Surface velocities, strain rates and force balance

Glacier surface motion was monitored from repeated surveys of a network of 17 stakes, drilled into the glacier surface, using a Geodimeter 410 Total Station. The stake network was set out on the glacier tongue, and positioned to cover longitudinal variations in basal thermal regime, and lateral variations in the likely structure of any subglacial drainage system (Reference RippinRippin and others, 2003). The resultant network was centred slightly west of the centre line (Fig. 1). Surveying took place from two fixed sites on the western valley side, with reference to two fixed points on the eastern side (Fig. 1). To investigate the components of the force balance, the stake network was devised to form 18 strain triangles, arranged to produce three interlinked hexagons (Fig. 1). This layout enabled ten theoretical ice blocks to be delineated on which the balance of forces could be determined (Fig. 1a).

Surveys were conducted between 5 July and 9 August 1999 on each day that weather conditions were favourable, typically every 2–3 days. A full survey consisted of two sightings to each glacier stake from each survey station, and three surveys to each of the fixed reference points from each survey station. For each survey, there were thus a total of four measurements to each glacier stake. Stake coordinates and positional errors were then determined using a least- squared estimation program (General Adjustment Program (GAP) version 6.11, developed in 1992 by J. Chandler and J.S. Clark of City University, London). Velocities in the along- flow direction and associated errors were then calculated (Reference Mair, Nienow, Willis and SharpMair and others, 2001; Reference RippinRippin, 2002).

Surface strain rates were calculated from the deformation of the 18 strain triangles, following the method of Reference MairMair (1997), who adapted the technique of Reference RamsayRamsay (1967). The technique involves determining the centre of each strain triangle as well as the two perpendicular principal strains and their angles of orientation for each period between two consecutive surveys (Reference Mair, Nienow, Willis and SharpMair and others, 2001). These strain rates were then used to calculate the components of the force balance operating on the ten theoretical ice blocks (Reference Van der Veen, Whillans and TheoryVan der Veen and Whillans, 1989a; Reference MairMair, 1997; Reference Mair, Nienow, Willis and SharpMair and others, 2001).

Reference Van der Veen and WhillansVan der Veen and Whillans (1989b) first discovered the presence of isolated sites of high basal drag (‘sticky spots’), unrelated to topographic highs, along the Byrd Station strain network, Antarctica, and suggested that basal drainage was important in controlling overall bed friction. Here, we employ their relatively simple plug-flow method, which avoids the numerical instabilities often encountered in more complex inverse methods (Reference BlatterBlatter, 1995; Reference Mair, Nienow, Willis and SharpMair and others, 2001). The measured surface strain rates and the area of the sides of each block are used to calculate longitudinal and shear forces. Basal drag is finally calculated by subtracting the driving stress. Force-balance analysis was carried out to identify the dominant controls on the resistance to flow. Driving stress is the dominant factor promoting glacier motion, while basal drag, longitudinal forces and lateral shear forces generally act to resist it. The ratio of basal drag (τ _b) to the driving force (τ _d) was used to identify the proportion of the driving stress resisted by basal drag. τ _d/τ _d 1 if some resistance to flow comes from non-local sources, and τ _b/τ _d> 1 if flow is driven by forces in addition to the driving force.

One of the main potential weaknesses of this method is the way in which the flow-law parameter A is dealt with. On temperate glaciers, where the ice temperature is close to 0°C throughout, A is usually assumed to be constant, and strain rates are usually assumed to be constant with depth (Reference Hooke, Calla, Holmlund, Nilsson and StroevenHooke and others, 1989; Reference Iken and TrufferIken and Truffer, 1997; Reference MairMair, 1997). However, on the polythermal midre Lovénbreen these assumptions are less valid, though there is a great degree of uncertainty surrounding the value of A due to factors such as melting, grain-size variations, impurities, ice density and ice fabric, many of which are affected by temperature (Reference HookeHooke, 1981; Reference Van der VeenVan der Veen, 1999).

We carried out a series of experiments to investigate the sensitivity of the force-balance components to changing values of the parameter A. Although a uniform change in ice stiffness affects the magnitudes of the resistive force-balance components, it does not change the direction in which they act, or significantly change the relative role of the forces in controlling the ice-block dynamics. The situation is, however, more complicated where cold and warm ice exists within the same ice block, as is thought to be the case on midre Lovénbreen in the middle and upper tongue (where cold ice overlies temperate ice towards the bed). Consequently, ideally, the stiffness parameter should perhaps be allowed to vary with depth. However, the sensitivity analysis presented above suggests that only the magnitude of forces is affected by the flow parameter. Due to the general uncertainties and limitations inherent in the force- balance technique (Reference MairMair, 1997), the uncertainty in the value of A for a given ice temperature and the fact that there are no field data to support the idea of depth-dependent strain rates on midre Lovénbreen, we use a fixed value of A of 3.96 × 10^–24 s^–1 Pa^–3, which equates to an ice temperature of between –1 °C and –2°C. This provides for slightly stiffer ice than would perhaps be employed for temperate ice, and is thus considered to be a suitable compromise for the polythermal state dominant over much of the glacier tongue. The potential error resulting from applying a spatially homogeneous flow parameter is considered to be as unknown as the uncertainties surrounding the flow-law formula itself, and, given other uncertainties here, a depth- independent flow parameter is considered acceptable (Reference MairMair, 1997; cf. Reference RippinRippin, 2002).

Our sensitivity tests indicate that it is the magnitudes of the resistive force-balance components that are changed as A varies, but that it does not change the direction or role of each force significantly. It is therefore patterns, and not magnitudes, of force-balance components that will be discussed, and no significance will be attached to the actual force-balance values. A more detailed overview of the procedures for calculating surface strain rates and force- balance components can be found in Reference Mair, Nienow, Willis and SharpMair and others (2001), and full details of its application here are given in Reference RippinRippin (2002).

Errors in stake location and surface velocities were determined from four measurements to each stake on each day (two from each survey station). Strain-rate errors were calculated by deriving three sets of possible coordinates for each stake on each day: the measured stake coordinates;the measured stake coordinates plus the maximum possible locational errors; and the measured stake coordinates minus the maximum possible errors. By calculating strain rates using all possible combinations of stake coordinates, nine possible calculations of strain rates for each stake in each time period were made. The mean strain rate and standard deviations were then determined from these nine measurements for each stake. The force-balance errors were determined using the same procedure. In this way we were able to propagate the calculated errors in velocity measurements through to strain rates and force-balance measurements.

Meteorological and hydrological data

Hourly air-temperature data were recorded at an automatic weather station located on the glacier at ~110ma.s.l., ~350m from the terminus (Fig. 1). Precipitation data were collected by the Norske Meteorlogiske Institutt, in the nearby settlement of Ny-Ålesund, ~4km northwest of the glacier terminus, at 0700 and 1900 h each day.

Proglacial stream discharge and turbidity data were recorded every hour at two temporary gauging sites, on streams draining the western and eastern parts of the glacier. The temporary gauging stations were located at stable stream cross-sections as close to the glacier snout as possible (~700 and 1000 m from the front respectively) (Fig. 1). For each station, water stage and turbidity were measured every hour using, respectively, a Druck pressure transducer and a Partech IR-15C infrared turbidity sensor wired to a Campbell Scientific data logger. Stage data were converted to discharge using a stage–discharge curve derived from daily measurements of discharge made using the standard velocity–area method. Discharge errors were typically <10% under typical flows but rose to ~15% during maximum flows (Reference Hodson and FergusonHodson and Ferguson, 1999; Reference Hodson, Tranter and VatneHodson and others, 2000).

Additionally, the discharge of the subglacially routed water that emerged proglacially as a pressurized upwelling, and entered the eastern stream upstream of the gauging station, was calculated using a chemically based mixing model:

where Q and C represent discharge (m³s^–1) and Si concentrations (mgL^–1) respectively, and subscripts gauge, sub and lat denote total (at the gauging station), subglacial and lateral components respectively (i.e. total runoff in the eastern river and the two drainage components that supply it). Assuming conservation of mass, Equation (1) can be rearranged to calculate the discharge of the proglacial upwelling. This approach was adopted because the channel immediately downstream of the upwelling was unstable, and made use of dissolved Si determined from filtered samples collected at the various sites. It was assumed that mixing of the turbid, chemically concentrated subglacial water with the dilute lateral stream water was conservative with respect to Si in transit through the proglacial region to the gauging station downstream. The use of chemically based mixing models can be problematic in glacier basins when used to split bulk meltwaters into subglacial and englacial components (Reference Sharp, Brown, Tranter, Willis and HubbardSharp and others, 1995). In our study, however, Si has been shown to be conservative throughout the proglacial plain due largely to the limited residence time of waters in transit through it (Reference MumfordMumford, 2002). Furthermore, we were able to allow the Si content of the subglacial and lateral stream waters to vary in our mixing model, as both sites were readily accessible and sampled on a daily basis for their dissolved Si content (Reference MumfordMumford, 2002).

Survey data and errors

Fourteen full surveys were undertaken during summer 1999. The mean error in the positions of all glacier stakes was 0.011 m in x, and 0.013 m in y. Preliminary calculations of xy coordinate positions showed that for surveys conducted only 1 or 2 days apart, displacements could not be resolved above the position errors. Combining adjacent periods where displacement patterns were similar resulted in a significant reduction in the errors. Therefore six periods were defined as follows: (1) 5–9 July, (2) 9–15 July, (3) 15–20 July, (4) 20–23 July, (5) 23 July—6 August and (6) 6–9 August, covering a total of 35 days.

The total displacement of the 17 stakes showed they could be separated into three groups, each displaying similar characteristics, representing the lower tongue (stakes 1 and 2), the middle tongue (stakes 3–10) and the upper tongue (stakes 11–17) (Fig. 2; cf. Fig. 1). The two lowest stakes moved only ~Ό.08m in a northwest direction over the 35 days;stakes 3–10 moved ~0.5 m in a northeast direction; and stakes 11–17 moved on average ~0.8m in a northerly direction, although stake 13 (close to the glacier’s western margin) was an exception and moved only ~0.45 m.

Fig. 2. Displacement of velocity stakes throughout the summer 1999 season for all stakes (numbered). Crosses and solid lines indicate northwest-trending stakes in the lower tongue (1 and 2); squares and dotted lines indicate northeast-trending stakes in the middle tongue (3–10); triangles and dashed lines indicate northerly- trending stakes in the upper tongue (11–17).

Surface velocities

Mean velocity errors in the middle and upper tongue were small compared with xy velocities in all six periods, averaging ~28% of measured velocities. However, in periods 1–4, errors were larger than velocities in the lowest two stakes constituting the lower tongue (~192% of measured velocities), due to very small rates of movement here. Where errors are large, we discuss only broad trends affecting large regions of the tongue.

Patterns of surface velocity for the six periods are shown in Figure 3. Velocities were generally higher during period 1 than in any subsequent period (Fig. 3a). The whole tongue slowed down in period 2, although velocities remained relatively high in the upper tongue (Fig. 3b). In period 3, velocities generally increased across the tongue (Fig. 3c). A slight slow-down in the upper tongue, and a speed-up in the lower tongue resulted in more similar velocities across the whole tongue in period 4 (Fig. 3d). Period 5 saw a substantial velocity drop (Fig. 3e), before a significant increase over most of the tongue, particularly the upper part, in period 6 (Fig. 3f). Thus, periods 1, 3, 4 and 6 were times of relatively high velocities, while periods 2 and particularly 5 were times of glacier slow-down.

Fig. 3. Horizontal xy surface velocities (md ¹) in the tongue. Arrows indicate xy velocity azimuth, and their size is proportional to the measured xy velocity magnitude (size guide shown in (a)). A linear kriging interpolation scheme was used to derive velocity contours. Contours are marked every 0.005 m d^–1 and labelled every 0.01 m d^–1. At the ice margin, xy velocities are set to zero. (a) Period 1 (5–9 July); (b) period 2 (9–15 July); (c) period 3 (15–20 July); (d) period 4 (20–23 July); (e) period 5 (23 July–6 August); and (f) period 6 (6–9 August).

Surface longitudinal strain rates

Errors in longitudinal strain rate were, on average, 35.11% of calculated strain rates. However, by excluding stake 13, where errors were particularly high, errors are reduced to 21.94%. In order to avoid over-interpretation where errors are significant, we do not attach any significance to individual measurements, and adopting the approach of Reference Mair, Nienow, Willis and SharpMair and others (2001), discuss only the broad patterns of surface strain rates influencing large areas of the tongue.

Patterns of surface longitudinal strain rates for the six periods are shown in Figure 4. The high velocities in periods 1, 4 and 6 were associated with longitudinal compression over most of the tongue, with possible longitudinal extension in the upper tongue (Fig. 4a, d and f). Compression increased towards the lower tongue in periods 1, 4 and 6. Conversely, the high velocities in period 3 and the low velocities during periods 2 and 5 were all associated with smaller rates of longitudinal compression (Fig. 4b, c and e).

Fig. 4. Longitudinal strain rate in the tongue. Strain rates were calculated for the centre of each strain triangle (marked with crosses), and then interpolated across the region covered by the stake network. Positive longitudinal strains indicate extension and are marked with fine dotted contours. Negative strains indicate compression and are marked with fine solid contours. The thick dashed contour is where strain rates are zero. (a) Period 1 (5–9 July); (b) period 2 (9–15 July); (c) period 3 (15–20 July); (d) period 4 (20–23 July); (e) period 5 (23 July–6 August); and (f) period 6 (6–9 August).

Force balance

The errors associated with τ _b and τ _b/τ _d are small in all cases (averaging ~2.08% of calculated τ _b/τ _d). Patterns of τ_b/τ_d for the six periods are shown in Figure 5. During periods 1 and 3, when velocities were high, patterns of τ _b/τ _d were similar, with a slippery region dominating in the middle and upper tongue and a sticky spot in the lower tongue (Fig. 5a and c). In periods 4 and 6, when velocities were also high, patterns of τ _b/τ _d were similar (but contrasted with those in periods 1 and 3), with a large slippery spot in the lower tongue, a large slippery region in the upper tongue, and a sticky spot between them in the middle tongue (Fig. 5d and f). In periods 2 and 5, which shared similarly low velocities and negligible strain rates, there were contrasting τ _b/τ _d patterns (Fig. 5b and e). Period 2 had fairly uniform τ _b/τ _d values close to unity across the tongue, indicating driving stresses were resisted almost entirely by basal drag (but with a slight sticky spot in the middle tongue). Conversely, τ _b/τ _d was highly variable in period 5, with a marked slippery spot in the middle tongue surrounded by sticky regions in the lower and upper tongue.

Fig. 5. Ratio of basal drag to driving stress (τ_b/τ_d) in the tongue. The glacier terminus is to the right (increasing northing). Solid lines and triangles represent the western line of force-balance blocks, while dotted lines and squares represent the eastern line of blocks. Where τ_b/τ _d <1, some resistance to flow comes from non-local sources (a slippery spot). Where τ _b/τ _d >1, flow is driven by forces in addition to the driving force (a sticky spot). Error bars representing the standard deviation are shown. (a) Period 1 (5–9 July); (b) period 2 (9–15 July); (c) period 3 (15–20 July); (d) period 4 (20–23 July); (e) period 5 (23 July–6 August); and (f) period 6 (6–9 August).

Meteorology and hydrology

Air-temperature and rainfall data are shown in Figure 6. This 35 day period was the warmest period of the year. Discharge and turbidity in the eastern and western proglacial streams are shown in Figure 7a, while Figure 7b shows total daily runoff from the glacier, and the subglacial discharge estimated from the mixing model. Figure 7a shows that ~76% of the total discharge drains from the east, while ~24% drains from the west. The western stream discharge showed little response to changing temperatures other than diurnal variations and showed only small daily turbidity peaks. This stream carries mainly supraglacial and ice- marginal discharge (Reference Irvine-FynnIrvine-Fynn and others, 2005). Discharge in the eastern stream was always greater. There are clear diurnal discharge and turbidity variations in the eastern stream, but more marked fluctuations also occur in response to longer-term, day-to-day temperature fluctuations. One of the most notable features of the eastern stream is the large increase in both discharge and turbidity after day 198 (Fig. 7a). This was associated with a rapid increase in discharge from the subglacially fed upwelling, which reached seasonal maximum values on day 201 (Fig. 7b).

Fig. 6. Mean hourly air temperature (°C) (thick black line) and 12 hourly precipitation totals (mm) (thin bars) during summer 1999. Air temperatures never fell below 0°C, so all precipitation was rain. The six intra-summer periods are marked.

Fig. 7. (a) Proglacial discharge (in m³ s ¹) and turbidity (in mV, logged directly from the turbidity sensor) time series in 1999 for the eastern and western outlets. The six intra-summer periods are marked. (b) Total (measured) daily runoff and (modelled) subglacial discharge (both in m³ d^–1).

The air-temperature data were used to drive a simple degree-day melt model in order to calculate the daily meltwater inputs to the glacier. Degree-day factors were determined following an analysis of 6 years of meteorological and ablation data by Reference BrinkhausBrinkhaus (2003). This work found that the ablation conditions in 1999 were best and most simply represented by the application of an average degree-day factor for all summers from 1997 to 2001 (8.75mmd^–1 °C^–1) and a lapse rate of 0.0065°CITΓ¹. The average degree-day factor compares favourably with degree- day factors found by other workers for glaciers in Switzerland, France, Norway, Arctic Canada, Patagonia, North Greenland and Sweden (see Reference Braithwaite and ZhangBraithwaite and Zhang, 2000, table 4).

Modelled daily melt was calculated by multiplying the degree-day factor by the mean daily temperature in a number of 50 m elevation zones. These zones represent those adopted by the Norsk Polarinstitutt mass-balance monitoring programme (e.g. Reference Hagen and Liestöl.Hagen and Liestøl, 1990). The modelled inputs to the glacier were then compared with measured discharge in the proglacial streams in order to estimate patterns of daily water storage (Fig. 8). In the western stream, modelled inputs match measured outputs reasonably accurately (Fig. 8a). Small discrepancies are probably due to the weaknesses in the model rather than any subtle variations in water storage (Fig. 8c). However, there were significant differences between inputs and outputs in the eastern stream, which we interpret in terms of water- storage variations (Fig. 8b). Virtually all of the stored water release may be accounted for by subglacial drainage (Fig. 8c;cf. Fig. 7b). Thus the switch to net runoff release is coincident with the rapid increase in subglacial drainage (around day 198).

Fig. 8. (a, b) Modelled and measured daily water inputs to and outputs from the western stream (a) and eastern stream (b). (c) The difference between inputs and outputs in the eastern and western streams. A positive difference indicates less water exiting the glacier than would be expected given the recorded temperatures, and thus water going into storage; while a negative difference indicates more water exiting the glacier than would be expected, and thus water being released from storage. Water balance is shown up to day 210 only, since there was no eastern discharge record after this.

The air-temperature and proglacial stream discharge data were also used to investigate the lag times between diurnal temperature and discharge fluctuations. Lag times give some indication of the speed at which meltwater is routed through the glacier’s drainage system and therefore the hydraulic efficiency of the drainage pathways (Reference Gurnell, Clark and HillGurnell and others, 1992). In each period, the lag with the highest correlation coefficient is used as a surrogate for the time lag between melting at the surface and the arrival of that water in the proglacial streams (Fig. 9). In the western stream, there was a reduction in lag between periods 1 and 3 from 8 to 3 hours. Thereafter, the lag remained at 3 hours except for the temporary increase to 6 hours in period 5. This implies a gradual increase in the hydraulic efficiency of the glacier drainage system feeding the western stream during the first half of the summer, as air temperatures and discharges generally rose, with a temporary reduction in efficiency during the low air temperatures and discharges of period 5. In the eastern stream, the lag was already only 3 hours in period 1. This increased to 7 hours during the low air temperatures and discharges of period 2, before returning again in period 3 to 3 hours. Thereafter, there was a lack of any clear lag in periods 4 and 5. Thus, the drainage system feeding the eastern stream did not evolve through the early summer and did not route the meltwater in a systematic way during the late summer as it had done on the western side.

Fig. 9. The correlation coefficients between air temperature and discharge associated with different lag times in the western outlet (a) and eastern outlet (b), in periods 1–5, when a full discharge record was available. Data were differenced to remove trends, and smoothed over a 5 hour period, using a centred 5 hour moving average. For each period, the greatest correlation coefficient indicates the average time lag between melting at the surface and discharge in the proglacial streams.