Transformat ion of the Snow Crysta l to a Part ic le of Ice

Elena Guseva-Loz inski

Immenhoferstr. 38, 70180 Stuttgart, Germany

Abs t rac t . To study the physical properties of snow under different meteorological conditions a mathematical model and numerical computer program were created and applied for some numerical modelling estimates. The non-linear mathematical model consists of partial differential equations and can be subdivided into a thermal part with phase changes in porous media, diffusion, structural transformation and mechanical parts. The model was applied to simulate the evolution of structural, thermal and mechanical parameters in a snow profile subject to meteorological parameters (air temperature and moisture, wind velocity, precipitation, density). The snow structure is very sensitive to the temporal variations of all external para- meters: temperature, humidity, precipitation and wind-pumping. Snow deposited in cold weather conditions is transformed through densification, metamorphism and recrystallisation. Snow crystals have unstable shapes. The tendency for mass and heat to be redistributed through sublimation is to minimise the surface free energy. The result of these processes is to change the shape of a snow crystal to that of a sphere. The transformation of the initial singular stellar crystal to a number of small grains with the same mass as the original crystal is described mathematically. It gives the rates of the transformations. Based on this mathematical approach we can predict changes of the crystal shapes, number of crystals and other physical properties inside a snowpack subject to different meteorological conditions.

1 I n t r o d u c t i o n Snowflakes falling on the Earth 's surface have a mono-crystalline, idiomorphic form (dendrite, for example) or polycrystalline elements with crystals ranging in sizes from 0.1-0.4 mm [1] at very low air temperatures (-50°C - -70°C) to several millimetres at air temperature around 0°C. Depending on the weather conditions (air temperature, moisture, wind velocity) the snowpack is formed under windless conditions from lamellar snowflakes with an initial snow dens- ity of 10-80 kgm -3 and idiomorphic contours, or from snowflake 0.2-0.3mm sized fragments formed under windy conditions with a density of about 200- 300 kgm -3.

The crystallisation process involves a change of phase of the initial sub- stance to a phase with lesser free energy. The main conditions controlling the crystallisation are the water vapour oversaturation or supercooling of the initial medium (water vapour in air). Both these processes are necessary for crystallisation and are characterised by the departure from thermodynamic equilibrium. The simplest growth form is polyhedral. The sizes of separate facets strongly depend on growth conditions. Oversaturation is higher around

388 Elena Guseva-Lozinski

apices and ribs and lesser in the central parts of facets. Tha t is one reason for the formation of the skeletal and arborescent, dendritic grain forms; another is rapid growth of crystals.

The initial state of the freshly deposited snow depends on the combin- ation of the wind condition and crystal type at the moment of deposition. A freshly deposited snowpack consists of different crystal forms. Most of the crystals deposited under windless conditions, have a dendritic form. If the oversaturation above the central facets part of the crystal is less then the vertices come to play a dominant role in the crystal growth process. This is a primary reason for the skeletal and dendritic crystal appearance. Dend- ritic and skeletal crystal construction can be divided into smaller elements (Fig. 1).

a) b) c)

Fig. 1. a) A dendritic crystal; b) A skeletal crystal; c) A crystal branch of multi- layered construction. It consists of several small elements. Arrows show rimed droplets at edges or corners [17]

The snowpack can be visualized as a porous medium composed of a network. The snow crystals lose their stability in the upper layers of the snowcover. As a result of the crystal instability, sublimation processes begin leading to internal transformation of the ice crystal network. The result of these metamorphic transformations is the establishment of the thermody- namic equilibrium of the snow crystals within the snowpack and decrease of the surface free energy. The changes of grain form and grain size within the snowpack results from superposition of both isothermal metamorphism and tempera ture metamorphism. A roundish form of snow grains, roundness of thin ring of the separate dendritic crystal branches are the direct result of this metamorphism. The grains draw together the interior snow-pack, and bonds are formed in the crystal contacts area. These metamorphic processes are accompanied by snow densification and by the increasing of the strength of the grain bonds.

Transformation of the Snow Crystal to a Particle of Ice 389

2 C r y s t a l S h a p e s a n d M e t a m o r p h i s m

Metamorphic processes occur very quickly within a freshly deposited layer of snow. Within 1-3 days, depending on t empera tu re and initial conditions, after deposition, snowflakes are t ransformed into polyhedral and oval grains due to high degree of supersaturat ing, t empera ture gradients and destruct- ive metamorphic processes. In the case of a stand-alone crystal more t ime is needed, about 60 days [2,3]. The velocity of me tamorph i sm slows down appreciably during subsequent stages. From crystallization theory we know, tha t #c -- #0 is the equilibrium condition in close proximity to a flat interface surface, where #c, Ito are the chemical potentials of the substance in vapour and in crystal forms respectively,

" c = ' ° + E (1) where S) is the specific volume of one particle, a is a specific free interface surface energy, R1,R2 are the main surface curvatures at a point of the inter- face surface, ~1,~2 are angles defined through the crystallographic oriented surface. (#c - #0) is positive, for crystal growth, and it is negative, when the crystals are undergoing sublimation. The real crystal form with its volume V is t ransformed to its equilibrium crystal form. This tendency to minimise the surface energy Ev=~o~st is described by the Gibbs-Curie-Wolf principle

n

Ev=const = ~ ~ S i = min. (2) i = 1

According to the Glaciological Vocabulary [2] the initial diameters of snow- flakes depend on the air temperature: -6°C r -- 1.5 - 2 mm; -8°C r -- 1 mm; -12°C r = 0.5 - 1 mm. The flakes were most ly stars, with plates and prisms being rare above -20°C. Snowflakes are modified during their fall through the a tmosphere due to over-riming. In other words, the snowflake surface is covered by numerous small crystals. The size of these new crystals is one hundred times smaller than the host snowflake. The number of small crystals so formed can reach to several hundreds. Thus the simple form of a snow- flake can convert to the very complicated dendritic form of snowflakes due to over-riming. Those surfaces with small crystals a t tached can be described as a combination of trihedral fracture angles.

The geometry of the initial crystal surface can be defined as a combinat ion of polyhedron faces with edges and vertices. Using Eq. (1) and principle (2), the average value of water vapour around crystal g with n faces and m edges and k vertexes can be writ ten as follows:

S 9 Se S~ = eon--sc + edm~c + ezk-~c' (3)

where e0, ed, ex are the water vapour pressures above the fiat ice surface, above edges and above vertices and where Sc, Sg, Se, Sx are the surface area

390 Elena Cuseva-Lozinski

of a polyhedron, of the flat par t of a face, of the area near edge and of the area above a vertex respectively (Fig. 2). The shaded area in Fig. 2 is the area of higher water vapour pressure due to surface heterogeneity. According to the Physical Encyclopedical Vocabulary [4], the thickness ld of this area is

i l0 -2, where [ = 10-5m is the average free pa th length of the gas (water vapour) molecules. The water vapour pressure above an edge can be rewri t ten as ed = e0 + Aed and above a ver tex as e~ = e0 + Aex. I t is known [5], t ha t the cohesion work a = 2a0 for light mobile phase boundaries, where a0 is the tension surface of flat ice Cro = 1 3 0 d i n / c m . Using the Thomson-Kelvin Equat ion for the case of not too small radius of curvature and Eq. (1), the following expressions can be wri t ten for the water vapour pressure and the ice,

Aee -- P0 ae and Aex -- P0 2ax Pi ld Pi ld ' (4)

where Po is the water vapour sa tura t ion pressure and p~ is the ice density.

Fig. 2. Hexagonal crystal. Sg is a facet surface area, ld is the thickness with the higher water vapor pressure near the edges, Sv is the area above a vertex. The shaded areas are the areas with higher water vapor pressure above a surface het- erogeneity

An est imate of the average value of water vapour around a polyhedral crystal ~ can be writ ten as

10_6,~ @=e0 l + n - - ~ ) , (5)

where n > 4, e0 is the water vapour pressure above the flat ice surface at t empera tu re T a n d / ~ is the average size of the polyhedron.

The roughness mono-crystal shapes have a big density of incoming tri- hedral fracture angles. In these points the addition of new particles to a crys- tal is very easy. I t did not request the overcoming of the potent ia l barrier. The growth limit is defined by the degree of oversaturat ion and t empera tu re gradients. Some simple forms of snowflakes give the very complicated forms of snowflakes due to over-riming, r l , m I are the radius and mass of a dend- ritic crystal, rc is the radius of the central hexagonal plate, r , is the size of

Transformation of the Snow Crystal to a Particle of Ice 391

crystal element (Fig. 3). We est imate the fracture number kt due to the form of the snowflake as follows:

1(rn/ /r/ - 47rpir~c) . 10 -6 "~ kt= rt 2 Pi ( 1 + 1 0 - 2 k ~ ) a n d ~ e 0 l + / z t - - - ~ ) , (6)

where k~ is a rime droplet parameter . The surface over-riming process in- creases the fracture number kt. Most common snowflakes are a flat plate ten times larger in diameter than in thickness [6]: do = 10-1r ] . The water vapour pressure e~ above elements of the s t ructure at t empera tu re T is defined by the Magnus relation

(17T ~( F) ev(T,r')=eoexP\T+235] I + V ' (7)

where r ~ is the radius of surface curvature of the elements of the ice mat - rix, where the water vapour pressure above a fiat surface at T = 0°C is e0, and where F is a shape parameter . Using (3)-(6), this pa ramete r was calcu- lated as F = 2 x 10 -5 for depth hoar layers and for fresh snow layers. The shape paramete r is corrected for an excess of water vapour pressure bo th above an inhomogeneous surface and above a smooth surface. Snowflakes incorporated in the snow-cover become rounded and the branches of dend- rites undergo evaporation. The rapid sublimation of the dendritic branches can be explained by sublimation due to surface curvature effects [7,8]. Col- beck [1], Hobbs and Mason [9] show, tha t the main process of bond growth between grains is vapour diffusion due to curvature difference of the different s t ructural elements.

In the initial stage grain possess an oval form; in the second stage they become spherical. The grains within the snowpack are close packed. Bonds originate in the contact areas of grains due to capillary processes and to over- sa turat ion near these contacts. Depth hoar is the result of post-deposit ional

1

a) b) c) d)

Fig. 3. a) Here is shown one of the typical forms of dendritic snowflake form with a circumscribed circle around the dendritic crystal, b) r / i s the radius of a circum- scribed circle around the dendritic crystal and rc is the radius of a central hexagonal plate, c) rt is the surface size of a crystal element, d) a part of the dendritic crystal from Fig. 1

392 Elena Cuseva-Lozinski

metamorphic processes. It is characterised by secondary idiomorphic and skeletal crystal forms. Surfaces of crystals have a step composition. The reason for step formation is the screw dislocation. In this widespread case the crystal growth is defined through the step moving. Steps interact to- gether through their own thermo- and diffusion- fields and intermolecular forces. The result of the step confluence is higher steps. Sometimes macro- scopic steps can also fall apart.

3 M a t h e m a t i c a l Approach Every layer of snow within the snowpack corresponds to individual snowfall. The mathematical model for a non-uniform stratified snow-cover with wind- pumping, snow densification and transformation of its thermo-mechanical and structural properties was investigated in the papers [10-1a]. The present mathematical model focuses on the modelling of the mass-transfer and struc- tural transformation of freshly deposited snow in the snowpack. Temperature T is determined as follows:

OT Cs--~ - C~V~ . V T = V . (KsVT) + f ( z ) + nj , (8)

where K~,C~ are the coefficients of heat conductivity and capacity, t is time, f (x) is the absorbed sun radiation distribution (Buger-Lambert-Beer Equa- tion), L is the heat of sublimation for transition from vapour to ice, j is the sublimation rate, Va is the Darcy flow. Heat conductivity and capacity coef- ficients depends linearly on the snow density p~. Determination of temporal density changes, due to viscous snow-pack densification, were based on pa- pers by Yosida [8]. The radial growth rate of the ice-matrix elements is given by the Arrhenius equation [14,15]

d-T = K0 exp

where r t, r0 are the current and initial curvature radii of different s t ructural elements. Each layer of snow within the snowpack consists of N snowflakes with centres and Nkt elements. We can consider a snowflake as consisting of a central part with radiate rays. These rays consist of elements. We can operate with two kinds of grains: grains with radius r~ and smaller grains with rt. Using the Arrhenius relation for the ice-matrix elements growth in snow and ice, in conjunction with the Darcy relation for airflow velocity and the Magnus condition, it is possible to write the conditions for the rate of sublimation/condensation on the surface of the grains j~ and jt as follows:

jc = S~(ev - ec)Ko exp( - E / R T ) N + S~V~ ~ z N, (10)

jt = St(ev - et)Ko e x p ( - E / R T ) N K t + S t V ~ z NKt , (11)

(12) J = Jc + Jr"

Transformation of the Snow Crystal to a Particle of Ice 393

The rate of sublimation-evaporation is defined through the processes of dif- fusion and wind-pumping in the porous space

O T ) Oev (13) Oev t- 0_~ De] Oe. ~rc Oev Oft ~ Oev J - at az are az ~-Def~[rt-O-~z +L'8 aT -5-;z +S~Va-5--S'

where j = j + + j - is the sum of the positive part of the mass transfer (corres- ponding to condensation) and negative part of the mass transfer (correspond- ing to evaporation) above the common sublimation/condensation surface Sf of the grains per one unit of volume. The water-vapor pressure above grains ec and et is described by (7). Here we use the hypothesis tha t the sublima- tion rate on the surface of ice-matrix elements is the result of equalizing the difference between the average water-vapor pressure in the pore space and the concentration above the structural elements and subl imat ion/evaporat ion through wind-pumping. Grain diameter changes (Magnus relation) occur as a result of the growth of coarse grains through evaporation of fine parts of the ice matr ix and sublimation rate (Eqs. (10)-(13))

dt - 4piN-ktTrr 2 + K0exp , (14)

drt jt dt - 4rp iNktr 2" (15)

The grain number per unit volume changes with time, and by using (6), is determined as

kt(t) = 1 0.75ps - Npir 3 Ir Np~r3t and rt(t) > 0. (16)

The boundary conditions for (8) and (13) are similar to those given in [12]. The initial da ta and surface boundary conditions are defined from meteoro- logical data.

Mathematical modeling of the connection between the snow strength and temperature, density and structural parameters for two-dimensional stratified snow/t im with bonded structure within the snowpack was done in paper [13].

4 Conclusion The thermo-mechanical mathematical model of structure formation and its changes allows study of the main characteristics of the non-uniform strat- ified snow and firn. This approach and mathematical model describes the temporal evolution of the properties of freshly deposited snow. The present mathematical model describes the recrystallisation processes and structural changes occurring within a fresh snow layer inside the snowpack and must be extended through the semi-empirical equations. This model is based on the mathematical model for snow/t im [12], the theory of mass- and heat transfer

394 Elena Guseva-Lozinski

in the polycrystalline solids [16], thermodynamic rules and on semi-empirical over-crystallisation theory of the snow structure. For the study of the phys- ical properties of snow subject to different meteorological conditions some numerical modelling calculations were made using the mathematical model and numerical computer program given in papers [10-13].

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Subsidiary of Harcourt Brace Jovanovich Publishers. 2. V.M. Kotljakov (Ed.). 1984 Glaciological vocabulary, Gidrometeoizdat, Lenin-

grad. 3. E.R. LaChapelle. 1992 Field Guide to Snow Crystals, International Glaciolo-

gical Society, Cambridge. 4. 1983 Physical encyclopedical vocabulary, Nauka, Moscow. 5. V.Gnielinski, A.Mersmann, F.Thurner. 1993 Verdampfung, KristaUisation,

Trocknung, Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig, Wiesbaden.

6. G.Seligman, 1980 Snow structure and ski fields, Printed in England by Foister &: Jagg LTD, Cambridge.

7. R.Perla, 1978 Temperature-gradient and Equi-temperature Metamorphism of Dry Snow, Paper presented at Deuxieme Rencontre Internationale sur la Neige et les Avalanches. Assoc. Pour l'Etude de la Neige et des Avalanches, France

8. Z.Yosida, 1963 Physical properties of snow. In: Kingery, W.D., (ed.) Ice and snow: properties processes, and applications. Cambridge, Massachusetts, The M.I.T. Press, 485-527.

9. P.V.Hobbs and B.J. Mason, 1964 , Philos.Mag. [8] 9, 181-197. 10. E.V.Guseva, V.N.Golubev. 1989 Thermomechanical mathematical model of the

formation of the structure and properties of the snowcover, Geojournal 19(2), 193-200.

11. E.V.Guseva, V.N.Golubev. 1990 Matematicheskaja model formirovanija stroy- enijya i svoystv snezhnogo pokrova [Mathematical model of the properties and structure of the snowcover], Issued., 68, 18-26. [In Russian with English sum- mary.]

12. E.V.Guseva, V.N.Golubev. 1997 Mathematical modelling of temporal changes in snow-tim properties in cold period, Annals of Glaciology, 24, 89 - 92

13. E.V.Guseva, 1998 Evolution of Snow-Firn Properties: A thermomechanical ap- proach, Proceedings of the 9 International Symposium on Continuum Models and Discrete Systems, 29-38.

14. P.J.Stefenson. 1967 Some considerations of snow metamorphism in the Antarc- tic Ice sheet in the light of ice crystal studies. Phys. Snow and Ice., 2, 725-740

15. A.J.Gow. 1969 On the rates of growth of grains and crystals in South Polar firn, Journal of Glaciology 8(53), 241-252.

16. U.GSsele, W.Frank, A.Seeger. 1980 Mechanism and Kinetics of the Diffusion of Gold in Silicon, Appl. Phys. 23, 361-368.

17. A.Yamashita , A.Asano, T.Ohno, 1985 Comparison of ice crystals grown from vapour in varying conditions, Annals of Glaciology 6,242-245.

(Received 22 Feb. 1999, accepted 15 June 1999)