U. S. DEPARTMENT OF AGRICULTURE
ALTA AVALANCHE STUDY CENTER
Progress Report No. 1
CRITIQUE ON HEAT AND VAPOR TRANSFER IN SNOW
Avalanche Hazard Forecaster
The flow of heat through snow is a complex process. Snow, particularly at low densities, is a good thermal insulator. Like other insulating materials, it transfers heat by a combination of molecular conduction through the solid framework, by conduction, diffusion and convection of the entrapped air, and by internal radiation exchange. Unlike other insulators, it also transfers heat by sublimation of the solids--in this case ice--and associated diffusion and convection of the vapor phase. The latent heat of sublimation of ice is high, 680 calories per gram at the freezing point; consequently water vapor can be an efficient means of heat transport within snow which introduces complicating factors in the bulk thermal conductivity. Owing to this contribution of water vapor to heat flow, the bulk, conductivity becomes dependent on absolute temperature as will be demonstrated below, Moreover, recrystallization of ice which has passed through the vapor stage rearranges the solid framework and effects a permanent time- dependent change of conductivity, In addition-to thermal effects, this latter change also profoundly alters the mechanical properties of snow: the stabilizing and strengthening effects of destructive crystal metamorphism are reversed and the snow becomes mechanically weaker.
Early workers [Abels (1), Jansson (2) and Devaux (3) are frequently cited] obtained values for the bulk conductivity of snow, and related these simply to the snow density. Abels obtained the expression
Jansson found that
while Devaux concluded that
In each case these expressions represent the average of data with considerable scatter, and later investigations [for instance, Kondrat'eva (4)) have generally found a similar scatter. It appears that the dependency of conductivity on density may be broadly described by the expressions given above, or similar ones, but within this dependency there is considerable variation from one snow sample to another. This variation may be attributed to differences in the ice framework (crystal character) for a given density and to the variable influence of water vapor transfer mentioned above.
More recently the separate elements of heat transfer in snow have come under investigations. Using precisely characterized snow samples in a laboratory apparatus, de Quervain (5) was able to compare the relative amounts of heat transferred by conduction in the ice framework and by the air in the pores. This was done by substituting gases of different known heat conductivity's for the air in the snow and extrapolating the accompanying change in snow conductivity to that which would accompany a theoretical gas of zero conductivity. For natural snow of density 0.33 g/cm3, it was found that about 75% of the beat flow could be accounted for by conduction in the ice. Working later with the same apparatus and similar snow samples, Mueller (unpublished report.) showed that this heat flow might amount to 85%. Both of these investigations intentionally disregarded the role of water vapor in the heat flow. From the described experimental conditions, it appears that the contribution of water vapor would be included in that portion of the heat flow attributed to conduction in ice.
Various Russian workers have recognized the importance of water vapor transfer to the thermal and mechanical properties of snow, but few reports of definitive studies are available. Many of the papers are speculative, Much of the Soviet thinking along these lines has been summarized for the immediate post-war period by Shakhov (6) who presents an extensive bibliography.
Probably the most thorough investigation to date, both theoretical and experimental, of the role of water vapor in snow behavior has been that by Yosida and his colleagues (7). Rather than measuring directly, Yosida measured temperature conductivity, K, and found from the expression
where c - specific heat of snow.
In these experiments, snow samples were exposed to sudden temperature changes and a plot obtained of mean sample temperature versus time to determine K. The mean sample temperature was measured by observing changes in pressure of the entrapped air within the sample. A large experimental scatter of versus p was found, and the following average relation obtained:
When samples were stabilized at some temperature T1, changed to a different temperature T2, and then returned again to T1, the two values of K thus determined often did not agree. This was attributed to crystalline changes effected by presence of the temperature gradients.
Starting with the general diffusion equation for water vapor,
where p = partial pressure of water vapor
Do = diffusion coefficient of water vapor in air
Yosida was able to show that a transport of mass (and hence heat) in the snow cover takes place by sublimation, and that the mass divergence at any point should be proportional to the second derivative of temperature with respect to height when the above expression is reduced to
the single-dimensional case found in a normal winter snow cover. He was able to demonstrate experimentally the general nature of this transport by measurement of mass changes in snow samples inserted in the snow cover.
Turning next to laboratory experiment, Yosida precisely measured mass transport in a column of snow subjected to a fixed and known temperature gradient, and concluded that the coefficient of diffusion of water vapor in snow, D, was
1) almost independent of density
2) apparently was unaffected by direction (i.e., by gravity)
3) was four to five times larger than Do
This latter conclusion is especially important, for it is a key to the character of vapor transfer in snow. Instead of acting as obstacles to diffusion of water vapor through intercrystalline spaces, the snow crystals contribute to it by a process of "handling on" vapor molecules across the small intercrystalline distances. Yosida pointed out that if lengths of air spaces and ice particles are given respectively by a and b the apparent microscopic temperature gradient, G is related to the actual microscopic gradient in the air spaces Go, as follows:
and thus Go can be much larger than G.
By delicate cold laboratory experiments on a microscopic scale, this actual transfer between adjacent crystals was demonstrated, each individual crystal showing growth opposite to the direction of heat flow.
Finally, Yosida showed strictly by theoretical reasoning that the heat transfer in snow due to water vapor must necessarily be as great as that due to air. Referring to the work of de Quervain and Mueller previously cited, this suggests that water vapor can account for 15% to 25% of the heat transfer in snow with densities associated with a metamorphosed snow cover. Yosida goes on to demonstrate that this percentage is also related to the crystalline structure of snow, which may vary widely for a given density. He appeals to the "Formzahl" concept of Weiner (8) to characterize this structure, and shows the range of Formzahl which can be expected for a given density,, For snow of density 0.1 g/cm3 under optimum conditions, the water vapor contribution to heat flow is calculated to be 37%. Presumably it would be even greater for snow of lower density.
Murcray and Echols (9) have recently suggested than an appreciable part of the heat transfer in snow at low temperature and high temperature gradient may be due to radiation exchange between the individual ice particles. This conclusion appears to be based on rather scanty observational evidence, and requires critical examination.
An estimate of radiation transfer may be made by postulating a simplified theoretical model. Consider an idealized snow layer made up of equally spaced thin ice laminae parallel to the surface and separated from one another by a distance, d, which is large compared with their thickness. (This corresponds to the extreme case where conduction through the ice framework is zero.) A temperature difference exists between top and bottom of the snow cover, imposing a uniform temperature gradient within the snow. Assume this gradient to be 1 degree C/cm, a large value. The ice laminae will have an emissivity for heat radiation, e, less than 1.0, but the latter value will be assumed, giving the maximum radiation transfer. The radiation transfer between two adjacent laminae at absolute temperatures T1 and T2 (and, since the temperature gradient is uniform, between top and bottom of the snow layer) is then given by:
where Qr = radiant heat flux
Let d = 0.5 mm and T1 = 273 degrees K (0 degrees C); then T2 = 273.05 degrees and Qr = 0.5cal/cm2/day.
For a given temperature difference, the radiation transfer diminishes with absolute temperature:
Let d = 0.5 mm and T1 a 243 degrees K (-30 degrees C); then T2 = 243.05 degrees and Qr = 0.3 cal/cm2/day.
These figures probably represent a low value,, for the mean intercrystalline distance can well be greater, On the other hand, it is highly unlikely that it exceeds more than a very few millimeters.
Let d = 3.0 mm and T1 = 273 degrees K; then T2 = 273.3 degrees and Qr = 2.9 cal/cm2/day.
This latter figure probably represents an extreme; ordinarily the radiation transfer might fall below 3 cal/day. Certainly under the temperature regine discussed by Murcray and Echols (-10 to -30 degrees C.) it must be very much less. The radiation heat transfer of 15 to 20 cal/cm2/day which these authors suggest seems quite out of the question. Water vapor transfer is a much more likely explanation for the amount of heat flow they observed. The variation in with different layers most probably is associated with differences in crystal structure (Formzahl).
The above analysis of internal radiation transfer receives support from the thorough treatment of the subject by Van der Held (10), who demonstrates that for particulate insulators in general, the radiation transfer becomes significant only at elevated temperatures (above 100 degrees C), and is negligible below 0 degrees C.
The mechanical and crystallographic effects of a temperature gradient in snow have also been investigated by de Quervain (11). This study is particularly noteworthy because it was carried out under carefully controlled laboratory conditions and because the crystallographic changes were observed in great detail by microscopic examination of snow thin sections at beginning and end of the experiment. The bulk mechanical characteristics of the snow were also quantitatively measured, Twelve different samples of homogeneous snow, prepared by sieving natural snow, were stored under various temperature and pressure environments for forty days and the changes then examined, The results of this study are summarized as follows:
1) In the absence of a temperature gradient fresh snow is transformed into a fine granular material. Hardness and strength as well as specific gravity increase considerably with a rise of temperature and load.
2) In the presence of a temperature gradient fresh snow is transformed into a course granular material--under the conditions of the experiment mainly to depth hoar. The development of hardness and strength is checked by a temperature gradient. An increase of strength does not occur unless a certain substantial load is applied. Otherwise even a drop of strength may result. The higher the temperature gradient, the sooner the transformation is achieved.
3) Density, strength and crystalline features are interrelated to a certain extent. There are, however, snow samples which have similar specific gravity, yet quite different strength, and samples of different crystalline appearance may agree in more than one mechanical property.
4) From experiments with plastic foils inserted as barriers in the snow it is concluded that a considerable vapor transfer is produced by the temperature gradient. As for the formation of depth hoar, a certain local turnover of incoming and outgoing vapor is necessary,, but in a strong gradient depth hoar can develop between foils (or ice sheets in nature) without convection of large scale.
This investigation provided quantitative illumination of two points which are pertinent to the present discussion. One is the result of experiments with vapor varriers (foils) in the snow sample which explicitly demonstrated the difference between large-scale and small-scale vapor transport: the small-scale transport is able to effect extensive crystal alteration (form depth hoar) without large-scale transport of mass from one part of the snow cover to another. The other point is the demonstrated complex interrelation among density, strength and crystal form (and presumably heat properties such as conductivity).
There are two aspects of heat and water vapor transfer in snow which are mentioned or implied in results of the researches discussed above, but which have not been explicitly treated by these authors, These are the temperature and pressure dependencies of vapor flux and its associated latent heat flux.
Referring to Yosida's microscopic air temperature gradient, Go, it is seen that the microscopic vapor pressure gradient between two adjacent ice particles in the snow cover may be determined if Go is known and the vapor pressure in contact with each particle is assumed to be the saturation vapor pressure over ice at the appropriate temperatures. From the character of the saturation vapor pressure curve for ice, it is immediately apparent that the magnitude of this vapor pressure gradient will be dependent on the absolute temperature as well as on Go. The wide variation in this vapor pressure gradient for a given temperature gradient which can be expected over the range of temperatures found in natural snow covers is illustrated in Figure 1.
Empirical observations in the field confirm this dependency. Rate of depth hoar formation is commonly found to be highest at the bottom of a shallow winter snow cover, where the temperature is highest, and diminishes toward the surface.
The decrease in air density accompanying decrease in the ambient air pressure reduces the relative contribution of the air to heat transfer in snow and allows freer diffusion of water vapor. This effect becomes significant at high altitudes, and presumably contributes to the empirically observed fact that depth hoar formation is more common at high altitudes. (A severe temperature environment and shallow snow cover can also be responsible for this fact.) In order to obtain conveniently the magnitude of this altitude dependency reference is made to Bowen's (12) analysis of relative heat transfer by diffusion of air and water vapor, which showed that
Qn = sensible heat transfer
Qv = latent heat transfer
T = temperature difference, degrees C
p = vapor pressure difference, mm
P = atmospheric pressure, Mm
(There is some question about the applicability of this expression to internal processes in a snow cover, but for the sake of the present discussion it, or a similar relation, is assumed valid.) The ratio of Qn to Qv is seen to be linearly dependent on pressure, and the corresponding non-linear dependency on altitude is graphed in Figure 2.
It must be recognized that there are two distinct processes Of crystal metamorphism which may take place within a snow cover. Both depend on vapor transfer. The first of these, occurring when temperature gradients are small or absent, is termed destructive metamorphism. Water vapor is transferred from one part of single snow crystal to another along the vapor pressure gradients set up by differences in curvature of different parts of the crystal. The ultimate result is to reduce the complex deposited snow crystals (such as stars or dendrites) to rounded, isometric ice grains. A process of homogenization takes place whereby crystals originally of widely different form all approach this common and condition, and the snow cover shrinks and becomes mechanically stronger. The second type of change is termed constructive metamorphism, and occurs when the temperature gradient becomes large enough to over-ride the intracrystalline surface energy gradient and cause water vapor transport through the snow cover. The vapor so transported is redeposited around new centers of crystallization, forming entirely new crystals which are large, angular, layered and mechanically fragile (depth hoar). Shrinkage in this case is small.
When the temperature gradient is removed, and especially when the snow temperature rises to the freezing point, destructive metamorphism quickly reverses the depth hoar formation, and the snow crystals once more tend to approach isometric ice grains. If depth hoar formation has been strong, these grains often are larger than those produced purely by destructive metamorphism. Even where they are indistinguishable, the altered gross mechanical character of the snow persists, and old depth hoar layers can often be recognized by their lower strength properties long after they have lost their crystallographic identity (see LaChapelle (13)).
It is not necessary that the mass divergence be large for depth hoar to form; it may, in fact, be very small or absent throughout most of the snow cover. Moreover, the mass flux through a given plane need not be great, The actual occurrence may be termed a large mass translation. A small amount of ice is lost at the bottom of the snow cover and a corresponding gain appears higher up through a displacement process in which the intervening layers may in their entirety pass through the vapor stage and be reformed as depth hoar, As noted above, this effect is most pronounced in the lower layers of the snow cover, but in the case when steep temperature gradients are persistent, the entire snow cover from top to bottom may be turned to depth hoar. (Snow conditions in south-east Switzerland prior to the catastrophic avalanche cycle of January 1951 were an excellent example of this latter occurrence.)
There remains considerable scope for investigation of the vapor transfer phenomenon in snow. From the standpoint of basic physics, it can provide additional understanding of snow behavior, and from the practical standpoint there is much to be learned about prediction and artificial modification of the depth hoar formation which is so important mechanically to such diverse problems as deterioration of compacted roadways and the formation of snow avalanches. The following specific studies could well prove worthwhile:
1) Measurement of thermal properties in snow where depth hoar is forming and where formation has been carried to completion.. It is possible that the accompanying crystallographic changes can be detected from alteration in the radiation properties of a given snow layer (see Giddings and LaChapelle (14)).
2) Collection of quantitative data on the rate of depth hoar formation in natural environments. It will be necessary here to solve the rather difficult problem of establishing criteria for quantitative measurement of such formation, especially at the early stages when the crystal changes are visible only on a microscopic scale.
3) Investigation of crystal metamorphism at high latitudes where shallow snow covers and permafrost are the rule. There appears to be a dearth of material on this aspect of snow behavior in the literature.
4) Checking under the field conditions the actual magnitude of the temperature and altitude dependency of vapor transfer described above.
5) Field experiments to determine quantitatively the duration and magnitude of loads necessary to reverse the weakening effect of depth hoar formation. (It is known from a practical standpoint that depth hoar on avalanche paths can be stabilized by compaction, usually under foot, and sometimes under ski.) It is expected that the required load will vary with temperature.
6) Both theoretical and experimental investigation of means of inhibiting depth hoar formation. The investigation with plastic foils by de Quervain, cited above, might serve as one starting-point. Chemically modifying the crystal habit of ice might be another.
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