International Snow Science Workshop (2002: Penticton, B.C.)

A Microstructural Dry Snow Metamorphism Model Applicable to Equi­Temperature and Temperature Gradient Environments

D.A. Miller*, E.E. Adams** and R.L. Brown***US Air Force Academy, Colorado

**Montana State University, Bozeman, Montana

Abstract: Dry snow metamorphism has traditionally been classified by the thermal gradient encountered in thesnowpack. Snow experiencing a predominantly equi-temperature environment develops different microstructurethan snow that is subjected to a temperature gradient. As such, previous research has evaluated snow metamorphismbased upon select thermal gradient dependent processes, when in reality, there is a continuum of physical processessimultaneously contributing to metamorphism. In previous research, a discrete temperature gradient transitionbetween the two thermal environments has been used to activate separate morphological analyses. The cUrrentresearch focuses on a unifying approach to dry snow metamorphism that is applicable to generalized thermalenvironments. The movement ofheat and mass is not prescribed, but is allowed to develop naturally throughmodeling of physical processes. Equi-temperature, transition to kinetic growth, and kinetic growth are consideredwith this model. Density, grain size, bond size, and temperature dependencies are each examined in an equi­temperature metamorphism environment. The same physical model is then used to defme a smooth transitionbetween equi-temperature and temperature gradient environments. Microstructural and environmental parametersthat influence the transition to kinetic growth are examined. The correlation with established trends and experimentsin each environment is excellent. The microstructural model is a new tool capable of evaluating metamorphism fora broad range ofmicrostructural parameters and thermal environments.

Keywords: snow and ice, temperature gradient, kinetic growth, snow crystal growth, snow crystal structure, snowmetamorphism

Introduction

Snow is a unique granular material. From thetime it touches the earth until it either transitions toliquid (melts) or to vapor (sublimates),t~J:Uig:ostmctureis Gontinugll changIDg...in.a.pwcesscalled metamomhism. The time-varyingrIDcrostructural quantities result in nonlinear materialresponses, making the study, understanding, andprediction of snow metamorphism vital to nearly allareas of snow science. The critical dependence ofsnow properties on microstructure makesmetamorphism an extremely important area of study.

Two categories ofmetamorphism are classicallydiscussed, depending upon the thermal environment.In a snowpack where the temperature is nearlyuniform, metamorphism is termed "equi­temperature". Ifa significant temperature gradient isapplied (frequently found in alpine snow), then

*Corresponding author address: D. A. Miller,Department of Astronautics, US Air Force Academy,CO 80840; tel 719-333-4088; emaildan.miller@usafa.af.mil

'---/

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"temperature gradient" metamorphism results.Traditionally, each ofthese conditions has beentreated as separate and distinct since each results in aunique microstructure (Sommerfeld and LaChapelle,1970). Use of the temperature environment todescribe the metamorphic process is not universallyaccepted. In fact, the categorization ofmetamorphism has resulted in much discussion anddebate. Colbeck (1980, 1982, 1983a) used crystalforms and driving forces to describe metamorphism.The current research reported here does not require adistinction between the different types ofmetamorphism, rather it uses physical processescommon to both. The resulting unified approachmakes the discussion somewhat obsolete andirrelevant.

In the absence of a macroscopic temperaturegradient, ice particles are observed to sinter bygrowing the bonds between the individual grains.This sintering process is commonly referred to asequi-temperature metamorphism. Colbeck (1980)correctly pointed out that truly equi-temperaturemetamorphism is impossible for any processinvolving phase change. Heat must flow for phase

change to occur, thereby requiring microscopictemperature gradients. Colbeck (1980) introducedthe term "radius-of-curvature" metamorphism due tothe curvature differences behind this process. Theresulting microstructure is commonly referred to asequilibrium form, rounds, or equi-temperature.

In seasonal snow, it is common for snowpacktemperature gradients to develop, resulting in"temperature gradient" or "kinetic growth"metamorphism. ~ combination ofheat stored inthe ground over the summer months and geothermal~ kee;:p th~la)::~t1le-~U:Jhe..o ou d close to O°Cfor most oUh~e.win1e.LS-eas.on (McClung andsehaerer:1993),~the uIlOO surfacl<..OLthesnowpackis. s-ubjected.to.diqm£ll tem~.llJureflliCfilaBQns. The diffusion ofwater vapor through theinterstitial pore space changes the nature ofmetamorphism significantly. A temperature gradient(and resulting vapor pressure gradient) in the poreproduces unique snow morphologies. ~ete~p~~ture graQient incr_~s, the snowmicrostructure can Q..hange from smooth roundedgrains with smooth interconnections to large, highlyfaceted, angular crystals with large surrounding porespaces. These crystal forms are often referred to asfacets, temperature gradient metamorphism forms,kinetic forms, recrystallized snow, and depth hoar.

Several theories have been developed thatadequately explain the physics of snow sintering inan equi-temperature environment and faceted graindevelopment in the presence ofa temperaturegradient. To date, no one has presented a unifiedtheory of metamorphism that includes detailedmicrostructure yet is not restricted by the temperatureenvironment. In fact, Arons and Colbeck (1995)comment "we still lack a model that integratesfundamental elements ofgeometry and the transfer ofsensible and latent heat in a physically soundinductive model". The objective of this research isto develop a unifYing dry snow microstructuremetamorphism theory that accounts for importantphysical parameters and processes yet is applicable ingeneral thermal environments. The approach willtransition smoothly between isothermal andtemperature gradient conditions.

1. Model Development

The approach begins with the definition ofsimple microstructural parameters. Conservation of

285

Mountain Snowpack

heat and mass are then defined with phase changethermodynamics as a coupling constraint. Theresulting non-linear coupled differential equations aresolved numerically by finite difference techniques.The development is only briefly summarized herewith the details available in Miller (2002).

1.1 Equilibrium Form Geometry

A simple grain and neck geometry is used for therounded form of snow. To model the ice geometry,some simplifications are made. First, the grains forthis morphology are assumed to be spherical. Theremay be several different sized grains, but theequilibrium forms are modeled as perfectly round Auniversally accepted intergranular neck definitiondoes not exist. Therefore, smooth concave necksspanning either side of crystal bonds are used tomake the interparticle connections. Edens (1997)developed a stereological approach defining thegrain-neck transition where the surface curvaturereverses sign. Geometric definitions used here aresimilar to Brown et al. (1999). The pore sizes aredefined based on the grain/neck sizes and snowdensity. While the detailed arrangement of grainsand necks is complex, some simplifications are made.For snow in an isothermal environment, it is assumedthat the material is macroscopically isotropic, ie, themicrostructure does not depend upon the observationdirection. When a temperature gradient is applied,metamorphism can result in directionally orientedmicrostructure. During long exposure to significanttemperature gradients, channels or mesoporesdevelop parallel to the temperature gradient. Icechains may also develop with large, striated, facetedgrains appearing. Due to this preferred orientation, itis assumed that the snow geometric properties aresymmetric with respect to an axis ofrotation aboutthe temperature gradient, making the problemtransversely isotropic. Symmetry is assumed in anyplane perpendicular to the temperature gradient. Tomodel equi-temperature and temperature gradientenvironments, a microscopic geometry is used bylinking a large number of ice spheres and neckstogether. The configuration of two grains and oneneck is given in Figure 1. Any number of grains andnecks may be defined.

International Snow Science Workshop (2002: Penticton, B.C.)

where Aec is the surface are undergoing phase change(m2

) and Vpore is the volume of the pore (m3). If the

vapor is treated as an ideal gas and using thedifferential form of the Clausius-Clapeyronrelationship, Fick's Law for vapor diffusion may beexpressed as

(2)

(3)

(1)

MS = J Aec

ec Vpore

ap + V.(J) = MSat - -where p is the vapor density (kg/m3

), J is themolecular flux vector (kg/(sm2

)), and and MS is themass supply or sink from phase change (kg/m3s).MS is given by

5. Liquid water in the snow is notconsidered. Solid ice and water vapor are the onlyphases of water considered for dry snow.

Figure 1 shows graphically the heat and massfluxes available given the geometry and assumptionsdiscussed. Mass can diffuse vertically through thepore spaces as well as to and from the ice surfaceduring phase changes. Heat can flow verticallythrough the ice network as well as to or from the icesurface due to the latent heat exchange associatedwith phase change.

The first consideration is the movement of vaporin the pore spaces. The mass conservation equationwith mass sources/sinks from phase change is givenby

Before the governing equations can bedeveloped, there are several assumptions that aremade:

1. Heat conduction in the pore space can beneglected, so that only heatflow through the icenetwork is considered. This is a common assumptiongiven the fact that the thennal conductivity ofsaturated vapor at cold temperatures is two orders ofmagnitude smaller than the thennal conductivity ofpure ice (Incropera and DeWitt,1985).

2. Saturated vapor, for a given temperature,exists everywhere in the pore spaces. Even thoughno measurements of relative humidity in pore spaceshave been published, this is a classic assumptiongiven the large surface area of the ice available tointeract with the pore.

3. The vapor saturatedpore may be treatedas an ideal gas. This is a common thennodynamicassumption for low pressure water vaporapplications. The snowpack absolute pressure shouldbe very close to atmospheric.

4. The primmy mass transport mechanismis vapor diffusion through the pore. As discussed inchapter I, substantial research has shown convectionto be a significant mass transport mechanism undervery select conditions. In general, convection isneglected for metamorphism modeling. Otherdiffusion mechanisms associated with the ice, such aslattice, surface, and grain boundary, are consideredsmall and neglected in the primary development.

1.2 Theoretical Development

Figure 1. Heat and mass flows during metamorphism.Q represents the heat flow through the ice network, Qec

is the heat flow from evaporation or condensation, J isthe mass flow in the pore, Jec is the mass flow fromevaooration or condensation.

where D is the water vapor diffusion coefficient in air(ml/s), R is the water vapor gas constant (J/(kgK)), Tis the pore temperature (K), L is the water latent heatof sublimation (J/kg), and P is the pore pressure (Pa).Ifequations (1), (2), and (3) are combined, a non­linear differential equation that is coupled to the icethrough phase change (MS) results.

Heat conduction through the ice is considerednext. Energy conservation in the ice accounting forthe latent heat associated with phase change leaves

d 2e 1 dA de HS-+----=- (4)dy 2 A dy dy kA

where k is the ice thermal conductivity (W/mK), A isthe cross sectional area for heat diffusion (m2

), and eis the ice temperature (K), and HS is a heat source

PoreIce

Q..

Q J:.......

X ···..

286

term that is coupled to the pore through the phasechange at the boundary. HS is given by

Phase change is controlled by the local heattransfer at the vapor/ice interface. Heat must flow toor from the interface to either add or remove latentheat. The heat balance at the interface can beexpressed as

HS = JeeL dAee . (5)dy

The heat conduction and mass conservationequations are coupled through the phase change flux,J

ec. This flux results from a vapor pressure gradient

between the ice surface and the surrounding pore.The vapor pressure over the ice surface is correctedfor temperature and curvature (Adams and Brown,1983; Gubler, 1985). The phase change vapor flux isgiven by

-DdPJec=----v.

RT dx

dB k dT_- kiee dx + pore dx - JeeL .

(6)

(7)

Mountain Snowpack

influence nearly all physical attributes such asstrength, stability, thermal properties, opticalproperties, elastic response, inelastic response etc.The broad influence of grains and theirinterconnections requires that their interactions beincluded in any reasonable metamorphism theory. Inthe current research, bonds, necks, and grains havebeen described and are an integral part ofthe theory.While the absolute sizes of grains and bonds areimportant, their relative size is commonly used todescribe the level of bonding or maturity ofmetamorphism. The bond to grain ratio is the ratio ofthe bond size to the grain radius. During themetamorphism of equal sized grains in an equi­temperature environment, several parameters such asgrain geometry, intergranular neck geometry, poregeometry, density, and temperature influence the rateand nature of metamorphism. The influence ofseveral parameters is examined by defining a baselineconfiguration, given in Table 1. Grain size, bondsize, density and temperature are then independentlyvaried to see the sensitivity of metamorphism to each.

Grain Sizes rg=0.125, 0.5,1.0 romRatio 0.4Snow Density 150 Wm'Temperature -5°C

In equation (7), the difference in conducted heatbetween the ice and pore is the latent heat. The phasechange temperature appears in the gradients inequation (7).

The mass conservation equation, energyconservation equation, and phase change constraintsresult in three non-linear differential equations thatare coupled through the phase change flux, Jec• Theequations are solved numerically after the ice andpore network are discretized into elements and nodes(nodes shown in Figure 2). Finite differenceequations are developed for each of the differentialequations. The resulting systems of equations aresolved using various iterative numeric techniques.The details of the solution are found in Miller (2002).

3. Equi-temperature Results

During snow metamorphism, severalmicrostructural parameters direct metamorphictrends. Traditionally, grain size has been the primarytime varying parameter considered. Grains are themost readily visible structure when examining snow.In addition to grain size, the interconnecting neck~ize has recently been recognized as an equallylll1portant feature. As with grains, bonds and necks

287

Table 1. Equi-temperature metamorphism studyparameters.

Figure 2 pictorially represents the heat and vaporflux paths that result from this analysis. Small vaporpressure gradients between nodes result in vapor fluxfrom the convex surfaces, to the pore, through thepore, and onto the concave necks. The localtemperature gradient in the pore was typically on theorder of 0.05 °C/m to initiate and maintain themetamorphism. While this temperature gradientappears very small, metamorphism in amacroscopically isothermal environment could notcommence without it.

International Snow Science Workshop (2002: Penticton, B.C.)

IceHeat Flow

PoreMass Flow

Figure 2. Heat and mass flow duringmetamorphism in equi-temperature conditions.Nodes for numerical solution are shown in the iceand Dore.

'''---...

" - ___....r=O.5mm-'--~'"."'. r-I Omm.",

10.13 L-----L----'--_____:~_____::'c---=-_'c:____=_'=--___=_'=:--_:_'::__---.J02 0.25 0.3 0.35 0.4 045 0.5 055 0.6 0.65

Bond to Grain Ralio [rblrg)

.•..••. ,Mass Flo'\v

-Heat flow

•...........••

.............

10.14L-----L=-----'--_____:~_____::-'-:---=-_'c:____=_'=--___=_'==--=~~ ~ ~ ~ ~ ~ M ~ % ~

Bond to Grain ratio (rblrg)

Figure 3. Bond growth and grain decayrates vs bond to grain ratio. T=-5°C, p=150

kg/m3

r-l.0mm~ --......

-____ r=O.5mm

-~----~-~

Ii': ~--f10·" -----

'"=>~

~ 10"3

i3

dramatically in time. As the bonds grow, the vaporpressure gradient decreases as the neck radius ofcurvature decreases. It is evident from Figure 3 whyvery large bond ratios are rarely, if ever, observed inseasonal snow.

Using the parameters in Table 1 but now varyingtemperature, Figure 4 shows the grain decay rate andbond growth rate for snow comprised of equallysized grains in an equi-temperature environment. Asexpected, the metamorphic rates are sensitive totemperature. The bond growth rate decreases by afactor of approximately 8 for all three grain sizes asthe temperature is lowered from 0 to -20°C,confirming another well established experimentalmetamorphic trend (Hobbs and Mason, 1964). Aspreviously discussed, microstructural temperaturegradients do develop in the pore and ice, but here theaverage macroscopic temperature is varied.

The vapor and heat flow depicted in Figure 2result from the following process. The convex grainsurface curvature creates a vapor pressure gradientbetween the ice surface and pore. Vapor sublimatesfrom the surface and becomes a vapor source,causing vapor to diffuse to the pore node. As vaporis added at the node, the temperature increasesslightly, creating a gradient to the neighboring nodes.Vapor diffuses in the pore toward the necks therebyincreasing the vapor pressure at this node. A vaporpressure gradient therefore develops between thepore node and the concave neck surface, causingvapor to diffuse toward the neck. Vapor is allowed tocondense and release the associated latent heat to theice neck's central node. This slightly raises the necknodal temperature resulting in heat conduction to theneighboring nodes. At the central grain node, heat isremoved thereby providing the latent heat ofsublimation required, and the cycle is complete. Thegeometry is updated as time integration proceeds byadding or removing mass from ice elements.

The first study examines metamorphism as thebond to grain size ratio is varied. Figure 3 shows thevariation ofgrain decay and bond growth with bondto grain radii ratio. As the bond to grain ratioincreases from 0.2 to 0.6, the bond growth ratedecreases by two orders of magnitude for each grainsize. The reduction in surface curvature, increase inneck length, neck surface area, and neck volumeduring metamorphism combine to reduce the growthrate with increasing bond size. An analogous processis taking place during grain radius reduction. Thesintering rate decreases dramatically with increasedbonding, a simple well understood yet importantresult (Hobbs and Mason, 1964; Maeno andEblnuma, 1983). Observations have shown that snowsinters very rapidly initially, then decreases

288

-12 ~ ~ ~-

10 -20 -18 -16 -14

r-:O.5m~ __ --­------12 -lO -8 -6 -4 -2 0

Temp (e)

Mountain Snowpack

the neck, the size of the pore is not a limitingparameter due to the local vapor movement. Given aspecifIc ice geometry, the diffusion distance betweengrains and necks does not vary substantially withdensity, only the pore size changes. Since the poresize decreases with increased density, the pore'svapor storage capability decreases, slightlydecreasing the metamorphic rate. The size of the poreis not the critical factor in this metamorphic regime.In the extremely dense snow where the pore becomesvery small, pore size could become a limiting factor.For common snow densities less than 450 kg/m3

,

density is not signifIcantly limiting local vaportransfer between grains and necks.

In order for this result to be valid, most of the

FO.125 mm

10·'~l-__c'::18--:'_16:---_1;';-4-.-:'12:--~.1;';:0--;'_8~--6;--~_4:--_~2--;'0Temp (C)

~ 10.12

~

~

~ ---~--------

g10.13

r=O.5 mm ___---~

--- llO·!D.g-"

'"'"or-"

~

~ 10.11

<D

r=O.5 mm----- --------------

F1.0mm~---------------~----.. _-- -- - - ----- --

--~------~--

Figure 4. Bond growth and grain decay rates vstemperature. p=150 kg/m3

, bond/grain ratio=O.4

10-12 '-----J'------'_--:-:':-::__~::__~::____=::---:=-_=_______::50 100 150 200 250 300 350 400 450 500

Density (kglm3)

10.13

'::-----,-,'.,,-----,-:,--==:----=::----=::----=:--:=--=--------::50 100 150 2CJJ 250 300 350 400 450 500

Densilv fka/m3,

Figure 5. Bond growth and grain decay ratesvs temperature. T=-5°C, bond/grain ratio=O.4

vapor leaving a grain should deposit on neighboringnecks. If this were not the case, the pore would eithersupply vapor or remove vapor locally throughdiffusion. If signifIcant diffusion to or away fromneck and grain localities existed, the pore size wouldinfluence metamorphic rates. When a temperature

The snow density in Table 1 is now varied whileholding the temperature and bond ratio fixed for allthree grain sizes. The ice geometry is fIxed for aparticular problem and the pore sizes are varied withdensity. As the density increases, the size of the porespace for vapor movement decreases. Figure 5 showsthe variation of bond growth rate and grain decay ratewith density. Metamorphism in an equi-temperatureenvironment appears to be relatively insensitive tovariations in density. The bond growth rate changesare generally less than 20% over a large range ofdensities. Grain decay rate changes with density areslightly larger, but still small compared to bond sizeand temperature affects.

This result may seem surprising, but uponfurther thought and investigation, is logical. Duringvapor metamorphism in an equi-temperatureenvironment, it is believed that vapor moves the veryshort distance from grains to neighboring necks(Kingery, 1960; Hobbs and Mason, 1964; Maeno andEblnuma, 1983). As the vapor leaves the grainSurface, diffuses through the pore, and condenses on

289

'"roor~

'"u'"~ 10. 12

'ro2;

F0125 mm

FO.5 mm--- - - -- - - - --- - - --~ ~

Fl.0mm-.- ... _------------ --- .. _------.----- --- .. -- - ---

International Snow Science Workshop (2002: Penticton, B.C.)

Figure 6. Vapor flux definition of temperaturegradient at onset ofkinetic growth. (a) TG=O, Equi­temperature vapor fluxes. (b) TG>O, TG less thanonset ofkinetic growth. (c) TG>O, TG at onset ofkinetic growth

The transition from grain decay to graingrowth is shown for three grain sizes as a function ofthe applied temperature gradient in Figure 7. J\§JheteQ1P~J:aturegr~qient ir!£reastls.., Jhe gr~ilLgt:m~thratemCLe.ases..Jhe onset ofkinetic growth is clearlyvisible as the grain radial growth rate transitions fromnegative to positive. Throughout the temperaturegradient range, the changes are smooth andcontinuous. Previous models required a discretetransition as new physical models were required.Also, kinetic growth transitions have generally beenbased on the lOoC/m criteria, with little or no regardto any physical parameters that may effect that value.Since the temperature gradient shown did not exceed

commencing simultaneously in both environments,but which is predominant when? In an equi­temperature environment, the excess vapor pressureis negative resulting in a negative kinetic growthvelocity. When the kinetic growth velocity isnegative, that process is inactive when the grain isdecaying, resulting in no kinetic growth. The smoothrounded forms are allowed to develop as previouslydescribed. As the temperature gradient increases tothe point where grains begin to grow, this is the firstopportunity for kinetic growth. Near the transitiontemperature gradient, the smooth and faceted formsmay be equally viable, resulting in combinedmorphologies. Mixed rounded and kinetic forms arerare, but have been observed (Colbeck, 1982; 1983a).As the temperature gradient continues to increase, thegrain growth due to kinetic metamorphism becomespredominant. The model includes both, and allowseach to commence as required. Figure 6 gives apictorial representation of this process and thedefinition of the transition temperature gradient. Asthe temperature gradient increases, the flux leavingthe grains decreases. At the transition temperaturegradient, the grains begin to grow.

gradient is applied, the density (pore size) willbecome a significant factor. The mass leaving thegrains and the mass arriving at the necks werecompared to see if a mass balance exists. While notpresented here, a mass balance between the grainsand surrounding necks was very good, supporting thelocal mass transfer.

The results so far have used equally sized grains.While not presented here, the model did demonstratethe growth of large grains at the expense of small.

4. Transition to Kinetic Growth

The transition between the smooth roundedgrowth forms common in an equi-temperatureenvironment and the sharp faceted forms found in atemperature gradient en~t.has-oot-9€enSffiaie~~nsivel~Mostresearch surrounding thistransition is experimental. It is commonly believedthat the critical temperature gradient for facetedgrowth is around 1O-20°C/m (Akitaya, 1974;Armstrong, 1980; Colbeck, 1983a, 1983b) or greaterthan 25°C/m (Marbouty, 1980). It is recognized thatthis transition is a function of density, pore size, grainsize, and temperature (Colbeck, 1983b), but littleanalytical work has been done to show the relativeimportance of each. It is recognized that low densitysnow transitions at a lower temperature gradient thanhigher density snow, but microstructurally basedanalytical approaches explaining this phenomenonare lacking. The current approach may be used tostudy this transition and various parametersinfluencing the transition. First, a definition ofkinetic growth transition temperature gradient isrequired.

There is clearly no agreed upon point whereequilibrium forms dominate and then suddenlyfaceted forms appear. A smooth transition based onphysical constraints is desired instead ofrelying onincomplete empirical data. Grains cannot grow in akinetic form when they are decaying, as may be thecase in an equi-temperature environment. When atemperature gradient is applied, vapor diffuses in thepore, and heat conducts through the ice. As thetemperature gradient increases, the grain decay ratedecreases, and the neck growth rate increases due tothe excess vapor pressure of the pore. Kinetic growthprocesses are allowed to begin when the temperaturegradient is sufficient to transition the grains fromdecay to growth.

For this study, the temperature gradientwhere a "majority" of the grains start to grow isdefined as the onset or transition temperature gradientfor kinetic growth. One can think of the equi­temperature and temperature gradient processes

290

a b c

- 25 0 C/m, the 0.5mm grains did not yet reach theirkinetic growth transition.

Mountain Snowpack

45c--~--~--c--~--~_---,

40

35

Figure 7. Grain growth rate vs temperaturegradient. The smooth transition to kineticgrowth is shown. T=-3°C, p=150 kg/m3

,

rJr.=O.4.- In a temperature gradient environment, the

size of the pore spaces, and therefore the density,should be a major factor in the onset of kineticgrowth. It is common1y-~~ed that 100~er densitysnow (larger pore spaces) is muCh more likely toshow kinetic growth with a smaller temperaturegradient. In addition to the pore size for vaportransport, large faceted crystals need sufficient spaceto grow. Marbouty (1980) showed an increase in thegrain size with decreasing density when subjected toa temperature gradient sufficient for kinetic growth.He also showed a kinetic growth upper limit densityof 350 kg/m3

, above which kinetic growth was notobserved even under large temperature gradients. Inhigher density snow, Marbouty found the lack of pore<.space obstru'21u;IYstal owth. The crystals becamesharp, but failed to grow appreciably in size. Thesecrystals formed under large temperature gradientsand dense conditions were referred to as "hard" depthhoar (Akitaya, 1974). Figure 8 shows the onset ofkinetic growth as a function ofgrain size and density.As expected, the onset of kinetic growth is a strongfunction of density. As the density decreases, thelarger pore spaces allow for easier vapor movement,resulting in a lower onset temperature gradient.

Using this model, the onset of kineticgrowth was strongly influenced by bond to grainratio, but relatively insensitive to temperature. Grainsize and density were the significant parametersdetermining the onset ofkinetic growth.

-1 o 10 15 20Temperature Gradient (degC/m)

25

Growth

Decay

291

30

t:E. 25(!)

~ 20~0

15

10

100 150 200 250 :OJ

Density (kg/m1

Figure 8. Onset temperature gradient for kineticgrowth vs density, T=-3°C, rJrg=0.5. Boxedregion represents experimental temperature gradientrequired for kinetic growth.

The model does include a kinetic growthmodule that is used to examine crystal growth afterthe transition temperature gradient is achieved. Thedetails and results of that module are given in Adamsand Miller (2002).

5. Conclusions

The current effort uses simplemicrostructural configurations and fundamentalphysical processes to simulate dry snowmetamorphism unlimited by the thermalenvironment. The model is useful for examining thesensitivity of several parameters in metamorphism.Sintering in an equi-temperature environment wasexamined and various sensitivities were discussed.The transition to kinetic growth was defined andstudied. The model provides a very flexible tool forexamining metamorphism at a microstructural level.Numerous parameters may be simultaneouslyexamined in a complex environment.

6. References

Adams, E.E and RL. Brown, 1983, Metamorphismof dry snow as a result of temperature gradient andvapor density differences, Annals o/Glaciology, vol4,3-9.

Adams, EE and D.A. Miller, 2002, Update on themorphology of crystal growth and grain bonding insnow, ISSW 2002.

Akitaya, E, 1974, Studies on depth hoar,Contributions from the Institute 0/Low TemperatureScience, Series A, vo126, 1-67.

International Snow Science Workshop (2002: Penticton, B.C.)

Armstrong, R.L., 1980, An analysis of compressivestrain in adjacent temperature-gradient and equi­temperature layers in a natural snow cover, JournalofGlaciology, vol 26, no 94, 283-289.Arons, E.M. and S. C. Colbeck, 1995, Geometry ofheat and mass transfer in dry snow: a review oftheory and experiment, Reviews ofGeophysics, vol33, no 4,463-493.

Brown, R.L., M.Q. Edens and M. Barber, 1999,Mixture theory of mass transfer based uponmicrostructure, Defence Science Journal, vol 49, no5,393-409.

Colbeck, S.C., 1980, Thermodynamics of snowmetamorphism due to variations in curvature, JournalofGlaciology, vol 26, no 94, 291-301.

Colbeck, S.c., 1982, An overview of seasonal snowmetamorphism, Reviews of Geophysics and SpacePhysics, vol 20, no 1,45-61.

Colbeck, S.C., 1983a, Theory of metamorphism ofdry snow, Journal of Geophysical Research, vol 88,no C9, 5475-5482.

Colbeck, S.c., 1983b, Ice crystal morphology andgrowth rates at low supersaturations and hightemperatures, Journal ofApplied Physics, vol 54, no5,2677-2682.

Edens, M., 1997, An experimental investigation ofmetamorphism induced microstructure evolution in a"model" cohesive snow, Ph.D Thesis, Montana StateUniversity, Bozeman, MI.

Gubler, H., 1985, Model for dry snow metamorphismby interparticle vapor flux, Journal ofGeophysicalResearch, vol 90, no D5, 8081-8092.

Hobbs, P.V. and B.J. Mason, 1964, The sintering andadhesion of ice, Philosophy Magazine, vol 9, 181­197.

Incropera, F. and D. DeWitt, 1985, Introduction toHeat Transfer, John Wiley & Sons, New York.

Kingery, W.D., 1960, Regelation, surface diffusion,and ice sintering, Journal ofApplied Physics, vo131,no 5, 833-838.

Maeno, N. and Takao Eblnuma, 1983, Pressuresintering of ice and its implication to thedensification of snow at polar glaciers and ice sheets,The Journal ofPhysical Chemisl1y, vol 87, no 21,4103-4110.

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Marbouty, D., 1980, An experimental study oftemperature-gradient metamorphism, Journal ofGlaciology, vol 26, no 94,303-312.

McClung, D. and P. Schaerer, 1993, The AvalancheHandbook, The Mountaineers, Seattle, WA.

Miller, D.A, An integrated microstructural study ofdry snow metamorphism under generalized thermalconditions, Ph.D Thesis, Montana State University,Bozeman, MI.

Sommerfeld, RA andE. LaChapelle, 1970, Theclassification of snow metamorphism, Journal ofGlaciology, vol 9, no 55, 3-17.