Velocity measurements

In May and June 1982, a network of 27 stakes was established on the glacier. Surveys of these stakes began on 3 June but the net was not completed until 23 June. This original net covered the greater part of the glacier; only the upper half of the accumulation area and the lowermost part of the ablation area were omitted in the interest of keeping the net small enough so that it could be surveyed by four people in one good day. In four places, up-glacier from stakes 02, 1.35, 188, and 263 on Figure lb, four stakes were set to form a roughly square strain diamond. In late March 1983, five additional stakes were added - the strain diamond up-glacier from stake 05 and the stake in the middle of the 02 diamond — in order to provide additional detail in an area where other experiments were in progress.

The 47 mm diameter stakes were initially 6 m long and were set in drill holes so that −0.5 m of the stake projected above the snow surface. As melting progressed, the stakes were cut. Then, when accumulation began in the autumn, extensions were added to keep the tops above the snow surface. Periodically, stakes melted out and had to be replaced. Ideally, the new stake was set and surveyed before the old one was lost, thus providing continuity. This ideal was not always achieved, so there are gaps in the record.

The stakes were normally surveyed at 7–10d intervals from mid-May to mid-September, weather permitting, and at 45–90 d intervals during the remainder of the year. During a survey the coordinates of the tops of the stakes were determined by measuring distances to two of four fixed points (three of which — 16.5, 18.5, and 20 — are shown in Figure la) using an AGA Geodimeter 12, and by measuring vertical angles to the stake tops from a third fixed point (either SM or NM; Fig. 1) with a Wild T-2 theodolite. The Geodimeter was placed directly on top of the stake and held in position by means of a short length of pipe that slid into or over the stake top. The reflecting prisms for the Geodimeter were permanently mounted on the fixed points. The Geodimeter measurements were adjusted for atmospheric pressure and temperature effects with the use of meteorological data from nearby Tarfala Field Station ( Fig.1).

Because the fixed points used for vertical angle measurements were different from those on which prisms were mounted, an iterative procedure was necessary to establish stake coordinates. A vertical coordinate was assumed and used to calculate initial stake coordinates based on the measured slope distances. A revised vertical coordinate was then obtained with the use of the vertical angle. The calculation converges rapidly.

At intervals of several weeks, a Brunton compass was used to measure the inclinations of stakes to the nearest 10' in two mutually perpendicular directions, parallel to the two axes of the net. Inclinations on the days of surveys were then obtained by linear interpolation or extrapolation. These measurements, together with the known length of the stake (including extensions), were used to determine the coordinates of the bottoms of the stakes. It was these coordinates that were used to calculate displacements and velocities. This procedure was adopted to accommodate changes in length of the stakes as they were cut or extended. All such changes in length were made in increments of 1.000 ± 0.002 m to simplify calculations.

Fig.1b.

Uncertainty in velocities

The uncertainties in the velocity measurements are discussed in Appendix I. For surveys made 7 d apart, as attempted during the last two summers of the study, the uncertainties in coordinates result in standard errors in the horizontal and vertical velocities of ±4.5 mm/d and ±1.5 mm/d, respectively, for the average stake and ±7 mm/d and ± 4 mm/d, respectively, for the most distant stakes. During the winter, when velocities were determined over periods of 6–10 weeks, the standard error is negligible.

By definition, some measurements will be in error by amounts substantially in excess of these estimates of the standard error. However, the overall consistency of the velocity variations among nearby stakes suggests that the large-scale pattern of velocity variation is correct, although some of the small-scale details may be due to measurement error.

Water-pressure measurement

Bore holes were drilled with a hot-water drill. We inferred that they had encountered the englacial or subglacial drainage system if the water level in the hole dropped well below the glacier surface during drilling and then began to oscillate diurnally. Pressure sensors, originally designed to measure oil pressure in automobile engines, were placed in these holes and connected to a chart recorder. Unless otherwise noted, there was no water flowing into the holes. Pressure sensors were also placed at depths of 35–40 m in some moulins.

Some of the sensors were calibrated as they were being lowered into the holes. Others were not, as we assumed, incorrectly, that sensors of a given model manufactured by the same company would have essentially the same calibration curve. Results from the uncalibrated sensors are subject to calibration errors of unknown magnitude. In addition, there is some uncertainty as to the depth at which some sensors were emplaced. Thus, there is considerable uncertainty in the absolute water pressures, but the direction of pressure changes and their approximate magnitude are believed to be reliable.

In holes drilled up-glacier from stake 1.35, water levels usually drop during drilling but rarely oscillate appreciably thereafter. We originally thought that these holes had not made contact with the glacial drainage system, and hence often did not instrument them. We now believe that subglacial water pressures are consistently high in this part of the glacier, averaging ˜ 90% of overburden pressure, and that this is somehow related to the existence of the till layer (Reference Hooke, Miller and KohlerHooke and others, 1988).

Horizontal velocities

There are a number of interesting features in the pattern of horizontal velocities shown in Figures 2 and 3a. First, there is a marked increase in seasonality with distance down-glacier. This is attributed, in part, to the contrast between the uniformly high and relatively constant water pressures in the overdeepening and the large diurnal pressure variations further down-glacier. The down-glacier decrease in ice thickness, particularly in the lower four strain diamonds, may be a contributing factor. With smaller ice thicknesses, a given increase in water pressure will be a larger fraction of the overburden pressure, and will therefore have a larger influence on the velocity (e.g. Reference BindschadlerBindschadler, 1983; Reference Iken and BindschadlerIken and Bindschadler, 1986).

Secondly, comparison of the temperature and precipitation records with the velocity variations reveals that each of the latter corresponds to a period of rising temperatures (resulting in increasing ablation rates) or with rainstorms of appreciable size or duration. Where data on water pressure are available, high water pressures generally accompanied the velocity peak. Several specific examples will be cited to illustrate these points:

1. In late June and early July 1982, a sharp increase in velocity in all strain diamonds then in use is correlated with an increase in temperature. Water levels in bore hole 82–1 increased slightly and water levels were generally high in moulin M2. Slightly after mid-July, a comparable increase in temperature, accompanied by rain, caused an increase in water pressure in moulin M2, but this had a noticeable effect on velocity only in diamond 1.35. Another increase in temperature and water pressure in late July had no effect at all.

2. In mid-June 1983, a low velocity peak in strain diamonds 03 and 05 is correlated with a temperature peak and high water levels in moulin Ml.

3. In late June 1983, the first major peak in velocity is correlated with a period of increasing temperature and rainy weather. (The subsequent temperature peak had no noticeable effect.) No water-pressure measurements are available for this time period.

4. The velocity peak in late July 1983 is again correlated with a temperature peak and accompanying rain. The peak in the 5 d running mean of the daily maximum water pressures actually precedes the velocity peak. However, in the more detailed record of the pressure variations (Fig. 4a), it will be seen that high water levels prevailed during the first part of the time interval spanned by the velocity peak.

5. Two subsequent smaller temperature rises in August and September 1983, the second accompanied by a good deal of rain, had appreciable effects on water pressure, but only in diamond 1.35 was there a corresponding peak in velocity.

6. Water pressures in bore holes 83–6 and 83–8 generally increased during the winter of 1983–84, but velocities decreased.

7. The first velocity peak in 1984 occurred in response to rising temperatures in late May. The increased water input resulted in rising water levels in bore holes prior to the velocity peak. By the time of the peak, however, water levels had dropped and were oscillating (Fig. 4b). The temperature peak occurred still later.

8. The second velocity peak in 1984 also occurred in response to sharply rising temperatures, which again resulted in high water levels in bore holes (Fig. 4b). (The more gradual rise in temperature and the accompanying rainfalls that followed during the next 50 d or so resulted in high water levels in bore holes, but this had practically no effect on velocity.)

9. The sharp rise in water pressure in September 1984 occurred in response to rain during the 7 d immediately preceding the last survey of the summer, and is visible in records from two moulins and three bore holes. Although this event occurred after a period of over 2 weeks with low water pressures, it did not result in a velocity peak. However, a distinct strain event did occur during this time, with compression between stakes 3.35 and 1.35, where the average strain-rate is 0, and with more than normal extension between stakes 07 and 05.

Noteworthy is the fact that not all periods of high temperature, rainfall, or water pressure resulted in increases in velocity. In the case of temperature and rainfall, it seems that preceding periods of high water input to the glacier probably opened the channel system sufficiently that renewed high inputs could pass through it without raising the water pressure. It is less obvious why some periods of high water pressure did not result in increases in velocity. Among possible explanations are: (1) use of 5d running means of peak pressures obscures important relations; (2) water pressures measured in bore holes located down-glacier from the riegel are not representative of those controlling the velocity; (3) velocities respond more to pressure changes than to absolute pressures.

The azimuths of the flow vectors also appear to vary systematically under some circumstances. In an experiment in 1987 on the south side of the glacier, near bore hole 87–1 ( Fig.1), Reference RydènRydén (1988) noted a systematic counterclockwise rotation of velocity vectors with increase in velocity. The vectors therefore became more nearly parallel to the glacier center line.

We examined the present data to see whether similar changes could be detected. In most cases, azimuths appeared to vary randomly, with deviations from the mean typically being less than 3°. As the standard error in azimuth is estimated to be ±3.3° for stakes moving 35 mm/d surveyed at 7 d intervals, these variations are assumed to be dominated by random errors in measurement. However, in a few cases there were systematic changes that are consistent either with expectation or with Rydén's observations. For example, during the 3 years of the study, there was a systematic counter-clockwise rotation in the azimuth of flow of stake 208. This is consistent with the overall flow pattern in this part of the glacier. Similarly, the flow vectors of stakes 198₁ and 2N rotated systematically inward to become more nearly parallel to the center line during the summer months. There is a suggestion of a similar change in azimuth for stake 2.1 and of an outward rotation of the flow vector of stake I98₄. However, the records for the latter two stakes are too noisy to be sure that the change is real. During the five major velocity peaks (early July 1983; late July 1983; late May 1984; mid June 1984; and late

June 1985), the azimuths at stakes 208, 03₂, and 03₃ tended to rotate counter-clockwise, and those of stakes 198₃, 3.35, 2.45, 1.35, 07, 06₂, and 06₃, clockwise. The changes at stakes 208 and 198₃ are generally consistent with a longitudinal stretching of the flow and a tendency towards necking during these special events. The changes at the other locations suggest that during such events the surface flow tends to reflect the bed contours more than normally is the case, with flow accelerating into the overdeepening down-glacier from the riegel.

Vertical velocities

There is also a seasonal variation in vertical velocity in some places on the glacier. In the overdeepening (strain diamonds 188 and 1.35), vertical velocities are systematically ˜3 mm/d higher during the summer, whereas down-glacier from the riegel (strain diamonds 05 and 02) they are systematically ˜1 mm/d lower (Fig. 3b).

The vertical velocity at the surface of a glacier, w_s , arises from a combination of flow on a sloping bed, vertical strain, and possibly englacial or subglacial cavity formation, thus:

where u_b is the horizontal component of the sliding velocity, ß is the bed slope, taken as positive when the bed slopes upward in the direction of flow, is the vertical strain-rate averaged over the glacier thickness, Z, and c is the vertical rate of cavity opening. Of these parameters, ß and Z can be obtained from radio echo-soundings, and , u _h, and c must be considered to be unknown for the purposes at hand, although measured surface strain-rates can be used to constrain , and there are some bore-hole data that constrain u _b (Reference Hooke, Holmlund and IversonHooke and others, 1987; paper in preparation by V. Pohjola and R. LeB. Hooke).

Reference Hooke, Brzozowski and BrongeHooke and others (1983b, 1987) (see also Reference Iken and BindschadlerIken and Bindschadler, 1986, p. 118–19) incorrectly used the horizontal component of the surface velocity, u _s, instead of u _b in Equation (I), and incorrectly believed that they could therefore estimate the rate of cavity opening at the bed.

Available data suggest that

varies with depth in valley glaciers (Reference RaymondRaymond, 1971; Reference HarrisonHarrison, 1975). We will now show that, in most cases, the vertical velocities are best understood by appealing to such a depth-dependence of e... Assume that 0 > u _b > u _s. Assume further that, if there is any cavity formation, it will occur during the summer. Thus, c 0 during the summer and c= 0 during the winter. Finally, assume that c = 0 in the overdeepening. This assumption is supported by the recent results suggesting: (1) that the overdeepening is underlain, at least in part, by a layer of deformable till ˜0.5 m thick (Reference Brand, Pohjola, and and HookeBrand and others, 1987); and (2) that drainage through the overdeepening is englacial rather than subglacial (Reference Hooke, Miller and KohlerHooke and others, 1988). Changes in the degree of dilation of the till layer could result in . However, an increase in porosity from say 30 to 40% would probably be more than sufficient to account for the observed variations in velocity (Reference JonesJones, 1979; Reference Alley, Blankenship, Bentley and RooneyAlley and others, 1987). Thus, unless the layer is considerably thicker than we presently believe it is, changes in porosity could not be large enough to effect w_s appreciably.

If c and u _b are both 0,

. If is 0 and u _b=u _s, . Thus, in the absence of cavity formation, w_s and w_s − u_stanβ represent limiting values of . Furthermore, if were independent of depth, would equal where is the value of at the surface. is calculated from measured values of and , assuming incompressibility. Thus, when does not lie between the limiting values for , it is likely that is varying with depth.

Values of w _s, , averaged over the 3 years of the study (2 years for strain diamond 05), are given in Table I. We have made separate calculations for the summer (July + August) and winter (September-May). June was omitted as it is a transition month, having characteristics of the winter regime in some years and the summer regime in others. The uncertainties shown in Table 1 are the standard deviations of the values from the 3 (or 2) years of measurement. To the extent that there were real inter-annual differences in the flow field, these should overestimate the standard error.

Table.1.

Interpretation of vertical velocities

*

may change sign at depth.

Detailed examination of the data in Table I reveals that

lies between its limiting values in only two instances, designated “no inconsistency” in the right-hand column of the table. In six of the remaining cases, the sense of the discrepancy is such that w _s would be higher (less negative or more positive) if e_zz were, in fact, independent of depth. We infer that i_zz , which is positive or extending at the surface in these cases, probably decreases with depth. In the remaining three cases the sense of the discrepancy is such that W _s would be lower (more negative or less positive) if were independent of depth. is negative at the surface “in these cases”, so we infer that it becomes less negative at depth.

In five instances, simply reducing to zero at depth is insufficient to explain the discrepancies. For example, beneath strain diamond 263 the bed slopes downward in the direction of flow and w_s is negative. It appears that only part of this negative w_s can be attributed to the bed slope, so

must be negative. However, is positive, so it appears that must change sign it depth.

Two points in Table I merit further comment. First, in strain diamond 05, is negative in 1983 and positive in 1984. While this mäy be due to errors in measurement, it is noteworthy that 1983 was a year of rather high summer velocities. Secondly, over the riegel somewhat up-glacier from strain diamond 1.35, bore-hole deformation measurements have suggested that the velocity at the bed may exceed that at the surface by a factor of four to five in late July and early August (Reference Hooke, Holmlund and IversonHooke and others, 1987). Thus, in this case, w _s - u _stan ß may not be a limiting value of . However, bore-hole deformation measurements in 1985 in the middle of strain diamond 1.35 suggest that the extrusion flow does not extend this far up-glacier.

Given the initial assumptions, there is only one instance in which the above results could be explained by a non-zero value of c. This is during the summer of 1983 in strain diamond 05, where the data could be interpreted as indicating that

was positive.

We return now to the problem of seasonal variations in W _S noted in the first paragraph of this section. The decrease in W _S down-glacier from the riegel during the summer suggests that here the longitudinal strain-rate,

, becomes less compressive or the transverse strain-rate, more extending at this time of year. The seasonal increase in velocity of strain diamond 05 tends to be greater than that of diamond 02, so becomes more compressive. On the other hand, velocity vectors near the margins (stakes 03₁ and 03₄) tend to rotate outward, so becomes more extending. The change in i_xxs is larger. Possibly increases (becomes less compressive) with depth, and does so more rapidly during the summer than during the winter. However, bore-hole deformation measurements (Reference RaymondRaymond, 1971; paper in preparation by V. Pohjola and R.LeB. Hooke) suggest that becomes more extending with depth in the ablation area of a valley glacier. If this effect increases seasonally here, it could explain the decrease in w_s near the center line. Noteworthy in this regard is the fact that w _s increases during the summer at stake 03₄.

The seasonal increase in w _s throughout the overdeepening, on the other hand, suggests an increase in longitudinal compression or a decrease in transverse extension. The summer increase in u_s is slightly larger in strain diamond 1.35 than in diamond 188, so

tends to become less compressive (Table I). However, the values for strain diamond 188 in Table I imply that during the summer, may not decrease with depth, whereas during the winter it very probably does. This suggests the possibility of an increase in horizontal compression at depth during the summer. Additional analyses presented below suggest that this is due to a decrease in basal drag near the middle of the overdeepening at this time of year. In addition, surface-velocity data and bore-hole deformation measurements indicate that here, in contrast to the situation further down-glacier, becomes less positive during the summer both at the surface and at depth. Consistent with such a change in is the fact that vertical velocities along the sides of the glacier (stakes 198₁₍ 198₄, 2.1, and 2N) increase an average of only ˜2 mm/d during the summer, rather than ˜3 mm/d as in the central part.

In the case of the overdeepening, we can make a rough calculation of the magnitude of these changes. Over a control area 400 m wide in the center of the glacier and extending 700 m from stake 188 to stake 1.35, an increase in vertical velocity of 3 mm/d would require an extra influx of ice of ˜850 m³/d. A summer decrease in basal drag beneath stake 188 would straighten the velocity profile and thus increase the flux; assuming laminar flow (Reference NyeNye, 1952, Equation (2)) and holding the surface velocity constant, a decrease from 0.8 to 0.6 bar (0.08 to 0.06 MPa) would increase the flux here by ˜400 m³/d. At the down-glacier end of the control area, the ice is only about half as thick, so a comparable change in basal drag would increase the loss of ice from the control area by only ˜100m³/d. Along the sides of the control area, the surface-velocity data suggest an inward rotation of velocity vectors of ˜1.3° during the summer (ignoring a rotation in the opposite direction at stake 198₄, which is in a somewhat special position at the base of a steep slope leading down from the accumulation area). u_s is ˜30 mm/d here, so such a change would correspond to a decrease in outward flux of ˜170m³/d. Bore-hole deformation measurements suggest that

may decrease even more at depth. For example, relative to vectors at the surface, the outward rotation of velocity vectors in the lowermost 50 m of hole 87–1 ( Fig.1) was 2–3 less during the last two-thirds of July 1987. Such a change, if it occurred on both sides of the control area, would decrease the outward flux of ice by an additional −150m³/d. Thus, with due consideration for the uncertainties in the calculations, it appears that seasonal changes in the variation of longitudinal and transverse strain-rate with depth probably can account for the increased vertical velocity in the overdeepening.

Longitudinal strain-rates along glacier center line

Mean annual longitudinal strain-rates were calculated between stakes lying along or near the center line of the glacier. (Stakes 06₂ and 06₃ were considered to be near the center line for this purpose; their average velocity was used in the calculations.) These strain-rates are shown in Figure 5a, superimposed on a cross-section of the glacier.

Fig.5.

a. Mean annual longitudinal strain-rate superimposed on longitudinal cross-section of the glacier, b. Differences between longitudinal strain-rates over six selected lime periods and mean annual longitudinal strain-rate. Positive values indicate that over the relevant time period is higher (more extending or less compressive) than the mean annual value. Gaps occur where data are missing for certain surveys. Dashed sections of these curves are less reliable than solid sections, either because a slake was cut just before or just after the event, or because the azimuth of the flow deviated significantly from the norm during the event, suggesting a possible survey error.

As expected, strain-rates are generally extending in the accumulation area and compressive in the ablation area. The most notable exception to this is over the riegel beneath stake 07, where the increase in slope of the glacier surface leads to extending strain-rates that are high enough to result in crevassing. The rather high compression beneath the equilibrium line is also contrary to normal expectations. This results from the combination of the down-glacier decrease in surface slope in this area and the downward thickening of the glacier as it enters the overdeepening.

Longitudinal strain-rates were also calculated over shorter time periods, and the difference between the mean annual longitudinal strain-rate and that during these time periods has been plotted in Figure 5b. To provide greater resolution in strain diamonds 1.35 and 188, the velocity at the centers of these diamonds was estimated by averaging the velocities at the two stakes on either side of the center; thus strain-rates were also calculated between, for example, stake 208 and the 198 line and between the 198 line and stake 188 (Fig. 1b). The absolute values of these strain-rates are biased because velocities decrease towards the margins, but differences between these values and the corresponding mean annual values are probably meaningful.

In the early winter (late November 1984 to late January 1985), strain-rates between stakes 246 and 188 are more compressive than average (Fig. 5b). Apparently, the winter deceleration is faster in the overdeepening than on the steeper slope up-glacier therefrom. There is also, over both riegels, a pattern of relative (with respect to the mean annual) compression above the riegel (stakes 1.35 → 07 and 04 → 02), and a compensating extension below the riegel, as if the glacier had grounded on the riegel.

Changes in longitudinal strain-rate during the high-velocity events are discussed in the next section.

Response of glacier to short periods of high water input

To study the dynamic response of the glacier to high water inputs resulting from warm weather or rain, the five major high-velocity events between June 1983 and June 1985 (Fig. 3a) were investigated in greater detail. First, we studied the difference between the strain-rate during the event and the mean annual longitudinal strain-rate (Fig. 5b), and secondly, we used a force-balance calculation (Reference Van der Veen and WhillansVan der Veen and Whillans, 1989) to investigate changes in basal drag during the events.

In interpreting these calculations, two points should be borne in mind. First, because, by definition, some velocity measurements will be in error by an amount considerably in excess of the estimate of standard error given earlier, we do not rely upon any single measurement; instead, we seek patterns of behavior that are consistent among many or all of the five events. Secondly, the actual duration of a velocity peak may have been much shorter than the time interval between surveys, or may have spanned parts of two survey periods. Thus, it is unlikely that the timing of the surveys would be such as to catch any two events at precisely the same stage.

Two consistent patterns in the changes in longitudinal strain-rate during the events (Fig. 5b) merit comment. First, between stakes 208 and 5.3 there is always a region over which strain-rates are less compressive than the mean annual rate. In general, strain-rates remain compressive during the velocity peaks but between stakes 188 and 5.3 they do, in fact, become extending in two cases. In contrast, down-glacier from stake 5.3, in some cases extending down to the 2 line, strain-rates during the velocity peaks are generally more compressive than normal. Down-glacier from the upper riegel (stake 07) there is, in most cases, a similar pattern of abnormal extension (or less than normal compression) followed down-glacier by a zone of abnormal compression.

High water inputs are likely to result in a reduction in basal drag, thus increasing the sliding speed; the observations just discussed suggest that in the central part of the glacier the most effective reduction occurs along the adverse slope leading out of the overdeepening, and in particular in the vicinity of stake 5.3. The zone of enhanced compression down-glacier from this interval rules out the possibility of explaining the velocity increase by tensional pulling from accelerating down-glacier ice. Likewise, the change from compression to extension between stakes 188 and 5.3 in two cases rules out the possibility of explaining it by push from up-glacier.

Secondly, in the area just up-glacier from the upper riegel, there was abnormal extension in all five of the events under consideration. In three of these, the extension continued down-glacier to or below stake 07, but in the other two the strain is compressive between stakes 1.35 and 07. This latter suggests that the acceleration in the vicinity of stake 1.35 is not due to pull from the more obvious accelerations that occur below the riegel. Rather, we hypothesize that subglacial water pressures may increase more here, relative to the area between stake 5.3 and the 2 line, because the bottom topography ( Fig.1) forces water toward the glacier center line as it passes over the riegel. Immediately after these events the strain-rate between stake 1.35 and the 2 line becomes either appreciably less extending or, in three of the five cases, strongly compressive. This suggests that the effects of a high water input last longer in the overdeepening.

The force-balance analysis further elucidates the first of these points. In this analysis, a block of the glacier is selected (e.g. Fig. 6), and measured strain-rates are used to estimate the longitudinal forces on its up-glacier and down-glacier ends and shear forces on its sides. Subtracting the body force leaves the drag on the bed as a residual. Summing forces in the χ or down-glacier direction yields:

where Rs are resistive stresses (using Reference Van der Veen and WhillansVan der Veen and WhiIIans' (1989) terminology), ∆x, ∆y, and ∆z are the average dimensions of the block, n is the outwardly directed unit vector normal to the face, ρ is the density of ice, g is the acceleration due to gravity, 5 is the surface slope (which is negative in the +x-direction), and τ_b is the desired drag stress. The subscripts x and y refer to the horizontal coordinate directions; following normal convention, the first subscript in a pair gives the normal to the plane on which the stress acts, and the second gives the direction of the stress (Reference JaegerJaeger, 1969, p. 3). The subscripts u, d, n, and s refer to the up-glacier, down-glacier, north, and south sides of the block.

Dividing by ∆x∆y and letting ∆z = Z yields:

The resistive stresses are defined by R_ij = τ_ij − δ_ijL (i,j, = x,y,z) where L = −ρgZ, τ_ij is the total (not deviaioric) stress, and δ_ijis the Kronecker delta. We assume that τ_zz = L which means that there are no “bridging effects” resulting, for example, from extensive separation of ice from the bed. Thus R_ZZ = 0. Then, using a Glen-type flow law,

, we find, after some manipulation, that:

Fig.6.

Three-dimensional sketches of blocks used in force-balance calculation. Locations of blocks are shown in Figure 1.

(Reference Van der Veen and WhillansVan der Veen and Whillans, 1989) where τ_e and

are the effective stress deviator and strain-rate, respectively. Given the strain-rates required to evaluate Equations (4) and the dimensions of the block, Equation (3) can be solved for

In the present application, strain-rates were determined from their definitions in terms of velocity derivatives. The latter were estimated from first differences after first determining the components of the velocities of the relevant stakes parallel to the sides of the block. Appendix II gives details of the specific stakes used to obtain the required velocity derivatives.

Calculations of the absolute value of the drag in this way make the critical assumptions that strain-rates are: (1) independent of depth, and (2) sufficiently uniform over the ends and sides of the block that the values determined above can be applied over these respective surfaces. However, here we are concerned primarily with changes in drag between different time periods. Thus, for our purposes, we assume only that any such lateral or vertical variations in strain-rate were sufficiently similar between the different time periods studied to make comparisons meaningful.

The essential result of these calculations is that, during the high-velocity periods, the basal drag in the overdeepening seems to have been reduced anywhere from 16 to 40% relative to its winter (31 October 1983–3 March 1984) value. In contrast, below the riegel the basal drag seems to have increased significantly during two of three events, and to have been reduced only slightly during the third. (Unfortunately, because data for some stakes are missing or believed to be unreliable, calculations could not be done for this area during the other two time periods.)

The changes in drag appear to result principally from changes in the longitudinal stresses. This can be seen in columns 8 and 10 of Table II, where the values of terms A and B in Equation (3) are given. Compared with the winter values, the values of term A (longitudinal forces on ends) vary by up to 0.25 bar (25 kPa), while those of term B (shear forces on sides) vary less than 0.1 bar (lOkPa). However, the longitudinal stress is not necessarily the independent variable.

In the case of the block in the overdeepening, the independent variable is probably T_B; decreases in drag, presumably caused by increases in water pressure, result in an acceleration. At the up-glacier end of the block, as already noted (Fig. 5b), this results in a decrease in compression and in some cases even extension. At the down-glacier end, in most cases, there is a corresponding increase in compression. Shear stresses along the sides increase modestly in all cases.

As might be expected from the above discussion, there is a reasonably good correlation between the percentage change in drag in the overdeepening and the change in

at the up-glacier end of the block ( Fig.7). Correlations were also sought between the velocities of stakes and the percentage reduction in drag. However, longitudinal strain-rates within the block apparently vary sufficiently among the various events (Fig. 5b) to render such correlations weak.

Fig.7.

Relation between percentage change in basal drag in the overdeepening and t_xxs near slake 188.

On the smaller down-glacier block the situation is quite different. In two of the three cases, the normal extension at the up-glacier end of the block changes to compression, indicating push from up-glacier, and in these two cases the normal compression at the down-glacier end of the block is reduced, in one case even becoming extension. The clear implication is that changes in basal water pressure have affected areas over the riegels up-glacier and down-glacier from this block more than the area directly beneath the block. In response to the resulting push from up-glacier and decreased resistance down-glacier, the block has accelerated, increasing shear strain-rates on the sides and base. Due to the proportionality between stress and strain-rate, the latter has resulted in an increase in drag at the base.

Table.2.

Force-balance calculations

* Upper number is south side and lower number is north side.

During the late May 1984 event, the situation in the vicinity of the down-glacier block was slightly different. In this case, it appears that drag was decreased beneath the block and this, in conjunction with a little less resistive “pull” from up-glacier, resulted in an acceleration of the block, which then rammed into the still lower part of the tongue, producing a large compression (Table II, col. 5). It is perhaps significant that during this event, which occurred earlier in the melt season than any of the others, compressive strain-rates at the down-glacier ends of both blocks increased markedly. Relevant is the fact that the two major areas of water input to the glacier are areas of crevassing just up-glacier from the two blocks. The change-over from winter to summer drainage conditions may progress down-glacier from these areas of water input.

The extent to which variations in strain-rate with depth might affect these calculations should be addressed. From the preceding discussion, it would appear that

at the up-glacier ends of the blocks is most critical. Were we to assume that decreased linearly with depth here, and thus, for example, use one-half of its surface value in the calculations for the larger block, the winter value of τ_b would decrease from −0.82 to −0.63 bar, and the May 1984 value, as an example, would be reduced from −0.58 to −0.49 bar. The reduction in basal drag would then be 22% instead of 29%. probably decreases with depth, and the decrease is probably less in the summer than during the winter, judging from the discussion of vertical velocities above. Thus, the reduction in drag is probably overestimated in Table II. However, the decrease in the depth dependence of E._ during the summer strongly suggests a decrease in drag at that time of year. Thus, the errors in the estimates of τ_b in Table Il are unlikely to be so high as to negate the qualitative conclusions reached.