Title Physical modeling of the densification of / and in the upper part of polar ice sheets

Author(s) Arnaud, Laurent; Barnola, Jean M.; Duval, Paul

Citation Physics of Records, 285-305

Issue Date 2000

Doc URL http://hdl.handle.net/2115/32472

Type proceedings

Note International Symposium on Physics of Ice Core Records. Shikotsukohan, Hokkaido, Japan, September 14-17, 1998.

File Information P285-305.pdf

Instructions for use

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Physical modeling of the densification of snow/firn and ice III the upper part of polar ice sheets

Laurent Arnaud, Jean M. Barnola and Paul Duval

Laboratoire de Glaciologie et de Geophysique de l'Environnement, CNRS, B.P. 96, 38402 St. Martin d'Heres Cedex, FRANCE x

o

20

40

60 I .r:: CL -OJ so o

100 Age at .84 density

SAE t5 data 2850 yrs Herron·Langway 2850 yrs '... ' ", Pimienta 2950 yrs , II 120 Alley-Arzt 2800 yrs , II , 1\ ----- Herron·Langway 7400 yrs ', \ ----- Pimienta 6750 yrs , 140 ----- Alley-Arzt 6450 yrs ,

0.3 0.4 0.5 0.6 0.7 o.s 0_9 density

Plalc 8. L. Arnaud el al. , Figure 3. X l

Close-off Age 3000

4000 "'" .'!l'" 0; 0 5000

6000 a) Vc=I(T), OO=I(T) ---- b) Vc=I(Gas Cont.), OO=I(T) ---- c) Vc=I(Gas Cont.), OO=I(Gas Cont. ---- d Pimienta, VC=f(T)

Close-Off depth 90

100 5 .<:: a.OJ 110 "C

120

130

o 500 1000 1500 2000 2500 Depth (m)

Plate 9. L. A rn aud et al.. Figure 7. 285

Physical modeling of the densification of snow/firn and ice in the upper part of polar ice sheets

Laurent Arnaud, Jean M. Barnola and Paul Duval

Laboratoire de Glaciologie et de Geophysique de l'Environnement, CNRS, B.P. 96, 38402 St. Martin d'Heres Cedex, FRANCE

Abstract: We propose a model for the densification of snow, fim, and ice that includes the following: the densification of isothermal snow by grain boundary sliding [Alley, 1987], the microscopic approach of Arzt [1982] for the geometrical description of porous materials during densification, and plastic deformation with a power law creep for fim and ice densification. Current density profiles are well reproduced by this physical model assuming variations of the relative density at the snow-fim transition. This density, Do, corresponds to the change from densification by deformationless restacking to densification by power law creep. Do is always lower than the theoretical packing density for mono-sized spheres (Do = 0.64) and decreases with temperature; its low values are unlikely to be an effect of the particle size distribution, but instead arise from nonuniform densification. Variations of Do with temperature are correlated with variations of the load between sites for a given density. Do could also be modified by differences in porosity structure during snow metamorphism due to different surface conditions (for instance, wind, surface temperatures, and temperature gradients). Applied to past climatic conditions of Vostok, Antarctica, the densification model predicts close-off ages and depths during the last climatic cycles. Through variations of Do parameters, uncertainties in these predictions were estimated.

1. Introduction information in the air to that in the ice it is crucial to know precisely the conditions Environmental information preserved that snow transforms into ice. The in polar ice is associated either with the ice following are needed: (a) the close-off matrix, for instance the temperature which density, (b) the age of the gas at this density, depends on the isotopic content of the ice; (c) the age of the ice, and (d) the close-off or it is associated with the air trapped in the depth. ice, as is the atmospheric composition. (a) The close-off density can be estimated Because the air is trapped at the fim-ice from measurements of the evolution of transition (the close-off depth, approxi­ closed porosity on fim samples mately 80 m below the surface), to link the [Schwander et aI., 1984, 1993; Kameda

Physics alIce Core Records Edited by T. Hondoh Hokkaido University Press, 2000, Sapporo 286 Physical modeling ofthe densification ofsnowlfirn and ice

and Naruse, 1994]. However, these and their age range can smooth the measurements were from small samples concentration of the trapped gas over a that need corrections to get the effective few centuries when the accumulation close-off density. Another way is to rate is very low (as it is at Vostok). measure the total gas content of the ice; (c) The age of the ice at the close-off is the then, when the atmospheric pressure main parameter to link the ice and gas and the temperature are known, it records. When no density measure­ allows to estimate the density at which ments are available, densification the air is isolated from the atmosphere models have to be used as discussed (in terms of pressure) [Martinerie et aI., below. 1992]. This method has been applied to (d) The close-off depth can be estimated by a large number of different , and an measuring the gravitational enrichment empirical relationship between the firn of 15N of N2 trapped in air inclusions temperature (present-day climate) and [Sowers et aI., 1992], or it can be the pore volume Vi when the air is estimated usmg firn-densification isolated has been found: models. When no data are available, the precision of the last two parameters depends on quality of the firn-densification However, gas content profiles of the model. Furthermore, recent studies use Vostok core suggests that this empirical independent information to determine the relationship might not be valid for all air dating or trapping depths, and then use a the past climatic conditions, and also densification model to get climatic that wind could affect the pore volume information. For instance, Schwander et ai. at the isolation depth [Martinerie et aI., [1997], working on the GRIP core, used

1994]. CH4 to get a gas chronology for GRIP and (b) The age of the air in the firn, and thus at 15N to get close-off depths, and then ran a the close-off depth, has been studied by densification model with different climatic diffusion models validated by measure­ parameters to fit the data and thus calibrate ments of the composition of interstitial the glacial to interglacial temperature firn air. The gravitational separation of change m Greenland. Similarly, the

different elements that causes an Greenland CH4 records of GISP or GRIP, enrichment of the heavier compounds which are of global significance, are now as a function of firn depth were widely used to synchronize Antarctic and included. This allows one to reproduce Greenland ice cores through densification the concentrations of atmospheric models [Blunier et aI., 1998; Steig et aI., compounds measured in the fim air 1998]. These examples show that firn­ [Schwander et aI., 1993; Trudinger et densification models should both reproduce aI., 1997], and also to reconstruct their the present- day situations and predict firn atmospheric history using an inverse profiles under climatic conditions having no method [Rommelaere et aI., 1997]. present-time analogues. The Vostok total These models showed that the air is few gas content profile [Martinerie et aI., 1994] tens of years old at the base of the firn, shows that an empirical relationship based L. Arnaud et al. 287

on present-day data might not be valid for and the depth at which this density is the past. This stresses the need for a reached increases as the temperature physical model instead of an empirical one. decreases [Anderson and Benson, 1963; We present a physical model based on Gow, 1975; Herron and Langway, 1980]: at pressure sintering and apply it to past -57°C it is approximately 30 m [Alley, Vostok climatic conditions. 1980] and at -23°C it is approximately 12 m [Herron and Langway, 1980]. In addition to the dependence on temperature, the 2. Densification model accumulation rate for a given temperature can also affect the depth of the critical The formation of ice in polar ice sheets density [Herron and Langway, 1980; results from the densification of dry snow Nishimura et aI., 1983] and firn. This densification is similar to the The second stage is firn (consolidated hot pressing procedure in ceramics and snow). Because seasonal variations of metals in which pressure is applied during surface temperature disappear, the trans­ sintering at high temperature, except that on formation of firn into ice can be considered ice sheets the pressure is due to the as isothermal. The end of this stage is accumulation of snow at the surface. characterized by the growth of the Three stages of densification are impermeable pore space fraction under the generally distinguished. During the first increasing overburden pressure. The stage, the densification of snow is mainly a average number of bonds per grain structural rearrangement of grains by grain­ (coordination number) increases with boundary sliding. Sintering is driven by the density from approximately 6 at the start of temperature gradient in the first few meters this stage to 16 for ice [Gow, 1969, this and by surface tension. Transport work]. Plastic deformation increases this mechanisms such as evaporation- coordination number and causes neck condensation and surface diffusion growth; particle motion is negligible. contribute to rounding of the grains and to During the final stage, atmospheric air intergranular bonding [Benson, 1962; Gow, is trapped in cylindrical or spherical pores; 1975]. The rearrangement of unbounded further densification of this bubbly ice is grains dominates the densification of highly driven by the pressure lag between the ice porous snow. Alley [1987] proposed a first matrix and the air in bubbles [Maeno and model for densification by grain boundary Ebinuma, 1983; Pimienta, 1987; Alley and sliding for snow below 2-m depth where Bentley, 1988; Salamatin et aI., 1997]. temperature-gradient effects are un­ Numerous empirical models de­ important. From Anderson and Benson scribing the densification of snow, firn and [1963], this particle rearrangement is ice that fit experimental density profiles complete at a critical density of about 0.55 have been developed [Bader, 1962; g/cm3, corresponding to a relative density of Anderson and Benson, 1963; Kojima, 1964; 0.6, which can be regarded as the maximum Herron and Langway, 1980; Kameda et aI., packing density. Moreover, according to 1994]. They were established and validated Benson [1962] and Arnaud [1997], this with depth-density data from many sites in critical density increases with temperature, Greenland and Antarctica. 288 Physical modeling ofthe densification ofsnowlfirn and ice

In the last few years there was a major contacts and the growth of the average advance in our understanding of the contact area was developed by Arzt [1982] mechanisms which contribute to densi­ and Fischmeister and Arzt [1983]. The fication during hot-powder compaction. continuous increase in coordination is Hot-isostatic pressing maps were made for modeled by assuming a random dense­ various materials. Dominant mechanisms of sphere packing of monosized spheres with a densification were identified and densi­ radial distribution function [Scott, 1962; fication rates were calculated as a function Mason, 1968]. Densification is modeled by of pressure, temperature and powder allowing each (spherical) particle to characteristics [Ashby, 1990]. It was shown increase in radius around fixed centers. As that for the densification of firn and ice, each particle grows, both the number of power-law creep is the controlling bonds per grain and the average area per mechanism in the second stage and for the contact increase. The coordination number first part of the final stage [Maeno and Zo for a random dense packing of spheres Ebinuma, 1983; Arzt et ai., 1983; Ebinuma equals 7.3 with Do (the relative density) and Maeno, 1987; Wilkinson, 1988]. equals 0.64. At Do the area of each contact Newtonian creep becomes the dominant is assumed to be zero. For the final stage densification mechanism of bubbly ice as with closed pores, densification is soon as the effective pressure, defined as determined by the creep of the thick shell the difference between the ice pressure and surrounding each hole [Wilkinson and the pressure within bubbles, becomes lower Ashby, 1975]. than 0.3 MPa [Pimienta and Duval, 1987; One purpose of this work is to validate Lipenkov et aI., 1997]. Concerning the approach proposed by Arzt [1982] for structural evolution during the second stage simulating the depth-density profiles of firn of densification, the Wilkinson and Ashby and ice in polar ice sheets. This model is [1975] model assumed a constant size and used with the snow densification model of number of contact per particles and Alley [1987] to describe densification in the spherical pores which are questionable entire upper part of polar ice sheets. This assumptions [Swinkels et ai., 1983]. This physical densification model helps to problem was partially solved for polar ice understand how the initial snow structure studies by Barnola et ai. [1991] by and climate causes differences of firn introducing an empirical function into the densification between sites. In the last Wilkinson and Ashby model that includes section, this model is used to reconstruct all structural variations during densification. past density profiles. However, the physical meaning of this function is not clear, and the extrapolation 2.1 Density profiles in polar snow, firn to other past climatic conditions (e.g., and bubbly ice climatic or glacial transitions) might not be We will focus the analysis of valid. densification profiles on two Antarctica A microscopic approach for the geo­ sites: Byrd [Gow, 1968] and Vostok metrical theory of pressure sintering of [Barkov, 1973]. The main characteristics of mono-sized spherical powders that includes these sites are given in Table 1 and their the continuous formation of particle density profiles are plotted in Fig. 1a. For a 1. Arnaud et al. 289

Table 1: Main characteristics of the studied sites in Antarctica and Greenland.

DeOS Site 2 D47 Byrd DomeC Vostok Mean Surface Temperature -19°c -23.3"c -25.4°c -28°c -54°c -57°c (Oc)

Accumulation rate 1.1 0.41 0.25 0.16 0.034 0.022 (Mg m· 2 yr')

Load pressure at 0.55 Mg m'; 0.035 0.06 0.05 0.063 0.109 0.123 (MPa)

Location 66° 43' S 76" 59' N 67° 23' S 79° 59' S 74° 3~' S 7So 2S' S 113°11'E 56° 04' W 138° 43' E 120° OI'W 123° 10' E 106°48'E References (Etheridge amI (Langway. 1967) (Arnaud and (Gow.1968) (Alley. 1980) (Barkov. 1973) Wookey. 1989) others. 1998) given density, the overburden pressure is depth, the shallowest depth where higher at Vostok station than at Byrd. Table temperature gradient effects become 1 shows similar results for other sites; for a unimportant [Alley, 1987). The den­ given density of 0.55 g/cm3, the overburden sification of isothermal snow is mainly a pressure is higher at the colder sites. structural rearrangement of grains by grain The effect of the accumulation rate is boundary sliding. Alley [1987] proposed a shown in Fig. Ib, which gives the first physical model for densification by relationship between the density and the grain boundary sliding. To be consistent age of ice. At Vostok, the transition from with the fim densification model (which fim to ice, corresponding to the end of the has one parameter), the model of Alley has pore closure process, was observed for a been slightly modified to have also only 3 density of about 0.84 g/cm . The age of the one parameter. (See §2.4 for the snow to ice at this transition is around 2800 years. fim transition.) At Byrd, the fim to ice transition is The densification rate given by Alley observed for a density of 0.825 g/cm3 and [1987] for grain boundary sliding can be the age of ice is only 250 years [Bamola et written, aI., 1991). dD/dt = y(AID2)(1 - (5/3)D)t , (2) 2.2 Equations governing densification of isothermal snow where y= R/(vr) . R and r are the grain and Although the snow metamorphism at the bond radii, respectively, and can be the surface due to temperature gradients or assumed constant during this first stage of mechanical effects (wind) must cause some densification. The parameter y includes all differences of densification between sites constant quantities: the viscosity of the (formation of the initial structure of the grain boundaries v (assuming isothermal material before densification), this study is conditions), and the geometrical parameters limited on densification modeling. Then, Rand r. The constant y is set such that the the densification model starts at the 2-m densification rate is continuous at the 290 Physical modeling ofthe densification ofsnowlfirn and ice

Density (g/cm3) 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0

25

50 E'-' ..c::..... 75 0.. + ~ Cl 100 • 125 a • • 1000 •• •• .... -----C":j •• ~ 2000 •• » •• '-"' • ~ c bll 3000 + Byrd T= ·28 c ct 11(= 16 g/cm'/yr •• <:t:: ••• 4000 • Vostok T= ·57"<, et ;\c= 2.2 g/cm2/yr \ 5000 0.30 0040 0.50 0.60 0.70 0.80 0.90 1.00 Density (g/cm3)

Figure 1: Depth-density (l.a) and time-density (l.b) profiles for two sites: Vostok and Byrd. transition between snow and firn (see §2.3 Second stage: tirn (0.6 < D < 0.9) and 2.4). Equations for the densification rate of firn of initial density Do are directly derived 2.3 Equations governing densitication of from the geometrical model developed by tirn and bubbly ice Arzt [1982]. This rate is: When grain sliding becomes negligible in the model, the densification rates are obtained by assuming that power-law creep IS the densification mechanism. The where p* = 41iP/aZD, a is the average constitutive equation is: contact area in units of R2 (R is the initial particle radius), P equals the ice load (3) pressure, p* is the real pressure acting on an average contact area, and Z the coordination number at the relative density where e and (j are the equivalent strain rate D. The driving force due to surface tension and stress, respectively. A and n are the is always very low compared to that creep constant and the stress exponent, resulting from the external pressure [Gow, respectively. 1968]: for a typical pore radius of 0.1 mm, the pressure due to the surface tension is L. Arnaud et al. 291 about 1 kPa (i.e., very low compared to the dD/ dt = 2A {o(I- D )/[1- (1- D yin t j2Peff In r . ice pressure). (5) The stress exponent n of the flow law is assumed to be 3 [Ebinuma and Maeno 1987]. The temperature dependence of th~ The effective pressure, Peff, now equals creep constant A is Arrhenius; we assumed P - Pb, where Pb is the pressure in the bubbles. This pressure is calculated from A(1) = 7.89x 103exp(-QIR1) , the conditions at the close-off (i.e., relative density Dc and atmospheric pressure PJ and where Q, the activation energy, was the Boyle-Mariotte equation: deduced from the densification of bubbly 1 ice [Pimienta, 1987], to equal 60 kJ-mol- . R and T are the gas constant and the absolute temperature, respectively. As the density increase, the ratio of the We could not use the approximation cylindrical to spherical bubble number proposed by Helle et aI. [1985] because it decreases continuously. At Vostok station predicts Zo = 7.7 with Do = 0.64 and Z = 12 (East Antarctica), cylindrical bubbles are when full density is reached; whereas in our observed down to 150 m (Fig. 4). Also, the case, the relative density at maximum pressure in the bubbles increases and the packing was always less than 0.64 and effective pressure decreases with depth. likely dependent on temperature [Benson, Consequently, densification can be modeled 1962; this study]. Therefore, we used the as deformation of spherical pores during the more complete equations of Fischmeister last part of the final stage (D>0.95), and and Arzt [1983] to calculate variations of Z because the stress will become lower than and a versus D (see Appendix). To obtain 0.3 MPa, the stress exponent of the flow the initial coordination number Zo for law equals 1 [Pimienta and Duval, 1987]. various values of Do, we assumed that p* Thus, the densification rate becomes approaches P as D tends towards 1. dDldt = (9/4)A(l - D)Peff . (6) Final stage: bubbly ice (0.9 < D < 1) At the bottom part of the fim, the open During the transition from n = 3 to n = pores begin to be isolated and can be 1 (or cylindrical to spherical pores), the assimilated to cylinders. Just below the model calculates the two densification rates close-off depth, the effective pressure is and uses the more rapid one. greater than 0.3 MPa [Lipenkov et aI., 1997]; consequently, the stress exponent Comparison with experimental data can be assumed to be 3 [Duval and From equations (4), (5) and (6), the Castelnau, 1995]. The densification rate is densification rate can be expressed as a given by the deformation of ice shells function of the effective pressure as surrounding cylinders [Wilkinson and follows: Ashby, 1975]: dD/dt =A· f(D).Peff . (7) 292 Physical modeling of the densification ofsnowljirn and ice

10000

0 0 DE08

1000 - 0 Site 2

X D47

6- Byrd 100 ~ DomeC

Vostok ~ + Q '-' 10 '+-<

Do =0.64

Do=0.57 0.1 Do =0.51

0.01 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 Relative Density

Figure 2: Experimental function f versus relative density for six sites from Greenland and Antarctica (Table 1). The theoretical curves giving the variation ofj{D) with density were obtained for three values of Do.

The experimental values of j(D) were data is lower than that for the random, calculated from equation (7) and dense-packing of mono-sized spheres (0.64). densification profiles of several sites in As shown in Fig. 2, variations of Do Antarctica and Greenland (c.f. Table 1). between 0.51 and 0.64 are necessary to Equations (4), (5) and (6) for firn and reproduce the densification profiles of firn bubbly ice were used to model the variation between -19°C to -57°C and Do appears to ofj(D) (Fig. 2). decrease with temperature. The experimental data (Fig. 2) agree Density-depth profiles simulated for fairly well with the predictions of the theory Vostok present-day and glacial conditions of densification behavior of Fischmeister with the empirical model from Herron and Arzt [1983] and Wilkinson and Ashby Langway [1980], the semi-empirical model [1975]. However, there are small deviations of Pimienta [Barnola et aI., 1991] and the from the theoretical curve at the beginning geometrical model (this work) are of the second stage. Furthermore, the initial compared in Fig. 3. For the present-day density Do that best fits the experimental conditions, the best agreement with the L. Arnaud et al. 293

o

20

\,,,, 40 \\ \ "\\ \ \\ \ \\ \ \ \ \ \ \ , \ \, 60 \\, .- \ \, E \ " -.c \" c- ,\ -al 80 0 \ '\

100 Age at .84 density

SAE 15 data 2850 yrs Herron-Langway 2850 yrs "\ Pimienta 2950 yrs 120 "\~ Alley-Arzt 2800 yrs \~ \ ~ \ \ Herron-Langway 7400 yrs ----- \ ----- Pimienta 6750 yrs \ 140 ----- Alley-Arzt 6450 yrs \

0.3 0.4 0.5 0.6 0.7 0.8 0.9 density

Figure 3: Density profiles simulated by different models for present-day (continuous lines) and glacial conditions (dotted lines) compared to the field SAE 15 data [Barkov, 1973]. (See color plate 8.) measured profile [Barkov, 1973] is ob­ last glacial maximum (T = -67°C and ac tained by the geometrical model with Do = 1.25 g/cm2yr). For the latter conditions, 0.53. However, at the density of 0.84, the which have no present equivalent, the depth three models predict similar depths and of the 0.84 density varies from 140 m with ages for the present-day conditions, the empirical approach of Herron and although this is not the case when the Langway to 127 m with the geometrical simulations are done with conditions of the model. These correspond to ages of ice 294 Physical modeling ofthe densification ofsnow/jirn and ice ranging from 7400 years to 6450 years. The atthe surface of the ice sheet. difference between predictions from the The technique used to characterize the Pimienta and the geometrical models comes structure of snow, firn, and ice is based on mainly from the choice of Do; in the former, photographs of polished surfaces of thick Do is fixed at 0.57, but in the latter it varies samples using reflected light [Arnaud et aI., with temperature. 1998a]. This new observation method simul-taneously allows the determination of 2.4 Structure of snow, tirn and ice the porous network and grain boundaries An advantage of the geometrical from a single image. As a second step, model is the possibility of linking image processing of these photographs densification to the structure of the material allowed determination of the parameters to understand the influence that climate defined above [Arnaud, 1997; Gay, 1999]. parameters have on densification processes Figure 4 shows the evolution of the porous

-=:~.---=-~- - (a) 19.8 m. 0.5 g/em'

(e) 97.R 111,0.1\4 giCln\ (J) t 20 111, 0.88 g/em'.

Figure 4: Binary images obtained by image processing of photographs obtained in reflected light from 3 3 3 sections of the Vostok core. (a) 19.8 m, 0.5 g/cm ; (b) 45m, 0.65 g/cm ; (c) 97.8 m, 0.84 g/cm ; (d) 120 m 0.88 g/cm3 L. Arnaud et al. 295 structure from snow to ice in the Vostok ice The change in the average fraction of core. the surface area of grains involved in bonds Variations of packing density within a (fJ) with density at Vostok (Fig. 5) agrees compact powder are common; low density with data obtained by Alley and Bentley regions containing large pores can develop [1988] and complements the direct during the early stage of sintering as a observations shown in Fig. 4. More than result of nonuniform densification. The 40 % of the surface area of grains is non-uniform structure of fim is illustrated involved in grain boundaries at the relative in Fig. 4. The average contact area between density of 0.6, whereas the theoretical value ice is significant at the theoretical must equal zero at the beginning of the fim initial density Do. Some grains without any densification. contact with pore can be seen in the two­ Snow grains are generally composed dimensional structure offim in Fig. 4. of polycrystalline ice grains and this texture In addition to these qualitative is retained during the metamorphism [Fuchs, observations, a quantitative validation of 1959; Sommerfeld and La Chapelle, 1970; the geometrical densification model of Arzt Arons and Colbeck, 1995]. The formation was obtained from this structural of grain bonds during the densification of characterization of polar fim [Arnaud et al., snow could result from its structure either at 1998b]. The parameters necessary to test deposition time or just after the densification models include the average metamorphism in the top few meters depth. number of bonds per grain (Z) and the In fact an explanation for the presence of average contact area relative to the grain grain bonds at the stage of snow size (a). These parameters cannot be compaction is plastic deformation of ice calculated directly with two-dimensional grains under the overburden pressure. Both structural analysis. Therefore, a different deformation and grain boundary sliding structural parameter was chosen for this would occur concurrently at the beginning study: the surface fraction of the average of compaction. There will certainly be grain involved in grain bonds (fJ). This small regions where particle rearrangement parameter is also known as the contiguity without deformation is prevalent, but also factor [Underwood, 1970], and can be other regions where flat areas form by determined using two-dimensional analysis. plastic deformation. Thus, the transition In this case, f3 is related to the specific from snow to fim corresponds to a change surface area of the grain-pore interface in the dominant densification mechanism. (Sv(g-p» and the specific surface area of the Particle rearrangement by sliding is the grain boundary (Sv(g-g»: dominant densification mechanism in snow, and power-law creep is the dominant one in fim, but both mechanisms must operate in a density range close to the transition Do. The From the model structural parameters (Z physical meaning of the initial density Do is and a), f3 is given by: described in more detail by Arnaud et al. [1998b]. f3 = Za/41r. Figure 5 also suggests that the particle in the geometrical densification model of 296 Physical modeling ofthe densification ofsnowlfirn and ice

l.00

Envelope of data for 3 sites from Alley and Bentley (1988) 0.80 • Vostok

/ /

/' / / / 0.60 .. ",./. / / '" 111. /

/' 0.40 "'. / / • • /

0.20

model with Do= 0.53 I , 0.00 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Relative density

Figure 5: Average fraction of the surface of crystals involved in bonds (ft) for Vostok, and envelopes of the data from three other sites from Alley and Bentley [1988]. Model curves with Do = 0.53.

Arzt [1982] should contain more than one the parameter (1 - f3a), which represents the grain. From the success of this densification fraction of free surface area of the average model for describing firn (Fig. 2), we have aggregate, is assumed that at Do the particle is not a grain, but an aggregate of grains to make the contact area between aggregates equal zero at Do. This is physically compatible with In Fig. 6, the dependence of this the occurrence of power-law creep during parameter on relative density is compared the densification of snow, even if this with that obtained by assuming there is no densification mechanism (power-law creep) aggregate. These structural data are also is not yet dominant. compared with model curves. The results, Assuming that densification by plastic shown in Fig. 6, are compatible with a deformation involves groups of crystals description of the firn as a group of (aggregates), not single grains, the aggregates. Moreover, the structural parameter f3a for the aggregate must equal evolution observed during densification is zero at Do. Svo(g-p) is the specific surface well reproduced by the geometrical model area of the grain-pore interface at Do; this of Arzt. value also equals the specific surface area of the aggregate-pore interface. Assuming a constant average surface area of aggregates, L. Arnaud et al. 297

1.00 • Vostok

0.80 • -•• cs: 0.60 .. I P ...-< ~) () '--' ',) ;-) ~~ OAO <. i

C~ (j -. C) 0.20 c· --- model with Do= 0.53

0.00 0.60 0.70 0.80 0.90 1.00 Relative density

Figure 6: Fraction of free surface area of grain (e) and aggregate (0) versus relative density for Vostok (Bh7). Model curves with Do = 0.53.

2.5 The snow-tiro transition: Do density profiles as a function of depth The value of Do = 0.64, deduced shows that the depth where the relative assuming a random dense-packing of density equals 0.6 increases as the mean mono-sized spheres for the initial particle annual temperature of the site decreases structure, is probably too high for polar fim. [Nishimura et aI., 1983]. For a 40°C The distribution of particle sizes and shapes decrease of the mean annual temperature affects the parameters Do and Zo [Bouvard (i.e., the difference between Vostok and and Lafer, 1990]; for instance, Do should be De08 sites), the overburden pressure larger with spherical particles of different corresponding to the relative density of 0.6 sizes [Aparicio and Cocks, 1995]. With the is multiplied by about 4 (Table 1). This model of Artz, densification by power-law change is well reproduced by the model of creep is supposed to occur as soon as the densification by grain boundary sliding dense packing of spheres is reached. The established by Alley [1987] for the first packing density Do that best fits the model stage of densification. Grain-boundary corresponds to the end of grain-boundary sliding is proportional to stress (Newtonian sliding as the significant densification process), which contrasts with the flow law mechanism. As a result, the theoretical exponent of 3 for particle deformation. maximum packing density cannot be Thus, concerning the effect of pressure, the reached. density Do at the transition between the first The observed variation of Do with and the second stage should decrease with location would also result from the temperature. Hence, power law creep competition between grain boundary sliding should prevail at a lower density in cold and power-law creep. The comparison of sites. Probably, this effect partly causes the 298 Physical modeling ofthe densification ofsnowlfirn and ice decrease of Do with site temperature. But, it 3. Reconstruction of the past density is also necessary to include the variation of profiles ice viscosity with temperature. The activation energy for the grain boundary A common application of a densi­ viscosity is about 42 kJ-mol-1 [Alley, 1987], fication model is to determine the and about 60 kJ-mol"1 for the power-law difference between the age of ice and that creep. The variation of the activation of the gas, and the close-off depths during energy with densification processes the past to compare the information trapped therefore partially counteracts the load in the bubbles to those linked with the ice. effect. In support of this, including both the Thus we have to extrapolate the empirical effects of load and temperature, Arnaud et links between temperature and close-off ai. [1998b] showed that in the cold site the density or critical density to previous power law creep process becomes the periods. dominant densification mechanism at lower In Fig. 7 are plotted the age of the ice density. The final effect is a decrease with and the close-off depth as reconstructed for temperature of the relative density Do, the first 2500 m of the Vostok core. (This corresponding to the transition between includes the last two glacial-interglacial snow and firn. This agrees with the results climatic cycles.) Results from the in Fig. 2. Moreover, Benson [1962], who densification model of Pimienta and from directly measured the "critical density" our geometrical model with different between the densification stages, observed a assumptions are shown. For all cases, the range of values from 0.62 to 0.56 for the temperatures and accumulation rates are temperature range from -16°C to -30 0c. those adopted for the GT4 chronology From the available data [Arnaud, 1997], Do [Petit et aI., 1999]. Calculations were made could be related to the temperature by: in a Lagragian reference, and the temperature perturbations were assumed to Do = 0.00226T (K)+ 0.03 . (8) propagate rapidly through the firn layers. For the geometrical model, we assumed the This relationship is empirical; thus as following for cases a) - c): for the close-off density, it could have been a) Do varies with temperature following different in the past. Also, Arnaud [1997] equation (8), and the close-off density proposed that the structure of the material at changes according to equation (1). this density influences the structural b) Do varies with temperature following properties of the firn, particularly those equation (1), and the close-off density determining the close-off porosity. In this follows the total gas content changes case, the empirical relationship between from Martinerie et ai. [1994]. In this this last parameter and the temperature case, we assumed that equation (1) was (equation (1» would come from the not valid during the past and that all physical relation between Do and T, and we changes in the total gas content should have: recorded in the Vostok core were due to changes in the close-off density. Vc = 0.319Do - 0.059. (9) (Similarly, to explain the Vostok gas content profile, Martinerie et ai. [1994] L. Arnaud et al. 299

Close-off Age 3000

4000 co Jl1

6000 a) Vc=f(T), ---- b) Vc=f(Gas Cont.), DO=f(T) ---- c) Vc=f(Gas Cont.), DO=f(Gas

Close-Off depth 90

100

'-'E ..c -as- 110 "0

120

130

o 500 1000 1500 2000 2500 Depth (m)

Figure 7: Close-off age (upper panel) and depth (lower panel) reconstructed for the [lIst 2500m of the Vostok core. a, b and c curves correspond to different assumptions in the geometrical model (see text) and d to the Pimienta model. (See color plate 9.) 300 Physical modeling ofthe densification ofsnowljirn and ice

evoked a change in altitude and in the about 25 %; similarly, the depth variations temperature-close-off density relation­ between the different assumptions are also ship.) However, for a sensitivity test we relatively large. Therefore, it appears that assumed that the atmospheric pressure the close-off depths are relatively less remained constant during the past constrained than the close-off ages. 220,000 years, even though this is One advantage of the geometrical unlikely. model for comparing gas and ice c) Do and the close-off density follow the information is that it predicts uncertainties gas content changes with Do linked to in the close-off ages and depths of about the close-off density by the equation (9). 10 %. To reduce this value it will be In this case, we assumed that equation necessary to better understand the influence

(8) also was not valid in the past. of climate on Do, Vc, and more generally, on The age of the ice at the close-off the firn structure. In this context, the follows pretty well the climatic variations development of tools to get independent and varies between 2000 and 3000 years estimates of close-off ages and depths are during the interglacial periods (0 to 300-m very promising. For example, Nitrogen and and 1650 to 1900-m depths) up to 6750 Argon isotopic measurements allow, years during the coldest periods. Without through the thermal diffusivity, comparison additional constraints, it is difficult to of the same temperature signal both in ice decipher between the different cases; thus, and in bubbles, and thus can indicate the the maximum uncertainty in the close-off firn thickness [Severinghaus et ai., 1998]. age determination is about 650 years (during the full glacial periods), and is always less than 10 % of the close-off age. Conclusion However, we argue that the geometrical model is better suited than the Pimienta We constructed a physical densi­ model for predicting close-off ages under fication model of snow, firn, and ice on the different climatic conditions. In this case, basis of grain sliding in the first stage and the age from our model is about 400 years then by plastic deformation and structural younger during the coldest periods than that evolution of a porous material in later previously assumed, and the associated stages. Differences in initial firn structures uncertainty, taken as the shaded area in Fig. are included through the density of the 7, is slightly less than 10 %. snow-firn transition, Do. Empirically, this The close-off depths also follow the density depends on temperature for which climatic variations. However, with the we propose an explanation. We have also assumption in the geometrical model that suggested a link between this density and Do and Vc follow the gas content changes the close-off density, and reconstructed past (case c), the close-off depth during some close-off depths and ages for the Vostok cold periods (around 900 m and 1500 m) core. However, the development of new can be as shallow as those during warm measurements to recover past structural periods. Also, while the age increases by a parameters, or new tracers based on factor of two between interglacial and isotopic gas composition are needed to glacial periods, the depth change is only L. Arnaud et al. 301 assess the validity of the empirical new mean radius of the truncated sphere R" relationships based on present conditions. (in units of the initial particle radius R) is calculated from this excess volume assuming mass conservation. It is Appendix R"=R' Average contact area (a) and [4Zo (R' -1)2(2R' + 1)+ c(R' _1)3 (3R' + 1)] coordination number (Z) versus relative + . density (D) [12R'{4R' -2Zo(R' -1)-c(R' _1)2}] The modeling of the particle geometry (A-3) and calculations are from Arzt [1982] and Fischmeister and Arzt [1983]. Powder The average contact area a (in unit of particles are assumed spherical with the R2) is obtained by averaging over all same radius, and the initial geometry existing contacts: Zo contacts of maximum approximates that of random dense packing. size, plus some progressively smaller ones. The treatment of homogeneous den­ This average area is sification is easier if, instead of letting the center-to-center distances shrink, we let the particles grow around their fixed centers. a(D)=a(R") The new particle radius R' (in units of = (n/3ZR,z13(R"z -1)z0 + RN2 c(2R" - 3) + c ]. the initial particle radius R) after this (A-4) fictitious growth is

(A-I) Acknowledgements After growth to this new radius, some particles overlap. The number of these This work was supported by the overlaps is calculated using the radial Programme National d'Etude de la density function (RDF) for random, dense­ Dynamique du Climat (PNEDC), CNRS, packing [Scott, 1962; Mason, 1968]. The and by Environment Program of the average number of sphere centers located Commission of European Communities within radius r = 2R' from the reference (CEC). We acknowledge the Russian sphere is equal to the average coordination Antarctic Expeditions, the Division of Polar number Z: Programs of the US National Science Foundation (NSF), and IFRTP for their Z(D) = G(2R') = Zo + C(R' - 1), (A-2) logistic support at Vostok station. We thank A. Manouvrier for the ice drilling and M. where G(r) integrated radial density Gay for the image processing. function. Moreover, the overlap produces an excess volume of material that is distributed uniformly over those parts of the sphere surfaces that are not in the contacts. So, the 302 Physical modeling ofthe densification ofsnowlfirn and ice

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