Introduction

Some dry-slab avalanches that threaten people and property initiate in weak layers of faceted crystals on crusts within the mountain snowpack. The associated slab avalanche hazard may persist since these faceted layers can remain weak for weeks or months. There are various snowpack processes that cause kinetic growth (faceting) of crystals. We focus on the formation of faceted crystals directly above wet layers buried by dry snow. A dry snow layer overlying a wet layer is typically subjected to a temperature and vapour gradient caused by the upper surface of the dry snow layer being below the freezing point, while latent heat in the wet layer holds the lower boundary of the dry layer near the freezing point until the wet layer freezes.

In this paper, we derive an equation for the freezing time of the wet layer, compare the calculated values with measured freezing times from seven cold-laboratory experiments (Reference Jamieson and HerwijnenJamieson and Van Herwijnen, 2002), identify the faceting that occurred while the wet layer was unfrozen, assess the variables that influence the freezing time and hence sustain the temperature gradient in the overlying dry snow, and model the formation of faceted crystals in a dry layer overlying a wet layer with the Swiss model SNOWPACK (Reference Bartelt and LehningBartelt and Lehning, 2002; Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others, 2002a, Reference Lehning, Bartelt, Brown and Fierzb).

Literature Review

Reference BirkelandBirkeland (1998) described the various processes that lead to the formation of faceted crystals near the surface of the snowpack, including the formation of faceted crystals in dry snow overlying a wet layer. Reference Fukuzawa and AkitayaFukuzawa and Akitaya (1993) observed that a 60 mm wet layer under a 20 mm dry layer required 6 hours to freeze during which grains in the dry layer became faceted. Reference Jamieson, Geldsetzer and StethemJamieson and others (2001) used weather data to argue that latent heat from a wet layer buried by dry snow in November 1996 contributed to the formation of faceted crystals in an overlying dry snow layer, which released over 700 large slab avalanches by 31 March 1997 in the Columbia Mountains, Canada. Also using weather data, Reference Greene and JohnsonGreene and Johnson (2002) argued that latent heat from a wet layer may have contributed to the formation of a faceted layer below a crust at lower elevations in the Wasatch Mountains, Utah, U.S.A. This faceted layer released over 200 slab avalanches over 70 days. In seven of seven cold-laboratory experiments, Reference Jamieson and HerwijnenJamieson and Van Herwijnen (2002) observed the growth of faceted crystals in a dry layer sieved onto a wet layer and subjected to subfreezing air temperature, but did not distinguish between faceting that occurred before and after freezing of the wet layer.

Reference ArmstrongArmstrong (1985) modeled a 1 m thick layer of rain-wetted snow that took 6 days to freeze during which the temperature gradient in the overlying layer of dry snow was favourable to faceting. Based on Reference Carslaw and JaegerCarslaw and Jaeger’s (1959) equations for heat flow in solids, Reference Colbeck and JamiesonColbeck and Jamieson (2001) derived an equation for the unsteady temperature profile in dry snow overlying a wet layer, and an equation for the heat required until the temperature profile became constant. Using Reference ColbeckColbeck’s (1983) grain-growth model, they showed that substantial kinetic grain growth at the interface was likely within hours.

In this paper, we derive the freezing time of a wet layer buried by dry snow and compare calculated freezing times with freezing times measured during cold-laboratory experiments. The freezing times and observed faceted crystals are also compared with those modeled by SNOWPACK (Reference Bartelt and LehningBartelt and Lehning, 2002; Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others, 2002a, Reference Lehning, Bartelt, Brown and Fierzb).

Theoretical Heat-Flow Model

The model consists of two snow layers (Fig. 1). The lower wet layer of thickness h has liquid-water content (volume fraction) θ and is insulated at its base. The upper dry layer of thickness H has a density ρ and effective thermal diffusivity κ. While the wet layer remains unfrozen, the temperature of the interface between the two snow layers is T₀ = 0°C. The latent heat per unit area in the wet layer is Lθh, where the latent heat per unit volume of water is L = 3.3 × 10⁸ J m^–3. The temperature of the dry layer is initially T_H < 0°C, and that of its upper surface is constant T_H < 0°C, causing heat to flow from the lower wet layer to the upper surface of the dry layer.

Fig. 1. Model and nomenclature used for heat flow from a wet to a dry snowpack layer.

The temperature T at time t and depth 0 ≤ z ≤ H is given by Reference Colbeck and JamiesonColbeck andJamieson (2001) based on Kondrat’eva (1954) or Reference Carslaw and JaegerCarslaw and Jaeger (1959):

(1)

Differentiating with respect to z yields the unsteady temperature gradient in the dry layer

(2)

where (T_H - T₀)/H is the average temperature gradient across the dry snow layer.

Evaluating T′ at the z = 0 interface where the cosine term in Equation (2) becomes one, the wet layer will freeze at time t_f when the latent heat has flowed across the interface:

(3)

where the effective thermal conductivity of the dry layer is k = κρC_p and C_p is the specific heat of ice or dry snow, approximately 2100 J kg^–1. Thermal conductivity can be estimated from density or grain type (Reference Sturm, Holmgren, König and MorrisSturm and others, 1997).

Evaluating the integral in Equation (3) yields

(4)

This equation implicitly gives the freezing time t_f, in terms of the properties of the two layers and the boundary conditions.

To obtain an explicit approximation for t_f, we seek simplification of the two infinite series in Equation (4). The first series equals π₂/6. To show the second series is small compared to the first, we consider its range for 1hour ≤ t_f ≤ 24 hours and 0.05 m ≤ H ≤ 0.5m. Since freezing times will increase with the thickness of the dry layer H, we constrain t_f/H₂ >250 000 sm^–2, approximately 70 hoursm^–2. For a 0.05 mthick dry layer, this constraint restricts freezing times to be at least 10 min, and for a 0.5 m thick dry layer freezing times must be at least 17 hours, values which are consistent with the field observation of Reference Fukuzawa and AkitayaFukuzawa and Akitaya (1993) and cold-laboratory experiments of Reference Jamieson and HerwijnenJamieson and Van Herwijnen (2002). Subject to this constraint, the second series ranges between 0 and 25% and averages 5% of the first series for 1hour time-steps and H = 0.05m, 0.1m, 0.2 m, 0.35 m and 0.5 m. Dropping the second and smaller series in Equation (4), the freezing time is approximated

(5)

This approximation is, in fact, a lower bound since the second series in Equation (4) is always positive. Equation (5) can be compared with experimental values or field values since all the variables can be measured. However, boundary conditions are unlikely to be constant as assumed for the derivation of Equations (1–5). In particular, the theoretical model is insulated below the initially wet layer, whereas in most realistic snowpack conditions there are more snow layers below the initially wet layer, which can both drain liquid water and draw heat from the wet layer.

The effect of slab thickness and wet-layer thickness on freezing time (Equation (5)) for ρ = 80 kgm^–3, θ = 0.048 and T_H = –10°C is shown in Figure 2. The thin wet layer, h = 0.01m, freezes within 4–8 hours and with little dependence on slab thickness, H = 0.05–0.2m. The thicker wet layer requires 25–90 hours to freeze, with freezing time increasing with slab thickness.

Fig. 2. Effect of thickness of slab and wet layer on freezing time for θ = 0.048 by volume; ρ = 80 kg m^–3, T_H = -10°C. The freezing time is only weakly dependent on the thickness of the slab for the thin wet layer.

Cold-Laboratory Experiments

Experimental methods

Reference Jamieson and HerwijnenJamieson and Van Herwijnen (2002) describe seven experiments in the Rogers Pass (British Columbia, Canada) cold laboratory, which varied in length from 2 to 9 days. In each experiment a wet layer was located between dry layers, and the upper surface cooled below the freezing point. These three snow layers were placed in a box (0.50m×0.50m×0.42m high) insulated on the sides and the base with 70 mm thick foam (Fig. 3). One wall of this box was removable to facilitate placing the basal block of snow and subsequent observations of crystals. The base layer for the experiments ranged in thickness from 0.10 to 0.21 m and consisted of a block of natural snow with a density of 200–300 kgm^–3 that was slid laterally into the bottom of the box where it fit tightly. The middle layer consisted of wet snow made from mixing dry snow with liquid water and spreading the mixture onto the base layer. The thickness of the wet layer ranged from 0.015 to 0.09 m (Table 1), and was taken as the average thickness of the crust at the end of the experiment. The liquid water was calculated from the volume of liquid water added to the mixture divided by the cross-sectional area of the box. This was in excess of the irreducible liquid-water content. The upper layer (subsequently referred to as the slab) consisted of natural snow sieved onto the wet layer and ranged in thickness from 0.055 to 0.14 m.The thickness of the slab was measured at the start and end of each multi-day experiment, and the initial thickness used for calculations. The density of the dry slab was measured at the end of each experiment with a 10^–4 m³ cylindrical sampler and digital weigh scale. Twelve thermistors, calibrated to within 0.1°C of 0°C in slush, were placed through the side of the box and 0.010–0.055m vertically apart in the wet layer and overlying dry snow. Depending on the thickness of the wet layer, there were one to three thermistors placed in it. The temperature of each thermistor was measured every minute and the average temperature of each thermistor was recorded every 30 min by a datalogger. During each observation, the removable wall of the box was detached to facilitate extraction of crystals from the base and middle of the slab. For each observation, sampling of crystals was at least 0.05 m from the site of previous samples and from any wall. The grains at the base and middle of the slab were observed through an 8× hand lens and a microscope, and classified according to Reference ColbeckColbeck and others (1990) at least once per day for the first few days. For the first two experiments, a control block consisting of dry snow sieved onto a dry base layer was prepared, and the crystals observed in the upper dry layer. The air temperature in the cold laboratory varied from –8° to –16°C several times per day according to the cycles of the compressor. The average air temperature during the freezing time was used as the surface temperature in Equation (5).

Fig. 3. Perspective diagram of insulated box used for measuring temperature profiles and changes in snow crystals in the dry snow (slab) above the wet layer. The inside dimensions of the box are 0.50 m by 0.50 m by 0.42 m high.The insulated front door of the box is removable and not shown.

Table 1. Observations for cold-laboratory experiments

Observations

Settlement of the dry slab over the multi-day experiments averaged 0.01m. No gaps formed between the three-layer snowpacks and the fixed walls of the box, although ice forming at the base of the box tended to push the removable wall away from the snowpack, often resulting in a gap of 0.02–0.04m.

Except for the experiments that started on 14 February and 11 March 2002, frozen percolation channels in the base layer and/or ice at the bottom of the base layer were observed, or warming to near the freezing point occurred in the base layer, indicating that liquid water drained from the wet layer.

The temperature profiles show the middle layer was at 0°C in all experiments for at least 2 hours. In Figure 4, we show six temperature profiles after 1–41 hours from the experiment that started on 25 March 2002 to exemplify the evolution of heat flow. The thermistors located at z = -0.02m and z = -0.03m were at 0°C after 1 and 4 hours, but only the thermistor at z = -0.03m was at 0°C after 9 hours. During this period the strong temperature gradient in the slab was drawing heat upwards; the base layer was at –1° to –2°C and had little gradient, likely because liquid water from the wet layer drained through percolation channels into the base layer. The wet layer was deemed to have frozen after 9 hours, when the last thermistor in the wet layer dropped below –0.5°C. The temperature profiles at 12 and 18 hours show that the initially wet layer had frozen into a crust and was cooling. The temperature gradient in the slab was decreasing but was still sufficient for faceting (<10–20°Cm^–1) (de Quervain, 1958; Reference AkitayaAkitaya, 1974; Reference ColbeckColbeck, 1983), and the temperature gradient in the base layer had increased, likely because of liquid water at the bottom of the base layer. After 41 hours, the base layer was cooling and the temperature gradient in all three layers was decreasing.

Fig. 4. Six temperature profiles after 1–41 hours from the experiment started on 25 March 2002. The profiles show that part of the middle layer (“crust”) was wet for approximately 9 hours.

In all seven experiments with the initially wet layer, evidence of faceting was observed at the base of the slab after the first post-set-up observation. In the last four experiments this was within 5 hours or less and before the wet layer froze (Table 1). In the first three experiments, faceting was observed in the first post-set-up observation after 17– 21 hours, but at least 6 hours after the wet layer froze. Faceting was not observed before the wet layer froze in the first three experiments either because it did not occur or because the first post-set-up observation was too late. The initial and final grain types and their size along with the thickness of the layers, the density of the dry layer and the freezing time of the wet layer are shown inTable 1for each experiment. As the dominant grain form, faceted crystals were observed at the interface before they were observed in the middle of the slab in four of the seven experiments and during the same observation in the remaining three experiments.

For the first two experiments in which control blocks were prepared without the middle wet layer, no faceting was observed in the upper dry snow layer, indicating that the wet layer was necessary for the observed faceting of the slab.

Comparison with Theoretical Model

We compare the freezing times from the experiments with values from the theoretical model since both approaches provide the freezing time, and these were measured to within 0.5 hours in the experiments. However, several of the conditions for these experiments were not the same as assumed for the theoretical model. The air temperature in the cold laboratory, and hence the surface temperature of the slab, varied during the experiments. The initial temperature of the slab was likely not T_H throughout its thickness. The wet layer likely froze at its upper boundary first, causing the interface temperature to fall below 0°C before the wet layer froze throughout its thickness. Most importantly, since the liquid water added was well above the irreducible water content θ_IRR of approximately 0.048 by volume (Reference Coléou and LesaffreColéou and Lesaffre, 1998), drainage was expected in all experiments.We assume the latent heat associated with the irreducible liquid-water content flowed upwards and the excess liquid-water content drained, warming the base layer. Consequently, we use θ = θ_IRR in Equation (5). Thermal diffusivity κ of the slab was calculated from k/ρC_p after thermal conductivity k was calculated from the density of the slab (Reference Sturm, Holmgren, König and MorrisSturm and others, 1997). The calculated freezing times are plotted against the measured freezing times in Figure 5. Excluding the experiment that started on 21 March 2002, the remaining calculated freezing times are comparable to corresponding measured values (coefficient of determination, R² = 0.83; average error 59%). The wet layer in the experiment that started on 21 March 2002 froze much faster than the calculated freezing time; perhaps the wet layer was locally thinner near the thermistors. Including this experiment increases the average error to 87% and R² drops to 0.47. Given the substantial differences in boundary and initial conditions between the theoretical model and the experiments, approximate freezing times from Equation (5) show reasonable agreement with measured freezing times. Also, six of the seven measured freezing times show the same trend as calculated freezing times, indicating that the effect of the variables H, h, k, ρ, T_H and θ = θ_IRR is appropriately represented in Equation (5).

Fig. 5. Comparison of measured freezing times from seven cold-laboratory experiments with freezing times simulated by SNOWPACK and calculated freezing times assuming that latent heat associated with the irreducible liquid-water content flows up from the wet layer into the dry layer. The 1:1 line shows that calculated and simulated freezing times are comparable to measured values for six of the seven experiments and exhibit the same trend.

Using the data fromTable 1, freezing times from the approximate solution (Equation (5)) can be compared with times from the implicit solution (Equation (4)). The differences are <0.2% except for the first experiment, which differs by 7.6%. This shows the utility of the approximate solution for a range of conditions similar to those found in the experiments.

Comparisons with Simulation

Simulation model

The one-dimensional snow-cover model SNOWPACK solves the unsteady heat-transfer and creep/settlement equations using a Lagrangian finite-element method. A full description of the model is beyond the scope of this paper, and the reader is referred to the publications by Reference Bartelt and LehningBartelt and Lehning (2002) and by Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others, 2002a, Reference Lehning, Bartelt, Brown and Fierzb.

In the present application, temperatures measured at the air/surface interface as well as T_B recorded by the lowest thermistor were used as Dirichlet boundary conditions for the model. Regarding thermal conductivity, Reference Fierz and LehningFierz and Lehning (2001) showed that SNOWPACK’s implementation follows quite closely the density-dependent equation given by Reference Sturm, Holmgren, König and MorrisSturm and others (1997). Finally, note that grain form was coded as a combination of the model parameters sphericity and dendricity (Reference Lesaffre, Pougatch and MartinLesaffre and others, 1998; Reference Fierz and BaunachFierz and Baunach, 2000). Sphericity is 0 for fully faceted grains and 1 for fully rounded grains (e.g. small rounds or wet grains). A dendricity larger than 0 is assigned to new-snow and decomposing particles, the size of which is set to 0.3 mm by default. Those assignments are somewhat user-dependent and may substantially affect grain evolution. For instance, grain growth does not start before dendricity reaches 0.

To initialize the model, the bulk properties of each layer as given in Table 1 were assigned to a suitable number of finite elements. Accordingly, for all experiments except the one started on 25 March 2002, initial dendricity and initial sphericity in the slab were both set to values greater than zero. Bulk properties for the base layer as well as for the wet layer were assumed to be the same for all experiments (Table 2) except for the initial liquid-water content in the wet layer (Table 3). Irreducible water content was fixed to 0.048 by volume for all layers. Element temperatures at the beginning of a simulation run were inferred from corresponding measured temperature profiles. Simulated freezing time was defined as the point at which all liquid water present anywhere in the snowpack freezes, which in most cases corresponds to the bottom of the wet layer. In any case, though, this time was found to be shorter by no more than an hour as compared to the temperature of the corresponding model element dropping below –0.5°C. Thus, for the sake of comparison, the longer times are shown in Table 3 along with the experimentally determined freezing times.

Table 2. Element properties in the wet and basal layer for SNOWPACK simulations

Table 3. Comparison of cold laboratory experiments with SNOWPACK simulations

Comparison of simulations with approximate solution

Assumptions of the theoretical model presented above are that no heat may flow downwards from the wet layer, and that it holds all the liquid water added. We simulated this situation by setting the irreducible water content close to the value corresponding to 100% of water added (Table 3). Resulting freezing times were two to four times longer than the observed freezing times, though. Still adding 100% of water to the wet layer but setting the irreducible water content to 0.048 by volume, SNOWPACK allowed the excess water to percolate instantaneously into the lower layer, thereby warming the whole base to the freezing point (see Fig. 6). However, this approach led to even longer simulated freezing times than previously found.

Fig. 6. Comparison of measured with simulated temperature profiles for the experiment started on 25 March 2002: (a)4 hours after experiment started (phase 1); (b) 18 hours after experiment started (phase 2). Open circles: measurement; solid line: simulation with θ = 0.05 and θ_IRR = 0.048 by volume; dashed line: simulation with θ = 0.28 and θ_IRR = 0.048 by volume; dot-dashed line: simulation with θ = 0.28 and θ_IRR = 0.27 by volume.

Finally, we sought a“best-fit” solution not only by matching observed and simulated refreezing times but also by considering a closest agreement of simulated with measured temperatures at various heights within the three-layer pack. Best results were obtained for the liquid-water contents θ shown inTable 3. Note that θ amounted to at most twice the fixed irreducible water content of 0.048 by volume corresponding to the dry density of the wet slab (Reference Coléou and LesaffreColéou and Lesaffre, 1998). This agrees well with the assumption made for the theoretical model above, but it also means we had to assume that most of the liquid water drained through to the bottom even before the first recorded temperature profile. The reasonable agreement of “best-fit” simulations with both measured and approximated freezing times is shown in Figure 5. The consistent overestimation of freezing time even by “best-fit” simulations may be due to the fact that heat removed by evaporation of some liquid water present in the wet layer was not properly taken into account.

Discussion

Equation (5) can be used to assess the effects of different variables on the freezing time and hence on the time during which a strong temperature gradient favourable to the kinetic growth of crystals exists in the overlying dry snow. The first term is proportional to the dry-snow thickness H. For dry-snow density ρ = 80 kgm^–3, surface and initial slab temperature T_H =–10°C and h = 0.095m, the freezing time increases five-fold as H increases six-fold from 0.05 m to 0.3 m.

Model simulations take account of all layers involved as well as of their characteristics, which are inferred from observations (Table 1). One advantage of simulations, however, is the possibility of running the same experiments while varying either θ or θ_IRR or both in order to corroborate simpler solutions such as the approximate one for calculating freezing times. Thus we may explore the consequences of extreme settings. Obviously, putting and holding 100% of water in the wet layer does not reproduce the experiment. It may mimic an impeding horizon such as an ice lens underlying the wet layer, though. In that case, simulations show to what extent faceting and kinetic growth could be enhanced (see Table 3). As stated previously, grain growth will only start once dendricity is zero. Thus, final grain-size will depend on how efficiently dendricity is reduced, which in turn depends strongly on the temperature gradient at the interface. Here observed trends are qualitatively well reproduced by “best-fit” simulations.

Since faceting at the base of the slab was observed prior to freezing in four experiments and in the first observation after freezing in the remaining three experiments, the kinetic growth process is initially dominant. While the assumptions behind the theoretical model were often not fulfilled in the experiments, the strong temperature gradient was sufficient for faceting during the measured freezing time or, in most cases, during the approximated freezing time. Hence, Equation (5) provides a simple estimate of the time during which the temperature gradient is conducive to faceting.

Conclusions

The temperature regime after a wet layer is buried by dry snow can be divided into two phases: freezing of the wet layer (phase 1), and subsequent decay of the temperature profile to a quasi-steady-state temperature profile (phase 2). Strong temperature gradients conducive to faceting occur, especially at the interface between the dry and wet layers, during the heat flow from a freezing wet layer into an adjacent dry layer. The duration of the strong heat flow can be estimated from Equation (5). In addition to the factors in this equation, the freezing time will depend on the time during which the wet layer is exposed to cold air prior to burial by dry snow, and on the time over which the dry layer accumulates.

Application of Equation (5) to practical problems such as avalanche forecasting will require measurement of the thickness of the wet and overlying dry layer, the density of the dry layer and the surface temperature. Partial freezing of the wet layer and gradual accumulation of the overlying dry layer will limit the utility of this equation for freezing time. However, given reliable meteorological input data, snow-cover models such as SNOWPACK will obviate these estimations and the assumed constant boundary conditions. Indeed, subject to the limitations of a series of cold-laboratory experiments, either the approximate solution developed in this paper or SNOWPACK simulations can be used to estimate the freezing time during which kinetic growth of crystals at the base of the dry layer is intense.

Four of the seven cold-laboratory experiments showed faceting within 5 hours or less during which the wet layer was freezing. All seven experiments showed faceting during the decay of the temperature gradient after the wet layer froze. These observations were qualitatively reproduced by SNOWPACK too, showing the usefulness of snow-cover models to provide information beyond the approximation of freezing time only.