3.1. Methods of velocity observations

Observations of glacier surface velocity are derived from in situ GPS surveys and three satellite sources using feature tracking between repeat-pass pairs of images: (1) Envisat Advanced Synthetic Aperture Radar (ASAR) with a nominal spatial resolution of 30 m, (2) Landsat Enhanced Thematic Mapper (ETM) band 8 (panchromatic) with a nominal spatial resolution of 15 m and (3) TerraSAR-X (Stripmap mode) with a nominal spatial resolution of 2 m. Each satellite source has its own advantages and disadvantages for this application, but together they provide appropriate spatial and temporal resolution and coverage of velocity fields.

Envisat ASAR provides a noisy retrieval of surface velocity because of its relatively coarse spatial resolution and its long revisit time (35 days), but its all-weather capacity and long archive allows plenty of observations through the modelled time period (2000-10). Landsat ETM is limited to the summer period only, and suffers from the scan-line correction (SLC) problem, which sometimes limits the glacier areas to which feature tracking can be applied. TerraSAR-X is the most valuable source of data for velocity retrievals because of its excellent repeat-pass period (11 days) and very high spatial resolution, but is limited because the only data collected are those requested by investigators.

The feature-tracking method used for ASAR has been adequately described by, for example, Reference Strozzi, Luckman, Murray, Wegmuller and WernerStrozzi and others (2002) and Reference Luckman, Murray, de Lange and HannaLuckman and others (2006) and relies on finding the positions of best match between patches from two images, which are then converted to a surface displacement over the whole glacier. For this study the size of patches depends on the satellite sensor and was chosen to maximize the coverage of velocity measurements while accommodating the maximum flow between images separated by different time periods. The error associated with these measurements depends mainly on the spatial resolution and revisit time, both of which vary according to the sensor and/or the dates of usable archived data. In all cases, however, the error in speed is significantly less than the interannual variability in speed (<40ma^-1). When comparing surface velocities from image pairs from different sensors, the complicating issue is the time period over which the displacements are measured, and how they sample the flow, which can vary on different timescales in response to tidal, meltwater and other factors. Hence, the data appear to be quite noisy where in fact this variability is an artefact of accumulating a variable flow rate over a variety of time periods.

Figure 1b shows an example of a TerraSAR-X velocity field, illustrating the quality of velocity data derived from this very high spatial resolution satellite sensor. Velocity data for three TerraSAR-X image pairs from along the flowline transect (Fig. 1b) are presented in Figure 2a. To illustrate how the velocity changes in time in response to seasonal forcing and the 2010 calving event, data are extracted from two points (purple markers in Fig. 1b), one on the ice shelf and one upstream of the grounding line. In July 2011 we installed three single-frequency GPS units, which transmit their position via the Advanced Research and Global Observation Satellite (ARGOS) every 3 hours. The GPS units are located near the front, and below and above the grounding line (black circles in Fig. 1b). The 3 hourly data are used to calculate velocities at the longer time intervals of the satellite data. The velocities from all datasets including Envisat, Landsat, ten TerraSAR-X pairs and GPS units are presented along with the numerical experiments and are used to constrain the results of the numerical modelling.

Fig. 2. (a) Satellite-derived surface velocities along the flowline during midsummer (red) and early winter (green) in 2009, and late summer 2010 (blue) shortly after the large calving event. (b) Modelled velocities for seasonal basal lubrication experiment. (c, d) The observed (c) and modelled (d) glacier tongue geometries (blue profiles). The black line is the fjord basal topography. The dashed red profile illustrates the modelled glacier tongue geometry after the 2010 calving event, and the grey line illustrates the prescribed submarine melt rate pattern under the shelf (m a–1, ten times exaggerated). (e) Modified lateral drag coefficient along the flowline. (f) Driving stress, basal drag, lateral drag and the gradients in longitudinal stresses along the flowline. The driving stress is plotted as negative to make a clear distinction from the other curves shown. Thus, a negative driving stress acts to drive the ice in the downslope direction; positive values for the other stresses shown correspond to resistance to flow. The inset shows detail near the shelf front.

3.2. Results of velocity observations

The flow speed of Petermann Glacier is highest near the grounding line and more-or-less constant along the floating ice shelf (1100-1400 m a^-1) as determined from observations (Fig. 2a). The seasonal variations in flow speed of ~25% are almost uniform along the whole ice shelf and extend several kilometres upstream of the grounding line. Even in the most densely sampled year (2009), the temporal resolution of our data does not allow a very precise assessment of the start and end of the summer peak in flow, and we can neither detect nor rule out a post-summer slowdown that might be associated with the seasonal evolution of the subglacial drainage system (Reference Howat, Box, Ahn, Herrington and McFaddenHowat and others, 2010). However, the summer speed-up is of short duration and seems to be linked to the period of surface melt (˜3 months). Above the grounding line the velocities are generally ˜20% lower than below, but undergo the same summertime acceleration. The observed velocities since 2006 also show a multi-annual increasing trend on the shelf of ˜30 m a^-2, which is shown and discussed in Section 5.4.

After the 2010 calving event, only a very minor acceleration near the terminus is observed (Fig. 2a), with a magnitude well below the seasonal variations. The high spatial resolution TerraSAR-X velocity data (Fig. 1b) reveal only small cross-sectional variations in ice flow, in particular in the lower part of the ice shelf and an abrupt drop in flow speed at their lateral margin, which both suggest limited lateral drag is being provided by the sides.

5.1. Pre-2010 calving event flow

In order to reproduce the pre-calving-event flow pattern and geometry, observed in 2000 (Reference Rignot and SteffenRignot and Steffen, 2008), a very low resistive lateral stress in the lower part of the floating ice tongue is required in the flow model. This finding is strongly supported by the satellite images and the feature-tracking velocity results that reveal only weak (above-water) attachment of the floating ice tongue to its lateral margins and slight cross-flow variations in speed in the frontal region, in particular in the lowest 25 km that calved away in 2010 (Fig. 1). Such lateral weakening is included in the model by multiplying the ice rheology in the lateral stress term by a lateral drag coefficient. An asymptotic reduction of this lateral drag coefficient by a factor of 1-0.4, first applied for the lower 50 km from the grounding line and then repeated for the last 25 km of the shelf (corresponding to the extent of the calving event), allows a successful reproduction of the along-flow geometry and velocity pattern as observed in 2000 (Fig. 2e). Note that this adjustment leads to very low lateral stresses (<1 kPa; Fig. 2f) in the lower 25 km. The corresponding adjusted geometry (Fig. 2d) is used as the initial geometry for the following model experiments.

5.2. Response to the 2010 calving event

In the first model experiment, we investigate the response of Petermann Glacier to the 2010 calving event. The large calving event is introduced in the model by simply removing the last 25 km of the floating tongue. To account for the seasonal background variations in flow speed, we additionally apply a seasonal forcing by regulating the basal lubrication as a linear function of surface runoff. This seasonality is explored further in Section 5.4, and here we focus merely on the effect of the calving event on the glacier flow.

Figure 2 compares the modelled flow against the satellite- derived glacier velocity along the flowline. The model emulates well both the observed spatial pattern of velocity and seasonal speed-up for pre-calving-event conditions, suggesting that it is adequately calibrated and performing well (Fig. 2b). The main difference between observations and model output occurs above the grounding line and is explained by the fact that the observations reflect the maximum surface velocity while the model calculates the width-averaged velocity. Where lateral resistance is high, such as above the grounding line, the applied parameterization of lateral drag should indeed result in width- averaged modelled velocities that are 80% of those observed along the centre line.

Despite the significant reduction in ice-shelf extent caused by the large calving event of 2010, the model results indicate only a minor speed-up after this event, in agreement with the observations. This insensitivity to shelf retreat is not surprising, as it is a direct consequence of the very low along-flow resistive stresses for the front part of the ice shelf resulting from the limited attachment to the fjord walls, and of the fact that this part of the shelf is thin (˜100 m) and wide (15 km). The observed and modelled geometry before and after the calving event are illustrated in Figure 2c and d, the lateral drag coefficient is shown in Figure 2e, and all relevant stresses along the flowline are summarized in Figure 2f. The driving stress and resistive forces along the shelf are <10 kPa, which is more than an order of magnitude smaller than stresses estimated for the front region of Jakobshavn Isbr* (Reference Van der Veen, Plummer and StearnsVan der Veen and others, 2011). Consequently, and in strong contrast to Jakobshavn Isbr* (Reference Vieli and NickVieli and Nick, 2011), the removal of the front part of the shelf did not cause a large change in buttressing forces. The close correspondence between observations and model output implies that the parameterization of geometry and drag coefficients is appropriate, and confirms that for Petermann Glacier, at least in the lower part of the ice shelf, the effect of buttressing is limited.

5.3. Transient response and future stability

In the next set of modelling experiments, we impose perturbations corresponding to the processes and forcings discussed in Section 1, and investigate their impact on the dynamics of the glacier. Three main external forcings are considered: shelf retreat, enhanced melt-induced basal lubrication and enhanced submarine melt.

Time evolution of the glacier ice-front position, grounding-line position and ice flux at the grounding line is shown in Figure 3a and b. At the start of the first experiment (year 1), we remove 25 km of the shelf by increasing the water level in the surface crevasses from 10 m to 70 m. This leads to a very slight increase in ice flux and flow, but little change in the grounding line position. Setting back the water level to its original amount of 10m after 4 years allows the glacier to readvance and reach its pre-retreat length after ˜30 years. The advance speed of the terminus is close to the flow speed of the glacier, which indicates that calving is suppressed during this readvance. As we lower the water level, the surface crevasses become too shallow to penetrate through the full thickness at the glacier front and fulfil the calving criterion, hence calving is halted until the glacier front thins sufficiently.

Fig. 3. Response of the shelf front and grounding line to different model perturbations. (a) Grounding line positions (dotted lines) and front position (solid lines). (b) Time evolution of flux at the grounding line for different perturbations (colours match the legend in (a)). The black and blue lines correspond to the shelf retreat by setting the water level (10-70 and 70-130m, respectively). The red and green lines show the results for the increased basal lubrication and increased submarine melt rate experiments. (c, d) The geometry and velocity profiles for the increased submarine melt rate experiment at different time-steps indicated by arrows in (a). The dark-blue and black profiles display the geometry before and after the large calving event at year 1, respectively. The colour of each profile corresponds to the colour of the arrows. (e) The glacier width along the flowline.

In an additional experiment we forced a further shelf retreat of 20 km after 4 years by setting the water level in surface crevasses to a higher value (130 m). By immediately setting back the water level to its earlier value (70m), the glacier starts advancing and forms its previous shelf in ˜15 years. This additional retreat experiment results in a larger increase in velocity and ice discharge than in the first shelf- removal experiment. This enhanced sensitivity to calving can be explained by the greater loss of lateral buttressing as the ice shelf is thicker closer to the grounding line and the lateral drag coefficient is larger. However, this further retreat still does not lead to a grounding line retreat, indicating that calving events of this scale will not be sufficient to destabilize Petermann Glacier. Note that the resulting ice flux anomaly is rather limited and diminishes within a few years.

In order to explore the effect of other perturbations on glacier evolution, we keep the water level in crevasses constant at 70 m after the first calving event (corresponding to the 2010 event) and apply the other perturbations in year 5.

In the second experiment we simulate a ten-times increase in the surface melt rate and doubling of the melt- season duration. Assuming that surface melt penetrates to the glacier base, the basal sliding coefficient is reduced by a factor which in our model decays from 1 to 0.1 from 30 km upstream to the grounding line. Note that in the model the basal sliding coefficient is already inversely proportional to the effective pressure at the bed and thus asymptotically approaches zero near the terminus as the glacier comes close to flotation. This additional prescribed reduction in the basal sliding coefficient causes a net reduction in basal resistance and leads to a flow acceleration and shelf front advance of ˜-10km over 40 years but leaves the grounding line position stationary. Ice flux at the grounding line increases initially from ˜12 to ˜-26km³a^-1. The increased ice flux then drops to 50% of its initial anomaly within a few years and then gradually decreases as the glacier geometry adjusts to the reduced basal resistance. The flow acceleration results in thinning upstream of the lubricated region, which diminishes towards the grounding line where the ice is close to flotation and basal resistance is already very low due to reduced effective pressure at the bed. Consequently, this perturbation results neither in a significant change in glacier thickness at the grounding line nor in grounding line retreat, as similarly demonstrated by Reference Nick, Vieli, Howat and JoughinNick and others (2009) from modelling experiments on Helheim Glacier. In an additional experiment (not shown here), decreased lateral drag along the fjord walls in the vicinity of the grounding line causes behaviour similar to that produced by increased basal lubrication.

In the third modelling experiment, we perturb the model by increasing the submarine melt rate underneath the floating shelf by a factor of three, a value prompted by the observation that the amount of heat flux entering Petermann Fjord is enough to accommodate such a melt rate (Reference Johnson, Munchow, Falkner and MellingJohnson and others, 2011). Our experiment shows that increased submarine melt under the shelf results in further retreat of the shelf and grounding line. The higher melt rates underneath the shelf cause it to thin, reducing the lateral resistance. As most melting takes place close to the grounding line, the higher melt rate also steepens the shelf near the grounding line, resulting in a higher gravitational driving force there. This, together with a lower lateral resistance, facilitates acceleration of flow in this section of the glacier, leading to faster ice discharge across the grounding line (Reference Dupont and AlleyDupont and Alley, 2005;Reference Goldberg, Holland and SchoofGoldberg and others, 2009). Increased discharge initiates thinning at the grounding line, which propagates upstream and eventually leads to grounding line retreat.

Figure 3c and d illustrate geometry and velocity profiles of the glacier in different years for the enhanced submarine melt rate experiment, including before and after the 2010 calving event. After applying the perturbation in year 5, there is an initial gradual retreat until year 20, as the shelf front thins and retreats to comply with the calving criterion. The shape of the melt rate profile (Fig. 2d) means that most of the thinning occurs in front of the grounding line ˜-570km from the ice divide. In year 20, the shelf becomes too thin to sustain the calving criterion in this region, resulting in a sudden further retreat of 20km. This new retreat event reduces the resistive stresses as it removes a significant part of the shelf, which is more firmly attached to the fjord walls than the part that calved in 2010. As a result, there is an increase in ice velocity and discharge across the grounding line and subsequent gradual thinning.

In year 55, the shelf starts advancing while the grounding line is still retreating at a slightly faster pace as it passes through a basal deepening (˜540 km). At this point, the ice discharge from the grounding line rapidly increases (second sudden change) as the grounding line retreats into a basal depression and wider area (Fig. 3e). The higher discharge drains more ice from upstream and transports it to the shelf region, resulting in a thicker and flatter ice shelf and a temporary frontal advance. Note that this additional discharge leads to a remarkably fast surface drawdown above the grounding line (red and orange profiles in Fig. 3c). As the shelf lengthens, submarine melt becomes more effective and again thins the shelf close to the grounding line, causing a large retreat and subsequent increase in ice flux (around year 93).

In summary, the irregular behaviour at the front and grounding line retreat is the consequence of the interplay between several basal depressions, the upstream increase in glacier width and the spatial and temporal fluctuation of the submarine melt. On the other hand, this experiment demonstrates that terminus-position changes may be highly asynchronous to grounding line motion and therefore may not be a valuable diagnostic for the glacier’s stability.

Our model result indicates that in the case of three times larger submarine melt rate, there will be an extensive retreat of the shelf front and grounding line (˜40 and 80 km, respectively), and ice discharge will increase by ˜300% in <100 years (Fig. 3a and b). In additional experiments, we examined the sensitivity of the glacier to different multipliers of submarine melt rates (not shown here). Results show the same pattern of grounding-line and shelf-front retreat for lower melt rates, but at different timescales. For example, increasing the submarine melt rate by factors of 2.5 and 2 shows the extensive retreat of the grounding line (similar to what happened in year 50 when applying a factor three, shown in Fig. 3a) significantly later (in years 80 and 185, respectively). Applying even lower factors only results in shelf front retreat and does not trigger the grounding line to retreat >2 km (i.e. into the overdeepening part of the bed) after 1000 years.

5.4. Seasonal dynamics

The observed seasonal variations in flow speed show a 25% increase of flow during the brief summer period (Fig. 4a and c). This seasonal pattern in velocity is very similar at the shelf front and above the grounding line, which therefore offers an opportunity to further explore environmental forcing factors.

Fig. 4. Observed (a, c) and modelled (b, d) surface ice velocity on the ice shelf (a, b) and above the grounding line (c, d) from 2006 to 2011. Velocities from Envisat, Landsat and TerraSAR-X are shown with stars, circles and triangles, respectively. The black lines show the velocity record of GPS units deployed in July 2011 and estimated for the same locations as the satellite data. Model results from experiments with seasonality in basal lubrication (red), frontal back pressure (blue) and submarine melt rate (green) are illustrated in (b) and (d). The vertical dashed lines indicate midsummer (1 July) for each year.

We compare three plausible seasonal forcings using our model: (1) seasonal variations in submarine melt beneath the shelf driven by the fjord water temperature cycle and enhanced convection from subglacial discharge of summer surface meltwater;(2) seasonal variation in back-stress and lateral drag resulting from concomitant variations in sea ice and ice melange, and;(3) seasonal variation in basal resistance due to surface meltwater availability.

For the first experiment, we keep the submarine melt pattern along the ice shelf constant and fixed to the grounding line, but apply an exaggerated increase (by a factor often) in melt rate from April to September each year (Fig. 2d). Such temporal variability in submarine melt produces only small seasonal variations in velocity both at the grounding line and at the ice front, and these are far below the observed magnitudes (Fig. 4b and d). Submarine melt is a gradual process;even a very high melt rate does not result in significant changes in velocity on seasonal timescales. We conclude that enhanced submarine melt is not a potential forcing of the observed seasonality in the surface velocity.

Sea ice in Petermann Fjord may exert a back pressure at the ice front and increases lateral resistance between the fjord walls and floating ice where there is open water in summer. To investigate whether such forcing can explain the observed seasonality in velocity of Petermann Glacier we apply a seasonal reduction in frontal back pressure and side resistance (along the last 20 km of the shelf, where there is open water in summer). The reduction rate is controlled by the air temperature in the fjord from RACMO simulations (Reference Ettema, Van den Broeke, Van Meijgaard, Van de Berg, Box and SteffenEttema and others, 2010). Such forcing is able to reproduce the observed variations in velocity near the ice front (Fig. 4a and b), but the amplitude of the seasonal response experienced above the grounding line is significantly smaller (Fig. 4d), which is not what we observe in the satellite data (Fig. 4c). Thus, this modelling experiment only to a certain extent supports variations in sea-ice or ice melange cover as the governing forcing for the observed seasonal cycle in velocity.

In the last experiment, we modify the basal lubrication by reducing the basal resistance coefficient from 1.0 to ˜0.6, which is a tuned value to produce the observed summer velocities. The value of the resistance coefficient is regulated by the amount of surface meltwater runoff from RACMO simulations. Surface meltwater runoff in the vicinity of the grounding line varies from zero in winter to a maximum of ˜0.043 m³ m^-2 d^-1 in summer. Applying a seasonal variation in basal lubrication (Fig. 4b and d) causes velocity changes similar to those observed above the grounding line as well as at the ice front. This suggests that processes initiated at or near the grounding line are the dominant control on the observed seasonal velocity variations of Petermann Glacier, but not fully eliminating the influence of variations in lateral resistance and back pressure on the floating ice tongue.

The observed increasing multi-annual trend of flow speed of ˜30 m a^-2 between 2006 and 2010 (Fig. 4a) is not reproduced by this model experiment because none of the climate parameters used to force the seasonal variation exhibit such a trend. We speculate that this increase is associated with reduced ice-marginal coupling, and can reproduce this in the model by lowering the friction parameter (not shown); a more careful analysis of the 2-D velocity field on the floating tongue is needed to determine the cause of the observed speed-up.