Calculating cumulative strain

In order to calculate the cumulative strain in a glacier, one first must know the velocity field. One then calculates the path that a particle of ice would follow through the glacier and velocity derivatives at discrete points along the path. To obtain the incremental strain as the ice moves from one point to the next, strain rates are then calculated from the velocity derivatives and multiplied by the time needed for this movement

(a) Particle paths and isochrons

Divide T020

(b) Direction of maximum cumulative extension Figure 13.5. (a) Particle paths calculated from the velocity field described in the text. Ages of isochrons are shown in years. (b) Orientation of maximum cumulative extension direction shown by bars at 200 m intervals along the particle paths in (a). (c) Contours of foc. (From Hudleston and Hooke, 1980, Figures 6 and 7. Reproduced with permission of Elsevier Scientific Publishing.)

Figure 13.5. (a) Particle paths calculated from the velocity field described in the text. Ages of isochrons are shown in years. (b) Orientation of maximum cumulative extension direction shown by bars at 200 m intervals along the particle paths in (a). (c) Contours of foc. (From Hudleston and Hooke, 1980, Figures 6 and 7. Reproduced with permission of Elsevier Scientific Publishing.)

(Ramsay and Graham, 1970, Equations 7-10). Finally, these incremental strains are added to get the cumulative strain.

Hudleston (Hooke and Hudleston, 1980; Hudleston and Hooke, 1980) made such a calculation using a comprehensive set of data on velocities and mass balance along the flow line on Barnes Ice Cap illustrated in Figure 13.5a (see also Figure 12.17). To estimate horizontal velocities at depth, he used measured surface velocities and adjusted the value of B in Equation (5.7) to get zero velocity on the bed, where the temperature is well below the melting point. Rather than use measured vertical velocities, he used the mass balance data to estimate the long-term steady-state vertical velocity at the surface and assumed that it decreased linearly with depth. Transverse strain rates are small, so he assumed that they could be neglected.

The results of his calculations are shown in Figures 13.5 and 13.6. By following particles starting at nine points in the accumulation area, he first mapped nine flowlines (Figure 13.5a). He then calculated the Figure 13.6. Calculated variation of yoc, yi0c, and yc with depth at locations of Barnes Ice Cap boreholes (a) T020 and (b) T081. Locations of boreholes are shown in Figure 13.5. (From Hooke and Hudleston, 1980, Figures 8a and 9a. Reproduced with permission of the International Glaciological Society.)

Figure 13.6. Calculated variation of yoc, yi0c, and yc with depth at locations of Barnes Ice Cap boreholes (a) T020 and (b) T081. Locations of boreholes are shown in Figure 13.5. (From Hooke and Hudleston, 1980, Figures 8a and 9a. Reproduced with permission of the International Glaciological Society.)

orientations of the axes of maximum cumulative extension (X-axes: Figure 13.5b). Noteworthy in this figure is the fact that these axes are nearly parallel to the bed throughout most of the glacier. The increased upglacier dip at the surface near and downglacier from borehole T061 is a consequence of the increase in the ratio of simple shear to pure shear as the equilibrium line is approached. As just noted, in simple shear the axis of maximum elongation dips 45° initially; with increasing deformation it is gradually rotated toward parallelism with the plane of the shear.

The cumulative strain magnitude, yoc, is shown in Figure 13.5c. These numbers do not appear significant until one realizes that yoc is proportional to the natural logarithm of the axial ratio of the strain ellipse. Thus, yoc = 8, found in the most basal ice, corresponds to an elongation of ~18 000:1. A 1 m cube would be stretched into a 1-m wide ribbon 134 m long and 7.5 mm thick!

Figure 13.6 shows the variation of yoc, 0c, and yc with depth in boreholes T020 and T081 (Figure 13.5a). Because the dominant strain pattern at T020, particularly in the upper part of the glacier, is nearly pure shear with vertical compression and longitudinal extension, the axis of maximum cumulative extension is nearly horizontal. Thus, 0c remains close to 0. On the other hand, yc is 0 at the surface and initially increases gradually with depth as du/dz increases. However, with increasing depth in the glacier, the ice arriving at T020 has passed through a larger and larger region dominated by pure shear (Figure 13.4). Thus, <c reaches a maximum (~12°) at a depth of ~240 m and then decreases at greater depth.

The pattern at the site of borehole T081 is different in several respects. Because this hole is in the upper part of the ablation area, ice at the surface accumulated some strain as it moved from higher in the glacier. Thus, Yoc > 0 at the surface. With increasing depth, the ice has traveled a greater distance and accumulated more strain so Yoc increases to about 7, representing an axial ratio of over 5000. Because ice near the surface has experienced a modest amount of mixed simple and pure shear (Figure 13.4), 0c « 10° and <c « 25° here. With increasing depth, <c first increases, reaching a maximum at a depth of ~ 120 m and then decreases, reflecting the early history of pure shear that this deeper ice experienced.

To further quantify the influence of the early history of pure shear, Hudleston calculated <c for a particle of ice that experienced a total strain, Yoc, of 3.75 entirely by simple shear (Hooke and Hudleston, 1980). In this case, <c is 80° and increases toward 90° as Yoc increases further. For comparison, in holes T020 and T081 the actual rotations at this strain magnitude are 9° and 57°, respectively

Let us now use our understanding of cumulative strain to study the origin of foliation.