Title Snow/Firn Densification in Polar Ice Sheets
Author(s) Salamatin, Andrey N.; Lipenkov, Vladimir Ya.; Barnola, Jean Marc; Hori, Akira; Duval, Paul; Hondoh, Takeo
低温科学, 68(Supplement), 195-222 Citation Physics of Ice Core Records II : Papers collected after the 2nd International Workshop on Physics of Ice Core Records, held in Sapporo, Japan, 2-6 February 2007. Edited by Takeo Hondoh
Issue Date 2009-12
Doc URL http://hdl.handle.net/2115/45449
Type bulletin (article)
Note III. Firn densification, close-off and chronology
File Information LTS68suppl_016.pdf
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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Snow/Firn Densification in Polar Ice Sheets
Andrcy N. Salamatin •.•••• , Vladimir Va. Lir.cnkov ••.••••• Jean Marc Bamola .'., Akira Hori ..... Paul Duval" ., Takco I-I ondoh ••••
• Kazan Slate University, Kazan, Rlissia, [email protected] .. Arctic and An/arctic Research Instilule, SI. Petersburg. Russia ••• Laborafoire de G/aciologie el Geophysique de I'Envirolll1emelll, Grenoble, France un Hokkaido University. Sapporo, Japan
Abstract: A sophisticated physical model of the dry e, Ra te of ice thin ni ng due to global ice-sheet snow/fi m densification process in ice sheets is motion proposed. Macroscopicnlly, snow and rim un dergo F Rheological funct ion in Eq. (7) vertical uniaxial compression wi th non-zero deviatoric g Gra vity acceleration stresses and strain ra tes. The present mathematical G Rheological fu nct ion in Eq. (7) description of dcnsificntion includes dilatancy and h Dept h "force-chain" eITects in snow and develops previous k Ra te constant concepts of ice-pnrticlc rearrangement by grain p Ice pressure produced by normal components boundary sliding and sintering by power-law creep of contact force s under overburdcn pressure. Both densification p, Load pressure (hydrostatic overburden stress) mechanisms work together during the firs t snow stage p Macroscopic pressure in ice skeleton until the closest pncking of ice grains is reached at Q Activation energy criticnl dcnsi ties of 0.7-0.76 and the fim stage controlled R(R) Mean equivalent-sphere current radius of ice only by thc dislocation creep sets on. In addition to the grains (nonnalized by surface size Rs) ice-grain coordination number and the slope of the R'.R" Fictitious normalized radii of plasticall y radial distribution func ti on, a new structural parameter defomled iee grains in the densification scheme is introduced to lIccount for gram bonding [lO] (agglomcr.ttion) effects. The model is constrained and R, Gas constant validllted on direct stereological observations of ice core , Fraction of free ice-grain surface not involved structures and a representat ive set of snow/fim density in plastic contacts depth profiles coveri ng a wide range of present-day Fraction of gmin surface area involved in grain cl imatic conditions (- 57.5 to - 10 °C and ice bonds 1 accumulation at 2.15 to 330 cm yr- ). Si mple eq uat ions I Timc arc derived fo r predic ting the depth of pore closure in T Tcmperalure (in K) lirn and the ice lIge at close-ofT. The paleocli matic "0 Principa l macroscopic deviatoric stresses evolution of quasi-stationary density-depth profiles and (j - 1,2,3) close-off characteristics at Vostok Station (East Verlicll l velocity with respect to ice-sheet Antarcticll) are simu lated and discussed. surface x Fraction of deviatoric deformations due to ice Key words: ice shect, snow/firn dcnsification, physical grain rearrangement model, paleoclimatic implication Z Coordinat ion number a Power-creep exponent in Eq. ( 13) p Dilatancy exponent in Eq. (18) List of symbols r Lateral deviatoric Slrain rates r Coefficient in Ihe grain boundary sliding a Mcan relative bond area in units of R! model (12) in [2] A Mcan relati ve area of the contact segment Ii Grain bond thickness bases in units of R! Ice-equi vll ient glacier thickness Mcan ice-grain lIrea measured in thin sections Correction cocfficient in Eq. (15) Accumulation rate in ice equivalent " Fract ion offrce grai n surface occupied by extra Dimensionless fo nn fac tors of the density neck volumc due to diffusive ice mass transfer dcpth profite in Eqs. (25) (bonding fac tor) Relative slope of the cumulative ice-gmin c ,,(,,') Kinematic (bu lk) viscosi ty in Eq. (6) radia l distribut ion function (RDF) Stra in rate tensor (principal strain rates, A Di latancy ra te parameter in Eq. (3) I' Non-linel1r viscosity in power-creep law (13) j - I ,2,3) v Li ncllr grain-boundary viscosity
-195- P Relat ive density of snow/fim deposits primary step related to p:lleorcconstructions from ice PI Density of pure ice core data. Predictions of the surface and close-off 11 Principal macroscopic stresses (j = I, 2, 3) densi ties as well as ice-age/gas-age difference lire (t) Densitication (compression) rate considered liS separate problems out of the scope of the present study. Sliperscripls The mechanical properties of snow li nd its Fict itious geometric characteristics of ice densification at loading were originally studied and groins in the densification scheme [10] theoreticall y interpreled by Voytkovskiy (73]. Herron Value al reference tempemlure r' (Vostok and Langway (39J subsequently proposed a density Station) depth (or age) relmionship that is now widely employed as a phenomenological fimificlltion model. However, SlibscriplS the applicmion of this rellltionship, as well as olher c Ice crystal growth semi-empirical approximations (e.g., [12,19,43, 71J) is of! Close-off characteristic of limited val idity, bei ng con fi ned to the ra nges of p Ice-grain plasticity chllracteristic present-day environmental conditions covered by the r Ice-grain rearnmgemenl characteristic experimental data. An ultemative, although much mo re S Ice sheet surface complicated approach is to develop a physical theory o Critical point (snow-to-fim transition) relating microstmctuml changes in the snow compact to its general macroscopic behavior during compress ion. In a series of papers [16, 17, 26. 38, 50], a suite of I. Introduction microstructural constitutive models were constmcted specifically for the case of snow under high 1001din~ 5 Fresh snow deposited on :1 dry glacier surface is rates (strain rates exceeding 10 s I). It was assumed subjected withi n a few uppemlOst meters to vll rious that the defonnation process and fracturing in the deposi tional, diagenetic, and meteorological processes granular stmcture took pillce predominantly in relatively [3]. Even in cold natural conditions, ice narrow necks connecting grain bodies. Such cond itions thermodynamically is relat ively close to melt ing, being differ substantia lly fro m those met in the natural at high homologous temperature. There exist a variety snow/fi m densification process at the glacier surface, of mass lransfer mechanisms in the pressureless where the strain ra tes are much lower. on the order of sintering of surface snow (e.g., [49]). Among them, 10 11 to 10--9 S- 1 and the intercrystalline contacts are well evapor.lIion-condensution (vapor tmnsport) and surface developed even at shallow depths [35]. The surface diffusion playa primary role in the snow metamorphism fraction of grain bonds is large [4.7.9), with neck-to [37,58.64] largely through the rounding and bonding gmin radii rat io of 0.6-0.7 [1 , 2]. It is therefore of ice crystals [14, 35]. At depths of 2-3 m, snow is a conventionally accepted after Alley (2] that particle low-density (porosity 50-65%) polydisperse compacted rearrangement, which dominates in the highly porous powder of ice gmins. The propcrti es of this stmclurc arc snow, is controlled by li near-viscous grain-boundnry considered admittedly 10 origi nate from the surface sliding. This microstructural phys ical description of the snow characterist ics which arc ultimately cl imate repacki ng mechanism can be directly incorpomted into controlled. Further on, particle rearra ngement [6] and snow/fim densification models [8, 20]. sintering under overburden pressure become the The number of bonds per grain (i.e .. coordination principal mechanisms of the snowlfim densification. number) in snow increases with density, restricting leading to the formation of bubbly ice upon pore closure intergranular sliding. As a result, the creep of ice gmins in deeper strata (50- 150 m). Although studies [16, 22, in contact gradually prevails over, becoming the sole 23, 49] show that plastic deformation (i.e., power-law mechanism of fimification beyond a critical snow dislocation creep) dominates in the development of densi ty and coordination number at which particle intergmnulllr contacts under loads exceeding 0.01- rearrangement essentially stops. This general scenario is 0.1 M Pa, the pressureless sintering effects driven by the commonly simplified by the ussulllption that the critic'll excess surface free energy of ice crystllis still can not be density separates two successive regimes of ignored. densificlltion, ei ther by grain boundary slidi ng (snow The transformation of snow into glacier ice, being a stage) or by plastic deformation (fi rn stage) (e.g., [8. 9]). fundamental glaciological phenomenon. is also a key Usua lly the transition between the two densification process Ihat links paleoclimatic records of ice properties zones is identified after Anderson and Benson (6] with in glaciers 10 those of atmospheric gases trapped in the the spt'Cific bend found in the density-deplh profile at a ice (e.g., [12 , 15, 33, 65, 67]). The most important relative density of approximately 0.6. However, as chamcteristics of the snow/fim densification process arc suggested by Ebinuma and coworkers [23-25], this first the close-off depth and ice age as well as the grain size critical point of sharp decrease in the dcnsi fi cation rate that ult imately determines the geometrical properties may simply manifest the onset of an intennedillie (size and number) of ;lir inclusions (i.e., bubbles and regime in which particle rearrangement and plus ticity hydmtes) in polar ice sheets [45, 47]. Here we work together, whi le the dislocation creep takes over at concentrate on snowlfim densification modeling as a higher relative densi ties of about 0.75.
-196- A physical microstructural model for the tim stage of in axial compression tests [63] may, at least in part, be densification by power+law creep was constructed [9] caused by the dilatant motion of ice grains as rigid based on a geometrical description of the dense random particles. packing of monosize ice spheres [10, 29] and an 3. Overburden pressure, increasing from the very approximate solution for the initial phase of plastic beginning of snow densification, acts as intergranular deformation of two contacting spheres [II, 76]. This contact fo rces that result equally in the rearrangement model was linked to Alley's model [2] for ice+grain and plastic deformation of ice particles. In principle, rearrangement in the snow stage (neglecting dislocation these two timitication mechanisms operate creep) by introducing the critical density as a variable simultaneollsly until restacking ceases at the critical (tuning) parameter. The principal problem encountered coordination number (i.e., critical density). This in such an approach was that the initial densely packed suggests that Anderson and Benson's interpretation [6) structure assumed in [10] had zero contact areas of the bend around a relative density of 0.6 in the between particles at the critical density, whereas Alley's density-depth profiles should be re-examined. scheme described gram sliding over developed 4. Grain bonds in snow and tim are well developed interfaces (i.e., grain boundaries). Hence, it was [4,9], and the mean bond radius in the snow stage suggested [7-9] to represent fim densitication by the remains largely constant [1,2] despite the relative plastic defonnation of groups of Ice crystals motion of grains. This suggests that water-vapor (aggregates, or agglomerates) rather than single grains. transport (pressureless sintering mechanisms), even if Clusters of ice crystals are distinguished in natural ice negligible as a densitication factor, still acts in cores, and structures of ice-grain agglomerdtes can be combination with grain creep in the neck-fonnation considered as more realistic. However, in Arnaud's process and should thus be incorporated directly into approach [8], the aggregates are defined without inner any physical model of dry snow/tim densitication. Such voids having the same specitic surface area as the an improvement can not be achieved within the original ice skeleton at the snow-to-tim transition. framework of [8], and additional reconciliation with Although this assumption avoids discontinuity in the Ar"L:t's theory [10] is necessary. parameterization of the ice-grain structure, crystal 5. According to [36], the granular ice skeleton carries bonding and neck development are reduced to pure the applied load in "force chains" and a certain frac tion agglomeration, which, in a monosize approximation, is of gr"dins in the polydisperse snow structure is equivalent to simple rescaling of micro-dimensions of essentially stress~free. This means that the effective the ice-grain compact. Nevertheless, the model [8] pressure at the contacts of defomling ice spheres is allows the direct extension of computational simu lations much higher than in a uniformly loaded structure. and theoretical predictions to various paleoclimatic and Furthennore, as the mean grain coordination number, thermodynamic conditions (e.g., [15, 33]). equal to - 7 at the critical density, increases by a factor In the present study, aiming at a more complete and of 2 in the tim stage [4,7], the plastic defonnation of sophisticated representation of the microstructural ice in the vicinities of numerous contacts distributed picture of ice-grain kinematics and stress-strain over the grain surface is more simi lar to the creep of distributions, we continue the elaboration of the relatively thin contacting spherical segments than that snowlfim densitication model recently started by for a single contact between two spheres (agglomerates) Salamatin and coworkers [60]. The following aspects as assumed in [8,9] after [10, I I]. Thus the plastic are of particular interest. deformation of grains also requ ires a more accurate I. As there is no possibility for independent lateral simulation. (horizontal) constriction of snow deposits on an With this in mind, we begin with the improved infinitely large ice sheet surface, naturdl densitication of description of snowlfim densitication, extending on the snow, tim, and bubbly ice strata is, macroscopically, [60]. The resultant physical mode l is evaluated and a process of uniaxial (vertical) compression with non validated on available ice core texture measurements zero deviatoric stresses and strain rates superimposed on and a representative set of snow/tim density profiles the global (deviutoric) deformation impacted by glacier covering the full range of present-day climatic motion [61]. This densification process can not be conditions (temperature, accumulation rate, wind speed, adequately modeled simply on the basis of hydrostatic and insolation). Finally, possible applicHiions of the compression as usually considered 111 powder developed theory to analysis of the snow/tim metallurgy (e.g., [10, 76]). densitication process in changing climate are considered 2. Ice+grain rearrangement in snow under uniaxinl with special emphasis on predicting the close·ofT (depth compression in accordance with general concepts of and ice age) characteristics. granular media and soil mechanics (e.g., [18, 55, 56]) can not be geometrically arbitrary on the microscopic 2. Theory of snowlfirn densifi cation level and is thus subjected to certain kinematic constraints relating volumetric compression rates to 2.1 . General notions and phenom enological effective deviatoric strain rates of particle restacking. equations Th is phenomenon is known as the dilatancy effect. For Let us consider the process of densitication of instance, the radial expansion of snow samples observed snow/tim deposits under the load pressure (hydrostatic
-197- overburden stress) Pi on a glacier surface in dry, cold deviatoric strain, y, > CtIr, and the total defonnational climatic conditions. Due to lateral constraints, compatibility in the ice compact presumes that the macroscopically it is a confined vertical (uniaxial) difference y, - (u,. is compensated for by extra plastic compression. We designate the corresponding compression &p - YP' Although the dilatancy in snow is component of the strain-rate tensor I:: by subscript" I ". generally small with respect to total deviatoric The total defonnations in the two other principal defonnation, this effect controls the interaction between orthogonal (horizontal) directions "2" and "3" are equal the two densification mechanisms. A conventional to zero, and I:: is determined as linear kinematic relation is assumed here between the densification rate due to grain rearrangement CtIr and the £, =-3& , £ 2 =£) =0 corresponding deviatoric deformation y,.. i.e., w~- " (E)/3~-(£, +£, +£, )/3 I The snow/fim densification (compression) rate & coincides with the lateral deviatoric strain rate Parameter A determines the rate of dilatancy and, p , y = El - Ir(£)/3 = E - lr(£)/ 3, and, by definition, being a function of is one of the principal structural J and deformational characteristics of snow as a granular material. It varies from 0 for ultimately friable, highly I dp = 3& , ( I) porous snow, when (u,. = y" to I in the critical (dense) p dl packing state, when CtIr = O. Simultaneous equations (2) and (3) yield expl icit expressions for the constituents of where p is the relative density (ice volume fraction) of the strain rates via the total densification rate (I) and the the ice structure; I is the time, and dldl is the particle fraction x of the deviatoric deformations due to the ice derivative. crystal rearrangement: yp= (l - x)&, (4) (u,. = x(I-A)w, "V [1-(1 -;!)x]w. The macroscopic stress tensor Ii is conventionally expressed via the isotropic pressure P and the vertical deviatoric stress T, (T2 = T3 = -TII2): : '---I'. ; ..../ : , , , ,,' I.'" ''', I (5) " 'j r + ':,- ',: I I , , , + r + , ,~ ~ - 3m, ' [ w,~(I-A)Y, ) Consequently, the problem of the snow/firn densification modeling is reduced to the construction of constitutive equations relating the defonnatioll rates y,. Figllre I: Interaction betweell dilatancy effecls and and YP(or t4 and '0" or x and w) to the averaged stresses plastic deviatoric dejormations ill snow dellsiJlcatioll by P and T, in the granular ice material. In accordance with grain rearrallgemellt. Ollly e.!(cess deviatoric creep the general concepts, we write the rheological law for compensatillgjor lateral di/atallt expamioll is depicted grain rearrangement by linear-viscous boundary sliding ill the laslfragment. " The overall macroscopic defonnation in the ice P =p+3,l'wr , (6) compact is the sum of two constituent parts due to (a) rearrangement (i.e., sliding) of grains as rigid Here ,,. and" are the coefficients of bulk and kinematic particles and (b) grain plasticity by dislocation creep viscosity, and p is the pressure produced by the normal under contact forces. These two mechanisms are components of the contact forces (i.e., by the force distinguished by the subscripts ",." and "p" , respectively. interactions between grains not related to their motion Thus, with res pect to each other). For low-density snow, J. - 0 and (I.\.- = y" while 12 = Xl = -P - O. In this case Eqs. (5) and (6) lead to 'I' = 2 ,,/3. These equations do not allow unique division For plastic deformation, one can envisl1ge the between the restacking and plastic strain rates. The following non-linear analogues of Eqs. (6): dilatancy [18, 55, 56J of the granular ice structure (see Fig. I) should additionally be taken into account. This (7) means that the uniaxial compression of ice powder, as an ensemble of rigid particles, can occur only at excess The apparent viscosity of ice-grain rearrangement" and functions F(lIJp) and G(yp) in Eqs. (6) and (7) (to be -198- determined below) are the principal rheological Further, in accordance with Arl.t's approach (see Fig. 2), characteristics of the ice compact, relating the the development of the ice-grain structure due to macroscopic behavior to the processes occurring on the dislocation creep is described as the concentric microstructural level. By definition, the vertical stress expansion of centre-fixed particles with the current II is equal to -PI with its deviatoric part TI given 'fictitious' equivalent-sphere radius R' measured in units identically by Eqs. (6) or (7). Based on Eqs. (3)-(7), of R. The basic shape of the plastically defomled grain these conditions can be wrillen as bodies is thus the sphere of relative radius R" truncated by the developing and newly-fonned contact faces with the free surface fraction s. The coordination number in fim is then Together with Eqs. (4), they deliver a general form of the snow/fim densification model with respect to {o and Z = Zo + C(R"-I) . (10) x. The snow/fim stmcture, including the effects of The critical value 20 of the coordination number for pressureless sintering in grain bonding, and the random dense packing and the slope C of the radial densification mechanisms by ice-particle rearrangement distribution function estimated in [10] for monosize and creep are sequentially considered in the following sphere powders are - 7-7.5 and - 15.5. Accordingly. R' subsections in order to explicitly transform Eqs. (8) into is expressed via the relative density p , and R" is related a complete physical model. to R', linking 2 to p (see Appendix A). 2.2. Densification stages and snow/firn structure p., description <} , <} Hereafter we conventionally introduce the two successive stages of snowlfim densification and distinguish them for clarity after [8,9] as "snow" and ---- ' "fim", respectively. However, in contrast to previous 77 :,-- \ studies, we assume after [60] that the first (snow) stage, dom inated by ice particle restacking, is simultaneously influenced by a gradual increase in the plastic strain of --:R'OI -- A'--- , ! grains. The fim (consolidated snow) stage starts when , particle rearrangement ceases at the closest, dense ~~ packing and is controlled only by grain dislocation creep under growing overburden pressure. In this / R/.,"\ ~- _~\ - context, a shift of the snow-to-fim transition (critical : surface""" point) to greater depths and higher relative densities can be expected. The number of contacts (bonds) per grain (coordination number, Z) is one of the principal microstructural characteristics of sllowlfim build-up. The increase in Z with density during the snow Figure 2: Modification of AI"Zt's geometric scheme [IOJ dellsification stage occurs primarily as a result of ice of ice-grain compactioll by dislocatioll creep coupled grain restacking. In accordance with ice core with neck formatioll aroulld plastiC faces due to measurements [4, 7], snow structure modeling [32], and diffusive water-vapor trallsport. Inset: Deformatioll of (J direct simulations of microscopic snow densification single spherical cOlllact domaill 011 a referellce graill. [41], we assume after [2] that Z in snow is a linear function of the relative density p, Although the restacking and dislocation creep of grains are considered to be the major pressure-sintering (9) (densification) mechanisms in snow and fim , the water vapor transport and other pressureless-sintering The subscript "0" designates the critical values at the processes in the granular ice material at relatively high critical depth of dense packing, where particle homologous temperature may also contribute to grain rearrangement stops and the fim stage begins. bonding and neck fonnation [1 ,2,35], enlarging the Consistently, fim densification is realized to occur by grain contact faces and slowing the rates of plastic centre-to-centre approach of ice crystals due to the deformation. To account for these effects, Arl.l's dislocation creep without sliding. Following [10], we geometrical description of the fim structure is extended represent the fim structure on average as a random to the snow stage where the creep of ice particles is dense packing of incompressible monosize grains of subservient in densification and the cumulative plastic equivalent-sphere radius R and use a linear strain of grains is, at least in part, annihilated by grain approximation of the cumulative particle radial rearrangement. Thus, without any loss of generality it distribution function (RDF) with relative slope C. can be assumed that the area fraction of plastically -199- formed intergmnular contacts I - s in snow is negligibly Hence, Eq. (10) fo r pressure sintering [10] in tim small, that is, s = R'= R"= I with Eq. (9) substituted for remains valid without substitution of R' for R" as Eq. (10) at p < PJ. A new microstructuml characteristic suggested in [29]. At the same time, it should be noted of the snow/fim build-up, the bonding factor,;;: is then that the polydispersity of natural snow and the primary introduced to describe the fmc tion of grain surface not redistribution of ice across the grain surface in the snow consumed by the plastically formed contacts but stage can result in a denser closest packing with higher occupied by excess neck volume (see Fig. 2) created 20 and much higher RDF slope C than the values due to pressureless sintering. The remainder of the conventionally accepted for spherical monos]ze reference grain surface exposed to voids ( I - (Is is powders [10]. Thus, 20 and C should be regarded as shared among the contact domains which can be tuning model parameters in addition to so. realized on average as spherical segments with flat bases A' (Fig. 2). This generalized geometry of 2.3. Alley a pproxima tion for snow densification by snow/fim structure is qualitatively described III grain-bounda ry sliding Appendix A. The equations for R", s, the mean relative Alley's theory [2] of the snow densification by grain bond area a', and the average area of the contact boundary sliding gives segment base A' (i.e., (J = a'/R i.!. and A = A'/Ri). in units of R2) are derived in the appendix. The fraction of gmin surface area involved in the grain bonds Sb is also obtained to link these microstructural characteristics via R' to p. The critical density PJ is shown to be uniquely where g and I' are the grain-bond thickness and expressible in tenns of 20 and C. viscosity, respectively. R is the mean equivalent-sphere Observations fI, 2J revealed that ice-grain bond radius of grains, and r is the mean bond radius. The development is counterbalanced to a large extent by coefficient r(2) can be expressed as a linear function of grain-boundary sliding in snow, resulting in a constant the coordination number (or density via Eq. (9)), mean bond area 00. This means, as confimled by decreasing from I to 0 as Z increases from 0 to 20. measurements [4, 7, 9J, that in the snow stage, the grain Comparison of this relation to the general bond fraction S b is a linear function of 2 or p (see equations (I), (4), and (8), neglecting the plasticity Eq. (9». As shown in Appendix A, to satisfy the effect (i.e., x - I and A. - 0) yields the following explicit condition a = 0o. the bond parameter; must also be a expression for ,,: linear function of 2, i.e., 8m'T(Z) (II) " 3~aRp" 2 (12) The critical value;o uniquely determines ao. Eq. (II) is extended below to the Am stage and will be validated on the basis of ice-core data. Finally, only three 2 microstructural parameters 20, C and so of dense Here, by definition, 0 = m.2/R and in accordllnce with packing control the evolution of snowltirn build-up with [I), (1 clln be regarded as II constant value o =ao in increasing density. Eqs. ( 12). In relation to the specific surface area, the present approach is quantitatively equivalent to the snow/tim 2.4. Plastic deforma tion in ice compact structural representation of ice crystal lIgglomerlltes Densitication by dislocation creep occurs under suggested by Anwud lind coworkers in [8,9J. [n this external pressure p, which is transfonned in the ice framework, in the snow stage, ; is equal to the mean skeleton to an effective pressure peff acting on grain fraction of the grain surface occupied by bonds and can contacts. On average, for uniformly distributed forces, it thus be regarded as a measure of agglomeration. ]s conventionally accepted (e.g., [10,50)) that However, Arnaud's approach assumes that ice crystal p = paZp~/4ff. However, according to Gubler [361, the aggregates initially have zero contact faces at the snow external stresses imposed on the polydisperse ice grain to-tim transition, precluding direct incorporation into structure lITe conducted by "fundamental units" (groups the ice grain rearrangement scheme [2J, which is of gwins) through "force chains" (series of single force essentially based on grain sliding over developed bearing gwins). As not all gains are stressed and interfaces. In contrast, Eqs. (9)-(11) explicitly introduce undergo the creep, the effective pressure on the grains an average description of intergrain contllcts that is cOlllrolling macroscopic defonnation, particularly in continuously consistent in snow with Alley's model [2) snow and low-density tim, is higher than the value and a[lows direct application of ArLt's theory [10) for averaged over all ice particles. Gubler's measurements fim densification. [36] of the force-chain lengths for coordination numbers Considering pressureless sintering as an important of 3-10 predict an extra increase in Peff in inverse factor in the fonnation of intergrain contacts, we still proportion to 2. Phenomenologically, this can be assume that the transportation of ex Ira mass of ice 10 accounted for by introducing an additional enhancement necks does not change the density or R" substantially. factor z/Zo into the above correlation betweenp and P4J- -200- In the framework of Ant's description [10] of the where '" is the snow/fim structural characteristic geometrical granular ice structure, it is assumed here assumed to be a small vlliue. that in plastic defonnalion, ice is squeezed from under Eqs. ( 14) and (15) together with Eq. (II) and other each contact area a' on a reference grain by power~law microstructural characteristics detennined In creep of the uniaxially compressed contac t ~segment Appendix A are equally valid in snow for p < PJ at layer (see Fig. 2, inset). The central part of the grain is R' = R"= I and in fim for P > PJ with Z given by subjected to uniform forcing from all surrounding Eqs. (9) and ( 10), respectively. segments and is thus in hydrostatic equilibrium. The rheological law for ice as a non-linear viscolls 2.5. Phys ical model for snow/firn densific ation rates incompressible body, relating effective strain rates eo to Simultaneous equations (4), (8), (12), (14), and (15) form a theoretical basis for physical modeling of the effective stresses TO, is given by snow/fim densification process. Substituting Eqs. (4), (14), and (15) into the first of Eqs. (8) for both snow and (13) fim stages yields where a is the creep index and J1 is the coefficient of non-linear viscosity. The problem of plastic deformation of the contact segments under effective pressure is considered in Appendix B. The obtained solution transforms the ice flow law ( 13) into a relationship between P"ff and wp, and the correlation between P and Pelf leads directly to The principal novelty of this relation is that it extends the following expression of F(at,) in Eqs. (7) and (8): now 10 plastic defonnation in snow, accounting for the dilatancy effects and ice-grain rearrangement described ~ F(", ) • .[k;Apz ' { 2-13ffl -201- increasing function of Zo and C (see Appendix A), ranges respectively from ~ = 0. 704 to 0.754, and in )""( P-Pmm)' . p < P, . (\8) agreement with the observational data, the slope of the Po - Pm .. theoretical curves of Z = Z(P) in Fig. 3 changes at p = Po with switching from Eq. (9) to Eq. (10). These values of where P is the dilatancy exponent and Pmi~ - 0.3 is the Po are substantially higher than the relative density (ca. minimum surface-snow (threshold) density below which -D.6) of the first uppennost bend point of the sharp the dilatancy effects are assumed 10 be negligible. decrease in densification mtes found in the density In snow, R'= R"= I, and Eq. (9) for coordination proliles by Anderson and Benson [6] and suggested to number Z completes the model ( 16)-(18). In tim, A '" I be the boundary between the snow and Jim stages. and r(Z) =: 0 for p > Po (2 ) 20). Correspondingly, Eq. (17) reduces 10 x = 0, and Eq. (16) determines directly the densitication rate ()) at Z given by Eq. (10). •• In both stages, the bonding factor (is given by Eq. (11). All other microstructural characteristics are specified in " ,'1"" 4, II", I. Appendix A " <>• "". The developed physical description of the snow/fim E 12 • UpB,Be • " I 0 •,• densification process involves three geometrical 0 • Site A • parameters (2o• C, (0), two grain interaction parameters .~ (p, e), and two ice properties (p, k.), all of which should m 2 0 8 be constrained or validated by ice core density measurements and texture analyses. ~ U , 3. Model constrai ning and va li dation a 3. 1. Snow/firn structure characteristics 0 Ice-grain connectivit y In snow and urn IS quantitatively described by the coordination number Z 0 02 0' 06 08 Relative density and the bond area fraction Sf> . Alley and Bentley [4] convincingly demonstrated, on the basis of a , stereo logical study of Be and UpB ice cores from the 3, ,' Siple Coast, West Antarctica, and data from Site A, " '6 ",'., . Greenland [5], that these microstructural characteristi cs 1: 2 If "" are well correlated w ith density for various types of E J!..', ' f 0 .,j' • snow and urn. Later measurements by Arnaud [7] of 0 • Vostok ,'-, 0 12 East Antarctic ice cores from Vostok Station and from .Q • Km200 o. m '. two sites (KM 105 and KM200) at 105 and 200 km 0 along the traverse from Mirny observatory to Vostok, 'E also confirmed this conclusion. The details of these and 8 8 U all other sites discussed in the present paper are listed in • b Table I. • • • In the proposed model, Z(P) is expressed by Eqs. (9) , and (10) in snow and urn, respectively. As detailed in 0.' 0.6 0.8 Appendix A, this relationship is controlled by the two Relative density geometrical parameters Zo, C, and by the resultant critical density Po. Fig. 3 compares the theoretical Figure 3: Coordination number versus relative density curves with available ice core data. The observed (symbo/~) for ice coresfi'om (a) UpB, Bc' alld Site A [4. geneml trends and the scalier of the measurements fall 5] alld (b) Vostok and KM200 [7] compared to model within the minimum uncertainty range of the critical approximatiolls (solid lines) for various ice-core coordination number of Zo = 6.5-8.0, and the slope of structures specified by critical microstl'llctural the radial-distribution function is constrained to the parameters: (a) (1) Zo = 6.5. C = 30. (2) Zo = 7. C = 40. expected enhanced values of C = 40-60 increasing with (3) Z." 7.5. C " 50. (4) Z." 8. C " 60; (b) (/) z, " 7. increase in Zo. lmportantly, for the Vostok and KM200 C " 40. (2)Z, ~ 7.5. C " 50. (3) Z." 8. C " 60. ice cores, representing two cl imatic extremes of low temperature with mild wind and higher temperature Another important peculiarity is that, even for with strong wind (see Table I and [461), the relatively sma ll variations of Zo and C, the coordination numbers in the lim stage in Fig.3b are corresponding changes in Po are sufficiently large to systematically lower at Vostok (consistent wi th curve I cause a noticeable shift of the critical depth 110 at which for Zo = 7, C = 40) than at KM200 (e.g., curves 2 and 3 the transition [rom snow to Jim occurs. These features for Zo = 7.5-8, C = 50-60). The critical density, as an will be addressed in more detail below. -202- The deduced estimates of Zo and C can be used to , conSlmin the critical value of the bonding factor ~ in ",-. Eq. (1 1) on the basis of bond-area frac tion 'I 0 0.8 , measurements Sb' As shown in Appendix A, the mean .Q • UpS, Be contact area a is directly related to S (i. e., ~) , and Sb is U 2 ~ 0 .• ... SiteA determined as aZ/(41T). Calculations by Eq. (I I) - performed at Zo =< 7 and C "" 40 are also in general ~ agreement with the grain-boundary surface • OA ~ measurements [4] for ~ = O.S±O.1 (see Fig. 4a). These 0 0 estimates correspond to mean relative bond area of " 0.2 a Go"" 0.S3±0.IS in snow and a relative bond radius of 0.52±0.OS, which is also close to observations [ I, 2]. 0 Figs.3a and 4a and add itional computational 0 0.2 0.4 0.6 08 experiments indicate lo wer values of Zo - 6.S-7 and Relative density higher values of ~ - 0.6 for Site A core, whil e the stmcture of the Be core is characteri zed by Zo - 7 and ~ - O.S at C = 40. Sno w/fim texture analyses of Antarctic ice cores from Vostok Station, KM105, and 0 • Vostok K.M200 [7-9] are sim ilarly consistent among the cores 0 0.8 n ... Km200 (Fig.4b) wi th a possible range of ~ - O.S-O. SS at ~ .:.. Km105 Zo = 7-7.5 and C = 40-S0. Ice-grain specific surface -• 0 .• measurements [S4] on the ice core from Mizuho Station, •~ after transfonnation into bond area fraction values, ~ '" suggest ~-va ILl es of - 0.4S-0.S (Fig. 4c) and confirm a 0 • ..• 0 OA tendency to denser fim stmctLlres with Zo - 7.S-S and " b C - SO-60 under the strong wind condit ions typical for Mizuho area [74]. 0.2 As summarized in Tab le 2, for the 6 si tes on OA 0.6 O.S Antarctic and Greenland ice sheets which cover a wide Relative density range of present-day ice-formation conditions, the microstructural parameters 20, C, and soare reliably constrained with in ±1O% uncertainty limits by direct ste reological observations of ice cores. More accurate 0 0.8 tuning is needed and can be performed on the basis of .2 ice core density measurements in order to obtain a better U • Mizuho understanding of the impact of weather and climate on ~ 0 .• the characteristics and evolution of the snow/tim • stmcture. •" ~ 0 0.4 '" 0 • 3.2. Modeling the density-depth profil e " • 3 Aside from the geometrical characteri stics of the 0.2 • c microstructure, no other parameters of the snow/fim densification model can be measured directly and/or OA 0 .• 0.8 separately in laboratory experiments. The best and, most Relative densrty probably only, way of constraining the model further is to infer the parameter va lues by filling theoreti cal Figure 4: BOlld area fraction versus relative dellsiry density-depth profiles to ice core density measurements. (symbo/s) for ice cores frolll (a) UpB. BC. Gild Sire A [4. In the general case, the non-stationary distribution of 5]. (b) Vostok, KM200, and KMI05 f7. 9], and the relative density p of sno wlfim deposits on an ice (e) Mizllho {54] compared to model predictions (solid sheet versus depth II is govern ed by the basic lines) for variOllS ice core sfruClllres specified by equat ion (\) where the particle derivative in a quasl different critical bonding factors: (a) 20 = 7, C = 40. one-dimensional approximati on is expressed as (/)' 0~ 0.4. (2) t;" ~ 0.5, (3) , . ~ 0.6; (b) (I) 2. ~ 7, ~ 40, •• ~ 0.5, (2) 2. ~ 7.5, ~ 50, t;" ~ 0,5, d a a e e = + 11' (3) 20 ~ 7.5, e ~ 50. t;" ~ 0.55; (e) (/) 2. ~ 7.5, e ~ 50, dl at all •• ~ 0,5, (2) 2. ~ 7.5, e ~ 50, t;" ~ 0,4, (3) 20 ~ 8, e ~ 60, t;" ~ 0.45. Here, II' is the downward ve rtical velocity of the porous ice medium with respect to the ice sheet surface. The -203- compression rate win Eq. (I) is related by Eqs. ( [6) and Here, b IS the Ice accumulation rate (in Ice ([ 7) to the load pressure equivalent), and in accordance with [61], el is the rate of thinning induced by the globa[ motion of the ice • sheet. In central parts of thick polar ice sheets, this P, = gp, [{XIII (19) - quantity is of the order of el bill, where II is the ice " equivalent glacier thickness. Consequently, the last term where g is the gravity acceleration and Pi is the density III Eq. (20) IS usually small (i.e., of pure ice. eIP/I(gPi ) - e1h - bh/ll « b), representing a second The velocity IV is governed by the general mass order correction. The latter correction may become conservation equation, which after integration with substantial in marginal areas, ice streams, and mountain respect to II takes the form glaciers at high now rates and intense global deformation when elll Ib - I within the snow/lim (20) densilication layer. However, in most of stich cases the quasi-one-dimensional approximation breaks down and more general Ice now schemes and snow/tim as derived in Appendix C. rheological models should be considered. Table 1: Site locations and ice-formation conditions. Drilling site Location P. T(IO m), b, Pi, Group·· Reference °C cm yr- 1 kg m-.l Dome du Gouler 45°55'N.6°55'E 0.43 -10 330 918 H 130] Ushkovsky 59°04'N, 1600 28'E 0.38 -15.8 59 919 H [68] H 72 69" 12'S. 41 °05'E 0.41 -20.3 34.5 919 H [57] KM60 67°05'S,93°19'E 0.46 -20.8 50.3 920 H 146, 47}, this work Milccnt 700 18'N, 45°35'W 0.38 -22.3 54.4 920 H [53] KM 105 67"26'S, 93°23'E 0.46 -24.5 34.1 t 920 H {46. 47J, this work Site 2 76°59'N,56°04'W 0.375 -25 42.4 920 L {44] BC 82°54'S. 136°40'W 0.33 ·26.5 9 920 L 14) KM 140 67°45'S,93°39'[ 0.485 -27 43.9 t 920 H [46, 47J, this work 825 79°37'S, 45°43'W 0.38 -27 15.2 920 L [31] Byrd 80"oo'S,120"OO'W 0.39(0.41) -28.7 17.4 921 L [34] Site A 700 45'N, 35°58'W 0.37 ·29.5 29 921 L [5J Crete 71 °07'N,37°19'W 0.38 -29.7 29.3 921 L [53] KM200 68"15'S, 94°05'[ 0.49 -30.5 28.7 t 921 H (46.47], thiswork Summit 72°35'N,37°38'W 0.42 -31.7 23 921 L [66.67] Mizuho 700 42'S, 44°20'[ 0.44 -33 7.5 921 H [54.74] PioneTSkaya 69°44'S.95°30'[ 0.47(0.46) -39 19.4 922 H [46, 69], (K.E. Smimov, pers. com.) Vostok- l 72°08'S,96°35'E 0.42(0.41) -47 8.9 923 L {46], (K.E. Smimov, pCTS. com.) Komsomolskaya 74"06'S.97°30'E 0.39(0.36) ·53.8 6.9 923 L [47] EPICA DC 75°06'S, 123"21'E 0.37 ·54.5 2.7 923 L {28. 59], this work DomcFuji 77°19'S, 39°42'E 0.39 .57 ; 3.2 : 924 L 121 , 42, 75].thiswork Vostok 78°28'S, 106°48'E 0.38(0.35) -57.5 ! 2.15 : 924 L 127, 47.48] • Values in parentheses are direct surface snow density measurements in the upper 20-50-cm layer ""See subsection 4.1 for explanations t Values corrected for the site movement and accumulation-rate changes along the icc flow line ; Values additionally constrained and vcrified through ice flow modcling [40, 62. 72] -204- Table 2: Microstructural parameters and densification characteristics deduced from ice core data. Drill ing site z, c p hOI m h ~ffi m loff' ky r B, (B~ ) Group Dome du Gotller 7.' '0 0.55 9.' 0.736 30.8 64.5 0.018 2.29(2.30) H Ushkovsky 7.5 '0 0.54 9.' 0.736 24.0 52.6 0.072 2.34 (2.36) H H72 7.' '0 0.56 9 0.736 24.6 54.9 0.116 2.42 (2.44) H KM60 8 60 0.52 9 0.754 26.9 57.1 0.084 2.27 (2.32) H Mi1cent 7.' '0 0.58 8.5 0.736 29.6 65.! 0.087 2.46 (2.49) H t KMI05 7.' '0 0.55 9.5 0.736 27.0 61.4 0.132 2.44 (2.44) H Site 2 7 40 0.57 8 0.714 29.6 68.4 0.116 2.58 (2.57) L Be ' 7 40 0.53 6 0.714 22.2 49.7 0.391 2.54 (2.57) L KM140 7.' '0 0.56 !O 0.736 30.1 69.0 0.116 2.47 (2.46) H B25 7 40 0.54 7 0.714 24.8 57.0 0.268 2.56 (2.56) L Byrd 7 40 0.54 7.5 0.714 26.4 61.3 0.253 2.57 (2.56) L Site A t 6.' 40 0.6 1 8 0.704 31.3 77.3 0.191 2.81 (2.75) L Crete 7 40 0.' 7.5 0.714 29.7 69.6 0.170 2.53 (2.5) L KM200 t 7.' 50 0.53 9.5 0.736 29.6 68.3 0.176 2.46(2.45) Ii Summit 7 40 0.59 6 0.714 32.9 73.4 0.227 2.65 (2.66) L Mizuho t 8 60 0.48 10 0.754 23.4 51.2 0.506 2.31 (2.35) H Pionerskaya 7.5 '0 0.53 8 0.736 35.9 81.4 0.308 2.52 (2.53) H Vostok-I 7 40 0.5 7 0.714 38.3 90.5 0.732 2.65 (2.63) L Komsomolskaya 6.' 40 0.55 ,.5 0.704 49.0 116.9 1.203 2.92 (2.90) L EP[CA DC 6.5 40 0.62 4 0.704 43.9 [01.0 2.626 3.0 (3.01) L Dome Fuji 6.5- 400.53- 4- 0.704- 47.1- 110.1- 2.427- 2.88(2.90) L 7 0.6 4.5 0.714 49.5 108.2 2.382 7 40 0.' 3.5 0.714 44.8 97.5 3.[73 2.73 (2.78) L t Microstructural charactcristics arc constraincd by dircct stercological measurements in subsection 3.1 Eq. (I) is solved stal1ing from a given surface snow boundary sliding, respectively. The parameters marked density P~ with an asterisk denote those at the reference temperature TO, and RII '" 8.314 J (mol Kf' is the gas P lh - O'" p~ . (21) constant. The ice crystal size is often characterized by the mean As the snow/fim densification is a temperature crystal area Ac measured in ice-core thin sections. For controlled process, the rheological parameters JI and kr nonnal grain growth, it is well established that at fixed in Eqs. (16) and (17) (the principal temperature temperature, Ac is a linear function of time. dependent properties of ice) are conventionally assumed Accordingly, in a variable climate, we have to be Arrhenius-type functions of absolute temperature T(in K): (23) k ~ k • exp[Q. (_' --'-)] , <~ RT• ' T (22) where A". is the ice crystal size at the ice sheet surface, k, ~k, ' -205- rO = 215.7 K. These estimates and a constant averaged with the observed textures (Vostok, Mizuho, Km200, 2 crystal size at the glacier surface, A c~ '" 0.7 mm , are Site A, BC, and KM lOS) fonn a representative assumed in our study in Eqs. (23) after [47] to simulate uniformly distributed set of data. These basic the relative ice-grain radius R = -..j{Ae/Ae, ) in Eq. (17). experimental profiles, ploned by open circles in Figs.5a-f, are used to constrllin the temperature The model (I) and (16)-(23) governs the nOI1- dependent rheological properties k,., and J.l, and the stationary density-depth distribution in the dry snow/fim index Other densi ty profiles from ice cores in Table I stratum on an ice sheet. These equations should be p. are employed to validate the model and to study the coupled with the heat transfer equation to account for development of the snow/fim structure during the temporal changes in temperature (e.g., [33]). Water densifiClltion process under difTerent climatic and vapor transport may also become important as a mass weather conditions. transfer mechanism in Eq. (\) at high temperature gradients (e.g., [37, 58, 64]). The model describes the densitication process above a certain depth h,>jJ at which atmospheric air becomes trapped in the ice matrix and 0.9 tim is transfortned into bubbly ice. The prediction of a pore closure is regarded as a separate problem that is beyond the scope of the present paper. In the .~ ~ applications considered below, based on [47, 5[, 52], a Q) 0.7 linear empirical correlation between the close-oIT ~ density Poff and firn temperature is conventionally .~ro employed, as given by ~ 0,5 o. Poff= 0.9 - 5.39.1O-I(T - 235) . (24) f--~~---1 Vostok o " An interactive computer system was developed to solve Eqs. (I), ([6), and (17) completed by Eqs. ([8) o 20 40 60 80 100 (24) and simulate density-depth profiles in the snow/fim Depth, m layer of an ice sheet. The program allows ready adjustment of model parameters and filling of the 0.9 computational predictions to Ice core density measurements. b .~ 00 3.3. Model con straining and ice core density data ~ 0.7 analys is A series of prel iminary computational tests was . ~ perfomled in order to study the sensitivity of the ro ~ 0.5 00 densification model t,o the rheological parameters 1:, k,., J.l, and the dilatancy exponent p. They revealed that Mizuho largely identical dengity-depth profiles can be obtained for a range of small values of I: in Eqs. (\5) and (16), 0.3 +-~---'--~----r-~---'- assuming that the non-linear viscosity decreases with o 20 40 increllsing 1:. Thus, the correction factor is fixed at Depth, m t: = 0.1, corresponding to an Ice pressure p approximately 7% lower than the load pressure PI at the 0.9 close-ofT depth, similar to the predictions for bubbly ice [6\]. It was also confirmed that, in accordance with the c o physical meaning, three other parameters k" p, and J.l selectively control the density distribution in the :: 0.7 respective depth intervals of near-surface snow, an ! ~ intemlediate zone of ice-grain rearrangement noticeably •• inOuenced by dilatancy effects, and the fim stage of ro ~ 0.5 densification by dislocation creep. A total of 22 ice core density profiles were selected for model constraining and validation. They are mainly Km 200 from the Antarctic and Green land ice sheets (Table \) 0.3 +-~--,-~--,---~---,--+ and cover a temperature range of almost 50 "C (- 57 to o 20 40 60 - 10 "q, ice accumulation rates of 2.2 to 330 cm yr 1, Depth, m and various weather (wind) conditions (e.g., [46]). Among them, density measurements for the 6 ice cores -206- In all simulations it is assumed that snowlrirn strata 09 have been fonned at constant present-day (or mean d Holocene) temperatures, accumulation rates, and relati ve surface snow densities Ps (Table I), and the density-depth relationships are modeled as quasi stationary distributions. If several measurements of accumulation rates at a certain site are available for a number of periods [46], the weighted mean value is calculated, corrected if necessary for site movement along the ice now line [47]. It should be emphasized Site A that the surface density Ps is not regard ed as a tuning 0.3 +~--,-~-,--~-,-~-I- parameter, but is understood as a long-term mean of the o 20 40 60 80 near-surface snow layer consistent with the general Depth, m trends of the density-depth profiles extrapolated to the surface [46]. [n some cases, these "eITective driving" densities can diITer slightly [rom the snow densities 0.9 measured directly in the 20-50 cm thick surface layer e (Table I, values III parentheses). The use of these present-day values may slightly perturb the simulated ·f densit y-depth curves within the upper 3-6 meters, but Q) 0.7 ~ does not change the general conclusions. The strain . ~ rates of vert ical thinning caused by "external" Ice motion are small in most cases, and = 0 in Eq. (20). ro e1 &0.5 The exceptions are the ice cores from the Dome du Gouler glacier at the summit of Mont Blanc [30] and the Be Gorshkov ice cap 111 the crater of the Ush ko vs ky volcano at Kamchatka [68]. Computational experiments indicate that plastic o 10 20 30 40 50 Depth, m deformation of ice grains in snow becomes important at effective pressures P."1T at intergrain contacts substantially grell\er than ca. - 0.1 MPa, th e threshold 0.9 value at which the power-law creep with constant f exponent a takes place in the densification process [48, 61]. To be consistent with these works, a value of .~ w c 0.7 a = 3.5 is employed here. Sensitivity tests con finned .1J that variations of a with in its uncertainty range (2.5 to .~ 4) can be compensated for by corresponding changes in :;;; the non-linear viscosit y /-I. The simulated density-depth ~ 0.5 curves obtained through minimizing the standard deviations from ice core measurements at Vostok, Km105 Mizuho, Km200, Site A, BC, and KM 105 (Fig. 5, solid lines) are in excellent agreement with the ice core data. 0.3 +-~--'---~--'--~-r- o 20 40 60 The corresponding best-fit values of p, k" and /-I (at Depth, m a = 3.5) are inferred at the micrOSlmctural characteri stics Zo, C, and sothat are detennined from the Figllre 5: Bw"ic ice-core dellsily profiles (open ci rcf~) direct stereological observations and additionally from six Amarctic and Greenland sites (a) Vostok. validated (within respecti ve uncertainty limits) through (b) Mizuho. (c) K.M200. (d) Site A. (e) BC, Gild the minimization procedure. The deduced (f) KMI05 wilh model predictions (solid lines) fitted to microstmclural parameters are summari zed in Table 2. COIls/rain rheological parameters. Dashed lines ill (a. h. These results suggest that the dilatancy index p varies c and e. j) are rhe profiles simulated al recommended [rom 3.5 to 10 over the present-day range of ice parameters from Table 3. Results ill (d) are compared fonnation conditions. In the Arrhenius relations (22), with those for L- and H-microstruclures of (I) Crete. the grain-rearrangement rate constant kr and the non (2) Milcellf. and (3) Mizuho ice cores (dashed lines). linear viscosity /-I can be represented by 1 I a Dolled lines ill (a) and (J) represent profiles calcillated kr' = 0.022±0.003 MPa- yr- and /-I' = 21 ± 1 MPa yr at for alternative vailies ofP (illcreasefrom 3.5 to 7 ill (a) the reference (Vostok) temperature of r" = 215.7 K and decrease from 9 10 6 in (f)), dOlled lille in (d) with respective activation energies of Qr = 70 kJ mo\-l corresponds to reduced apparent activation energy of and Qp = 58 kJ mol-I. The only inconsistency IS l Qr = 40 kJ mor of ice-gl"(lill rearrallgemellf. observed in the deduced (best-fi t) k," value for the -207- Mizuho ice core, wh ich is a third of the mean estimate. (dotted line). Despite the uncertainties discussed here, This result can be attributed at least in part to an artifact the deduced rheological properties of snow and firn caused by a recent short-tenn increase til ice appear to be quite reasonable and realistic. As further accumulation [74]. Overtuning may also be responsible, verificat ion of these results, the model predictions can since even for the Mizuho core the density-depth profile be compared to the ice core density data from other simulated at the mean values of k,' and J./ (Fig.5b, drilling sites listed in Table I. dashed line) does not deviate much from the best-tit Thus, fixing the rheological parameters in Eqs. (22) curve. The question of overtulling is also discussed in as the best-fi t mean values inferred on the basis of the 6 the context of sensitivity tests below. ice cores considered above (Vostok, Mizuho, Km200, The obtained power-creep parameters are well SiteA, BC, and KMI05) and applying the deduced constrained with accuracy of a few percent and are constraints on the snow/firn build-up, the closely coincident with the conventionally accepted microstructural characleristics 20, C, ~, and the activation energy Qp = 60 kJ mol- I and dilatancy exponent /3 are tuned to fit the simulations p = 4 1±18MPa"' yr deduced at 2l7K for bubbly ice separately to each of the remainder observational densitication [48]. It is also important to remember Ihat density profiles. As shown in Appendix 0 , the best-fit the phenomenological enhancement factor 2/4 used in density-depth curves (solid lines) match closely Ihe ice Eqs. (14) and (15) renormalizes the non-linear viscosity, core measurements (open circles). The corresponding decreasing slightly the value, since 20 is substi tuted for estimates for the adjustable parameters (4, C, SQ, and /l) a higher (unknown) coordination number at which the are summarized in Table 2. Although these values intergrain contact forces become unifomlly distributed include uncertainties in terms of the prescribed in the tim structure. snow/firn rheology and environmental ice-fornlation A noticeably lower activation energy of conditions, they are in fu ll agreement with the direct 41 ±2 kJ mol-1 was previously found for ice-grain stereological observations of the 6 basic ice cores. This rearrangement [2], consistent with the assumption that resu lt is a convincing confimlation of the validity of the grain sliding may be controlled by molecular diffusion constrained rheological equations (22) and demonstrates around obstacles through the intercrystalline boundary. the much broader signiticance and utility of available In addition to the linear viscosity v, the grain ice-core structural data. The sensitivity of the simulated rearrangement rate constant kr defined by Eqs. (\ 7) density-depth profiles to the values of the 4 tuning contains a number of possible temperature dependences, parameters is discussed in further detail in the next including the correction factor c, the grain-bond sect ion. thickness 0, the total contact area aoZo, and the initial It should be noted here that the Dome du Gouter and grain size R. [47]. This complexity may, at least in part , Us hkovsky si tes both experience windy conditions and be responsible for the higher best-fit estimate of Q, the highest temperatures, at which surface snow melting obtained here as an apparent activation energy. begins in summer [30, 68]. For these ice cores, the rales FurthemlOre, this treatment is the first time that kr has of snow/ fi m strata thinning due to glacier motion are 2 3 l estimated to be e _ 2.7.10- and 3·1O- yr- , been explicitly introduced and quantified to model the l ice-structure repacking mechanism coupled with grain respectively, and appreciably innuence the downward plasticity and di latancy effects thai hinder the relative veloci ty in Eq. (20). Even in such limiting cases, the motion of ice particles. These peculiarities of snow model predictions are in good agreement with the densification were not distinguished in Alley's theory measured density profiles. [2]. In addition to boundary diffusion, grain sliding involves neck shearing by dislocation creep, quasi 4. Applications. Discussion liquid layer development, and other factors that may participate equally in the ice-crystal rearrangement 4.1. Impact of ice-formation conditions on snowlfirn process, increasing the activation energy. As revealed structure by the preliminary computations, kr controls the The inferred microstructural parameters summarized densification of snow on ly within the upper 15-20 m of in Table 2 reveal an important peculiarity that the snow deposits subject to short-tenn climatic impacts microstructure of snow/firn deposits varies according to and characterized by high natural nuctuations of density the conditions of ice formlllion. As a first (see Fig. 5). Sensitivity tests on the associated approximation, the examined ice cores can be uncertainty of the second of Eqs. (22) indicate that the empirically divided in two groups on the basis of the snow-density profile is relatively stable with respect to dense packing characteristics (the critical coordination changes in kr' , allowing for a range of 0.01- number Zo and the RDF slope C). One group, 1 I 0.04 MPa- yr- without exceeding the data scatter. distinguished as L-group, exhibits relatively low 4 of Correspondingly, activation energies of 55-60 to - 6.5-7 and C of - 40, while the other group (H-group) 75 kJ mol- I do not contradict the measurements. is characterized by higher Zo of - 7.5-8 and C of - 50- However, Alley's estimate [2] appears to be too low. For 60. Correspondingly (see Table 2), the respective example, even qualitative similarity is lost between the critical densities are relatively low, ,l\I - 0.704-0.714, observational profile and the density-depth curve and higher, PtJ - 0.736-0.754. The L- and H-structural simulated at Q, = 40 kJ mol-I for Site A in Fig. 5d ty pes are typical for low- and high-temperature -208- extremes (see Table I). However, it should be parlicles, resulting in smaller values for Zo. C and Po. emphasized that there is no direct correlation with For example, direct measurements of the specific pore temperature in the intermediate interval from -40 to surface in Vostok and KM200 ice cores [7, 9J have - 24"C. The extreme cases (e.g., Mizuho and Summit, confinned Ihat pores in SIlOW at Vostok are essentially KM200 and Site A) occur in simi lar temperature larger (low void surface area) than at KM200. In this conditions. As illustra ted in Fig. 6a, the critical bonding context, the dominance of the H-group at high fac tor does no t vary from one group to the other, and temperature (> - 25 °C) can be attributed not only to both groups cover the same range of scaner windiness but also, at least in part, to high (.;;0 - 0.55±O.05) when plolted against temperature. Only accumulation, masking the insolation and/or efTects of low surface densities of Ps - 0.33-0.42 are met in the L· high temperature gradients [19]. group, and the di latancy exponent P fo r this group te nds I I to increase with temperature from 3.5 to 8 (Fig. 6b). A 0 value of P- 4 appears to be well established fo r central - - 06 f--o~ 0 - Antarctic cores of the L-group under low-temperature N 0 • conditions (circled cluster in Fig.6b). The H-group g> , • • • o~ • is 0 • • exhi bits much higher surface de nsities Ps - 0.38-0.49 0 • 0.5 1- - ~ -- and dilatancy exponents p- 8-10. The results of S -- • a computational tests (dOlled curves in Figs. Sa and f) demonstrate the high sensitivity of the density-depth curves to P in the intennediate range of depths, allowing , th is parameter to be re liably constrained. To illustrate the maximum divergence in density 10 profi les for dilTerent snow/fim structures, the curves simulated for 3 other sels of microstructural parameters ~, C, .;;0 are compared with the graph for Site A (L group) in Fig.5d. The curve I for the Crete ice core structure demonstrates the maximum deviation from the Site A core within the same L-group, while the Mi lcent and Mizuho ice core structures (curves 2 and 3) represent extreme cases from the H-group. Although the c , o -l . , . -H density profiles and close-ofT characteristics fa ll within b a range of ± I 0% of that for the average structure, the , , profi les from the L- and I'I-groups do not overlap, -so -40 -30 -20 -10 Te mperature, cC revealing a continuous transit ion between geometri cal and physical properties from one structure group to the othel'. It can clearly be seen that fo r similar temperaturcs Figure 6: Critical bonding ftlcfor ~o (a) alld dilatancy and accumulation rates, the L-group has distinctly lower exponellf P (h) for L-group (open symbols) (llId H-group de nsificat ion rates (i.e., a "harder" structure) in (solid symbols) ploued againsl sllrfilce temperalllre. comparison with the H-group. This suggests that other Sq1lares alld circles corre!)pond 10 microslruclllral factors, such as wind speed [ 19, 46] and/or insolation parameters constrained by direct stereologiclli [l3}, may also play important roles in the formation of measl/remenrs and inferred frolll the density-depth snow/fim strata and parlicipate in switching between the profiles. respeclil·ely. Dashed lines show /he range of.;;o densification regimes under changing cl imate. deviations ill (a) alld tendencies ofP groll'lh ill (b). In accorrumce with [46], the surface density is primarily correlated with wind speed, and the Antarctic As grain restacking in snow can destroy necks and si tes (r ioncrskaya, Mizuho, KM200, KM140, and reduce bond areas due to the inevitable shearing of KM105) characterized by intennediate surface intergranular contact zones, the dynamic equilibrium temperatures (- 39 to - 24.5 "C) and strong winds (mean between grain rearrangement and neck growth due to wind velocity > 9- 10 m s I) all fall into the H-group. It water-vapor dilTusion (ice-mass transporl) maintains a can thus be speculated that in this case, the initial certain degree of bonding in excess of dislocation creep surface-SIlOw stnlcture will be closer to perfect, without a noticeable direct in nue nce of temperature and resul ting in a denser closest packing at the snow-to-fim snow structure on the critica l bonding factor .;;0 (see transi tion. In contrast. at low wi nd speed, typical fo r Fig. 6a). sites in the L-group at temperat ures below - 25 "C, [n the soner snow structures of the H-group, with a surface snow is less dense [46] and looser. In addition, more well-developed spec ific pore-space surface, the insolation [13] and surface tempera ture can playa greater intensity of dilTusive water-vapor transport signi ficant role in snow structure fo nnation, faci li tati ng interferes wi th and minim izes the dilatancy efTects in (especiall y at relat ively low accumulation rates) grain the SIIOW densification slage, leading to the highest bonding and growth at the expense of smaller ice values of p. The interplay between insolation (surface temperature) and accumulation most li kely controls the -209 - growth of the dilatancy index in the L-group, in general Komsomolskaya (see Tables I and 2) reveal a tendency correlation with the increase in ice accumullllion rates, for p to grow with accumulation rate. The depths which reduces the influence of insolation. The BC ice corresponding to the critical density (Table 2) also show core is an obvious example of a highly dilatant snow that, in both groups, ice accumulation significantly structure (i.e., low fJJ fonned at the enhanced contrast interferes in the densification process and perturbs the between high temperature (insolation) and low general tendency of 110 to decrease with temperature accumulation. For comparable accumulation rates but at (compare Mizuho and Summit, BC and Site 2, very low temperatures, the central Antarctic stations of Us hkovsky and Dome du GOllter cores in Tables I and Vostok, Dome Fuji, EPICA DC, Vostok-I, and 2). Table 3: Densification-model parameters and their recommended values. Parameter Notalion Value L-group H-group Ellvirollmelllal ice-jormatioll conditions Relative surfacc snow dcnsity A < 0,42 > 0.38 Surface tcmpcralUre, ~c T, < -24 > -40 Ice accumulation rate, cm yr - I b Microstrlle/llral characteristics Critical coordination number z. 6.75±0.25 7.75±O.25 RDFsiopc c 40 55±5 Critical bonding factor (0 0.55±O.05 Dilatancy exponcnt t p 3.5-8 8-10 Rheological parameters Grain rearrangement-rate constant :, MPa 1 yr- I 0.022±0.003 Activation energy of grain rearrangement, kJ mol- I Q, 70 Creep index a 3.5 Non-linear viscosity of ice ~, MPa" yr !' 21±1 Activation energy of dislocation crecp, kJ mol l Q, 58 Ratio of dcviatorically dcfonncd grains , -D. I Close-ofT dcpth factor B. 2.75±0.25 2,42±O.11 Close-ofT ice-age factor B, 2.76±O.24 2,40±0.012 t Correlation with temperarure is specified in Fig. 6b : At Ihe reference temperature ofT" = 2 15.7 K Although the present analysis reveals rather complex 4.2. Characteristic featu res of snow/firn densification relationships between the structure of the snow stratum as revealed by modeling at the surface of ice sheets and the conditions of ice A properly constrained snowlfim densification model fonnation, demonstrating the necessity for further study, makes it possible to discuss the physics of fimification the principal tendencies appear to be quite clear. The in more detail. Fig. 7 shows the predicted variations of introduction of two (L- and H-) types of snow build-up the densification rates and the corresponding with characteristic microstructural parameters components due to ice-particle rearrangement and essentially reduces the uncertainty of snow/fim dis location creep with respect to depth under diITerent densification modeling. The results of constraining the climatic conditions at Vostok and KM200. A densification model are summarized in Table 3, where temperature increase of up to 30 °C (Table I) and the recommended model parameters are given wi th the changes in the snowlfim structure (Table 2) result in a corresponding estimated uncertainties. The density more than ten-fold increase in w(note the vertical scales profiles simulated lIsing these parameterizations are in the figure). At low temperatures and relatively small presented in Figs.5a-c, Figs. 5e, f, and Appendix D k,., appreciable plastic deforma tion at, of the porous-ice (dashed lines). The deviation of these profiles from the skeleton develops from the beginning of the snow best-fit curves (solid lines) is within ±2-3%, and is densification stage (Fig.7a, dolled curve 2). The comparable to the uncertainty of measurements. fraction x of deviatoric defonnation by grain-boundary -210- sliding in Eq. (4) for the Vostok core thus gradually in Eq. ([ 7). As a result, ice-grain rearrangement diminishes with depth from unity to zero (Fig.7a, practically remains the sole mechanism of deviatoric dashed curve), as the density and coordination number deformation (Le. , x ::::: I) during the entire snow stage increase, strengthening the dilatancy efTects through (Fig. 7b, dashed curve). However, dilatancy dominates parameter ..t. in Eq. (16). Dilatancy determines the III Eqs. (3) and (4), still controlling ice-crystal impact of plastic defonnation on snow densification, rearrangement and plastic defonnation in the snow and an increase in A controlled by p in Eq. (18) (as compression (Fig. 7b, dotted curves) described by revealed by Eqs. (16) and (\7» results finally in a n1pid Eqs. (16) and (17). In the K.M200 core, an abrupt drop in wdue to the decrease in ice-grain rearrangement decrease in the densification rate occurs at higher rates £t+ in the middle of the snow stage (Fig. 7a, dotted densities, where grain-boundary sliding ceases, and the curve I). firn stage begins at depth of around 30% shallower than for the Vostok core (see Table 2). [n both cases, the 0.' , critical depth 110 below which densificatioll occurs solely \ by the dislocation creep of ice grains can be clearly \ a identified in Fig. 7 as the point at which the fraction of "~ OA I 0.8 deviatoric deformation by grain rearrangement x •0 becomes zero. ~ I p=O.55 The model predictions indicate that the snow-to-firn 0.3 - r- 0.6 c 11i'" , I 11 transition is characterized by a local perturbation in the "c , 1- compression rates. This peculiarity can be clearly 0 ,, l' discerned for the low-temperature core from Vostok ." 02 - , I f-o.4 X • , (Fig. 7a) as a drop in OJ caused (due to the interplay ~ I ~ 1 ...... \ between x and A) by an abrupt increase in the dev iatoric E ,, Vostok 0 0.1 , I 0.2 stress in Eq. ( 16) as x falls to zero. At the higher 2" _'0(-- I temperatures and relatively low load pressures of the " -- , , , I K.M200 core, the compression rate in the firn stage 0 0 starts to grow with depth (with PI), passing through a 0 20 40 60 80 100 local maximum (Fig.7b). Both eITects become more Depth, m prominent if the deviatoric-stress factor I: in Eq. (16) is increased. Similar changes in the slope of the snow/lirn density profiles around relative densities of 0.72-0.78 6 I were observed and explained as due to the emerging ~ b I "~ 0.8 dominance of dislocation creep in a series of papers [24, to I 25 , 54]. The critical densities deduced in the present ~ p=O.6 I study and given in Table 2 lie within the same range of 4 I c '"~ I 0.6 0 0.7-0.76. As shown in Appendix A, Po is related to the ~ I ·u firn structure by the geometrical characteristics Zo and c 1- 0 l' C, increasing with these values (Table 2) from the L OA X ."• group to H-group. The characteristic bend (upper ~ 2 , "~ , critical point) observed after [6] in the densi ty-depth E 1 ...... \ profiles around relative densities of 0.55-0.6 at 0 , Km200 0.2 approximately half 110 can be identified here (Fig. 7) " 2" ',_-' with the maximum decrease in snow densification rate 0 -- -- ' , 0 due to the dilatancy efTects triggering the dislocation 0 20 40 60 creep of ice gT -211- Table 3 are compared in Fig. 8 (thin solid lines) to those ±5-7% between the two simulated profiles can be (bold lines) predicted by the model [8] and the observed, attributable to the ice-core structural correspond ing density measurements. Besides the variability between the L- and H-groups, which was not obvious difference in shape, a noticeable mismatch of taken into account in previous studies. 0.9 0.7 0.5 Vostok Site A 0.3 0 20 40 60 80 100 0 20 40 60 80 0.9 'w '"c 0.7 "0" ~ ~ 0.5 "'" Mizuho Be 0.3 0 20 40 60 0 10 20 30 40 50 0.9 0.7 0.5 Km200 Km105 0.3 o 20 40 60 o 20 40 60 Depth, m Depth, m Figllre 8: Comparison of dellsity-deplh profiles for basic ice cores from Vostok. Mizllho KM200. Site A. BC, alld KMJ05 (lhill solid lines) wilh lhe predictiolls ofrhe //Iodel [8J (bold lilies). Opell circles dellote experimelltal datel (see Table I). It is also interesting to compare the results of the densification process. As shown in the inset of Fig. 5a, present model with detailed snow density measurements and as expected [3], systematically higher densification near the ice-sheet surface, where diITerent depositional, rates are observed at Vostok within the uppennost 3- diagenetic and meteorological processes (not taken into 5 m compared to the computations. However, the deeper account in modeling) are superimposed on the modeled part of the simulated density profile closely follows the pressure sintering lind may even dominate in the observational data. -212- 4.3. Paleoclim atic implications of the snow/lim B, and Bb are considered to reflect the intrinsic dcnsificatio n model peculiarities of the densification phenomenon and its One of the most important applications of a snow/firn relationship with the snow/fim structure and ice densification model is in providing the basis for formation conditions. Sensitivity tests show that the predicting the diITerence between the ice age and the variations in B, and Bh within each group do not exceed age ofatlllospheric gases occluded in the ice (e.g., [12, ± 1. 5% and ±3% over a ±50% range of accumulation 15, 33, 65, 67]). Accurate simulation of the gas age - rate and ± 10 °C range of temperature, respectively. ice age relationship is a key question of deriving reliable Once original mircostructural properties of snow/fim paleoclimatic histories from ice cores. In this context, deposits and local quasi-stlilionary ice-fonnation the depth h,>/! and ice age toff in fim at the close-off conditions have been specilied in temlS of the B-factors density P~ff are the principal characteristics related to (e.g., based on present-day ice core measurements), trapping atmospheric gases in the snow/fim Eqs. (25) together with the first of Eqs. (22) become a densification process. As the pore closure problem is useful tool for ice core data analyses and can be not considered explicitly here, a simplified empirical employed directly for predicting quasi-stationary close correlation (24) between Puff and temperature (47, 51, oIT characteristics under different climates within a 52] is used in calculations. The best-fit present-day certain group. close-ofT characteristics fo r all selected ice cores are listed in Table 2. A reliably constrained physical model can be further employed as a robust instrument to study -50 , 2.4 relat ionships between Puff' h"ff' and foff as well as their , , ' \ a dependences on climate. \ \ -55 2 " Under quasi-stationary ice-formation conditions, a \ "~ certain similarity between difTerent density-depth \? E l' __ T, 0 profiles scaled by the typical values of Al and hoJJ can be 0 ca ;; -60 I .' envisaged, with the product Pohoff being correlated --- b ~ (approximately equal) to btuJJ. Direct calculations based ~ ., E E on the data from Tables I and 2 confirm this expectation "ID >- § and s how that the critical density Al is very close to the -65 , 1.2 mean snow/lim density. The same approach can be '" applied for scale analysis of Eq. ( 16), which pennits -- approximate integration with respect to depth from 0 to -70 0.8 lI'iff afier neglecting deviatoric stresses (c= O) , substituting Eqs. (I) and (20) at e, = 0, and assuming 10 140 PI "" gp;Poh instead of Eq. (19). This leads to two b simplified expressions fo r toJJ and hoff 130 8 E ~" oi 120 0> ~ID m , ,-,0 ,- u I , 110 '9 (25) i I , / - •a , / •a , ~ ,/ a 4 , I a -- \" 100 '- ' ~ --- hOIf where 8, and Bh are the dimensionless fonn factors of 2 90 the density-depth profile and Al is determined by Zo and 0 10 20 3D 40 Ti me SP, kyr C (see Appendix A). For non-:£ero val ues of e, ' the accumulation rate b in Eqs. (25) should be replaced by Figure 9: Paleoclimatic variations of close-off the difference b - 0.5Ale hoff l clwrac/eristics at Vostok Station for the last 45 kyr. The va lues of 8 , and 81> calculated for the deduced (a) Ice-formation conditions (Ts and b) [72J and close-off characteristics are given in Table 2 and (b) quasi-stafionOlY close-off depth h"ff and ice age roJ! summarized in Table 3. As might be expected, the calclliated ji'om Eqs. (25) (bold lillej) and ji'om the coefficients B, and BI> are practically identical and can model [8J (thin lines), respectively. Open and solid be regarded as generalized attributes of difTerent types squares denote hoJJ alld t,>jJ predicted by the proposed of snow, both ranging from 2.3 to 2.5 for the H-group general model. and from 2.5 to 3.0 for the L-group. As these findings are originally based on the fitting of theoretical profiles to field observations and thus are independent of The central Antarctic sites of deep drilling projects possible interpretations derived from model parameters, are of special interest. They are characterized by very - 213- similar present-day B-factors, B(;::; 2.87±0.13 and artilicially diminishes the mean snow/fim density and as B it ;::; 2.89±O.II, which in accordance with sensitivity follows from the second of Eqs. (25), increases the tests are not expected to vary by more than 3% during close-off depth in comparison with more sophisticated glacial periods. Recent 45 kyr histories of surface description of snow densification proposed in the temperatures and accumulation rates (Fig.9a) present work (Fig. 10, thin curves). It should be noted reconstructed at Vostok Station [72] have been used to that independent studies [33, 70] of isotopic separation illustrllle the possible paleoclimlllic impl ications of the process of pennanent atmospheric gases (,sN of developed densification model and the simplified molecular nitrogen and 40Ar of argon) in polar lim also equations (25). The corresponding variations in hojJ and support shallower close-off depths than those predicted toJ! simulated in the quasi-stationary ap proximation are by the model [8]. shown in Fig. 9b. The bold curves for hojJ and tojJ are calculated from Eqs. (25) at the mean values of 5. Conclusion Bt = 2.76 and Bit = 2.82 estimated for the assumed climate changes at Vostok, using the best-lit The densification process of snow/fim deposits at the microstructural parameters from Table 2. The respective ice sheet surface is a vertical (un iaxial) compression close-ofT characteristics predicted by the general model with non-zero deviatoric stresses and strain rates are shown in Fig.9b for 9, 13, 15, and 21 ky r before superimposed on global glacier motion. The overall present (B. P.) by open and solid squares. As predicted macroscopic defonnation in the granular ice compact is above, the deviation does not exceed 2-3%. The attributable to a combination of the rearrangement of corresponding quasi-stationary density-depth profiles at grains as rigid particles, and the plastic deformation of the climatic extremes of the Holocene optimum (9 kyr grains. Dilatancy efTects are revealed in the kinematic B.P.) and the Last Glacial Maximum (2 1 kyr B.P.) are relationship between the macroscopic compression and plotted in Fig. 10. deviatoric strains in ice-grain restacking. The increasing overburden pressure from the very beginning of the snow densification sl -214- (15), (17) for effective pressure and deviatoric stresses Acknowledgements on grain contacts. The creep index a and the non-linear viscosity II in the ice flow law (13) together with the The authors are grateful to J. Kipfstu hl, H. Narita, grain-rearrangement rate constant kr and the dilatancy F. Nishio, T. Shiraiwa, K. Smirnov, and F. Wilhelms for exponent p introduced in Eqs.(17) and (18) are the providing the original ice core density data employed in principal model parameters responsible for the this study. Valuab le discllssions with R. Alley, macroscopic rheological behavior of the snow/fim A.V. Kosterin, A.G. Egorov and usefu l comments by compact. T. Blunier, O. Gagliardini, V.N. Golubev, SA Sokratov The model is constrained and validated on a are also gratefully acknowledged. representative set of ice core data (see Table I) which This collaborative research is a contribution to covers a wide I1Inges of present-day temperature (- 57.5 Project 4 of the Subprogram "Study and Research of the to - IO °C) and Ice accumulation rate (2.2 to Antarctic", the Federal Target Program "World Ocean" 330 em yr-I) conditions. The measured snow/fim of the Russian Federation. The present work was density profiles and available data on ice core structures supported in Russia through Grant No. 05-05-64797 allow the rheological parameters to be reliably from the Russian Foundation fo r Basic Research and in 1 I Japan through the Japan Society for the Promotion of determined, giving kr' = 0.022±0.003 MPa- yr- . and j.t" = 21 ±1 MPa" yr at the reference (Vostok) Science under the FY2004 JSPS Invitation Program for temperature of T' = 215.7 K with the respective Research in Japan. activation energies Qr = 70 kJ mol- 1 and Qp = 58 kJ mol-I for the creep exponent a = 3.5. The References critical densities deduced from the calculations (Tab le 2) fall within the range of 0.7~0.76 , and are [I] R.B. Alley, "Texture of polar firn for remote explained by the onset of power~creep pressure sensing", Ann. Glaciol., 9, 1987, pp. 1-4. sintering. The characteristic bend observed aner [2] R.B. 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Schwander, "The transfonnation of snow to ice and the occlusion of gases", The Environmental Appendix A. Firn-structure characteristics Record in Glaciers and Ice Sheets, H. Oeschger, and c.c. Langway, Jr. (Eds.), John Wiley and Sons, New In accordance with Ar.tt's approach [10], as York, 1989, pp. 53-67. schematically illustrated in Fig. 2, the ice-grain stmcture [66] J. Schwander, J.M Bamola, C. Andrie, development is described as concentric expansion M. Leuenberger, A. Ludin, D. Raynaud and (,growth') of centre-fixed spherical particles. The B. Stauffer, "The age of the air in the tim and the ice current 'fictitious' equivalent-sphere radius R' at Summit, Greenland", 1 Geophys. Res., 98(02), nonnalized by the initial radius R. is defined as 1993, pp. 2831-2838. (A I) [67] J. Schwander, T. Sowers, J.M. Bamola, T. Blunier, A. Fuchs and B. Malaize, "Age scale of the air in the Summing after [10] the excess volumes of spherical summit ice: Implication fo r glacial-interglacial caps "cut" from the reference spherical grain of radius temperature change", J. Geophys. Res. , 102(016), R' and distributed evenly across its free surface, as 1997, pp. 19,483-19,493. shown in Fig. 2, we arri ve at the transcendental ice [68J T. Shiraiwa and 8 others, "Characteristics of a volume conservation equation relating the radius RI' of crater glacier at Ushkovsky volcano, Kamchatka, the obtained tmncated sphere to R'. Russia, as revealed by the physical properties of ice cores and borehole thennometry", J. Glaciol., II" = II" _ Z. (II' - 1)' (2/1' + 1) _ C (II' -1)' (311 ' + 1) . 47( 158), 2001 , pp. 423-432. 4 16 (A2) [69] K. E. Smimov, "Issledovaniya razreza lednikovogo pokrova v royone stantsii Pionerskaya (Vostochnaya Correspondingly, the fraction of the free surface of Antarktida) [Studies of the cross-section of the ice the tmncated sphere of radius R" is sheet in the vicinities of Piorenerskaya Station (East Antarctica»)", Mater. GlyatsioJ. Issled. [Data of Glaciological Studies], 46, 1983, pp. 128·132. s = 1_ 2 0 R" -l_ C(R* -IY (A3) 2 R" 4 R" [70J T. Sowers, M. Bender, D. Raynaud and 15 Y.S. Korotkevich, "The & N of N2 in air trapped in An average cylindrical neck fonned at the plastic polar ice: A tracer of gas transport in the tim and a contact on the reference grain "consumes" the truncated possible constraint on ice age - gas age differences", spherical surface, the area of which IS 1. Geophys. Res. , 97(014), 1992, pp. 15,683- 15 ,697. 4nR" \ 1 - (I - (5)/Z, yielding directly the relative bond [71J M.K. Spencer, R.B. Alley and T.T. Greyts, Mea in units of R2 "Preliminary tim-densitication model with 38-site dataset", J. GlacioL, 47(159) , 2001 , pp. 671-676. [72] E.A. Tsyganova and A.N. Salamatin, "Non- stationary temperature field simulation along the ice flow line "Ridge B - Vostok Station", East The fraction of grain surface area involved in the Antarctica", Mater. Glyatsiol. Issled. [Data of grain bonds is thus Glaciological Studies], 97, 2004, pp. 57-70. [73J K.F. Voytkovskiy, Mekh -218- Consequent ly, the axial strain rate averaged over the segment thickness can be expressed via ~ as Accordingly, the total spherical surface (of ra dius R') per contact is 4/tR'''-/Z. The equivalent spherical cup has the relative base area An infinitesimal segment layer of radius r'(h') at distance h' fro m Ihe contact face defonns with strain A = The segment cut fro m this cap by the bond plane (see Fig. 2) is regarded as the icc grain contact zone in which plastic deformat ion predorninmll ly takes place. In the framework or And's scheme [10], the process The power-creep law (13) then gives of dcnsification stops at ru ll density P = I when the gra in packing is converted into a space- fill ing stack of polyhedrons. This li mit is reached at s = 0 with the correspondi ng maximum value or R" = R"~, which can be easil y ca lculated fro m Eq. (AJ) as Integration of this equation over the segment thickness with respect to h' from 0 to H' yields The corresponding max imum value of R' = R'-. obtained from Eq. (A2) dctemlines the critical density (82) in Eq. (AI) at P '" I, Le., Finally, combining Eqs. (81) and (82), the '" ~ (l1R'_ )' . relat ionship between p~ff an d ~ is obtai ned as V Appendix B. Power-law creep of contact p, ~ 3A[2 3HI'!" (I- (I - ,),.) V'~ - 219- . 011 Ov e =-+-, I at" ay -+pap, (a-+-u a'l +gplJflv(_ -b )= 0· a, i ax ay I Eq. (20) is arrived at directly. Noting that , by deflnition, Appendix D. Comparison of simulated and measured ice-core density profiles 0.9 o 0 0'lq, o o 0.7 o 0.5 0 Dome du Gouter KmSO 0 0.3 0 20 40 60 0 20 40 60 0.9 0 0 ,., 0 ·ili- 0 c 0 Q) 0.7 u 0 ~ '" Qi'" 0.5 (Y Ushkovsky Milcent 0.3 0 20 40 0 20 40 60 0.9 "" 0 0 0.7 0.5 H72 Site 2 0.3 +-~--,-~--,--~ o 20 40 o 20 40 60 Depth, m Depth, m -220- 0.9 o 0 o 0.7 0. 5 Km14 0 Crete 0.3 0 20 40 60 0 20 40 60 0.9 ,~ "'c 0.7 "'" '" ."!!1 0.5 IY'" Summit 0.3 0 20 40 60 0 20 40 60 0.9 0.7 o 0.5 Byrd Pionerskay a 0.3 +~---,-~---,-~~-,- o 20 40 60 o 20 40 60 80 Depth, m Depth, m -221- 0 0.9 0 0 ,r -» !Y o c: 0 !? '"OJ 0.7 '0 !7 OJ > "' '"OJ 0.5 0:: Vostok-1 EPI CA-DC 0.3 0 20 40 60 80 0 20 40 60 80 100 o • 0.9 0.7 00 0 0 0 0.5 0 Komsomolskaya Dome Fuji 0 0.3 0 40 80 120 o 40 80 Depth, m Depth, m -222-