A mechanism for differential frost heave and its implications for patterned-ground formation Rorik A. Peterson1∗, William B. Krantz2y 1Department of Chemical Engineering, University 1 Notation of Colorado, Boulder, Colorado, 80309-0424 2Institute of Arctic And Alpine Research, Univer- sity of Colorado, Boulder, Colorado, 80309-0450 Cdf group of constants in the Taylor Series expansion ∗Present address: Institute for Arctic Biology, Uni- for(@f=@Wl) versity of Alaska, Fairbanks, Alaska, 99709-7000 Cp heat capacity CT η constant in the perturbed temperature-gradient solution yPresent address: Department of Chemical Engi- CT ζ constant in the perturbed temperature-gradient solution neering, University of Cincinnati, Cincinnati, Ohio, d0 length scale corresponding to the maximum depth of freez- 45221-0171 ing 0 e1;2 group of constants in the definition ofN E Young's Modulus (1−χ)p fl χq f 0 @fl Abstract l @Wl g gra vitational constant Gf i temperature gradient atz ; z The genesis of some types of patterned ground in- ; f i h dimensional heat-transfer coefficient cluding hummocks, frost boils and sorted stone cir- h thickness of the frozen region(z − z ) cles has been attributed to differential frost heave 1 s f H dimensionless heat-transfer coefficient (DFH). However, a theoretical model that adequately k overall thermal conductivity describes DFH has yet to be developed and validated. k unfrozen soil hydraulic conductivity In this paper we present a mathematical model for 0 kf;u thermal conductivity of frozen/unfrozen soil the initiation of DFH, and discuss how variations γ Wl in physical (i.e. soil/vegetation properties) and en- kh k0 φ vironmental (i.e. ground/air temperatures) proper- L heat offusion for water ties effect its occurrance and length scale. Using the P −p1 N σ Fowler & Krantz multidimensional frost-heave equa- p exponent in the soil characteristic function tions, a linear stability analysis (LSA) and a quasi- P total pressure steady state real-time analysis are performed. Re- Pb pressure due to bending sults indicate that the following conditions positively p1 pressure in the far field (e.g. basal plane) effect the spontaneous initiation of DFH: silty soil, q exponent in the soil characteristic function small Young's modulus, small non-uniform surface t time heat transfer or cold uniform surface temperatures, T temperature and small freezing depths. The initiating mechanism T0 dimensionless reference temperature in Newton's Law of for DFH is multi-dimensional heat transfer within Cooling boundary condition the freezing soil. Numerical integration of the lin- Tair dimensionless air temperature ear growth rates indicate that expression of surface Tb temperature at the basal reference plane, zb patterns can become evident on the 10-100 year time Tf;u temperature of the frozen/unfrozen soil scale. Ts temperature at the ground surface 1 ∆T temperature scale typical for the freezing pro- drawing water through a soil matrix towards a freez- cess ing domain is known as cryostatic suction. Primary Vf freezing front velocity frost heave is characterized by a sharp interface be- vi ice velocity tween a frozen region and an unfrozen region. In w deflection of the plate midplane in thin plate contrast, secondary frost heave (SFH) is character- theory ized by a thin, partially frozen region separating the Wl unfrozen water fraction of the total soil poros- completely frozen region from the unfrozen region. ity at the lowest ice lens Within this region, termed the frozen fringe, discrete 0 Wl derivative of the unfrozen water content in the ice lenses can form if there is sufficient ice present to downward direction at zl support the overburden pressure. The theory of pri- x; y horizontal space coordinates mary frost heave cannot account for the formation of z vertical space coordinate multiple, discrete ice lenses. Miller (1980) states that zb basal plane, permafrost table in the field ice-lens formation probably only occurs by the pri- zf location of the freezing front mary heave mechanism in highly colloidal soils with zl location of the lowest ice lens negligible soil particle-to-particle contact. z location of the ground surface s Frost heave that is laterally nonuniform is referred α dimensionless wavenumber in the x direction to as differential frost heave. Differential frost heave β~(1+δ)+µ(φ−Wl) αfk (1+β~) (DFH) requires that the freezing be both significant fl−N and slow. Adequate upward water flow by cryostatic β~ βl 0 −Wlfl −δγ(fl−N) 2 suction requires either saturated soil conditions or a γρihL kh i βl gkT0 very high water table because unsaturated soil condi- χ unfrozen-water fraction of the void space = tions can suppress the heaving process. Suppression Wl/φ occurs due to insufficient interstitial ice, which is re- − ρi quired to support the overburden pressure and allow δ 1 ρ w for ice-lens formation (Miller, 1977). η perturbation in the freezing front location γ exponent for hydraulic-conductivity function The DFH process is not fully understood. The Γ exponential growth rate according to linear complex interactions that occur in freezing soil have theory not been fully determined or described and there are λ wavelength of the perturbations as yet no predictive models for this phenomenon. λG constant in the Gilpin thermal regelation the- However, secondary frost heave, which is often associ- ory ated with DFH, has received a certain amount of at- µ λGLρw tention from the scientific community. Several math- ku ν Poisson's ratio ematical models and empirical correlations have been developed to describe one-dimensional SFH. Most of ρf frozen soil density the mathematical models represent variations and re- ρi;w ice/water mass density σ pressure scale of one bar finements of the original one-dimensional SFH model φ unfrozen soil porosity of O'Neill and Miller (1985). A set of describing equa- ratio of η/ζ tions for three-dimensional SFH based on the O'Neill ζ perturbation in the ground-surface location and Miller model has been developed by Fowler and Krantz (1994). Here these equations are used as a basis for investigating the DFH process. 2 Introduction DFH has been cited as a possible cause for some forms of patterned ground by Taber (1929) and Frost heave refers to the uplifting of the ground sur- Washburn (1980) among others. Patterned ground face owing to freezing of water within the soil. Its refers to surface features made prominent by the seg- typical magnitude exceeds that owing to the mere regation of stones, ordered variations in ground cover expansion of water upon freezing (∼ 9%) because of or color, or regular topography. Since patterned freezing additional water drawn upward from the un- ground is a manifestation of self-organization in na- frozen soil below the freezing front. The process of ture, its formation is a question of fundamental sig- 2 nificance. In this paper we present a mathematical must be solved numerically. One-dimensional, nu- model for a mechanism by which DFH can occur and merical solutions to the model by Miller (1977, 1978) may cause the initiation of some types of patterned have been presented by O'Neill and Miller (1982, ground. Using this model, a limited parameter space 1985), Black and Miller (1985), Fowler and Noon of environmental (i.e. temperature, snow cover) and (1993), Black (1995) and Krantz and Adams (1996). physical (i.e. soil porosity, hydraulic conductivity) In several instances, numerical results agree favor- conditions is defined within which DFH can initiate ably with laboratory results such as those of Konrad spontaneously. (1989). The predictive capability of the M1 model has also been demonstrated (Nakano and Takeda, 1991, 1994). 3 Prior Studies Lewis (1993) and Lewis and others (1993) were the first to use a LSA in an attempt to predict the occur- Semi-empirical models correlate observable charac- rence of DFH. Using the model equations developed teristics of the SFH process with the relevant soil by Fowler and Krantz (1994), they found that DFH properties and climatic variables. The more preva- occurs spontaneously under a wide range of environ- lent models include the segregation potential (SP) mental conditions. Lewis (1993) included the nec- model of Konrad and Morgenstern (1980,1981,1982) essary rheology by modeling the frozen region as an and that of Chen and Wang (1991). Typically, semi- elastic material and using thin plate theory. How- empirical models are calibrated at a limited number ever, this \elastic condition" was incorrectly inte- of sites under unique conditions. Since these unique grated into the analysis giving unconventional results conditions are accounted for by using empirical con- that predicted a single unstable wavenumber. stants, it can be difficult to generalize the models for Independently, Noon (1996) and Fowler and Noon use with varying environmental conditions and field (1997) reported results for a LSA using the Fowler sites. Semi-empirical models are easy to use, com- and Krantz model with a purely viscous rheology. putationally simple, and provide fairly accurate pre- Noon's results indicated that one-dimensional heave dictions of SFH (Hayhoe and Balchin, 1990; Konrad, is linearly unstable using two different ground-surface Shen and Lidet, 1998). However, they provide lim- temperature conditions: constant temperature and ited insight into the underlying physics and cannot snow cover. However, the constant-temperature re- predict differential frost heave. sults are not physical due to unrealistic parameter Fully predictive models are derived from the funda- choices. The remaining condition necessary for in- mental transport and thermodynamic equations. In stability, the presence of snow cover, appears too re- theory, they require only soil properties and bound- strictive. Patterned ground is observed both in the ary conditions in order to be solved.