Snow flux

Snow structure, heat, and mass flux through snow

M.R. de Quervain

Federal Institute for S n m and Avalanche Research, Weissj%hjoch/Davos, mitzerZand

ABSTRACT: A general survey is presented on the basic phenomena gov- erning metamorphism of snow and its permeability to heat, vapour and water. The behaviour of several elementary models accessible to calculation reveals the effects of the isolated processes and points to their complex interaction in real snow.

RESUME:

élémentaires, qui déterminent la métamorphose de la neige et sa perméabilité 3 la chaleur, 2 la vapeur et 2 l'eau. Le comportement

d'un nombre de modèles simples et accessibles au calcul révèle les effets des processus isolés et leur concours complexe dans la neige réelle.

Un tour d'horizon est présenté sur les phénomènes

1. INTRODUCTION

This introductory survey paper offers a general picture of pro- cesses acting between snow surface and ground insofar as they are relevant to hydrological problems.

heat flux whereas snow structure represents primarily a passive me- dium. As structural metamorphism is a result of mass transfer gov- erned by temperature, however, there is a mutual and complex inter- action between all these elements. Simple models with numerical ex- amples are introduced rather to elucidate problems than to present solutions adequate to the real conditions. A considerable number of authors have built up our present knowledge on the whole matter. As this is not intended to be a bibliographic paper, only selected ref- erences are given. The author has taken the liberty of inserting some of his early observations which are not easily accessible.

These are primarily mass and

2. DRY SNOW PROCESSES 2.1 Mass Flux and S n m Structure

ture, are one visible result of mass flux in snow cover. Many au- thors have contributed to the description and understanding of snow metamorphism: W. Paulcke [l], G. Seligman [Z], H. Bader [3], A.S.

Kondrat'eva [4], H.P. Eugster [SI, Z. Yosida [6, 71 and his pupils, and E.R. Lamapelle [ 8 ] , to mention a few.

tive or isothermal metamorphism, very local transformations are ob- served. They tend to reduce the free energy installed in the large specific surface of dendritic crystals and result in a fine granular material. As no proper mass flux is connected to this process, it is not given further attention in this paper.

As soon as a temperature gradient is built up in a snow layer (which may occur immediately after deposition) constructive or (tem- perature) gradient metamorphism sets in, characterized by

Snow metamorphism, ie., the morphological changes in snow struc-

In a first process after deposition of new snow, called destruc-

203

-

growth of selected crystals,

-

decrease of crystal number per unit volume,

-

development of even crystal facets, and re-entrant angles.

With respect to bulk properties of snow, it results in reduced den- sity and strength and increased air permeability, viscosity, and brittleness compared with snow developing under isothermal conditions,

Gradient metamorphism is primarily based on diffusion and de- pends on a pore system.

into the pore space and from there condensed to a neighbouring colder particle.

saturation pressure with respect to the mean temperature.

rection z) of vapour pressure is

Ice is evaporated from a warmer particle Vapour pressure between the two particles is roughly at Mass transport by diffusion in an unidirectional gradient (di-

f Do aP m = - - . -

RVTk aZ

with m mass flux (g/cm2s), p vapour pressure (mbar), in the absence of ventilation practically identical with saturation pressure ps, Do diffusion constant of water vapour in air (0.22 cm2/s), f structural parameter characterizing the available pores (discussed later), Rv specific gas constant for water vapour (4.62 mbar cm3/goK), Tk mean absolute temperature (OK).

perature gradient, aps/az

=

aps/aT e aT/az, and introducing satura- tion pressure as a function of temperature T,

In expressing the gradient of saturation pressure by the tem-

ps

=

A exp [BIT

-

To)] (2) with To

=

273'K, A

=

6.42 mbar, B

=

0.0857°K-1, mass flux becomes

A change of mass flux along the direction of z produces a change in density p

Considering Tk to be constant, density changes in time according to

These are the basic equations governing gradient metamorphism, as derived by Giddings and Lamapelle [8], Yosida [9] and the author

[lo] with slight adjustments.

(aT/az

=

C), density increases everywhere in a snow sample, ie.,

@/at is positive for all values of z, with the highest turnover at the warm end. As an effective source of vapour flux only this warm boundary is identified.

According to equation (5) for a constant temperature gradient

If snow consisted of a block of ice perforated by a system of cylindrical pores parallel with the temperature gradient (model I , Fig. 13, the factor f would be represented by porosity n.

sisting of ice lamellae perpendicular to the temperature gradient, follows the same equation with the factor f now being roughly l/n so that mass flux is increased by a factor l/n2 against model I.

enhanced temperature gradient

-

^l/n aT/az exists in the air space.

Ice is evaporated on the cold side of each lamella and condensed to a higher extent on the warm side, which results in a thickening of the lamellae and in a migration towards the positive gradient.

Yosida [9] has described this process as "hand to hand delivery".

Real snow behaves somehow between the two models, but there are a number of effects to be considered that are beyond the potentials of simplified models.

Another complementary extreme model (model II, Fig. l), con- An

Example: Assuming porosity n

=

0.7

=

f, and a constant temper- ature gradient aT/az

= -

0.2"C/cmY T

=

0°C

(Ti( =

273"K), equa- tions (3) and (5) yield for model I a maximum mass flux of m

=

1.2

-

^{/cm2s (%} g/cm2 day) and a density change dp/dt

=

2 * g/cm3s (% 1.7 g/cm3 day).

Observed mass flux is obviously higher than that calculated with the quoted value of the diffusion coefficient. By introducing an effective diffusion coefficient several times higher than the stan- dard value, as proposed by Yosida, a heuristic solution for this dis- crepancy is offered but not an explanation.

assume the factor f to be of the order of 1, as structural inhibition of the flux and local rise of temperature gradient cancel each

other's effect. As to the elevated vapour diffusion proposed by Yosida they point to some possible source in his experiments. A special feature of gradient metamorphism not reproduced by the basic models I and II is selective growth of crystals and increase of air permeability with advancing metamorphism.

(de Quervain, [lo]).

Giddings and Lamapelle

In this respect, the following points have been raised

-

There must be specifically warm and cold crystals in an im- mediate neighbourhood according to their point of attachment to the structure (Fig. Z), whereby the cold crystals grow at the expense of the warm ones.

-

In addition to diffusion, convective currents have to be ex- pected if the bottom of the snow cover is warmer than the surface.

to loe4 mbar/cm may appear and cause, in snow of medium air Due to thermally unstable stratification, air pressure gradients of

permeability (Ka

=

ZOO cm/s), moisture currents of lou6 to

g/cm2s, which are indeed two or three orders of magnitude higher than those calculated for model I. It should be mentioned, however, that gradient metamorphism does not depend so much on the direction of the temperature gradient with respect to that of gravity.

Pressure variation caused by wind gusts is another probable source of convective currents, preferably in the uppermost snow layers. Ventilation as a whole is probably the main reason why sub- stance is lost in certain snow layers in contrast to equation (5).

Its importance has been clearly demonstrated by Yen [ll, 121.

Dense fine granular snow with low permeability will admit little mass flux and be subject to minor gradient metamorphism only, thus be- coming more dense (by settling) and impermeable. Loose snow with

Mass flux and air permeability are linked in a mutual way.

205

high permeability, on the other hand, is strongly metamorphosed in a gradient and will preserve its low density and high permeability.

A new aspect has been added to the process of vapour flux by the fact that water vapour is not a fully uniform matter but com- posed of a small amount of isotopes leo, 2H (deuterium) and 3H (trit- ium). Water molecules containing these elements differ slightly in their saturation pressure so that sublimation processes lead to a change of concentration in the solid and gaseous phase. Due to their small concentrations mass flux and metamoqhism are hardly influenced in a quantitative manner, by the isotopes, but these permit tracing the origin of mass flux and dating deposits.

been investigated with respect to melting and refreezing (Arnason 2.2 Heat Conductivity

Similar effeyts have [i31

I .

Energy may penetrate dry snow in three ways:

-

molecular heat conduction through the ice skeleton;

-

-

radiation transfer.

transport of sensible heat through the pores by diffusion, convection, and mass flux;

All three components are linked together by snow structure as the dominant parameter.

defined by

An overall thermal conductivity, excluding radiant transfer, is Q

=

k e grad T

Q

=

heat flux (cal/cm2s)

ke

=

thermal conductivity (cal/cm soc)

Measurements of thermal conductivity have been published by many au- thors and expressed by empirical formulae as functions of density.

Mellor [14] has given a compilation in his excellent survey of snow properties. As density does not completely define snow structure, a scatter of measured conductivities for identical densities in the range of -I: 50 per cent is not surprising (Fig. 3).

2.2.1 Thermal conductivity of the ice skeleton

Usually the conductivity of the ice skeleton is evaluated as the difference between the measured effective conductivity of snow ke and a calculated conductivity of the pore system kp, both related to 1 cm2 of snow.

k,

=

ke

-

^kp

A direct experimental check of this component of the conductivity was tried in replacing the air of the pores by gases of other thermal conductivities (carbon dioxide and hydrogen) and extrapolating the measured effect to a hypothetic gas of zero conductivity.

the skeleton for snow of density 0.33 g/cm3 and 55 per cent for a density of O. 15 g/cm3 (de Quervain [15]).

for structures composed of spherical ice particles led to similar relations (de Quervain [16], Woodside [17]).

By this means 75 per cent of the conductivity was attributed to Attempts to calculate ks

2.2.2 Heat flux through the pore system

As Yosida [6, 91, Yen [18, 11, 121, and others have pointed out, heat flux through the pore system is not only a matter of thermal diffusion in moist air but is also connected to vapour flux and in certain cases to convection.

pore space is composed of thermal diffusion and flux of latent heat in a gradient of vapour pressure. Considering equations (1) and (6) we get for the respective conductivities :

In the absence of convective flow, thermal conduction in the

kpo

=

total conductivity within pore space (f

=

l), ka

=

conductivity of air by thermal diffusion, Ls

=

heat of sublimation.

Introducing standard values for Tk

=

273°K: Ls

=

676 cal/g, aps/aT

=

0.52 mbar/"C, ka

=

0.57

-

(1)) one obtains

cal/cm soc; (Do and Rv see equation kpo

=

(0.57 + 0.62)

- ⁼

^{1.2 *} cal/cm s o c Thus both components are of the same magnitude.

Assuming that the structural parameter f introduced in equation (1) for adjusting mass flux to the characteristics of the ice skele- ton is also effective for thermal diffusion, the real conductivity of the pore system related to 1 cm2 of snow would be kp

=

f

-

^kpo,

thus with equation (7):

ke

=

ks + f kpo (9)

As previously stated the factor f is of the order of 1, but mostly f > 1 as shown below. Recalling the simplest possible combined model of pores and lamellae (de Quervain. [15]) a further geometrical inter- pretation of f is presented (model III, Fig. 41. Structural

parameters of this model are

a

=

Fa/F; b

=

Fb/F; C

=

Fc/F; w

=

S/L F = F a + Fb + F c ; a + b + c

=

1

Let the hatched space be ice with a conductivity ki and the open space moist air with a total conductivity kpo.

three columns Fa

-

Fc are parallel and independent, the overall con- ductivity of the system is

Thus, provided the

In this form bki is identical with the conductivity ks of the skele- ton, and the term in large rounded brackets corresponds to the factor f. This latter may amount to values smaller or, preferably, larger than 1. "Hand to hand transfer" is represented by the term

c

-

^{ki/[kpo +}

^...I.

Porosity n is expressed by n

=

a + c (1 ⁺ w).

207

Example: With the assumptions: a

=

0.1; b

=

0.1; c

=

0.8;

w

=

0.7; ki

=

5

-

characteristics of model II are:

1.2

-

cal/cm SOC, calculated

k!fo =

porosity n

=

0.66; density p

=

0.31 g/cm3 7.4 * 10-4 cal/cm s

ke

=

0.1 ki + 2.04 k 5 10-4 + 2.4 10-4

=

=

Direct conduction through ice (column Fb): 67 per cent of total.

Conduction through pores (columns Fa and Fc): 33 per cent. This re- sult is close to possible real conditions.

It should be noted that, in the three models, mass flux and thermal conductivity do not depend on an absolute scaling factor such as grain size or pore width. In natural snow of given density, however, transfer rates certainly increase with grain size. This

effect points to the importance of convection or enforced ventila- tion. Yen [12] has paid particular attention to these processes in extending equation (4) by a term for moisture exchange due to ventilát ion.

am M,

a

Ps

(1 1) - - - - + G - - ap

- -

at Ma az (P

-

^Ps)

G

=

current of (dry)air g/cm2s,

h Y a =

molecular weight of vapour or air, p

=

air pressure.

From experiments the relation between G and ap/at was established and an effective diffusivity De and a resulting thermal conductivity kv were derived as functions of the air flow G:

De

=

^{95.38 (G +} 0.456 10-4)1/2 (cm2/s) (12) (13) kv

=

0.77

-

^{+ 0.314 G}

^-

^{89.40 * G2 + 8640} G3 (CGS-units) k, stands only for the component depending on vapour transport (right term) in equation (8).

Example:

kv

=

0.77

-

^{(CGS), for G}

⁼

g/cm2s results De

=

1.15 cm2/s; kv ^{For G}

⁼

O results De

=

1.08

- =

0.65 cm2/s;

(CGS).

Thus, slight ventilation produces a significant effect on thermal conductivity and a stronger one on mass exchange, ie., metamorphism.

In summarizing heat conduction in dry snow, we conclude that heat is transferred in similar proportions by the ice skeleton, by

thermal diffusion and by diffusive mass flux. Conduction through ice increases progressively with density and is certainly dominant in the range of medium and higher density.

temperature stratification, variation in air pressure or wind gusts, may effect substantial heat flux primarily near the surface.

these transport mechanisms depend on a temperature gradient.

Ventilation caused by unstable All

3. WET SNOW PROCESSES 3.1 Water Storage in Wet Snow

Some phenomena and their definition about free water in snow

-

(Free) water content: W. Relative quantity of free water present in snow,

W

=

100

.

C,/(Gi + Gw) %

&, Gi

=

mass o f water and ice respectively per unit volume

-

Equilibrium (free) water content: We. Free water-holding capacity without runoff outside a zone of saturation of capillary rise.

is conpletely filled with water.

of wet snow.

-

Saturation water content: Ws. Water content when pore space

ps

-

pi pw 1200(ps

-

0.917) 1 O0

ws = -. - =

or % (15)

Pw

-

^{Pi Ps PS 0.083 + 0.917/n}

pi, pw, p s

=

density of ice, water, and saturated wet snow respect ive ly ,

n

=

porosity without water.

-

Relative water content: w. Water content related to the state of saturation.

w

^* 100 % w =

'VS

-

Capillary saturation: Water content in a zone of capillary rise hc above a water table (not uniform). In cylindrical pores of diameter dc, capillary rise is roughly

hc % 30/dc IMI.

Wet snow crystals are coated with a thin layer of water, and in re-entrant angles at contacting grains, water pockets are found. To what extent this water will remain as equilibrium water content and represent a free water-holding capacity has not been unanimously agreed on. Some authors accept only free water in the order of 1 to 6 per cent as being in equilibrium (Gerdel [19]), (U.S. Corps of Engineers [ZO]) whereas others assume a wider range from 5 to 25 per cent (de Quervain [21]) or even up to 55 per cent (Moskalev [22]).

Wakahama [23] reports snow layers with 20

-

30 per cent water with- out referring to a state of equilibrium.

Equilibrium water content, whatever its range, depends on snow density, grain size, and grain shape in a complex manner. The spe- cific inner surface of snow, specific number of grain contacts, and pore width are important.

than that with coarse grains; therefore, high equilibrium water con- tent is related to new SEOW, snow of felt-like structure, and fine granular material, whereas low values are found in metamorphic coarse old snow.

exists .

water may exist in a transient state during and following a melting period or a rainstorm.

3.2 Water Percolation through Wet S n m

Water movement through wet snow, studied among the first charac- teristics of snow by Ahlmann, Lutschg, Oechslin, Seligman, and Church (1925-39), follows one of the following patterns :

-

Vertical flow in a water film along the surface of the grains Snow with fine grains holds more water

An optimum dry density for maximum storage obviously In addition to equilibrium water, a considerable amount of free

defined as film flow by Yosida (see Wakahama [23]).

209

-

Vertical flow in isolated saturated pores (channel flow, see

-

^Wakahama) Darcy-flow in saturated snow under a pressure head.

^.

Some measurements 'illustrate water content in an equilibrium state and capillary effects above a water table (de Quervain

~ 4 1 1 .

hc dc z P S W w

Type of snow mm mm cm g/cm3 % %

New snow

-

Equilibrium zone

- ^-

^{15.5 0.320 20.4 28}

-

Capillary zone 75 0.4 6.5 0.775 49.2 83 Old snow, rounded

grains, D 2-3 mm

-

Capillary zone 32 0.9 1.5 0.930 48.3 95 old snow, rounded

grains, D < 1 mm

-

Equilibrium zone

- -

^{26.5 0.410 18.7 25}

-

Equilibrium zone

- -

¹³ O. 495 5 9

-

Capillary zone 55 0.55 3 0.950 50 98 Legend: hc

=

capillary rise,

dc

=

ayparent capillary diameter (dc

=

30/h,), z

=

level of sample above water table, P s

=

density of wet snow,

W

=

free water content, w

=

relative water content.

3.2.1 Film-flow

Film-flow may be represented by a model IV of vertical ice cyl- (A com- inders coated with a water film moving under gravity force.

plementary model with tubes was proposed by Wakahama [23].) With the following specifications

N

=

number of cylinders per cm2, r

=

radius of cylinders,

6

=

thickness of the water film (assumed 6 ^<< r), pi, pw

=

density of ice and water,

model IV yields a density of

p s

=

Nra (rpi + 26pw) and a free water content of

Supposing an equilibrium water content We

=

5 per cent as ob- served in natural snow of 2 mm grain size and a radius r

=

0.1 cm, the thickness of the stable film, calculated with equation (18), would be 6,

=

24p. Because in a real snow structure free water is

also caught in pockets and corners, the stable film is probably much thinner than 2411 on convex grain surfaces.

thickness 61 is superposed to the stable film, and movement is cal- culated for laminar viscous flow, a water flux M is obtained

If by current meltwater production or rain a moving film of

M

=

2rlrN~,g61~/3v (g/cm2s) (19) (v

=

kinematic viscosity of water)

and a mean flow velocity ü of

Ü

=

M/2arNpwG1

=

g612/3v

= (+)'I3 (A)

^{2 m N 2/3} (cm/s) (20) 3VPW

Thus, the mean flow velocity depends on water flux according to u

=

const.

-

(M/x-N)~/~, whereas the filtering velocity mo is just W p W *

Total water content is found in a rather questionable manner by Example: Let water flux be (meltwater rate) M

=

g/cm2s,

(corresponding to 43 mm H20 per 12 hours), r

=

0.1 cm (as above), N

=

10/cm2 and v

=

1.8

-

cm2/s (viscosity of water at O°C) . Calculated values are :

substituting the sum 6, + 61 for 6 in equation (18).

Thickness of moving film

-

6 1

=

9 . 6 ~ (eq. 19) Mean flow velocity u

=

16.7

-

^cm/s

⁼

^{1 cm/min (eq. 20)}

Surface velocity us

=

25. cm/s

=

1.5 cm/min Filtering ve locity mo

=

cm/s

=

6 ^* cm/min

Density pS

=

0.31 g/cm3 (eq. 173

Free water content W

=

6.8 % (including We

=

5%) Ceq. 183 A realistic meltwater rate or rain produces in the given model a moving film of only 1 0 ~ and a transient water content of only 1.8 per cent. Of course, the model should not be overstressed. It may reveal only very general features and trends of the processes.

3.2.2 Channel flow

The complementary model with capillary tubes as applied by Wakahama acts the same way as long as the water film is thin compared to the radius of the tubes. Once a tube is filled with water a Hagen-Poiseuille-flow takes place (channel flow) which is very effi-

cient. The quantity of water draining from a single ice cylinder as calculated in the example (r

=

1 nun, 6,

=

lop) is transported by a single capillary tube of 0.1 mm diameter only.

In real snow, percolation is not spread evenly over the whole structure. Water from film flow is collected by capillary pores and drained through isolated channels in an irregular manner. This has been demonstrated by Gerdel [19], Wakahama [23], and many others.

It seems that in isothermal snow preferred channels are gradu- ally opened, a process that needs an energy source. Besides pene- trating radiation, heat production by the percolating water may be considered.

3 * M l o e 7 g/cm3s snow is melted as a local average. If the energy is focussed to specific bottle necks, a substantial widening of the pores should result. Provided all generated heat is instantly used 211 With a vertical water flux of M g/cm2s an amount of

for melting ice a cylindrical pore of diameter do is widened with time t by gravitational flow of water of 0°C according to

A

=

heat of fusion of ice (33.4

-

lo8 erg/g) .

As seen from Figure 5 a pore of initial diameter of 10a2cm (0.1 mm) would not change substantially within a month, whereas a pore of 0.3 mm width would be widened by 30 per cent within 10 days and soon become unstable.

meltwater

-

apparently under isothermal conditions

-

induces consid- erable growth of grains. This phenomenon deserves further attention.

Somehow in contrast to this statement is the observation that

3.2.3 Darcy-flow

In a system of predominantly saturated pores and under a hy- draulic head Darcy-flow takes place according to

-

vx

=

filtering velocity ( cm/s J

Kw

=

hydraulic conductivity (cm/s) , h

=

hydraulic head (cm H20)

*I

")

=

cm H20 as pressure unit is common in hydraulics.

Ka, provided laminar flow conditions are maintained in both cases.

Theoretically Kw

=

Ka/" ( a

=

pw/Ua ratio of dynamic viscosities of water and air

=

104).

(Ka). de Quervain [21].

Hydraulic conductivity of snow corresponds with air permeability

Comparison of measured conductivities for water (K,) and air

Snow type KW Ka/ 104

Fine granular (D .:1 mm) p s

=

0.35 0.57 0.52 Medium granular (0.8 < D < 1.5 mm) 0.38 1.20 1.19 Coarse granular (D > 2 nun) 0.385 2.24 2.16

D average grain diameter (estimate), Ps in g/cm3, Kw and Ka in cm/s .

Calculations of Kw based on structural parameters have been proposed by various authors. Moskalev [ZZ] discusses these problems extensively and offers a formula valid for laminar conditions

Kw

=

2.88

-

^{n *} D1.63 (cm/s) (22) D

=

mean grain diameter (mm), n porosity

On a homogeneous snow slope with impermeable base, for example, a roof, steady Darcy-flow conditions are easily demonstrated. Let slope angle be $, length of slope s, width b, and vertical perco- lation of water due to melting or rain mo (corresponding to vertical filtering velocity). The index o indicates steady state. Provided the depth z of the Darcy-flow layer is small compared to the distance

x (see Fig. 63, the flow q (x) in cm3/s at any cross-section x is equal to the total vertica? flux above x:

Thus zo(x)

=

(mo x cot$)/Kw, in particular zso

=

(mo s * cot$)/Kw.

The Darcy-layer is triangular shaped in a longitudinal section.

Example: With $

=

lo", s

=

2 104cm (200 m), b

=

100 cm, mo

=

10-4cm/s (as in a previous example), Kw

=

2.3 (cm/s), depth of the Darcy-layer at the lower end of the slope turns out zso

=

4.9 cm.

If vertical percolation mo suddenly ceases at the upper surface of the Darcy-layer, runoff is nourished by the water stored in the Darcy zone and dies out in following approximately an exponential

1 aw

2Kws in$

qs(t)

=

qso exp (-X.t) w i t h x

= -

s .n (n

=

porosity)

Because changes of water production actually originate at the snow surface and time lapse of percolation from surface to ground is not negligible in a deep snow cover, equation (24) is modified by a decreasing vertical percolation. Assuming a similar exponential law m(t)

=

mo expl-<t) for the dying out base flow, runoff at the

point s would be

This function is bell shaped with

<

depending inversely on snow depth (see Fig. 7).

effects such as water permeability of the ground, formation of gaps between snow and ground and, in particular, presence of channels in the ground surface. Therefore in an alpine environment it is seldom observed over large areas. Favourable locations are vast permafrost areas, smooth snow-covered glaciers below the firn line, rock plates and impermeable ice layers within the snow cover. Moskalev stresses the importance of saturated layers with respect to formation of wet snow avalanches by buoyancy and lubrication [22].

3.3 Penetration of Water in Dry S n m

Energy transfer through wet snow is effected only by air circu- lation, water migration, and short wave radiation. Beyond the range of radiation percolating water above all is able to advance a zero degree isotherm into dry snow.

movement from the ground.

one cm3 of dry snow an amount Pz

=

Ps ci ^* T/Lf is frozen in bringing the dry snow to 0°C (ps density of dry snow, Ci specific heat of ice, Lf heat of fusion, T negative temperature of the dry snow). Another quantity of water remains captured as equilibrium water content in the conquered zone and only surplus water is able to advance the frontier. It is interesting to note that by this mechanism of thawing, no ice is melted in the interior of the snow cover at all.

Under natural conditions Darcy-flow is disturbed by various

Capillary rise may effect an upward From percolating water at 0°C penetrating

2 13

Further percolation is often obstructed at the isothermic level.

A lateral expansion takes place and ice lenses are formed.

horizons are predetermined to a certain extent by the stratification of the snow.

and closer to the surface due to lower snow temperatures. Subsequent meltwater may break through the ice layers at distinct points and drain through separated vertical channels often characterized by ice plugs. As a whole thawing of a snow cover often takes place in such an irregular manner (Fig. 8).

3.4

thawing up because it is a matter of normal heat conduction in the frozen part with phase change at the O°C-isotherm.

well known [25] and characterized for a given constant surface tem- perature by

Such Their sequence is denser early in the thawing season

Refreezing of a Wet Snow Cover

The process of refreezing of wet snow is not symmetrical to The treatment is

(263 Lf * W

-

^{Ps v?T y}

⁼

^{200 * b T} e~p(-y~/4a)/erf(y/2a~/~) With W

=

free water content

e),

y

=

Zft-1/2 (Zf position of freezing level below surface at time t) ,

a

=

ke/ciPs (temperature .diffusivity in the refrozen snow) b

=

(Lf

T

=

surface temperature O C (positive sign).

ci * Ps)1/2,

For a set of relevant parameters y is calculated (graphical and the depth of the freezing level is found as Y)

Zf

=

y t1/2 ₁₂₇₎

For example with Ps

=

0.4 g/cm3, ke

=

cal/cm°Cs, W

=

lo%, T

=

^1-5'1, we find y

=

0.053 cm s-'/~.

Zf is at a depth of 15.6 cm.

values are exceeded if standard figures for ke are applied.

reasonably explained by ventilation ie., penetration of heavy cold air from the surface.

3.5 Metamorphism in Wet snow

Metamorphism of wet snow is not fully investigated yet. In the active melting process protruding parts of snow crystals are liq- uefied first, primarily because of exposure to heat flux, and grains become rounded, but under quasi-isothermal conditions they tend to

grow.

described under 3.3 is only one partial explanation. Slight differ- ences in the melting point between edges and re-entrant angles effect local transfer of ice, but not necessarily grain growth.

respect is something that should be examined more closely. At the bottom of a snow cover several metres deep peak stresses of the order of 10 Njcm2 (1 kp/cm2] may exist in certain grain boundaries, pro- ducing temperature differences near to O. Ol°C against stress-free parts. These latter will become colder than the adhering stress- free water and form a freezing source.

may play a similar role, whereas isotopic effects are probably too weak for lack of concentratilon.

After 24 hours freezing level Experiments have shown that theoretical This is

Freezing of meltwater in its first contact with cold snow as

Whether or not local stress conditions are important in this

Soluble chemical components

4. RADIANT TRANSFER

Energy transfer by radiation within the snow cover affects dry and wet snow in a similar manner. As it is concentrated on the uppermost layers and certainly treated extensively with the problems of the snow surface, only a short summary is given here.

4.1 Penetration of VisibZe Sun Radiation

Incoming radiation is first sorted out by the snow surface acting as a valve.

ation, but in the granular form it returns a high percentage of energy by direct or multiple reflection at outer or inner crystal surfaces and by scattering at edges, corners, bubbles, etc. The same processes effect attenuation of penetrating radiation with gradual absorption. Albedo, the coefficient of backscatter into the upper half space is not related to the very surface only, but implies a certain depth. In this range penetration of the remaining radiation should deviate from the exponential law

Bulk ice is highly transparent for visible radi-

which is generally accepted to hold within the snow cover.

The overall extinction coefficient E is composed of an absorp- tion term a and a scattering term B according to

E

=

^{a(1 +} 2 B/a)l/' (Dunkle and Bevans [26]) (29) Absorption is governed by snow density, and scattering depends primarily on the number of crystal surfaces penetrated by a straight

line per unit length, ie., on density and grain size (roughly ci + p s

and ß + PS/D).

sity and decreasing grain size.

size of non-scattering grain bonds are enhanced, so that in the highest density range extinction diminishes toward its final value

in bulk ice. Mellor [14] offers a survey on this problem, completed by his own measurements and discusses the dependence on wave length extensively.

From the point of view of energy transfer, absorbed radiation is an internal spacial energy source independent of conduction prn- cesses. Absorbed energy

Therefore extinction increases with increasing den- At high density, however, number and

(301 -&Z

dJ/dz

=

JO E

-

^e

is heating up the snow at a level z, whereas thermal conductivity tends to equalize temperature differences. Mellor [14] gives the differential equation for both effects.

1.7 cm-l. Taping a value of 0.2 cm-l as an example, energy flux at a depth of 25 cm is 0.67 per cent of the flux penetrating the albedo- layer. Assuming an incident global radiation of 50

-

^cal/cm2s

and an albedo of 0.8, energy flux at z

=

25 cm amounts to 6.7

-

(density 0.3 g/cm3) would be heated 0.03"C per hour. The same energy flux is found in a temperature gradient of 1°C/m. Hence, energy gain by short wave radiation is practically confined to the uppermost 50 cm of snow.

ground by radiation becomes active.

Extinction coefficients are measured in the range O. 017 to

cal/cm2s, and without any conductive loss 1 cm3 of snow

In a snowpack shallower than that, heating of the

2 15

4.2 Effects of Long Wave Radiation

Long wave radiation, as emitted by the atmosphere and the snow itself according to Stefan-Boltzman (uT4) is restricted to ice layers several tenths of a millimetre thick, as ice is practically "black"

in the pertinent range of wave length (> 511).

A striking combined effect of penetrating short wave radiation and superficial emission of long wave radiation is the occasional formation of a "film crust" (firn mirror) above a melting snow layer, assisted probably by the cold of evaporation (de Quervain [21]).

Within the snow cover there is theoretically an exchange of long wave radiation between neighbouring snow crystals. The re- sulting energy flux for small temperature differences TI

-

^{T, is}

-

with E (Emissivity)

=

0.9; u

=

1.36 *

T

=

270°K ( - 3 O C ) the flux is AT * cal/cm2s.

a temperature difference of about 0.01"C and results in a flux J Q

ductivity of vapour in the pores.

exchange can be neglected within the snow cover.

cal/s cm0K4 and

A temperature gradient of 10°/m appears in a pore 1 mm wide as cal/cm2s, which is of the order of 1 per cent of the con-

Consequently long wave energy

5. CONCLUDING REMARK

With this short survey isolated basic phenomena have been out- lined by means of oversimplified models, solely for our under- standing. Real snow is much more complicated. The processes linked together and, above all, the actual structure of snow, introduce many additi onal parameters to be cons idered.

RE FE RENCES

PAULCKE W. (1934). Eisbildungen I. Der Schnee u. seine Diagenese. Z. F. Gletscherkunde XXI 4/5, 259-282.

SELIGMAN G. (1936). Snow structure and ski fields. McMillan Ltd., London.

BADER H. et al. (1939). Der Schnee und seine Metamorphose.

Beitr. z. Geol. d. Schweiz. Hydrologie 3 (Kümmerli u. Frey, Bern).

KONDRAT'EVA, A. S. (1945). Teploprovodnost' snegovogo pokrova i fizicheskie protsessy, proiskhodiaschie v nom pod vlianiem temperaturnogo gradienta. (Thermal conductivity of the snow cover and physical processes caused by the temperature gradient.) Akademii Nauk SSSR. (U.S.A. SIPRE Trans. 22, 1954, 13 pp.)

EUGSTER, H.P. (1952). Beiträge zu einer GefÜgeanalyse des Schnees. Beitr. z. Geol. d. Schweiz. Hydrologie 5, 6 4 pp.

(KÜmmerly u. Frey, Bern.)

YOSIDA, Z. (1950). Heat transfer by water vapor in a snow cover. Low Temp. Sci. 5, pp 93-100.

[71 YOSIDA, Z., and KOJIMA, K. (1954). Acceleration of metamor- phism of snow by temperature gradient. IUGG Congress, Rome, GIDDINGS, J.C., and LaCHAPELLE, E. (1962). The formation of depth hoar. J. Geophys. Res. 67/6, pp 2377-2383.

YOSIDA, Z. et al. (1955). Physical studies on deposited snow.

Thermal properties. Contrib. Low Temp. Sci. 7, pp. 19-74.

de QUERVAIN, M.R. (1963). On the metamorphism of snow. In

"Ice and Snow" (Ed. W.D. Kingery), M.I.T. Press, Cambridge Mass., pp. 377-390.

YEN, YIN-CHAO. (1962). Effective thermal conductivity of ven- tilated snow. J. Geophys. Res. 67/3, pp. 1091-1098.

YEN, YIN-CHAO. (1963). Heat transfer by vapor transfer in ventilated snow. J. Geophys. Res. 68/4, pp. 1093-1101.

ARNASON, B. (1969). Equilibrium constant for the fraction- ation of deuterium between ice and water. J. Phys. Chem. 73, MELLOR, M. (1964). Properties of snow. Cold Regions Science and Engineering III, A 1. U.S. Cold Regions Research and Engineering Lab.

de QUERVAIN, M. R. (1954). Zur Warmeleitung von Schnee. (On thermal conductivity of snow.) IUGG Congress, Rome, AISH Publ.

39, IV, pp. 26-32.

de QUERVAIN, M.R. (1946). Zur Temperaturdynamik der Schneedecke. Int. Ber. Eidg. Inst. f. Schnee u. Lawinen- forschung, Weissfluhjoch/Davos, No. 25.

WOODSIDE, W. (1958). Calculation of the thermal conductivity of porous media. Can. J. Phys. 36/7, pp. 815-823.

YEN, YIN-CHAO. (1967). The rate of temperature propagation in moist porous mediums with particular reference to snow. J.

Geophys. Res. 72/3, pp. 1283-1288.

GERDEL, W.R. (1954). The transmission of water through snow.

Trans. Am. Geophys. Un. 35/3, pp. 475-485.

U.S. CORPS OF ENGINEERS. (1956). Summary report on snow in- vestigations. North Pacific Division. U.S. Corps of Engineers

(30, June).

de QUERVAIN, M.R. (1948). Ueber den Abbau der alpinen Schnee- decke. (On ablation of the alpine snow cover.) UIGG Congress, Oslo, IASN Rep. II, pp. 55-67.

MOSKALEV, YU. D. (1966). Voznidnoveniye i Dvizheniye Lavin.

(Avalanche mechanics.) Hydromet. Publ. House, Leningrad, 152 pp.

WAKAHAMA, G. et al. (1968). Infiltration of melt water into snow cover. Low Temp. Sci. Ser. A, 26, pp. 54-76 and 77-85.

de QUERVAIN, M.R. (1946). Zur Messung der Schneefeuchtigkeit.

Int. Ber. Eidg. Inst. SLF, 26.

GRöBER-ERK. (1955). Gmdlagen der Wärmeübertragung. Verl.

Springer, Berlin.

IASH Publ. 39, IV, pp. 92-97.

pp. 3491-3494.

217

[26] DUNKLE, R.V., and BEVANS, J.T. (1956). An approximate analysis of the solar reflectance and transmittance of a snowcover. J.

Met. Vol. 13.

I

N4

M .li U

2 19

I B

Fig. 2. Local variation of temperature gradient and mass flux in a snow structure (schematic)

DENSITY p (G/CM’)

Fig. 3. Thermal conductivity of snow as a function of density (after M. Mellor, [14])

F,- - I\

221

d

6

5

4

3

2

1

O

I

I I I I

I I

I I l I I I

---+

I

I

7 ^I

I I

I 130d

I I I

I

d a y s

I

P

to

²⁰ 30

Fig. 5. Widening of pores by meltwater.

d diameter of pore

223

l.

a

O, 8

O, 6

O6 4

o,

2

0°C ISOTHERME

\ \

\

\

\

\

\,

\

\

\

\

\

\

\

\

\

\

\

\

t

\

\

\

\

t

\

-.

-.

‘ . .

4 2

Fig. 7. Runoff from snow (see text p. 215)

-T[“C]

r71

O

I

5 0

- E

o

N

Fig. 8. Penetration of the D’-isotherme in cold snow and formation of ice lenses and ice cones. See paragraph 3.3

DISCUSSION

E.J. Langham (Canada)

-

In your comments on percolation in un- saturated snow you made reference to the preferred paths of flow through the snow.

interaction between the water and snow such that the channel be- comes enlarged and as a result the preferred flow becomes even stronger.

the graph used during these remarks is not given in the preprint.

would like to look at this data more carefully and I wish to ask if the ideas expressed on this form of structural modification have been reported?

I might add that I have observed the existence of these chan- nels during some experiments in which coloured ice crystals were inserted in the snowpack1. The paths began at points where melt- water flowed through an ice layer near the snow surface. (The holes

in the ice layer were at crystal boundaries and were like those pro- duced when ice is rotted by radiation.)

The modification of the structure of the snowpack resulting from this kind of flow pattern may be responsible for other observ- able effects. For example, it would explain the columnar structure that forms within the snowpack when it refreezes after a period of me 1 tin g.

During the temporary cessation of melting conditions the water would drain out of such channels and leave a vertical structure of higher permeability. Your emphasis on the importance of the circu- lation of air within the snowpack leads one to ask whether the de- -Jelopment of vertical structure might not play a part in this process also?

You mentioned that in such a case there may be

In your presentation you developed this idea further; however I

M.B. de Quervain (Switzerland)

-

This phenomenon is simply an idea inserted in this paper.

am sure that glaciologists concerned with the percolation of water must have come across this idea of the opening of pores by the energy of the water, the water being a power plant.

I have not seen it anywhere else but I

C. ObZed (France)

-

Je voudrais remercier d'abord le professeur de Quervain pour la remarquable synthzse des échanges thermiques internes qu'il nous a proposés, et regretter ensuite que les données de structure nécessaire (porosité, crystallographie) soient rarement disponibles, en dépit du role important qu'ils peuvent avoir dans la chronologie de l'onde de fusion. En conséquence, je voudrais

savoir si:

un modèle

B

plusieurs couches (respectant la stratification), et s'il faut en attendre une nette amélioration par rapport au modèle simple en considérant qu'une couche homogène de caractéristiques moyennes?

lière, étant donné sa position-clé entre les échanges atmosphéri- ques et ceux qui sont propres au manteau?

I) Le professeur de Quervain a déjà effectué des calculs avec

2) I1 faut considérer la couche de surface de fason particu-

"A new technique for the use of dye in the study of meltwater movement in the snowpack. " E. J. Langham, National Research Council of Canada, 8th Symposium on Hydrology, 1971.

2 2 5

3) En période de fusion Btablie, il n'y a pas un phénomène d'homogénéisation des caractéristiques de la couche, permettant l'emploi d'une loi empirique pour ks

=

f(ps)?

thermique de la neige.)

Enfin, en ce qui concerne l'écoulement superficiel de l'eau de fonte, le professeur de Quervain a-t-il des connaissances d'expe- rience analogues 2 celles du Dr. Yevdjevich (University of Colorado, Fort Collins) pour le ruissellement de la pluie?

(Conductivité

M.R. de Quervain (Switzerland)

-

With regards to the strati- fied model, I think that it is the next step and should be done be- cause every layer has its own density and permeability but I have not made such a model.

from the Devil, meaning the surface of a solid body. But I think that the snow surface comes from God; it is rather nice to have a snow surface. The surface is really important; it is influenced by the influx and outflux of radiation and heat, and is transformed into the loose layer we find in the interior of the snowpack, I think that everything that is inside has to pass through the surface and the properties of the surface layer (of several mm or cm

thickness) are often different from anything inside the snowpack.

The homogenization of the snowpack by melting does occur.

sure the differences are largest in a dry snowpack and that when it gets wet the heat conductivity, density, etc. , will be more homo- geneous.

With regards to the runoff at the base of the pack, I have spoken of Darcy-flow with the understanding that it hardly ever exists in nature at the base of the pack because either the water goes into the ground or it stays between the ground and the snow or it runs through a channel without touching the snow.

As for surface, the physicist Pauli once said the surface comes

I am

J.R. Meiman (U.S.A.)

-

Dr. de Quervain impressed me with the number of times he gave us the theoretical model and then said, llnow

in real snow.'I I think this attests to his great experience.