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 1f 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 10^{2}
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 10^{2}).
To a fair approximation in highporosity 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 aplane 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 D_{w} 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 D_{w} 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, D_{w}.
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 + 0.006p^{2} (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 sixfold 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/T^{2 }preceding the exponential is nearly cancelled
by the dependence of D_{w} on T^{3/2}, 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 C1.)
The mass flux, J_{m}, in grams/cm^{2}sec can be obtained as MJ/RT, and
thus
Now D_{w} 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:
T_{c} = mean snow temperature in degrees C.
T_{o} = 273' K
L = 12,200 cal/mole
R = 82.1 cm^{3} atm/degrees Cmole
M = 18.0 gm/mole
p = 0.00603 atm at O degrees C.
D_{w}= 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 F1). 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/cm^{3}.
(This crystal form appears to originate with difficulty in snow of density greater than
0.30 g/cm^{3}.) Porosity decreases only 18% from 0.10 g/cm3 to 0.25 g/cm^{3},
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 temperaturesensing 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 freshlyfallen 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 225239, 1958.
5. Yosida, Z. and others Physical Studies on Deposited Snow: IThermal Properties.
Contributions from the Institute of Low Temperature Science, Hokkaido University. No. 7,
pp 1974, 1955.
6. Jansson, M. Ueber die Warmeleitungsfahigkeit des Schnees. Ofversigt Af Kgl.
Vetenskahsakademiens, 1901
