U. S. DEPARTMENT OF AGRICULTURE
FOREST SERVICE
ALTA AVALANCHE STUDY CENTER
Project C
Progress Report No. 1
CRITIQUE ON HEAT AND VAPOR TRANSFER IN SNOW
E. LaChapelle
Avalanche Hazard Forecaster
December 1960
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
Where:
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.
References Cited
1. Abels, G. Daily variation of temperature in snow and the relation between
the thermal conductivity of snow and its density. Rep. fur Met. de St. Petersb. Akad. 16
(1892).
2. Jansson, M. Ueber die Warmeleitungsfahigkeit des Schnees, Ofversigt Af Kgl.
Vetenskahsakademiens, 1901.
3. Devaus, J. L'Economie radio-thermique des champs de neige et des glaciers,
Annales de Physique, 20 (10): 5- 67, 1933.
4. Kondrat'eva, A.S. Thermal conductivity of the Snow Cover and Physical Processes
Caused by the Temperature Gradient. SIPRE Translation 22, March 1954 (originally published
1945 in Russia).
5. de Quervain, M. Zur Warmeleitung von Schnee. Extrait de Publication No. 39 de
l'Association Internationale d'Hydrologie Assemblee generale de Rome, tome IV, 1954.
6. Shakhov, A. A, Physical Processes in a Snow Cover. SIPRE Translation 15, January
1952. (Originally published 1948 in Russia).
7. Yosida, Z. and others Physical Studies on Deposited Snow: I--Thermal Properties.
Contributions from the Institute of Low Temperature Science, Hokkaido University. No. 7,
pp 19-74, 1955.
8. Wiener, 0. Zur Theorie der Refraktionskonstanten. Bertichte u.d. vorles d.
Konichlicken Sachs. Gesell. d. Wise zu Leipzig, Math.-Phys. Klasse 62, 256. 1910.
9. Murcray, W. B., and C. Echols, Some Observations on the Flow of Heat Through
Cold Snow. J. of Meteorology, Vol. 17, No. 5. pp 563-566, October 1960.
10. Van der Held, E.F.M. The Contribution of Radiation to the Conduction of Heat.
Appl. Sci. Res., Sect. A, Vol. 3, pp 237-249, 1952.
11. de Quervain, M. On Metamorphism and Hardening of Snow under Constant Pressure
and Temperature Gradient. Extrait des Comptes Rendus et Rapports, Assemblee Generale de
Toronto 1957, Tome IV, pp 225-239, 1958.
12. Bowen, I. S., The Ratio of Heat Losses by Conduction and by Evaporation from
any Water Surface. Physical Review, Vol. 27, pp. 779-787, June, 1926.
13. LaChapelle, E. Snow Studies on the Juneau Ice Field, Juneau Ice Field Research
Project Report No. 9, Amer. Geograph. Soc. of New York, March, 1954.
14. Giddings, J. C. and E. LaChapelle Diffusion Theory Applied to Radiant Energy
Distribution and Albedo of Snow. J. of Geophys, Research, Vol. 66, No. 1, January 1961.(in
press).
