1. ©2010 Elsevier, Inc. 1 Chapter 3 Cuffey & Paterson

2. ©2010 Elsevier, Inc. 2 FIGURE 3.1 Figure 3.1: The crystalline structure of ice (adapted from drawings in Hobbs 1974). In the top-most diagram, dashed lines represent hydrogen bonds, the solid lines covalent bonds. Circles indicate oxygen atoms. Molecules arranged as tetrahedra combine to form hexagonal rings in two closely spaced planes. Atoms numbered 1 through 6 form one such ring. Viewed along the c-axis, the rings make a hexagonal mesh. Viewed obliquely, the ring layers are stacked as shown in the bottom sketch.

3. ©2010 Elsevier, Inc. 3 FIGURE 3.2 Figure 3.2: Schmidt diagram. Each point in (a) represents the intersection of the c-axis of one crystal with the surface of a hemisphere when each c-axis passes through the center of the sphere. The contours in (b) represent 1, 3, and 5% of points within 1% of area.

4. ©2010 Elsevier, Inc. 4 FIGURE 3.3 Figure 3.3: Change of concentration (number per volume) of bubbles and clathrate inclusions at depth in (a) the Greenland Ice Sheet at North-GRIP (data from Kipfstuhl et al. 2001) and (b) the East Antarctic Ice Sheet at Dome Fuji (data from Ohno et al. 2004).

5. ©2010 Elsevier, Inc. 5 FIGURE 3.4 Figure 3.4: Variation of grain diameter with depth in the central Greenland Ice Sheet (GISP-2 core). Filled circles show the apparent mean grain diameter seen in the plane of thin sections (data from Alley and Woods 1996). Open squares show the mean diameter of the largest 50 grains seen in thin sections (data from Gow et al. 1997). Grain sizes increase through the upper few hundred meters. The transition from coarser to finer grains between 1500 and 1700 m corresponds to increased impurity concentrations between Holocene ice and underlying ice from the Last Glacial Maximum. Very large grains (up to 60 mm) were found in the basal layers, but are not shown here.

6. ©2010 Elsevier, Inc. 6 FIGURE 3.5 Figure 3.5: Variation, with age, of mean cross-sectional area of crystals in firn at Byrd Station, Antarctica. Redrawn from Gow (1971).

7. ©2010 Elsevier, Inc. 7 FIGURE 3.6 Figure 3.6: Plot of crystal growth rate against reciprocal of the absolute temperature for the data in Table 3.1.

8. ©2010 Elsevier, Inc. 8 FIGURE 3.7 Figure 3.7: C-axis fabrics produced in simple stress systems by (a) rotation, by itself or accompanied by polygonization, and (b) rotation accompanied by migration recrystallization and nucleation. These two classes of fabrics are sometimes referred to as deformation fabrics and recrystallization fabrics, respectively. Each diagram is the projection on a horizontal plane. Figure 3.8: Sketch showing grain rotation from deformation of polycrystalline ice. In (a), tension τxx causes resolved shear stresses on the basal planes of two grains, which deform by basal glide. Mutual interference leads to rotation of c-axes away from the axis of tension, as shown in (b). Sliding along the grain boundary must also occur, as shown. Adapted from Alley (1992a).

9. ©2010 Elsevier, Inc. 9 FIGURE 3.8 Figure 3.9: Variation with depth of fabric and average crystal size in the Greenland Ice Sheet (Camp Century) and West Antarctic Ice Sheet (Byrd Station). Adapted from Herron and Langway (1982).

10. ©2010 Elsevier, Inc. 10 FIGURE 3.9 Figure 3.10: How a block of ice (shaded gray) deforms in pure shear and simple shear as it moves in a glacier. Vectors in both cases show the pattern of ice flow in a glacier constrained by a rigid boundary, such as the bed. In these examples, the ice slides along the boundary in the pure shear case but not in the simple shear case.

11. ©2010 Elsevier, Inc. 11 FIGURE 3.10 Figure 3.11: Typical shape of creep curves for polycrystalline ice. The minimum strain rate, which usually occurs at a strain of about 0.01, is commonly used as the reference for comparisons between different experiments and samples. Adapted from Budd and Jacka (1989).

12. ©2010 Elsevier, Inc. 12 FIGURE 3.11 Figure 3.12: Covariation of effective stress and strain rate for locations on spreading ice shelves. Circles: Ross Ice Shelf data from Jezek et al. (1985). Squares: data for five ice shelves, reported by Thomas (1973b). Broken line corresponds to creep exponent n = 2.5.

13. ©2010 Elsevier, Inc. 13 FIGURE 3.12 Figure 3.13: Comparison of different types of creep relations. The power-law relation with n = 3, appropriate for ice, is intermediate between linear viscous (n = 1) and perfectly plastic behaviors. Here, plastic yield stress is 100 kPa (1 bar) and viscosity for the linear relationship is about 1014 Pa s.

14. ©2010 Elsevier, Inc. 14 FIGURE 3.13 Figure 3.14: Variation of minimum octahedral shear strain rate with initial density. The tests used unconfined, uniaxial compression with an octahedral shear stress of 200 kPa, at −3 °C. Adapted from Hooke et al. (1988); data from experiments reported by Jacka (1994).

15. ©2010 Elsevier, Inc. 15 FIGURE 3.14 Figure 3.15: Covariation of measured strain rates with ice properties in the basal layers of Meserve Glacier. Left panel: variation with soluble impurity concentration C. Right panel: variation with grain diameter raised to the power −0.6. The coefficient γ and the exponent of D have both been optimized to give the best possible relationships. Adapted from Cuffey et al. (2000a and 2001) and used with permission from the American Geophysical Union, Journal of Geophysical Research.

16. ©2010 Elsevier, Inc. 16 FIGURE 3.15 Figure 3.16: Local in situ measured values of strain rate compared to strain rates predicted from the recommended “base” creep relation. Solid line indicates a perfect match. Broken lines correspond to increases of measured strain rates by 3 and 9 times. Open symbols refer to polar ice; filled symbols to temperate ice. Open squares: simple shear in polar ice sheets and glaciers. Open diamonds: spreading of polar ice shelves. Filled triangles: strain rate values of temperate glaciers at a stress of 100 kPa. Filled circles: simple shear in temperate glaciers. Dark filled circle: tunnel closure in a temperate glacier. Data given in Appendix 3.2.

17. ©2010 Elsevier, Inc. 17 FIGURE 3.16 Figure 3.17: Measured variation of ˙ (deformation rate divided by the cube of the stress) for ice with a strong single-maximum fabric deformed in uniaxial compression applied at different angles to the axis of the sample. The sample axis was within a few degrees of the fabric axis. Data from Shoji and Langway (1988).

18. ©2010 Elsevier, Inc. 18 FIGURE 3.17 Figure 3.18: Measured creep curves for ice deformation experiments, showing softening by development of preferred orientation fabrics, in various combinations of compression and shear (magnitudes shown at right, in units of MPa). The increase of strain rate from the secondary minimum to the tertiary value is greater if shear dominates than if compression dominates. The combined octahedral stress for all tests was 0.4 MPa. Adapted from Li et al. (1996).

19. ©2010 Elsevier, Inc. 19 FIGURE 3.18 Figure 3.18: Measured creep curves for ice deformation experiments, showing softening by development of preferred orientation fabrics, in various combinations of compression and shear (magnitudes shown at right, in units of MPa). The increase of strain rate from the secondary minimum to the tertiary value is greater if shear dominates than if compression dominates. The combined octahedral stress for all tests was 0.4 MPa. Adapted from Li et al. (1996).

20. ©2010 Elsevier, Inc. 20 FIGURE 3.19 Figure 3.19: For ice with an isotropic fabric, the variation of shear strain rate due to a shear stress (τo) combined with a uniaxial compressive stress (σo) perpendicular to the shear plane. Solid line shows the prediction of the Nye-Glen relation; circles show calculations with the ensemble-average relation. In both cases, the increase is linear at low values for τo/σo but becomes cubic at high values.

21. ©2010 Elsevier, Inc. 21 FIGURE 3.20 Figure 3.20: Softening of ice due to preferred orientation fabrics, calculated with ensemble-average anisotropic creep relation. Circles show the theoretical change in shear strain rate for simple shear of ice with a single-maximum fabric – with “characteristic angle” referring to the cone-angle (90° is isotropic ice). Also shown is theoretical change in compressive strain rate in uniaxial compression of ice with a girdle fabric – with “characteristic angle” being halfway between the inner and outer angles. Inner and outer angles are 30° apart.