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A Mechanism for Differential Frost Heave and Its Implications for Patterned-Ground Formation

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A Mechanism for Differential Frost Heave and Its Implications for Patterned-Ground Formation Journal of Glaciology , Vol. 49, No.164, 2003 Amechanism fordifferential frost 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 Arctic 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. soil/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 correspondingto 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 scale typicalfor 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 porosity 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, permafrost 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. Patterned ground 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 scale of1bar ¿ 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 models arecalibrated 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 models are 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 ifthere 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 that ice-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 soils 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 tions areused asabasis forinvestigating the DFH process. conditions.Lewis
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