Recrystallization

Crystals of glacier ice vary in size and also in the degree to which they are interlocked. If there were no bonding across grain boundaries, for example, some polycrystalline ice samples would fall apart into a pile of roughly equant grains, up to a few millimeters in maximum dimension, while others would hang together like a three-dimensional jigsaw puzzle. We will use the term texture to refer to these characteristics of crystal size

Figure 4.10. Stress-strain rate data for ice at -10 °C. (Adapted with permission from Duval et al, 1983, Figure 2. Copyright 1983 American Chemical Society.)

10-2 10-1 100 101 Axial stress, MPa

Figure 4.11. Reverse creep due to release of stored elastic strain energy upon unloading of a sample. The original loading of the sample was 1 MPa. At time 0, the stress was reduced by 0.06 MPa (upper curve) and by 0.13 MPa (lower curve), respectively. Horizontal parts of the curves reflect a balance between further creep under the reduced stress and delayed release of elastic strain energy. (Adapted with permission from Duval et al., 1983, Figure 5. Copyright 1983 American Chemical Society.)

Time since stress reduction, min

Figure 4.11. Reverse creep due to release of stored elastic strain energy upon unloading of a sample. The original loading of the sample was 1 MPa. At time 0, the stress was reduced by 0.06 MPa (upper curve) and by 0.13 MPa (lower curve), respectively. Horizontal parts of the curves reflect a balance between further creep under the reduced stress and delayed release of elastic strain energy. (Adapted with permission from Duval et al., 1983, Figure 5. Copyright 1983 American Chemical Society.)

and shape. In addition, under prolonged strain the c-axes of the crystals develop a variety of preferred orientations, or fabrics. Both texture and fabric affect the rheology of ice.

In order to study these processes, glaciologists, like petrologists, use thin sections. The thin sections are typically somewhat less than a millimeter thick and 60-80 mm across. When polarized light is passed through a thin section and then observed through another polarizing filter oriented at right angles to the first, the individual crystals can be seen because the crystal structure rotates the light as it passes through the crystal, and the amount of rotation depends on the orientation of the crystal. When thus viewed, the different crystals have different colors (or grayscale tones in a black and white image - Figure 4.12). With the use of a universal stage on which the thin section can be both rotated around a vertical axis and tilted about either of two mutually perpendicular horizontal axes, crystals can be oriented so their c-axes are vertical. In this orientation, the crystal remains black as the stage is rotated around its vertical axis. The orientation of the crystal is then noted and plotted on an equal-area net (Figure 4.13). To interpret such a plot, visualize a hemisphere with its convex side down and with a crystal in its center. The c-axis of the crystal intersects the hemisphere. A point on a fabric diagram like those in Figure 4.13 is the projection of this point of intersection onto the flat surface. Thus, a vertical c-axis plots at the center

Figure 4.12. Photographs of thin sections of ice from the Greenland ice sheet near Thule. Photographs were taken under crossed polarizers. The different grayscale tones of the grains reflect different orientations of the c-axes. (a) Initial texture formed by compaction of snow with addition of small amounts of melt water. The c-axes have a weak preferred orientation, with a preference for vertical orientations. (b) Texture resulting from grain growth with little or no deformation. The c-axes still have a weak vertical preferred orientation. (c) Texture resulting from polygonization. Adjacent grains with nearly the same grayscale tone (arrows) have c-axes that are nearly parallel to one another. The grain in the lower center is bent; in the one to left of center, distinct boundaries have formed between parts with slightly different orientations. (d) Texture following significant deformation. Grains are interlocked, and c-axes have a strong preferred orientation. (From Hooke, 1970.)

Figure 4.12. Photographs of thin sections of ice from the Greenland ice sheet near Thule. Photographs were taken under crossed polarizers. The different grayscale tones of the grains reflect different orientations of the c-axes. (a) Initial texture formed by compaction of snow with addition of small amounts of melt water. The c-axes have a weak preferred orientation, with a preference for vertical orientations. (b) Texture resulting from grain growth with little or no deformation. The c-axes still have a weak vertical preferred orientation. (c) Texture resulting from polygonization. Adjacent grains with nearly the same grayscale tone (arrows) have c-axes that are nearly parallel to one another. The grain in the lower center is bent; in the one to left of center, distinct boundaries have formed between parts with slightly different orientations. (d) Texture following significant deformation. Grains are interlocked, and c-axes have a strong preferred orientation. (From Hooke, 1970.)

of the circle, and a c-axis dipping "south" plots between the center and the bottom of the circle. The points are normally plotted on a Schmidt equal-area net; this net is designed so that a unit area on the hemisphere plots into a unit area on the net. Consequently, a c-axis dipping at 45° actually plots about 55% of the distance from the center of the net to the boundary.

W"

Figure 4.13. Examples of crystallographic fabrics observed in ice. Plots are projections on lower hemisphere of an equal-area net. Triangles on edges show direction of bubble elongation. In fabrics produced by simple shear, direction of shear, shown by arrows, is presumed to be parallel to bubble elongation. All fabrics except (c) were measured on cores from boreholes in Barnes Ice Cap (Figure 4.15); (c) is schematic. (a) Fabric with weak preferred orientation of c-axes in superimposed ice. (b) Fabric resulting from uniaxial compression normal to the plane of the diagram. (c) Schematic fabric formed in pure shear. (d) Broad single-maximum fabric. (e, f, g) Fabrics resulting from simple shear in plane of diagram. ((a) and (d)-(g) from Hooke and Hudleston, 1980; (b) from Hooke and Hudleston, 1981.)

Ice that forms from compaction of snow, perhaps with some addition of percolating meltwater, usually consists of crystals that are 2-4 mm in diameter (Figure 4.12a). Through a series of processes that we will refer to, collectively, as dynamic recrystallization, the texture and fabric of this ice are altered during deformation. Dynamic recrystallization, or simply recrystallization, is a consequence of the high local internal stresses mentioned above, and the resulting widely differing internal energies in adjacent grains.

One or more of three processes may be involved in recrystallization. In order of increasing energy difference between adjacent grains, these are grain growth, polygonization, and nucleation of new grains (Duval and Castelnau, 1995). (Note that the terminology for these processes differs among authors.) Grain growth results from relatively slow migration of grain boundaries. This migration is driven by the decrease in free energy that accompanies the reduction in total area of grain boundaries (Montagnat and Duval, 2000). Typical rates range from ~10-3 mm a-1 at -30 °C and 1 kPa (local driving stress) to ~10 mm a-1 at -10 °C and 300 kPa (Duval et al., 1983). The migration is driven by the curvature of the grain boundaries. Higher pressures occur on the concave sides of such boundaries, which is commonly the side of the smaller grain, and molecules tend to move from the high-pressure side to the low-pressure side ofthe boundary (Alley, 1992). Thus, smaller crystals disappear. The result is a characteristic texture with equant crystals of relatively uniform size (Figure 4.12b). Because temperatures in the accumulation zones of polar ice sheets are relatively constant to depths of a few hundred meters (see Figure 6.6a), grain boundary migration occurs at relatively constant rates and grain size thus increases nearly linearly with depth. At greater depths, grain size becomes approximately constant because polygoniza-tion, which decreases grain size, balances grain growth (Alley et al., 1995).

Polygonization (also called rotation recrystallization) involves the alignment of dislocations to form a new grain boundary within a bent crystal. The crystal is thus divided into two grains with nearly the same orientation (Figure 4.12c, arrows). Under relatively high strain rates, polygonization begins at strains of ~1% (Duval and Castelnau, 1995), but at the much lower strain rates found in the central regions of continental ice sheets, cumulative strains can approach 100% without causing polygonization (Alley, 1992). Thus, polygonization occurs at relatively shallow depths in temperate glaciers, but is normally found only at depths greater than a few hundred meters in polar ice sheets.

Nucleation of new grains entails the appearance of small grains that are oriented for easy glide, with their basal planes parallel to the maximum resolved shear stress. When they first appear, such grains are relatively unstrained in comparison with adjacent older deformed grains. As the free energy of the system can be lowered by migration of boundaries of the new grains into the adjacent ones, the new grains grow at the expense of the older ones (Alley, 1992). This is probably partly responsible for the interlocking textures seen in highly deformed ice (Figure 4.12d).

Such nucleation and grain growth (also called migration recrystallization) occurs at relatively high strain rates and predominantly at temperatures above 10 °C. It is thus characteristic of basal ice in temperate glaciers and of ice in the lowermost few hundred meters of polar ice sheets.

Development of fabrics with preferred orientations of c-axes

As just noted, newly nucleated unstrained crystals tend to grow at the expense of older deformed ones. However, as they grow, the continued straining gradually rotates them into orientations that are no longer optimal, and they begin to accumulate strain energy. Consequently they, in turn, are eventually consumed by even newer grains. It is the resulting preference for orientations with basal planes parallel to the maximum resolved shear stress that results in the increase in creep rate, in laboratory experiments, after about 1% strain (Figure 4.8b). Preferred orientations of c-axes are the manifestation of this preference (Figure 4.13). Thus two basic processes are involved in the development of these c-axis fabrics: recrystallization and grain rotation.

Let us illustrate these processes by tracing the development of such fabrics, starting with ice near the surface in the accumulation area (Figure 4.12a). The c-axes of the crystals are either uniformly distributed or have a weak preference for vertical orientations (Figure 4.13a). The latter probably results from the orientation of snowflakes that have thin disklike shapes, and thus, like a pile of poker chips, tend to lie flat as they accumulate. In addition, the vertical temperature gradient may have had some influence during sintering.

As the ice becomes buried, it is compressed vertically and stretched longitudinally and sometimes also laterally. Where longitudinal and lateral strain rates are comparable in magnitude, the stress field is referred to as uniaxial compression (Figure 4.14a), whereas if lateral strain rates are negligible it is pure shear (Figure 4.14b). In such stress configurations, slip occurs most readily on the basal planes of crystals whose c-axes are inclined at ~45° to the compression axis (Figure 4.14d). Thus crystals are nucleated in this orientation, and these crystals grow at the expense of adjacent more highly stressed ones, leading to a conical

Uniaxial compression

Compression

^Resolved ^ shear

Extension

Pure shear

Compression

Simple shear

Compression

~ 30o - 35o after rotation

~ 30o - 35o after rotation

Extension

Pure shear

Figure 4.14. Stress configurations and their relation to orientations of c-axes. (a) Uniaxial compression. (b) Pure shear. There is no strain normal to the plane of the diagram. (c) Simple shear. (d) The c-axes and basal planes in a newly nucleated crystal in uniaxial compression or pure shear. With continued compression, the c-axes rotate toward the axis of compression. (e) Simple shear viewed parallel to the shear direction with basal planes also parallel to the shear direction as in the fabric of Figure 4.13f. The symbols (+) and Q signify stress vectors directed into and out of page, respectively. (f) Simple shear viewed normal to the shear direction with basal planes inclined to the shear direction as in the fabric of Figure 4.13g. In (d)-(f) short-dashed lines show orientations of basal planes.

distribution of c-axes in uniaxial compression (a small-circle fabric: Figure 4.13b.) and to two maxima aligned in the direction of extension in pure shear (Figure 4.13c). (Small-circle fabrics are also commonly referred to as girdle fabrics, although "girdle" implies a great circle.) The vertical compression and lateral extension, however, have the effect of rotating basal planes clockwise on the left and counterclockwise on the right in Figure 4.14d. Thus, as the crystals grow, the c-axes rotate toward the compression axis (Alley, 1992), with the result that the mean angle between the compression axis and the c-axes is typically only ~30°-35°, not 45° (Kamb, 1972; Hooke and Hudleston, 1980). Crystals that have been rotated too far, and thus become highly stressed, are resorbed, while nucleation develops new crystals in more favorable orientations.

If the ice at this depth is close enough to the bed, drag exerted by the bed results in a stress configuration approximating simple shear parallel to the bed (Figure 4.14c). Crystals with vertical c-axes are then preferred. The resulting fabrics, which are common in ice sheets (Gow and Williamson, 1976; Hooke and Hudleston, 1980), have single maxima that range from relatively broad (Figure 4.13d) to quite tight (Figure 4.13e).

The fabrics in Figures 4.13b,c,d all seem to form under roughly equivalent cumulative strain. The differences among them are primarily caused by stress configuration. As a class, we will refer to them as broad single-maximum fabrics.

Although the increase in creep rate associated with recrystallization usually begins at effective strains, ee (see Equation (2.11)) of ~0.01 in the laboratory (Figure 4.8b), broad single-maximum (and equivalent) fabrics are not particularly evident until ee = 0.04 and only become well developed at ee = 0.4 (Kamb, 1972; Jacka and Maccagnan, 1984). In the field, Hooke and Hudleston (1980) found that such fabrics first appeared at ee = 0.7-0.8. For reference, circles that have been deformed into ellipses by strains of these magnitudes have axial ratios of 1.02, 1.08, 2.22, and ~4.5, respectively. Thus, creep rates increase long before a detectable preferred c-axis orientation develops.

In simple shear at cold temperatures or high strain rates (or high cumulative strains), the single-maximum fabric strengthens (Figure 4.13e). However, at lower strain rates and temperatures above -10 °C, an unexpected fabric appears. First, the single maximum splits in two, with a maximum on either side of the shear direction (Figure 4.13f). The basal planes corresponding to these c-axis orientations are still parallel to the shear direction, but do not have optimal orientations (Figure 4.14e). Then, with increased cumulative strain, strain rate, or temperature, first one and then a second maximum appears inclined to the direction of shear (Figures 4.13g and 4.14f). These planes are definitely not well oriented for glide, and thus must stiffen the ice, at least slightly. These multiple-maximum fabrics appear at ee = 1.3 (Hooke and Hudleston, 1980). The corresponding axial ratio of the strain ellipse is ~15.

The origin of these multiple-maximum fabrics is not understood. They have been attributed to annealing under conditions of near stagnation (Budd and Jacka, 1989) and have been reproduced in the laboratory

Small

Circle__Weakly (

.Boreholes

Weakly oriented Single-maximum"

Multiple-maximum

800n

Single-maximum 4

.Boreholes

800n

600-

Weakly oriented Single-maximum"

Multiple-maximum

Single-maximum 4

600-

Distance from divide, km

Bubbly white (Pleistocene) ice | | Deformed superimposed ice Vertical exaggeration, 4x

Figure 4.15. Vertical cross section along a flow line on Barnes Ice Cap showing zones characterized by particular fabrics. Arrows show locations of cores used to determine fabric type. (After Hooke and Hudleston, 1980.)

by repeatedly compressing a sample and then annealing it (Huang et al., 1985). However, these processes are not consistent with the occurrence of such fabrics in ice that is actively deforming, as in Barnes Ice Cap (Figure 4.15) (Hooke and Hudleston, 1980).

Matsuda and Wakahama (1978) measured the orientations of a-axes as well as c-axes in ice with multiple-maximum fabrics. They did this by observing etch pits in the thin sections. In ice with four-maximum fabrics, they found that the a-axes of adjacent crystals were systematically aligned in a way that suggested mechanical twinning. Noting that strong shear deformation under high temperatures is required to produce such fabrics, they suggested that the large amount of plastic strain energy thus produced can be absorbed by propagation of twin boundaries without changing the relative structural relation between crystals or the crystal-boundary structure, and without resulting in strong bubble elongation.

Because the various fabrics appear to form under fairly specific conditions of cumulative strain, strain rate, and temperature, and because these parameters all tend to increase systematically with depth in the accumulation area of a glacier, fabric type also varies with depth. For example, in Barnes Ice Cap transitions from weakly oriented to broad single-maximum (or equivalent) fabrics occur at depths of 80-140 m, and the broad single maximum gives way to multiple-maximum fabrics at 140-200 m (Figure 4.15). At Byrd Station in Antarctica, the transition to broad single maximum fabrics (small circle variety) occurs at a depth of ~350 m. Then, a strong single-maximum fabric appears at ~1200 m and multiple-maximum fabrics show up at ~ 1830 m. Differences in temperature are probably largely responsible for the difference in depths to the transitions, although cumulative strain may also play a role; Barnes

Ice Cap is near or above -10 °C throughout, whereas at Byrd Station the temperature exceeds -10 °C only below 1900 m. In Barnes Ice Cap, as the various layers are advected outward they become exposed at the surface in the ablation area (Figure 4.15).

Summary

Given these various processes of recrystallization and crystal deformation, one may well ask how we should visualize the deformation of poly-crystalline ice on an intergranular scale. Available evidence suggests that stresses are heterogeneous, that intracrystalline glide takes place on basal planes within individual grains, that this glide results in internal rotation of the crystal structure, and that nucleation of grains with basal planes parallel to the maximum resolved shear stress and resorption of grains that have rotated out of this orientation results in the development of fabrics with preferred orientations. Mismatches between adjacent grains resulting from unequal slip at grain boundaries are accommodated by grain-boundary migration and by rotation and translation of grains. These grain-boundary processes are thus likely to be rate limiting. Computer models incorporating these principles successfully simulate many characteristics of fabric evolution in ice sheets (Etchecopar, 1977; Van der Veen and Whillans, 1994).

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