Introduction

It is usually supposed that the strength of ice varies in a simple way with position in a glacier. Ice strength decreases with depth and perhaps also close to the lateral margins of a glacier. This view allows large-scale modelling of ice flow using concepts of continuum mechanics (e.g. Reference Paterson,Paterson, 1981: Reference Hutter,Hutter, 1983). Many tests of these concepts have been made on glaciers and ice streams. Examples of such studies on temperate glaciers include a straight section of Athabasca Glacier, Alberta, Canada (Reference Raymond,Raymond, 1971) and over a riegel under Storglaciären, Sweden (Reference Hooke,, Calla,, Holmlund,, Nilsson, and Stroeven,Hooke and others, 1989). Tests have been made on polar glaciers at Dye-3 in southern Greenland (Reference Van der Veen, and Whillans,Van der Veen and Whillans, 1990); along the Byrd Station strain network in the interior of the West Antarctic ice Sheet (WAIS; Reference Whillans, and Johnsen,Whillans and Johnsen, 1983); and on Barnes Ice Cap, Canada (Reference Hooke, and Hanson,Hooke and Hanson, 1986). A new test, by Reference Whillans, and Van der Veen,Whillans and Van der Veen (1993b) on Ice Stream B, West Antarctica (Fig. 1), produced unexpected results that called into question some assumptions about spatial variations in ice strength and flow, especially that these variations are primarily in the vertical direction.

Fig. 1. Map of West Antarctica. Ice streams are shaded and labeled, A-E. The study is conducted near the “UpB camp” on Ice Stream B. Surfaсе contour interval is 200 m.

Patterns in surface strain rate and surface slope on Ice Stream B are unusual (Reference Hulbe, and Whillans,Hulbe and Whillans, 1994). The patterns do not appear to be caused by flow over or around basal topographic features or sticky spots. Force-budget analysis of the surface strain rates produces negative basal drag in some areas, which is physically improbable (Reference Whillans, and Van der Veen,Whillans and Van der Veen, 1993b). Furthermore, topographic features are not advecting downstream at the speed of surrounding ice, as would be expected if they were similar to rafts identified elsewhere in West Antarctica (Reference Bindschadler,, Stephenson,, MacAyeal, and Shabtaie,Bindschadler and others, 1987). Instead, the surface topographic features are slowly migrating up- and across-glacier. None of the usual explanations for horizontal variations in strain rate seem appropriate for the Ice Stream B data.

Here, precision Global Positioning System (GPS) measurements of an extensive strain grid near the UpB camp are used to address more thoroughly the nature of unusual flow patterns first identified by Reference Whillans, and Van der Veen,Whillans and Van der Veen (l993b). The work is more thorough because the new grid is much larger and because relative vertical velocities are obtained in addition to horizontal velocities. The earlier strain grid of Reference Whillans, and Van der Veen,Whillans and Van der Veen (1993b) is incorporated into the present grid, allowing time-changes in strain rate and topography to be observed.

Some analyses of the present data are given in Reference Hulbe, and Whillans,Hulbe and Whillans (1994). That report describes the GPS-survey method and presents the strain rates and vertical velocities. It seeks, but fails, to explain the measured strain-rate gradient and Surface slope patterns in terms of Sticky spots (sites of extra basal friction or partial blockage of flow). Here those results are re-evaluated and new comparisons are made with older strain-rate and surface topographic observations. A more Straightforward model is proposed. It involves horizontal variation in ice strength, and accounts for more of the observations than did Reference Hulbe, and Whillans,Hulbe and Whillans (1994).

Flow Characteristics

Velocity gradients and surface topography from surveys conducted in late 1984 and late 1985 (used by Reference Whillans, and Van der Veen,Whillans and Van der Veen (1993b), referred to as 1984–85), and late 1991 and late 1992 (1991–92) are used to analyze ice flow. Surveying methods and errors are discussed in Appendix 1. Topographies for the large strain grid measured in 1991 and for the smaller grid measured in 1984 are shown in Figure 2. Gridpoint velocities and positions are tabulated in Reference Hulbe,Hulbe (1994). Plots of the velocity gradients for the entire strain grid are in Reference Hulbe, and Whillans,Hulbe and Whillans (1994). Here attention is directed toward the two sites of most extreme velocity variation, shown in Figures 3 and 4.

Fig. 2. Ice surface topography in 1991 and 1984, with respect to the same portion of the bed. Dots indicate the positions of survey markers in the strain grid. Contour interval is 2 m. Contours in the 1984 map are difference in height from the “0” meter contour. Contours in the 1991 map are height of the snow surfасе above the ellipsoid (WGS84). The “A”, “B” and “C” lines are along y ≈ −8, −9 and −10 km, respectively.

Fig. 3. Surface topography and velocity patterns in the up-glacier hill and valley area. A hypothetical aligned-crystal ice band is shaded in each panel. (a) 1991 surface topography, as in Figure 2. (b) Vertical velocity anomaly, the difference between observed vertical component of velocity and the product of ice speed and local surface slope. Isotach interval is 0.5 m a⁻¹. Positive velocities are upward. (c) Longitudinal stretching (∂u_t /∂l). Isoline interval in all velocity gradient maps is 0.5 × 10⁻³ a⁻¹. (d) Transverse extension (∂u_t /∂l). (e) Lateral shearing (∂u_t /∂l). (f) Flowline turning (∂u_t /∂l). Variations in turning with respect to the regional leftward turning of 2 × 10⁻³ a⁻¹ are indicated by arrows. Stippling indicates sites that the flow tends to “avoid”.

Fig. 4. Topography and velocity patterns in the down-glacier valley area. Panel descriptions are the same as in Figure 3.

Local reference frames are defined such that comparisons are made with respect to the same portion of the bed. The X axis of the coordinate system used to plot the data lies along the long axis of the grid and is positive down-glacier. The Y axis is perpendicular to the X axis and is positive toward the southern (glacier-left) shear margin. Velocity gradients are calculated in a coordinate system (l, t) that is more significant to glacier flow. The longitudinal (l) axis is rotated to be parallel to ice flow, 10° clockwise from the X axis. The transverse (t) axis is perpendicular to the l axis and is rotated 10° clockwise from the Y axis. Both coordinate systems are fixed and do not translate or rotate with ice motion.

Longitudinal stretching (δu/δl);

Longitudinal (along-flow) normal strain rates are extensive. There are large variations on the several kilometer scale (Figs 3c and 4c). The variations do not correspond to changes in surface slope, as would have been expected were they due to flow over a basal obstruction (Reference Whillans, and Johnsen,Whillans and Johnsen, 1983). Time-changes in longitudinal stretching are shown in Figure 5.

Fig. 5. Profiles of longitudinal stretching, ∂u_t /∂l, in 10⁻³ a⁻¹, along transects between the A and B, and B and C lines. Gradients are calculated at the centers of quadrilaterals formed using four adjacent grid paints, and are plotted at the x direction midpoints of the quadrilaterals. Dashed and solid lines represent the 1984–85 and 1991–92 observations, respectively. The x axis (in this and Figs 4 and 6) corresponds to the x axis on the topographic maps (Fig. 2). There is an overall decrease in stretching of about 20%. This is due to the larger decrease in along-grid velocity occurring at the down-glacier end of the grid. The pattern of relatively more and relatively less stretching has remained fairly constant. An exception, at ≈−14 km, is on the glacier-right flank of the up-glacier hill.

The grid-wide pattern of relatively more and relatively less extension has remained fairly constant over time (Fig. 5). Magnitudes have decreased by about 20%. An exception is found along the glacier-right flank of the up-glacier hill (Α-B lines ~−13.5 km; B-C lines ~−13.5 to −15.5 km). The overall decrease in stretching is due to the larger decrease in along-grid velocity occurring at the down-glacier end of the grid.

Transverse Compression(δu_t/δt)

Transverse (across-flow) normal strain rates are negative due to across-flow compression as the ice stream narrows (Reference Vornberger, and Whillans,Vornberger and Whillans, 1990). Superimposed on this are local variations, most notably the large compression concentrated in narrow bands at the sites of the surface valleys (Figs 3d and 4d). A simple topographic relaxation model suggests that the topographic variation does not account for the local variations in compression (Reference Hulbe, and Whillans,Hulbe and Whillans, 1994). That model sought, but failed, to reproduce the surface topography from observed strain rates using both linear and non-linear relationships between topography and transverse compression and a range of ice viscosities. Strain rates in the valleys are too compressive to be caused by surface relaxation alone.

Time-changes in topography and in transverse compression are correlated. In the down-glacier end of the grid, the location of maximum compression (most negative ∂u_t /∂l) has moved up-glacier (Fig. 6, B-C lines, from ~−0.5 to −1.5 km). This is the same as the movement of the down-glacier valley (Fig. 2). Only the glacier-rightward edge of the upstream band of rapid compression is measured in both the 1984–85 and 1991–92 Studies. Here too, migration of topography and sites of rapid transverse compression appear to be connected. The up-glacier hill moved to the glacier-left between surveys, and compression increased leftward (ahead) of it (Fig. 6, A-B lines ~−14 through −16.5 km) and decreased rightward of it (in its wake) (Fig. 6, A-B lines ~−10 through −14 km; B-C lines ~−9 through −16.5 km). In both the down-glacier and up-glacier cases, the correlation between surface valleys and extreme compression is preserved as the valleys migrate.

Fig. 6. Profiles of transverse compression, ∂u_t /∂l), in 10⁻³ а⁻¹. Local compression maxima have migrated in the same sense as surface topographic features (see Fig. 2).

Lateral Shearing(δu_t/δt)

Lateral shearing, the across-stream gradient of along-flow velocity, is largest (most negative) near the margin, and approaches zero at the kinematic center line of the ice stream. The gradient of the shear strain rate is not simple, as would be expected if ice strength and basal resistance were spatially homogeneous. Instead, there are several zones where the spatial gradient in shearing is very small: from (x, y) ≈ (−16, −6) km to (x, y) ≈ (−10, −5) km in Figure 3e, and from (x, y) ≈ (−2, −10) km to (x, y) ≈ (2, −8) km in Figure 4c. A possible interpretation is that ice within these narrow (about 2 km wide) zones is especially strong in the direction of shear.

Time-change in lateral shearing is shown in Figure 7. This velocity gradient becomes faster over time, which is Curious in light of the grid-wide decrease in ice speed. Evidently, time-changes in ice velocity are not spatially uniform.

Fig. 7. Profiles of lateral shearing, ∂u_t /∂l), in 10⁻³ a⁻¹. In general, lateral shearing is more rapid (more negative) in 1991–92 than in 1984–85. Exceptions are found along the A and B lines profile. Up-glacier (more negative x), where a glacier-left facing slope has become a glacier-right facing slope, shearing has slowed. Down-glacier, where a surface valley has deepened, shearing has remained about the same.

Flowline Turning(δu_t/δl))

The along-flow gradient of across-flow velocity describes the change in across-flow motion as one proceeds down-glacier. This flowline turning is generally to the glacier-left (counterclockwise) within the strain grid area. The pattern of relatively more or relatively less glacier-leftward turning suggests that flow avoids two sites, (x, y) ≈ (−15, −8) km in Figure 3f and (x, y) ≈ (3, −9) km in Figure 4f. Glacier-left of the sites, turning is relatively more leftward (more positive values of ∂u_t /∂l). Glacier-right of the sites, turning is relatively more rightward (less positive ∂u_t /∂l).

Time-change in flowline turning is shown in Figure 8. This velocity gradient also becomes faster over time but with the opposite sign to the change in lateral shearing (Fig. 7). Together, these changes indicate little difference in the shear Strain rate. ∂u_t /∂l, but a faster counterclockwise rotation of ice within the ice stream.

Fig. 8. Profiles of flowline turning, ∂u_t /∂l, Glacier-leftward (southward) turning has increased for both profiles.

Horizontal Variations in Ice Strength

Spatial variations in strength in an ice stream may be due to variations in such properties as ice temperature or crystal orientation fabric. Shear margins located up-glacier could cause either kind of variation. Horizontal temperature gradients can result from strain heating within a margin. Simple calculations for the glacier-left shear margin of Ice Stream B (the Dragon) indicate a warming of 3–4 K that would persist for hundreds of years after the shear margin becomes inactive (Reference Whillans,, Jackson, and Tseng.Whillans and others, 1993). A second effect of shear is ice crystal alignment. Laboratory experiments and numerical modelling (Reference Kamb,, Heard,, Borg,, Carter, and Raleigh,Kamb, 1972; Reference Van der Veen, and Whillans,Van der Veen and Whillans, 1994) find that simple shear creates a fabric in which the с axes of ice crystals are aligned perpendicular to the plane of dominant shear. Field observations made near nunataks of the S⌀r-Rondane Mountains, East Antarctica, differ from those results (Reference Fujita, and Mae,Fujita and Mae, 1994). Ice samples collected there have c axes that are aligned but oriented ~45° from the down-glacier direction and presumed dominant shear direction. Regardless of which type of alignment occurs, the alignment will persist until new strains develop a new fabric; the threshold strain for recrystallization is 40–50% (Reference Jacka, and Maccagnan,Jacka and Maccagnan, 1984). Loss of crystal alignment by recrystallization would require ~100 years at the present study area, where strain rates are typically 2–3 × 10⁻³ а⁻¹. Thus, long after a margin becomes inactive, elevated temperature or crystal alignment can persist within the ice.

Warmer ice is weak in all directions. It is subject to faster transverse compression and lateral shearing than adjacent, colder ice. No special longitudinal (along-flow) stretching is expected in a warm ice band, because colder (stiff) ice blocks on either side would act as stress guides, carrying the longitudinal normal stress. If the weak ice band is not vertical, transverse compression may cause stiffer ice on the hanging-wall side of the band to be displaced upward with respect to the block on the other side of the band.

A band of ice with an aligned crystal fabric is weak only to shear along the basal planes, perpendicular to с axes. The band is easily deformed in that direction and is strong in others. For example, if с axes are aligned across flow (basal planes parallel to flow), the band is weak to lateral shearing but strong to transverse compression. If the c axes are not quite horizontal, transverse compression may result in relative vertical displacement of adjacent blocks.

Two bands of special deformation are identified in the study area (shaded in Figures 3 and 4). Transverse compression is especially fast (Figs 3d and 4d) and the gradient in lateral shearing is very small across the up-glacier and down-glacier surface valleys (Figs 3e and 4e). That combination implies ice that is weak in the transverse (across-glacier) direction, while it is strong to shear (Fig. 9). The interpretation may be that the bands mark ice with a crystal fabric in which с axes are diagonal to the flow direction, similar to the situation found near the Sr-Rondane Mountains (Reference Fujita, and Mae,Fujita and Mae, 1994).

Fig. 9. Cartoon of a band of aligned-crystal ice. The analysis indicates crystals are aligned such that the band is weak to transverse compression and strong to lateral shear. Transverse compression causes the upper block to move upward (up-in-cline) relative to the lower block.

Changes in ice-stream topography over time indicate that the bands of aligned-crystal ice must be inclined. Hills will grow both vertically and laterally as displacement proceeds along an inclined band. In the present study area, the glacier-left side of each valley is moving relatively downward while ice on the glacier-right side is moving relatively upward. It appears that ice to the glacier-right of the bands is being “pushed” up and to the left, over the aligned-crystal ice bands. This is similar to displacement of lithospheric blocks along a reverse fault, although in the ice stream there may be no discrete fault discontinuity. The direction of displacement implies that the bands dip toward the center of the glacier. Strain-rate changes between 1984–85 and 1991–92 are compatible with this interpretation. In both locations, the rates of transverse compression have decreased where stronger overriding ice supplanted an aligned-crystal band (Fig. 6). At the upstream site, lateral shearing has decreased slightly (Fig. 7) where the boundary between aligned-crystal and Stiff ice has been replaced by a stiff ice hill.

The surface expression of an aligned-crystal band should change with time and down-glacier motion. The portion of the band exposed at the surface should narrow due to progressive overriding as adjacent ice is pushed up-dip. Consistent with this prediction, both the upstream and downstream surface valleys, which overlie the bands, narrow downstream, as do the regions of rapid transverse compression and slow lateral shearing. The proposed bands of aligned crystals seem to be overridden by adjacent ice.

Deep Strain Rates

Deep strain rates can be inferred by considering volume continuity. This approach is practicable because the GPS surveying techniques used in this study yield precise vertical velocities. The principle of the calculation is that in order to conserve volume, transverse compression, for example, must be balanced by longitudinal and vertical extension. In general, these quantities, measured at the surface, do not balance one another. The imbalance is a measure of how much deep strain rates differ from measured surface strain rates. A method is developed in Appendix 2 to compute ice motion at depth using measured surface strain rates, vertical velocity, slope, and ice thickness. The resulting “deep divergence difference”,

The deep divergence difference so calculated is shown in Figure 10. It is limited to the portion of the strain grid for which precise ice-thickness data are available (from Reference Novick,, Bentley, and Lord,Novick and others, 1994). Variations in transverse compression dominate the calculation. This is deduced from the similarity between Figures 3d and 4d and Figure 10. Large positive values of deep divergence difference around (x, y) = (−1, −9) km and (x, y) = (−15, −5.5) km indicate sites of deep spreading. The negative anomaly at (х, y) = (−13, −6.5) km in Figure 10 suggests a destination for ice moving away from the positive anomaly to its glacier-left. That is, beneath the zones of rapid surface compression there is deep spreading.

Fig. 10. Deep divergence difference, computed as discussed in Appendix 2. Isoline interval is 0.5 × 10⁻³ a⁻¹. Sites of large divergence (large positive values) are beneath the hypothetical aligned-crystal ice bands.

The pattern of deep divergence difference is most readily interpreted in terms of lateral volume transfer. Relatively large spreading is occurring beneath both of the proposed aligned-crystal ice bands. To the glacier-right of the upstream aligned-crystal band, which is a site of anomalous upward motion at the surface, the calculated deep divergence difference is negative, indicating deep convergence. This site of negative divergence is the likely destination for ice moving away from the positive divergence beneath the band. This action supplies volume to the upward displacement of the adjacent ice block. The pattern is possibly the same for the downstream band, but that band is too close to the edge of the grid to determine the destination of deep ice. Deep ice seems to be moving from beneath the special bands to beneath uplifting ice.

Discussion

The proposed model of aligned crystal bands explains most of the relationships among strain rates, vertical velocities, and ice-surface topography and changes over time. The bands suggested in Ice Stream B are weak in the across-flow direction and strong to lateral shear. They are possibly inclined toward the center of the ice stream, leading to along-incline displacement of ice and the creation of hills and valleys in the surface topography. The bands may have originated with the coalescence of up-glacier tributaries. Bands with aligned-crystal fabrics acquired in tributary shear margins would now be incorporated into the main body of the ice stream. A strain history like that at S⌀r-Rondane would account for the present crystal orientation: the bands are soft to transverse compression because crystals have с axes oriented about 45° to the transverse direction.

The importance of aligned-crystal bands to ice flow, and ice-flow models, depends on scale. In this study, bands of aligned ice crystals appear to dominate local variations in strain rate and surface slope (over several-kilometer distances). Analyses of the budget of forces on a horizontal scale similar to the ice thickness require that these bands be identified and accounted. Reference Whillans, and Van der Veen,Whillans and Van der Veen (1993b) did not do that in their Ice Stream B force-budget calculation but assumed, as has been usual, that ice strength varies only with stress and temperature, which is taken to vary only with depth. Neglect of the possibility of spatial variations in ice-crystal alignment may account for the unusual values of basal drag, including negative values, they computed.

On the broad scale, bands of aligned ice crystals also affect assessment of the budget of forces. The shearing rate is at least two times slower and transverse compression is at least two times faster in the bands than in the surrounding ice (Figs 3 and 4). A reasonable guess for the fraction of special-strength ice is 10% (the fraction of the strain grid that overlies such ice). The gross-scale shearing rate is thus about 5% smaller (1O% of two times larger strain rates) than if the aligned-crystal bands were absent. This error is small compared with other uncertainties in the use of a constitutive-relation for ice. Thus, the bands do not dominate the broad-scale mechanics, so the usual approaches taken in modeling studies are not affected.

There is some similarity between the bands discovered here and the variable-strength banding proposed elsewhere for deep ice (Reference Hooke, and Hanson,Hooke and Hanson, 1986; Reference Cunningham, and Waddington,Cunningham and Waddington, 1990). Though the banding in those examples is thought to be in near-horizontal layers and at a much finer scale than the bands identified in Ice Stream B, the effect is similar. Deformation is concentrated in weaker zones, while boudins of stiffer ice survive with much less strain. Some of the topographic highs in the ice stream, like the up-glacier hill (Fig. 3) or the hill near the UpB camp (Fig. 4), may be large boudins that have formed and continue to be deformed by the processes suggested here.