A mechanism for differential heave and its implications for patterned-ground formation

Rorik A. Peterson1∗, William B. Krantz2†

1Department of Chemical Engineering, University 1 Notation of Colorado, Boulder, Colorado, 80309-0424 2Institute of 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 †Present 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 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 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/ 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 −p∞ 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 p∞ 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 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, 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 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 of O’Neill and Miller (1985). A set of describing equa- ψ 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. Based on ideas presence and the absence of snow cover (Williams due to Hopke (1980) and Gilpin (1980), Miller (1977, and Smith, 1989). Furthermore, since only a single 1978) proposed a detailed model for SFH. This model soil was investigated, the influence of soil type on the was later simplified by Fowler (1989), extended to in- predictions is difficult to evaluate. In light of these clude differential frost heave by Fowler and Krantz points, further investigation of the linear stability of (1994) and extended further to include the effects differential frost heave is necessary. of solutes and compressible soils by Noon (1996). Nakano (1990, 1999) introduced a model called M1 for frost heave using a similar approach. Fully pre- 4 The Differential Frost Heave dictive models do not require calibration and can be applied to anywhere the relevant soil properties and Model thermal conditions are known. These models also provide insight into the underlying physics involved. The model for secondary frost heave that we use in These attributes facilitate model improvements and this paper is that due to Fowler and Krantz (1994), extensions. Due to the complexity of some models, which is a modified version of the O’Neill and Miller they can be computationally difficult and inconve- model. We chose to use this model because it con- nient to use. tains all the necessary physics to describe differential Except under asymptotic conditions, these models frost heave while maintaining a numerically tractable

3 form (Fowler and Noon, 1993). Most importantly, Ground surface this model properly takes into account thermal rege- Zs lation in the partially frozen region termed the frozen fringe. Thermal regelation within the fringe allows for differential heaving. The original O’Neill and Ice lenses unfrozen region Miller model (1982, 1985) used a rigid-ice approxima- tion that assumes all the ice within the frozen fringe Zl and lowest ice lens moves as a uniform body with a Frozen fringe { Zf constant velocity. In order for differential heave to occur, this cannot be the case. Fowler and Krantz Z (1994) demonstrate that thermal regelation within frozen region the fringe allows for differential frost heave. Our Permafrost table

model uses the Fowler-Krantz formulation. Readers Zb should refer to Fowler and Krantz (1994) and Krantz and Adams (1996) for details. Figure 1: Diagram of saturated soil undergoing top- down freezing. The region between zl and zf is the 4.1 Physical Description frozen fringe. Ice lenses occur in the frozen region giving rise to frost heave. Above zi there is negligible The physical system in which occurs is further freezing of liquid water. shown in Figure 1. Top-down freezing occurs over a time-scale of weeks to months. The upper, frozen region is separated from the lower, unfrozen region for equilibration of the pressure and temperature pro- by the frozen fringe. The frozen fringe is a partially files within the frozen fringe are much shorter than frozen transition region containing ice, liquid water the time scale for the freezing process. Thus, the and soil. The thickness of the frozen fringe in sea- entire frost-heave process can be modeled as semi- sonally frozen ground is on the order of 1-10 mm. continuous where some parameters are evaluated at The frozen region is characterized by regular bands the base of the warmest (lowest) ice lens, zl. of pure ice called ice lenses. It is the existence of ice For one-dimensional frost heave, the surfaces zs lenses that gives rise to surface heave greater than and zf are parallel planes. The frozen region in- the 9% associated with the volume change of water cluding ice lenses moves upwards relative to zb but upon freezing (Washburn, 1980). In typical field situ- does not deform. In this instance, the rheology of ations where frost heave occurs, permafrost underlies the frozen material is irrelevant. However in differen- the unfrozen region (Williams and Smith, 1989). tial frost heave, the frozen region deforms as it moves Because the frozen fringe is relatively thin, it can and the rheology of the material must be specified so be reduced mathematically to a plane across which the mechanical problem can be solved in conjunction jump boundary conditions are applied (Fowler and with the thermal and frost-heave problems. Krantz, 1994). This plane is located at zf in Fig- ure 1. Freezing of saturated soils is fairly slow as can be inferred from the large Stefan number (St ≡ 4.2 Thermal Problem −1 −1 LCp ∆T ) associated with this process. Because the time scale for freezing is much longer than that Due to the large Stefan number for freezing soil, the for heat conduction, the temperature distributions thermal problems in both the frozen and unfrozen re- in the unfrozen and frozen regions are quasi-steady gions above and below the frozen fringe, respectively, state. The thermal gradients in the unfrozen and are greatly simplified. The two-dimensional, steady- frozen region are obtained by solving for the temper- state temperature distributions in both regions are ature distributions in each region. obtained from the solution to the Laplace equation: Because ice lenses occur in a discrete manner, the frost-heave process is inherently a discontinuous pro- ∇2 cess. However, Fowler and Krantz (1994) demon- Tu = 0 zb < z < zf (1) 2 strate using scaling arguments that the time scales ∇ Tf = 0 zf < z < zs (2)

4 The boundary condition at the ground surface is Newton’s Law of Cooling. For both problems, the boundary condition at zf specifies the temperature as the STP freezing point of water. For this paper, the temperature at zb will also be specified. ∂T f = −H(T − T ) at z = z (3) ∂n f air s Tf = 0 at z = zf (4)

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where H is a heat-transfer coefficient, Tair is the di- mensionless air temperature at the ground surface and Tb is the dimensionless temperature of the un- derlying permafrost. In the absence of significant so-

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mFm the geometry of the frozen region as well as its me- Z nFn f oFo possible ice chanical properties. The frozen fringe is, mathemat- ically, reduced to a plane across which jump bound- ary conditions are applied (Fowler and Krantz, 1994). Figure 2: Diagram of the frozen fringe showing the This plane is the lower boundary of the frozen region, initiation of differential frost heave. Relative dimen- sions are not to scale. The dashed line in the left fig- zf . However, because it will be perturbed and dis- torted, albeit infinitesimally, and will no longer be a ure shows the location of zf which is defined where plane in a strict mathematical context, we refer to it the ice content is zero. A new ice lens forms within as an interface. the water pressure boundary layer below the lowest Figure 2 shows a schematic of the frozen fringe be- ice lens. After initiation of a new ice lens, the lo- fore and after inception of differential frost heave due cation of zf will move either upwards or downwards relative to its previous position. The two possibili- to initiation of a new ice lens. In fact, the location 0 00 ties are shown by zf and zf . The ice crystals labeled of the newly forming ice lens is very close to the 0 00 base of the lowest ice lens relative to the thickness “possible ice” will exist for zf and not for zf . The of the frozen fringe. There is a boundary layer in the new location of zf is predicted by the linear stability frozen fringe near the lowest ice lens wherein most of analysis and is not prescribed in the model. the pressure drop causing upward permeation occurs. The new ice lens will form within this boundary layer (Fowler and Krantz,1994). The dashed line labeled zf is the bottom of the frozen fringe, defined as where the ice content is zero. At the inception of differential frost heave, the loca- tion of zf can move either up or down. Because of the quasi-steady state assumption, any movement of zf is relative to the stationary, one-dimensional bound- ary. Although zf is always moving downwards on the time scale of freezing-front penetration, quasi-steady

5 state must be assumed in order to conduct the lin- frozen water at 0◦C are the cause of this “weakness”. ear stability analysis. While this is an abstraction Thus, we would expect the newly accreted material from reality, it is an expedient and useful practice to have a very low Young’s modulus and thus neglect- for linear stability analyses of systems with unsteady ing it completely introduces only small error. In fact, basic states (Fowler, 1997). Two possible locations as a first approximation the entire frozen region is for zf are shown in the right side of Figure 2: one assumed to have a constant Young’s modulus. This for upward movement and one for downward move- approximation is crude since there is a nonzero tem- ment. Which of these two possible outcomes actually perature gradient within the frozen region. In reality, occurs is not prescribed in the model but is a re- the Young’s modulus monotonically increases with sult of the linear stability analysis. However, linear decreasing temperature and has a maximum value at theory does prescribe that zf is either in-phase or the ground surface where it is coldest. anti-phase with the ground-surface perturbation, zs. Due to the simplifications discussed above, the me- Any relative displacement other than 0 or π is not al- chanical model reduces to the bending of a thin, lowed. The location of zl in Figure 2 will be in-phase elastic plate with a constant Young’s modulus. If with the ground surface at zs since the lowest ice lens the deformation is small relative to the thickness of precludes water migration above it. Thus, Figure 2 the plate, the deflection of the plate’s midplane from demonstrates how zs and zf can be either in-phase equilibrium, w, can be approximated as (Brush and or anti-phase due to the formation of a new ice lens. Almroth, 1975): Mechanically, when nonuniform ice-lens formation 4 begins, the frozen region will begin to deform and Pb = D ∇xyw (7) bend. We model this as the bending of a thin, elastic E h3 plate, though a viscous, viscoelastic, or viscoplastic D ≡ 1 (8) 12(1 − ν2) model might also be used. Laboratory experiments ∇4 ≡ by Tsytovich (1975) demonstrate that frozen soil does xyw w,xxxx + w,xxyy + w,yyyy (9) behave elastically on short time scales. The necessity where Pb is the pressure, E is Young’s modulus, h1 is for including the rheology of the frozen material in the frozen region thickness, and ν is Poisson’s ratio. the model is to provide a “resistive force” to deforma- In this analysis, ν = 0.5. tion of the frozen material. As will be demonstrated Boundary conditions for Equation 7 must be spec- later, when the rheology is neglected, a linear stabil- ified in the x and y directions. Because the problem ity analysis predicts that the most unstable mode has is unbounded in these directions, periodic boundary a wavelength approaching zero, which is intuitively conditions are used. For the x direction, incorrect. w(x = 0) = 0 (10) A conceptual difficulty is encountered when assum- −→ ing an elastic rheology in this problem since frozen w,xx(x = 0) = 0 no torque (11) material is being accreted. This problem can be w(x = 0) = w(x = nπ/2) −→ periodicity(12) avoided if we neglect the addition of newly frozen w,xx(x = 0) = w,xx(x = nπ/2) −→ periodicit(13)y material in the geometry of the elastic material be- ing bent. What this means physically is that when a There is a corresponding set for the y direction nonuniform ice lens begins to form, only the material (however only the two-dimensional problem will be spanning from the ground surface, zs to the lowest considered here). The mechanical problem is coupled ice lens, zl, is actually being bent. In the sense of to the frost-heave problem through the load pressure the elastic model, the newly accreted material has an at zl. In the one-dimensional basic state, the frozen elastic modulus of zero. This approach is convenient region is planar and there is no bending (Pb = 0). because it avoids some inherent complexity, and there In the two-dimensional state, the non-zero Pb is a is considerable justification for its use. Experimental function of x. The load pressure is the sum of the data for Young’s modulus of frozen soil as a function weight of the overlying soil and the pressure due to of temperature show that the modulus approaches bending. zero as the subfreezing temperature approaches zero (Tsytovich, 1975, pg. 185). Large amounts of un- P = ρf gh1 + Pb (14)

6 4.4 Nondimensionalized Frost-Heave heave. This in turn will aid in understanding the Equations evolution of some types of patterned ground that are due to differential frost heave. The Fowler and Krantz frost-heave equations are two The LSA performed here follows standard linear coupled vector equations that describe the velocity of stability theory (Drazin and Reid, 1981). A com- both the upward-moving ice at the top of the lowest plete detailed analysis is available in Peterson (1999). v ice lens, i, and the downward-moving freezing front, The describing equations 15 and 16 are perturbed V f . Specifically, the equations are: around the one-dimensional basic state solution and

k linearized. Because the equations are linear in the − − f Gi − ˜ Gf [1 + δ (φ Wl) µ] k 1 + β perturbed variables, the solution to these perturbed V u f = variables will take the general form: δWl + φ + β˜ (φ − Wl)− µ (φ − Wl) W l (15) X0 = X(z, t) eiαx+iβy (18) kf vi = −αfk Gi − WlVf (16) where α and β are the wavenumbers of the distur- k b  u   bance in the x and y directions, respectively. Because where Gi and Gf are the thermal gradients in the un- one-dimensional frost heave occurs in the z direction, frozen and frozen regions, respectively, φ is the soil perturbations in both the x and y directions cause a porosity, Wl is the unfrozen water volume fraction three-dimensional shape to evolve. Initially, however, at the top of the frozen fringe (base of the lowest only the propensity for a two-dimensional instability ice lens), and kf and ku are the frozen and unfrozen will be investigated by setting β = 0. The functional thermal conductivities, respectively. Vectors are de- form of X in Equation 18 depends on whether the noted in bold and the coordinate system is oriented problem is steady-state or transient. Although the with z pointing up as shown in Figure 1. The value frost-heavbe process is inherently transient, it will be of Wl is determined using the lens-initiation criterion treated as quasi-steady state. If the instabilities grow (Fowler and Krantz, 1994) that involves the charac- very rapidly relative to the basic state, this assump- teristic function. tion is justified. A successful example using this tech- nique for alloy solidification is presented in Fowler p+1 1 − Wl (1997). In this instance, assuming quasi-steady state N Wl φ = 1 − fl = q (17) yields results somewhat close to experimental obser- φ φ  Wl    φ vations. Thus, the solution to the perturbed variables   takes the general form: The solution to the one-dimensional problem 0 iαx Γt is fairly straightforward. The steady-state, one- X = X0(z) e e (19) dimensional Laplace equation for temperature is where Γ is an exponential growth coefficient. solved to determine the relevant temperature gradi- The perturbed temperature distribution is deter- G G ents i and f . Work by Fowler and Noon (1993) mined in terms of the basic-state temperature distri- and Krantz and Adams (1996) has shown that this bution. First, the solution to Equation 2 subject to model predicts the one-dimensional frost-heave be- boundary conditions 3 and 4 yields the unperturbed havior observed in the laboratory quite well. temperature in the frozen region:

z − zf Tf = H Tair (20) 5 Linear Stability Analysis 1 + H (zs − zf )

A linear stability analysis (LSA) is performed here Solution of Equation 1 subject to boundary condi- to determine under what circumstances the one- tions 5 and 6 is trivial when Tb = 0. dimensional solution to the secondary frost-heave Tu = 0 (21) equations 15 and 16 is unstable to infinitesimal per- turbations. The conditions under which the one- The frost-heave equations (15) and (16) require dimensional solution is unstable will provide insight the thermal gradient at zf . Steady-state is also as- into the mechanisms responsible for differential frost sumed in the perturbed region in anticipation that

7 the time-scale for perturbation growth is greater than The perturbed variables are substituted into Equa- the time-scale for heat conduction. Starting with tions 15 and 16 and the equations linearized. Two the two-dimensional form of Equation 2, the tem- equations result that describe the perturbed ice and perature is perturbed and the differentiation carried freezing-front velocities in terms of perturbed quan- out. Boundary conditions are applied at perturbed tities.: locations, where the Taylor series is truncated after k the linear term. The perturbed temperature field is G0 f 1 + δ − φ − W µ − 1 + β˜ G0 − β˜0G = i k l f f solved and then differentiated again to yield:  u       V 0 δW + φ + β˜ φ − W − µ φ − W W + ∂T 0 f l l l l f ≡ 0 Gi = CT ηη + CT ζ ζ (22) h 0 0 0 i 0 0 ∂z Vf δW − β˜W + β˜ φ −Wl − µ φ −Wl W + µWlW z=zf l l l l

h   (29) i where and e2αzf + e2αzs CT η = −αGi (23) kf 2αzf 2αzs ˜ 0 ˜0 0 0 0 e − e 1 + β vi + β vi = −µφ Gi − WlVf − Wl Vf  ku α(zf +zs) 2αGiH e      CT ζ = (24) kf kf 2 2αzf 2αzs 0 0 0 0 α + H e − e +µ Wl Gi + WlGi − Wl Vf − 2WlWl Vf ku ku The thermal gradient below the freezing front is      − kf 0 ˜ kf ˜0 − ˜ 0 assumed to be zero in this analysis. This condition (1 + δ) Giβ + Giβ βWlVf ku ku is representative of freezing above per-     0 − ˜ 0 − ˜0 mafrost. Thus, Gf = Gf = 0. βWl Vf β WlVf The purturbed mechanical problem begins by per- i (30) turbing the pressure condition, Equation 14. The pressure is nondimensionalized with the pressure The perturbed temperature gradients and pressure scale σ. were determined in the previous sections. The re-

0 maining perturbed variables can be cast in term of P (z) + P (z, x) ρ gh + P 0 N + N 0 = = f 1 b (25) N by use of their definitions. Then using Equation σ σ 26, the two equations are only functions of η and The linearized form of N 0 is: ζ. This process is straight forward but algebraically complicated (Peterson, 1999). 0 N = e1η − e2ζ (26) The perturbed velocities are determined using the kinematic condition. The kinematic condition at the where freezing front is slightly complicated because the flux of material through the boundary zf must be ac- −ρf gh1 e1 = (27) counted for. However, because the accreted material σ is neglected in the elastic model, both conditions are − E 3 4 ρf gh1 + 9 h1 α straight forward: e2 = (28)  σ  ∂ζ v0 = = Γζ (31) i ∂t h1 is the thickness of the frozen region and E is Young’s Modulus. 0 ∂η Vf = = Γζ (32) The frost-heave equations (16) and (15) are com- ∂t plicated functions of the temperature gradients and Solution of the perturbed frost heave equation is the pressure at the top of the frozen fringe. Each now possible. Equations 29 and 30 can be expressed potentially-varying parameter must be perturbed in terms of Γ, α, η, ζ and parameters that are con- (e.g. replacing β˜ with β˜ + β˜0). Thus, the follow- stant for the specific case being analyzed. An eigen- ing parameters are perturbed: β, fl, Gi, Gf , N, Vf , value problem for the growth coefficient Γ as a func- vi, Wl, zf , and zs. tion of α results.

8 The environmental conditions include the temper- Table 1: Parameter values for frost-susceptible soils ature boundary conditions at the ground surface, zs,

p q φ k0 γ ρs and at the permafrost table, zb. The general New- [m [kg ton’s Law of Cooling boundary condition is applied at −2 −3 s ] m ] the ground surface. The effects of varying degrees of Chena Silt 1.48 0.66 0.48 1.22× 6.4 1378 −8 snow cover (including none) and differing vegetative 10 cover can be explored using this boundary condition. Illite Clay 2.44 7.27 0.67 1.69× 10.0 875 10−9 For simplicity, the gradient in the lower, unfrozen re- Calgary Silt 2.00 6.58 0.45 2.50× 8.9 1529 gion is zero for this analysis. 10−9 The final environmental condition is the freezing depth, h1. This parameter has a range in dimen- sional form of 0 → d0 where d0 is the depth of the active layer. It should be apparent that the freezing depth is a function of time. As freezing progresses, A(α)Γ2 + B(α)Γ + C(α) = 0 (33) h1 increases. Thus, by specifying a value for h1 in the reference set of parameters, the stability of one- The algebraically cumbersome expressions for A, B dimensional frost heave at a certain, frozen instant of and C can be seen in Peterson, 1999. An explicit time is being analyzed. The choice of h for the ref- expression for the two roots of Γ is easily obtained. 1 erence set was chosen to correspond to a time early in the freezing process when it is believed differen- 6 Linearized Model Predic- tial frost heave is most likely to be initiated. It will be shown in Section 6.7 that the mechanism for in- tions stability is related to differential heat transfer. The temperature gradient in the frozen region is greatest This section presents the predictions obtained from early in the ground-freezing process, and thus, DFH the linear stability analysis. It was shown in Section is more likely to initiate at early times in the freezing 4 that there are a large number of parameters that process. The quasi-steady-state assumption allows arise from the frost-heave model. These parameters for specification of h1. are divided into three groups: soil parameters, en- The physical constants that arise in this problem vironmental conditions, and physical constants. We are primarily concerned with the properties of ice define a reference set that comprises a set of envi- and liquid water. These values are summarized in ronmental conditions about which we will vary some Table 2. A constant temperature boundary condi- parameters and analyze their effect on the stability of tion of −10◦C was chosen for the top surface (Smith, one-dimensional frost heave. These conditions were 1985). This is accomplished in the thermal problem chosen to be representative of areas when differential by allowing H → ∞. Chena silt was chosen as the frost heave is observed. There are an additional six soil type because, as will be demonstrated, it has the parameters that describe the particular soil being an- greatest range of unstable behavior. A depth of freez- alyzed: p, q, φ, k0, γ and ρs. The first two come from ing of 10 cm, or about 10% of the maximum depth the characteristic function that relates the difference of freezing was also chosen. There is no surface load in ice and water pressures to the amount of unfrozen because in field situations the only overburden is the water. The values of p and q are determined em- weight of the frozen soil. A Young’s Modulus of 5 pirically by fitting the characteristic function to ex- MPa is used for the elastic modulus (Andersland and perimental data. In this paper three different types Ladanyi, 1994). of frost-susceptible soils are analyzed: Chena Silt, Illite Clay, and Calgary Silt. These soils were cho- 6.1 Effects of Parameter Variations sen because values for all six parameters could be obtained from previously published data (Horiguchi In the following sections the effects of varying dif- and Miller, 1983) with curve fitting when necessary. ferent parameters from their reference-set values are Table 1 lists the values for all six soil parameters for investigated. The results are presented by plotting the three soils investigated. the dimensionless growth rate, Γ, as a function of the

9 There are three values that are of most importance Table 2: Physical constants for the linear stability in each plot: Γ , α , and α , indicated in analysis model. max max ntl Figure 3. The maximum value of the growth rate, parameter symbol value Γmax, and the corresponding value of the wavenum- 3 −3 ber, α , indicate which mode grows fastest in the ice density ρi 0.9 × 10 kg m max 3 −3 water density ρw 1.0 × 10 kg m linearized model. Also, αmax indicates the lateral heat of fusion L 3.0 × 105 J kg−1 spacing of the most highly amplified, two-dimensional −1 −1 ice thermal conductivity ki 2.2 W m K −1 −1 mode through Equation 35. The neutrally stable water thermal conductivity kw 0.56 W m K −1 −1 wavenumber, αntl, indicates the maximum wavenum- heat capacity Cp 2.0 kJ kg K gravitational acceleration g 9.8 m s−2 ber for which modes can be linearly unstable. These values are not explicitly marked in subsequent figures to reduce clutter. 0.002 Γ max 6.2 Effect of Soil Type

0.001 The type of soil is the most influential parameter in the model when determining the stability of one- dimensional frost heave. It is important to note Γ+ that only a small fraction of soils exhibit frost heave 0.2 0.4 (Williams and Smith, 1989). A delicate balance α between the hydraulic conductivity of the partially α α frozen soil, particle size, and its porosity is necessary -0.001 max ntl for frost heave to occur. Silts and silty-clays meet these requirements in general. Furthermore, differ- Figure 3: Plot of the positive root of the linear stabil- ential frost heave is not necessarily observed in all ity relation as a function of the wavenumber for the frost-susceptible soils. Whether this observation is reference set of parameter values. The soil is Chena due to the specific soil or its environmental condi- Silt. Wavenumbers greater than αntl are stable and tions is a major objective of this analysis. Figure 4 αmax is the wavenumber with the maximum growth plots the growth rate as a function of wavenumber rate under these conditions. for the three soil types specified earlier in Table 1. All three soils are considered frost susceptible. However, for the reference set of parameter values, dimensionless wavenumber, α (see Equation 19). The only Chena Silt has a propensity for differential frost scaling factors for these parameters are heave (i.e. positive value of Γ+). It should be noted that these results do not indicate that Calgary Silt 2 1 d0ρwL −1 −1 and Illite clay will never heave differentially. Other Γs = = ≈ 350 days (34) [t] ku∆T parameters can have a significant effect on the stabil- ity predictions, and under the correct conditions, the 2π ≈ 6 −1 αs = meter (35) soils may heave differentially. λd0 λ The utility of a model such as this one hinges on For comparison purposes, the results will be pre- whether it can predict observations made in the field. sented in dimensionless form. The stability criterion, Unfortunately, there is a limited number of soil types Equation 33, is quadratic, and therefore there are to analyze due to a dearth of adequate characteriza- two roots Γ+, Γ−, with Γ+ > Γ−. In this analysis, tion data including hydraulic conductivity and sub- the lower root Γ− was always negative. The posi- freezing temperature as functions of unfrozen water tive value, Γ+, is plotted for the reference set using content. In fact, the work by Horiguchi and Miller Chena Silt in Figure 3. In all cases both roots are (1983) is one of the few comprehensive data sets for real numbers for the range of α values presented, and frost-susceptible soils available in the open literature. the principle of exchange of stabilities is valid (Drazin Fortunately, the predictions shown in Figure 4 do and Reid, 1981). agree with observations made in the field. In almost

10 0.01

0.008 E = 1.0 MPa 0.006

0.004

0.002 0.002 5.0 Chena Silt Γ 10.0 0.001 + 0.2 0.4 0.6 0.8 1 -0.002 α

Γ+ 0.1 0.2 0.3 0.4 0.5 Figure 5: The linear growth coefficient, Γ+, as a func- -0.001 Calgary Silt tion of wavenumber, α, for three values of the Young’s Illite Modulus, E. The soil type is Chena Silt. Clay α -0.002

Figure 4: The linear growth coefficient for three soil and others 1998), there is a large range of reported types using the reference set of parameter values. values. Andersland and Ladanyi (1994) report that for frozen silts, sands and clays, a Young’s Modulus of about 5 MPa is typical for the temperature range of interest: −10 → 0◦C. The effect of Young’s Modu- all instances, hummocks are observed in silty-clayey lus on the growth rate predictions is shown in Figure materials. The value of αmax for Chena Silt is of the 5. order unity. This wavenumber corresponds to a di- As might be expected, the range of unstable modes mensional wavelength on the order of meters, which decreases with increasing Young’s Modulus. The is also observed in the field (Williams and Smith, value of αmax decreases indicating that a “stiffer” 1989). The value of Γmax is several orders of magni- soil results in less closely spaced patterning. Be- tude less than unity indicating that differential frost cause Γmax decreases with increasing E, it would also heave will take (dimensionally) several years to de- take longer for patterns to become expressed in stiffer velop. It is commonly believed that most types of soils. It appears that increasing elasticity is a stabi- patterned ground form on the 10-100 year timescale lizing mechanism. As it becomes more difficult to (Hallet, 1987; Mackay, 1980). “bend” the frozen region, the propensity for differen- tial heave is reduced. 6.3 Effect of the Elastic Constant It is worth noting that only in the limit of E → ∞ does the propensity for differential frost heave at Here and in subsequent sections, we investigate the some value of α disappear. Both αntl and αmax ap- effects of parameter variations away from the ref- proach zero only when E → ∞. One major cause erence set for Chena Silt, which has the greatest for larger values of E for a particular soil would be propensity for differential frost heave. Although cal- colder temperatures. However, colder temperatures culating the effect of the elastic constant on linear also effect the thermal regime in the frozen region. stability is not difficult, interpreting the results can This effect is explored separately in Sections 6.4 and be confusing due to some of the assumptions made in 6.5. the mechanical model. The complication arises due to the difficulty in determining an appropriate value 6.4 Effect of the Ground Surface for the Young’s Modulus, E. While there are sev- Heat-Transfer Coefficient eral references that address the Young’s Modulus for frozen soils (Andersland and Landanyi, 1994; Puswe- In all the previous results we have assumed a con- wala and Rajapankse, 1993; Tsytovich, 1975; Yuanlin stant temperature of −10◦C at the ground surface.

11 T = -20 oC 0.003 air 0.2 H = 10 0.002 -10 oC

0.1 0.001 o Γ 100 -1 C + Γ+ 0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 α -0.05 100,000 α -0.001

Figure 7: The effect of warmer constant temperature Figure 6: The linear growth coefficient, Γ+, as a func- conditions on the growth of differential modes for the tion of wavenumber, α, for three values of the dimen- reference set of parameter values. The growth rate, ≡ −1 sionless heat-transfer coefficient H h do kf . The Γ is plotted in dimensional form. value of Γmax for H = 10 is two orders of magnitude greater than for a constant temperature boundary condition (H → ∞) changes in soil moisture after the hummock shape has matured. It is reasonable to assume that the differential height between the crest and the trough However, a constant-temperature condition is unre- would cause water drainage from the top that col- alistic for prolonged periods of time (i.e. > 1-2 days). lects in the inter-hummock spaces, thus propagating A lumped parameter boundary condition with an further organic growth in the depressions. overall heat-transfer coefficient is more reasonable for long-term analysis. The constant temperature con- 6.5 Effect of Ground-Surface Tem- dition is obtained from the heat-transfer-coefficient perature condition by allowing H → ∞. Figure 6 plots Γ+ as a function of α for three values of the dimension- Snow cover, which is almost always present to some less heat-transfer coefficient. A dimensional value of degree in arctic and subarctic regions, also has an the heat-transfer coefficient, h, can be obtained by insulating effect. However, snow is more likely to ≡ −1 reversing the scaling H h do kf . maintain a constant temperature at the ground sur- It is readily apparent that the “insulating” effect face due to its ability to function as a “heat sink” represented by smaller values of H causes a greater because of the relatively large latent heat associated range of unstable modes. Furthermore, as H de- with freezing. To analyze the effect of snow cover, it creases from ∞ to 10, Γmax increases two orders of is more useful to return to the constant temperature magnitude. Insulation obviously has a destabilizing boundary condition and vary the value of Tair. Since effect. H → ∞ for the constant-temperature condition, the Both αmax and the corresponding value of Γmax ground-surface temperature is equivalent to Tair. A increase as H decreases. There are several scenar- smaller (in magnitude) value of Tair corresponds to a ios in the field in which small values of H might be thicker and/or less thermally conductive snow layer. applicable. The most obvious instance is that of in- In fact, the only limitation in this case is Tair < 0. creased ground cover in the form of grasses, mosses The effect of the ground-surface temperature is shown and shrubs. It is noteworthy that the extensive sur- in Figure 7. vey of earth hummocks by Tarnocai and Zoltai (1978) In Figure 7 it can be observed that as the ground- supports this prediction. In essentially all places surface temperature warms, Γmax decreases. Fur- where earth hummocks are found, there is a mod- thermore, in the limit of Tair → 0, the growth co- erate amount of organic ground cover. The fact that efficient for all unstable modes tends to zero. These the tops of hummocks are sometimes more barren results indicate that the occurrence of differential than the trough regions is most likely the result of frost heave should be more prevalent in less snow-

12 0.002 The fact that there is a critical value of the freezing h1 = 0.1 depth is important when attempting to gauge what wavenumber might commonly be most expressed in 0.001 the field. In the limit h1 → 0, all wavenumbers have a positive growth rate. As freezing progresses and h1 0.2 increases from zero, the largest wavenumbers get “cut off” (i.e. Γ < 0) while the smaller wavenumbers Γ+ 0.1 0.2 0.3 0.4 0.5 + continue to grow, albeit at a slower rate. As h1 con- 0.3 α tinues to increase, more and more wavenumbers get -0.001 cut off. Eventually when h1 ≈ 0.23, all wavenumbers get cut-off and no differential modes are unstable. It can be seen in Figure 8 that very small wavenumbers Figure 8: The linear growth coefficient, Γ+, as a func- tion of wavenumber, α, for three depths of freezing, always have small positive growth rates until cut-off. Large wavenumbers have large growth rates but get h1. There are no positive roots (Γ+, Γ− < 0) at cut-off early. Mid-range wavenumbers have moder- freezing depths greater than h1 ≈ 0.23. ate growth rates for intermediate periods of time. Thus, mid-range wavenumbers might become most expressed in the field since they have a moderate covered areas than those where it accumulates, and length of time to grow and moderate growth rates in fact this is observed. A good example is the Toolik for most of the time. Lakes, Alaska, field site where D. A. Walker (personal Although linear theory is not strictly valid once communication) has observed that frost boils tend to differential modes grow to a finite amplitude, it is be more prevalent in the wind-blown areas. Frost insightful to see what wavenumber the model pre- boils are small mounds of soil material, ∼ 1 meter dicts would become most expressed during the entire in diameter, presumed to have been formed by frost period of freeze-up. This assumption is not exces- action (CAPS, 1998). Ground-surface temperature sively crude since the dimensionless growth rates are measurements in both regions confirm that surface much less than unity in this case,and the perturba- temperatures are indeed colder in the barren regions. tions are going to grow relatively slowly. As a simple approximation, we define the normalized growth rate as follows: 6.6 Effect of Freezing Depth . 1 ∂ζ ∂ ln ζ ζ ≡ = (36) It is appropriate to investigate the effect of freezing ζ ∂t ∂t ≡ − depth, h1 zs zf , on the stability theory pre- where ζ is the perturbation in the ground-surface lo- dictions. The nondimensionalization establishes that cation. The magnitude of ζ at time t0 is determined h1 is of the order one. When ground-freezing first by integration. commences, h1 begins at zero and increases (approx- 0 imately with the square root of time (Fowler and t . Noon, 1993)). By examining the behavior of αmax ln ζ = ζdt (37) 0 and Γmax for increasing values of h1, we are able to Z 0 ζ t . observe how the stability of the system changes as = exp ζdt (38) ground-freezing progresses. Figure 8 shows the effect ζ 0 "Z0 # of freezing depth on the stability analysis predictions. As one might expect, the largest range of unstable where ζ0 is the magnitude of the original perturbation modes occurs at the smallest values of the freezing at t = 0. depth and slowly decreases as h1 increases. However, Figure 9 shows the numeric results of integrating contrary to previous results where unstable modes Equation 38 for several values of the wavenumber α, existed for all values of a parameter, stabilization oc- indicated by the dots. The integration is performed 0 curs at a finite value of h1. For the reference set of until a time t at which point all modes become sta- parameter values shown here, that value is h1 ≈ 0.23, ble (i.e. h1 ≈ 0.23). The most expressed wavenumber which corresponds to a dimensional value of 23 cm. under these conditions is α = 2.3 which corresponds

13 3

2 2 ζ ζ 1 ζ0 1 ζ0

0 2 3 4 5 0.1 0.2 0.3 0.4 0.5 α h1 Figure 10: Evolution of the normalized ground- Figure 9: Relative amplitude of the ground-surface surface perturbation as a function of the freezing perturbation, ζ/ζ , for several values of the wavenum- 0 depth, h . ber, α, at the time t0 when all modes become sta- 1 ble. Soil type is Chena Silt and H = 10. At t = t0, h1 ≈ 0.23 atively small in magnitude. to a dimensional wavelength of 2.7 meters. This re- 6.7 Mechanism for Instability sult is quite encouraging since the spacing of hum- In the previous sections it has been demonstrated mocks is typically 1-3 meters (Tarnocai and Zoltai, that there is a range of modes that are unstable for 1978). Because all modes are stable for times greater most cases of interest. In fact, if the Young’s Modulus 0 than t (or h1 > 0.23), a further increase in frost is allowed to approach zero (E → 0), all modes are depth will result in all modes decreasing in ampli- unstable for some soils while no modes are unstable tude. for other soils. In this limit, the bending resistance To determine the evolution of ζ/ζ0 in time as h1 of the frozen soil is essentially removed (see Equation increases, Equation 38 is integrated in time up to t0 7). Figure 11 shows the LSA prediction for the ref- for a given wavenumber. The results for α = 2.6 are erence set of parameter values and Chena Silt except shown in Figure 10. This figure shows the evolution that E = 0. It is evident that the elasticity of the of ζ/ζ0 as freezing progresses (i.e. h1 increases). The frozen layer is providing the stabilizing mechanism normalized perturbation amplitude reaches a maxi- for differential frost heave. mum at h1 ≈ 0.08 and then begins to decrease. At In order to help understand the mechanism that h1 = 0.2, ζ/ζ0 ≈ 2.65 as expected from Figure 9. The causes the instability, Figure 12 shows a schematic amplitude returns to its initial value at h1 ≈ 0.36 and of the in-phase perturbed surfaces. Also sketched continues to decrease as h1 increases further. These are approximate isotherms for the constant temper- results indicate that an initial perturbation of mag- ature boundary condition at the top surface. The nitude ζ0 will have increased in magnitude at the end large arrow pointing to the right shows the net direc- of the freezing season for active-layer depths less than tion of differential heat flow that is occurring from a 0.36 meters. crest region to a trough region. Before the perturba- There is no assumption in this derivation that the tions occur, heat transfer occurs only in the vertical perturbation grows as exp[Γ t] as linear theory re- direction. However, once the system is perturbed, quires. In this sense, this is a real-time analysis. heat transfer can occur in both the horizontal and However, other assumptions have implications that the vertical directions. Solving Equation 33 for η/ζ must be examined. Perhaps most importantly, all indicates that the magnitude of the bottom surface product terms in perturbed variables are assumed perturbation, η, is less than the top surface pertur- small and neglected. Hence, this real-time analysis bation, ζ, albeit a small difference. However, linear is still only applicable when all perturbations are rel- stability theory predicts perturbations grow exponen-

14 tially. Thus, although the difference between η and ζ is small initially, that difference will grow expo- nentially fast (at least until nonlinear terms become significant). In fact, it is difficult to speculate about 60 what form the perturbations will take once nonlinear terms become significant. Figure 12 shows that the thickness has increased from the basic state value un- 40 derneath a crest, and decreased underneath a trough. Γ Because the temperatures at the boundaries zf and + zs are fixed at the basic-state values and the conduc- 20 tion path-length has changed, the temperature gradi- ent at the bottom surface, Gi, has also changed. Un- derneath a crest the gradient has decreased in mag- nitude and, according to Equation 15, the velocity 40 α 80 120 of the freezing front will also decrease. Continuing this line of reasoning, since the conduction path un- derneath a trough has decreased, the gradient at zf Figure 11: Stability predictions for the fundamental has increased and the velocity of the freezing front case when the Young’s Modulus is set to zero (E = 0). has correspondingly increased. Since there is a posi- tive feedback mechanism occurring at this point, the perturbations will continue to grow. In order to understand the increase in Γ with in- creasing wavenumber observed in Figure 11, it is nec- essary to consider the magnitude of differential heat flow shown in Figure 12. As the wavenumber in- creases, the magnitude of heat flux from a crest region to a trough region increases. In order for frost heave to occur, the latent heat of the incoming water at zl ζ must be removed. Since there is differential flow from Z a crest to a trough, additional heaving can occur un- s derneath the crest region. This also corresponds to less heave occurring in the trough region.

6.8 Implications of the Steady-state Assumption

This analysis is based on the quasi-steady state as- η Z sumption that perturbations grow much faster than f changes in the basic state. However, predictions indi- cate that the growth rate Γ is actually less than unity Figure 12: Schematic of the perturbed surfaces show- in most cases, and thus perturbations grow slowly rel- ing isotherms and the direction of differential heat ative to changes in the basic state. For some time pe- flow. The basic state surfaces z and z are shown s f riodic problems with slow growth rates, Floquet the- by the dashed lines and the perturbed surfaces are ory has been successfully applied when solving the the thick solid lines. linear stability problem (Drazin and Reid, 1991, p. 354). However, the entire freeze-thaw cycle (or pe- riod) must be modeled. This analysis has only ad- dressed the frost-heave process that occurs during freezing, which is only part of a larger cycle. The

15 thawing process involves solifluction, soil consolida- (Washburn, 1980; Williams and Smith, 1989). How- tion, water percolation and evaporation, and possibly ever, the mechanisms discussed here apply only to other processes. the freezing process. Active-layer thaw is itself a The present analysis indicates that the frost-heave unique process and may possess additional mecha- process alone is capable of causing spontaneous pat- nisms that could give rise to patterning. One possi- tern generation, albeit, on a 10-100 year timescale. bility is the buoyancy-induced soil circulation theory Combination of the frost-heave model with a thaw presented by Hallet and Waddington (1991). Cou- model could possibly yield more conclusive theoreti- pling the DFH model with Hallet and Waddington’s cal evidence that patterns are generated by the pro- model might provide a more complete explanation of cesses discussed here. It may be that the mature pattern formation and soil circulation by accounting periglacial patterns observed are actually the result for both the freezing and thawing processes. of a complex combination of all the processes men- The use of thin plate theory to describe the up- tioned above. per, frozen region as a purely elastic material is re- stricted to the linear region. Vertical displacements 7 Conclusions are assumed very small. Furthermore, thin plate the- ory assumes h1  2πλ, or that the thickness of the region being deformed is much less than the charac- In this paper we have demonstrated that the frost- teristic lateral dimension. This assumption is obvi- heave model due to Fowler and Krantz (1994) is lin- ously not valid for larger freezing depths. To describe early unstable under a range of environmental con- the evolution of DFH after initiation when these as- ditions and for several (but not all) soil types. Dif- sumptions break down, a more comprehensive elastic ferential heat flow coupled with the delicate balance constitutive relation needs to be implemented. Al- between hydraulic conductivity and cryostatic suc- ternatively, a different rheology can be used such as tion potential act as a destabilizing mechanism for viscous, viscoelastic, or viscoplastic. These rheolo- one-dimensional frost heave. When the upper frozen gies may in fact describe the long-term evolution of region is modeled as a purely elastic material, the DFH more accurately than a purely elastic one. force required to deform the frozen region acts as a stabilizing mechanism to differential frost heave. Pre- The steady-state assumption can only be used as vious stability analyses in the literature (Lewis and a first approximation and is obviously invalid near others, 1993; Fowler and Noon, 1997) have failed to marginal stability. This drawback points to the ne- describe accurately the mechanisms at work that can cessity of solving the nonlinear equations as a time- give rise to differential frost heave. The analysis pre- evolution problem. While a nonlinear stability anal- sented here has corrected errors in how Lewis (1993) ysis can provide further information about the ob- coupled the mechanical problem to the frost heave served planform and sub- or supercritical stability, problem, and identified a source of instability that it must also use a frozen-time assumption. Current was overlooked by Noon (1996) in the case of a con- stability theory cannot adequately account for the ef- stant temperature boundary condition. fects of a changing basic state. Thus, a numerical so- The mechanisms responsible for differential frost lution of the time-evolution problem is probably the heave are dependent on the instantaneous depth of most valuable continuation of this investigation. freezing and can effectively be shut off when depths of freezing exceed a critical value. Numerical inte- gration of the predicted growth rates as freezing pro- gresses beyond this critical value indicates that there 8 Acknowledgements is an intermediate-valued wavenumber that is most expressed. We greatly appreciate the helpful comments from The dimensionless growth rates predicted by the Stephen Jones, Mathew Sturm, and an anonymous theory presented here are small,  1, indicating that reviewer. This work was support by the National pattern expression would take tens to hundreds of Science Foundation Grant OPP-9321405. RAP ac- years to mature. This predicted length of time is knowledges the American Geophysical Union for fi- similar to the scale reported by many field observers nancial support.

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17 Konrad, J. M. and N. R. Morgenstern. 1982. Pre- Nakano, Y. 1999. Existence of travelling wave solu- diction of frost heave in the laboratory during tions to the problem of soil freezing described by transient freezing. Canadian Geotechnical J., a model called M1. CRREL Report, 99-5. 19, 250-259. Noon, G. C. 1996. Secondary frost heave in soils. Konrad, J. M., M. Shen and R. Ladet. 1998. Predic- (D. Phil. thesis, Oxford, Oxford University.) tion of frost heave induced deformation of Dyke KA-7 in Northern Quebec. In Permafrost: Sev- O’Neill, K. and R. D. Miller. 1982. Numerical so- enth Int. Conference, Yellowknife, Canada, 595- lutions for a rigid-ice model of secondary frost 83-13 599. heave. CRREL Technical Report, . Krantz, W. B. and K. E. Adams. 1996. Validation O’Neill, K. and R. D. Miller. 1985. Exploration of a of a fully predictive model for secondary frost rigid ice model of frost heave. Water Resources 21 heave. Arctic and Alpine Res. 28 284-293. Research, , 281-296. Lewis, G. C. 1993. A predictive model for differen- Peterson, R. 1999. Differential frost heave manifest tial frost heave and its application to patterned as patterned ground: modeling, laboratory and ground formation. (M.S. Thesis, University of field studies. (Ph.D. thesis, University of Col- Colorado, Boulder.) orado, Boulder.) Lewis, G. C., W. B. Krantz and N. Caine. 1993. A Puswewala, U. G. A. and R. K. N. D. Rajapankse. model for the initiation of patterned ground ow- 1993. Computational analysis of creep in ice and ing to differential frost heave. In Proceedings of frozen soil based on Fish’s unified model. Cana- the Sixth Int. Conference on Permafrost. 1044- dian J. Civil Eng., 20, 120-132. 1049. Smith, M. W. 1985. Observations of soil freezing Mackay, J. R. 1980. The origin of hummocks. Cana- and frost heave in Inuvik, Northwest Territories, dian J. of Earth Sciences, 17, 996-1006. Canada. Canadian J. Earth Science, 22, 283- 290. Miller, R. D. 1977. Lens initiation in secondary heaving. In Proceedings of the Int. Symp. Frost Taber, S. 1929. Frost heaving. J. Glaciol., 37 428- Action in Soils, 68-74. 461. Miller, R. D. 1978. Frost heaving in non-colloidal Tarnocai, C. and S. C. Zoltai. 1978. Earth hum- soils. In Third Int. Conference on Permafrost, mocks of the Canadian arctic and subarctic. Edmonton, Canada, 708-713. Arctic and Alpine Res., 10, 581-594. Miller, R. D. 1980. Freezing phenomena in soil. In Tsytovich, N. A. 1975. The Mechanics of Frozen D. Hillel, ed. Applications of Soil Physics, Aca- Ground. McGraw-Hill, Washington DC. demic Press, New York, 254-299. Washburn, A. L. 1980. Geocryology, a Survey of Nakano, Y. 1990. Quasi-steady problems in freezing Periglacial Processes and Environments. John soils: I. Analysis on the steady growth of an ice Wiley and Sons, New York. layer. Cold Regions Science and Tech., 17, 207- 226. Williams, P. J. and M. W. Smith. 1989. The Frozen Earth. Fundamentals of Geocryology Cambridge Nakano, Y. and K. Takeda. 1991. Quasi-steady University Press. problems in freezing soils: III. Analysis on exper- imental data. Cold Regions Science and Tech., Yuanlin, Z., H. Ping, Z. Jiayi, and Z. Jianming. 19, 225-243. 1998. Effects of temperature and strain rate on the constitutive relation of frozen saturated silt. Nakano, Y. and K. Takeda, 1994. Growth condition In Permafrost: Seventh Int. Conference. Yel- of an ice layer in freezing soil under applied loads: lowknife, Canada. 1235-1239. II. analysis. CRREL Report, 94-1.

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