The ice microstructure (grain boundaries) is a key feature used to study ice
evolution and to investigate past climatic changes. We studied a deep ice core,
in Dome Concordia, Antarctica, which records past mechanical deformations. We
measured a "texture tensor" which characterizes the pattern geometry and
reveals local heterogeneities of deformation along the core. These results
question key assumptions of the current models used for dating

Anderson M. P.

Arnaud L.

Azuma N.

F Graner

G Durand

Gay M.

Hammer C. U.

J Weiss

Mecke K.

Weiss J.

English

arXiv

The ice microstructure (grain boundaries) is a key feature used to
study ice evolution and to investigate past climatic changes. We
studied a deep ice core, in Dome Concordia, Antarctica, which
records past mechanical deformations. We measured a “texture
tensor” which characterizes the pattern geometry and reveals
local heterogeneities of deformation along the core. These
results question key assumptions of the current models used
for dating

G. Durand

F. Graner

J. Weiss

EDP Sciences OAI-PMH repository (1.2.0)

Deformation of grain boundaries in polar ice

The ice microstructure (grain boundaries) is a key feature used to study ice evolution and to investigate past climatic changes. We studied a deep ice core, in Dome Concordia, Antarctica, which records past mechanical deformations. We measured a "texture tensor" which characterizes the pattern geometry and reveals local heterogeneities of deformation along the core. These results question key assumptions of the current models used for dating

Durand, Gaël

Graner, François

Weiss, Jérôme

HAL-INSU

Deformation of grain boundaries in polar iceGae¨l Durand, Franc¸ois Graner, Je´roˆme WeissTo cite this version:Gae¨l Durand, Franc¸ois Graner, Je´roˆme Weiss. Deformation of grain boundaries in polar ice.2004. <hal-00000573v3>HAL Id: hal-00000573https://hal.archives-ouvertes.fr/hal-00000573v3Submitted on 22 Jul 2004HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00000573, version 3 - 22 Jul 2004Europhysics Letters PREPRINTDeformation of grain boundaries in polar iceG. Durand1, F. Graner2[∗], and J. Weiss11 Laboratoire de Glaciologie et de Ge´ophysique de l’Environnement (FRE 2192 CNRS- Universite´ J. Fourier Grenoble 1), BP 96, F-38402 Saint Martin d’He`res cedex, France2 Laboratoire de Spectrome´trie Physique (UMR 5588 CNRS -Universite´ J. Fourier Greno-ble 1), BP 87, F-38402 Saint Martin d’He`res cedex, FrancePACS. 83.80.Nb – Geological materials: Earth, magma, ice, rocks, etc... .PACS. 81.70.-q – Nondestructive materials testing and analyzis: optical methods.PACS. 62.20.Fe – Deformation and plasticity (including yield, ductility, and superplasticity).Abstract. –The ice microstructure (grain boundaries) is a key feature used to study ice evolutionand to investigate past climatic changes. We studied a deep ice core, in Dome Concordia,Antarctica, which records past mechanical deformations. We measured a “texture tensor”which characterizes the pattern geometry and reveals local heterogeneities of deformation alongthe core. These results question key assumptions of the current models used for dating.Motivations. – Polar ice cores are the focus of many investigations because they recordthe history of climatic changes. Owing to snow accumulation, snow to ice transformationand slow ice sheet flow (∼ 10−12s−1), a journey down to the deep layers of the ice sheet is ajourney back to several hundred of thousands of years into the past [1].A crucial step of paleoclimatic studies from ice cores is dating. In Antarctica, countingannual layers is impossible [2]: absolute dating is only possible for the very top of the ice coreswhere ice layers containing volcanic impurities can be related to historical volcanic eruptions.Below, dating relies on ice sheet flow models of the evolution of ice layer thinning with depth[2]. Such models are loosely constrained by the identification of large climatic transitions. Forthe sake of simplicity, these models assume a smooth and monotonous increase of the thinningwith depth, hence ignore any possible localization of the deformation [2, 3].In this letter, we question this essential assumption. We present a method to extractgeometrical information (such as thinning, shear, localization of the deformation) from picturesof a cellular pattern using local spatial averages of the “texture tensor” [4]. We apply thisanalyzis to the grain boundaries (the so-called “microstructure”) of ice samples from a deepice core.Samples. – Dome Concordia, Antarctica (75◦ 06’ 04” S, 123◦ 20’ 52” E, elevation 3233m a.s.l) is at the summit of an Antarctic ice dome. It has been chosen because it is usuallyassumed that the ice flow is axisymmetric around the vertical (z) axis, and isotropic within thec© EDP Sciences2 EUROPHYSICS LETTERShorizontal plane. In February 2003, the European ice core drilling program (EPICA) reachedthe depth of 3201.65 m below the ice surface, close to the ground (depth 3309 ± 22 m).Samples have been extracted and transported according to standard procedures [5]. Weanalyzed ∼11 cm high, ∼ 0.1 mm thin sections (Fig. 1), with a sampling interval varyingfrom 2 to 25 m. The vertical axis on a thin section always corresponds to the true ice corevertical axis z; its horizontal axis, labelled x in what follows, has an unknown and variableorientation within the horizontal plane. The tilt of the core axis from the in situ vertical axiswas below 0.5◦ above z = 2300 m, then increased up to 5.2◦ at 3100 m.Data analyzis. – By processing pictures of ice under crossed polarizers, we determine thegrain boundaries [5], then the sites (“vertices”) where three boundaries meet, see Fig. (1a).For each pair of neighbouring sites, we draw the vector which links them and denote it by~ℓ. We then construct the tensor ~ℓ ⊗ ~ℓ: its coordinates are (ℓiℓj), where i, j are here x or z;this tensor is not sensitive to the sign of the vector (it is invariant under ~ℓ → −~ℓ), but itcharacterizes its length ℓ and its direction.To statistically characterize the pattern, ref. [4] proposed to define the texture tensor M ,as the average of ~ℓ ⊗ ~ℓ over a box of fixed size. Such box should be smaller than the image(in order to visualize local details), but still large enough to include a number N ≫ 1 of wallvectors (to have relevent statistics) [6]. Here, grains have variable size (grains grow with time,hence old grains in deep ice are much larger than young grains near the dome surface): itwould be difficult to select such a fixed box size.We thus chose a local averaging and define M at each given site as:M =〈(ℓ2x ℓxℓzℓzℓx ℓ2z)〉p=1NN∑k=1~ℓ(k)⊗ ~ℓ(k) = R(λ1 00 λ2)R−1. (1)Here, R is the rotation which diagonalizes M , and (λ1, λ2) are the corresponding eigenvalues;〈·〉p denotes the average over N vectors, up to the p-th neighbours (Fig. 1a). Hence, at eachsite we include approximately the same number of vectors, N ≈ 3 + 6 + ... + 3 × 2p−1. Asmaller p explores local details, a larger p (hence a larger scale) improves the statistics.This tensor, independent from cristallographic information, does not require any knowledgeor hypothesis regarding the past history of the material. The diagonal components Mxxand Mzz of this tensor are both of order of the average square distance〈ℓ2〉between sites.Conversely, the off-diagonal component Mxz = Mzx is much smaller, and even vanishes whenthe pattern is isotropic. Hence λ1, λ2 > 0: the largest eigenvalue corresponds to the directionin which grains are most elongated (Fig. 1 b and d).While many other quantitative descriptors exist [7], few are adapted to the determinationof the anisotropy of such microstructure pattern. Previous studies on polar ice calculate theaspect ratio of the grains from the linear intercept method [5, 8]. Other use the eigenvaluesof the grain inertia tensor [9, 10]; this is a true tensor, with mathematical advantages: forinstance, its value does not depend on the particular choice of axes, hence undergoes lessartifacts when examining the 2D cut of a 3D pattern.The tensor M has the same advantages; but also an additional one, thanks to its definition(eq. 1) being quadratic in ~ℓ: it has a physical signification in terms of mechanical deformations[4]. More precisely, its variations (with respect to a reference M0 measured in an isotropic,relaxed state) define a statistical strain tensor U :U =log(M)− log(M0)2, (2)G. Durand et al.: Deformation of grain boundaries in polar ice 3Fig. 1 – Two thin sections of ice imaged between crossed polarizers in white light (a and c) andtheir corresponding M tensor analysis (b and d). Each grain has an almost uniform crystallographicorientation, visualized by its color. (a) At 362 m depth, the microstructure looks isotropic; (c) at2629 m depth the microstructure is visibly anisotropic. Superimposed on (a): notations used in themeasurement of M , here e.g. at the site labeled by a black dot. We note ~ℓ a vector linking it to oneof its neighbours. There are 3 such vectors in the first shell, p = 1 (black); 6 in shell p = 2 (grey);12 in shell p = 3 (white). The texture tensor M measured at each site is represented as an ellipse(362 m: (b) and 2629 m: (d)), with its axes along the eigenvectors, and its half-axes proportionalto eigenvalues λ1, λ2 respectively. A site around which the pattern is isotropic is represented by acircle; conversely, a strong anisotropy is represented as an elongated ellipse. The size of the ellipse(same scale for each ellipse) represents the local length of the grain boundaries, i.e quantifies the localgrain size. Due to the definition of M (eq. 1), we exclude sites closer than p grains from the imageboundary; here p = 3.4 EUROPHYSICS LETTERSFig. 2 – Histogram of Uxz (left) and of Uzz(right) measured with p = 3 on Fig. (1b): 362 m deepsample (dots), and the corresponding gaussian distribution (same mean and standard deviation) (thinsolid line); and on Fig. (1d): 2629 m deep sample (thick solid line).which exists even during large deformations and, at least during elastic deformations, coincideswith the classical definition of strain [4].For the 2D pattern studied here, M0 is unknown, and processes other than deformation(such as grain growth) modify the microstructure. We first assume that the (viscoplastic)deformation is isochore (incompressibility of ice), so that TrU = 0 and λ01λ02 = λ1λ2. Wealso assume that the reference state is isotropic; hence M0 is isotropic too, so that λ01 = λ02 =√λ1λ2. We then can measure U through the following equation:U =12R log(λ1√λ1λ2)00 log(λ2√λ1λ2)R−1 = 14R log(λ1λ2)00 log(λ2λ1)R−1. (3)Hence U has two independent components Uzz and Uxz, both largely insensitive to the grainsize fluctuations.To estimate the statistical variability of the measure, we generate 150 isotropic microstruc-tures from a 2D Potts model of normal grain growth [11]. We check that (i) each componentof U fluctuates around zero; (ii) its distribution is gaussian (see an example with p = 3 onFig. 2); (iii) its standard deviation σ only depends on the number N of vectors, i.e on thescale of observation p; and (iv) σ(N) ∝ N−1/2: more precisely, σ ≈ 0.34 N−1/2.Results. – We first measure the deformation and its heterogeneity within one sample.Fig. (1b) shows that the upper sample (depth 362 m) is almost homogeneous. Its anisotropy issmall and, as expected from a sample taken at a dome, it mainly reflects a uniaxial compressionalong the vertical axis. Conversely (Fig. 1d), the anisotropy of the lower sample, at 2629 m:is stronger; is localized in space; and breaks the axisymmetry around z (see Discussion). Thissymmetry breaking cannot result from the tilt of the ice core, as the grains are tilted in averageby 13.5◦ from the vertical, much more than the core tilt at 2629 m (2.6◦).G. Durand et al.: Deformation of grain boundaries in polar ice 50 500 1000 1500 2000 2500 30000.000.050.100.150.200.250 500 1000 1500 2000 2500 3000-0.30-0.25-0.20-0.15-0.10-0.050.00 (c)|Uxz|(a)(b) depth (m)Uzz depth (m)50010001500   NFig. 3 – Localization of deformation inside the ice core. For each point σ is deduced from the numberof vectors N (a). Each point corresponds to one of 329 samples; same scale for both figures. (b) |Uxz|versus depth. Closed circles highlight samples where |Uxz| > 3σ. (c) Uzz versus depth. Two largeclosed circles linked by a thick solid line indicate two successive measures which 3 σ confidence intervalsdo not overlap. A large open circle indicates a sample with a large significant shear (Uxz > 3σ, from(b)). Vertical lines indicate zones, associated to climatic transitions, where the grain growth ratechanges significantly (see Discussion).Quantitatively, the distributions of Uxz (vertical compression) and Uzz (horizontal shear)of both samples differ significantly (Fig. 2). The upper sample is not sheared (〈Uxz〉 = 0.00),and is slightly flattened onto the horizontal plane (〈Uzz〉 = −0.05). The lower sample issignificantly sheared (〈Uxz〉 = 0.13) and strongly flattened (〈Uzz〉 = −0.25). Note that forp = 3 (N = 21) the intrinsic value of σ (see above) is 0.074; here, the statistical deviation(0.1) is comparable, but slightly larger, reflecting the heterogeneity of deformation at smallscales (10−2 − 10−1 m).We now turn to heterogeneities at large scale (1 − 102 m), by measuring the global U(integrated over all the grain boundaries of a given sample), versus the sample depth (Fig. 3).Whereas Uxz remains around zero, as expected in a dome situation, Uzz increases (in absolutevalue) with depth.The essential point is the localization: namely, the observation of a deformed (shearedor compressed) layer immediately close to less deformed ones. As mentioned, the grain sizeincreases with depth, therefore the number of vectors per sample decreases 3a, hence σ in-creases. The localization is nevertheless significant, and is increasingly frequent at increasingdepth (closed circles on Fig. 3b and c).Discussion. – Since the pictures are 2D cuts of actual 3D grains, the apparent averagegrain size could be misestimated; however, since U is dimensionless, it should remain unaf-fected. When, and only when, the pattern is axisymmetric, we can refine the above analyzis(eq. 3). In that case, the eigenvectors are (x, y, z), and the corresponding eigenvalues are6 EUROPHYSICS LETTERS(λ1, λ1, λ2). The eigenvalues of U are now log(λ1/λ2)/6 and log(λ2/λ1)/3. We have checkedthat the results corresponding to the 2D analyzis (Figs. 2, 3) and the 3D one (data not shown)are completely similar, with even slightly more heterogeneities using the latter analysis. Ofcourse, care is required for the interpretation of an individual sheared sample (for instance re-garding Uzz at 2629 m, Fig. 2b). But on average, the breaking of the axisymmetry hypothesis(large open circles on Fig. (3c)) does not introduce significant artifacts on the compressionheterogeneities we recorded (closed circles on Fig. 3c).A smooth shear deformation increasing with depth, sampled with a random azimuth, couldlead to an apparent heterogeneity of both Uzz and Uxz. However, this would imply an increas-ing anisotropy of the microstructure on the horizontal xy plane, whereas six horizontal thinsections were also analyzed (very few were available) and showed very small anisotropy. Thisisotropy of the horizontal thin sections analyzed, combined with the Uxz values highlightedon figure 3b is actually another expression of strain heterogeneity.Processes such as normal grain growth counteract the anisotropy of grains induced bydeformation. Hence the strain recorded by the microstructure U underestimates the strain εactually experienced by the material. The growth rate itself could undergo some fluctuactions.At few depths, near the bottom of the core, a correlation has been observed between smallgrain sizes (related to a large amount of dust in the ice [12]) and large strains. In these fewcases, highlighted by the thick vertical lines in figure 3c, the large differences of Uzz observedbetween two adjacent layers could be partly explained by the difference in the grain growthrate. However, this effect is unable to explain most heterogeneities (closed circles on Fig. 3c).In addition, it does not explain the large number of shear layers (closed circles on Fig. 3b).We note here that such a correlation between grain size and Uzz would be difficult to explainin case of a strong effect of a random azimuth. We thus argue that the heterogeneity ofanisotropy, observed at both small and large scales, cannot be entirely an observation artifactand has a mechanical origin.Conclusion and perspectives. – A dating chart is the relation t(z) between depth z andage t of ice: it requires a model, and hypotheses. The ice core is drilled exactly at the summitof a dome, in order to assume that the flow is axisymmetric; so that the ice thinning resultsfrom vertical compression only, without horizontal shear. Current models further assume asmooth and monotonous increase of the thinning of ice defined as the ratio e(z)/e(0), wheree is the thickness of the annual ice layer at depth z.The second assumption has already been shown to be wrong in the Greenland ice coresGRIP and GISP2 ([13, 14, 15]). Flow disturbances have been reported within at least the 30%deepest part of the GISP2 core, based on the observations of wavy ash layers, crystal stripingsvisible by eyes, and anomalous fabrics [13]. The dating has been particularly questioned bythe observation of a folding, i.e. local inversion of ice layers [16].Here, our method is more accurate and applies to ice itself, without requiring markersnor extreme events. We show that flow disturbances are detectable from almost the top ofthe ice sheet, and increase in number and intensity with depth. In fact, both dating modelassumptions are contradicted by our results: the strain gradient is variable and not evenalways positive (Fig. 3c); and although the flow at a dome is axisymmetric on average, indetail there is a symmetry breaking due to shear, especially in deep layers (Fig. 3b). Thissuggests to reconsider current standard dating charts.In the future, we expect to correlate the grain boundary pattern with the c-axis orientation,to improve our understanding of their coupling.∗ ∗ ∗G. Durand et al.: Deformation of grain boundaries in polar ice 7We thank M. Aubouy for his support and useful discussions, M. Glock for reading themanuscript, and M. Krinner and M. Manouvrier for their technical support. This work waspartially supported by CNRS ATIP 0693. This work is a contribution to the “European projectfor Ice Coring in Antarctica” (EPICA), a joint ESF (European Science Foundation)/EC sci-entific program, funded by the European Commission under the Environment and ClimateProgramme (1994-1998) contract ANV4-CT95-0074 and by national contributions from Bel-gium, Denmark, France, Germany, Italy, the Netherlands, Norway, Sweden, Switzerland andthe United Kingdom.REFERENCES[∗] Author for correspondence at graner@ujf-grenoble.fr. Fax: (+33) 4 76 63 54 95.[1] Epica community, Nature, 429 (2004) 623.[2] Schwander J., Jouzel J., Hammer C.U., Petit J.R., Udisti R. and Wolff E. , Geophysicalresearch letters, 28 (2001) 4243.[3] Hammer C.U., Clausen H.B., Dansgaard W., Gundestrup N., Johnsen S.J. and ReehN., J. Glacio., 20 (1978) 3.[4] Aubouy M., Jiang Y., Glazier J. A. and Graner F., Granular Matt., 5 (2003) 67[5] Gay M., and Weiss J., J. Glaciol., 45 (1999) 547.[6] Asipauskas M., Aubouy M., Glazier J. A., Graner F. and Jiang Y., Granular Matt., 5(2003) 71.[7] Mecke K. and Stoyan D. (Editors), Morphology of Condensed Matter - Physics and Geometryof Spatially Complex Systems, Lecture Notes in Physics, Vol. 600 (Springer, Heidelberg) 2002[8] Arnaud L., Weiss J., Gay M. and Duval P., Annals Glacio., 30 (2000) 8.[9] N. Azuma, Y. Wang, K. Mori, H. Narita, T. Hondoh, H. Shoji and O. Watanabe, Ann.Glaciol., 29 (1999) 163[10] Wilen L.A., Diprinzio C.L., Alley R.B. and Azuma N., Micro. Res. and Tech., 62 (2003)2.[11] Anderson M.P., Grest G.S. and Srolovitz D.J., Philos. Mag. B, 59 (1989) 293.[12] Weiss J., Vidot J., Gay M., Arnaud L., Duval P. and Petit J.R., Annals Glacio., 35(2002) 552.[13] Alley R.B., Gow A.J., Meese D.A., Fitzpatrick J.J, Washington E.D. and BolzamJ.F., J. Geophys. Res., 102 (1997) 26819.[14] Gow A.J., Meese D.A., Alley R.B., Fitzpatrick J.J., Anandakrishnan S., Woods G.A.and Elder B.C., J. Geophys. Res., 102 (1997) 26559.[15] Dahl-Jensen D., Thorsteinsson T., Alley R. and Shoji H., J. Geophys. Res., 102 (1997)26831.[16] Grootes P.M., Stuiver M., White J.W.C., Johnsen S. and Jouzel J., Nature, 366 (1993)552.

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