Modelling environmental influences on calving at Helheim Glacier , East Greenland

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Introduction
Calving is an important mass loss process for both ice sheets and smaller tidewater glaciers (e.g., Church et al., 2001;Rignot and Thomas, 2002;Blaszczyk et al., 2009) and plays a strong role in tidewater glacier dynamics (Meier and Post, 1987).In the past there has been debate about whether changes in glacier dynamics and terminus retreat, observed at a number of locations worldwide, were triggered by changes in calving rate or changes in glacier velocity, a debate made more complicated by the potential feedback between the two mechanisms (van der Veen, 2002).Modelling work has since shown that changes at the terminus are able to trigger retreat (Nick et al., Figures Back Close Full 2009), and it is now generally thought that calving rates are affected by a number of external environmental variables.The four main hypothesized mechanisms for tidewater glacier retreat are: 1. Increased water in crevasses, which increases calving by increasing the depth of fractures around the calving front (Benn et al., 2007a).
2. Increased basal water pressure, leading to faster sliding, which can also enhance fracturing by increasing longitudinal strain rates (van der Veen, 2002).
3. Undercutting of the terminus by submarine melt, which alters the stress field around the terminus and is hypothesised to enhance calving (Motyka et al., 2003;O'Leary and Christofferson, 2013).
4. Reduction in backstress from proglacial ice mélange, which can otherwise provide a stabilising force to the calving face and inhibit calving (Nick et al., 2009;Amundson and Truffer, 2010).
These mechanisms depend on atmospheric and ocean temperatures driving either ice surface or submarine melt.This means that the environmental signals involved are strongly linked, as increased air and ocean temperatures are typically strongly correlated (Deser and Blackmon, 1993).In addition, the physical mechanisms relating ice dynamics to oceanic and atmospheric warming are also inter-related.For example, the plume models of O' Leary (2011) and Xu et al. (2012) show that undercutting rates at the terminus are affected not only by the surrounding water temperature, but also by the rate of subglacial discharge which is directly linked to glacier surface ablation and hence air temperatures.The interaction between different forcing variables and mechanisms makes it difficult to distinguish the precise causes of tidewater glacier retreat by observation alone.
By including calving in an ice flow model it is possible to experiment with applying these environmental forcing factors separately and deduce the likely sensitivity of tidewater glaciers to them.Such a study has previously been performed by Vieli and Figures

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Full Nick (2011) using a physically realistic calving criterion based on the modelled depth of crevasses in the ice.However, this type of calving criterion is best applied with an accurate representation of the stress field around the terminus, whereas the vertically averaged model used in previous studies (e.g., Nick et al., 2010;Vieli and Nick, 2011) is likely to lead to large inaccuracies in modelled crevasse penetration.A previous publication presented a two-dimensional, vertically-resolved flowline model of Columbia Glacier, Alaska, using a similar calving criterion, which provides a more detailed solution of the stress field and therefore more reliable estimates of crevasse penetration (Cook et al., 2012).That study was limited in scope due to the time-consuming, manual implementation of calving.In this paper we present an extension of the previous model which now automates the calving process, making model runs significantly faster and allowing a greater scope of experiments.Some areas of the physics involved have also been made more robust, although the model is still currently limited to representing grounded ice.
The chosen study location is Helheim Glacier, East Greenland (Fig. 1a), which was selected due to its recent interesting behaviour.The mean annual terminus location of Helheim Glacier was stable throughout the late 1980s and early 1990s, although subannual variations of between 2 and 4 km are typical (Bevan et al., 2012).After a period of thinning in the 1990s (Abdalati et al., 2001), the front began to retreat in 2001.Over the period 2001-2005 the front retreated by 6 km, the main trunk sped up by around 3.0 km a −1 and the surface lowered by 40 m (Howat et al., 2005;Luckman et al., 2006).By 2006, thinning had stopped near the front and the glacier slowed, advancing approximately 4 km from its most retreated position.Since 2007 the glacier has once again shown a stable mean annual terminus position, although somewhat further retreated than the pre-2004 position (Howat et al., 2007;Bevan et al., 2012).Although large annual variation in terminus position is typical at Helheim Glacier, the rapid and consistent retreat of 2001-2005 was exceptional and a similar retreat was also observed in other glaciers in the region, leading to the conclusion that it was likely the result of a single environmental event (Howat et al., 2008).The cause of this synchronous retreat has Figures

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Full been widely debated, with high air and sea temperatures hypothesized as potential triggers (e.g., Joughin et al., 2008b;Murray et al., 2010;Christofferson et al., 2012).The bed elevation data available for Helheim Glacier are limited to ice-covered areas, where airborne radar measurements can be made; therefore, this retreat has also provided a modelling opportunity with data available to set up a model which has sufficient bed data available to experiment with a full range of terminus behaviour: retreating, steady and advancing.In this paper our time-evolving, flowline model of Helheim Glacier is tested for sensitivity to the four main environmental inputs outlined above.

Model setup
This study uses a vertical two-dimensional (2-D) ice flow model to analyze the stresses along a flowline within the glacier, which can then be used to determine whether and where calving occurs.The model is allowed to evolve through time, with calving applied according to the crevasse depth calving criterion proposed by Benn et al. (2007b) and used in previous modelling studies (Nick et al., 2010;Otero et al., 2010;Cook et al., 2012).The criterion states that calving will occur where the surface crevasse field penetrates below sea level.The ice flow solution is calculated using the open source Elmer/Ice finite element modelling software (Gagliardini et al., 2013).Calving is applied using additional code which examines the output from the Elmer/Ice software and updates the geometry to allow for any calving events.If the model is run with a sufficiently short time-step, multiple time-steps will occur between each calving event, thus allowing individual calving events to be distinguished.In this case the model also provides data on the modelled size and frequency of calving events which can be compared to observed data.It should be noted that the calving events do not necessarily represent individual icebergs, but instead a rapid change of terminus position which could also encompass the release of numerous smaller ice volumes.Figures

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Data sources
The study used data from Helheim Glacier at its point of furthest retreat in July 2005, since the terminus position at this date allowed 2 km of bed elevation data for the model to advance into; this enabled an investigation of a full range of glacier behaviour.Elevation and velocity were measured along a flowline close to the center of the glacier identified using surface debris features and velocity data (Fig. 1a).A flowline model is not well suited to representing areas of convergent flow, such as where the outlet glacier meets the Greenland Ice Sheet.Therefore, a point was identified at 152 km along the flowline where the behaviour of the glacier changes from ice-sheet-type, funneling ice from a wide range of angles, to an outlet glacier where the flow is channelised and the velocity vectors are roughly parallel; the model was limited to locations downstream of this point.
Bed elevation was taken from a gridded data product available from the Center for Remote Sensing of Ice Sheets (CReSIS) (Gogineni, 2012); this product is derived from radar depth measurements made by CReSIS from 2006-2009 and NASA's Operation IceBridge from 2009-2011 (Leuschen et al., 2011).These data provide a horizontal resolution of 500 m and vertical error of around 20 m in areas of good bed signal, while areas of poor radar return use an interpolated bed with higher associated error.Surface elevation data were taken from a pair of ASTER images (July 2005), with photogrammetry providing a DEM with a vertical accuracy of 10 m (Murray et al., 2010).Surface velocity was measured by feature tracking between consecutive Landsat-7 ETM+ images (July, August and September 2005), the method and errors are described in detail in Bevan et al. (2012).The model output was compared to observed terminus position; this was measured at the intersection between the flowline and a manually digitised ice front, mainly using images from Landsat-7 ETM+, but also supplemented by ASTER, SPOT 5 and airborne lidar images where available (Fig. 2b).Errors on these terminus positions have been estimated to be less than 100 m (Bevan et al., 2012).The resulting initial geometry of the model is shown in Fig. 1c.Introduction

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Ice flow model
The geometry is used to find the velocity and stress fields within the glacier using Stokes flow for an incompressible fluid: where u is the 2-D velocity vector in the vertical plane, τ is the deviatoric stress tensor, p is pressure, ρ i = 910 kg m −3 is the ice density and g is gravitational acceleration.The deviatoric stress is related to the strain rate ( ˙ ij ) by the constitutive relation: where the effective viscosity µ is given by n=3 is the most commonly used exponent in glaciology, and is adopted here.The Arrhenius factor A depends on the temperature of the ice relative to the pressure melting point (T )according to the Arrhenius equation: where R is the universal gas constant, and Q, the creep activation energy, and the constant A 0 depend on the ice temperature.For this study standard values of these constants were taken from Paterson (1994).In the absence of observational data, the ice temperature profile with depth was approximated from a thermo-mechanical modelling study of Jakobshavn Isbrae, a similar outlet glacier on the West coast of Figures

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Full Greenland (Funk et al., 1994).This temperature profile gives temperatures between −5 and −10 • C at the surface and bed, and approximately −25 • C in the centre of the glacier.The precise values depend on the local ice thickness; however, the basal ice temperature never reaches the pressure melting point.

Boundary conditions
The glacier surface is assumed to be stress-free , and elevation changes are calculated by the equation for the kinematic boundary condition: where s is the surface elevation and M the surface mass balance.Mass balance values were taken from Andersen et al. (2010), who directly observed melt from stakes placed along the glacier in two consecutive melt seasons in 2007 and 2008.These measurements were assumed to approximate the entire summer melt.Since the model uses only a two-dimensional geometry, technically there are no lateral boundary conditions to apply.However, this neglects important 3D processes such as the confluence of tributaries, the effect on volume conservation of changes in channel width, and lateral drag from the glacier margins, which has been suggested to have a stabilising effect on glacier termini (O'Neel et al., 2005).Little can be done to address the effect of tributaries without switching to a 3-D geometry, but by measuring the channel width at the surface and making assumptions about the channel geometry we are able to estimate the lateral drag and flux contribution from changes in channel width.We assume a trapezoidal channel shape with a width at the base two thirds of that at the surface, a constant lateral velocity gradient and sliding at the margins of half the centreline velocity.This leads to a volume contribution from lateral spreading or constriction of ice (φ): where H is the ice thickness and U x and W are the velocity and channel width at the surface.This was added as an additional surface mass balance term.The geometry can also be used to derive a vertically varying lateral drag term: where w(z) is the channel width at an elevation z and u x the velocity along flowline.The vertically varying force (F ) was imposed on each element in the glacier body, opposing flow.
To initialize the model and remove the unstabilising effects of errors in the DEMs used, a spin-up run was performed allowing the surface elevation to evolve until its rate of change came into line with annual mass balance values.This process provides a stable starting point for experiments, although it also means that there are some areas of difference between the modelled and observed geometry and surface velocity profile (Fig. 1b and c).These differences occur where the model fails to reproduce the full flow behaviour of the ice, such as regions where 3-D flow features are significant, or where there are errors in the bed DEM, which have been shown to have a significant effect on modelled velocities (Zwinger and Moore, 2009;Seroussi et al., 2011).
Basal sliding was implemented in the model using the non-linear scheme proposed by Gagliardini et al. (2007).Basal water pressure was taken to be equivalent to depth below sea level, and the sliding parameters were adjusted to provide a best fit to observed surface velocity when the model reached its relaxed state.At the inflow boundary a constant velocity of 3500 m a −1 was applied (an average of the observed surface velocity of 4006 m a −1 and the modelled sliding velocity of approximately 3000 m a −1 ).
See Fig. 1b for the surface velocity profile after spin-up.At the terminus, water pressure was applied to the calving face, which was allowed to evolve according to the modelled velocity profile.In experiments where an additional backstress was applied to the calving face to replicate the effect of ice mélange, this acted parallel to the x axis, in the opposite direction to ice flow.In experiments Introduction

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Full designed to replicate submarine melt, nodes on the calving face were negatively displaced, parallel to the x axis and according to the assumed melt rate.Because the applied water pressure acts perpendicular to the boundary, in these experiments the model was able to fully represent the buoyancy forces arising from the undercut cavity.However, the model is unable to represent floating ice caused by detachment of the basal ice boundary from the underlying bedrock, and model runs in which the glacier reached the flotation point were terminated.

Model experiments
The model was first allowed to run freely with no environmental inputs, in order to produce a base set of terminus position and calving event size data.The model was then tested using the four external variables hypothesised to affect terminus behaviour: depth of water in crevasses, basal water pressure, undercutting of the terminus by submarine melt and backstress from ice mélange.The results from the crevasse water depth experiments were also used to identify cases with steady and retreating terminus behaviour; these settings were then used to test the other variables in scenarios with differing terminus behaviour.Each variable was applied in a seasonally varying manner, with maximum values occurring in a summer season, assumed to last from May to September in most cases.The only exception was experiments using backstress from ice mélange, which was applied over a reduced winter season of January-May, effectively creating an additional autumn season (September-January) in which no surface ablation or undercutting occurred and no backstress was applied.The precise values used for each variable are discussed separately in the relevant results section for the sake of clarity, and a list of all experiments performed is provided in Table 1.In each experiment, the time-step was chosen to ensure that multiple time-steps occurred between calving events.This process is important, as if calving were allowed to occur at every time-step there would be a spurious dependence of calving rate on the time-step.The time-step necessary depends on the environmental forcing used; in cases where the model produced more frequent calving events, a time-step of 0.001 yr Introduction

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Full (8.8 h) was used, while for cases with less frequent events (such as those with low crevasse water depth) a maximum of 0.003 yr was chosen.The model used a variable mesh size of 100 m near the upstream boundary and 40 m at the calving front.

Sensitivity analysis
To test if the chosen mesh density and time-step affected the results, terminus position and calving event size were examined using a range of shorter time-steps and finer grids.Increased mesh resolution was found to cause a change of up to 0.2 % in mean calving event size, and a maximum change in terminus position at the end of a 5 yr run of 1.3 %.Changes in time-step had a more significant effect on the mean size of icebergs produced, as smaller time-steps allowed smaller calving events to be resolved, but had a less than 1.0 % effect on modelled terminus position.
The model was also tested for sensitivity to errors in the input data used, by examining the terminus position over a 5 yr run in advancing, steady and retreating experiments (Cook, 2012).No observable effect on terminus behaviour arose from changes in the length of the surface relaxation period, the precise formulation of the lateral drag term, the applied surface mass balance or minor changes in the bed elevation profile.Minor changes in terminus behaviour were caused by changes in the inflow velocity boundary condition, the shape of the basal water pressure profile and the mass balance term used to correct for 3D flux properties.However, these were concluded to be unlikely to have a significant impact on the model results.The only input variable to have a substantial effect on model results was the ice temperature profile used.A change of ±5 • C in the glacier ice produced a significant difference in the calving behaviour of the glacier and also on the modelled evolution of its terminus.

Results
In our initial experiment the model was allowed to run freely for a 5 yr period, with no water in crevasses or other external forcing variables.Over the 5 yr period the model produced 130 calving events (or one every 14 days, on average), with the size distribution roughly normally distributed around a mean length of 479 m (Fig. 2a).Comparing the modelled and observed terminus behaviour over this period, we see that both the model and the real glacier advanced substantially from their position in July 2005, although the real glacier advanced somewhat sooner and further than the model (Fig. 2b).After 1 yr the model's terminus position stabilised, whereas the true glacier continued to fluctuate in length, probably in response to external climate factors.

Crevasse water depth
Few data are available on depths of water in crevasses; therefore, a range of crevasse water depths (henceforth abbreviated as cwd) from 0 to 50 m was selected to represent a wide range of terminus behaviour (see Table 1).As would be expected, larger water depths had the effect of deepening the base of the surface crevasse field (Fig. 3a), causing more frequent calving events to occur.The higher calving frequency had a significant effect on the terminus evolution (Fig. 3b), with greater crevasse water depths causing the terminus to retreat significantly, while for lower water depths the terminus advanced.
In some experiments the terminus reached the flotation point, after which point the experiment was terminated as the model cannot represent floating ice. Figure 3b also shows that the terminus position at times became fixed at a particular point.These locations typically coincided with a local bedrock maximum, on which the terminus was pinned as advance of the model into deeper water triggered calving.This behaviour echoes results using empirical calving laws which also inhibit advance into deeper water (Nick and Oerlemans, 2006).However, unlike the experiments using empirical calving laws, as the terminus geometry evolved our model was able to advance Figures

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Full past bedrock overdeepenings, although this often coincided with the model reaching flotation point.This behaviour mirrors the observation by Joughin et al. (2008b) that Helheim Glacier advanced past an overdeepening by forming a floating ice tongue to "bridge" the gap.

Basal water pressure
Observations at Helheim Glacier have linked basal water pressure and velocity; a study by Andersen et al. (2010) used a surface mass balance model to find a correlation between observed surface velocity and estimates of melt water availability.The relationship between surface meltwater and velocity is expected to be complicated; velocities tend to increase with increasing basal water pressure, however, the basal water pressure depends not only on the flux of water to the bed but also on the type of drainage system at the bed.If sufficient water is added to the system it may trigger a change from a distributed to a more efficient channelized drainage system, thus decreasing basal water pressure (Bartholomew et al., 2010;Howat et al., 2010).Generally, percentage seasonal changes in velocity appear to be higher on the ice sheet than on outlet glaciers, but even outlet glaciers can show seasonal variations in speed of up to 15 %, thought to be caused by changes in basal water pressure (Joughin et al., 2008a).Satellite observations of Helheim Glacier have shown sub-annual velocity changes of approximately 20-30 % (Bevan et al., 2012); however, it it unclear whether these arise mainly from changes in basal sliding conditions or coincident changes in terminus po- using cwd fixed at 0 m and secondly using a seasonally varying cwd, with a summer value of 30 m. Seasonal variation in basal water pressure had a significant effect on the modelled terminus behaviour in both sets of experiments (Fig. 4), with higher basal water pressures causing an increase in calving rates and hence a relative retreat of the terminus.The effect was strongest when applied in conjunction with a varying cwd (Fig. 4e), where the terminus behaviour altered from a strong retreat to a slight advance when the variation in basal water pressure was applied.

Undercutting by submarine melt
Undercutting of the calving face by submarine melt is hypothesized to affect the surrounding stress balance by removing supporting ice, and hence affect calving rates (O'Leary and Christofferson, 2013).Recent observations around Greenland suggest that submarine melt rates in this region may be sufficient to have an effect on calving; studies have used either measurements of fjord temperature and water velocity to estimate the heat flux transported to the glacier terminus, or used the difference between ice flux to the grounding line and surface ablation to infer average melt rates across the calving face.Estimated mean melt rates range from 11 to 1424 m a −1 for glaciers across Greenland (Rignot et al., 2010;Enderlin and Howat, 2013) and between 205 and 1238 m a −1 for Helheim Glacier (Sutherland and Straneo, 2012;Enderlin and Howat, 2013).Based on these estimates, a range of maximum undercutting rates from 1000 to 5000 m a −1 were applied parallel to the flowline, with a vertical profile of zero melt at the waterline, increasing linearly towards the bed.During the winter season a fixed maximum melt rate (m.m.r.) of 150 m a −1 was used, based on plume modelling results by O'Leary (2011).A full list of values used is given in Table 1.
The effect of undercutting on the longitudinal deviatoric stress distribution around the front is shown in Fig. 5a.There are areas where the stress is clearly affected, particularly at the bed around a kilometre behind the front where the stress is strongly affected by increased buoyancy forces acting on the ice.However, in general these differences represent a change of only around 5 % from the original profile, and the Introduction

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Full change in crevasse field depth is relatively small.As might be expected from this, the undercutting applied had relatively little effect on the modelled terminus behaviour (Fig. 5b).In order to test this further, a wider range of undercutting rates was tested to see at what point the terminus would begin to retreat.As shown in Fig. 6, the rates required to cause terminus retreat were between 10 and 20 km a −1 , around an orderof-magnitude higher than published estimates of melt rates.

Ice mélange
Ice mélange is a mixture of calved ice and sea ice, often observed in front of Greenland glaciers.In winter months this mélange can freeze solid, and provide a stress opposing the flow of the glacier, which has been hypothesised to inhibit calving.Relatively little is known about the mélange due to the difficulties in performing studies in the proglacial fjord environment.In order to include the effects of the ice mélange in the calving model both its thickness and the stress it applies on the calving face must be known.Using LiDAR data from July 2007, the ice mélange in front of Helheim Glacier was measured to have a typical freeboard of 10-20 m, discounting isolated larger icebergs (T.James, personal communication, May 2012).We use a mean freeboard value of 15 m, equivalent to a full thickness of approximately 135 m.The only study to measure the stress exerted on the calving face used a coincident change in velocity of 15 % upon the break up of the ice mélange at Store Gletscher, West Greenland to estimate a backstress of 30-60 kPa over the full calving face, equivalent to a force of 1.8-3.6 × 10 7 Nm −1 .Initial experiments showed that values lower than this had no observable effect on the model, therefore experiments were performed using a range of backstress from 0.0 to 50.0 × 10 7 Nm −1 over the vertical range of 15 to −120 m a.s.l., applied during the winter season (see Table 1).
The effect of two of the backstress values on the modelled longitudinal deviatoric stress profile can be seen in Fig. 7.A backstress of 5.0 × 10 7 Nm −1 caused very little difference to the stress field around the front, while for a backstress of 25.0 × 10 7 Nm −1 the effect on the modelled crevasse depth was more significant, showing a clear reduc-Introduction

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Full tion in crevassed extent (Fig. 7).When the modelled terminus evolution was examined, only the very highest backstress was seen to be able to affect terminus behaviour, while the lower values had very little effect at all (Fig. 8).

Model analysis
The nature of these experiments makes it difficult to use normal validation methods to compare the model's performance against observed measurements.The calving model was found to be potentially sensitive to small errors in the bed DEM, as the model often became pinned on bedrock highs.It was also possible to tune it to a very wide range of terminus behaviour by altering the crevasse water depth; therefore, comparing terminus behaviour was not an appropriate validation method.Instead we chose two methods of comparing the model output to known glacier behaviour: comparing calving event size distribution and the glacier's dynamic response to calving.During its retreat, Helheim Glacier was observed to produce distinct calving episodes days to weeks apart, during which the terminus retreated by 0.5 to 1 km (Joughin et al., 2008b).This qualitatively agrees with the style of calving produced by the results presented here, with a typical calving length of 479 m and frequency of every 14 days.A similar calving event size was also produced by a recent particle-based model of Helheim Glacier (Bassis and Jacobs, 2013).The approximately normal calving event size distribution produced by the model disagrees with other published studies which observe a power-law type distribution of iceberg sizes, where small ice losses are the most common (Dowdeswell, 1989;Chapuis, 2011;Åström et al., 2013).This difference is to be expected given the mesh resolution of 40 m and time-step of 0.003 yr used in most experiments -this is insufficient to model very small calving events, but able to capture large changes in terminus position which are likely to dominate long-term behaviour.The discrepancy is also affected by the fact that each calving event predicted by the model does not necessarily occur by the release of a single iceberg but could also indicate a rapid disintegration into smaller ice masses.In fact, this second possibility may be the most likely; in most of the modelled calving events, areas of ice 4422 Figures

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Full The other possibility for comparing modelled to observed behaviour is to examine the effect of calving events on glacier dynamics.It has been reported by Nettles et al. (2008) that large calving events at Helheim Glacier occur at the same time as rapid increases in flow speed of up to 15 % near the calving front, decreasing further upglacier.When the modelled velocity was examined immediately after the calving of large icebergs (400-450 m), a similar increase in velocity was observed of up to 10 % near the front, decreasing to around 1 % 20 km upstream.This behaviour provides further evidence that the model provides a reasonable representation of the interaction between calving and glacier dynamics.

Discussion
The modelled glacier behaviour is shown to be significantly more sensitive to environmental factors related to air temperature and surface run off than to oceanic forcing.This is of particular interest given that surface mass balance models generally agree that surface ablation has been steadily increasing in Greenland over the past two decades (Vernon et al., 2013).The strong effect of water in crevasses may be expected as this relationship arises axiomatically from the calving model used.Selecting the values of cwd to use was challenging as no measurements of observed water depth have been published, and likely values are difficult to predict given the changing volume of crevasses as ice advects downstream, and the probable high rate of drainage in the fractured area around the terminus.Observations have shown that water-filled crevasses can occur even near the calving margin (Danielson and Sharp, 2013), so it is at least known that high water depths are possible in this region, and may become more frequent if the trend of increasing surface ablation continues.and hence sliding velocity.Observations of seasonal velocity variations of 15 % on Greenland outlet glaciers had previously been concluded to be insufficient to have a large effect on terminus behaviour (Joughin et al., 2008a).The results presented here indicate that changes in basal water pressure leading to velocity changes of this magnitude can in fact have a strong influence on modelled calving rates.The finding supports previous tidewater glacier modelling work (Vieli et al., 2000) and observations at other tidewater glaciers (Kamb et al., 1994;Sugiyama et al., 2011;Danielson and Sharp, 2013) which had already observed a significant dynamic response to changes in basal water pressure, and highlights that changes in basal conditions should not be excluded as an important influence on tidewater glaciers.The link between changes in air temperature and runoff, and changes in velocity may, however, be complicated.Adaptations in the basal hydrology system can mean that an increased input of water to the bed results in a more efficient drainage system and hence reduces basal water pressure (Bartholomew et al., 2010).In the context of previous debate over whether dynamic changes in tidewater glaciers are triggered at the terminus or by changes in velocity (e.g., van der Veen, 2002), the results from these two sets of experiments (crevasse water depth and basal water pressure) suggest that either mechanism may have a strong influence on glacier dynamics.
In contrast to previous results, the insensitivity to undercutting by submarine melt shown in these experiments was surprising.Previous research has indicated that a link between glacier retreat in South East Greenland and ocean warming was likely (Murray et al., 2010;Vieli and Nick, 2011), with retreat coinciding with warm ocean temperatures.Previous modelling work has also indicated that undercutting has the potential to cause tidewater glacier retreat (Vieli and Nick, 2011), and in modelling work using a similar two-dimensional, high-resolution solution of stresses around the calving front, cavity formation has been shown to have a significant effect on the stresses around the terminus (O'Leary and Christofferson, 2013).The study by O'Leary and Christofferson (2013) was diagnostic, not time-evolving, and we hypothesize that this was the reason for the difference between their results and ours.Figures To test this hypothesis, we set up two experiments: one applied an undercut cavity of varying length to a fixed terminus while the second applied a fixed undercutting rate, but also allowed the ice around the terminus to evolve as the undercut cavity size increased.The results in Fig. 9 show that when the terminus geometry was fixed, our model reproduced the linear relationship between cavity length and stress displacement previously reported (O' Leary and Christofferson, 2013).However, when the ice was allowed to deform it acted to counterbalance this stress displacement, creating a significantly reduced response to undercutting in the stress field.It may be, therefore, that the fixed geometry causes the O'Leary and Christofferson (2013) model to overestimate the effect of undercutting.
The insensitivity to undercutting would be significantly different if the undercutting were to be concentrated at the waterline (i.e.notching), as has previously been observed in lake-terminating glaciers (Kirkbride and Warren, 1997;Benn et al., 2001) and tidewater glaciers in Svalbard (Vieli et al., 2002).In that case the remaining ice block would be unsupported and highly likely to shear off from the glacier, leaving behind a submarine ice foot.This style of calving was not considered in our model.The terminus of the model remained on a downward sloping bed throughout the experiments, which may also have contributed to the observed insensitivity to undercutting.Previous work on ice sheets has shown that glaciers may be stable even under grounding line retreat on downward facing slopes, but much more sensitive on reverse bed slopes (Schoof, 2007).To fully determine the sensitivity of tidewater glaciers to undercutting future work should include experiments on a wider range of bed topography.
The experiments presented also showed little relationship between backstress and terminus behaviour, except in experiments in which the backstress was likely to be unrealistically high.Although little is known about the likely magnitude of backstress provided by a proglacial mélange, it is unlikely to exceed the shear strength of sea ice, which has been measured at 550 ± 120 kPa (Frederking and Timco, 1984), while our experiments used stresses of up to 3.7 MPa.Although these results show that backstress from ice mélange is unlikely to have a direct effect on the crevasse field Introduction

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Full around the terminus, it has been hypothesised by Amundson et al. (2010) to affect calving rates by exerting enough force to prevent a calved block of ice from tipping, thus keeping it pinned in contact with the main body of the glacier.Such large blocks of ice could feasibly have a much greater effect on the stress at the front of the glacier, so the winter ice mélange should not be discounted as having a possible effect on calving rates, and future modelling work should include this type of process.
With greater data availability a more rigorous validation of the model would be possible by comparing both the distribution of calving event size and frequency and the terminus behaviour of the model with observed data.This would require improved bed elevation data and an englacial ice temperature record for a more accurate ice flow solution, data on crevasse water depths to make the application of calving more rigorous and a long, high frequency record of terminus position against which the modelled calving record could be compared.If these were combined with a more sophisticated representation of ice mélange influences, as discussed above, then this type of model has real potential to produce not only a highly accurate method of predicting future tidewater glacier behaviour, but also to diagnose the likely causes of glacier retreat as it is observed.

Conclusions
This paper presents a flowline, tidewater glacier model, capable of representing the calving of grounded ice using a physically realistic calving criterion.The results demonstrate that it reproduces realistic calving behaviour, examined qualitatively by comparing the size and frequency of calving events, and the subsequent dynamic response.However, it would benefit from better data availability for more rigorous validation.In response to previous debate over whether glacier retreat is triggered first by glacier acceleration or increases in calving rate, the model results indicate that either process has the potential to cause terminus retreat and it may be more enlightening to consider the specific mechanisms involved.Introduction

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Full The results presented show a surprising insensitivity to oceanic influences and a noticeable sensitivity to the effects of surface ablation.The lack of sensitivity to simulated ice mélange may be because of the relatively simple representation of its effect on the calving front used in the model.However, the weak effect of submarine melt on calving rates appears to arise from the ability of ice deformation in the model to counteract the stress displacement caused by a cavity in the calving front.The strong effect of both crevasse water depth and basal water pressure on the model's behaviour raises the possibility that outlet glaciers may be strongly affected by recent trends in surface ablation.The results demonstrate the potential for this type of model to improve understanding of tidewater glacier dynamics, their response to external forcing, and to diagnose the causes of glacier retreat.Introduction

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Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | To reduce the impact of uncertainty in the ice temperature profile on future modelling work, an extension of the model to include full thermo-mechanical coupling should be used, preferably constrained by observations of ice temperature through borehole measurements.Figures Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | sition.The model was tested with a summer increase in basal water pressure of 50 kPa, equivalent to a change in velocity of approximately 15 % as observed byJoughin et al. (2008a), which was designed to represent a spring speed-up event.Although such a change had relatively little effect on the instantaneous longitudinal deviatoric stress, causing a percentage change of only around 0.5 % (Fig.4a), it increased both the mass flux reaching the calving front and longitudinal extension, thus having a long-term effect on the predicted crevasse field.Two sets of experiments were performed, first 4419 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | downstream of the predicted calving point also become crevassed below the water line in the same timestep.
If a hydraulic connection exists between the glacier surface and the bed, surface meltwater can also affect the terminus behaviour by altering basal water pressure, Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Table 1 .
Full list of experiments performed, arranged by results section.S and W refer to summer and winter values.D w is the crevasse water depth (cwd), dP is the change in hydrostatic head applied to the basal water pressure, M f is the maximum melt rate applied at the calving face and F f is the force applied to the calving front over a vertical range of −120 to 15 m a.s.l.In ice mélange experiments where both the cwd and the backstress vary, the length of seasons in cwd and backstress does not match, meaning that there is also an autumn season where both the cwd and the backstress are set at zero.