2.1. Ice thickness and surface elevation

Since 1993, the University of Kansas has collected ice-thickness measurements over the Greenland ice sheet using airborne radar (Reference Gogineni, Chuah, Allen, Jezek and MooreGogineni and others 1998) operating concurrently with the NASA Airborne Topographic Mapper (ATM) lidar instrument. The radar is capable of sounding ice 3 km thick with a range resolution of 4.5 m (Reference GogineniGogineni and others 2001) while the lidar sensors resolve surface elevations to an accuracy of 10 cm (Reference Krabill, Thomas, Martin, Swift and FrederickKrabill and others 1995). Co-registering the radar and laser data and subtracting the radar ice thickness from the laser surface elevation provides the elevation of the bed at each sounding location.Because of its significance in ice-sheet drainage and potential contribution to sea-level rise, considerable attention has been given to Jakobshavn Isbræ with respect to the spatial distribution of radar surveys. The high density of radar flight-lines allowed point measurements to be interpolated to a 750 m grid using the ordinary kriging method (Reference Plummer, Gogineni, van der Veen, Leuschen and LiPlummer and others 2008). This bed topography map is used here.

Driving-stress estimates require accurate surface elevations, which are derived from airborne lidar sounding. A relatively high density of laser flights over Jakobshavn in 1997 and 2005 allowed for a detailed description of surface elevation in the lower channel in these two years. Ordinary kriging was used to create a continuous surface digital elevation model (DEM) for both years, with error estimates σ ≤ 20 m for 1997 and σ ≤ 5 m for 2005. These surface elevations are used to calculate values of driving stress in both years.

Ice-thickness data are needed to calculate the depth-integrated resistive stresses. Bed topography data from the University of Kansas are combined with surface elevation data for each epoch in order to determine changes in ice thickness. Because there was no airborne flight over Jakobshavn Isbræ in 1995, the assumption is made that from 1995 to 1997, changes in surface elevation were small (Reference Csatho, Schenk, van der Veen and KrabillCsatho and others 2008) and the 1997 surface elevations are applied in the 1995 force-budget calculation. No lidar was flown in 2000, so to estimate the surface DEM for that year, the 1997 and 2005 DEMs were linearly interpolated through time. This interpolation scheme was validated by comparison with data collected from 1999 and 2001 lidar flights. While this procedure may introduce additional errors in ice thickness, these are relatively minor (<10m), and resulting errors in the resistive terms associated with gradients in longitudinal stress and lateral drag are small (<10 kPa) and do not affect the main conclusions. Errors in ice thickness associated with errors in surface elevation are relatively minor. For the driving stress, on the other hand, small errors in surface elevation translate into large errors in surface slope and the estimated driving stress. For this reason, driving stress is only calculated for the times where accurate surface elevations are available (1997 and 2005). Consequently, basal drag can only be estimated for 1995 (using the 1997 driving stress) and 2005.

2.2. Surface velocity

Ice-flow velocity was measured from satellite interferometric image pairs in October 1995, October 2000 and October 2005 (Reference Joughin, Abdalati and FahnestockJoughin and others 2004, Reference Joughin2008a). The velocities were determined using standard speckle-tracking techniques applied to RADARSAT image pairs separated by 24 days with estimated errors of ±3% and gridded with a spatial resolution of 500 m (Reference Joughin, Tulaczyk, Bindschadler and PriceJoughin and others 2002).

2.3. Flowline and transects

Owing to the curvilinear nature and the small aspect ratio of the lower channel, it is impractical to calculate a full three-dimensional force-balance analysis using the traditional method of superimposing a grid with a single axis oriented parallel to the direction of flow (e.g. Reference Whillans, Chen, van der Veen and HughesWhillans and others 1989; Reference O’Neel, Pfeffer, Krimmel and MeierO’Neel and others 2005). For this study, a width-averaged force balance is calculated, with the curvilinear central flowline delineating the x-axis and a series of cross-sectional transects representing local y-axes. The westernmost transect is ∼250 m upstream of the 2006 terminus position and is located at the westernmost radar sounding in a 2006 along-flow channel flight. Nineteen additional transects are plotted at ∼1900 m intervals on, and perpendicular to, the central flowline upstream and eastward to a distance of ∼38 km from the front. The transects extend 5 km in both directions of the central flowline (Fig. 1).

Fig. 1.

Map showing the central flowline and location of transects locally perpendicular to the flow of the ice stream. Dotted curves show the boundaries of the ice stream, defined as that part of the glacier where lateral drag provides resistance to flow.

Channel boundaries for each transect are defined by the two inflection points of the transverse velocity gradient, ∂U/∂y. These inflection points correspond to the minimum and maximum lateral shear stress, R_xy , indicating the transverse location on the ice stream where the role of lateral drag changes from resisting the driving stress on the fast-moving part to acting in conjunction with the driving stress in dragging outboard slower-moving ice downstream (Reference Whillans and van der VeenWhillans and Van der Veen, 1997; Reference Van der Veen, Jezek and StearnsVan der Veen and others 2007). Figure 2 illustrates the procedure for determining ice-stream width. Figure 2a shows the velocity perpendicular to the selected transect and Figure 2b shows the lateral shear stress. The minimum and maximum shear stress define the boundaries of the ice stream. The distance along the transect between these channel boundary points is taken to be the channel width. Using this technique, the channel boundaries for each transect were very similar for all three years considered in this study. Because of this, and in the interest of consistency, the average boundary position between all years was calculated (Fig. 1) and a uniform channel width at each transect was adopted.

Fig. 2.

(a) Surface and bed elevation (scale on the left) and 2005 surface velocity (bold curve; scale on the right) across transect 2. (b) Lateral shear stress. Vertical lines mark maximum and minimum in shear stress and are taken to be the positions of the lateral margins of the ice stream.

2.4. Force balance

For each transect and epoch considered, the width-averaged surface velocity, U, ice thickness, H, surface elevation, h, and their average errors are calculated from the measurements and used to evaluate force balance along the ice stream (following techniques outlined in Reference Van der Veen and WhillansVan der Veen and Whillans, 1989). The width-averaged driving stress, τ _d, is calculated from

where x represents the flow direction, ρ = 917 kg m⁻³ is the density of ice, and g = 9.8 m s⁻² is the gravitational acceleration. At any transect location, surface slopes are estimated from width-averaged surface elevations at the two bounding transects (roughly 4 km distance). Because the driving stress is sensitive to small variations in surface slope, τ _d is calculated only for 1997 and 2005 for which accurate surface elevations are available. Because elevation changes prior to 1997 were small (Reference Csatho, Schenk, van der Veen and KrabillCsatho and others 2008) the 1997 driving stress is applied to the 1995 force budget to infer basal drag. The lack of accurate surface elevations in 2000 precludes estimating basal drag for that time.

Neglecting the contribution of shear strain rates to the effective strain rate, the longitudinal resistive stress R_xx is related to the stretching rate by invoking the flow law

where n = 3 is the flow-law exponent and B is the temperature-dependent viscosity parameter (related to the rate factor as B = A ^−1/n ). Strain rates are calculated from measured surface velocities and taken to be constant with depth. The high surface speeds suggest that most if not all flow is associated with basal sliding or with enhanced deformation of a soft basal layer (Reference Lüthi, Funk, Iken, Gogineni and TrufferLüthi and others 2002), in which case the velocity is constant over all or most of the ice thickness. The implication of this assumption is that resistive stresses may be overestimated. Following Reference ThomasThomas (2004), the value B = 400 kPa a^1/3 is adopted here, corresponding to an effective ice temperature (depth-averaged) of −10°C. Resistance to flow associated with along-flow tension and compression is given by

Gradients are evaluated from values at two bounding transects. Negative values of F _lon act in conjunction with driving stress, while positive values resist driving stress.

Averaged over the width, W, of the ice stream, resistance associated with lateral drag, F _lat, is given by

where the subscripts n and s refer to values at the northern and southern shear margins, respectively (Reference Van der VeenVan der Veen, 1999, section 5.8). The lateral shear stress, R_xy , is estimated from the shear strain rate, , as

Again, the shear strain rate is assumed constant over the ice thickness. Using a viscosity parameter B _m = 400 kPa a^1/3 results in F _lat values for 2005 that exceed the combined effects of driving stress and longitudinal stress gradients in the lower channel, resulting in negative basal drag. Because negative basal drag is physically unrealistic, the value of the viscosity parameter at the shear margins, B _m, was lowered to 300 kPa a^1/3 for all three times, possibly accounting for softer ice due to localized strain heating and fabric development.

Drag at the glacier bed, τ _b, cannot be estimated independently and is obtained from the requirement that forces acting on the glacier must balance, i.e.

Basal drag is estimated for 1995 and 2005, making the approximation that the 1997 driving stress can be applied to 1995.

Because of the geometry of the bedrock channel, the term ‘basal drag’ may be somewhat misleading as this commonly refers to flow resistance on a bed with little topographic relief in the transverse direction. In this case, the channel is narrow with steeply sloping side-walls, and basal drag includes all resistance originating at the valley walls under the main part of the ice stream. Arguably, some of the resistance included should be considered lateral drag similar to valley walls impeding the flow of mountain glaciers. For brevity of the discussion, this distinction is not made here explicitly.

In addition to determining the force-balance terms using the above formulas, associated errors are calculated using standard formulas for error propagation. The resulting equations for errors in the results are rather lengthy and are not discussed in this contribution (see Reference TaylorTaylor, 1997, ch. 3).

3.1. Driving stress

Surface elevations averaged over the width of the glacier (Fig. 3a) reveal along-flow thinning between 1997 and 2005. Thinning is most pronounced towards the terminus, but does persist to ∼40 km inland. Thinning increased progressively from ∼34 m at 37 km from the 2006 calving-front position to ∼120 m close to the 2006 terminus (at x = 0 in Fig. 3). The corresponding increase in large-scale surface slope (2 × 10⁻³ over 37 km) resulted in an average increase in driving stress of ∼20 kPa.

Fig. 3.

(a) Width-averaged surface elevation for 1997 and 2005 and bed topography along the lower 35 km of the main channel. (b) Corresponding driving stress.

At each transect, the driving stress is calculated from the width-averaged surface elevation at the two neighboring transects and thus represents an average over about two to three ice thicknesses. Figure 3b shows that the driving stress is significantly higher than what has been found on other ice streams (10–20 kPa; Reference Whillans, Chen, van der Veen and HughesWhillans and others 1989) or previously estimated for Jakobshavn (208 kPa; Reference Truffer and EchelmeyerTruffer and Echelmeyer, 2003). Over the interval 1997–2005, the width-averaged driving stress averaged over the entire flowline segment increased from 276 to 296 kPa, corresponding to the overall steepening of the ice surface. Locally, changes in driving stress exceed 100 kPa but it is not evident that these local changes are real or reflect different resolutions of the surface DEMs. Fewer laser data are available for 1997 than for 2005 and it is possible that the earlier DEM does not fully capture some of the small-scale surface features present in the 2005 DEM. For the purpose of this study – investigating the possible causes for speed-up – these small-scale effects are not important. The main conclusion relevant to the large-scale dynamics of Jakobshavn Isbræ is that the average driving stress increased by only a small amount, <10% over the lower region.

Figure 3 shows the existence of a distinct pattern of alternating low and high values of driving stress that appears to be stationary. These waves are likely linked to bedrock topography that induces small-scale variations in surface elevation. We note the similarity between these waves and those observed by Reference Reusch and HughesReusch and Hughes (2003) on Byrd Glacier who attributed the waves to sites of higher and lower basal resistance.

3.2. Gradients in longitudinal stress

The longitudinal stress, F _lon, is linked directly to the stretching rate and thus to along-flow velocity gradients. Width-averaged velocities are shown in Figure 4 and indicate that at ∼37 km from the 2006 front, surface velocities increased slightly less than 50% between 1995 and 2005. In the lower 40 km, the three sets of velocities increasingly diverged in the downstream direction until reaching their maxima at the terminus. At the 2006 front position, velocities were 5423 m a⁻¹ in 1995, 8924 m a⁻¹ in 2000 and 10 980 ma⁻¹ in 2005, corresponding to a ∼100% speed increase.

Fig. 4.

(a) Width-averaged surface velocity for each of the three years considered. (b) Longitudinal resistive stress, R_xx . (c) HR_xx . (d) Gradients in longitudinal stress.

The large speeds and associated stretching lead to corresponding longitudinal stresses of several hundred kPa (Fig. 4b). Over the period considered, R_xx increased along the entire glacier segment considered in this study. However, gradients in longitudinal stress multiplied by the ice thickness (Fig. 4c), as called for in the force-balance equation, remained small throughout the speed-up (Fig. 4d). A linear regression on the curves in Figure 4c gives an average contribution to the large-scale force balance of ∼10 kPa. Because gradients in longitudinal stress are positive, this term acts together with the driving stress in moving the glacier forward.

Spatial variations similar to those observed in the driving stress are present in the curves for F _lon; however, over the 10 year period considered, maxima and minima appear to have remained stationary and we propose that these local variations are linked to bed topography.

3.3. Lateral drag

Near the terminus, lateral drag contributes ∼200 kPa to flow resistance, decreasing towards the interior to <100 kPa (Fig. 5). The greatest increase in lateral drag occurred over the period 1995–2000; the subsequent change from 2000 to 2005 is much smaller and within error limits.

Fig. 5.

Resistance to flow from lateral drag.

3.4. Basal drag

Resistance originating at the glacier bed is estimated from Equation (6) to satisfy the requirement that the sum of all forces acting on a section of the glacier must be zero. Near the calving front, basal drag is small and decreased by ∼50 kPa throughout the speed-up (Fig. 6). Maximum basal resistance occurs ∼20 km upstream of the 2006 calving front; farther inland, this source of resistance decreases as driving stress decreases. Considered over the entire segment, average basal drag decreased slightly from 186 kPa in 1995 to 181 kPa in 2005.

Fig. 6.

Resistance to flow from basal drag.

3.5. Large-scale force balance

Table 1 summarizes the results of the force-budget calculations and gives the various terms averaged over the length of the glacier considered in this study. Driving stress increased by 20 kPa and most of this increase was balanced by an increase in lateral drag.

Table 1.

Force-balance terms for 1995 and 2005. Note that the 1997 value of driving stress is used to estimate basal drag in 1995

4.1. Release of back-stress

According to the buttressing hypothesis (e.g. Reference HughesHughes, 1986; Reference Howat, Joughin, Fahnestock, Smith and ScambosHowat and others 2008; Reference JoughinJoughin and others 2008a), the speed-up of Jakobshavn Isbræ resulted from loss of back-stress as the floating ice tongue weakened and disintegrated. Resistance to flow arises when a floating ice tongue pushes against, over or past an obstruction and may be crucial to the stability of the grounded part of the glacier by reducing the stretching rate at the grounding line.

As originally described, back-stress (or ‘back-force’) applies to a floating ice shelf and represents ‘the total force transmitted upstream by forces, other than the pressure of sea water at the ice front, that oppose the spreading of the ice shelf’ (Reference PatersonPaterson, 1994, p.298). In other words, back-stress represents the amount by which the stretching stress R_xx is reduced compared with the value for a free-floating ice shelf. For a free-floating ice tongue spreading in one direction only, this stretching stress is given by (Reference WeertmanWeertman, 1957; Reference Van der VeenVan der Veen, 1999, section 5.6)

where ρ _w represents the density of sea water (1025 kg m⁻³) and the superscript (0) denotes the solution in the absence of lateral drag or pinning points. Where the ice tongue is bounded by fjord walls, part or all of the driving stress is balanced by lateral drag and this results in reduced along-flow spreading. At any point on the ice tongue, the back-stress σ _b is defined as the down-glacier integrated resistance associated with lateral drag (Reference Van der VeenVan der Veen, 1999, section 5.6):

in which τ _s represents the lateral shear stress at the fjord walls, W is the width of the fjord and x = G is the position of the calving front. Correspondingly, the stretching stress is reduced to

The magnitude of the back-stress depends on the strength and structural integrity of the floating part and is not known a priori. However, an estimate can be made using the present results, assuming that Equation (7) can be applied to the 2006 calving-front position in all three epochs. Note that, in doing so, the estimated back-stress also includes basal resistance on the terminal portion seaward of the 2006 calving-front position that was grounded in earlier years. Comparison of this free-floating solution to the resistive stress obtained from measured velocity gradients allows the back-stress to be estimated. Results are summarized in Table 2, noting that R_xx (0) is estimated from extrapolation of the curves in Figure 4b. To convert measured strain rate to resistive stress R_xx , a viscosity parameter B = 400 kPa a^1/3 was applied. This produces values that are greater than the stretching stress for a free-floating ice tongue. Selecting a value for B that corresponds to ice close to the melting point lowers R_xx values to below those predicted by Equation (7). The last column in Table 2 gives the corresponding back-stress for the lower value for B; the results suggest a reduction of 60 kPa from 1995 to 2000 and a further decrease of 90 kPa between 2000 and 2005. These estimates for σ _b(0) are likely to represent upper limits and are used next to evaluate the effect of release of back-stress on flow up-glacier.

Table 2.

Comparison between stretching stress estimated from measured surface speeds, R_xx (0), and the solution for a free-floating ice tongue, , and inferred back-stress at the grounding line, σ _b(0). H(0) represents ice thickness at the grounding line

4.2. Transmission of longitudinal stress

A decrease in back-stress at the grounding line must be balanced by increased flow resistance on the grounded part and either lateral drag or basal drag must increase to accommodate the increase in glacier speed. Howat and others (Reference Howat, Joughin, Tulaczyk and Gogineni2005, Reference Howat, Joughin, Fahnestock, Smith and Scambos2008) assume that this increase is equally distributed over the coupling length L taken to be about 15 times the ice thickness. Reference Kamb and EchelmeyerKamb and Echelmeyer (1986) suggest a shorter coupling length of about seven ice thicknesses (∼7 km), while the coupling-length formula derived from scaling principles in Reference HindmarshHindmarsh (2006, p. 1755) gives L = 4 km. Integrating Equation (3) from x = 0 to x = L, gives

Averaged over the distance L, resistance associated with gradients in longitudinal stress is

If the coupling length represents the distance over which changes in back-stress are transmitted up-glacier, R_xx (L) remains unchanged and, ignoring for simplicity variations in ice thickness over time, the average decrease in is

With H(0) = 1000 m (slightly greater than the measured 1995 grounding-line thickness), L = 15 km (Reference Howat, Joughin, Tulaczyk and GogineniHowat and others 2005, Reference Howat, Joughin, Fahnestock, Smith and Scambos2008) and ΔR_xx (0) = 60 kPa, this gives . In other words, a lowering of back-stress of 60 kPa at x = 0 requires an increase in resistive forces of 4 kPa averaged over the lower 15 km of the glacier. Using a smaller coupling length of L = 4km (Reference HindmarshHindmarsh, 2006), the required average increase in resistive forces is 15 kPa. Compared with a driving stress ranging from 200 to 400 kPa, such minor adjustments are inconsequential and cannot explain the observed large changes in glacier speed.

It could be argued that initially the increase in stretching stress is balanced by a greater increase in resistive stresses but over a shorter distance than the coupling length L. To investigate this possibility, consider the increase in lateral drag across the lowermost transect. In 1995, the width-averaged speed (at transect 1 in Fig. 1 or x = 0 in Figs 3–7) was 5.4 km a⁻¹ and lateral drag provided 175 kPa to flow resistance; in 2000, the speed had increased to 8.9 km a⁻¹ and lateral drag amounted to 209 kPa (Fig. 5). Assuming this increase in flow resistance from lateral drag acted over the 2 km distance separating the first and second transect, this would have been sufficient to balance a lowering of back-stress at x = 0 km of 68 kPa. At the same time, the driving stress decreased slightly (Fig. 3b). A greater reduction in σ _b(0) could have been compensated for in ∼2 km from the grounding line, and farther up-glacier R_xx would have remained unchanged. Yet our results show that the stretching stress increased over the entire length of the glacier considered in this study (Fig. 4).

Fig. 7.

Ratio of effective basal pressure in 1997 and 2005 for two values of the exponent m in the sliding relation (Equation (13)) and observed ratio of surface velocity in 2005 and 1995.

From these considerations, the hypothesis that the speedup and retreat of Jakobshavn Isbræ resulted from the breakup of its floating ice tongue should be rejected. The associated decrease in back-stress was too small to significantly affect the stress distribution of the region under consideration here. Perhaps the reduction in σ _b(0) allowed the speed at the grounding line to increase, but because gradients in longitudinal stress remained small, this stress perturbation was not transmitted up-glacier, except perhaps through adjustments of the glacier profile, if any. While the present study only considers three time periods separated by 5 years each, velocity determinations at higher temporal resolution show that changes in glacier speed occur almost instantaneously along the lower 30–40 km of the glacier (Reference JoughinJoughin and others 2008a). If indeed these speed changes resulted solely from changes in boundary conditions at the glacier’s lower end, one would expect a kinematic wave of adjustment to travel up-glacier.

4.3. Approach to flotation

Reference PfefferPfeffer (2007) proposed an instability mechanism based on the behavior of a sliding law that relates sliding speed inversely to the effective basal pressure. As the glacier thins the effective basal pressure decreases, resulting in an increase in sliding speed that may be substantial if the ice thickness approaches the flotation thickness and the effective basal pressure approaches zero. The available data can be used to investigate whether this mechanism could have played a role during the speed-up of Jakobshavn Isbræ.

It must be noted that it is not evident that the large speeds on Jakobshavn Isbræ are due to sliding over a well-lubricated bed. Reference Iken, Echelmeyer, Harrison and FunkIken and others (1993) and Reference Lüthi, Funk, Iken, Gogineni and TrufferLüthi and others (2002) suggest that most of the fast motion is due to enhanced deformation in a soft basal layer. In that case, lamellar flow may be a more appropriate model and the ice velocity would be independent of effective basal pressure. This model is explored further in section 4.4. Furthermore, Reference Luckman and MurrayLuckman and Murray (2005) found that the speed-up began in spring 1998, which would have preceded the onset of significant thinning (Reference ThomasThomas, 2004). However, here we explore all previously suggested explanations for the rapid increase in speed on Jakobshavn Isbræ, including the model proposed by Reference PfefferPfeffer (2007).

Following Reference PfefferPfeffer (2007), the sliding speed U _s is related to basal drag and effective pressure P_e as

where A _s is a sliding parameter and the value of the exponent m is between 1 and 3. If an easy connection exists between the subglacial drainage system and the ocean, the effective pressure equals the difference between ice overburden and hydrostatic pressure (Reference Van der VeenVan der Veen, 1999, section 4.6):

where D represents the depth of the glacier bed below sea level. The additional pressure head required to drive water from the subglacial system is ignored.

During speed-up, basal drag did not change in an important way, so it is considered constant in the following analysis. If the sliding parameter can also be considered constant, the ratio of sliding velocity in 1997 to that in 2005 follows from

The ratio of effective pressures is shown in Figure 7 for m = 1 and m = 2. Also shown is the ratio of observed surface speeds in 2005 and 1995. This latter ratio is assumed to be a proxy for the relative increase in sliding speed (left-hand side of Equation (15)).

Comparison of the curves in Figure 7 shows that, depending on the value chosen for the exponent m, the decrease in effective pressure due to glacier thinning can explain the increase in glacier speed over the lower 4–10 km. Farther up-glacier, however, the increase in surface speed is more than can be explained by the decrease in effective basal pressure. This suggests that, while glacier thinning and increased sliding may have contributed to glacier speed-up near the terminus, other processes must have played a role in causing the acceleration farther up-glacier.

4.4. Change in subglacial hydrology

An additional hypothesis is that the observed speed increase is related to changes in subglacial hydrology. The drainage model described by Equation (14) assumes an easy connection between the subglacial drainage system and the adjacent ocean. The associated subglacial water pressure represents the minimum pressure required to discharge water over a sloping bed and towards the grounding line (Reference Van der VeenVan der Veen, 1999, section 4.6). Where drainage is impeded, higher water pressures are needed for water to move down-flow. These higher water pressures lower the effective basal pressure and increase sliding speed according to the sliding relation (Equation (13)).

There is abundant evidence for changes in sliding speed associated with changes in subglacial hydrology caused by variations in the amount of water supplied to the glacier bed (e.g. Reference Lappegard, Kohler, Jackson and HagenLappegard and others 2006; Reference Harper, Humphrey, Pfeffer and LazarHarper and others 2007; Reference Bartholomaus, Anderson and AndersonBartholomaus and others 2008; Reference FlowersFlowers, 2010). In particular during spring and early summer, large volumes of water may exceed the drainage capacity of existing passageways, resulting in temporary water storage in a pressurized drainage system. However, subglacial conduits respond by becoming more efficient over the course of the summer. Consequently, there are seasonal variations in glacier speed but no long-term sustained elevated speeds have been observed to result from hydrological changes. In the case of Jakobshavn Isbræ, continued increase in speed suggests that the subglacial drainage system is becoming increasingly inefficient over time, which appears to be an unlikely scenario. In fact, Reference Van de WalVan de Wal and others (2008) show a positive relation between surface melt and ice velocity on a seasonal timescale in the ablation zone in West Greenland. Over the entire 17 year duration of their measurement campaign, however, annual ice velocities decreased slightly despite a significant increase in ablation rate, suggesting the drainage system adjusts continuously to maintain a more-orless constant annual water discharge.

Similarly, Reference Sundal, Shepherd, Nienow, Hanna, Palmer and HuybrechtsSundal and others (2011) observed slowdown of several glaciers in West Greenland after a critical threshold in surface melting and runoff is reached. These observations are consistent with the theoretical model of Reference SchoofSchoof (2010) that captures the dynamical transition between channelized subglacial drainage (low subglacial water pressure) and drainage through interconnected cavities (high subglacial water pressure). This model predicts that higher rates of steady water supply to the glacier base can result in a decrease in sliding speed.

4.5. Weakening of marginal and basal ice

Our results show that the dynamics of Jakobshavn Isbræ are controlled by basal and lateral drags balancing the driving stress. This agrees with the study of Reference Truffer and EchelmeyerTruffer and Echelmeyer (2003) who found that observed velocities on Jakobshavn Isbræ could be modeled by assuming that lateral and basal drags balance the local driving stress. Close to the terminus, lateral drag provides most of the flow resistance, while farther up-glacier basal drag becomes the dominant term and this partitioning did not change significantly over the time period considered in our study. The question is how to explain the near doubling in glacier speed. Near the terminus, speed may have increased as the glacier thinned and the effective basal pressure decreased (Fig. 7) but this cannot explain the acceleration farther inland. To address this we consider two end-member solutions, namely laminar flow in which the driving stress is balanced entirely by basal drag, and the model in which all or most of the driving stress is balanced by lateral drag. In reality, flow of Jakobshavn is a combination of these two, but the two extreme models allow us to investigate qualitatively the possible causes for the observed speed-up.

Assuming laminar flow with no contribution from basal sliding, the surface speed is given by (Reference Van der VeenVan der Veen, 1999, equation (5.1.7))

where B is the viscosity parameter and n = 3 is used for the flow-law exponent. The contour plot in Figure 8a shows isolines of surface speed (in km a⁻¹) as a function of basal drag and viscosity parameter, using an ice thickness of H = 1500 m, the average thickness for the upper part of the ice stream. The plot shows that for values of basal drag found for Jakobshavn Isbræ (200–300 kPa; Fig. 6), a very small value for the viscosity parameter is required to produce velocities of several km a⁻¹ as observed on the glacier. This is consistent with the suggestion that the lower ice layers consist of soft temperate ice that facilitates deformation (Reference Iken, Echelmeyer, Harrison and FunkIken and others 1993; Reference HorganHorgan and others 2008). In the laminar flow model, in which vertical shear is concentrated near the glacier base, this can be accounted for by using a small viscosity parameter.

Fig. 8.

(a) Contours of velocity (km a⁻¹; contour interval 2 km a⁻¹)as a function of basal drag and viscosity parameter for lamellar flow. (b) Contours of velocity (km a⁻¹; contour interval 2 km a⁻¹ up to 20 km a⁻¹ and 10 km a⁻¹ for higher values) as a function of driving stress and viscosity parameter.

The contour plot in the lower panel of Figure 8 shows contours of surface speed for the model in which the driving stress is balanced entirely by lateral drag. In that model, the center-line speed is given by (Reference Van der VeenVan der Veen, 1999, equation (5.5.16))

In these calculations, H = 1500 m and W = 4000 m (note the change in contour interval above 14 km a⁻¹). For a driving stress of 300 kPa (Fig. 3), a viscosity parameter of 220 kPa a^1/3 predicts a center-line speed of ∼6 km a⁻¹, comparable with the 1995 speed near the terminus.

The contour plots in Figure 8 show that an increase in glacier speed can be achieved by increasing basal drag or driving stress or by lowering the viscosity parameter. For τ _b = 250 kPa and B = 140 kPa a^1/3 , the predicted surface speed is ∼4 km a⁻¹ according to the laminar flow model. Doubling this speed to 8 km a⁻¹ requires a decrease in the viscosity parameter to 120 kPa a^1/3 or an increase in basal drag to ∼320 kPa. For the second model, with driving stress balanced by lateral drag, an increase in speed from 6 to 12 km a⁻¹ requires a reduction in the viscosity parameter from 220 to 170 kPa a^1/3 or an increase in driving stress from 300 to 380 kPa. For both models, explaining the observed speed-up in terms of increased resistive stress requires changes in the resistive stress that are, generally, greater than inferred from the observations (Figs 3, 5 and 6), leading to the conclusion that the likely cause for speed-up was weakening of basal or marginal ice. If this weakening resulted solely from warming of the stress-bearing part of the ice column, an increase in ice temperature of 2–3°C would be implied.

4.6. Ice viscosity

An important parameter affecting results of the force-budget analysis is the viscosity parameter in the flow law linking measured strain rates to resistive stresses. Two different values are used here, one applicable to longitudinal stresses (Equation (2)) and one for lateral shear stresses (Equation (5)). An initial value of 400 kPa a^1/3 was selected for both (following Reference ThomasThomas, 2004), but our analysis indicates a lower value may be more appropriate.

The viscosity parameters used here to infer resistive stresses from measured strain rates apply to the longitudinal stress R_xx and lateral shear stress R_xy . The basal ice layer is temperate (Reference Iken, Echelmeyer, Harrison and FunkIken and others 1993; Reference Lüthi, Fahnestock and TrufferLüthi and others 2009) and may be further softened by development of crystal-orientation fabric (Reference HorganHorgan and others 2008) making the ice softer by a factor of three compared with ice with a random fabric (Reference ReehReeh, 1985). Thus, for linking vertical shear strain rate to vertical shear stress, a (much) smaller viscosity parameter should be used. However, this has no effect on our calculations because vertical shear is not explicitly considered and, instead, is inferred from force balance.

Using B = 400 kPa a^1/3 in Equation (2) to estimate the longitudinal resistive stress R_xx from the along-flow stretching rate results in values of this stress that are significantly greater than values corresponding to an ice shelf spreading unimpeded in one direction (Table 2). That comparison suggests a value for the viscosity parameter about half the initially selected value, applicable to ice within a few °C of the melting temperature (Reference HookeHooke, 1981). It is possible that the ice in the lower parts of the glacier is warmed by meltwater percolating downward into crevasses and refreezing at depth. For the present analysis and interpretation this is not an important issue. Gradients in longitudinal stress are unimportant in the large-scale balance of forces (Table 1), and selecting a smaller value for the viscosity parameter would further diminish the role of this term.

Selecting an appropriate value for the viscosity parameter used in the calculation of lateral drag is more important. A requirement is that resistance from lateral drag cannot exceed the driving stress as this would imply negative resistance at the glacier bed. This led us to adopt the value B _m = 300 kPa a^1/3 . It may be noted that with continued speed-up after 2005, a smaller value may be needed for force-balance calculations in the years following 2005. Considering the intense lateral shear across the margins, it is entirely plausible that the ice is softened locally by meltwater, strain heating and/or fabric development, resulting in a lower value for the viscosity parameter.

In our force-balance calculations, the same value for the viscosity parameter is used for all three years. If indeed the speed-up was caused by weakening of the lateral margins, lower values should be used for later years. This would reduce resistance from lateral drag in 2000 and 2005 and increase basal drag accordingly. Increased basal drag would further contribute to velocity increase. Investigating this interaction in more detail requires a model that incorporates all relevant stresses and their interactions, such as that employed by Reference Truffer and EchelmeyerTruffer and Echelmeyer (2003).