-freezingtemperature asfunctions of dimensionless form.The stability criterion, Equation ( 33),is unfrozenwater content. In fact, the workby Horiguchi and quadratic,and therefore there aretwo roots ¡ , ¡ , with Miller (1983)is oneof the fewcomprehensive datasets for ‡ ¡ ¡ > ¡ .Inthis analysis,the lowerroot ¡ was always frost-susceptible soils availablein the openliterature. Fortu- ‡ ¡ ¡ negative.The positive value, ¡ ,isplottedfor the reference nately,the predictionsshown in Figure 4 doagree with ‡ set usingChena Silt inFigure 3 .Inall cases, bothroots are real observationsmade in the field.In almost all instances, hum- numbers forthe rangeof ¬ valuespresented, andthe principle mocksare observed in silty-clayey materials. Thevalue of ofexchange of stabilities isvalid(Drazin andR eid,1991). ¬max forChena Silt is of the order unity.Thiswavenumber Thereare three valuesthat areof most importancein correspondsto a dimensionalwavelength on the orderof meters, whichis alsoobserved in the field(Williams and each plot: ¡max, ¬max and ¬ntl,indicatedin Figure 3 .The Smith, 1989).Thevalue of ¡ is severalorders ofmagni- maximumvalue of the growthrate, ¡max,andthe corres- max tude less thanunity ,indicatingthat DFH willtake (dimen- pondingvalue of the wavenumber, ¬max,indicatewhich sionally)several years to develop. I tiscommonlybelieved modegrows fastest inthe linearizedmodel. Also, ¬max indi- cates the lateralspacing of the most highlyamplified, two- that most types ofpatterned groundform on the10^100year dimensionalmode through Equation ( 35).Theneutrally stable time-scale (Mackay,1980;Hallet, 1987). wavenumber, ¬ntl,indicatesthe maximumwavenum ber for whichmodes canbe linearly unstable. These values are not explicitlymarked in su bsequent figures,to reduce clutter.

6.2.Effect of soil type

Thetype of soilis the most influentialparameter inthe model whendetermining the stabilityof one-dimensional frost heave.I tis importantto note that only a smallfraction of soils exhibitfrost heave(Williams andSmith, 1989).Adelicate balancebetween the hydraulicconductivity of the partially frozensoil, particle size andits porosityis necessaryfor frost heaveto occur .Silts andsilty claysmeet these requirements ingeneral. F urthermore, DFH is notnecessarily observed in allfrost-susceptible soils.Determining whetherthis obser- vationis dueto the specific soilor its environmentalcondi- Fig.4.The linear growth coefficient for three soil types using tionsis amajorobjective of this analysis.Figure 4 plotsthe the reference set of parameter values. 75 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Journal of Glaciology

Fig.5.The lineargrowth coefficient, ¡ ,as afunction of wave- Fig.6.The linear growth coefficient, ¡ ,as afunction of ‡ number, ¬,for three values of theYoung’‡smodulus, E.The soil wavenumber, ¬,for three values of the dimensionless heat- 1 type is Chena Silt. transfer coefficient H h d0 k¡ .The value of ¡max for ² f H 10 is two orders of magnitude greater than for acon- ˆ stant-temperature boundary condition ( H ). 6.3.Effect of theelastic constant ! 1

Inthis andsu bsequent subsections, weinvestigate the effects of coefficient, h,canbe obtained by reversing the scaling 1 parameter variationsaway from the reference set forChena H h d k¡ . ² 0 f Silt,which has the greatest propensityfor DFH. Although Itis readilyapparent that the ``insulating’’effect repre- calculatingthe effect ofthe elastic constanton linear stability sented bysmaller valuesof H causes agreaterrange of isnotdifficult, interpreting the results canbe confusing due to unstablemodes. F urthermore, as H decreases from to 10, 1 some ofthe assumptions madein the mechanicalmodel. The ¡max increases twoorders ofmagnitude.Insulation obviously complicationarises dueto the difficultyin determining an hasa destabilizingeffect. appropriatevalue for the Young’smodulus, E.While there Both ¬max andthe correspondingvalue of ¡max increase areseveral references that address the Young’smodulusfor as H decreases. Thereare several scenarios in the fieldin frozensoils (Tsytovich,1 975;Puswewala and Rajapankse, whichsmall valuesof H mightbe applicable. The most 1993;Andersland and Landanyi, 1 994;Y uanlinand others obviousinstance is that ofincreasedground cover in the form 1998),there isalargerange of reported values.Andersland ofgrasses, mosses andshrubs. Itis noteworthythat the exten- andLadanyi ( 1994)report that forfrozen silts, sandsand sivesurvey of earth hummocksby Tarnocaiand Zoltai ( 1978) clays,a Young’smodulusof about 5 MPais typicalfor the tem- supports this prediction.In essentially all places where earth perature rangeof interest: ^10 0³C. The effect ofY oung’s hummocksare found, there isamoderateamount of organic ! moduluson the growth-ratepredictions is shownin Figure 5 . groundcover .Thefact that the tops ofhummocksare some- Asmight be expected, the rangeof unstable modes times more barrenthan the troughregions is most likelythe decreases withincreasing Y oung’smodulus.Thevalue of ¬max result ofchanges in soil moisture after the hummockshape decreases, indicatingthat a ``stiffer’’soilresults inless closely hasmatured. I tis reasonableto assume that the differential spacedpatterning .Because ¡max decreases withincreasing E, heightbetween the crest andthe troughwould cause water it wouldalso take longer for patterns tobecom eexpressed in drainagefrom the topthat collects inthe inter-hummock stiffer soils.I tappearsthat increasing elasticity is astabilizing spaces, thus propagatingfurther organicgrowth in the mechanism.Asit becomes more difficultto ` `bend’’the frozen depressions. region,the propensityfor differential heave is reduced. Itisworthnoting that onlyin the limit of E does 6.5.Effect of ground-surfacetemperature ! 1 the propensityfor DFH atsome valueof ¬ disappear.Both ¬ and ¬ approachzero only when E . One major Snowcover ,whichis almostalways present tosome degreein ntl max ! 1 causefor larger values of E fora particularsoil would be arctic andsu barcticregions, also has an insulating effect. coldertemperatures. However,coldertemperatures also However,snowis more likelyto maintain a constanttem- affectthe thermal regime inthe frozenregion. This effect is perature atthe groundsurface due to its abilityto function exploredseparately in section 6.4and 6.5 . asa ``heatsink’ ’becauseof the relativelylarge latent heat associatedwith freezing .Toanalyzethe effect ofsnowcover , 6.4.Effect of theground surface heat-transfer it ismore usefulto return tothe constant-temperaturebound- arycondition and vary the valueof T . Since H for coefficient air ! 1 the constant-temperature condition,the ground-surfacetem- Inall the previousresults, wehave assumed aconstanttem- perature is equivalentto Tair.Asmaller (inmagnitude) value perature of^1 0³C atthe groundsurface. However ,acon- of Tair correspondsto a thickerand/ orless thermallyconduc- stant-temperature conditionis unrealistic forprolonged tivesnow layer .Infact, the onlylimitation in this case is periodsof time (i.e. 41^2days).Alumpedparameter Tair < 0.The effect ofthe ground-surfacetemperature is boundarycondition with an overall heat-transfer coefficient shownin Figure 7 . ismore reasonablefor long-term analysis.Theconstant-tem- InFigure 7 it canbe observed that asthe ground-surface perature conditionis obtainedfrom the heat-transfer-coeffi- temperature warms, ¡max decreases. Furthermore, inthe cient conditionby allowing H .Figure6 plots ¡ as a limit of Tair 0,the growthcoefficient for all unstable modes ! 1 ‡ ! function of ¬ forthree valuesof the dimensionless heat- tends tozero .Theseresults indicatethat the occurrenceof transfer coefficient.A dimensionalvalue of the heat-transfer DFH shouldbe more prevalentin less snow-coveredareas 76 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Peterson and Krantz: Differentialfrost heave and patterned-ground formation

Fig.7.Theeffect of warmerconstant-temperature conditions on the growth of differential modes for the reference set of param- eter values.The growth rate, ¡ is plotted in dimensional form. Fig.9.Relative amplitude of the ground-surface perturbation, ±=±0,for several values of the wavenumber, ¬, at the time t0 when allmodes become stable.Soiltype is Chena Siltand thanthose wheresnow accumulates, and in fact this is H 10. At t t0, h1 0.23. observed.A goodexample is the ToolikLakes (Alaska, ˆ ˆ º U.S.A.)fieldsite whereD .A.Walker(personal communica- tion,2000 )hasobserved that frost boilstend tobe more number mightcommonly be most expressed inthe field.In the limit h 0,all wavenumbers have a positivegrowth prevalentin the wind-blownareas. F rost boilsare small 1 ! moundsof soil material, 1mindiameter ,presumed tohave rate. Asfreezingprogresses and h1 increases fromzero, the ¹ beenformed by frost action(NSIDC, 1998).Ground-surface largestwavenumbers are ` `cut off’’(i.e. ¡ < 0) while the ‡ temperature measurements inboth regions confirm that sur- smaller wavenumberscontinue to grow ,albeitat a slower facetemperatures areindeed colder in the barrenregions. rate. As h1 continuesto increase, more andmore wave- numbers arecut off.Eventually when h 0.23,all wave- 1 º 6.6.Effect of freezingdepth numbers arecut offand no differentialmodes areunstable. Itcanbe seen inFigure 8 that verysmall wavenumbers Itis appropriateto investigate the effect offreezing depth, alwayshave small positivegrowth rates untilcut off.Large h z z ,onthe stabilitytheory predictions .Thenon-di- 1 ² s ¡ f wavenumbershave large growth rates butare cut offearly . mensionalizationestablishes that h1 is ofthe orderone. When Mid-rangewavenumbers have moderate growth rates for groundfreezing commences, h1 beginsat zero and increases intermediate periodsof time. Thus,mid-range wavenum- (approximatelywith the squareroot of time (Fowlerand bers mightbecome most expressed inthe fieldsince they Noon,1 993)).Byexamining the behaviorof ¬max and ¡max havea moderatelength of time togrow and moderate forincreasing values of h1,weare able to observe how the growthrates formost ofthe time. stabilityof the system changesas ground freezing progresses. Althoughlinear theory is notstrictly validonce differen- Figure8 showsthe effect offreezing depth on the stability tialmodes growto a finite amplitude,it isinsightfulto see analysispredictions. As onemight expect, the largestrange whatwavenumber the modelpredicts wouldbecome most ofunstablemodes occurs atthe smallest valuesof the freezing expressed duringthe entire periodof freeze-up.Thisassump- depthand slowly decreases as h1 increases. However,con- tionis notexcessively crude since the dimensionless growth traryto previous results whereunstable modes existed forall rates aremuch less thanunity in this case,and the perturba- valuesof a parameter,stabilizationoccurs ata finite valueof tionsare going to grow relatively slowly .Asasimple approxi- h1.Forthe reference set ofparameter valuesshown here, that mation,we define the normalizedgrowth rate asfollows: value is h 0.23,whichcorresponds to a dimensionalvalue 1 º 1 @± @ ln ± of 23 cm. ±_ ; 36 Thefact that there is acritical valueof the freezing ² ± @t ˆ @t … † depthis importantwhen attempting to gauge what wave- where ± is the perturbationin the ground-surfacelocation. Themagnitude of ± at time t0 is determined byintegration:

t0 ln ± ±_ dt 37 ˆ … † Z0 ± t0 exp ±_ dt ; 38 ± ˆ … † 0 ÁZ0 ! where ± is the magnitudeof the originalperturbation at t 0. 0 ˆ Figure9 showsthe numeric results ofintegratingEqua- tion( 38)for several values of the wavenumber ¬, indicated bythe dots.The integration is performed untila time t0, at whichpoint all modes becomestable (i.e. h 0.23). The 1 º most expressed wavenumberunder these conditionsis ¬ ˆ 2.3which corresponds to a dimensionalwavelength of 2.7m. Fig.8.Thelineargrowth coefficient, ¡ ,asafunction ofwave- Thisresult is quiteencouraging since the spacingof hum- ‡ number, ¬,for three depths of freezing, h1.There are no positive mocksis typically1^3 m(Tarnocaiand Zoltai, 1 978). roots (¡ ; ¡ <0)at freezingdepthsgreater than h1 0.23. Becauseall modes arestable fortimes greaterthan t0 (or ‡ ¡ º 77 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Journal of Glaciology

Fig.10.Evolution ofthe normalizedground-surfaceperturbation Fig.11.Stabilitypredictions for the fundamental case when as afunction of the freezing depth, h1. theYoung’smodulus is set tozero ( E 0). ˆ

h1 > 0.23),afurther increase infrost depthwill result inall modes decreasingin amplitude. topsurface perturbation, ±,albeitby a small difference. Todetermine the evolutionof ±=±0 in time as h1 increases, However,linearstability theory predicts perturbationsgrow Equation( 38)isintegratedin time upto t0 fora givenwave- exponentially.Thus,although the differencebetween ² and ± number.Theresults for ¬ 2.6areshown in Figure 1 0.This is small initially,that differencewill grow exponentially fast ˆ figureshows the evolutionof ±=±0 asfreezingprogresses (i.e. (at least untilnon-linear terms becomesignificant) .Infact, it h1 increases).Thenormalized perturbation amplitude is difficultto speculate aboutwhat form the perturbations reaches amaximumat h 0.08and then beginsto decrease. willtake once non-linear terms becomesignificant. Figure 1 º At h 0.2, ±=± 2.65as expected from Figure 9 .The 12showsthat the thickness hasincreased from the basicstate 1 ˆ 0 º amplitudereturns toits initialvalue at h 0.36and con- valueunderneath a crest, anddecreased underneath a trough. 1 º tinues todecrease as h1 increases further.Theseresults indi- Becausethe temperatures atthe boundaries zf and zs are fixed catethat aninitial perturbation of magnitude ±0 will have atthe basic-state valuesand the conductionpath length has increasedin magnitude at the endof the freezingseason for changed,the temperature gradientat the bottomsurface, active-layerdepths 50.36 m. Gi,hasalso changed. U nderneatha crest the gradienthas Thereis noassumption in this derivationthat the per- decreased inmagnitude and, according to Equation ( 15),the turbationgrows as exp ¡ t aslineartheory requires. Inthis velocityof the freezingfront will also decrease. Continuing ‰ Š sense, this isareal-time analysis.However ,otherassump- this lineof reasoning ,since the conductionpath underneath tionshave implications that must beexamined. Perhaps atroughhas decreased, the gradientat zf hasincreased and most importantly,allproduct terms inperturbed variables the velocityof the freezingfront has correspondingly areassumed small andneglected. Hence, this real-time increased.Since there is apositivefeedback mechanism analysisis still onlyapplicable when all perturbations are occurringat this point,the perturbationswill continue to grow . relativelysmall inmagnitude. Inorder to understand the increase in ¡ withincreasing wavenumberobserved in Figure 1 1,it is necessary toconsid- 6.7.Mechanismfor instability er the magnitudeof differential heat flow shown in Figure 12.As the wavenumberincreases, the magnitudeof heatflux Ithasbeen demonstrated abovethat there is arangeof froma crest regionto a troughregion increases. Inorder for modes that areunstable for most cases ofinterest. Infact, if frost heaveto occur ,the latentheat of the incomingwater at theYoung’smodulusis allowedto approach zero ( E 0), all ! modes areunstable for some soils whileno modes are unstablefor other soils. In this limit, the bendingresistance ofthe frozensoil is essentiallyremoved (see Equation( 7)). Figure1 1showsthe LSAprediction for the reference set of parameter valuesand Chena Silt exceptthat E 0. It is evi- ˆ dent that the elasticityof the frozenlayer is providingthe stabilizingmechanism forDFH. Inorder to help explain the mechanism that causes the instability,Figure1 2showsa schematic ofthe in-phaseper- turbed surfaces. Alsosketched areapproximate isotherms forthe constant-temperature boundarycondition at the topsurface. The large arrow pointing to the rightshows the net directionof differentialheat flow that isoccurringfrom acrest regionto a troughregion. Before the perturbations occur,heattransfer occurs onlyin the verticaldirection. However,oncethe system is perturbed, heattransfer can Fig.12. Schematic of the perturbed surfacesshowing isotherms occurin boththe horizontaland verticaldirections. and the direction ofdifferential heat flow.The basic state sur- SolvingEquation ( 33)for ²=± indicatesthat the magni- faces zs and zf are shown by the dashed lines,and the per- tude ofthe bottomsurface perturbation, ²,isless thanthe turbed surfaces are the thick solid lines. 78 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Peterson and Krantz: Differentialfrost heave and patterned-ground formation

zl must beremoved.Since there is differentialflow from a culationtheory presented byHallet andW addington( 1992). crest toa trough,additional heaving can occur underneath Couplingthe DFH modelwith Hallet andW addington’s the crest region.This also corresponds to less heaveoccur- modelmight provide a more complete explanationof pattern ringin the troughregion. formationand soil circulation by accounting for both the freezingand thawing processes. 6.8.Implications of thesteady-state assumption Theuse ofthin-plate theory to describe the upper,frozen regionas apurelyelastic materialis restricted tothe linear Thisanalysis is basedon the quasi-steady-state assumption region.V ertical displacements areassumed verysmall. that perturbationsgrow much faster thanchanges in the Furthermore, thin-platetheory assumes h 2º¶, or that 1 ½ basicstate. However,predictionsindicate that the growth the thickness ofthe regionbeing deformed is muchless than rate ¡ isactuallyless thanunity in most cases, andthus per- the characteristic lateraldimension. This assumption is turbationsgrow slowly relative to changes in the basicstate. obviouslynot valid for larger freezing depths. Todescribe Forsome time-periodic problems withslow growth rates, the evolutionof DFH after initiationwhen these assumptions Floquettheory has been successfully appliedwhen solving breakdown, a more comprehensiveelastic constitutive the linearstability problem (Drazin andReid, 1991,p.354). relationneeds tobe implemented. Alternatively,adifferent However,the entire freeze^thawcycle (or period) must be rheologycan be used such asviscous,viscoelastic or visco- modeled.This analysis has only addressed the frost-heave plastic.These rheologies may in fact describe the long-term process that occurs duringfreezing, which is onlypart of a evolutionof DFH more accuratelythan a purelyelastic one. largercycle. The thawing process involvessolifluction, soil Thesteady-state assumptioncan only be used asafirst consolidation,water percolation and evaporation, and pos- approximationand is obviouslyinvalid near marginal sta- siblyother processes. bility.Thisdrawback points to the necessity ofsolving the Thepresent analysisindicates that the frost-heaveprocess non-linearequations as atime-evolutionproblem. While a aloneis capableof causing spontaneous pattern generation, non-linearstability analysis can provide further informa- albeiton a 10^100year time-scale. Combinationof the frost- tionabout the observedplanform and sub- orsupercritical heavemodel with a thawmodel could possibly yield more stability,it must alsouse afrozen-time assumption.Current conclusivetheoretical evidence that patterns aregenerated stabilitytheory cannot adequately account for the effects of bythe processes discussed here.Itmaybe that the mature achangingbasic state. Thus,a numericalsolution of the periglacialpatterns observedare actually the result ofacom- time-evolutionproblem is probablythe most valuablecon- plexcom binationof all the processes mentionedabove. tinuationof this investigation.

7.CONCLUSIONS ACKNOWLEDGEMENTS

Inthis paper,wehave demonstrated that the frost-heave Wegreatlyappreciate the helpfulcomments fromS. J .Jones, modeldue to F owlerand Krantz (1994)is linearlyunstable M. Sturm andan anonymousreviewer .Thiswork was sup- undera rangeof environmentalconditions and for several portedby U .S.N ationalScience F oundationgrant O PP- (but notall) soil types. Differential heatflow coupled with 9321405.R.A.P .acknowledgesthe AmericanGeophysical the delicatebalance between hydraulic conductivity and Unionfor financial support. cryostaticsuction potential act as a destabilizingmechan- ism forone-dimensional frost heave.When the upperfrozen regionis modeledas a purelyelastic material,the force REFERENCES required todeform the frozenregion acts asa stabilizing mechanism toDFH. Previousstability analyses in the litera- Andersland, O.B.andB .Ladanyi.1994. Introductionto frozenground engineering . NewYork,Chapman and Hall. ture (Lewis andothers, 1993;F owlerand N oon,1997)have Black,P .B.1995.RIGIDICEmodel of secondary frost heave. CRREL Rep.95-12. failedto describe accuratelythe mechanisms atwork that Black,P .B. andR. D .Miller.1985.Acontinuumapproach to modeling frost cangive rise toDFH. Theanalysis presented here hascor- heaving. In Anderson, D.M. andP .J.Williams, eds.Freezing and thawingof rected errors inhow Lewis (1993)coupled the mechanical soilwatersystems:astateof thepractice report .NewYork,American Society of Civil Engineers,36^45. problemto the frost-heaveproblem, and identified a source Brush, D.O.and B .O.Almroth. 1975. Bucklingof bars,plates,andshells. New ofinstabilitythat wasoverlooked by N oon( 1996)in the case York,McGraw-Hill Inc. ofaconstant-temperature boundarycondition. Chen Xiaobaiand Y aqingW ang.1991.Newmodel of frostheave prediction Themechanisms responsiblefor DFH aredependent on forclayey soils. Sciencein China, Ser .B , 34(10),1225^1236. Drazin,P .G.andW.H.R eid.1991. Hydrodynamicstability . NewYork,Cam bridge the instantaneousdepth of freezing and can effectively be UniversityPress. shut offwhen depths offreezing exceed a critical value. Fowler,A. C. 1989.Secondary frost heave in freezingsoils. SIAM J.Appl. Numericalintegration of the predicted growthrates asfreez- Math., 49(4),991^1008. ingprogresses beyondthis critical valueindicates that there is Fowler,A. C. 1997. Mathematicalmodels inthe applied sciences. Cambridge, CambridgeU niversityPress. anintermediate-valued wavenumber that ismost expressed. Fowler,A. C. andW .B. Krantz.1 994.A generalizedsecondary frost heave Thedimensionless growthrates predicted bythe theory model. SIAMJ.Appl.Math. , 54(6),1650^1675. presented here aresmall, 1,indicatingthat pattern expres- Fowler,A.C.andC .G.Noon.1993.Asimplified numericalsolution ofthe M iller ½ modelof secondary frost heave. Cold Reg.Sci.Technol. , 21(4),327^336. sionwould take tens tohundreds of years to mature. Thispre- Fowler,A. C. andC. G.Noon. 1 997.Differentialfrost heave in seasonally dicted lengthof time issimilar tothe scalereported bymany frozensoils. In Knutsson,S., ed.Ground freezing 97 :frost actionin soils . fieldobservers (Washburn,1 980;W illiamsand Smith, 1989). Rotterdam,A. A. Balkema,8 1^85. However,the mechanisms discussed here applyonly to the Gilpin, R.R. 1980.A modelfor the prediction of icelensing and frost heave freezingprocess. Active-layerthaw is itself auniqueprocess in soils.WaterRes .Res. , 16(5),918^930. Hallet,B .1987.On geomorphicpatterns with afocuson store circles viewed andmay possess additionalmechanisms thatcould give rise asa free-convectionphenomenon. In Nicolis,C. andG. Nicolis, eds. topatterning .One possibilityis thebuoyancy-inducedsoil cir- Irreversible phenomenaand dynamical systems analysisin geosciences .Dordrecht, 79 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 23:36:01, subject to the Cambridge Core terms of use. Journal of Glaciology

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MSreceived 23March 2000and accepted inrevised form 9December 2002

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