In this paper, we study the effects of basal friction, sub-aqueous undercutting and glacier geometry on the calving process by combining six different models in an offline-coupled workflow: a continuum–mechanical ice flow model (Elmer/Ice), a climatic mass balance model, a simple subglacial hydrology model, a plume model, an undercutting model and a discrete particle model to investigate fracture dynamics (Helsinki Discrete Element Model, HiDEM). We demonstrate the feasibility of reproducing the observed calving retreat at the front of Kronebreen, a tidewater glacier in Svalbard, during a melt season by using the output from the first five models as input to HiDEM. Basal sliding and glacier motion are addressed using Elmer/Ice, while calving is modelled by HiDEM. A hydrology model calculates subglacial drainage paths and indicates two main outlets with different discharges. Depending on the discharge, the plume model computes frontal melt rates, which are iteratively projected to the actual front of the glacier at subglacial discharge locations. This produces undercutting of different sizes, as melt is concentrated close to the surface for high discharge and is more diffuse for low discharge. By testing different configurations, we show that undercutting plays a key role in glacier retreat and is necessary to reproduce observed retreat in the vicinity of the discharge locations during the melting season. Calving rates are also influenced by basal friction, through its effects on near-terminus strain rates and ice velocity.

Accelerated discharge of ice into the oceans from land ice is a major
contributor to sea level rise, and constitutes the largest source of
uncertainty in sea level predictions for the twenty-first century and beyond

Recently, it has been suggested that ocean warming could play an important
role in determining glacier calving rate and acceleration, by impacting
submarine melt rates

In previous modelling work

These problems can be circumvented using discrete particle models, which
represent ice as assemblages of particles linked by breakable elastic bonds.
Ice is considered as a granular material and each particle obeys Newton's
equations of motion. Above a certain stress threshold, the bond is broken,
which allows the ice to fracture.

A compromise should be found by coupling a continuum model, such as Elmer/Ice, to a discrete model, such as HiDEM, to successively describe the ice as a fluid and as a brittle solid. Sliding velocities and ice geometry calculated with the fluid dynamic model are used by the discrete particle model to compute a new calving front position. The effect of subglacial drainage mixing with the ocean during the melt season is taken into account by using a plume model that estimates melt rates at the front according to pro-glacial observed ocean temperatures, subglacial discharge derived from surface runoff and ice front height.

In this paper, we use both the capabilities of the continuum model Elmer/Ice and the discrete element model HiDEM. We harness the ability of HiDEM to model fracture and calving events, while retaining the long-term ice flow solutions of a continuum approach. The aim is to investigate the influence of basal sliding velocity, geometry and undercutting at the calving front on calving rate and location. We determine the undercutting with a high-resolution plume model calculating melt rates from subglacial discharge. The simple hydrology model that calculates the subglacial discharge is based on surface runoff that is assumed to be transferred directly to the bed and routed along the surface of calculated hydrological potential. We illustrate the approach using data from Kronebreen, a fast-flowing outlet glacier in western Spitsbergen, Svalbard (topography, meteorological and oceanographic data, as well as horizontal surface velocity and front positions from 2013), to assess the feasibility of modelling calving front retreat (rate and position).

Kronebreen is a tidewater glacier that flows into Kongsfjorden in Svalbard,
one of the fastest glaciers in the archipelago. The glacier front position
undergoes seasonal oscillations, showing advance during the winter and spring
followed by retreat in the summer and autumn. Since 2011, the summer retreat
has outpaced the winter advance, with an overall net retreat of

Plumes of turbid meltwater, fed by subglacial discharge, are observed adjacent to
the glacier terminus during the melt season

The bed topography,

Ice surface velocities were derived from feature tracking of TerraSAR-X image
pairs in slant range using correlation windows of

We use surface velocity and frontal position data described above to test the
effects of sliding and undercutting on calving using different models in a
global approach. This one-way offline coupling approach is divided into three
parts using six models (see Fig.

Model scheme presenting the calculation of the sliding and geometry (Elmer/Ice) as well as the undercutting at the subglacial discharge as input to the glacier calving from the HiDEM.

Observation times of velocity acquisitions,

We set

First, we infer the sliding velocity at each observational time from surface
velocities using an adjoint inverse method implemented in Elmer/Ice with an
updated geometry from observations. At each iteration,

We call this approach an offline coupling because inputs to the HiDEM are output results from Elmer/Ice and undercutting model but not vice versa. In Elmer/Ice, we use the observed frontal positions. A completely coupled physical model would use the output of HiDEM, the modelled front position, as input to the ice flow model Elmer/Ice and the undercutting model. It would also calculate the basal friction from a sliding law rather than an inversion. In principle, such an implementation is possible using the same model components as this study.

At the base of the glacier, we use a linear relation for sliding of the form

Front position and surface elevation changes with Elmer/Ice during

After each inversion, the temporal evolution of the glacier is mathematically
described by the kinematic boundary condition defined at the surface,

We assume that the front is vertical above the water line so that the
observed front position (at the surface of the glacier) is the same at sea
level. We call

The surface mass balance,

The temporal subglacial discharge at the calving front is estimated from
integration of daily surface runoff assumed to be directly transferred down
to the glacier bed. Assuming the basal water pressure at over burden, the
flow path of the meltwater towards the glacier front is determined from the
hydraulic potential surface defined as

A high-resolution plume model is used here to simulate the behaviour of
subglacial discharge at the terminus of Kronebreen. The model is based upon
the fluid dynamics code Fluidity

The geometry of the model is adapted to Kronebreen by setting the water depth
to 100 m and initialising the model with ambient temperature and salinity
profiles collected from ringed seals instrumented with GPS-equipped
conductivity, temperature and depth satellite relay data loggers
(GPS-CTD-SRDLs)

The model is spun up for 1000 model seconds until the turbulent kinetic
energy in the region of the plume reaches a steady state and thereafter run
for 10 min of steady-state model time. Melt rates are extracted from the
duration of the steady-state period and then time-averaged and interpolated onto
a uniform 1 m

The high computational cost of the model means that it cannot be run
continuously over the study period, nor can the full range of discharges and
oceanographic properties be tested. Instead, representative cases

We assume a vertically aligned surface front at the beginning of the melt
season. We know the position of the front,

Three cases of undercutting

When the first discharge occurs, the melt rate calculated with the plume
model in 2-D is summed for the period of time between

At time

if the new observed position

if the new observed position

if the new observed position

The melt summed up between

The fracture dynamics model is described in detail in

At the beginning of a fracture simulation, the ice has no internal stresses
and contains a few randomly distributed broken beams, representing small
pre-existing cracks in the ice. The dynamics of the ice are computed using a
discrete version of Newton's equation of motion, iteration of time steps, and
inelastic potentials for the interactions of individual blocks and
beams. As the ice deforms under its own weight, stresses on the beams
increase, and if stress reaches a failure threshold the beam breaks and the
ice blocks become disconnected but continue to interact as long as they are
in contact. In this way cracks in the ice are formed. For computational
reasons, we initialise the glacier using a dense packed face-centred cubic
(fcc) lattice of spherical blocks of equal size. This introduces a weak
directional bias in the elastic and fracture properties of the ice. The
symmetry of the underlying fcc lattice is, however, easily broken by the
propagating cracks. The ground under the ice or at the seafloor is assumed
to be elastic with a linear friction law that varies spatially
(Eq.

The time step is limited by the travel time of sound waves through a single
block and is thereby set to 10

The basal friction coefficients,

HiDEM reads a file with surface and bed coordinates on a grid and a file with
surface and basal ice (to take into account the undercutting) coordinates.
For simulations with an undercutting at a discharge location and in order to
avoid complication in the HiDEM (position of the basal ice), we remove
particles below the maximum melt (no ice foot as shown by the dashed line in
Fig.

There is a clear separation of timescales between the velocities of
sliding (

In a fully coupled model, the altered ice geometry after calving could then be re-implemented in the flow model, acting as the initial state for a continued prognostic simulation with the continuum model. Here, this back-coupling is replaced by prescribing the next observed configuration.

The mean volumetric frontal ablation rate (or mean volumetric frontal calving rate) at the
ice front at time

We investigate the effect of three different parameters on calving activity:
the geometry,

Different configurations,

Basal friction coefficient obtained from inverse modelling and
observed frontal position for

The basal friction coefficient,

The hydrological model predicts that there are two main subglacial channels
with discharge exceeding 1

Total volume of subglacial discharge modelled per period of calving front recording.

The melt rate profiles calculated by the plume model for four different
volumes of subglacial discharge are shown in Fig.

Melt rates,

At a discharge of 1

The modelled frontal position is summarised in Fig.

The frontal submerged undercutting driven by the plume differs in shape from
one location to another. In the first 50 m below the surface, the
undercutting at the SD is not as abrupt as at the ND and is also smaller
(Fig.

The observed mean volumetric calving rate averaged over the entire calving
front volume of ice,

Observed and modelled mean volumetric calving rates,

Observed mean volumetric calving rate,

To assess the performance of the offline coupling, we evaluate the mean
volumetric calving rate averaged over the entire calving front volume of ice
(see Eq.

Basal velocity, advanced front before calving modelled with
Elmer/Ice,

At

Strain rates modelled with HiDEM for each configuration. Yellow colouring shows the crevasse pattern and is denser close to the front where the difference between each configuration for the four selected iterations can be observed.

With peak surface runoff, at

Vertical front configuration at

At the end of the melt season at

Two configurations vary the friction coefficient,

Our plume model uses a fixed, planar ice front to calculate submarine melt
rates rather than a time-evolving geometry. This assumption is supported by

By using ambient temperature and salinity profiles that do not vary in time,
we neglect the inter- and intra-annual variability in Kongsfjorden. This
variability can affect the calculated melt rate in two ways: (i) the
three-equation melt parameterisation explicitly includes the temperature and
salinity at the ice face, and (ii) the ambient stratification affects the
vertical velocity and neutral buoyancy height of the plume. The direct effect
of changes in temperature and salinity on the melt equations are well tested.
Past studies using uniform ambient temperature and salinity conditions have
found a linear relationship between increases in ambient fjord temperatures
and melt rates, with the slope of the relationship dependent upon the
discharge volume

For ND (Fig.

Because the imposed undercuttings are the product of melt during the whole
interval between observations, the model results should be treated with
caution.

Firstly, the HiDEM results show that undercutting associated with meltwater
plumes is an essential factor for calving during the melt season (

Secondly, the model results replicate the observed high calving rates at

In this study, we use the abilities of different models to represent different glacier processes at Kronebreen, Svalbard, with a focus on calving during the melt season of 2013. Observations of surface velocity, front position, topography, bathymetry and ocean properties were used to provide data for model inputs and validation.

The long-term fluid-like behaviour of ice is best represented using the
continuum ice flow model Elmer/Ice, which computes basal velocities by
inverting observed surface velocities and evolves the geometry, including the
front position. During the melt season, a subglacial hydrology system is
created and the water is eventually evacuated at the front of the glacier. We
used a simple hydrology model based on surface runoff directly transmitted to
the bed and routing the basal water along the deepest gradient of the
hydraulic potential. Two subglacial discharge locations have been identified
by this approach: the northern one evacuates water with a high rate
(

In this paper, we have shown that offline coupling of ice-flow, surface melt, basal drainage, plume-melting, and ice-fracture models can provide a good match to observations and yield improved understanding of the controls on calving processes. Full model coupling, including forward modelling of ice flow using a physical sliding law, would allow the scope of this work to be extended farther, including prediction of glacier response to atmospheric and oceanic forcing.

The Elmer/Ice software is an open-source finite element
for Ice Sheet, Glaciers and Ice Flow Modelling available at

DV contributed to the design of the study, the offline coupling, the development of the undercutting model, the Elmer/Ice and HiDEM set-ups and the writing of the manuscript. DB edited the manuscript. All other authors provided comments on the manuscript. JÅ developed the HiDEM model and used Kronebreen as the test and development case. AE developed the plume model. TZ contributed to the Elmer/Ice set-up. RP calculated the water discharge. DB and AL provided the observed surface velocity maps. WVP developed the coupled energy balance–snow modelling approach. JK provided the interpolated bed map.

The authors declare that they have no conflict of interest.

We thank CSC – IT Center for Science Ltd. for the CPU time provided under Nordforsk NCoE SVALI. Thomas Zwinger was supported by the Nordic Center of Excellence eSTICC (eScience Tools for Investigating Climate Change in Northern High Latitudes) funded by Nordforsk (grant 57001). Acquisition of the TerraSAR-X imagery was funded by the ConocoPhillips Northern Area Program, via the CRIOS project (Calving Rates and Impact on Sea Level). The lead author received an Arctic Field Grant from the Svalbard Science Forum to acquire radar lines for the basal topography in 2014. Finally we would like to thank the reviewers and the editor for their input and help to improve the paper. Edited by: Andreas Vieli Reviewed by: two anonymous referees