U. S. DEPARTMENT OF AGRICULTURE
FOREST SERVICE

ALTA AVALANCHE STUDY CENTER
Project C
Progress Report No. 2

THE FORMATION RATE OF DEPTH HOAR

J. C. Giddings
Dept. of Chemistry University of Utah

and

E. LaChapelle
Avalanche Hazard Forecaster

April 1961

INTRODUCTION

Once snow has been deposited from the atmosphere, its subsequent evolution is strongly influenced by the internal diffusion of water vapor. In the absence of strong temperature gradients, vapor diffusion from one part of a snow crystal to another along pressure gradients established by differences in surface curvature results in the reduction of complex crystals to isometric snow grains. This process is termed destructive metamorphism. on the other hand, a strong temperature gradient in the snow cover establishes a gross vapor gradient which overrides those due to surface tension differences. This causes snow crystals to sublime and redeposit around new centers of crystallization. The latter process is termed constructive metamorphism, and leads to mechanically weak snow layers consisting of cup shaped crystals, or depth hoar.

Precipitated snow thus represents an unstable form of ice crystals under the temperature regime normal to a temperate snow cover. It has been suggested (Bader, (1)) that depth hoar is an equilibrium form of crystalline ice, and the observed stability of these crystals, once formed, appears to bear this out. Mature depth hoar structure persists for extended periods of time with very little physical change as long as snow temperature remains below freezing and superimposed compressive loads are light.

The existence of depth hoar layers in the snow cover apparently was recognized in the polar regions over a century ago (Seligman, (2)). Its presence in Alpine snow covers was first described by (Paulcke (3)), who pointed out its significance in avalanche formation. Since then, the peculiarly weak mechanical structure of depth hoar has come to be recognized as an important factor in snow stability. While the effect of temperature gradient on depth hoar formation has been clearly recognized, and also demonstrated in the laboratory (e.g., de Quervain, (4)), few quantitative data have been published on formation rates.

Assuming that the growth of depth hoar crystals depends directly on diffusional transfer, it should be possible to predict growth rate as a function of temperature and temperature gradients, vapor pressure of ice, and the diffusion coefficient of vapor in air. These quantities depend very much on climate, and consequently the climatic variations of this phenomenon should be amenable to explanation. It is the object of this report to consider the physical and climatic factors influencing constructive metamorphism.


VAPOR TRANSFER IN SNOW

The flux of water vapor through a unit area normal to the z axis (where z is measured in the direction of the gradient of temperature) ignoring thermal diffusion, can be expressed as



where p is the vapor pressure and D is the apparent diffusion coefficient. The influence of snow structure on D will be discussed later. Assuming that the change in D with distance z is small, the rate of accumulation of vapor within a region can be written as



where r is the rate at which vapor is deposited in the solid form. Because of the rapid formation and condensation of vapor, p may be taken as the equilibrium vapor pressure with very little error. This quantity, ignoring the small effects due to high surface area volume ratios, may be taken as a function of temperature only, p = p(T), and may be approximated by



where L, the latent heat of vaporization, and the constant s are assumed to be independent of temperature. The temperature may be expressed as a function of coordinate z by means of the temperature profile, T = T(z). Since the temperature profile is a slowly varying function
of time, may be set equal to zero and may be written as.Equation (2) can consequently be written as


The above considerations make it possible to write



and



Employing equation (3) for vapor pressure, equations (5) and (6) become



and



where x = L/RT. The rate of accumulation of solid, r, becomes




The first term on the right hand side (r.h.s.) is proportional to the second derivative of temperature, and is thus equivalent to an expression obtained by Yosida (5). The second term on the r.h.s. is proportional to the temperature gradient squared. This term is especially important in the presence of shallow snow in cold climates. The necessity for this additional term can be established by considering the hypothetical case of a uniform temperature gradient, dT/dz = const. A plot of p (ordinate) against T or z (abcissa) is, like other vapor pressure curves, concave up, and the term



is finite and positive. Thus while the first term on the r.h.s. is zero, there is a finite accumulation rate r which must be assigned to the second term on the r.h.s. The latter quantity will always contribute to some extent whether the temperature gradient is constant or not.

Several factors must be considered both in estimating and measuring the apparent diffusion coefficient, D. One factor is the presence of a solid network which, for many diffusion processes, obstructs diffusion and reduces D. Another factor is the "hand to hand 11 passage of vapor molecules, described by Yosida, in which vapor is condensed on one side and released on the other. This is usually thought to increase D. Both of the above factors are accounted for in the following analysis. If a straight line is extended through a homogeneous snow cover in the z direction, a fraction f of the line will intersect void space and a fraction 1-f will intersect solid material. The quantity f is the porosity of the snow. Since the solid (ice) has a heat conductivity of the order of 102 larger than that of air, the temperature gradient along the line in the solid will be much smaller than that in the void space, (although the ratio will be much less than 102). To a fair approximation in high-porosity snow, the entire temperature drop along the line may be considered to take place within the void space. Consequently the average temperature gradient within the void space is related to the observed temperature gradient by



a relationship obtained previously by Yosida. This equation is obviously limited at very high densities (low porosities).

If one now imagines a-plane of unit area normal to the z axis intersecting the snow and fixed to the solid network, the flux through
the plane, from equation (1) is



The flux through the plane, however, is occurring entirely in the void space intersected by the plane since no diffusional transfer is occurring within the solid. The possibility of a large interface diffusion term is also considered slight. Thus the flux of water vapor through the plane may be written as



where Dw is the diffusion coefficient of water vapor in air not in the presence of solid. The porosity f appears in (11) because only the fraction f of the unit area is void space contributing to diffusion. Combining the above expressions we obtain



In practice D should be a slight amount less than Dw because of the approximation made in obtaining equation (10). It is seen that the two factors considered previously do not greatly influence diffusion. The "obstruction" factor is cancelled out because of the increased temperature gradient found in the void space due to the presence of solid, and because the vapor need not go around the obstructions, but can condense on them. The hand to hand transfer does not contribute to the flux because this transfer does not shift water molecules across a plane fixed in the solid network.

Yosida and colleagues have measured the diffusion of water vapor in snow by means of a column of consecutive, removable cans filled with snow. The snow was held intact by means of a fine wire gauze stretched across the bottom of each can. The accumulation of ice by diffusion was weighed directly. These investigators found that the apparent diffusion coefficient was four to five times as large as that for pure vapor diffusion, Dw. They attributed this to the "hand to hand" delivery of
water vapor, a conclusion the present authors find untenable in view of the above arguments. Several explanations of this increase can be proposed. It is very possible that a narrow air gap developed over part of the cross section between the wire gauze and the snow in the can below. The thermal conductivity in such a gap would be greatly reduced because of the absence of the highly conducting solid network. Consequently the temperature gradient would be greatly increased in the gap. As shown by equation (11), the flux through a given cross section is proportional to the temperature gradient and would thus be greatly increased. Since the weight change of the separate cans depends on the mass flux across the boundaries, this phenomenon could well account for anomalous diffusion coefficients. An estimate of the increase in the observed D due to the presence of an air gap can be made by assuming that the ratio of the temperature gradient in the gap compared to that in the neighboring snow is equal to the inverse ratio of the respective thermal conductivity's,



The expression of Jansson (6), 0.00005 + 0.0019p + 0.006p2 (P = density), can be substituted for. Thermal conductivity in air and the diffusion of latent heat as water vapor contribute almost equally to the conductivity in the gap,. This situation will, of course, change at higher altitudes since the latter quantity is inversely proportional to pressure. Ignoring this for the momentmay be approximated as twice the thermal conductivity of air at 0 degrees C;




This ratio must be multiplied by the fraction of the total cross section in which a gap occurs to calculate the expected increase in D. If this fraction is constant, the increase in D is seen to be very strongly dependent on density. This would be contrary to the results of Yosida where a six-fold increase in p did not lead to serious changes in D. The ratio in equation (13) varies between 2.15 and 23.17 for the densities employed by Yosida. The measured increase in D was found to range between 3.18 and 4.55. The calculated results are of the correct order of magnitude for this increase, but a variation in cross sectional gap area would have to be postulated to explain the constancy of measured values. If the least dense snow settled most strongly forming more gap area than the densest snow, the trend could be explained. More experimental data are needed to determine the role of the above factors.

Any experimental method in which weight changes depend on mass flux across a narrow gap is subject to the same considerations as discussed above. If the width of the gap is small compared to the container dimensions, the gap will not seriously disturb the normal temperature profile. The difference in mass flux from that normally expected will then be approximated by the ratio in equation (13). If one wishes to measure diffusion phenomenon without the disturbing effects of air gaps, it should be possible to record the weight at each end of a cylinder filled with snow and subject to a temperature gradient, and compare weight changes with those predicted by diffusion theory.


THE RATE OF DEPTH HOAR FORMATION

Here we shall apply the relations deduced in the foregoing section to calculating the rate at which depth hoar crystals of a given size might be expected to form in a given temperature regime.

Apparently depth hoar crystals grow from one side of existing snow crystals or grains (See Yosida, op. cit.) in a direction opposite to the vapor pressure gradient, while these same grains lose material by sublimation on the other side. We define depth hoar formation as complete hoar crystals deposited from water vapor. Size of these crystals may thereafter continue to increase.

The source of material for these crystals is the vapor flux through the snow, J, previously evaluated. In order to relate J to the gross temperature gradient, equation (10) is substituted in equation (11):



Using the expression for p from equation (3), we obtain



For a given temperature gradient, J increases very rapidly with temperature because of the exponential term. The term 1/T2 preceding the exponential is nearly cancelled by the dependence of Dw on T3/2, and the two together have no significant role in establishing temperature dependence. The exponential term (and the vapor pressure) is reduced approximately by a factor of two for each 8 degree C drop in temperature. Thus the rate of formation of depth hoar should be much larger in the lower parts of the snow cover where the temperature is close to 0 degrees C. (See previous discussion of this in Report C-1.)

The mass flux, Jm, in grams/cm2-sec can be obtained as MJ/RT, and thus



Now Dw is dependent on air density, and hence on altitude. In order to generalize this expression, we introduce the term

:



where Pa = atmospheric pressure at sea level
Pa = atmospheric pressure at some given altitude, a

In order to simplify evaluation of equation (17), we reintroduce p from equation (3) and express its temperature dependence in the form of an approximate empirical term which is valid over the range from 0 degrees to -20 degrees C commonly encountered in winter snow covers:



Tc =  mean snow temperature in degrees C.
To =  273' K
L = 12,200 cal/mole
R = 82.1 cm3 -atm/degrees C-mole
M = 18.0 gm/mole
p = 0.00603 atm at O degrees C.
Dw= 0.22 cm2/sec at 0 degrees C. and 760 mm pressure

Introducing the numerical values for the above constants yields:



With the mass flux approximated in terms of quantities measurable in the field, we next estimate the time required for depth hoar crystals of given size to form from this vapor supply. Assume a layer of depth hoar crystals equal in thickness to the average crystal size, d (centimeters), has formed from sublimation of this vapor supply. The amount of mass required can be calculated if the density of this layer is known. Field observations fortunately furnish a basis for estimating this density. Time profile records over several years at Berthoud Pass, Colorado, where depth hoar formation is common, show that such layers commonly approach a bulk density of 0.25 to 0.30 as this crystal type becomes fully developed. (See Report F-1). We adopt here a mean bulk density of 0.28 for the purpose of computation. The time required to form this layer of thickness is d is then given by



In reaching this general approximation, some considerations have necessarily been slighted. It will be noted that a term for the porosity, f , vanished on formation of equation (14). Field experience suggests that high porosity (low density) snow types are more amenable to depth hoar formation. Time profile records indicate that depth hoar formation commonly is initiated in snow layers of density 0.10 to 0.25 g/cm3. (This crystal form appears to originate with difficulty in snow of density greater than 0.30 g/cm3.) Porosity decreases only 18% from 0.10 g/cm3 to 0.25 g/cm3, hence this factor must actually play a rather limited role. Variations in initial density and crystal type apparently do have a more pronounced effect on the ultimate size of the depth hoar crystals.

Presently available evidence does not give any factual data on whether the formation rate of depth hoar is constant with time, other factors being constant. Qualitative field observations suggest that the initial formation of depth hoar crystals may be more rapid than the subsequent growth of these crystals, but this has not been confirmed by direct experiment. A variable formation rate would require further modification of Equation (20).


EXPERIMENTAL RESULTS

It is possible to test the validity of equation (20) by comparing calculated depth hoar formation times with those observed experimentally. Turning first to data reported in the literature, we note the results of laboratory experiments at the Weissfluhjoch reported by de Quervain, in the paper previously cited. Of these studies of snow metamorphism under controlled laboratory conditions, reference is made to de Quervain's Experiment No. 10, involving depth hoar formation under very light compressive load. Data reported from this experiment are sufficient to permit calculation of formation time.

Observations of depth hoar formation under controlled conditions were made at the Alta and Berthoud Pass avalanche stations during winter of 1960/61. In both cases the shallow natural snow cover early in winter was dug away back to the ground, and a light plywood frame installed to support temperature-sensing elements. These were thermocouples at Alta and Weston dial thermometers at Berthoud Pass. The frames were filled to a depth of 20 cm at Alta and 30 cm at Berthoud Pass with freshly-fallen snow shoveled off the surface of the snow cover. Frequent observations were made of temperatures at various levels in the snow, and at the end of the experiments, the snow was dug up for crystal examination and density measurements. The Alta experiment was terminated after 12 days by heavy falls of fresh snow. The one at Berthoud Pass was continued for 55 days by scraping fresh snowfalls off the test layer in the frame.

Data from these experiments are summarized in the following table. There are two separate entries for the Alta study, representing two separate layers in the snow.

The last column of corrected time in days is obtained by replacing the assumed density of 0.28 g/cm in equation (20) with the actual value for the final density of the depth hoar formed, which is available from each of these experiments.

Considering the wide range of altitude, climate, experimental conditions and initial snow types which these experiments encompass, the agreement between theory and observation is remarkably good. Equation (20,) which has been derived purely from theoretical considerations of water vapor diffusion and generalized geometry of snow cover structure, appears in fact to describe the influence of various climatic factors accurately. Further checks against observation no doubt will suggest some refinements, but the validity of the approach appears to be established.





References Cited
1. Bader, H. Mineralogical and Structural Characterization of Snow In: Snow and its Metamorphism (Der Schnee and Seine Metamorphose) Beitrage zur Geologie der Schweiz, Geotechnische Serie, Hydrologie, Lieferung 3, Bern, 1939. (in English as SIPRE Translation 14)
2. Seligman, G. Snow Structure and Ski Fields. MacMillan & Co. Ltd., London, 1936
3. Paulcke, W. (Title not given) Der Bergsteiger, No. 6, P. 340, Vienna, 1932.
4. 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.
5. 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.
6. Jansson, M. Ueber die Warmeleitungsfahigkeit des Schnees. Ofversigt Af Kgl. Vetenskahsakademiens, 1901

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