Dynamic controls on glacier basal motion inferred from surface ice motion

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jf000925, 2008 Dynamic controls on glacier basal motion inferred from surface ice motion Ian M. Howat, 1 Slawek Tulaczyk, 2 Edwin Waddington, 3 and Helgi Björnsson 4 Received 23 October 2007; revised 2 April 2008; accepted 1 May 2008; published 13 August [1] Current heuristic laws that relate the motion of glaciers due to sliding along the bed to the subglacial water pressure fail to reproduce variations in sliding speed on timescales of specific hydrologic events, such as lake drainage, rainfall, or surging. This may be due to the importance of subglacial cavity evolution and shifts in the glacier stress field, both of which are not accounted for in typical sliding laws. We use multiple time series of surface motion over a 66-day period at Brei*amerkurjökull, Iceland, to infer changes in bed separation and longitudinal force budget. We observe multiple, distinct periods of increased surface motion and uplift corresponding to periods of rainfall and/or increased temperatures. We find consistent hysteresis and lags between motion and variations in both the bed separation and longitudinal stress gradient that we attribute to the redistribution of normal stresses at the bed during cavity growth. Increases in the longitudinal stress gradient suggest a downglacier stress transfer during increased basal motion that is consistent with increased drainage system efficiency toward the terminus. Our results suggest that the transient evolution of the subglacial drainage system and shifts in the glacier stress field are important controls on basal motion. Citation: Howat, I. M., S. Tulaczyk, E. Waddington, and H. Björnsson (2008), Dynamic controls on glacier basal motion inferred from surface ice motion, J. Geophys. Res., 113,, doi: /2007jf Introduction [2] Many studies have documented a relationship between water input, from surface melting and rainfall, and increased ice-flow speed on temperate and polythermal glaciers and ice caps [e.g., Anderson et al., 2004; Iken and Bindschadler, 1986; Jansson, 1995; Kavanaugh and Clarke, 2001; Meier et al., 1994; Sugiyama and Gudmundsson, 2004] and the margin of the Greenland Ice Sheet [Zwally et al., 2002]. The observed increase in ice speed with increased water input is attributed to increased pressurization of the subglacial drainage system, leading to increased separation from the bed, loss of basal traction, and enhanced basal motion. Basal motion, either through deformation of subglacial till or sliding at the ice-bed interface, can account for much of the total annual ice throughput of glaciers and is therefore an important control on mass balance. However, the mechanics of basal motion remain poorly understood and are given only rudimentary empirical treatment in iceflow modeling. [3] Classical (i.e., hard bed ) sliding theory postulates that a layer of water at the ice-bed interface cannot support a 1 School of Earth Sciences and Byrd Polar Research Center, The Ohio State University, Columbus, Ohio, USA. 2 Department of Earth and Planetary Sciences, University of California, Santa Cruz, California, USA. 3 Department of Earth and Space Sciences, University of Washington, Seattle, Washington, USA. 4 Institute of Earth Sciences, University of Iceland, Reykjavík, Iceland. Copyright 2008 by the American Geophysical Union /08/2007JF shear stress, so that drag at the glacier sole is generated by undulations in the topography that obstruct ice flow [Fowler, 1981; Kamb, 1970; Weertman, 1957, 1979], as well as friction from basal debris that bridge the water layer [e.g., Iverson et al., 2003; Schweizer and Iken, 1992]. As long as these bed obstacles are small relative to the ice thickness [Fowler, 1981], the motion of basal ice, u s, along the bed can be related to the large-scale basal drag, t b, as: t b ¼ Cu m s where C and m are positive constants that are dependent on basal roughness and ice rheology. When water pressure increases to a threshold fraction of the overburden, cavities will form on the lee (down-glacier) side of bed obstacles, reducing the contact area between ice and bed and reducing further the basal drag exerted on the glacier sole [Lliboutry, 1968]. Since the size of these cavities should vary with water pressure, the subglacial effective pressure, p e, defined as the ice overburden minus the water pressure, is included in equation (1) as: t b ¼ Cu m s pn e ; C; m; n > 0 ð2þ which has been the standard sliding law used in ice-flow modeling for over three decades [Bindschadler, 1983; Budd et al., 1979; Paterson, 1994; Pattyn, 2002]. [4] Many attempts have been made to calibrate and validate equation (2) with observations [e.g., Bindschadler, 1983; Blake et al., 1994; Budd et al., 1979; Iken and Bindschadler, 1986; Jansson, 1995]. Typically, this ð1þ 1of15

2 involves comparison between ice surface velocity and borehole water levels, representing the glacier water pressure, over periods ranging up to months. On seasonal timescales, a general proportionality exists between sliding and water pressure, supporting equation (2), but the scaling parameters in this relationship can change from year to year [e.g., Hooke et al., 1989], or between different periods in a single melt season [e.g., Iken and Truffer, 1997]. [5] On the shorter timescales associated with distinct anomalies in ice speed induced by rainfall, lake drainage, intense melting, or during glacier surges, the temporal relationship between borehole-derived water pressure and sliding tends not to be in phase or temporally consistent [e.g., Harper et al., 2005, 2007; Iken, 1981; Iken et al., 1983; Kamb et al., 1994]. Most commonly, sliding increases with increasing water pressure, peaks hours to days before the peak in water pressure, and then decreases more quickly than water pressure [Blake et al., 1994; Iken and Bindschadler, 1986; Kamb et al., 1994; Sugiyama and Gudmundsson, 2004]. This transient pattern results in a hysteresis in the relationship between sliding speed and water pressure that is in conflict with the assumption of steady state implicit in equation (2). [6] Based on measurements at Columbia Glacier, Kamb et al. [1994] hypothesized that sliding speed should be more directly and consistently related to changes in water storage within the subglacial drainage system, rather than to water pressure measured in boreholes. This is because borehole water pressures may not be representative of the mean water pressure over the length of longitudinal stress coupling, and therefore not representative of the effective pressure in equation (2). A similar conclusion was drawn by Harper et al. [2005, 2007] using a large network of basal pressure sensors. Kamb et al. [1994] found an improved relationship between sliding speed and the rate of water storage, measured from the net rate of glacier filling, which they argue is more representative of the aerially averaged water pressure. This conclusion is supported by measurements of vertical ice motion (uplift) during increased basal motion. Since uplift results, in part, from increased subglacial cavity formation and water storage [e.g., Iken et al., 1983; Mair et al., 2002b], the hypothesis of Kamb et al. [1994] would predict a direct relationship between the rate of uplift and sliding. Such a relationship has been observed at several glaciers [e.g., Anderson et al., 2004; Harper et al., 2007; Iken and Bindschadler, 1986; Kamb and Engelhardt, 1987; Sugiyama and Gudmundsson, 2004]. However, to directly infer separation at the bed from the vertical motion of surface markers, the contributions of vertical strain and bed-parallel motion must be subtracted from the total uplift [Hooke et al., 1989]. It is unclear from existing records whether bed separation varies with sliding speed in a more direct and consistent way than water pressure varies with sliding speed. [7] Another cause for the hysteresis between sliding and water pressure may be the evolution of the subglacial drainage system. It is well known from observations that the subglacial drainage system and the distribution of water at the glacier bed vary with changing hydrologic conditions [Fountain and Walder, 1998; Harper et al., 2002, 2007; Kamb, 1987; Kavanaugh and Clarke, 2001; MacGregor et al., 2005; Mair et al., 2002a; Nienow et al., 1998]. Therefore in cases where the subglacial drainage system evolves primarily due to changes in hydrological conditions, rather than by ice deformation, it is likely that a transient relationship exists between sliding and water pressure. Using a numerical model of transient cavity evolution and sliding, Iken [1981] found that sliding should be greatest when the rate of increase in water pressure is the greatest and before cavities expand to their stable configuration. As cavities open, the overburden is supported by an increasingly small portion of the bed. These areas of high normal stress will be concentrated on the upstream side of bed obstacles, with lower normal stress on the downstream sides. Spatially integrating these normal stresses gives the total force acting in the upstream direction, the average of which is the basal drag. In this way, cavity growth leads to increased gradients in normal stress at the bed and the basal drag. Therefore basal drag will increase or sliding velocity will decrease under constant effective pressure as the cavities expand. If a s is the instantaneous fractional area of separation at the bed, a transient sliding law should then take the form: t b ¼ fðu s ; p e ; a s Þ ð3þ [8] Therefore assuming t b is constant, the observed hysteresis between sliding speed and water pressure implies lower values of a s during increases in water pressure and higher values when water pressure begins to decrease. While there is some field evidence supporting a function based on equation (3) [Harper et al., 2007], a new sliding law that incorporates transient effects of cavity development has not been constrained or parameterized. [9] Another potential problem with sliding laws of the type in equation (2) is that the assumption that t b can increase limitlessly with sliding and effective pressure may be invalid. This is because large horizontal compressive stresses on the upstream sides of obstacles must be balanced by lower stress on the lee side. However, as the area of separation grows and gradients in normal stress become very large, pressures on the lee side may exceed the threshold of cavity formation. This provides a negative feedback on cavity growth and therefore limits the generation of basal drag. This limit was first derived by Iken [1981] and generalized analytically by Schoof [2005] and numerically by Gagliardini et al. [2006] to be dependent solely on p e and the slope of bed obstacles. The limit on basal drag invalidates equations (1) and (2) because it is independent of sliding velocity. Therefore if basal drag falls below the driving stress, the difference must be balanced by increased gradients in along-flow (longitudinal) stress or lateral shear stress. In this way, a glacier resting on bedrock may display dynamics during increased basal motion similar to those of soft-bedded glaciers, such as Antarctic ice streams [Schoof, 2006a, 2006b; Tulaczyk et al., 2000b]. [10] Recent observational and theoretical evidence supports the hypothesis that basal motion is largely controlled by stress redistribution beneath and within glaciers. Fischer et al. [1999] and Kavanaugh and Clarke [2001] found that increases in basal water pressures at Trapridge glacier induced weakening and failure in the till, resulting in transfer of stresses to hydraulically unconnected regions of the bed on timescales of hours. Truffer et al. [2001] and 2of15

3 Amundson et al. [2006] used observations of surface motion at Black Rapids glacier and other data to constrain a crosssectional ice flow model, finding that variations in basal motion on timescales of hours to days could be explained only through till failure and stress-bridging to either stronger-bedded regions or the glacier margins, where the excess stress was accommodated through increased shearing. Mair et al. [2001] found that motion events at Haut Glacier were due to redistribution of basal sticky spots under changes in basal hydrology. They and others hypothesized that temporal and spatial variations in basal drag are related to the evolution in the subglacial drainage system from the inefficient mode, where increases in discharge increase water pressures and basal separation area, to the efficient mode, where increased discharge has the opposite effect [Mair et al., 2002a]. Since this evolution will be spatially heterogeneous due to variations in melt or rainwater penetration to the bed, the variation in basal drag should be spatially variable as well. Such short-term spatial and temporal variability would result in transfers of stress from weak to strong regions of the bed, leading to equally rapid variations in glacier speed. [11] Based on the evidence cited above, we hypothesize that the three-dimensional motion of ice at the glacier surface is controlled by basal and ice stresses, as well as bed separation, at large spatial scales. To test this hypothesis, we use previously established techniques to estimate concurrent changes in sliding speed, bed separation and longitudinal force budget from the surface motion of Brei- *amerkurjökull, Iceland, over a three month period. We compare these records to look for consistent patterns that either support or discount the roles of transient cavity evolution and stress redistribution in the relationship between glacier hydrology and sliding. 2. Field Site [12] Brei*amerkurjökull is a large outlet glacier of the southern margin of the Vatnajökull ice cap in southeast Iceland (Figure 1). It is a prime location for comparative studies of ice dynamics because of its spatially variable geometry and flow. The eastern third of the 12-km wide glacier follows a subglacial trough, reaching depths of over 200 m below sea level, ending at a calving front on the Jökulsarlon lagoon. Ice thickness within the trough exceeds 600 m [Björnsson, 1996; Björnsson et al., 2001]. The central portion of the glacier rests upon elevated bed, terminates on land, and has ice thicknesses up to 300 m. The western edge follows a shallower subglacial trough and has a narrow calving front in Lake Brei*asarlon. Glacier speeds were measured with optical feature tracking [see Scambos et al., 1992] using Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) image pairs acquired in July Speeds were between 2 and 3 m d 1 within the eastern trough, and less than 0.2 m d 1 on the central, elevated portion (Figure 1). [13] It is important to mention that while most of the classical sliding theory presented in section 1 applies to hard-bed (i.e., no motion through till deformation) glaciers, Brei*amerkurjökull has traditionally been assumed to be a soft-bedded glacier. However, an overwhelming number of recent studies find a plastic rheology for till [e.g., Hooke et al., 1997; Iverson et al., 1999, 1998, 1994; Tulaczyk et al., 2000a; van der Meer et al., 2003], indicating that most basal drag is accommodated by irregularities in the bed [Schoof, 2002, 2003], such as the numerous bedrock outcroppings found at Brei*amerkurjökull. Schoof [2005] has pointed out that this rheology yields a similar basal friction law to that for hard-bedded sliding. Additionally, the presence of a general upper boundary in the basal drag would suggest that hard and soft bed would behave similarly during increased basal motion at low effective pressures. Therefore while we acknowledge that the presence of viscously deforming till may influence our results, the evidence presented above suggests that our results should be generally applicable to hard and soft-bedded glaciers. 3. Methods [14] We recorded the surface displacement of Brei- *amerkurjökull at 5 Global Positioning System (GPS) receiver installations between March 26 and June 1, 2005 (Figure 1). We also installed a receiver on land 500 m in front of the central portion of the glacier to serve as the fixed reference for differential post-processing. Stations S1, S2 and S3 ( S for slow ice) were positioned 1 km apart along the slow-moving central flow line of the glacier, beginning 3 km from the land-terminating margin. Bed topographies at the GPS stations were measured by Björnsson [1996] and Björnsson et al. [2001] using icepenetrating radar. Subtracting these bed elevations from our GPS-measured surface elevations gives ice thicknesses of 300, 270 and 200 m for S1, S2 and S3, respectively, with the bed between 30 and 40 m above sea level. Stations F1 and F2 ( F for fast) were positioned along the fastermoving axis of the eastern subglacial trough 6 km from the calving front. Ice thicknesses were 550 and 530 m, respectively, with the bed at approximately 180 m below sea level at both locations. We were limited to a smaller separation distance between units F1 and F2 (500 m) due to heavy crevassing. A sixth GPS station (F3), installed lower on the fast-moving ice, provided no measurements. [15] We followed the GPS antenna installation procedure of Anderson et al. [2004]. Each antenna was mounted upon three support poles drilled 6 m into the ice, providing measurements of ice motion independent of ablation. To conserve power, the GPS receivers were programmed to simultaneously collect 1 hour of data at 15 s epochs every 4 hours. These data were postprocessed in kinematic mode using Trimble Geomatics Office Software to yield 6 baseline solutions per day, with an average baseline length uncertainties of 1 mm in the horizontal and 5 mm in the vertical. Displacements and speeds presented in this paper are calculated from differencing the 4-hourly solutions for the three horizontal and vertical components of the baseline length and are therefore not affected by systematic positioning errors [King, 2004]. [16] The University of Iceland/Utrecht University automatic weather station, located within 250 m of station S3, measured 30-minute temperatures at a height of 1 m above the glacier surface. The Iceland Meteorology Office station 3of15

4 Figure 1. Velocity map overlay of Brei*amerkurjökull derived from automated surface-feature tracking between repeat ASTER images (24 June and 28 July 2004). Coordinates are in UTM. The locations of the five DGPS stations are labeled and the white line traces the ice front for clarity. Inset shows glacier location. at Fagurhólmsmýri, approximately 5 km from the glacier terminus, provided daily precipitation totals. 4. Results 4.1. Ice Motion [17] The time series of horizontal and vertical ice motion show three distinct, multiday horizontal speedup and surface-lifting events corresponding with periods of intense rainfall and/or melting (Figure 2). While the onsets of speedups were nearly simultaneous at all stations, speedup durations and absolute magnitudes were greater at the stations located on the thicker part of the glacier above the calving front, with peaks in speed at stations F1 and F2 occurring up to 12 hours after stations S1 S3. The largest event, beginning on day 105, lasted 2 to 4 days at stations S1 S3 compared to 15 days at stations F1 and F2. During this event, peak speeds reached 0.87, 0.71 and 0.77 m d 1 respectively at stations S1 S3, or 25, 25 and 34 times the respective minimum speeds. At stations F1 and F2, maximum speeds reached 1.95 and 1.90 md 1 respectively, or 4 and 5 times their minimum speeds. [18] Stations S1 S3 gained elevation at an average rate of 3 mm d 1 (Figure 2), while, stations F1 and F2 lost elevation at a background rate of 8 mmd 1. Periods of rapid surface uplift were concurrent with the horizontal speedup events. Total uplift during the event beginning on 4of15

5 Figure 2. (a) The solid curve shows half-hourly temperature, and bars are daily precipitation. (b) The solid curves are 4-hour horizontal speeds, and dashes are vertical displacement at each station. Uncertainties in differentially post-processed, 4-hourly positions are 1 mm in the horizontal and 5 mm in the vertical. Therefore uncertainty in speed is 8.5 mm/day. day 105 was m at stations S1 3 and m at stations F1 and F Horizontal Strain Rates [19] Longitudinal (along-flow) strain rates, _e xx, are calculated from the change in distance, DD, between adjacent stations in each line over the time interval, Dt, as: _e xx ¼ DD DDt [20] Compressive strain rates have negative values. We refer to the location of each time series of _e xx by the adjacent station names (e.g., between S2 and S1 isp ffiffi S1/2). The quantifiable uncertainty in strain rates are 2 a(ddt) 1 where a is the uncertainty in baseline distance between stations (1 mm), giving roughly d 1 for the S line (D 1000 m) and d 1 for the F line (D 500 m), which are an order of magnitude lower than the average observed strain rates. We also note that the finitedifferencing involved in equation (4) is an approximation of the true strain rate field, introducing unknown errors. [21] Figure 3 shows that strain rates were compressive between every station excluding brief periods of extension at F1/2. The most sustained period of extension occurred at the beginning of the day 105 event, lasting through four observations. At all stations, peaks in rates of compression occurred immediately after peaks in the speed. Following the peak, rates of compression decreased at the same time as speed decreased at F1/2, while compression remained elevated as speed decreased between S-line stations. Rates ð4þ of compression at S1/2 were roughly twice as large as at S2/3, and comparable to F1/ Vertical Strain Rates [22] In order to relate ice motion observed at the glacier surface to basal motion we must first consider the contribution of vertical strain, due to ice convergence, to uplift of the surface. The vertical strain rate, _e zz, is related to the horizontal strain rate components by the continuity equation: _e zz ¼ _e xx þ _e yy [23] We calculate _e xx from the observed motion of ice at the glacier surface using equation (4) and approximate the rate of cross-flow spreading, _e yy, from the glacier geometry as [van der Veen, 1999, p. 136]: _e yy ¼ u W dw dx where u is the ice speed, W is the glacier width (12 km) and dw/dx in the region of the GPS stations is about 0.2. For the observed ice speeds, _e yy is an order of magnitude less than _e xx. Assuming vertical strain is uniformly distributed through the ice column [Anderson et al., 2004], the vertical displacement, z d, of the surface due to ice convergence is then: z d ¼ HDt _e xx þ _e yy ð5þ ð6þ ð7þ 5of15

6 Figure 3. Solid curves are average horizontal speed of adjacent stations, and dashes are horizontal strain rate between stations calculated from equation (4) in the text. Note that the right y-axes are reversed to show the timing of peaks in compressive strain rates (negative values) and horizontal speed. [24] We acknowledge that vertical strain rates may vary with depth. Work by Gudmundsson [2002], Sugiyama and Gudmundsson [2003], and Sugiyama et al. [2003] shows that vertical strain rates typically decrease with depth during basal motion events, so that the surface horizontal strain rates would overpredict the mean vertical strain rate of the ice column. This would result in overestimation of z d in equation (7) and an underestimation of basal separation. However, they observed much greater temporal variability in surface strain rates than we find at Brei*amerkurjökull, suggesting a larger englacial stress perturbation than encountered here. Since the assumption of constant vertical strain rate with depth may lead to large, unknown errors in the estimate of bed separation (section 4.4), we limit our analyses to broad discussion of the relative timing of variations these estimates. This is justified by the large rates of vertical ice motion, which suggests a significant contribution from cavity formation during the motion events despite the possible variability in strain rate with ice depth revealed by Gudmundsson [2002]. [25] Applying equation (7) to the data, Figure 4 shows that at S1/2, z d contributed 1.02 m to total vertical surface displacement over 65 days compared to 0.49 m at S2/3. Rates of increase in z d at S1/2 increased from approximately 1cmd 1 to over 7 cm d 1 during the major day 105 motion event, giving a total vertical of uplift of 0.22 m over three days. Rates of increase were also similarly greater during the events beginning on day 88 and day 132. In contrast, there was little short-term variability in the rate of change in z d at S2/3. However, at both S-line regions, the rate of increase z d was approximately twice as large for the period following the major day 105 motion event as it was before the event. At F1/F2, z d accounted for 0.55 m of total vertical displacement over the 44 days both stations were recording. Slightly over one-third (0.2 m) of this uplift occurred during the major day 105 motion event Bed Separation [26] The GPS installations measured the Lagrangian motion of a point just below the ice surface (i.e., at the base of the stakes) and are independent of ablation or accumulation. Therefore the observed vertical motion is the sum of (1) vertical strain, (2) motion parallel to the mean bed slope, (3) dilation of subglacial sediments, and (4) vertical motion of basal ice relative to the bed due to cavity formation (i.e., bed separation). [27] Our objective is to isolate terms 1 through 3 to yield term 4, the bed separation, which is attributed to the expansion and collapse of subglacial cavities. For this we employ a method first developed by Hooke et al. [1989] and applied in many subsequent studies [e.g., Anderson et al., 2004; Harper et al., 2007; Sugiyama and Gudmundsson, 2004]. The first term, vertical strain, is estimated from equation (7). The second term, bed-parallel motion, is obtained from the radar-sounded bed topography at the scale of the ice thickness and its contribution to vertical displacement is shown in Figure 4. The third term, vertical motion due to till dilation, cannot be isolated without borehole data. However, Boulton and Hindmarsh [1987] found the hydraulic diffusivity of the Brei*amerkurjökull till to be on the order of 10 7 m 2 s 1 and the maximum magnitude of porosity change to be 0.1. Since the characteristic depth for diffusion of water pressure into the till is of the order of decimeters [Tulaczyk et al., 2001], till 6of15

7 Figure 4. The solid curves are average observed vertical surface displacements of adjacent stations, and dashes are vertical displacement due to ice convergence, calculated from ice thickness and horizontal strain using equations (5) to (7) in the text. The gray dots are vertical displacement due to motion down the mean bed slope, as measured by ice-penetrating radar [Björnsson, 1996; Björnsson et al., 2001]. dilation can account for no more than several mm per day of glacier uplift, and no more than a few cm over the record. This contribution to vertical ice motion is therefore small relative to the other terms shown in Figure 4 and can be neglected. [28] By subtracting the contribution of the above components 1 and 2 from the observed surface displacement and neglecting component 3, we are left with an estimate of the temporal change in bed separation. Since horizontal strain rates are used in this calculation, we solve for the bed separation at the points halfway between adjacent stations. [29] Figure 5 shows that peaks in bed separation roughly corresponded with peaks in speed, but this pattern differed from region to region. At F1/2, peaks in bed separation occurred 12 and 60 hours after peaks in speed for the day 88 and 105 motion events, respectively, with the peaks in speed occurring with peaks in the rate of basal uplift. Between the S-stations, peaks in bed separation were much closer in time to peaks in speed, separated by less than 8 hours (one or two observation cycles) for all motion events. The changes in bed separation were 3 and 5 times greater between the F-stations than the S-stations for the day 88 and 105 motion events. [30] At all stations, bed separation decreased at a slower rate than it increased. Between stations F1 and F2, 5 days elapsed following the day 88 event before bed separation returned to its previous level and the bed separation was still over 0.25 m greater 20 days after the day 109 peak than it was before this peak. There was a contrast in the pattern of decreasing bed separation between different regions of the S-line. The down-glacier zone (between stations S2 and S3) behaved similarly to the F-stations, with gradual decreases in bed separation following the peaks. Decreases in bed separation at the up-glacier zone (between stations S1 and S2) appeared faster, with a decrease of over 0.1 m in 5 days following the day 133 event, reaching a level 0.05 m below its previous minimum for the preceding 52 days Longitudinal Resistive Stress [31] The downslope force of gravity acting on the glacier, termed the driving stress, t d, is resisted by drag t b at the glacier bed by along-flow gradients in longitudinal stress R xx and by across-flow gradients in the shear stress R xy generated at the sides of the glacier. Over horizontal length scales that are large relative to the ice thickness, the alongflow component of the glacier force budget can be written [van der Veen, 1999; van der Veen and Whillans, 1989a, p. 36]: 0 ¼ t dx þ t Z h h H R xx Z h h H R xy dz with Cartesian coordinates x, y and z in the along-flow, cross-flow and vertical directions, respectively. H is the ice thickness and h is surface elevation. R xx and R xy are the resistive stresses from longitudinal compression and lateral shearing, respectively. In this coordinate system, t dx is always positive because the stress acts along the direction of increasing x and t bx is always negative. Therefore positive ð8þ 7of15

8 Figure 5. The solid curves are average horizontal speed of adjacent stations, and dashes are the bed separation, which is the total vertical surface displacement minus the contributions of vertical strain and bed-parallel motion. (negative) values for the third and fourth terms indicate a stress gradient vectors pointing along (against) the direction of flow. Since our observations are limited to motion along the flow direction, our analysis only examines gradients in the depth-integrated longitudinal resistive stress (the third term on the right-hand side in equation (8)). Written in terms of deviatoric stresses, the longitudinal resistive stress is [van der Veen, 1999, p. 38]: R xx ¼ 2t xx þ t yy [32] In order to calculate deviatoric stress from the observed strain rates, an appropriate constitutive relation must be chosen. The observed driving stress at Brei*amerkurjökull is 100 kpa, giving a viscosity on the order of Pa s [Paterson, 1994]. Values for the shear modulus for elastic deformation of temperate mountain glaciers range from 3 to 6 GPa [Glen, 1975; Hobbs, 1974; Sugiyama et al., 2007] (values down to 1 GPa are typically used for polar ice [Vaughan, 1995]). The viscosity divided by the shear modulus, the Maxwell Time, provides the characteristic timescale of elastic deformation, which, in this case, ranges from 0.5 to 1 hour. Therefore the observed strain rates should be the result of viscous deformation and are related to the deviatoric stresses by [Hooke, 2005, p. 16; van der Veen, 1999, p. 15]: ð9þ t ij ¼ B_e 1 n 1 e _e ij i; j ¼ x; y ð10þ where B and n are the parameters from Glen s Flow law [Paterson, 1994, p. 85] and _e e the effective strain rate (see Appendix A) [Hooke, 2005, p. 14; van der Veen, 1999, p. 136]. [33] In order to calculate mean glacier stresses over the ice column using equation (10), we must assume that horizontal strain rates are constant with depth (i.e., blockflow). This assumption has been applied in many past force budget studies of temperate, and even polythermal, glaciers [e.g., Hooke et al., 1989; Mair et al., 2001; O Neel et al., 2005; van der Veen and Whillans, 1993] and may result in an overestimation of F lon if gradients in surface strain rates are significantly larger than those at the bed. However, the assumption of block-flow should be valid for this application because (1) Brei*amerkurjökull is isothermal and (2) temporal variations in F lon associated with motion events should result mostly from variations in basal motion since little change in deformation rate would occur at the timescales of our observations in regions away from the shear margins [Amundson et al., 2006; Truffer et al., 2001]. We assess the potential uncertainties resulting from this and other assumptions in our estimates of R xx and F lon in the Appendix A. [34] Under the assumption of depth-invariant, horizontal strain rate, the third term on the right-hand side of equation (8) is approximated from the gradient in surface strain rate along the direction of flow by: F HB_e1 n 1 e 2_e xx þ _e yy ð11þ where H is the mean ice thickness between adjacent stations and F lon is defined as the horizontal gradient in the depthintegrated, longitudinal resistive stress. Adopting the typical 8of15

9 values for B and n for temperate ice (100 kpa yr 1/3 and 3 [e.g., Hooke, 2005, pp ]) we solve for F lon using the strain rates along the S-line (Figure 3) and known glacier thickness. Quantifiable uncertainties in this calculation arise from uncertainty in strain rates (derived in section 4.2) and ice thickness (±5 m), which give a 1 sigma error in F lon of ±8 kpa. Larger values for B and n would result in larger amplitude of F lon and vice versa. The sensitivity of this calculation to the various independent variables is presented in the Appendix A. [35] Since F lon is the horizontal gradient of the vertically integrated resistive stress, its sign indicates the direction along which the stress is acting. According to the force budget equation (8), negative values of F lon indicate an against-flow gradient (i.e., stress acting in the opposite direction of the driving stress) while positive values indicate an along-flow gradient (i.e., acting in the direction of the driving stress). For simplicity, we refer to the sign and magnitude of F lon separately: changes in the magnitude of F lon are referred to as increases or decreases irrespective of the sign. [36] The time series of normalized values for the longitudinal resistive stress, R xx, are shown with the ratio of F lon to the driving stress in Figure 6. On average over the record, R xx was over twice as large in magnitude at S2/3 (downglacier) than at S1/2 (upglacier) and both were negative in sign for the duration. The larger negative value in upglacier R xx relative to downglacier resulted in an increasing alongflow gradient in resistive stress, so values of F lon were positive (i.e., the same sign as the driving stress). For the first 20 days of observation F lon averaged 10% of the driving stress, pointing in the direction of flow. Starting on day 106, F lon increased synchronously with speed and peaked at 35% of the driving stress 24 hours after the peak in speed. Following this peak, F lon decreased to zero over 10 days and then increased again, fluctuating between 5 and 25% of the driving stress. These fluctuations were the result of large fluctuations in the strain rates near the terminus. F lon peaked again during the day 133 motion event at 35% of the driving stress. Again, the peak in F lon lagged the peak in speed by approximately 24 hours. Following this event, F lon decreased and then became negative, peaking at 15% of the driving stress on day 143. Following this negative peak, F lon returned to positive values by day 144 and remained between 15 and 20% of the driving stress for the remainder of the record. 5. Discussion 5.1. Changes in Basal Separation During Motion Events [37] Bed separation increases as subglacial cavities expand when water pressure increases beyond a minimum pressure for cavity formation, which is dependent on the roughness of the bed and the ice overburden pressure [Iken, 1981]. The expansion of cavities is opposed by creepclosure of the cavity walls, which is dependent on cavity geometry and effective pressure [Nye, 1953]. Therefore decreases in water pressure lead to cavity contraction and decreased bed separation. Since these cavities are waterfilled, several studies have postulated that changes in cavity size should be directly related to changes in the volume of water stored within them [e.g., Harper et al., 2007; Iken and Bindschadler, 1986; Iken et al., 1983] and, subsequently, variations in basal separation should reflect changes in water storage. [38] Since cavities will expand at any water pressure above the separation pressure, and will close under any drop in water pressure, the transient relationship between water storage or basal separation and water pressure is not unique. In relating basal separation to subglacial water pressure we are therefore limited to the broad interpretation that increased bed separation implies cavity expansion and water storage at pressures above the separation pressure. Conversely, decreased bed separation implies cavity constriction and decreased water storage under a drop in water pressure. [39] The broad correlation between increased rates of melting and/or precipitation, bed separation and surface motion support these interpretations. However, the relationship between ice speed and bed separation is not temporally consistent, as would be predicted by equation (2) if variations in bed separation represent variations in the subglacial water pressure. In all cases, the peak in bed separation occurs after the peak in speed. As described in section 1, this lag is common in observations from borehole-derived water pressure. Furthermore, ice speed decreases much faster than bed separation, resulting in hysteresis between ice speed and bed separation (Figure 7). Again, this hysteresis is of the same form as observed between siding speed and borehole-derived water pressure [e.g., Sugiyama and Gudmundsson, 2004]. The similarity in temporal relationships between ice speed and bed separation and speed and water pressure support the conclusion that changes in bed separation are representative of changes in the subglacial water pressure, through cavity formation and water storage. [40] The hysteresis between bed separation and ice speed is consistent with the effect of transient cavity geometry predicted by Iken [1981] and discussed in section 1. If we take bed separation to be closely related to subglacial water storage and broadly representative of water pressures as presented above, it would follow from equation (3) that under constant basal drag, the fractional area of separation from the bed, a s, should increase as the vertical bed separation increases, leading to a decreased proportionality between bed separation and ice speed. [41] Consistent with the findings of Iken et al. [1983], the transient dependence on cavity evolution can also be seen in the lag between peaks in ice speed and peak in bed separation. Since we interpret decreasing bed separation as representing creep-closure of cavity walls under a decrease in water pressure, this lag indicates a decrease in ice speed before a decrease in storage and water pressure. Such a lag would be predicted by equation (3) if the peak in speed marks the point at which a s becomes large, resulting in a decrease in u s under steady p e and t b. The relatively long lag between peak speed and peak bed separation at the F-line suggests more sustained high water pressures despite large cavities, indicating less efficient drainage. Slower drainage at the F-line may be due to a connection between its drainage system and the lagoon into which it calves. The F-line is grounded roughly 200 m below sea level with an ice thickness of about 550 m, limiting the maximum effective pressure to 60% of the ice overburden if a direct 9of15

10 Figure 6. Plots of longitudinal resistive stress between S-line station pairs calculated from horizontal strain rates and equation (9) in the text (a) and the ratio of the gradient in longitudinal resistive stress, calculated from equation (11), to the driving stress (b). Error bars are obtained from the quantifiable uncertainties in measured strain rates and ice thickness. connection exists between the local basal drainage system and the calving front. This would limit the slope of the hydraulic gradient within the drainage system and, subsequently, the maximum rate of discharge Changes in Glacier Force Budget During Motion Events [42] Our results suggest that substantial variations (>10% of the driving stress) in the gradient in longitudinal resistive stress, F lon, coincide with ice speed anomalies at Brei*amerkurjökull. Since the driving stress, t d, does not vary substantially (<1%) over these short timescales, equation (8) requires that greater positive (negative) values for F lon be balanced by either an increase (decrease) in basal shear stress, t b, which is always negative, or a decrease (increase) in the across-flow gradient in depth-integrated, lateral resistive shear stress. [43] The amount of flow resistance transferred from the lateral shear margins to the glacier centerline scales with the ice viscosity and the ratio of the ice thickness to the glacier width [Raymond, 1996]. At the location of the GPS stations, the width of Brei*amerkurjökull is over 20 times the ice thickness, so that the ratio of thickness and the distance between shear margins is small relative to the ratio of ice thickness to the longitudinal stress coupling length (i.e., a few thicknesses). This suggests that the observed changes in longitudinal resistive stress are likely greater than that which could be attributed to changes in lateral shearing at the glacier margins. Therefore the observed changes in F lon are likely balanced by changes in the basal shear stress during speedup in the region of our observations. [44] While not all variability in speed correlates with variability in F lon, the two largest observed ice motion events correlate temporally with increasing, positive values for F lon, which we infer to mean an increase in basal shear stress. An increase in basal shear stress with ice speed would be consistent with equation (1) and equation (2) if effective pressure remains constant. However, the peak in F lon lags the peak in ice speed after both ice motion events (Figure 8). Furthermore, F lon remains elevated several days after termination of the ice motion events, resulting in a similar hysteresis to that between ice speed and bed separation or water pressure described above. Again, these lags and hysteresis are most simply explained by transient sliding under cavity evolution, as first put forth by Iken [1981] and described by equation (3). If cavities were large at the end of the basal motion event, a rapid drop in water pressure (increase in p e ) would result in a continued increase in t b. Such a drop in water pressure is suggested by the decrease in bed separation, which occurs just before the peak in F lon for both events (Figure 8). [45] The local, along-flow increases in F lon during increased basal motion necessitate against-flow increases in F lon upglacier to maintain force balance. Continuing with our assumption that lateral gradients in shear stress are not a substantial component in the force budget over the central region of Brei*amerkurjökull, an increase in against-flow gradients in longitudinal stress would imply a reduction in 10 of 15

11 Figure 7. Plots of bed separation versus sliding speed for GPS station pairs (above) S1/2 and S2/3 and (below) F1/2 with labels marking the date in day of 2005 and arrows indicating the progression of time. basal shear stress upglacier of our observations. This interpretation is consistent with the standard model for subglacial drainage system distribution [e.g., Fountain and Walder, 1998; MacGregor et al., 2005; Mair et al., 2002a]. Tunnelized (i.e., efficient) drainage will likely develop fastest and persist longer closer to the glacier front due to less overburden pressure and slower creep-closure rates of tunnels, as well as greater quantities of surface melt and 11 of 15

12 Figure 8. Plots of mean horizontal speed (top), F lon (middle), and mean bed separation (bottom) for the S-line. All data is smoothed using a 6-point moving average. Vertical bars mark peaks in horizontal speed. throughflow of water. In a tunnelized drainage system, basal water is channeled over a smaller area of the bed, so that the fractional area of separation remains small and basal drag can be maintained. Also, once a tunnelized system forms, drainage and effective pressure will increase rapidly due to an inverse relationship between discharge and water pressures within tunnels [Röthlisberger, 1972]. [46] In contrast, up-glacier regions are expected to have distributed (i.e., inefficient) drainage for more of the year, due to higher overburden pressures and less water flux [Mair et al., 2002a]. Increased penetration of meltwater to the bed would increase basal water pressures and cavity formation, leading to a greater fractional area of basal separation. If effective pressures become low enough, basal drag will not locally resist the driving stress [Gagliardini et al., 2006; Iken, 1981; Schoof, 2005]. To maintain force balance, the excess stress would then be transferred downglacier through gradients in longitudinal stress, where drainage is more efficient, water pressures are lower and basal drag in excess of the local driving stress can be supported. By this mechanism, the efficiently draining terminus would act as a barrier to inland ice during increased basal motion. If this is the case, force budget calculations further upglacier should show an increase in F lon of similar magnitude and opposite sign as observed downglacier. This pattern of down-glacier transfer of longitudinal stresses has been observed on several other glaciers of varying geometries and basal conditions on diurnal to seasonal frequencies [Amundson et al., 2006; Mair et al., 2002a; Mair et al., 2001; Sugiyama and Gudmundsson, 2003]. Our measurements suggest a similar mechanism may operate at Brei*amerkurjökull at a frequency dominated by the timescale of substantial hydrometerologic change (i.e., storm frequency) as opposed to seasonal or diurnal temperature variations. This is likely a function of Iceland s temperate, maritime climate. [47] Due to the failure of the third GPS station on the F-line, we lack the data to calculate changes in F lon in that region. However, the large variations in horizontal strain rates during the major day 105 event suggests that variations in longitudinal forces may play an important role in controlling motion there as well. Unlike on the S-line, Figure 3 shows a distinct period of extensional strain rate at the start of the ice motion event, quickly switching to compressive strain rates at the peak of ice speed. This suggests that increased basal motion propagated upglacier from the terminus, possibly due to changing gradients in longitudinal stress near the calving front. A high degree of variability in the speed of calving glaciers is well observed and is attributed to changes in resistive stresses near the front [e.g., Howat et al., 2005; Kamb et al., 1994; Meier et al., 1994; Vieli et al., 2000, 2004]. It is possible that the large rainfall and/or melting events that correspond with the motion events also temporarily increased water levels within the lagoon, which would have perturbed both the hydrostatic stress balance at the front and the effective pressures within the drainage system. 6. Conclusions [48] In this paper, we have used multiple time series of high-precision, three-dimensional surface motion to infer short-term changes in vertical bed separation and longitudinal force budget over a 66-day period at Brei*amerkurjökull. Our primary objective was to find consistent patterns between bed separation, force budget and ice motion that 12 of 15

13 would help elucidate the mechanisms controlling short-term variations in ice speed. We hypothesized that at timescales on the order of days or less, the transient evolution of cavity configuration should be important in controlling the relationship between sliding, effective pressure and basal shear stress because of cavity development due to changes in basal hydrology, rather than ice deformation. Also, at these timescales, we expected that along-flow gradients in resistive stress could play an important role in redistributing basal shear stress during increased basal motion. Both of these processes are excluded from typical sliding laws, which assume steady state cavity configurations and locally balanced glacier driving stress. [49] We observed multiple, distinct periods of increased surface motion and uplift corresponding to periods of rainfall and/or increased temperatures. From the records of surface uplift and vertical strain we isolated variations in bed separation, or the vertical motion of basal ice relative to the glacier bed. Increases in bed separation are interpreted to represent expansion of subglacial cavities at water pressures above separation pressure, while decreases in bed separation indicate cavity closure under decreasing water pressures. We found a hysteresis in the relationship between ice speed and basal separation that closely resembles the hysteresis often observed between ice speed and borehole-derived water pressures. We attribute this hysteresis, as well as the lag between peak ice speed and peak bed separation, to the redistribution of normal stresses at the bed during cavity growth. We attribute a longer lag between peak ice speed and bed separation above the calving portion of the glacier (F-line stations) to a lower hydraulic gradient and inhibited drainage. [50] We estimate a time series of the gradient in the longitudinal resistive (i.e., deviatoric) stress, F lon, from the observed horizontal strain rates on the surface of the landterminating portion of the glacier. Higher compressive strain rates upglacier relative to down resulted in a down-flowpointing F lon that increased markedly during the two largest motion events, reaching 35% of the driving stress. Lateral gradients in shear stress should be small in the center of this wide and thin glacier, suggesting that these increases are balanced locally by increased basal shear. As with bed separation, a hysteresis is apparent between F lon and ice speed, with peaks in F lon lagging peaks in speed. Increased basal shear stress under decreased speed can be attributed to continued cavity growth and/or increasing effective pressures, which is consistent with the record of bed separation. [51] An increase in F lon during increased basal motion implies increased longitudinal stress transfer from upglacier onto the study area. This suggests a reduction in basal drag during increased basal motion upglacier of the study area. Consistent with previous studies, we suggest that this stress transfer reflects a contrast in subglacial drainage systems: the terminus is expected to have efficient, tunnelized drainage, leading to a relatively low area of separation from the bed and higher basal drag. In contrast, the upglacier drainage system is expected to have inefficient, dispersed drainage, resulting in a high area of separation and a potential decrease in basal drag. [52] Our results corroborate other recent studies in suggesting that the transient evolution of the subglacial drainage system is an important control on basal motion on timescales of days, to possibly weeks in the case of poorly draining systems such as those connected to a calving front. At these timescales, steady state sliding laws are not adequate for constraining the relationship between water pressure and basal motion and the redistribution of bednormal stresses under expansion must be taken into account. Additionally, our data suggest that spatial and temporal variability in basal conditions impact glacier flow-speed through complex shifts in the large-scale distribution of glacier stresses. To better constrain these processes, which may have implications for glacier and ice sheet stability under climate changes, future field programs should focus on acquisition of spatially integrated, concurrent observations of changes in subglacial drainage system configuration and glacier dynamics. For example, large networks of GPS receivers could provide a full three-dimensional picture of surface motion and stress fields at depth using inversion modeling methods [e.g., Amundson et al., 2006]. Additionally, repeat radar or acoustic echo-sounding devices could provide data allowing for inference of bed characteristics over large areas. Appendix A [53] Uncertainties in the calculation of the longitudinal resistive stress, R xx, and along-flow gradient in the depthintegrated resistive stress, F lon, are the combination of (1) measurement error, (2) uncertainty in ice rheology, (3) uncertainty in the effective strain rate, and (4) the simplifying assumption of constant strain rates with depth. The only quantifiable component of uncertainty is (1), which is calculated formally in the text and shown by error bars in Figure 7. Here we assess the potential impacts of factors (2) (4) based on the expected behavior and the results of a sensitivity test presented in Table A1. [54] We use single values of the stiffness parameter B and exponent n in Glen s Flow Law to calculate deviatoric stresses from surface strain rates. Brei*amerkurjökull, like all Icelandic glaciers, is at the pressure-melting temperature throughout the ice column. Also, our measurements are away from shear margins so that shear softening of the ice likely did not occur. We therefore expect negligible temporal or spatial variability in B or n. According to our sensitivity analysis (Table A1), error in B results in an equal amount of relative error in F lon, but only a very small perturbation to R xx. Variability in n will have a very large impact on F lon and R xx. [55] The total effective strain rate is given by: 2_e 2 e ¼ _e2 xx þ _e2 yy þ _e2 xx þ 2 _e2 xy þ _e2 xz þ _e2 yz ða1þ [56] Error in _e e will lead to an error in the magnitudes of R xx and F lon by a power of 1/n 1 through Glen s Flow Law, or 2/3 for n = 3. Since we only observe motion along the flow line, we cannot calculate lateral shear strain rate, _e xy and this term is set to zero. The along-flow gradient in the shear strain rate in the vicinity of the stations calculated from the remotely sensed surface velocity measurements shown in Figure 1 are d 1, an order of magnitude less than rate of longitudinal compression. Due to its likely small value relative to other terms, perturbations 13 of 15