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Edited by: Timothy C. Bartholomaus, University of Idaho, USA

Reviewed by: Andrew John Sole, University of Sheffield, UK; Chris Borstad, University Centre in Svalbard, Norway

*Correspondence: Jaime Otero

This article was submitted to Cryospheric Sciences, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Calving is an important mass-loss process at ice sheet and marine-terminating glacier margins, but identifying and quantifying its principal driving mechanisms remains challenging. Hansbreen is a grounded tidewater glacier in southern Spitsbergen, Svalbard, with a rich history of field and remote sensing observations. The available data make this glacier suitable for evaluating mechanisms and controls on calving, some of which are considered in this paper. We use a full-Stokes thermomechanical 2D flow model (Elmer/Ice), paired with a crevasse-depth calving criterion, to estimate Hansbreen's front position at a weekly time resolution. The basal sliding coefficient is re-calibrated every 4 weeks by solving an inverse model. We investigate the possible role of backpressure at the front (a function of ice mélange concentration) and the depth of water filling crevasses by examining the model's ability to reproduce the observed seasonal cycles of terminus advance and retreat. Our results suggest that the ice-mélange pressure plays an important role in the seasonal advance and retreat of the ice front, and that the crevasse-depth calving criterion, when driven by modeled surface meltwater, closely replicates observed variations in terminus position. These results suggest that tidewater glacier behavior is influenced by both oceanic and atmospheric processes, and that neither of them should be ignored.

Iceberg calving is one of the most important and least understood mechanisms of ice loss at ice sheet and marine-terminating glacier margins accounting for about half of the mass loss from the Greenland and Antarctic ice sheets (Cuffey and Paterson,

Benn et al. (

Nick et al. (

Rather than introducing new calving criteria, other contributions to the calving problem—such as those by Amundson and Truffer (

Water filling crevasses is known to play an important role in calving processes, favoring calving through hydrofracturing (e.g., Scambos et al.,

Ice mélange, a heterogeneous mixture of sea ice and calved ice, can freeze solid and provide a stress opposing the flow of the glacier. This stress maintains the integrity of the calving margin, preventing calving (Amundson et al.,

Recently, Bondzio et al. (

In this paper, we present results from a numerical model developed using the open-source finite-element software Elmer/Ice (Gagliardini et al.,

Some previous modeling work has been applied to Hansbreen: Vieli et al. (

Hansbreen is a polythermal tidewater glacier which flows into Hornsund fjord in southern Spitsbergen (Figure ^{2} from 0 to 500 m above sea level (a.s.l.). The glacier terminus is about 2.5 km wide, the central 1.5 km of which sits in water. The ice thickness of the central flowline at the terminus is about 100 m, of which 55 m are submerged. The glacier lies on a reverse-sloping bed for the first 4 km up-glacier from the terminus and the center of the fjord lies below sea level as far as 10 km up-glacier. The maximum ice thickness is about 400 m. Further, details on the glacier surface and bed morphology can be found in Grabiec et al. (

To account for gentle surface slopes, glacier surface elevations were taken from the SPIRIT Digital Elevation Model (DEM) V1 (whose correlation parameters are set for gentle slopes), based on SPOT5 Stereoscopic Survey of Polar Ice imagery acquired on 1 September 2008. The DEM has a 40 m resolution and a 30 m root-mean-square (RMS) absolute horizontal precision (

Bed topography was derived from ground-penetrating radar (GPR) data (Grabiec et al.,

Surface velocities were measured daily at 16 stakes (Figure

The near-terminus surface velocities exhibit a seasonal pattern overlaid by strong interannual variability. Each year has a spring speed up, followed by a rapid slow down, followed sometimes by a gradual speed up through the winter (Figure

Interannual observations of Hansbreen's terminus position over the last decades reveal a generally smooth retreat with occasional abrupt changes (e.g., Vieli et al.,

The ice mélange in the glacier forebay was qualitatively evaluated as either “complete,” “partial,” or “free” from the same time-lapse photographs used to measure terminus position. For the remaining period, we used the values of the nearest cell in a 25-km resolution time series of sea ice concentrations derived from Nimbus-7 SMMR and DMSP SSM/I-SSMIS passive microwave data (Cavalieri,

We apply a dynamical downscaling method (which uses a modified version of the Polar WRF 3.4.1 model) to produce—from a regional climate dataset consisting of meteorological, sea-surface temperature and sea-ice concentration data—input data for glacier thermomechanical modeling of Hansbreen (Finkelnburg et al., in preparation).

Surface mass balance (SMB) was obtained from European Arctic Reanalysis (EAR) data, with 2 km horizontal resolution and hourly temporal resolution, constrained by automatic weather stations (one in Hornsund and two in Hansbreen) and stake observations (Finkelnburg,

Ice is treated as an incompressible viscous fluid. The Stokes system of equations describing the dynamical model is composed of equations describing the steady conservation of linear momentum and the conservation of mass of an incompressible continuous medium:

where

where the shape factor

where _{s}(

As the constitutive relation, we adopt Nye's generalization of Glen's flow law (Glen,

This equation links the deviatoric stress

where

The constitutive relation (4) is expressed in terms of deviatoric stresses, while the conservation of linear momentum (1) is given in terms of full (Cauchy) stresses. Both stresses are linked through the equation

where ^{−3}a^{−1}) were used in the model (Albrecht et al.,

We introduce a scalar damage variable

For undamaged ice (

In this study, we use a very simple function for

The time evolution of the glacier surface is calculated by solving the free-surface evolution equation

where _{S} is the surface elevation, _{S} and _{S} are the horizontal and vertical components of the flow velocity at the surface, respectively, and

The upper surface of the glacier is a traction-free zone with unconstrained velocities. At the ice divide at the head of the glacier, horizontal velocity and shear stresses are set to zero.

For boundary conditions at the bed, we use a friction law that relates the sliding velocity to the basal shear stress in such a way that the latter is not set as an external condition but part of the solution:

where

Since the inversion procedure requires a continuous function for the surface velocity, we calculate it as a sixth-degree polynomial regression for each 4-week period.

At the glacier terminus, we set backstress to zero above sea level and equal to the water-depth-dependent hydrostatic pressure below sea level. In model runs with ice mélange, the additional backstress is applied to the calving face, in the opposite direction of ice flow (negative X). In the absence of further data, we assumed the range of 30–60 kPa estimated by Walter et al. (

The CDw calving criterion (Benn et al.,

Following Todd and Christoffersen (

where σ_{n}, the “net stress,” is positive for extension and negative for compression. The first term on the right-hand side of Equation (10) represents the opening force of longitudinal stretching, adapted by Todd and Christoffersen (_{e} represents the effective stress, _{e} is multiplied by the sign function of the longitudinal deviatoric stress, τ_{xx}, to ensure that crevasse opening is only produced under longitudinal extension (τ_{xx} > 0). The second term on the right-hand side is the ice overburden pressure, which leads to creep closure, where ρ_{i} is the density of glacier ice,

At each time step, the glacier is divided into a rectangular mesh with 10 vertical layers and a horizontal grid size of ca. 50 m in the upper glacier and ca. 25 m near the terminus. The Stokes system of equations (1) is solved by a finite element method using Elmer/Ice and the 2-D stress and velocity fields are computed along the central flowline (Figure

At the terminus, the grid nodes are shifted down-glacier according to the velocity vector and the length of the time step and the terminus position is updated according to the calving criterion.

Prognostic model runs were carried out with a 1-week (1/48 of a year) time step, starting from the 2008 glacier geometry. Every 4 weeks (four time steps), we ran an initialization process which consisted of solving the Robin problem (Jay-Allemand et al.,

Our aim was to investigate the influence of ice mélange backstress and crevasse water depth on terminus position. Given the absence of field measurements, we parameterized crevasse water depth in terms of surface meltwater.

First, we analyzed the effect of crevasse water depth held fixed throughout the entire modeled period. Under this scenario, it was not possible to replicate the observed terminus position variations; instead, we constrained the magnitude of the crevasse water depth to that which best approximates the observed terminus positions. Using this best-fit crevasse water depth, we ran a similar sensitivity analysis for ice mélange backstress and determined the backstress that best fits the observed terminus positions. Finally, using this best-fit ice mélange backstress, we ran the model with a time-varying crevasse water depth _{w} expressed as a linear function of the surface meltwater _{w} (units meters per week) predicted by the SEB model, i.e., _{w} = _{w}, where

This experiment was repeated for a range of values for the linear coefficient, and the results corresponding to the best-fitting value very closely matched the observed terminus position variations.

Given the difficulties of measuring the depth of water in crevasses, we ran the model for a range of crevasse water depths (from 6 to 12 m) to evaluate the sensitivity of the model to this parameter. We found that calving rate is highly dependent on the depth of water in crevasses, with an increase of just a few meters causing the glacier to switch from advance to retreat (Figure

To test the effect of ice mélange on calving rate and terminus position, we varied ice mélange backstress from 0 to 70 kPa (based on Walter et al.,

We found that the effect of ice mélange backstress on glacier front position was significant, even under low stresses (Figure

As discussed by Todd and Christoffersen (

Use of such a parameterized time-varying crevasse water depth, in combination with the best-fit ice mélange backstress from the previous experiment, yielded terminus positions in very good agreement with the observations (Figure

In tidewater glacier modeling, it is common practice to use mean annual surface velocities (e.g., Cook et al.,

Our results demonstrate that our model is capable of reproducing the seasonal fluctuations of the terminus position of Hansbreen, provided that the key model variables are adequately tuned and parameterized.

The modeled terminus positions are shown to be highly sensitive to changes in crevasse water depth, in agreement with previous studies (e.g., Cook et al.,

The water in crevasses is mostly produced by melting at the glacier surface, so the mean water depth in crevasses can be parameterized in terms of surface melting, which can be modeled either using air temperature or temperature-radiation index models (e.g., Jonsell et al.,

In contrast with one previous modeling study (Cook et al.,

Both our model and the observations show frontal retreat beginning soon after the peak in surface meltwater, although the maximum calving occurs a few weeks later, suggesting a delayed response by the glacier system to meltwater input. The source of this lag could be two-fold. On one hand the cumulative effect of thinning by ablation in both the real glacier and the model, which helps the crevasses to penetrate down to the waterline. On the other hand, in the case of the real glacier there is a buildup of the water pressure in the crevasses as meltwater accumulates; alternatively, if the water escapes from the crevasses there will also be a cumulative weakening of the bulk of the terminal zone of the glacier due to the enlargement of the conduits and fissures by melting promoted by the escaping water.

Even though our model does a good job reproducing the observed front positions, it does not consider other possible controls on the calving process, specifically ocean-induced melting. Adequately incorporating this mechanism would require developing a fjord circulation model to estimate the subaqueous melt rates and couple the fjord and glacier systems.

In regards to possible shortcomings of the model, we note that we added a body force term to the Stokes system (Equation 1) to take into account, in our 2-D flowline model, of the lateral drag on the glacier sidewalls. However, this body force term does not take into consideration the effect of ice flow from tributaries on the central flowline dynamics. This effect is expected to be significant at Hansbreen, which has three tributary glaciers flowing into the main branch near the terminus (Figure

In this study, we investigate the relative importance of some proposed controls on calving—namely, crevasse water depth and ice mélange backstress—and evaluate their influence on the terminus position changes of a tidewater glacier.

Our results suggest that ice mélange backstress plays an important role in regulating the seasonal advance and retreat of the terminus, mostly by preventing calving when the mélange chokes the fjord. The model results also indicate that calving and the associated terminus position changes are highly sensitive to the amount of water filling near-terminus crevasses, itself a function of surface meltwater availability. The sensitivity of calving rate to crevasse water depth suggests that calving is strongly affected by atmospheric forcing. These results, taken together, show that tidewater glacier dynamics are influenced by both oceanic and atmospheric processes, and that neither of them should be ignored.

JO led the study. JO and FN designed the experiments. JO did the numerical modeling of glacier dynamics and RF provided regional climate modeling and SEB data. JL provided GPR data, DP ice velocity and AWS data, and EW terminus positions and sea ice coverage data. JO, FN, and JL contributed to the discussion of the results. JO and FN wrote the initial draft of the paper primarily, JL made the figures. All authors contributed to and approved the final manuscript.

This research was funded by Spanish State Plan for Research and Development (R&D) project CTM2014-56473-R. Field data collection was funded by grants EUI2009-04096 and CTM2011-28980 of the Spanish Programmes of Euro-Research and R&D, respectively, and the Polish National Science Centre within statutory activities No3841/E-41/S/2014 of the Ministry of Science and Higher Education of Poland. The regional climate modeling data was produced under grant no. SCHE 750/3-1 of the German Research Foundation (DFG) and grant no. 03F0623A of the German Federal Ministry of Education and Research (BMBF).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was carried out under the frames of the International Arctic Science Committee Network on Arctic Glaciology (IASC-NAG) and the European Science Foundation PolarCLIMATE programme's SvalGlac project. The satellite images used in this paper were provided by the SPIRIT Program CNES (2008), Spot Image, and ASTER METI and NASA (2011), all rights reserved. The surface velocity data used in the paper were collected based on Stanislaw Siedlecki Polish Polar Station in Hornsund.