Calving of icebergs is a major negative component of polar ice-sheet mass balance. Here we present a new calving model relying on both continuum damage mechanics and linear elastic fracture mechanics. This combination accounts for both the slow sub-critical surface crevassing and the rapid propagation of crevasses when calving occurs. First, damage to the ice occurs over long timescales and enhances the viscous flow of ice. Then brittle fractures propagate downward, at very short timescales, when the ice body is considered as an elastic medium. The model was calibrated on Helheim Glacier, Southeast Greenland, a well-monitored glacier with fast-flowing outlet. This made it possible to identify sets of model parameters to enable a consistent response of the model and to produce a dynamic equilibrium in agreement with the observed stable position of the Helheim ice front between 1930 and today.

The discharge of ice from the Greenland and Antarctic ice sheets increased
sharply in recent decades

Most studies dealing with iceberg calving follow the approach proposed by

Another approach to modelling calving uses discrete-element models

In the last few years, some authors have used continuum damage mechanics to
represent both the development from micro-defects in the ice to the
development of macro-scale crevasses, and their effects on the viscous
behaviour of the ice while keeping a continuum approach

Here we consider an approach that combines damage mechanics and
fracture mechanics. The proposed physically based calving model can cover
both the accumulation of damage as the ice is transported through the
glacier, and the critical fracture propagation that
characterizes calving events. The slow development of damage represents
evolution of viscous ice at long timescales, while the use of fracture
mechanics makes it possible to consider calving events that occur at shorter
timescales, during which the ice can be considered as a purely elastic
medium. The description of the physics used is presented in
Sect.

We consider an incompressible, isothermal and gravity-driven ice flow in
which the ice exhibits a non-linear viscosity. The ice flow is ruled by
Stokes equations (i.e. Navier–Stokes equations without an inertial
term), for the momentum and the mass balance:

The upper surface is defined as a stress-free surface. In
the coordinate system (

Similar to the upper free surface, the bottom surface evolution is described by

The ice front is defined as a third free-surface, which can
also undergo melting. In analogy to the other free surfaces this gives

The physical and numerical parameters used in this paper are listed in
Table

Continuum damage mechanics (CDM) was introduced by

The principle of CDM models is based on the use of a damage variable, usually
denoted

To describe the effect of damage on the ice flow, an effective deviatoric part
of the Cauchy stress tensor is introduced:

Our approach models the non-linear viscous flow of ice. The viscous behaviour of the ice is described using Glen's flow law, which links the deviatoric part of the Cauchy stress tensor to the strain rate tensor. Consequently, when accounting only for viscous deformation, damage to the ice only affects the deviatoric part of the Cauchy stress tensor, and not the cryostatic pressure.

Damage is a property of the material at the mesoscale. It is therefore
advected by the ice flow, and evolves over time depending on the stress
field. To take this evolution into account, we recommend an advection
equation:

The choice of the damage criterion is crucial for the representation of
damage increase, and its physical expression is a critical step in the
formulation of a CDM model. Commonly used criteria are the Coulomb criterion

Damage envelope in the space of principal stresses.

Instead, we use a pure-tensile criterion, described as a function of the
maximum principal Cauchy stress

The stress threshold for damage initiation

This formulation of the damage criterion implies some limitations to the
calving model. In particular, it does not account for shear and compressive
failure mechanisms. However, it remains consistent with the approach of

As pointed out by

Introducing the effective deviatoric stress tensor

By identification with Eq. (

Continuum damage mechanics can be used to deal with the
degradation of ice viscosity with increasing damage at long timescales. It
can be understood as a way to simulate subcritical crevasse nucleation and
propagation

The key physical parameter of LEFM is the stress intensity factor

However, when considering real cases, the opening term

This formula relies on the use of the superposition principle: in the case of
linear elasticity, the value of the stress intensity factor at the tip of the
crack can be seen as the sum of contributions of all individual point loads
along the crack length. In our case, instead of considering the value of the
along-flow component of the deviatoric stress tensor at the tip of the crack,
we multiplied it by the weight function

In LEFM theory, a fracture is able to propagate downward in the ice if the
stress intensity factor is greater than the fracture toughness

The weight function

Crevasse shape. See Table

From Eq. (

Compared to the work of

Once the conditions for fracture initiation are fulfilled, we consider that
the crevasse propagates vertically. In

In this simplified LEFM framework, calving would theoretically occur only if

This framework has two consequences. First, the stress profile

The calving model described in the previous sections is summarized in
Fig.

The CDM and LEFM models were implemented in the finite element ice sheet/ice
flow model Elmer/Ice. Further information regarding Elmer/Ice can be
found in

The model was applied to Helheim Glacier, a fast-flowing well-monitored glacier in Southeast Greenland. The abundance of observations there allowed us to compare and constrain our model parameters against past glacier evolution. For our model development, we focus on a two-dimensional flowline problem.

As stated by

Algorithm of the calving model where

As we focused on changes in the front and on representing calving, we only
needed a bedrock topography covering the last 10 km
up to the front to capture any basal features that could
influence the behaviour of the front. For this reason, we chose to follow
the work by

Following our notation system, the ice flows along the horizontal
“

The geometry covers the last 30 km of the glacier, with an initial
thickness varying between 900 m at the inlet boundary, and 700 m at the
front. Using the metric from

The specific boundary conditions used for the 2-D application are described
below, using the boundary conditions presented in Sect.

The basal friction

Glacier location and geometry.

For melting at the calving front, we chose an ablation function that
increased linearly with depth, from zero at sea level to 1 m day

The inflow boundary condition (

When dealing with a 2-D flowline representation of the ice flow, we had to
take some three-dimensional effects into account. In particular, lateral
friction along the rocky-margins of the glacier can play a significant role
by adding resistive stress to the flow. Here, we modified the gravitational
force using a lateral friction coefficient

Even if the velocity and the surface topography were known and correspond to
the state of the glacier observed in May 2001, some relaxation was necessary
to obtain a stable steady state

The model was calibrated by varying the three parameters

The stress threshold

The damage enhancement factor

The critical damage value

Positions of the calving front over the 10-year period. Each colour
corresponds to a set of parameters

Damage can be produced anywhere in the glacier. As we needed to obtain a steady state for the damage field, we had to let the damage created upstream be advected to the front. The model was therefore spun-up for 8 years: the front was kept fixed at its initial position, without submarine frontal melting, or calving. After 8 years, the front was released, and frontal melting and calving were activated.

Over the last century, Helheim Glacier has probably undergone several advance
and retreat cycles

This threefold classification of glacier behaviour was generalized to the
48 simulations. To eliminate aberrant behaviour, we ran a sanity check by
considering plausible sets of parameters as those that lead to a simulated
front position within the range [340, 350 km]. The results are
presented in Fig.

Parameters set for the damage model discriminated by

State of Helheim Glacier after 365 days of simulation for the set of
parameter (

The steady advance of the front with no, or with only a few calving events
(blue triangles) can be explained first by considering the parameters (

In contrast, the too rapid retreat of the front (yellow circles) can be
explained as follows: when

The chosen range for

Finally, parameter

However, discretization into three classes of parameters should not mask the
continuous behaviour of the glacier, depending on the value of
(

The acceptable parameter combinations are located on the 3-D diagonal
represented by the red diamonds in Fig.

Frequential response of variation in the calving front in the eight simulations
using the set of parameters (

Focusing on this parameter set, we highlighted the following main features of
the glacier front dynamics. First, at short timescales, the calving activity
can be described through a small-size/high-frequency event distribution.
This activity is characterized by quasi-periodic retreats of the front of
limited extent (between 50 and 150 m), associated with a period varying
from about 3 to 15 days (and only 1 day for very rapid front retreat, see
Sect.

The glacier front dynamics is also characterized by a larger-amplitude/lower-frequency oscillation (see Fig.

To investigate the strength of this feature, we ran this simulation (red
curve in Fig.

This observation, combined with the mechanism proposed above, could explain
the cycles in the position of the front, and also suggests that the surface
geometry (driven by the bedrock topography) controls the calving dynamics by
modifying the rate at which damage primarily develops in the ice. Thus, in
the experiments presented in this paper, the slow cycles in
Fig.

The analysis described in the previous section for one parameter set was
extended to the 12 simulations that successfully passed the sanity check, and
showed that the model's response remained qualitatively unchanged even when
the damage parameters changed. Altogether, we identified 6534 distinct
calving events. Characteristic calving event sizes emerged from the
distribution, similar to those visible in the inset in
Fig.

When

However, it should be recalled that this plot cannot be interpreted as an
iceberg's size distribution. Indeed, one has to distinguish between the front
retreat and the size of the resulting iceberg(s) which may differ
considerably, as the calved portion of ice may fragment into many icebergs
and/or capsize. Although the distribution of the distance the front retreats
could be an interesting parameter to calibrate the model, it would require
continuous tracking of the front position of the actual glacier, as discrete
determination of the position

In addition, we estimated the influence of the other parameters discussed
above. The results showed that the model is only slightly sensitive to the

Finally, model sensitivity to the initial heterogeneous distribution of
micro-defects introduced in Sect.

Several improvements could be made to the degree of physics included in the model. Their implementation should be straightforward, as soon as the knowledge of their underlying properties is sufficient to better constrain the resulting physical parameters.

Distribution of the size of the calving events corresponding to the 12 realistic simulations. Different modes can be related to the different combinations of damage parameters. The axis representing the calving event size is truncated at 200 m. However, a few events, up to 2000 m were recorded.

Among the improvements that could be made to the model, the presence of
surface melt water in crevasses is the most obvious. This has a major effect
on their propagation, as it represents a supplementary hydrostatic force in
the stress balance tending to favour crevasse opening.

The implementation of this feature in our modelling framework is
straightforward. To do so, we just have to add a water pressure term in the
expression of the Cauchy stress tensor, depending on the water level in the
crevasse

At the current stage of development, the model does not include the
representation of basal crevasses. These are cited as a possible explanation
for the production of large tabular icebergs, as they require a
full-thickness fracture. This basal propagation is only possible if the
glacier is near or at flotation, such that the tensile stress is high enough
to trigger a fracture

Considering the case of highly crevassed glacier surfaces and closely spaced
crevasses, our fracture propagation framework could not rely on a
parameterization that only considers lone crevasse propagation. In this case,
problems arise from the fact that the stress concentrations at crevasse tips
are reduced by the presence of neighbouring crevasses (

The major effect of this development would be a supplementary delay in the time and the position of calving events, probably resulting in calving events occurring nearer to the calving front, where the tensile stress is higher, and as such resulting in smaller size distribution of icebergs.

In this work, we combined continuum damage mechanics and linear elastic fracture mechanics to propose a process-based calving model. This model is able to reproduce the slow development of small fractures leading to the appearance of macroscopic crevasse fields over long timescales, while considering ice as a viscous material. It also allows for the elastic behaviour of breaking ice, consistent with the critical crevasse propagation that triggers calving events, characterized by very short timescales. The model was applied to Helheim Glacier, which allowed us to constrain the parameter space for the most important free parameters. Within these constraints, the simulated ice front led to cyclic calving events, and the position of the glacier front remained in the range observed over the last century.

The three decisive parameters are the damage initiation threshold

It should be borne in mind that this sensitivity test is based on the response of one specific glacier to a poorly known external forcing and with limited observations. Under these conditions, we show that some sets of parameters definitely produce reasonable behaviour, but these parameters should not be transferred to another configuration without being sure that the response of the model is realistic. Despite this limitation, this calving model based on realistic physical approaches gives reliable results and could be easily implemented in classical ice-flow models.

The calving process described in this paper is immediately driven by the
variation in longitudinal stretching associated with horizontal velocity
gradients, producing a first-order control of the calving rate, as stated by

As stated by

Physical and numerical parameters. Tunable parameters are in italic.

This study was funded by the Agence Nationale pour la Recherche (ANR) through
the SUMER, Blanc SIMI 6-2012. All the computations
presented in this paper were performed using the CIMENT infrastructure
(